diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzbahj" "b/data_all_eng_slimpj/shuffled/split2/finalzzbahj" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzbahj" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\label{Intro}\n\nMuch of the predictive power of perturbative Quantum Chromo Dynamics \n(pQCD) is contained in factorization theorems and in the universality \nof non-perturbative hadronic matrix elements \\cite{Collins:1989gx}. \nPredictions follow when processes with different hard scatterings \nbut the same matrix elements are compared. \nIn the case of leading power contributions, the universal matrix\nelements are interpreted as parton (quark or gluon) distribution\nfunctions (PDFs). With the PDFs extracted from a global QCD \nanalysis \\cite{Pumplin:2002vw,Martin:2002aw,Gluck:1998xa},\npQCD has been very successful in interpreting and predicting \nhigh-energy scattering processes. \n\nHowever, significant uncertainties still exist in the PDFs\ndue to the accuracy of experimental data and to unknown higher order \ncorrections to perturbative calculations. In particular, the PDFs\nare least constrained in the region where the parton momentum \nfraction $x>0.5$ for valence quark distributions and $x>0.3$ for\ngluon and sea quark distributions \\cite{Pumplin:2002vw,Martin:2002aw}. \nOn the other hand, \nprecise PDFs are needed for many reasons \\cite{Kuhlmann:1999sf}. \nFor example, the discovery potential of the \nLarge Hadron Collider (LHC) on new physics as excess \nin particle\/jet spectrum at large momentum requires \naccurate PDFs at large $x$ and large factorization scale $\\mu$. \nSince PDFs at a large $\\mu$ are obtained by solving DGLAP evolution \nequations with input PDFs at a lower factorization scale, and \nthe evolution feeds the large-$x$ partons at a lower scale to \nthose at a higher scale with smaller momentum fraction $x$, \nthe precision of PDFs at large $\\mu$ depends on the accuracy \nof PDFs at large $x$ and low factorization scale.\nFurthermore, reliable information on the ratio of \n$d(x)\/u(x)$ as $x\\rightarrow 1$ could provide very important insights\ninto the non perturbative structure of the nucleon\n\\cite{Melnitchouk:1995fc,Brodsky:1994kg,Isgur:1998yb,Farrar:1975yb} \nand references therein. \nHowever, because of the PDFs steeply falling shape as a function of\n$x$ as $x{\\rightarrow} 1$, and because of the convolution of two PDFs,\nmost observables in hadronic collisions do not provide tight enough \nconstraints to the PDFs at large $x$. On the other hand, inclusive \nlepton-hadron deep inelastic scattering (DIS) at large Bjorken $x_B$ \nis a more direct and clean probe of large-$x$ parton distributions.\nRecently, experiments at the Jefferson Laboratory have produced \nDIS data at large $x_B$ with high precision, but at relatively \nlow virtuality $Q^2$ of the exchanged virtual photon in \nlepton-hadron collisions \n\\cite{Tvaskis:2006tv,Melnitchouk:2004ep,Cardman:2006xt}.\nExperiments measure DIS \ncross sections, or, equivalently, the DIS structure functions, not PDFs.\nIn order to extract PDFs at large $x$ from these and other data at low\n$Q^2$, it is necessary to have theoretical control over power\ncorrections, such as the dynamical power corrections (or high twist\neffects), $\\propto \\Lambda_{\\rm QCD}^2\/Q^2$ with the non-perturbative\nscale \n$\\Lambda_{\\rm QCD}\\sim 1\/$fm \\cite{Ellis:1982cd,Qiu:1988dn}, \nthe target mass corrections (TMC), $\\propto x_B^2 m_N^2\/Q^2$ \nwith nucleon mass $m_N$ \\cite{Schienbein:2007gr}, \nand possibly, final-state jet mass corrections (JMC), \n$\\propto m_j^2\/Q^2$ \\cite{Collins:2007ph}.\nThese corrections become larger and larger as data approach the\nkinematic limit $x_B=1$. \nIn this paper, we examine the uncertainties in extracting PDFs at low\n$Q^2$ and large $x_B$ caused by the target mass and jet mass\ncorrections.\n\nAt the leading power, the perturbative QCD factorization treatment \nof DIS cross sections neglects all $1\/Q^2$-type power corrections.\nHowever, TMC play a somewhat special role.\nSince the mass of the target is a non-perturbative quantity, the\npartonic dynamics of short-distance factors in the QCD factorization\nformalism should not depend on it.\nTherefore, for any hadronic cross section that can be factorized \nin perturbative QCD, the effect of TMC should be implicitly included \nin the definition of the non-perturbative hadron matrix elements, and \nexplicitly accounted for in the kinematic variables of the\nobservables. In this sense, TMC are mostly of kinematic origin. \nAt large $x_B$ and low $Q^2$, the $x_B^2 m_N^2\/Q^2$-type\nTMC can be an important part of the measured cross sections, and \nshould be identified and removed before we extract the leading \npower PDFs at large $x$. \n\nFollowing the pioneering work by Georgi and Politzer (GP) in \nas early as 1976 \\cite{Georgi:1976ve}, many papers have been \nwritten on TMC, in particular, for lepton-hadron DIS.\nA recent review by Schienbein {\\it et al.} provides a \nnice summary of this effort \\cite{Schienbein:2007gr}.\nMost existing calculations use the technique\nof operator product expansion (OPE)\nto resum $m_N^2\/Q^2$ corrections to the structure function moments. \nA strong debate \nhas been centered on the inversion of the moment formula\n\\cite{DeRujula:1976tz,Gross:1976xt,Johnson:1979ty,Bitar:1978cj,Steffens:2006ds}. \nIf we keep the target mass in the DIS kinematics,\nthe Bjorken scaling variable $x_B$ for the DIS cross sections \nor structure functions needs to be replaced \nby the Nachtmann variable \\cite{Nachtmann:1973mr},\n$\\xi=2x_B\/(1+\\sqrt{1+4x_B^2 m_N^2\/Q^2}) \\rightarrow x_B$ as\n$m_N^2\/Q^2\\rightarrow 0$. \nIf the target mass cannot be neglected at low $Q^2$,\nthe Nachtmann variable $\\xi$ is less than 1 even at $x_B=1$.\nOnly if one ignores the $x_B=1$ kinematic threshold, and allows $\\xi$ to\nrun up to 1, does the inverse Mellin transformation of the structure\nfunction moments give back the structure functions in $x_B$ space\n\\cite{Johnson:1979ty,Bitar:1978cj}.\nAs a consequence, the inverted structure functions are finite in the \nunphysical $x_B>1$ region. \nThe unphysical region has been argued to disappear with the inclusion of\npower-suppressed higher-twist terms in the computation\n\\cite{DeRujula:1976tz}. Alternatively, many prescriptions have been\nsuggested to fix the moments inversion problem, or to\nphenomenologically eliminate the unphysical region, see\n\\cite{Johnson:1979ty,Bitar:1978cj,Steffens:2006ds}. None of these\nprescriptions is entirely satisfactory or unique. \n\nTo completely avoid the ambiguities in connection with the structure\nfunctions moments and their inversion, it is natural to \ninvestigate the TMC in the momentum space without using the OPE \nand taking the moments. \nThis is most easily done in the context of the\nfield theoretic pQCD parton model, as pioneered by Ellis, \nFurmanski, and Petronzio in Ref.~\\cite{Ellis:1982cd}. Recently, \nKretzer and Reno applied and compared both approaches in the case \nof neutrino initiated DIS experiments\n\\cite{Kretzer:2002fr,Kretzer:2003iu}. \nIn this paper, we revisit the TMC in DIS in terms of the perturbative \nQCD collinear factorization approach in momentum space and express \nthe long distance physics of structure functions and the leading \ntarget mass correction in terms of PDFs that share the same \npartonic operators with the PDFs of zero hadron mass. \nIn our approach, the momentum space structure functions have \nno unphysical region. \nMoreover, our approach can be generalized to semi-inclusive DIS and\nhadronic collisions, where the OPE is not applicable.\n\nIn the collinear factorization approach at the leading power in $1\/Q^2$, the \nshort-distance factors are perturbatively calculated with massless\nfinal-state light partons. As recently pointed by Collins, Rogers and\nStasto in Ref.~\\cite{Collins:2007ph}, the outgoing parton lines should\nacquire jet subgraphs\/functions to have correct kinematics. \nThe invariant mass in the jet subgraph leads to the before mentioned \n$m_j^2\/Q^2$-type JMC, which are particularly sensitive to the \nlarge-$x_B$ kinematics and the extraction of large-$x$ PDFs. \nIn this paper, we discuss the role of the jet functions in modifying\nthe DIS kinematics in the collinear factorization approach. \nWe neglect the soft interactions \nbetween the beam jet and the final-state jet functions, and \npresent a collinear factorization formalism for calculating \nDIS structure functions with a non trivial jet function. \nBased on a toy-model estimate, we argue \nthat the JMC has a significant effect on the extraction of \nPDFs when $x\\gtrsim 0.6$. The connection of the jet function with\nlattice QCD computations of the non-perturbative quark propagator is\nalso discussed. \n\nThe rest of our paper is organized as follows. In Sec.~\\ref{sec:TMC},\nwe drive the TMC in terms of QCD collinear factorization in momentum \nspace. We explicitly demonstrate that our result has no unphysical \nregion for the DIS structure functions. We compare our result with\nTMC predicted by other approaches. In Sec.~\\ref{sec:JMC}, we discuss\nthe JMC. Finally, we present our summary and thoughts on \nfuture extensions in Sec.~\\ref{sec:conclusions}. In the main text we\nlimit the discussion to light partons and the transverse and longitudinal\nstructure functions. In the appendices, we generalize our formulae. \n\n\n\n\\section{target mass corrections}\n\\label{sec:TMC}\n\nThe DIS cross section is determined by the hadronic tensor\n\\begin{align}\n W^{\\mu\\nu}(p,q) & = \\frac{1}{8\\pi} \\int d^4z\\, e^{-iq\\cdot z}\n \\vev{p|J^{\\dagger\\mu}(z)J^\\nu(0)|p} \\ ,\n \\label{eq:Wdef}\n\\end{align}\nwhere $p$ is the nucleon 4-momentum, $q$ is the virtual boson \n4-momentum, $J^\\mu$ is the electromagnetic or electroweak current, and\n$\\ket{p}$ is the hadron wave function.\nIn the impulse approximation the lepton-nucleon interaction proceeds\nthrough the scattering of the virtual boson with a parton (quark or\ngluon) belonging to the nucleon, and having 4-momentum $k$, see\nFig.~\\ref{fig:hadronictensor}. With these 4-momenta we can build the\nfollowing useful invariants:\n\\begin{align}\n x_B & = \\frac{-q^2}{2 p\\cdot q}, \\quad\n Q^2 = -q^2, \\quad \n m_N^2 = p^2, \\quad \n x_f = \\frac{-q^2}{2 k\\cdot q } \\ .\n \\label{eq:invariants}\n\\end{align}\nThe first 3 invariants, namely, the Bjorken variable $x_B$, the rest\nmass $m_N$ of the nucleon and the vector boson virtuality $Q^2$, are\nexperimentally measurable. We call them ``external invariants''. \nThe fourth invariant, $x_f$, is the Bjorken variable for a partonic\ntarget and is not experimentally measurable, so we call it\n``internal''. \n\nWe work in a class of frames, called collinear frames, \ndefined such that $p$ and $q$ do not have transverse momentum. \nThen we can decompose $p$, $q$ and $k$ as follows.\n\\begin{align}\n\\begin{split}\n p^\\mu & = p^+ {\\overline n}^\\mu \n + \\frac{m_N^2}{2 p_A^+} n^\\mu \\\\\n q^\\mu & = - \\xi p^+ {\\overline n}^\\mu \n + \\frac{Q^2}{2\\xi p^+} n^\\mu \\\\\n k^\\mu & = x p^+ {\\overline n}^\\mu \n + \\frac{k^2 + k_T^2}{2 x p^+} n^\\mu \n + \\vec k_\\perp^{\\,\\mu} \\ .\n \\label{eq:kinematics} \n\\end{split}\n\\end{align}\nThe light-cone vectors $n^\\mu$ and ${\\overline n}^\\mu$ satisfy\n\\begin{align}\n n^2 = {\\overline n}^2 = 0 \\qquad n\\cdot{\\overline n}=1 \\ ,\n\\end{align}\nand define the light-cone plus and minus directions, respectively. \nThe plus- and minus-components of a 4-vector $a$ are defined by\n\\begin{align}\n a^+ = a\\cdot n \\qquad a^-=a\\cdot{\\overline n} .\n\\end{align}\nIf we choose ${\\overline n} = (1\/\\sqrt{2},\\vec0_\\perp,1\/\\sqrt{2})$ and \n$n = (1\/\\sqrt{2},\\vec0_\\perp,-1\/\\sqrt{2})$, we obtain \n$a^\\pm = (a_0 \\pm a_3)\/\\sqrt{2}$. The transverse parton momentum $k_T$\nsatisfies $k_T\\cdot n = k_T \\cdot {\\overline n} = 0$. \nThe nucleon plus-momentum, $p^+$, can be interpreted as a \nparameter for boosts along the $z$-axis, connecting the target rest\nframe to the hadron infinite-momentum frame.\nThe parton fractional light-cone\nmomentum with respect to the nucleon is defined as\n\\begin{align}\n x = k^+ \/ p^+ \\ ,\n\\end{align}\nand is an internal variable. The virtual boson fractional momentum\n\\begin{align}\n \\xi = - \\frac{q^+}{p^+} \n = \\frac{2 x_B} {1+\\sqrt{1+4 x_B^2 m_N^2 \/ Q^2}} \n\\end{align}\nis an external variable, \nand coincides with the Nachtmann variable \\cite{Nachtmann:1973mr}. \nNote that in the Bjorken limit\n($Q^2 {\\rightarrow}\\infty$ at fixed $x_B$) $\\xi {\\rightarrow} x_B$ and we recover the\nstandard kinematics in the massless target approximation. \nIn this paper, we will consider light quarks $u,d,s$ only and set\n$m^2_{u,d,s}=0$. In Appendix~\\ref{app:strfns} we will extend our\nresults to heavy quarks.\n\n\\begin{figure}[tb]\n \\vspace*{0cm}\n \\centerline{\n \\includegraphics\n [width=\\linewidth]\n {fig.WA-factorized.eps}\n }\n \\caption[]{\n Collinear factorization of the hadronic tensor in the impulse\n approximation. The top blob represents the interaction of a\n virtual boson with a parton computed in pQCD at any order in\n $\\alpha_s$. \n }\n \\label{fig:hadronictensor}\n \\label{fig:DISfactorization}\n\\end{figure}\n\n\nCollinear factorization for the hadronic tensor can be obtained by\nexpanding the parton momentum $k$ \nin Fig.~\\ref{fig:DISfactorization} around its positive light-cone component, \n\\begin{align}\n \\widetilde k^\\mu = x p^+ {\\overline n}^\\mu \\ .\n\\end{align}\nCorrespondingly, we can define the collinear invariant\n\\begin{align}\n \\widetilde x_f = \\frac{-q^2}{2\\widetilde k\\cdot q} = \\frac{\\xi}{x} \\ .\n \\label{eq:tildexf}\n\\end{align}\nAccording to the QCD factorization theorem \\cite{Collins:1989gx}, \nthe nucleon hadronic tensor can then be factorized as follows: \n\\begin{align}\n\\begin{split}\n & W^{\\mu\\nu}(p,q) \\\\ \n & \\qquad = \\sum_f \\int \\frac{dx}{x} \\, \n {\\mathcal H}_f^{\\mu\\nu}(\\widetilde k,q) \\, \\varphi_{f\/N}(x,Q^2,m_N^2) \\\\\n & \\qquad + O(\\Lambda^2\/Q^2)\n \\label{eq:pQCDfactnolims}\n\\end{split}\n\\end{align}\nwhere ${\\mathcal H}_f^{\\mu\\nu}$ is the short-distance partonic \ntensor for scattering on a\nparton of flavor $f$, and $\\varphi_{f\/N}$ is the leading twist parton\ndistribution function for a parton of flavor $f$ inside a\nnucleon $N$, see\nFig.~\\ref{fig:DISfactorization}. For example,\nthe quark distribution at leading order in $\\alpha_s$ is defined as \n\\begin{align}\n \\varphi_q(x,Q^2,m_N^2) = \\int \\frac{dz^-}{2 \\pi} e^{-ix p^+ z^-}\n \\vev{p|\\overline\\psi(z^- n)\\,\\frac{\\gamma^+}{2}\\,\\psi(0)|p} \\ .\n \\label{eq:quarkPDFatLO}\n\\end{align}\nA proper gauge link between the two fermion field operators is\nrequired to have a gauge-invariant parton distribution, but drops out\nif one chooses the light-cone gauge $n\\cdot A = 0$, where $A$ is the\ngluon field.\nHigher orders in the Taylor expansion are suppressed by powers of\n$\\Lambda^2\/Q^2$, with $\\Lambda$ a hadronic scale, and contribute to\nrestore gauge invariance in higher twist terms \\cite{Qiu:1988dn}. We will\ndiscuss in detail how to obtain such a factorized form in\nSection~\\ref{sec:JMC}. \nIn Eq.~\\eqref{eq:pQCDfactnolims}, the partonic tensor ${\\mathcal H}^{\\mu\\nu}$ can be\ncomputed perturbatively to any order in $\\alpha_s$, and can depend on the\nnucleon mass only kinematically through the invariant $\\widetilde\nx_f$. Dynamical target mass corrections can enter only through the\nproton wave function $\\ket{p}$, whence the explicit dependence of\n$\\varphi$ on $m_N^2$ in\nEqs.~\\eqref{eq:pQCDfactnolims}-\\eqref{eq:quarkPDFatLO}.\nFrom now on we will suppress such dependence for ease of notation. \nFor higher twist terms, the situation\nis more complicated, because the equations of motion may induce\ndynamical correlations between lower- and higher-twist terms\n\\cite{Ellis:1982cd}, but we will not discuss this issue here. \n\n\\begin{figure}[tb]\n \\vspace*{0cm}\n \\centerline{\n \\includegraphics\n [width=0.6\\linewidth]\n {fig.WA-amplitude.eps}\n }\n \\caption[]{\n DIS in the impulse approximation, for the special case of an\n internal on-shell light parton, $k^2 = 0$, relevant to collinear\n factorization. The current jet has momentum\n $p_j$ and the target jet has momentum $p_Y$. The net baryon number\n is only shown to flow in the target jet (lower part of the graph).\n }\n \\label{fig:DISamplitude}\n\\end{figure}\n\n\nStructure functions are obtained by suitable projections of the\ntensors in Eq.~\\eqref{eq:pQCDfactnolims}, see Appendix~\\ref{app:strfns}. \nIn this paper, we choose the helicity basis to perform the projection\nof the $W^{\\mu\\nu}$ and ${\\mathcal H}^{\\mu\\nu}$ tensors. The transverse and\nlongitudinal structure functions read\n\\begin{align}\n\\begin{split}\n F_{T,L}(x_B,Q^2,m_N^2)\n = \\sum_f \\int \\frac{dx}{x} h_{f|T,L}(\\widetilde x_f,Q^2) \n \\varphi_f(x,Q^2) \\ .\n \\label{eq:FTL_naive}\n\\end{split}\n\\end{align}\nThe advantage of the helicity basis is that \nin the right hand side there are no kinematic prefactors, \nwhich would appear when considering the $F_{1,2}$\nstructure functions, as discussed in Ref.~\\cite{Aivazis:1993kh} and reviewed in\nAppendix~\\ref{app:strfns}. \n\n\\begin{figure*}[tb]\n \\vspace*{0cm}\n \\centerline{\n \\includegraphics\n [width=0.45\\linewidth]\n {fig.F2_NLO_xB.eps}\n \\hspace*{0.5cm}\n \\includegraphics\n [width=0.45\\linewidth]\n {fig.F2_NLO_Q2.eps}\n }\n \\caption[]{\n Comparison of prescriptions for NLO target mass corrections to the\n $F_2$ structure function. The ratio $F_2\/F_2^{(0)}$ is plotted as a\n function of $x_B$ and $Q^2$. The structure functions have\n been computed using MRST2002 parton distributions \\cite{Martin:2002aw}. \n }\n \\label{fig:F2_TMC}\n\\end{figure*}\n\nApplying the factorized Eq.~\\eqref{eq:pQCDfactnolims} \nwithout paying attention to the kinematic\nlimits on $x$, which have been understood in\nEq.~\\eqref{eq:pQCDfactnolims}, and using Eq.~\\eqref{eq:tildexf}, one\nwould obtain \nwhat we call the ``na\\\"ive'' TMC in collinear factorization: \n\\begin{align}\n F^\\text{nv}_{T,L}(x_B,Q^2,m_N^2) = F_{T,L}^{(0)}(\\xi,Q^2) \\ ,\n \\label{eq:FFnv}\n\\end{align}\nwhere $F_{T,L}^{(0)}$ are the structure functions as they would be defined\nand computed in the massless nucleon limit by setting $m_N^2=0$ from\nthe beginning. Indeed, the partonic structure functions $h_{f|T,L}$\nare independent of the hadron target, and are defined in the same way \nfor the massive and massless nucleon cases. \nAs a consequence of the fact that $F_{T,L}^{(0)}(y,Q^2)$ has\nsupport over $0 < y \\leq 1$, the target mass corrected\n$F^\\text{nv}_{T,L}$ can be different from zero in the kinematically\nforbidden region $1 < x_B \\leq 1\/(1-m_N^2\/Q^2)$. The appearance of\nsuch an unphysical region is also a feature of the OPE approach \n\\cite{Georgi:1976ve,DeRujula:1976tz}, as discussed in the introduction.\nEq.~\\eqref{eq:FFnv} has been introduced in Ref.~\\cite{Aivazis:1993kh}\nand compared to the OPE approach in\nRefs.~\\cite{Kretzer:2002fr,Kretzer:2003iu}. \n\n\\begin{figure*}[tb]\n \\vspace*{0cm}\n \\centerline{\n \\includegraphics\n [width=0.45\\linewidth]\n {fig.R_NLO_xB.eps}\n \\hspace*{0.5cm}\n \\includegraphics\n [width=0.45\\linewidth]\n {fig.R_NLO_Q2.eps}\n }\n \\caption[]{\n Comparison of prescriptions for NLO target mass corrections to the\n ratio of the longitudinal and transverse cross sections,\n $R=\\sigma_L\/\\sigma_T= F_L\/F_1$. The ratio $R\/R^{(0)}$ is plotted \n as a function of $x_B$ and $Q^2$. The structure functions\n have been computed using MRST2002 parton distributions\n \\cite{Martin:2002aw}. \n }\n \\label{fig:R_TMC}\n\\end{figure*}\n\nIn fact, a closer examination of the handbag diagram kinematics \nreveals that\nthere is no unphysical region. Let us consider the handbag diagram in\nthe right hand side of Fig.~\\ref{fig:DISfactorization}, and \nlimit the discussion to on-shell light quarks or gluons,\n$k^2=0$, in both the initial and final states.\nThe general case of off-shell partons, including heavy quark\nproduction is discussed in Appendix~\\ref{app:kinematics}.\nBecause of baryon number conservation, the net baryon number must \nflow either into the target jet or into the current jet. We shall\nseparately examine these two cases. \nIf the net baryon number flows into the target jet (bottom part of\nFig.~\\ref{fig:DISamplitude}), \nthe jet invariant masses satisfy $m_j^2 = p_j^2 \\geq m_f^2$ and \n$p_Y^2 \\geq m_N^2$.\nLet us consider the invariant momentum square of the process,\n$s = (p+q)^2 = (p_j + p_Y)^2$. Since the 2 jets are\nmade of on-shell particles, $p_j\\cdot p_Y \\geq0$. Hence,\n$s \\geq m_j^2 + m_N^2$. In summary, the current \njet mass must satisfy\n\\begin{align}\n 0 \\leq m_j^2 \\leq s - m_N^2 \\ . \n \\label{eq:mj_bdown}\n\\end{align}\nSince $s-m_N^2 = (1\/x_B -1)Q^2$, Eq.~\\eqref{eq:mj_bdown} \nguarantees that the handbag diagram is non-zero only when\n$x_B \\leq 1$, as it must be\non general grounds because of baryon number conservation, irrespective\nof the model used to compute the process. \nOn the other hand, if the net baryon\nnumber flows into the current jet (top part of\nFig.~\\ref{fig:DISamplitude}). The invariant jet masses satisfy \n$m_j^2 \\geq m_N^2$ and $p_Y^2 \\geq 0$, so that\n\\begin{align}\n m_N^2 \\leq m_j^2 \\leq s \\ ,\n \\label{eq:mj_bup}\n\\end{align}\nwhich again guarantees that the handbag diagram respects the \n$x_B \\leq 1$ limit. \nWithin the collinear factorization approach, \nthe momentum of the active quark entering the short-distance \nhard part that generates the current jet is approximated to be on mass\nshell, $\\tilde{k}^2=0 \\ll m_N^2$. \nThat is, the baryon number is very likely to flow into \nthe target jet for the factorized contribution to the \nDIS cross section, and \nEq.~\\eqref{eq:mj_bdown} gives the relevant limits on $m_j^2$.\nUsing $m_j^2 = (\\tilde k+q)^2 = (1\/\\widetilde x_f-1)Q^2$ and $\\widetilde\nx_f=\\xi\/x$ in Eq.~\\eqref{eq:mj_bdown}, we obtain \n\\begin{align}\n x_B \\leq \\widetilde x_f \\leq 1 \\ ,\n \\label{eq:xflimits}\n\\end{align}\nwhich implies the following limits on the $dx$ integration in\nEq.~\\eqref{eq:FTL_naive}: \n\\begin{align}\n \\xi \\leq x \\leq \\frac{\\xi}{x_B} \n \\label{eq:xlimits}\n\\end{align}\nEqs.~\\eqref{eq:xflimits}-\\eqref{eq:xlimits}\nexplicitly guarantee $F_{T,L} = 0$ if $x_B > 1$, so that there is no\nunphysical region for target mass corrected structure functions:\n\\begin{align}\n\\begin{split}\n & F_{T,L}(x_B,Q^2,m_N^2) \\\\\n & \\qquad = \\int_\\xi^{{\\xi}\/{x_B}} \\frac{dx}{x} h_{f|T,L}(\\widetilde x_f,Q^2) \n \\varphi_f(x,Q^2) \\ .\n \\label{eq:FTL_TMC}\n\\end{split}\n\\end{align}\nEq.~\\eqref{eq:FTL_TMC} is our formula for calculating DIS structure \nfunctions with the TMC. As expected, it has the hadron mass dependence \nexplicitly in the integration limits caused by the DIS kinematics and \nimplicitly from the hadron states in the definition of the PDFs. \nThe na\\\"ive structure functions \\eqref{eq:FFnv} are obtained when\nconsidering $x\\leq \n1$ as upper integration limit in Eq.~\\eqref{eq:FTL_TMC}. This limit is\na general and process-independent consequence of the \ndefinition of a parton distribution in the field theoretic parton\nmodel \\cite{Jaffe:1983hp}, but in DIS it is weaker than $x \\leq\n\\xi\/x_B$, which is induced by 4-momentum and baryon number\nconservation.\nIn the massless target limit, $m_N^2\/Q^2 {\\rightarrow} 0$, \nthe constraint \\eqref{eq:xlimits} reduces to $x_B \\leq x \\leq 1$,\nand we recover the massless structure functions as we should expect:\n\\begin{align}\n F_{T,L}(x_B,Q^2,m_N^2) \\xrightarrow[m_N^2\/Q^2{\\rightarrow} 0]{} \n F_{T,L}^{(0)}(x_B,Q^2) \\ .\n\\end{align}\n\nIn Fig.~\\ref{fig:F2_TMC}, we plot the ratio of the TMC corrected\n$F_2$ to the massless $F_2^{(0)}$, with TMC computed using the analog\nof Eq.~\\eqref{eq:FTL_TMC}, see Appendix~\\ref{app:colfactsfn}, \nthe naive prescription \\eqref{eq:FFnv}, and\nthe Georgi-Politzer prescription. The corrections are in general quite\nlarge at $Q^2=2$, but still non negligible at the generally considered\n``safe'' scale $Q^2=25$ GeV$^2$. \nFrom the right panel of the figure, one can estimate how large\n$Q^2$ should be to safely neglect TMC. At $x_B\\lesssim\n0.5$ the TMC are smaller than 5\\% if $Q^2\\gtrsim 10$ GeV$^2$. However,\nat larger $x_B$, one may need to go to $Q^2 \\gtrsim 100$\nGeV$^2$ for TMC to become small. Note also the difference between\n$F_2$ and $F_2^\\text{nv}$, which is smaller than 30-40\\% at $Q^2=2$:\nit gives the size of the contribution of the unphysical region\n$\\xi\/x_B < x \\leq 1$, which has to be subtracted from the na\\\"ive\nstructure function.\n\nThe difference between TMC of $F_2$ (and similarly of $F_{1,T}$)\nin collinear factorization and in the Georgi-Politzer formalism is\nsmaller than 15-20\\% at the lowest scale, and rapidly disappears at\nlarger scales. So one\nis tempted to brush aside the question of what formalism is correct,\nif willing to accept this level of uncertainty. However, the\nsituation completely changes when considering $F_L$, or the ratio $R$ \nof the longitudinal to transverse cross section, \n\\begin{align}\n\\begin{split}\n R = \\frac{\\sigma_L}{\\sigma_T} = \\frac{F_L}{F_1} \\ ,\n \\qquad\n R^{(0)} = \\frac{F^{(0)}_L}{F^{(0)}_1} \\ ,\n\\end{split}\n\\end{align}\nwhose TMC\/massless ratios are plotted in Fig.~\\ref{fig:R_TMC}.\n(Note a factor of $2x_B$ with respect to other common conventions, see\nAppendix~\\ref{app:invsfn}.) The TMC of $R$ \nare much larger than for $F_2$. Most importantly, the\ndifference between the collinear factorization and Georgi-Politzer TMC\nis huge, up to a factor 10 (5) at $Q^2$=2 (25) GeV$^2$! Therefore,\none has to decide which formalism to use. This is especially important\nfor a fit of the gluon PDF, to which $F_L$ is sensitive.\n\n\\begin{figure}[tb]\n \\vspace*{0cm}\n \\centerline{\n \\includegraphics\n [width=\\linewidth]\n {fig.WA-LO.eps}\n }\n \\caption[]{\n DIS handbag diagram at leading order in $\\alpha_s$. \n }\n \\label{fig:DIS_LO}\n\\end{figure}\n\nIt is also important to note that our formula for TMC \nin Eq.~\\eqref{eq:FTL_TMC} explicitly eliminates the kinematically \nforbidden region $1 < x_B \\leq 1\/(1-m_N^2\/Q^2)$ because of the \nintegration limits on the parton momentum fraction $x$. \nAs $x_B\\rightarrow 1$, structure functions calculated by \nusing Eq.~\\eqref{eq:FTL_TMC} approach to zero, the kinematic limit, \nsmoothly, except for the lowest order contribution, whose partonic \nstructure functions are derived from the tree level handbag \ndiagram in Fig.~\\ref{fig:DIS_LO}. \nIndeed, by explicit computation at tree level in the approximation of\nmassless quarks, we obtain \n\\begin{align}\n h_{q|T}(\\widetilde x_f,Q^2) \n = \\frac{1}{2}\\, e_f^2\\, \\delta(\\widetilde x_f - 1) \n = \\frac{1}{2}\\, e_f^2\\, x\\, \\delta(x - \\xi) \\ ,\n \\label{eq:hTLO}\n\\end{align}\nwhere $e_f$ is the electric charge of the parton $f$. \nSubstituting the tree-level partonic structure function into\nEq.~\\eqref{eq:FTL_TMC}, the lowest order contribution to \nthe transverse structure function, \n\\begin{align}\n F_T(x_B,Q^2,m_N^2) \n = \\left\\{\\begin{array}{cl}\n F_T^{(0)}(\\xi,Q^2) \\quad & x_B \\leq 1 \\\\\n 0 & x_B > 1 \\ , \\\\\n \\end{array} \\right.\n\\end{align}\nremains positively finite when $x_B\\rightarrow 1$ and does\nnot vanish as $x_B\\ra1$, as shown by the dashed line \nin Fig.~\\ref{fig:FTvsFT0}, \n \n\\begin{figure}[tb]\n \\vspace*{0cm}\n \\centerline{\n \\includegraphics\n [width=0.9\\linewidth]\n {fig.FT_JMC.2.eps}\n }\n \\caption[]{\n Transverse structure function plotted as a function of $x_B$,\n with and without target and jet mass corrections, computed with\n only light quarks at lowest order in $\\alpha_s$ using MRST2001LO\n parton distributions \\cite{Martin:2002dr}. The dotted line is the\n massless structure function. The dashed line includes only TMC,\n and corresponds to $Z=1$ in Eq.~\\eqref{eq:JMCnosoft}. The solid line\n corresponds to JMC coming only from the continuum part $\\rho$ of\n the jet spectral function, $Z=0$ in Eq.~\\eqref{eq:JMCnosoft}. JMC\n are computed using a log-normal spectral function with\n $\\vev{m_j^2} = 0.3$ GeV$^2$ and standard deviation\n $\\sigma_{m_j^2}=\\vev{m_j^2}$. \n }\n \\label{fig:FTvsFT0}\n\\end{figure}\n\nThis problem exists only at the lowest order and \narises because of the $\\delta$-function behavior of the partonic\nstructure function \\eqref{eq:hTLO} and the assumption that \nthe final-state is made of a massless quark, $m_f^2=0$, \nas shown in Fig.~\\ref{fig:DIS_LO}. The $\\delta$-function\nbypasses the kinematic constraint from the integration limits\nin Eq.~\\eqref{eq:FTL_TMC} and forces $x=\\xi(x_B)$, which exhibits \nthe mismatch between the phase space for $x$ at the parton level\nand that for $\\xi(x_B)$ at the hadron level.\nUnder the collinear approximation the momentum fraction\n$x$ for a massless parton can be as large as 1, while the plus\nmomentum fraction of the virtual photon $\\xi(x_B)$ smaller than 1 \nfor a finite target mass $m_N$.\nAs a result, the perturbatively calculated structure functions \ndo not vanish at $x_B=1$ because the PDFs are finite at\n$x=\\xi(x_B=1)=2\/(1+\\sqrt{1+4m_N^2\/Q^2}) < 1$. \nAs we will discuss in the next section, this explicit \nphase space mismatch at the lowest order could be improved\nif the single massless quark final-state in Fig.~\\ref{fig:DIS_LO}, \nwhich is not physical, is replaced by a jet function as shown \nin Fig.~\\ref{fig:DIS_JET_LO}.\n\nWe conclude this section by stating that \nif one is performing global QCD fits of the PDFs\nin the context of pQCD collinear factorization, our formalism in \nEq.~\\eqref{eq:FTL_TMC} might be the most consistent way to treat TMC, \nbecause it expresses the long distance physics of structure functions \nand the leading target mass correction in terms of PDFs that \nshare the same partonic operators with the PDFs of zero hadron mass. \nMoreover the structure functions calculated using our formulae \ndo not have the $x_B>1$ unphysical region and vanish at the \n$x_B=1$ kinematic limit except for the lowest order contribution \nthat will be discussed further in next section.\nThe same collinear factorization formalism can be easily and\nconsistently extended to semi-inclusive DIS measurements and hadronic\ncollisions, for which the OPE formalism is not applicable, but which\nare included in global QCD fits of parton distributions. Careful\nanalysis of kinematics and conservation laws will guarantee that no\nunphysical region appears in these observables, as well. \nThe obtained formulae will not merely be an approximation to the TMC\nfor those processes, as argued in\n\\cite{Kretzer:2003iu,Kretzer:2002fr}, but will give the correct answer\nin the context of pQCD collinear factorization.\n\n\n\\section{Jet mass corrections}\n\\label{sec:JMC}\n\n\\begin{figure}[tb]\n \\vspace*{0cm}\n \\centerline{\n \\includegraphics\n [width=0.7\\linewidth]\n {fig.WA-handbag-jet.eps}\n }\n \\caption[]{\n DIS handbag diagram at leading order in $\\alpha_s$ generalized to\n include a jet function $\\hat J(l)$ beside the target function\n $\\hat T(k)$. \n }\n \\label{fig:DIS_JET_LO}\n\\end{figure}\n\nIn this section we discuss the possibility to include \na jet function into the lowest order contribution to have a \nmore realistic kinematic constraint on the ``single quark'' \nfinal-state \\cite{Collins:2007ph}. \nHopefully, we can reduce the unphysical positive \nvalue of structure functions at $x_B=1$. As discussed in the last section,\nthis is caused \nby the $\\delta$-function behavior of the partonic structure \nfunctions, the assumption that the final-state is made of \na massless quark, $m_f^2=0$, and the mismatch between the \nphase space for $\\xi(x_B)$ and $x$. \n\nThe assumption that the leading order final-state is made of \na massless quark, $m_f^2=0$, is clearly unphysical \nbecause the quark has to hadronize due to color confinement, \nso that the current jet will have an invariant mass $m_j^2$.\nThen, we may heuristically set $m_f^2 =\nm_j^2$ for the cut quark line, and substitute $\\delta(\\widetilde x_f\n- 1)$ with $\\delta(\\widetilde x_f - 1\/(1+m_j^2\/Q^2))$ in\nEq.~\\eqref{eq:hTLO}: \n\\begin{align}\n h_{q|T}(\\widetilde x_f,Q^2) {\\longrightarrow} \n \\frac{1}{2}\\, e_f^2\\, x\\, \n \\delta \\big( x - \\xi(1+\\frac{m_j^2}{Q^2}) \\big) \\ .\n \\label{eq:hTLO_jet}\n\\end{align}\nFurthermore, we may assume that the current jet has\nan invariant mass probability distribution $J_m(m_j^2)$ normalized to 1,\nand accordingly smear the structure functions in \\eqref{eq:FTL_TMC}:\n\\begin{align}\n\\begin{split}\n & F_T^{JMC}(x_B,Q^2,m_N^2) \\\\\n & \\qquad = \\int_0^\\infty dm_j^2 J_m(m_j^2) \n \\int_\\xi^{{\\xi}\/{x_B}} \\frac{dx}{x} h_{f|T}(\\widetilde x_f,Q^2) \n \\varphi_f(x,Q^2) \\\\\n & \\qquad = \\int_0^{\\frac{1-x_B}{x_B}Q^2} dm_j^2 J_m(m_j^2) \n F_T^{(0)}\\big(\\xi(1+m_j^2\/Q^2),Q^2\\big) \\ .\n \\label{eq:FTL_TMC_JMC_heuristic}\n\\end{split}\n\\end{align}\nIf $J_m(m_j^2)$ is a sufficiently smooth function of $m_j^2$, we obtain \n\\begin{align}\n F_T^{JMC}(x_B,Q^2,m_N^2) \\xrightarrow[x_B{\\rightarrow} 1]{} 0 \\ . \n\\end{align}\nThe jet mass corrections (JMC) so introduced are of order\n$O(m_j^2\/Q^2)$. It is easy to see that in the limit $Q^2 \\gg\n\\vev{m_j^2}$, the massless $F_T^{(0)}$ decouples from the integration\nover the jet mass, and we recover the structure function with TMC:\n\\begin{align}\n F_T^{JMC}(x_B,Q^2,m_N^2) \\xrightarrow[Q^2 \\gg \\vev{m_J^2}]{}\n F_T(x_B,Q^2,m_N^2) \\ .\n\\end{align}\n\nIn the following, we will discuss how to put this Ansatz on a more\nfirm theoretical basis.\n\n\n\\subsection{Collinear factorization with a jet function}\n\\label{sec:jetfact}\n\nWe aim at including in the DIS handbag at leading order in\n$\\alpha_s$ a suitable jet function to take into account the\ninvariant mass of the jet produced by the hadronization of the struck\nquark, see Fig.~\\ref{fig:DIS_JET_LO}. \nNote that in computing the DIS cross section with the handbag diagram\nof Fig.~\\ref{fig:DIS_JET_LO}, we are making several assumptions. \nFirst, we are assuming that it makes sense to separate the final state\ninto a current jet and a target jet, respectively the top and the\nbottom blob. Because of color confinement, this separation can only\nmake sense as an approximation, and is justified for inclusive and\nsemi-inclusive cross sections \nif the rapidity separation between the 2 jets is large \nenough. This is in general the case at asymptotically large\n$Q^2$. However, at finite $Q^2$, the rapidity difference between the 2\njets tends to 0 as $x_B{\\rightarrow} 1$, and the struck quark may participate\nin the hadronization process together with the unstruck target\npartons. Thus, we need to take care in estimating the range in\n$x_B$ in which the handbag diagram is a meaningful approximation to\nthe DIS process. The second assumption we make, intimately related\nwith the first one, is that color neutralization of the current jet\nhappens via the exchange of soft momenta, which we can neglect when\ndiscussing 4-momentum conservation. \n\nIn order to obtain a\ncollinear factorization formula, we will closely follow the\nprocedure of Ellis, Furmanski and Petronzio \\cite{Ellis:1982cd}.\nThe hadronic tensor is\n\\begin{align}\n W^{\\mu\\nu}(p,q) = \\frac{e_f^2}{8\\pi} \n \\int \\frac{d^4k}{(2\\pi)^4} \n \\mbox{Tr} \\big[\\hat T(k) \\gamma^\\nu \\hat J(l) \\gamma^\\mu \\big] \\,\n {\\mathbb K}(k,p,q) \\ .\n\\end{align}\nwhere we considered only 1 flavor for simplicity. The sum over quark\nflavors will be restored at the end of the computation.\nWe use a hat to denote a matrix in Dirac space. The trace over color\nindexes can be easily factorized and included in the target function\n\\cite{Ellis:1982cd}. The remaining trace is over Dirac indexes.\nThe target function $\\hat T$ is defined as\n\\begin{align}\n\\begin{split}\n \\big[\\hat T(k)\\big]_{ij} \n & = \\sum_Y \\delta^{(4)}\\big(p-k-\\sum_{i\\in Y}p_i\\big)\n \\big|\\vev{p|k,Y}\\big|^2 \\\\\n & = \\int d^4z e^{iz\\cdot k} \\vev{p|\\overline\\psi_j(z)\\psi_i(0)|p} \\ ,\n\\end{split}\n\\end{align}\nwhere \n$\\bra{k,Y} = \\bra{k}\\bra{Y}$, \n$\\bra{Y}$ are all possible final states originating from the\ntarget fragmentation, \nand $\\bra{k}$ is a parton state of momentum $k$. \nAnalogously, the jet function $\\hat J$ is the \nnon-perturbative quark propagator: \n\\begin{align}\n\\begin{split}\n \\big[\\hat J(l)\\big]_{ij} \n & = \\sum_Y \\delta^{(4)}\\big(l-\\sum_{i\\in Y} p_i\\big)\n \\big|\\vev{l|Y}\\big|^2 \\\\\n & = \\int d^4z e^{iz\\cdot l} \\vev{0|\\overline\\psi_j(z)\\psi_i(0)|0}\n \\ , \n\\end{split}\n\\end{align}\nand $\\bra{l}$ is a quark state of momentum $l$. \nThe jet momentum is constrained by momentum conservation to $l=k+q$,\nbut it is useful to keep it explicit in our formulae. The function\n$\\mathbb K$ is included to impose the kinematic constraints, \nthe non-trivial one being \n$x_f^\\text{min} \\leq x_f \\leq x_f\\text{min}$, see\nAppendix~\\ref{app:kinematics}:\n\\begin{align}\n\\begin{split}\n \\mathbb K(k,p,q) & = \\theta(k^++q^+) \\theta(k^-+q^-) \\\\ \n & \\times \\theta(p^+ - k^+) \\theta(p^- - k^-) \\\\\n & \\times \\theta(x_f-x_B) \\theta(1-x_f) \\ ,\n\\end{split}\n\\end{align}\nwhere, for light quarks, \n\\begin{align}\n\\begin{split}\n x_f^\\text{min} & = \\frac{x_B}{1-x_B k^2\/Q^2} \\\\\n x_f^\\text{max} & = \\frac{1}{1-k^2\/Q^2} \\ .\n\\end{split}\n\\end{align}\nTo obtain the leading power contribution, we expand\n$\\hat T(k)$ in terms of Dirac matrices and neglect terms that \ndepend on the vector defining the direction of the gauge link\nin the PDFs, which are suppressed by powers of $1\/Q^2$ \n\\cite{Qiu:1988dn}, \n\\begin{align}\n\\begin{split}\n \\hat T(k) & = \\tau_1(k) \\hat {\\mathbb I} + \\tau_2(k) \\ksl\n + \\tau_3(k) \\gamma_5 + \\tau_4(k) \\ksl \\gamma_5 \\, ,\n\\end{split}\n\\end{align}\nand, analogously, we expand $\\hat J$ as :\n\\begin{align}\n\\begin{split}\n \\hat J(l) & = j_1(l) \\hat {\\mathbb I} + j_2(l) \\lsl\n + j_3(l) \\gamma_5 + j_4(l) \\lsl \\gamma_5 \\ .\\\\\n\\end{split}\n\\end{align}\nFor massless quarks, $\\tau_1 = 0$. The terms $\\tau_{3,4}$, which are\nproportional to $\\gamma_5$ cancel when computing unpolarized cross\nsections. In pure QCD, $j_{3,4}=0$ because of parity invariance, and\n$j_1$ only enters in traces with an odd number of $\\gamma$ matrices,\nhence does not contribute. We are left with the terms proportional to\n$\\tau_2$ and $j_2$. The dominance of the $k^+$ and $l^-$ components of\n$k$ and $l$ in the Breit frame suggests to define\n\\begin{align}\n&\\begin{split}\n \\tau_2(k) & = \\frac{1}{4 k^+} \\mbox{Tr}\\big[\\nsl \\hat T(k)\\big] \\\\\n & = \\frac{1}{4 k^+} \\int d^4z e^{iz\\cdot k}\n \\vev{p|\\overline\\psi(z)\\gamma^+\\psi(0)|p} \\, \\\\\n \\label{eq:tau2def}\n\\end{split} \\\\\n&\\begin{split}\n j_2(l) & = \\frac{1}{4 l^-} \\mbox{Tr}\\big[\\nbsl \\hat J(l)\\big] \\\\\n & = \\frac{1}{4 l^-} \\int d^4z e^{iz\\cdot l}\n \\vev{0|\\overline\\psi(z)\\gamma^-\\psi(0)|0} \\ .\n \\label{eq:j2}\n\\end{split}\n\\end{align}\nAfter these manipulations, the hadronic tensor reads\n\\begin{align}\n\\begin{split}\n W^{\\mu\\nu}(p,q) & = \\int \\frac{dk^+dk^-d^2k_T}{(2\\pi)^4} \\\\\n & \\times \\frac{e_f^2}{8\\pi} \n \\mbox{Tr} \\big[\\ksl \\gamma^\\nu \\lsl \\gamma^\\mu \\big] \\,j_2(l)\\,\n \\tau_2(k)\\, {\\mathbb K}(k,p,q) \\ ,\n \\label{eq:W-step0}\n\\end{split}\n\\end{align}\nwhere \n\\begin{align}\n k^\\mu & = x p^+ {\\overline n}^\\mu \n + \\frac{k^2+k_T^2}{2 x p^+} n^\\mu \n + k_T^{\\,\\mu} \\\\\n l^\\mu & = (x-\\xi) p^+ {\\overline n}^\\mu \n + \\big( \\frac{k^2+k_T^2}{2 x p^+} \n + \\frac{Q^2}{2\\xi p^+} \\big)n^\\mu \n + k_T^{\\,\\mu} \\ . \n\\end{align}\nFor later use, let us also define \n\\begin{align}\n \\frac{1}{\\pi} H_*^{\\mu\\nu}(k,l) = \\frac{e_f^2}{8\\pi} \n \\mbox{Tr} \\big[\\ksl \\gamma^\\nu \\lsl \\gamma^\\mu \\big] \\ .\n \\label{eq:H*def}\n\\end{align}\n\nOur goal is to obtain a factorized expression for the hadronic tensor\nin terms of collinear parton distribution functions, see for example\nEq.~\\eqref{eq:quarkPDFatLO}. For this purpose we need to let\n$\\int dk^-d^2k_T$ act only on $\\tau_2(k)$, which defines the collinear\nPDF modulo factors of 2. In doing this we will be forced to make\napproximations on the momenta entering and exiting the hard scattering\nvertex, viz., $k$ and $l$. In principle, one would like to avoid it and\nallow approximations in the computation of the hard scattering\ntensor only \\cite{Collins:2007ph}. In this way, one can ensure that\nthe final state obeys 4-momentum conservation, and avoid potentially\nlarge errors in region of phase-space close to the kinematic\nboundaries. While in most cases this is not a problem for inclusive\ncross sections, it might become very important for exclusive\nobservables. In our case, we want to compute the inclusive DIS cross\nsection at large $x_B {\\rightarrow} 1$ in collinear factorization: \nin order to extend the validity of our\ncomputation as close as possible to this kinematic boundary, we\nneed to pay attention to the approximations we will make, keep\nthem at a minimum, and estimate the range of validity in $x_B$ and \n$Q^2$ of the approximations we will have to make.\n\nThe first step in the collinear factorization of the hadronic tensor \n\\eqref{eq:W-step0}, is to expand $H_*^{\\mu\\nu}$ around the momentum of\na collinear and massless quark:\n\\begin{align}\n H_*^{\\mu\\nu}(k,l) = H_*^{\\mu\\nu}(\\widetilde k,\\widetilde l)\n + \\frac{\\partial H_*^{\\mu\\nu}}{\\partial k^\\alpha}\n (k^\\alpha - \\widetilde k^\\alpha) + \\ldots\n\\end{align}\nwhere\n\\begin{align}\n\\begin{split}\n \\widetilde k^\\mu & = x p^+ {\\overline n}^\\mu \\\\\n \\widetilde l^\\mu & = \\widetilde k^\\mu + q^\\mu \\ .\n\\end{split}\n\\end{align}\nThe higher order terms in the expansion are suppressed as powers of\n$\\Lambda^2\/Q^2$, where $\\Lambda^2$ is a hadronic scale, and contribute\nto restore gauge invariance in higher-twist diagrams \\cite{Qiu:1988dn}. In\nthis paper, we will retain only the leading twist term of the\nexpansion. Note that we did not yet make any kinematic\napproximation: in principle, one may sum over as many higher-twist\nterms as desired.\n\nThe second step involves using the spectral representation\nof $\\hat J$ \\cite{Weinberg:1995mt} to explicitly introduce the\ninvariant jet mass in the formalism:\n\\begin{align}\n \\hat J(l) = \\int_0^\\infty \\hspace*{-.3cm}dm_j^2\\, \n \\big[J_1(m_j^2) \\hat{\\mathbb I} + J_2(m_j^2) \\lsl \\big]\n 2\\pi \\delta(l^2-m_j^2) \\theta(l^0) \\ ,\n \\label{eq:Jhat}\n\\end{align}\nwhere the spectral functions $J_i(m_j^2)$ are positive definite and\nnormalized to 1:\n\\begin{align}\n \\int_0^\\infty \\hspace*{-.3cm}dm_j^2\\,J_i(m_j^2) = 1 \\ .\n \\label{eq:J2norm}\n\\end{align}\nIn particular, by substituting Eq.~(\\ref{eq:Jhat}) into (\\ref{eq:j2}),\nwe obtain\n\\begin{align}\n j_2(l) = \\int_0^\\infty \\hspace*{-.3cm}dm_j^2\\,J_2(m_j^2)\\,\n 2\\pi \\delta(l^2-m_j^2)\\, \\theta(l^0) \\ ,\n \\label{eq:j2-spectral}\n\\end{align}\nso that we can interpret $m_j$ as the jet invariant mass, and $J_2(m_j^2)$\nas its probability distribution. \n\nIn the light-cone\ngauge $n\\cdot A=0$, the jet spectral function is related to the non\nperturbative quark propagator: \n\\begin{align}\n\\begin{split}\n & \\int_0^\\infty \\hspace*{-.3cm}\n dm_j^2\\,J_2(m_j^2)\\,2\\pi \\delta(l^2-m_j^2)\\, \\theta(l^0) \\\\\n & \\qquad = \\frac{1}{4 l^-} \\int d^4z e^{iz\\cdot l}\n \\mbox{Tr} \\big[ \\gamma^- \\vev{ 0 | \\overline\\psi(z)\\psi(0)|0} \\big] \\ .\n \\label{eq:jet-qprop}\n\\end{split}\n\\end{align}\nComputations of the non-perturbative quark propagator have been\nperformed in lattice QCD \\cite{Bowman:2005vx} and using \nSchwinger-Dyson equations, see \\cite{Fischer:2006ub} for a review. \nHowever, there are several difficulties in extracting information\nrelevant to the jet spectral function from these computations: (i) the\nquark-antiquark correlator appearing in \\eqref{eq:jet-qprop} is \ntypically computed in the Landau gauge instead of the light-cone\ngauge, (ii) computations are performed in Euclidean space instead of\nMinkowski space, (iii) one needs to extract the spectral\nrepresentation from the computed correlator. \nThe biggest problem is that the analytic structure of the quark\npropagator is not sufficiently well known to either perform the\nanalytic continuation back to Minkowski space or to extract its\nspectral representation \\cite{Fischer:2006ub,Alkofer:2003jj}. \nAs a way to avoid this\nproblem, it would be interesting to see if it is possible to rotate\nthe whole handbag diagram, including its external momenta, to\nEuclidean space as done in \\cite{Blum:2002ii} for the computation of the\nhadronic contribution to the muon anomalous magnetic moment. \nIn this way one\nwould be able to directly use the lattice propagator in the\ncomputation of the forward Compton amplitude. Alternatively, one may \ntry to use the light-cone QCD formulation on the lattice discussed in\n\\cite{Grunewald:2007cy}, which exploits the Hamiltonian formulation of\nQCD in order to remain in Minkowski space. A more phenomenological\napproach to the spectral function will be discussed in the next \nsubsection.\n\nThe third step involves our first kinematic approximation. In order to\nfactorize $j_2$ from the $dk^-d^2k_T$ integrations, we need to\napproximate $l {\\rightarrow} \\widetilde l$, so that \n\\begin{align}\n j_2(l) \\,{\\longrightarrow}\\, j_2(\\widetilde l) \n = \\int_0^\\infty \\hspace*{-.3cm}dm_j^2\\,J_2(m_j^2)\\,\n 2\\pi \\delta(\\widetilde l^2-m_j^2)\\, \\theta(l^0) \\ .\n \\label{eq:approx-step3}\n\\end{align}\nThen, the hadronic tensor reads\n\\begin{align}\n W^{\\mu\\nu}(p,q) & = \\int_0^\\infty\\hspace*{-.3cm}dm_j^2\\,J_2(m_j^2) \n \\int dk^+\\, H_*^{\\mu\\nu}(\\widetilde k,\\widetilde l)\\,\n \\delta({\\widetilde l}^2-m_j^2) \n \\nonumber \\\\\n & \\times \n \\int \\frac{dk^-d^2k_T}{(2\\pi)^4} 2\\tau_2(k)\\, {\\mathbb K}(k,p,q)\n \\label{eq:W-step3}\n\\end{align}\nwhere $\\theta(l^0)=\\theta(k^0+q^0)$ is already included \nin the kinematic constraint function $\\mathbb K(k,p,q)$.\nWe can expect the approximation \\eqref{eq:approx-step3} \nto be reasonable in a region \nwhere $j_2(l)$ does not vary strongly with $l$. In terms of the\nspectral representation, this requirement is satisfied if the integral\nin Eq.~\\eqref{eq:W-step3} is dominated by values of $m_j^2$ close to where \nthe jet spectral function has a maximum. We will discuss below the\nconditions on $x_B$ and $Q^2$ for which this condition is satisfied.\nNote that this kinematic approximation only acts on the\n$\\delta$-function in Eq.~\\eqref{eq:W-step3} so that $J_2(m_j^2)$ has\nbeen left unapproximated: in this sense the approximation is the\nmildest possible compatible with collinear factorization.\n\nThe fourth step involves decoupling $\\mathbb K$ and $\\tau_2$ in\nEq.~\\eqref{eq:W-step3}. It can be achieved by\nreplacing\n\\begin{align}\n {\\mathbb K}(k,q,p) \\,{\\longrightarrow}\\, {\\mathbb K}(\\widetilde k,q,p) \n = \\theta(\\widetilde x_f-x_B) \\theta(1-\\widetilde x_f) \\ .\n\\end{align}\nNote that $\\theta(\\widetilde k^0+q^0)=\\theta(p^+-\\widetilde k^+)\n=\\theta(p^--\\widetilde k^-)=1$ because of the constraints on\n$\\widetilde x_f$. In terms of $x$,\n\\begin{align}\n \\xi \\leq x \\leq \\xi\/x_B \\ .\n\\end{align}\nThis is a delicate step because it involves approximating the\nkinematic constraints, such that the integration over $k_T$ and $k^-$\nare unbounded. This clearly is not a good approximation as $x_B {\\rightarrow} 1$\n\\cite{Collins:2007ph}, in which case the struck parton carries most of the\nnucleon plus-momentum, so that the minus and transverse components\ncannot be large. To appreciate this, consider\n\\begin{align}\n s = (p+q)^2 = (p_Y + l)^2 \\ ,\n\\end{align}\nwhere $p_Y$ is the total four momentum of the target jet, and we\ndefine $m_Y^2 = p_Y^2 \\geq 0$, see\nFig.~\\ref{fig:DISamplitude}. Using the full kinematics of\nEq.~\\eqref{eq:kinematics}, we obtain\n\\begin{align}\n s = \\frac{1-\\xi}{\\xi} Q^2 + (1-\\xi) m_N^2 \\ .\n\\end{align}\nOn the other hand, in the center-of-mass frame, $\\vec p_Y = - \\vec l$\nand $\\vec p_{Y,T} = - \\vec l_T = - \\vec k_T$, so that \n\\begin{align}\n s & = (p_Y^0 + l^0)^2 \\\\\n & = \\Big( \\sqrt{m_Y^2 + k_T^2 + (p_Y^3)^2}\n + \\sqrt{m_j^2 + k_T^2 + (l^3)^2} \\,\\Big)^2 > 4k_T^2 \n \\nonumber\n\\end{align}\nCombining these 2 results, we obtain\n\\begin{align}\n k_T^2 < \\frac{1-\\xi}{4\\xi} Q^2 \\big( 1 + \\xi\\frac{m_N^2}{Q^2} \\big)\n \\ .\n \\label{eq:ktbound}\n\\end{align}\nAs $x_B{\\rightarrow} 1$, $\\xi {\\rightarrow} \\xi_{th} \\lesssim 1$ so that the $(1-\\xi)$\nfactor tends to close the available $k_T$ phase space. \nIn Section~\\ref{sec:numest}, we will discuss\nin which region of $x_B$ and $Q^2$ we may in fact neglect this\nbound. Using the definition \\eqref{eq:tau2def} of $\\tau_2$,\nthe hadronic tensor reads\n\\begin{align}\n W^{\\mu\\nu}(p,q) & = \\int_0^\\infty\\hspace*{-.3cm}dm_j^2\\,J_2(m_j^2) \n \\label{eq:W-step4} \\\\\n & \\times \\int_\\xi^{\\xi\/x_B} \\frac{dx}{x} \n H_*^{\\mu\\nu}(\\widetilde k,\\widetilde l)\\,\n \\delta({\\widetilde l}^2-m_j^2) \\varphi_q(x,Q^2) \\ ,\n \\nonumber\n\\end{align}\nwhere the quark PDF $\\varphi_q$ is defined as in\nEq.~\\eqref{eq:quarkPDFatLO}.\n\nAs a last step, we define an on-shell and massless jet momentum \nfor the partonic tensor, \n\\begin{align}\n \\widehat l^\\mu = \\widetilde l^- n^\\mu = \\frac{Q^2}{2\\xi p^+} n^\\mu\n \\, \n\\end{align}\nand replace \n\\begin{align}\n H_*^{\\mu\\nu}(\\widetilde k,\\widetilde l) \\,{\\longrightarrow}\\, \n H_*^{\\mu\\nu}(\\widetilde k,\\widehat l) \n = \\frac{e_f^2}{8} \n \\mbox{Tr} \\big[\\widetilde\\ksl \\gamma^\\nu \\widehat\\lsl \\gamma^\\mu \\big] \n\\end{align}\nThis is needed: (i) to ensure that $q_\\mu H_*^{\\mu\\nu} = 0$, hence \nthe gauge invariance of the hadronic tensor, and (ii) to allow use of\nthe Ward identities in proofs of factorization \\cite{Collins:2007ph}. \nThis approximation, made on the hard scattering coefficient,\nis less critical then the kinematic approximations previously\ndiscussed because it does not change in itself the kinematics of the\nprocess. It is analogous to the approximation taken in considering the\nusual handbag diagram of Fig.~\\ref{fig:DIS_LO} \nwith a massless quark line joining the 2 virtual photon, \nexcept that it approximates only the computation of the Dirac traces.\n\nFinally, we define the LO hard scattering tensor\n\\begin{align}\n {\\mathcal H}_f^{\\mu\\nu}(\\widetilde k,q,m_j^2)\n & = H_*^{\\mu\\nu}(\\widetilde k,\\widehat l) \n \\delta(\\widetilde l^2-m_j^2) \\\\\n & = \\mbox{Tr} \\big[\\widetilde\\ksl \\gamma^\\nu \\widehat\\lsl \\gamma^\\mu \\big]\n \\delta(\\widetilde l^2-m_j^2) \\ ,\n\\end{align}\nwhich for $m_j^2=0$\ncoincides with the LO hard scattering tensor\ncomputed for a diagram without jet function, as in\nEq.~\\eqref{eq:pQCDfactnolims}. \nThe hadronic tensor can then be written in factorized form as\n\\begin{align}\n W^{\\mu\\nu}(p,q) & = \\int_0^\\infty\\hspace*{-.3cm}dm_j^2\\,J_2(m_j^2) \n \\label{eq:W-JET} \\\\\n & \\times \\int_\\xi^{\\xi\/x_B} \\frac{dx}{x} \n {\\mathcal H}_f^{\\mu\\nu}(\\widetilde k,q,m_j^2)\\, \\varphi_q(x,Q^2) \\ ,\n \\nonumber\n\\end{align}\nwhich is the central result of this section. \nThe transverse structure function reads\n\\begin{align}\n\\begin{split}\n & F_T(x_B,Q^2,m_N^2) =\n \\int_0^\\infty\\hspace*{-.3cm}dm_j^2\\,J_2(m_j^2) \\\\\n & \\qquad\\qquad \\times \\sum_f \\int_\\xi^{\\xi\/x_B} \\frac{dx}{x} \n h_{f|T}(\\widetilde x_f,Q^2,m_j^2) \\varphi_q(x,Q^2) \\ ,\n \\label{eq:FT-JET} \n\\end{split}\n\\end{align}\nwith $\\varphi_q$ defined in Eq.~\\eqref{eq:quarkPDFatLO}.\nThe longitudinal structure function $F_L=0$ because $h_L=0$.\nAn explicit computation gives $h_T(\\widetilde x_f,Q^2) = \\frac12 e_f^2\nx \\delta\\big(x-\\xi(1+m_j^2\/Q^2)\\big)$ so that at LO\n\\begin{align}\n & F_T(x_B,Q^2,m_N^2) \n \\label{eq:FTL_TMC_JMC_precise} \\\\\n & \\qquad = \\int_0^{\\frac{1-x_B}{x_B}Q^2} \\hspace*{-.2cm} dm_j^2 J_2(m_j^2) \n F_T^{(0)}\\bigg(\\xi\\Big(1+\\frac{m_j^2}{Q^2}\\Big),Q^2\\bigg) \\ , \\nonumber \n\\end{align}\nNote that when $Q^2 \\gg \\vev{m_j^2}$, where $\\vev{m_j^2} = \\int dm_j^2\\,\nm_j^2 J_2(m_j^2)$, the massless $F_T^{(0)}$ decouples from the integration\nover the jet mass, and we recover the TMC to the LO structure functions. \n\n\\begin{figure*}[tb]\n \\vspace*{0cm}\n \\centerline{\n \\includegraphics\n [width=0.45\\linewidth]\n {fig.FTratio.002.eps}\n \\hspace*{0.5cm}\n \\includegraphics\n [width=0.45\\linewidth]\n {fig.FTratio.025.eps}\n }\n \\caption[]{Effect of jet mass corrections on $F_T$, computed with a\n toy jet spectral function as described in the\n text. Plotted is the ratio of $F_T$ with both TMC and JMC to $F_T$\n with only TMC included, as a function of $x_B$ for $Q^2=2$ and 25\n GeV$^2$. The shaded band corresponds to a log-normal jet mass\n distribution with $\\vev{m_j^2} = 0.2-0.4$ GeV$^2$ and\n $\\sigma_{m_j^2} = \\vev{m_j^2}-2 \\vev{m_j^2}$. The dashed and dot-dashed\n lines corresponds to delta functions at $m_j^2=m_\\pi^2$ and\n $m_j^2=m_N^2$, respectively. \n }\n \\label{fig:FT_JMC}\n\\end{figure*}\n\n\n\\subsection{The jet spectral function}\n\nLet us discuss more in detail the properties of the nonperturbative\nquark spectral function $J_2$, defined in\nEq.~\\eqref{eq:j2-spectral}. Let us start from the definition of the \n$j_2$ component of the jet function, \n\\begin{align}\n\\begin{split}\n j_2(l) & = \\frac{1}{4l^-}\\mbox{Tr}\\big[ \\gamma^- \\hat J(l)\\big] \\\\\n & = \\sum_Y \\delta^{(4)}\\big(l-\\sum_{i\\in Y} p_i\\big)\n \\vev{0|\\bar\\psi_f(0)|Y}\\gamma^- \\vev{Y|\\psi_f(0)|0} \\ ,\n\\end{split}\n\\end{align}\nwith $f$ the quark flavor. For simplicity, we consider only light\nquark flavors with $m_f \\ll m_\\pi$.\nThe color $c$ of the quark operator $\\psi$ is not neutralized, so\nthat it must appear in the final state $\\ket{Y}$. \nIn the physical process, we are \nassuming that the struck quark's color is neutralized by a soft\ngluon exchange with the target's remnant.\nWe also assume that we can neglect\nthe soft exchange for the purpose of evaluating the change of \nkinematics induced by the inclusion of a quark jet function on \nthe lowest order contribution to the inclusive DIS. \nThis assumption is likely valid if the jet and \ntarget rapidities are sufficiently separated. \nHowever, this might not be the case \nclose to the kinematic limit $x_B=1$, and the approximation\nwill break down as we shall soon see. Because of color\nconfinement, we may assume that no more than 1 particle in the final\nstate is colored, all the other ones binding into colorless hadrons.\nThe colored particle must be a quark, to match the quark operator's\ncolor, and we denote it by $\\ket{q_{f'}^c}$.\nHence, the final state is made of 1 quark plus\nan arbitrary number of hadrons, the lightest of which is a pion:\n\\begin{align}\n \\ket{Y} = \\ket{q_{f'}^c}\\ket{h_1} \\cdots \\ket{h_N} \\ ,\n \\label{eq:finalstates}\n\\end{align}\nwith $N\\geq0$ and $f'=f$ when $N=0$. \nThe spectral function $J_2$, defined in Eq.~\\eqref{eq:j2-spectral},\ncan be written as \n\\begin{align}\n J_2(m_j^2) = Z \\delta(m_j^2 - m_q^2) + (1-Z) \\rho(m_j^2) \\ ,\n \\label{eq:J2general}\n\\end{align}\nwhere the $\\delta$-function is due to the contribution of the single\nparticle $\\ket{q_{f'}^c}$, and $00$ in Eq.~\\eqref{eq:finalstates} and\nis normalized to 1 because of Eq.~\\eqref{eq:J2norm}. Due to the\nassumption \\eqref{eq:finalstates}, $\\rho$ has a bell shape:\nit is equal to 0 up to $m_j^2 = (m_\\pi+m_q)^2 \\approx m_\\pi^2$,\nincreases up to a maximum and then tends to 0 as $m_j^2 {\\rightarrow} \\infty$ to\nsatisfy the normalization condition.\nUsing Eq.~\\eqref{eq:J2general} in \\eqref{eq:FTL_TMC_JMC_precise}, we\nobtain \n\\begin{align}\n\\begin{split}\n & F_T(x_B,Q^2,m_N^2) \n = Z \\, F_T^{(0)}\\big(\\xi,Q^2\\big) \\\\\n & \\quad + (1-Z) \\int_{m_\\pi}^{\\frac{1-x_B}{x_B}Q^2} \\hspace*{-.2cm} \n dm_j^2 \\rho(m_j^2) \n F_T^{(0)}\\bigg(\\xi\\Big(1+\\frac{m_j^2}{Q^2}\\Big),Q^2\\bigg) \\ .\n \\label{eq:JMCnosoft}\n\\end{split}\n\\end{align}\nSetting $Z=1$ is equivalent to calculating the standard handbag\ndiagram without the jet function, and one recovers the TMC formula.\n\nThe first term in Eq.~\\eqref{eq:JMCnosoft} shows that the introduction\nof the jet function in the handbag diagram goes some way toward\nsoftening the problem with the unphysically positive $F_T$ at $x_B=1$,\nbut does not solve it. The reason is that we cannot kinematically\nneglect the effect of the color neutralizing soft interactions when we\ncompute the hadbag diagram close to $x_B=1$, where the\nrapidity difference between the current and target jets\nis becoming smaller and smaller. A full solution\nto this problem is the inclusion of a ``soft function'', in\naddition to the target and jet functions, which describes the soft\nexchanges in the context of fully unintegrated\ncorrelation functions \\cite{Collins:2007ph}. \nThe soft function has essentially \nthe effect of smearing the jet function, avoiding the\nsingular behavior displayed by the $\\delta$-function. \nFor a phenomenological inclusion of the soft function in collinear\nfactorization, we can substitute $J_2$ with a continuous function \n$J_m$ such that \n\\begin{align}\n J_m(m_j^2) \\xrightarrow[m_j^2 {\\rightarrow} 0]{} 0 \\ ,\n\\end{align}\nbecause of phase space, and\n\\begin{align}\n J_m(m_j^2) \\xrightarrow[m_j^2 \\gg m_\\pi^2]{} J_2(m_j^2) \\ .\n\\end{align}\nIt can be physically interpreted as the (smeared) jet mass distribution,\nanalogously to $J_2$, and we will call it smeared jet spectral\nfunction. The structure function is then computed as in the Ansatz\ndiscussed at the beginning of the Section: \n\\begin{align}\n & F_T(x_B,Q^2,m_N^2) \n \\label{eq:JMCeffsoft} \\\\\n & \\qquad = \\int_{m_\\pi}^{\\frac{1-x_B}{x_B}Q^2} \\hspace*{-.2cm} \n dm_j^2 J_m(m_j^2) \n F_T^{(0)}\\bigg(\\xi\\Big(1+\\frac{m_j^2}{Q^2}\\Big),Q^2\\bigg) \\ .\n \\nonumber\n\\end{align}\nWe note that the jet spectral function $J_2$ is defined as a quark\ncorrelation function in vacuum, therefore it is process-independent.\nOn the other hand, the smeared jet function $J_m$ is process-dependent\nbecause it effectively includes the soft momentum exchange with the\ntarget. As a result, it's shape at $m_j^2 \\lesssim m_\\pi^2$ might\ndepend on $x_B$ and $Q^2$. However, the average jet mass squared,\n$\\vev{m_j^2}_m=\\int_0^\\infty dm_j^2 \\, m_j^2\\, J_m(m_j^2)$ should\nexhibit a small sensitivity on $x_B$ and $Q^2$ because we may expect \n$\\vev{m_j^2} \\gg m_\\pi^2$, see Section~\\ref{sec:numest}.\nEq.~\\eqref{eq:JMCeffsoft} is a reasonable approximation to the full\nhandbag diagram \ncomputation in the region of $(x_B,Q^2)$ phase space where the\nintegration over $dm_j^2$ in Eq.~\\eqref{eq:FTL_TMC_JMC_precise}\nextends well beyond the peak of the continuum $\\rho(m_j^2)$, \nnamely if \n\\begin{align}\n \\frac{1-x_B}{x_B}Q^2 \\gtrsim \\vev{m_j^2}_\\rho \\ ,\n \\label{eq:JmJ2validity}\n\\end{align}\nwhere\n\\begin{align}\n \\vev{m_j^2}_\\rho = \\int_{m_\\pi^2}^\\infty dm_j^2 \\, m_j^2\n J_2(m_j^2) \\ .\n\\end{align}\nIn these conditions, the structure function \\eqref{eq:JMCeffsoft} is\nnot much sensitive to the behavior of the jet function at small \n$m_j^2$, where $J_m$ may substantially differ from $J_2$.\n\nFor practical applications of JMC to global QCD fits of the PDFs, it\nis necessary to develop a flexible enough and realistic\nparametrization of the smeared jet spectral function $J_m$. For this\npurpose, one may try to use a Monte Carlo simulation of DIS events in\norder to generate enough data and test possible parametrizations. One\nmay also study the invariant jet mass distribution in $e^++e^- {\\rightarrow}\n\\,\\text{jets}$ events, where the same jet function $\\hat J$ discussed in\nthis Section appears in the LO cross-section. However, these studies\nlie outside the scope of this paper, and we leave them for the\nfuture. \n\n\\subsection{Numerical estimates}\n\\label{sec:numest}\n \nIn order to obtain an estimate for the magnitude of JMC and of the\npresent theoretical uncertainty, we employ a toy model for the jet\nspectral function. Let's consider a bell-shaped smooth function such\nas the log-normal distribution \n\\begin{align}\n f(x;\\mu,\\sigma) & = \\frac{1}{x\\sigma \\sqrt{2\\pi}} \n \\exp\\left[ - \\frac{(\\log x - \\mu)^2}{2\\sigma^2} \\right]\n\\end{align}\nwhere \n\\begin{align}\n\\begin{split}\n \\mu & = \\frac12 \\log \\left( \\frac{\\bar x^4}{\\bar x^2+\\sigma_x^2}\n \\right) \\\\ \n \\sigma & = \\left[ \\log\\left( \\frac{\\sigma_x^2}{\\bar x^2}+1\n \\right)\\right]^{\\frac 12} \\ ,\n\\end{split}\n\\end{align}\nand $\\bar x$ and $\\sigma_x$ are the average value of $x$ and its\nstandard deviation. Then, we can parametrize the continuum\npart $\\rho$ of the toy jet mass distribution in\nEq.~\\eqref{eq:FTL_TMC_JMC_precise} \nin terms of the average jet mass $\\vev{m_j^2}_\\rho$ and its standard\ndeviation $\\sigma_{m_j^2}$:\n\\begin{align}\n \\rho(m_j^2) = f(m_j^2-m_\\pi^2;\\mu,\\sigma) \\ ,\n\\end{align}\nwith \n\\begin{align}\n\\begin{split}\n \\mu & = \\vev{m_j^2}_\\rho - m_\\pi^2 \\\\ \n \\sigma & = \\sigma_{m_j^2} \n\\end{split}\n\\end{align}\nin units of GeV$^2$. \nFrom the typical particle \nmultiplicity of the current jet at the JLab energy, \nwe estimate $\\vev{m_j^2}_\\rho =0.2-0.4$ GeV$^2$, and assume \n$\\sigma_{m_j^2} = C \\vev{m_j^2}_\\rho$ with $C=1-2$.\n\nIn Fig.~\\ref{fig:FTvsFT0}, we plot the JMC to the transverse structure\nfunction as obtained in Eq.~\\eqref{eq:FTL_TMC_JMC_precise} by\nneglecting soft momentum exchanges.\nThe dashed line corresponds to $Z=1$, and is equivalent to computing\nonly TMC. The solid line corresponds to JMC coming only from the\ncontinuum part $\\rho$ of the jet spectral function, i.e., $Z=0$.\nFor comparison, the massless structure function is plotted as a dotted\nline. The true jet mass corrected $F_T$ should lie somewhere in\nbetween because $01$. \nWhen performing global QCD fits of the PDFs\nin the context of pQCD collinear factorization, the procedure\npresented in this paper might be the most consistent way to treat TMC, \nbecause it expresses the long distance physics of structure functions,\nand the leading target mass correction, in terms of PDFs that share the\nsame partonic operators with the PDFs of zero hadron mass. Hence it\nallows to unambiguously separate the kinematic effects of the target's\nmass from its dynamical contribution to parton matrix elements and the\nPDFs. \n\nOur formalism for TMC in Eq.~\\eqref{eq:FTL_TMC}\nis valid at leading twist and any order in\n$\\alpha_s$. Calculating TMC for the power-suppressed higher-twist\ncontributions to the structure functions is a non-trivial\n\\cite{Ellis:1982cd} but important issue for measuring the size \nof parton correlations in the nucleon wave-function, which we leave\nto a future effort. The leading-twist formalism can be easily\nextended to polarized DIS structure functions \\cite{AW}, for which a\ncorrect evaluation of TMC is even more important than in the\nunpolarized case because the bulk of available data is in fact in the\nlarge-$x_B$ domain. \nThe extension to semi-inclusive DIS and to hadronic collisions is also\nvery important, in order to fully include TMC in global PDF fits. An example is\nthe Drell-Yan cross-section at large Feynman $x_F$, which has the\npotential to further constrain large-$x$ PDFs \\cite{Owens:2007kp}. \nIt is also straightforward to extend the TMC analysis \nto DIS on nuclear targets, in order to include the effects\nof nucleon binding and Fermi motion \\cite{AQV}. This is especially\nimportant for studying the large-$x$ neutron PDFs and the $d\/u$ ratio,\nwhich are extracted from data taken with a Deuterium target.\n\nIn the second part of the paper, we examined the impact of a\nfinal-state jet function on the extraction of PDFs at large $x_B$. \nWe proposed to write the leading order hadronic tensor, \nhence the lowest order contribution to DIS cross section,\nin terms of the spectral representation $J_2$ of the jet function,\nwhich has the physical meaning of invariant jet mass distribution. \nWe evaluated the impact of JMC on the leading order DIS\nstructure functions, and found it to be \npotentially large even at not so small values of photon virtuality\nsuch as $Q^2 = 25$ GeV$^2$. \nIn the NLO cross-section, the impact of JMC is likely to be\nreduced, because a non-zero jet invariant mass can be produced in the\nhard scattering beyond tree level, but is still potentially large.\nWe also evaluated the range of validity in\n$x_B$ and $Q^2$ of the approximations we made.\n\nFor practical applications to global fits of PDFs, it is important\nto investigate the shape and properties of the smeared jet spectral\nfunction $J_m$, which effectively includes the neglected soft momentum\nexchanges in the final state.\nThis can be phenomenologically done using a Monte-Carlo simulation and\nthen trying several parametrizations of $J_m$. In a more fundamental\napproach, we noticed that the jet spectral function $J_2$ is related \nto the non-perturbative quark propagator, which can be computed in\nlattice QCD or using Schwinger-Dyson equations. To avoid the\ndifficulties connected to the analytic continuation to\nMinkowski space, one may try and rotate the whole handbag\ndiagram to Euclidean space, or use a Hamiltonian-based formulation of\nlattice QCD.\n\nIn conclusion, the obtained results on TMC and JMC will be very important\nwhen using large-$x_B$ and low-$Q^2$ data on DIS structure\nfunction (like those obtained at Jefferson Lab)\nto extract reliable PDFs at large-$x$, and to disentangle\nkinematic effects from the dynamically interesting higher-twist parton\ncorrelations. The discussed extensions of our formalism to other\nprocesses will allow a full inclusion of TMC and JMC in global QCD\nfits of parton distribution functions.\n\n\n\n\\begin{acknowledgments}\nWe thank C.~Aubin, T.~Blum, M.~E.~Christy, J.~C.~Collins, V.~Guzey,\nC.~E.~Keppel, P.~Maris, W.~Melnitchouk, D.~Richards, C.~Weiss\nfor useful discussions and suggestions. This work has been supported\nin part by the U.S. Department of Energy, Office of \nNuclear Physics, under contract DE-AC02-06CH11357\nand contract DE-AC05-06OR23177 \nunder which Jefferson Science Associates, LLC \noperates the Thomas Jefferson National Accelerator Facility, and \nunder grant DE-FG02-87ER40371 and \ncontract DE-AC02-98CH10886.\n\\end{acknowledgments}\n\n\n\\begin{appendix}\n\n\\section{Kinematic constraints at finite $Q^2$}\n\\label{app:kinematics}\n\nLet us consider the handbag diagram for a DIS process on a nucleon\ntarget, as depicted in the right hand side of \nFig.~\\ref{fig:DISfactorization}. We repeat the kinematic analysis of\nthe handbag diagram performed in Section~\\ref{sec:TMC}, but for the\ngeneral case of an off-shell bound parton of momentum $k$, and $k^2 \\lesssim\nm_f^2$. The limit of on-shell quarks of mass $m_f^2$,\nrelevant to collinear factorization, can be obtained setting\n$k^2=m_f^2$ and $x_f = \\widetilde x_f$ in the formulae below. \n\nWe consider the scattering of a\ngeneric vector boson ($\\gamma, W^\\pm, Z$) on a parton of flavor $f$\nof mass $m_f$. The lowest order couplings are displayed in\nFig.~\\ref{fig:LOcouplings}. The masses of the quarks (other than $f$)\ncoupled to the vector boson are $m_1$ and $m_2$. The current jet mass\nmust satisfy \n\\begin{align}\n m_j^2 \\geq s_\\text{th}\n\\end{align}\nwhere\n\\begin{align}\n s_\\text{th} = (m_1+m_2)^2 \\ .\n\\end{align}\nAs discussed in Sec.~\\ref{sec:TMC}, the net baryon number is likely \nto flow through the bottom of the handbag diagram for the leading \nDIS contribution that is given by the collinear factorization formalism.\nTherefore,\n\\begin{align}\n s_\\text{th} \\leq m_j^2 \\leq s-m_N^2 \\ .\n\\end{align}\nUsing $m_j^2 = (q+k)^2 = k^2 + (1\/x_f-1)Q^2$ we obtain\n\\begin{align}\n \\frac{x_B}{1-x_B k^2\/Q^2} \n \\leq x_f \\leq \n \\frac{1}{1+(s_\\text{th}-k^2)\/Q^2} \\ .\n \\label{eq:xflims}\n\\end{align}\nUsing $m_j^2=(Q^2+\\frac{\\xi}{x}k^2)(\\frac{x}{\\xi}-1)$,\nEq.~\\eqref{eq:xflims} can alternatively be expressed as limits over the\nfractional momentum $x=k^+\/p^+$: \n\\begin{align}\n x^{min} \\leq x \\leq x^{max}\n\\end{align}\nwhere\n\\begin{align}\n & x^{min} = \\xi \\frac{Q^2+s_\\text{th}-k^2+\\Delta[k^2,-Q^2,s_\\text{th}]}{Q^2} \n \\nonumber \\\\\n & x^{max} = \\xi \\frac{Q^2+s-m_N^2-k^2+\\Delta[k^2,-Q^2,s-m_N^2]}{Q^2} \n \\nonumber \\\\\n & \\Delta[a,b,c] = \\sqrt{a^2 + b^2 + c^2 - 2(ab+bc+ca)} \\ .\n \\label{eq:xlims}\n\\end{align}\nWe finally note that\n\\begin{align}\n x_f & = \\frac{\\xi}{x} \n \\frac{1}{1-\\frac{\\xi^2}{x^2}\\frac{k^2}{Q^2}} \\ .\n\\end{align}\n\n\n\\begin{figure}[tb]\n \\vspace*{0cm}\n \\centerline{\n \\includegraphics\n [width=0.9\\linewidth]\n {fig.LOcouplings.eps}\n }\n \\caption[]{\n Lowest order couplings of a generic vector boson ($\\gamma, W^\\pm,\n Z$) to a parton of flavor $f$ and mass $m_f$. The masses of the\n produced quarks are $m_1$ and $m_2$. Left: boson-quark\n scattering ($m_2=0$). Right: boson-gluon fusion. \n }\n \\label{fig:LOcouplings}\n\\end{figure}\n\n\n\\section{Invariant and helicity structure functions}\n\\label{app:strfns}\n\n\\subsection{Helicity structure functions}\n\\label{app:helicitystrfns}\n\nWe work in a collinear frame and for generality we keep the quark mass\ndifferent from zero.\nFor collinear on-shell partons we have from Eq.~\\eqref{eq:kinematics}\n\\begin{align}\n\\begin{split}\n p^\\mu & = p^+ {\\overline n}^\\mu \n + \\frac{m_N^2}{2 p_A^+} n^\\mu \\\\\n q^\\mu & = - \\xi p^+ {\\overline n}^\\mu \n + \\frac{Q^2}{2\\xi p^+} n^\\mu \\\\\n \\widetilde k^\\mu & = x p^+ {\\overline n}^\\mu \n + \\frac{m_f^2}{2xp^+} n^\\mu \\ ,\n\\end{split}\n\\end{align}\nwhere $m_f$ is the mass of the parton of flavor $f$. For later use, we define the shorthands\n\\begin{align}\n \\rho_B^2 = 1+4x_B^2\\frac{m_N^2}{Q^2} \\qquad \n \\rho_f^2 = 1+4x_f^2\\frac{m_f^2}{Q^2} \\ ,\n\\end{align}\nwhere, as in Eq.~\\eqref{eq:invariants},\n\\begin{align}\n x_B = \\frac{-q^2}{2 p\\cdot q} \\qquad\n x_f = \\frac{-q^2}{2 \\widetilde k\\cdot q } \\ .\n\\end{align}\n\nFollowing \\cite{Aivazis:1993kh}, we define the longitudinal, transverse and\nscalar polarization vectors with respect to the virtual photon\nmomentum $q$ and a reference vector $p$,\n\\begin{align}\n \\varepsilon_0^\\mu(p,q) \n & = \\frac{-q^2p^\\mu+(p\\cdot q)q^\\mu}{\\sqrt{-q^2}[(p\\cdot q)^2-q^2p^2]}\n = \\frac{-q^2p^\\mu+(p\\cdot q)q^\\mu}{\\sqrt{-q^2}(p\\cdot q) \\rho^2(p,q)} \n \\nonumber \\\\\n \\varepsilon_\\pm^\\mu(p,q) \n & = \\frac{1}{\\sqrt{2}} (0,\\pm 1,-i,0) \\nonumber \\\\\n \\varepsilon_q^\\mu(p,q) \n & = \\frac{q^\\mu}{\\sqrt{-q^2}} \\ ,\n \\label{eq:polvect}\n\\end{align}\nwhere\n\\begin{align}\n \\rho^2(p,q) = 1-p^2q^2\/(p\\cdot q)^2 \\ .\n\\end{align}\nIt is immediate to verify that $\\rho^2(p,q) = \\rho_B^2$ and\n$\\rho^2(\\widetilde k,q) = \\rho_f^2$.\nThe polarization vectors satisfy the following conditions\n\\begin{alignat}{2}\n & \\varepsilon_\\lambda \\cdot \\varepsilon_{\\lambda'} = 0 \n & \\qquad & \\text{for\\ } \\lambda\\neq\\lambda' \\nonumber \\\\\n & \\varepsilon_\\lambda \\cdot \\varepsilon_\\lambda = 1 \n & \\qquad & \\text{for\\ } \\lambda = 0,+,- \\\\\n &\\varepsilon_q \\cdot \\varepsilon_q = - 1 \\nonumber\n\\end{alignat}\nand, in particular, $q\\cdot \\varepsilon_0 = q\\cdot \\varepsilon_\\pm = 0$.\nThe helicity structure functions $F_\\lambda$ are defined as\nprojections of the hadronic tensor:\n\\begin{align}\n F_\\lambda(x_B,Q^2) = P_\\lambda^{\\mu\\nu}(p,q) W_{\\mu\\nu}(p,q)\n\\end{align}\nwith $\\lambda=L,T,A,S,\\{0q\\},[0q]$. The longitudinal, transverse,\naxial, scalar, and mixed projectors $P_\\lambda^{\\mu\\nu}$ are\n\\begin{align}\n\\begin{split}\n P_L^{\\mu\\nu}(p,q) & = \\varepsilon_0^\\mu(p,q) \\varepsilon_0^{\\nu*}(p,q) \\\\\n P_T^{\\mu\\nu}(p,q) & = \\varepsilon_+^\\mu(p,q) \\varepsilon_+^{\\nu*}(p,q) \n + \\varepsilon_-^\\mu(p,q) \\varepsilon_-^{\\nu*}(p,q) \\\\\n P_A^{\\mu\\nu}(p,q) & = \\varepsilon_+^\\mu(p,q) \\varepsilon_+^{\\nu*}(p,q) \n - \\varepsilon_-^\\mu(p,q) \\varepsilon_-^{\\nu*}(p,q) \\\\\n P_q^{\\mu\\nu}(p,q) & = \\varepsilon_q^\\mu(p,q) \\varepsilon_q^{\\nu*}(p,q) \\\\\n P_{\\!\\!\\{0q\\!\\}}^{\\mu\\nu}(p,q) \n & = \\varepsilon_0^\\mu(p,q) \\varepsilon_q^{\\nu*}(p,q)\n + \\varepsilon_q^\\mu(p,q) \\varepsilon_0^{\\nu*}(p,q) \\\\\n P_{[0q]}^{\\mu\\nu}(p,q) & = \\varepsilon_0^\\mu(p,q) \\varepsilon_q^{\\nu*}(p,q)\n - \\varepsilon_q^\\mu(p,q) \\varepsilon_0^{\\nu*}(p,q) \\ . \n \\label{eq:helicityprojectors}\n\\end{split}\n\\end{align}\nUsing \n\\begin{align}\n\\begin{split}\n & \\varepsilon_+^\\mu(p,q) \\varepsilon_+^{\\nu*}(p,q) \n - \\varepsilon_-^\\mu(p,q) \\varepsilon_-^{\\nu*}(p,q)\n = \\frac{-i\\varepsilon^{\\mu\\nu\\alpha\\beta} p_\\alpha q_\\beta}\n {(p\\cdot q) \\rho_B} \\\\\n & \\varepsilon_+^\\mu(p,q) \\varepsilon_+^{\\nu*}(p,q) \n + \\varepsilon_-^\\mu(p,q) \\varepsilon_-^{\\nu*}(p,q) \\\\\n & \\qquad= -g^{\\mu\\nu} + \\varepsilon_0^\\mu(p,q) \\varepsilon_0^{\\nu*}(p,q)\n - \\varepsilon_q^\\mu(p,q) \\varepsilon_q^{\\nu*}(p,q) \\ , \n\\end{split}\n\\end{align}\none easily sees that \n\\begin{align}\n F_T(x_B,Q^2) & = - W^\\mu_\\mu(p,q) + F_L(x_B,Q^2) - F_q(x_B,Q^2) \n \\nonumber \\\\\n F_A(x_B,Q^2) & = \\frac{-i\\varepsilon^{\\mu\\nu\\alpha\\beta} p_\\alpha q_\\beta}\n {(p\\cdot q) \\rho_B} W_{\\mu\\nu}(p,q) \\ .\n \\label{eq:helicityids}\n\\end{align}\nEven if not apparent from Eq.~\\eqref{eq:polvect}, a consequence of the\nnormalization conditions is that the reference vector has the only\nfunction to define the $t-z$ and transverse planes in conjunction\nwith $q^\\mu$: as long as it lays in the $t-z$ plane, a different\nreference vector defines the same polarization vectors\n\\cite{Aivazis:1993kh}. For example, $\\varepsilon_\\lambda^\\mu(p,q) =\n\\varepsilon_\\lambda^\\mu(\\widetilde k,q)$.\nAs we will see, choosing $\\widetilde k$ instead of $p$ is convenient when\ndefining the parton level helicity structure functions, which read\n\\begin{align}\n h_\\lambda(x_f,Q^2) = P_\\lambda^{\\mu\\nu}(\\widetilde k,q) \n {\\mathcal H}_{\\mu\\nu}(\\widetilde k,q)\n\\end{align}\nand satisfy identities analogous to Eq.~\\eqref{eq:helicityids}, with $p\n{\\rightarrow} \\widetilde k$.\n\n\n\\subsection{Invariant structure functions}\n\\label{app:invsfn}\n\nFor a generic lepton-hadron scattering, \nwe define the hadronic $F_i$ and partonic $h_i$ \ninvariant structure functions with $i=1,\\ldots,6$ by the following\ntensor decomposition of the hadronic tensor:\n\\begin{align}\n\\begin{split}\n & W^{\\mu\\nu}(p,q) \n = \\Big( -g^{\\mu\\nu} + \\frac{q^\\mu q^\\nu}{q^2} \\Big) F_1(x_B,Q^2) \\\\\n & \\quad + \\Big(p^\\mu - q^\\mu\\frac{p\\cdot q}{q^2}\\Big)\n \\Big(p^\\nu - q^\\nu\\frac{p\\cdot q}{q^2}\\Big)\n \\frac{F_2(x_B,Q^2)}{p\\cdot q} \\\\\n & \\quad - i \\varepsilon^{\\mu\\nu\\alpha\\beta} p_\\alpha q_\\beta \n \\frac{F_3(x_B,Q^2)}{p\\cdot q} \n - \\frac{q^\\mu q^\\nu}{q^2} F_4(x_B,Q^2) \\\\\n & \\quad \n - \\frac{p^\\mu q^\\nu + q^\\mu p^\\nu}{2 p\\cdot q} F_5(x_B,Q^2) \n + \\frac{p^\\mu q^\\nu - q^\\mu p^\\nu}{2 p\\cdot q} F_6(x_B,Q^2)\n \\label{eq:invariantF}\n\\end{split}\n\\end{align}\nand\n\\begin{align}\n\\begin{split}\n & {\\mathcal H}^{\\mu\\nu}(\\widetilde k,q) \n = \\Big( -g^{\\mu\\nu} + \\frac{q^\\mu q^\\nu}{q^2} \\Big) h_1(\\widetilde x_f,Q^2) \\\\\n & + \\Big(\\widetilde k^\\mu - q^\\mu\\frac{\\widetilde k\\cdot q}{q^2}\\Big)\n \\Big(\\widetilde k^\\nu - q^\\nu\\frac{\\widetilde k\\cdot q}{q^2}\\Big)\n \\frac{h_2(\\widetilde x_f,Q^2)}{\\widetilde k\\cdot q} \\\\\n & - i \\varepsilon^{\\mu\\nu\\alpha\\beta} \\widetilde k_\\alpha q_\\beta \n \\frac{h_3(\\widetilde x_f,Q^2)}{\\widetilde k\\cdot q} \n - \\frac{q^\\mu q^\\nu}{q^2} h_4(\\widetilde x_f,Q^2) \\\\\n & - \\frac{\\widetilde k^\\mu q^\\nu + q^\\mu \\widetilde k^\\nu}\n {2 \\widetilde k\\cdot q} h_5(\\widetilde x_f,Q^2) \n + \\frac{\\widetilde k^\\mu q^\\nu - q^\\mu \\widetilde k^\\nu}\n {2 \\widetilde k\\cdot q} h_6(\\widetilde x_f,Q^2) \\ .\n \\label{eq:invarianth}\n\\end{split}\n\\end{align}\nThese 2 definitions differ from the notation of \nRef.~\\cite{Aivazis:1993kh} in\nthe chosen denominators. Our definitions have the advantage of\ndisplaying a duality between the hadron and parton level, which\ncan be obtained from each other by exchanging $p \\leftrightarrow\n\\widetilde k$, and lead to\na lesser degree of mixing between the hadron and parton structure\nfunctions under collinear factorization, see Eq.~\\eqref{eq:Fi_hi}.\nBy applying the projectors \\eqref{eq:helicityprojectors} \nto Eqs.~\\eqref{eq:invariantF}-\\eqref{eq:invarianth}, \nit is straightforward to show that\n\\begin{xalignat}{2}\n &F_L = -F_1 + \\frac{\\rho_B^2}{2x_B} F_2 & \\quad\n &h_L = -h_1 + \\frac{\\rho_f^2}{2\\widetilde x_f} h_2 \\nonumber \\\\\n &F_T = 2F_1 &\n &h_T = 2h_1 \\nonumber\\\\\n &F_A = \\rho_B F_3 &\n &h_A = \\rho_f h_ 3 \\nonumber\\\\\n &F_S = F_4 - F_5 &\n &h_S = h_4 - h_5 \\nonumber\\\\\n &F_{\\{\\!0q\\}}\\! = -\\rho_B F_5 &\n &h_{\\{\\!0q\\}}\\! = -\\rho_f h_5 \\nonumber \\\\ \n &F_{[0q]} = -\\rho_B F_6 &\n &h_{[0q]} = -\\rho_f h_6 \\ ,\n \\label{eq:helvsinv}\n\\end{xalignat}\nwhere we understood the dependence of $F_{i\\lambda}$ on $(x_B,Q^2)$\nand of $h_{i\\lambda}$ on $(\\widetilde x_f,Q^2)$ for ease of notation.\nNote that $F_L$ differs by a factor of $2x_B$\nwith respect to other common conventions. In our notation, the ratio $R$\nof transverse and longitudinal electron-nucleon cross sections reads\n\\begin{align}\n R = \\frac{\\sigma_T}{\\sigma_L} = \\frac{F_L}{F_1} \\ .\n\\end{align}\n\n\n\\subsection{Collinear factorization for structure functions}\n\\label{app:colfactsfn}\n\nAs discussed in Section~\\ref{sec:TMC} and\nAppendix~\\ref{app:kinematics}, the collinear factorization theorem\nstates that \n\\begin{align}\n\\begin{split}\n W^{\\mu\\nu}(p,q) & = \\sum_f \\int \\frac{dx}{x} \\, \n \\theta(\\widetilde x_f^{max}-\\widetilde x_f) \n \\theta(\\widetilde x_f-\\widetilde x_f^{min}) \\\\ \n & \\times {\\mathcal H}_f^{\\mu\\nu}(\\widetilde k,q) \\, \\varphi_{f\/N}(x,Q^2) \n \\label{eq:pQCDfactlims}\n\\end{split}\n\\end{align}\nwhere\n\\begin{align}\n \\widetilde x_f & = \\frac{\\xi}{x} \n \\frac{1}{1-\\frac{\\xi^2}{x^2}\\frac{m_f^2}{Q^2}} \\\\\n \\widetilde x_f^\\text{min} & = \\frac{x_B}{1-x_Bm_f^2\/Q^2} \\\\ \n \\widetilde x_f^\\text{max} & = \\frac{1}{1+(s_\\text{th}-m_f^2)\/Q^2} \\ .\n\\end{align}\nThe corresponding limits of integration on $dx$, namely\n$x^\\text{min}$ and $x^\\text{max}$, can be read off\nEq.~\\eqref{eq:xlims} setting $k^2=m_f^2$.\nAs discussed in Appendix~\\ref{app:helicitystrfns},\n$P_\\lambda^{\\mu\\nu}(p,q) = P_\\lambda^{\\mu\\nu}(\\widetilde k,q)$, hence\nthe factorization theorem for helicity structure functions reads\n\\begin{align}\n\\begin{split}\n & F_\\lambda(x_B,Q^2,m_N^2) \\\\\n & \\quad= \\sum_f \\int_{x^\\text{min}}^{x^\\text{max}} \n \\frac{dx}{x} \\, h^f_\\lambda(\\widetilde x_f,Q^2) \\,\n \\varphi_{f\/N}(x,Q^2) \\\\\n & \\quad = \\sum_f \\int_{\\widetilde x_f^\\text{min}}^{\\widetilde x_f^\\text{max}} \n \\frac{d\\widetilde x_f}{\\widetilde x_f} \\, h^f_\\lambda(\\widetilde x_f,Q^2) \\,\n \\varphi_{f\/N}\\Big(\\frac{\\xi}{\\xi_f},Q^2\\Big) \\ ,\n \\label{eq:helF_TMC}\n\\end{split}\n\\end{align}\nwhere\n\\begin{align}\n \\xi_f = \\frac{2\\widetilde x_f}\n {1+\\sqrt{1+4\\widetilde x_f^2m_f^2\/Q^2}} \\ .\n\\end{align}\nThe last line of Eq.~\\eqref{eq:helF_TMC} is particularly interesting,\nbecause the Nachtmann variable $\\xi$ only appears in the argument of\n$\\varphi$, without touching the integration limits.\nIn shorthand notation, where we highlight the dependence on $x_B$ and\n$\\xi$ and suppress that on $m_N^2$ and $Q^2$, and understand the sum\nover $f$, the helicity structure functions read \n\\begin{align}\n F_\\lambda(x_B) & \\equiv h_\\lambda^f \\otimes \\varphi_{f\/N} (\\xi) \\ .\n \\label{eq:Flambda_hi}\n\\end{align}\nFor the invariant structure functions, kinematic prefactors often\nappear: \n\\begin{align}\n\\begin{split}\n F_1(x_B) & = h_1^f \\otimes \\varphi_{f\/N}(\\xi) \\\\ \n F_2(x_B) & = \\frac{x_B}{\\widetilde x_f} \\frac{\\rho_f^2}{\\rho_B^2}\n h_2^f \\otimes \\varphi_{f\/N}(\\xi)\\\\ \n F_{3,5,6}(x_B) & = \\frac{\\rho_f}{\\rho_B} h_{3,5,6}^f \n \\otimes \\varphi_{f\/N}(\\xi) \\\\\n F_4(x_B) & = h_4^f \\otimes \\varphi_{f\/N}(\\xi) \n + \\big( \\frac{\\rho_f}{\\rho_B} - 1 \\big) h_5^f \\otimes \\varphi_{f\/N}(\\xi) \\ .\n \\label{eq:Fi_hi}\n\\end{split}\n\\end{align}\n\nThe ``massless structure functions'' can be obtained by setting $m_N^2=0$,\nhence, $\\xi=x_B$ in Eqs.~\\eqref{eq:Flambda_hi}-\\eqref{eq:Fi_hi}:\n\\begin{align}\n F_{\\lambda,i}^{(0)}(x_B) = F_{\\lambda,i}(x_B)|_{m_N^2=0} \\ .\n\\end{align}\nIn this definition we left the quark mass $m_f$ arbitrary.\n\nThe ``na\\\"ive'' target mass corrected structure functions\n$F^\\text{nv}$ are obtained by using $x\\leq 1$ as an upper limit of \nintegration over $dx$ in Eq.~\\eqref{eq:helF_TMC}. \nThis limit is a general and process-independent consequence of the\ndefinition of a parton distribution in the field theoretic parton\nmodel \\cite{Jaffe:1983hp}, but in DIS it is weaker than $x \\leq\nx^\\text{max}$, which is induced by 4-momentum and baryon number\nconservation as discussed in Section~\\ref{sec:TMC}. In detail, the\nna\\\"ive helicity structure functions read \n\\begin{align}\n & F_\\lambda^\\text{nv}(x_B) = \\\\\n & \\quad= \\sum_f \\int_{x^\\text{min}}^1 \n \\frac{dx}{x} \\, h_\\lambda(\\widetilde x_f,Q^2) \\,\n \\varphi_{f\/N}(x,Q^2) \n \\label{eq:Fnv_hi}\n\\end{align}\nUsing the definition of massless structure functions, one finds\n\\begin{align}\n\\begin{split}\n F_{1,\\lambda}^\\text{nv}(x_B) & = F_{1,\\lambda}^{(0)}(\\xi) \\\\\n F_2^\\text{nv}(x_B) & = \\frac{1}{\\rho_B^2} \\frac{x_B}{\\xi}\n F_2^{(0)}(\\xi) \\\\\n F_{3,4,5}^\\text{nv}(x_B) & = \\frac{1}{\\rho_B} F_{3,4,5}^{(0)}(\\xi) \\\\\n F_4^\\text{nv}(x_B) & = F_4^{(0)}(\\xi) \n + \\frac{1-\\rho_B}{\\rho_B} F_5^{(0)}(\\xi) \\ . \\\\\n\\end{split}\n\\end{align}\nThese formulae have already appeared in\n\\cite{Aivazis:1993kh,Kretzer:2003iu,Kretzer:2002fr}, modulo the change\nof notation discussed in Appendix~\\ref{app:invsfn}. As already noted\nin the main text, they are non-zero in the unphysical region $x_B>1$.\n\n\\subsection{Structure functions with Jet Mass Corrections}\n\nAt LO, the helicity structure functions with jet mass corrections read\n\\begin{align}\n\\begin{split}\n & F_\\lambda(x_B,Q^2,m_N^2) \\\\\n & \\qquad = \\int_0^{\\frac{1-x_B}{x_B}Q^2} \\hspace*{-.2cm} dm_j^2 J_2(m_j^2) \n F_\\lambda^{(0)}\\big(\\xi(1+m_j^2\/Q^2),Q^2\\big) \\ .\n \\label{eq:FhelJMC}\n\\end{split}\n\\end{align}\nThe JMC to invariant structure functions can be obtained from\nEqs.~\\eqref{eq:FhelJMC} and \\eqref{eq:helvsinv}. Suppressing the $Q^2$\nand $m_N^2$ dependence of the structure functions for ease of\nnotation, we obtain: \n\\begin{align}\n\\begin{split}\n F_1^\\text{JMC}(x_B) \n & = \\int_0^{\\frac{1-x_B}{x_B}Q^2} \\hspace*{-.2cm} dm_j^2 J_2(m_j^2) \n F_1^{(0)}\\big(\\xi(1+m_j^2\/Q^2)\\big) \\\\\n F_2^\\text{JMC}(x_B) \n & = \\int_0^{\\frac{1-x_B}{x_B}Q^2} \\hspace*{-.2cm} dm_j^2 J_2(m_j^2) \n \\frac{1}{\\rho_B^2} \\frac{x_B}{\\xi(1+m_j^2\/Q^2)} \\\\\n & \\times F_2^{(0)}\\big(\\xi(1+m_j^2\/Q^2)) \\\\\n F_{3,5,6}^\\text{JMC}(x_B) \n & = \\frac{1}{\\rho_B}\n \\int_0^{\\frac{1-x_B}{x_B}Q^2}\\hspace*{-.2cm} dm_j^2 J_2(m_j^2) \\\\\n & \\times F_{3,5,6}^{(0)}\\big(\\xi(1+m_j^2\/Q^2)\\big) \\\\\n F_4^\\text{JMC}(x_B)\n & = \\int_0^{\\frac{1-x_B}{x_B}Q^2}\\hspace*{-.2cm} dm_j^2 J_2(m_j^2) \n \\Big\\{ F_4^{(0)}\\big(\\xi(1+m_j^2\/Q^2)\\big) \\\\\n & + \\frac{1-\\rho_B}{\\rho_B} F_5^{(0)}\\big(\\xi(1+m_j^2\/Q^2)\\big)\n \\Big\\} \\ .\n\\end{split}\n\\end{align}\n\n\\section{Target mass corrections in the OPE formalism}\n\\label{app:GP}\n\nWe collect here for completeness the target mass corrections to\nthe electromagnetic structure functions obtained in the operator\nproduct expansion formalism of De Rujula, Georgi and Politzer\n\\cite{Georgi:1976ve,DeRujula:1976tz}, see also\n\\cite{Schienbein:2007gr} for a thorough review and discussion: \n\\begin{align}\n F_1^{GP}(x_B,Q^2) & \n = \\frac{x_B}{\\rho_B} \\Big[\n \\frac{F_1^{(0)}(\\xi,Q^2)}{\\xi} + \\frac{m_N^2 x_B }{Q^2 \\rho_B}\n \\Delta_2(x_B,Q^2) \\Big]\n \\nonumber \\\\\n F_2^{GP}(x_B,Q^2) & \n = \\frac{x_B^2}{\\rho_B^3} \\Big[\n \\frac{F_2^{(0)}(\\xi,Q^2)}{\\xi^2} + 6 \\frac{m_N^2 x_B }{Q^2 \\rho_B}\n \\Delta_2(x_B,Q^2) \\Big]\n \\nonumber \\\\\n F_L^{GP}(x_B,Q^2) & \n = \\frac{x_B}{\\rho_B} \\Big[\n \\frac{F_L^{(0)}(\\xi,Q^2)}{\\xi} + 2 \\frac{m_N^2 x_B }{Q^2 \\rho_B}\n \\Delta_2(x_B,Q^2) \\Big]\n \\label{eq:GP} \n\\end{align}\nwhere \n\\begin{align}\n \\Delta_2(x_B,Q^2) & = \\int_\\xi^1 dv \n \\Big[ 1 + 2\\frac{m_N^2 x_B}{Q^2 \\rho_B} (v-\\xi)\n \\Big] \\frac{F_2^{(0)}(v,Q^2)}{v^2} \\ ,\n\\end{align}\nand $F_i^{(0)}$ are the perturbative structure functions computed in\nthe massless target approximation $m_N^2\/Q^2 {\\rightarrow} 0$. Formulae for the\n$F_{3-6}^{GP}$ structure functions can be found in\nRef.~\\cite{Blumlein:1998nv}. \n\nNote that in the notation of Appendix~\\ref{app:strfns}, differently\nfrom Ref.~\\cite{Schienbein:2007gr}, the longitudinal structure\nfunction is defined such that \n\\begin{align}\n F_L(x_B) = \\frac{\\rho^2}{2x_B} F_2(x_B) - F_1(x_B) \\ ,\n \\label{eq:FLdef}\n\\end{align}\nand\n\\begin{align}\n R(x_B) \\equiv \\frac{\\sigma_L(x_B)}{\\sigma_T(x_B)} \n = \\frac{F_L(x_B)}{F_1(x_B)}.\n \\label{eq:Rdef}\n\\end{align}\nThis notation is explained in detail in the Appendices of\nRef.~\\cite{Aivazis:1993kh}. Combining Eqs.~\\eqref{eq:FLdef} and\n\\eqref{eq:Rdef} we obtain\n\\begin{align}\n F_1(x_B) = \\frac{\\rho_B^2}{2x_B} \\frac{F_2(x_B)}{1+R(x_B)} \n\\end{align}\nin agreement with Ref.~\\cite{Schienbein:2007gr}.\n\nEquations~\\eqref{eq:GP}\nhave been used to compute the OPE target mass corrections in\nFigs.~\\ref{fig:F2_TMC} and \\ref{fig:R_TMC}.\nNote that both $F_L^{GP}$ and $F_1^{GP}$ receive a correction\nfrom an integral of $F_2^{(0)} \\gg F_{1,L}^{(0)}$.\nThis explains the large size of the\ntarget mass corrections for the OPE curves of Fig.~\\ref{fig:R_TMC}. \n\n\n\\end{appendix}\n\n\n\\vfill\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe standard model in particle physics was very successful in giving\na unified description of strong, weak and electromagnetic processes.\nHowever, it was always considered a problem by many theorists\nto understand properly the {\\it mass giving algorithm} in this\ntheory, i.e.\\ to answer the question what lies behind the\nso-called ``Higgs phenomenon''. Is this so-called spontanous\nsymmetry breaking by the vacuum\nexpectation value of a scalar quantum field --- yielding at the same\ntime the gauge boson and the fermion masses in the theory --- the\ncorrect way to account for the appearance of nonzero masses\nin physics? In other words, is the Higgs mechanism of the\nstandard model only a convenient algorithm to generate masses\nof the gauge and fermion fields without loosing the renormalizability\nof the theory, or has this enigmatic phenomenon a deeper understanding \nwith the present formulation of the theory yielding only a \n{\\it particular parametrization} of a so far not very well understood\naspect of the theoretical description: the origin of nonzero masses\nin nature? Related to this is the question whether the scalar\nquantum field $\\phi$ needed to formulate the Higgs mechanism is, indeed, a true\nspin zero matter field which materializes as a spinless particle\nof a certain mass showing up in high energy processes.\n\nRecently there have appeared several suggestions of a different\nnature and interpretation for the mass-giving mechanism in \nparticle physics which would not require a Higgs particle \nto exist. The proposal of Pawlowski and R\\c{a}czka (see \\cite{1} and\nreferences therein) stresses the conformal invariance of the\noriginal theory without actually breaking the conformal symmetry\nin the course of introducing masses, but rather considering the\nappearance of nonzero mass ratios as a certain choice of gauge.\nThe other method which was proposed by the present author in\ncollaboration with H. Tann \\cite{2} aims at accounting\nfor the origin of nonzero masses by using a Weyl-geometric\nframework starting from an originally massless Weyl-symmetric\ntheory and a subsequent explicit breaking of the Weyl symmetry with the help\nof a term in the Lagrangean involving the curvature scalar $R$ of the\nWeyl space $W_4$ and the mass of the scalar field. In this framework\nthe mass generation in an originally massless and scaleless theory is\nconsidered to be due to an interplay between the ambient\ngeometry --- a Weyl-geometry --- and a universal scalar quantum field.\nThe breaking $W_4\\to V_4$ yielding a pseudo-Riemannian, i.e.\\\n$V_4$, description in the limit is formulated as a condition on the\nWeyl vector fields $\\kappa_\\rho$ which in turn\nrepresent an aspect of the Weyl-geometry (see Sect. II\nbelow) being given in the broken case\nas a derivative of the scalar field\nyielding thus, finally, zero length curvature, i.e.\\ $f_{\\mu\\nu}=0$.\n[The Weyl vector fields $\\kappa_\\rho$ are the $D(1)$ or\ndilatation gauge potentials and $f_{\\mu\\nu}$ are\nthe corresponding curvature components.] At the same time the\nvalidity of Einstein's equations for the metric in the $V_4$ limit is\nrequired to be satisfied with the energy-momentum tensors of the\nnow massive fields as sources and with --- as it turns out ---\na gravitational coupling\nconstant depending on the modulus $\\Phi$ of $\\phi$.\nOn the one hand, i.e.\\ as far as gravitation is concerned,\nthe squared modulus, $\\Phi^2$, of the scalar field plays the\nrole of a Brans-Dicke-type field in this\nformalism \\cite{3}. On the other hand, i.e. as far as the generation\nof nonzero masses for the gauge and fermion fields is concerned,\nthe field $\\phi$ plays a role of a Higgs-type field in this\nbroken Weyl theory.\n\nThe idea of our geometrically motivated method basically\nis that in accounting for the\norigin of nonzero masses in nature the theoretical framework\nshould include gravitation from the outset starting thus from the\ninvestigation of a dynamics of massless boson and fermion fields\nformulated in a Weyl space $W_4$ containing the dynamics of\na metric modulo conformal rescalings. Subsequently the Weyl-symmetry\nis broken explicitly yielding a Riemannian description together\nwith a definite length (and mass) scale being established and\na set of field equations for the metric being required to be satisfied.\n\nThe plan of the paper is as follows. After some introductory remarks\non Weyl geometry and Weyl spaces we briefly review the theory\npresented in \\cite{2} in which the geometry of a Weyl space of\ndimension four and its use in elementary particle theory was studied \nin detail. In this work the scalar field --- called $\\vphi$ there ---\nwas a complex quantum field with nonzero Weyl weight carrying\nbesides its transformation character under Weyl transformations\n[see below] no further representation properties. This Weyl\ncovariant theory is then generalized in Sect. II~by including\nthe electroweak gauge group $\\buildrel\\sim\\over{G} = SU(2)_W\\times U(1)_Y$ in\nthe description with the scalar field --- now denoted by $\\phi$ ---\npossessing in addition representation properties with respect\nto the weak isospin group $SU(2)_W$ as well as the hypercharge\ngroup $U(1)_Y$. After some general remarks about symmetry breaking\nin gauge theories at the end of Sect. II, this Weyl-electroweak\ntheory (WEW theory) is explicitly broken in Sect. III~and the\nrole of electromagnetism (Subsection A) and gravitation\n(Subsection B) in the resulting theory, formulated in a Riemannian\nspace, is investigated in detail. Subsection C deals with\nthe wave equation for the scalar field $\\phi$, and Subsection D, finally, is\ndevoted to the determination of the free parameters of the theory.\nAn essential point in the\npresented discussion is that the breaking of the Weyl-symmetry\nand the appearance of a length and mass scale in the theory\nis qualitatively different from the so-called spontanous\nsymmetry breaking in the electroweak theory yielding the masses\nof the gauge boson and fermion fields there. We show, using the\ncoset representation of the scalar field $\\phi$ derived and\ndiscussed in Appendix A and B, that the so-called spontanous\nsymmetry breaking is a particular choice of gauge within\nthe electroweak theory which is characterized by the choice of\nan origin $\\hat\\phi$ in the coset space, with $\\hat\\phi$ being invariant\nunder the {\\it electromagnetic} gauge group $U(1)_{e.m.}$ exhibiting\nthus, finally, a residual $U(1)_{e.m.}$ gauge symmetry of the theory\nwhich is, in fact, a nonlinear realization of the original\n$\\buildrel\\sim\\over{G}$-symmetry on the subgroup $U(1)_{e.m.}$. We end in Sect. IV~with \nsome concluding remarks and discussions of the results obtained\ndue to the true symmetry breaking occurring in this\nWeyl-electroweak theory when a definite intrinsic unit of lengths\nis established.\n\n\n\\section{Electroweak Theory in Weyl-Symmetric Form}\n\n\\noindent{\\bf A. Geometric Preliminaries.}\n\nIn this section we investigate a unified electroweak\ntheory in the presence of ``gravitation'' formulated in a Weyl space.\nSince Einstein's metric theory of gravitation is neither\nconformally invariant nor a theory which could be formulated in\na Weyl space we have to break the Weyl-symmetry in a second step,\nas mentioned, in order to recover Einstein's theory in this framework.\nOn the other hand, we want to generate masses for the matter fields of\nthe theory by starting from a gauge dynamics involving at first only massless\nfields in order to see how this mass generation obtained by\nWeyl-symmetry breaking compares to the Higgs mechanism in the\nstandard model. To facilitate our discussion we shall, however,\ndisregard the $SU(3)$ gauge group of colour and shall treat\nonly the electroweak part of the standard model together with gravitation\nneglecting thus the strong interactions. Moreover, we treat only\none generation of fermions for simplicity, i.e. the leptons\n$e$ and $\\nu =\\nu_e$. Our main interest here is to see how the\nelectron mass $m_e$ and the $SU(2)$-gauge boson masses\nappear due to Weyl-symmetry breaking.\nThe fact that a second fermion generation with $m_\\mu > m_e$ introduces\n{\\it another} and {\\it different} mass scale given by the myon mass\n$m_\\mu$ --- opening up, moreover, the possibility of $\\mu_{e3}$-decay ---\nand, similarly, for the third generation with a tau-mass\n$m_\\tau > m_\\mu$, cannot be explained by the present model.\nIn this respect the broken Weyl theory is, unfortunately,\nnot better than the standard model.\n\nIn order to stress the role of gravitation in the present context,\nwe like to make the following remarks. The usual Higgs\nmechanism yielding nonzero masses for the fields in the standard model\ndoes not limit the actual size of the masses obtained: The mass value\nfor the fermion fields could be shifted to arbitrary large values.\nOf course, the gauge boson masses are related to the strength of\nthe Fermi coupling constant for charge changing weak currents.\nHowever, the feature that the elementary fermion masses could be\narbitrary large seems to be an unphysical one since the generation of nonzero\nrest masses is accompanied by the generation of gravitational fields\nand gravitational interactions. If gravity were included in the\nstandard model one would expect in a consistent theory that elementary\nmasses cannot be shifted to arbitrary large values due to\ndamping effects resulting from the consideration of\ngravitational interactions. This is reminiscent of Hermann Weyl's\nremark that a theory which tries to account for the\norigin of masses in nature cannot be formulated consistently\nwithout considering gravitation at the same time. The standard\nmodel should thus be formulated in a general relativistic setting.\nIn starting from a massless conformally invariant scenario\na formulation of the original dynamics in a Weyl space\nor Weyl geometry would thus be very suggestive. Let us, therefore,\nbegin our investigation by formulating a gauge dynamics of a massless\nspin zero and a single generation of massless spin $\\frac{1}{2}$\nfields in a Weyl space $W_4$.\n\nIn \\cite{2} the geometry of a Weyl space was investigated\nin detail. We shall use the notation defined there (see in particular\nAppendix A of this paper). We shall refer to \\cite{2} as to I in\nthe following and refer, for example, to Eq.(1.2) of I\nas to (I, 1.2) etc..\n\nA Weyl space $W_4$ is characterized by two differential forms:\n\\begin{equation}\nds^2= g_{\\mu\\nu}(x) dx^\\mu\\otimes dx^\\nu\\\/;\\qquad\n\\\/\\kappa =\\kappa_\\mu (x)dx^\\mu \\,.\n\\label{21}\n\\end{equation}\nA $W_4$ is equivalent to a family of Riemannian spaces\n\\begin{equation}\n(g_{\\mu\\nu}, \\kappa_\\sigma),\\\/(g'_{\\mu\\nu}, \\kappa'_\\rho),\\\/(g''_{\\mu\\nu}, \\kappa''_\\rho)\\ldots\n\\label{22}\n\\end{equation}\nwith metrics $g_{\\mu\\nu}(x),~\\\/ g'_{\\mu\\nu}(x)\\ldots$ and Weyl vector fields\n$\\kappa_\\rho (x),\\kappa'_\\rho (x)\\ldots$ related by\n\\begin{eqnarray}\ng'_{\\mu\\nu}(x)&=&\\sigma(x)\\, g_{\\mu\\nu} (x) \\\\ \\label{23a}\n\\kappa'_\\rho(x)&=&\\kappa_\\rho(x) +\\partial_\\rho \\log\\sigma (x)\\,,\\label{23b}\n\\end{eqnarray}\nwhere $\\sigma (x)\\!\\in\\! D(1)$, $\\sigma (x)=e^{\\rho(x)}>0$, with\n$D(1)$ denoting the dilatation group which is isomorphic\nto $R^+$ (the positive real line). The transformations \n(2.3) and (2.4) are called {\\sl Weyl-transformations}\ninvolving a conformal rescaling (2.3) of the metric\ntogether with the transformation (2.4) of the Weyl\nvector fields. In the following discussion we shall consider\na Weyl space of dimension $d=4$ possessing Lorentzian\nsignature $(+,-,-,-)$ of its metrics. (For reference to the\nearlier history of Weyl spaces and Weyl geometry see the\nreferences quoted in I.)\n\nA $W_4$ reduces to a Riemannian space $V_4$ for\n$\\kappa_\\mu =0$; a $W_4$ is equivalent to a $V_4$ if the\n``length curvature'' associated with the Weyl vector field\n$\\kappa_\\mu$ is zero, i.e.\\ for\n\\begin{equation}\nf_{\\mu\\nu}=\\partial_\\mu\\kappa_\\nu -\\partial_\\nu\\kappa_\\mu =0\\\\.\n\\label{24}\n\\end{equation}\n\nIn I we studied a Weyl invariant dynamics of massless\nfields involving the metric $g_{\\mu\\nu}$, the Weyl vector or\n$D(1)$ gauge fields $\\kappa_\\rho$, and the electromagnetic, i.e.\\\n$U(1)=U(1)_{e.m.}$\\ gauge fields $A_\\mu$, as well as the ``matter''\nfields $\\varphi$ (spin zero) and $\\psi$ (spin $\\frac{1}{2}$, Dirac spinor) with\nWeyl weight $w(\\varphi)=-\\frac{1}{2}$ and $w(\\psi)=-{3\\over 4}$~[see I]. It\nturned out in the discussion given in I that $\\varphi$ is not\na bona fide matter field but could better be characterized\nas a universal Bans-Dicke-type field or a Higgs-type field related\nto symmetry breaking. On the other hand, the spinor field\n$\\psi$ {\\it is} a true matter field representing\nleptons in the present formalism.\n\nThe local group structure of the theory studied in I was\n$SO(3,1)\\otimes D(1) \\otimes U(1)$ for the nonfermionic fields and\n$Spin(3,1)\\otimes D(1)\\otimes U(1)$ for the Dirac\nspinor field $\\psi$ with $Spin(3,1)$ denoting the universal\ncovering group of the orthochronous Lorentz group \n$SO(3,1)\\equiv O(3,1)^{++}$ acting in the local spin space ${\\cal C}_4$\nrepresenting the standard fiber of the spinor bundle on\nwhich the field $\\psi(x)$ is defined as a section (see I\nand the discussion below), and with $U(1)$ denoting the\nelectromagnetic gauge group.\n\nThe pull back of a connection on the corresponding frame bundle\nwas denoted by the one-forms of Weyl weight zero (Latin indices\nare local Lorentzian indices):\n\\begin{equation}\n(w_{ik}=-w_{ki}\\,,\\kappa\\,, A)\n\\label{25}\n\\end{equation}\nwith coefficients with respect to a natural base\n$dx^\\mu$ in the dual tangent space $T_x^*(W_4)$ to $W_4$ at $x$\ngiven by:\n\\begin{equation}\n(\\Gamma_{\\mu ik}(x)=-\\Gamma_{\\mu ki}(x)\\,,\\kappa_\\mu (x)\\,, A_\\mu(x))\\,.\n\\label{26}\n\\end{equation}\nThe first entry in (\\ref{25}) is Lorentz-valued (i.e.\\ antisymmetric\nin $i$ and $k$), the second is $D(1)$-valued (corresponding to a\nreal, noncompact, abelian gauge group), and the third is $U(1)$-valued \n(corresponding to the complex, compact, abelian electromagnetic\ngauge group). The fully covariant derivative of a \ntensor quantity $\\phi^{(n,m)}$ of type $(n,m)$, i.e.\\ covariant\nof degree $n$ and contravariant of degree $m$, possessing Weyl\nweight $w(\\phi^{(n,m)})$ and charge $q$, i.e.\\ transforming under\nWeyl transformations (2.3) and (2.4) as\n\\begin{equation}\n{\\phi^{(n,m)}}'(x)=[\\sigma(x)]^{w(\\phi^{(n,m)})}~\\phi^{(n,m)} (x)\\,,\n\\label{27}\n\\end{equation}\nand under electromagnetic gauge transformations as\n\\begin{equation}\n{\\phi^{(n,m)}}'(x)=e^{-{iq\\over\\hbar c}}~\\phi^{(n,m)} (x)\\,,\n\\label{28}\n\\end{equation}\nis given by\n\\begin{eqnarray}\n\\tilde D\\phi^{(n,m)} &=&D\\phi^{(n,m)} + {iq\\over \\hbar c} A\\phi^{(n,m)}\\nonumber\\\\\n &=&\\nabla\\phi^{(n,m)} -\\omega(\\phi^{(n,m)})\\kappa\\phi^{(n,m)} +{iq\\over \\hbar c}\n A\\phi^{(n,m)} .\\label{29}\n\\end{eqnarray}\nHere $D = dx^\\mu D_\\mu$ denotes the Weyl-covariant derivative \n(I, A3), and $\\kappa =\\kappa_\\mu dx^\\mu$, $A=A_\\mu dx^\\mu$. Furthermore,\n$\\nabla =dx^\\mu\\nabla_\\mu$ with $\\nabla_\\mu$ denoting the covariant\nderivative with\nrespect to the Weyl-connection $\\Gamma_{\\mu\\nu}{}^\\rho$ defined by\n\\begin{equation}\n\\Gamma_{\\mu\\nu}{}^\\rho=\\bar\\Gamma_{\\mu\\nu}{}^\\rho +W_{\\mu\\nu}{}^\\rho =\\frac{1}{2} g^{\\rho\\lambda}\n (\\partial_\\mu g_{\\nu\\lambda}+\\partial_\\nu g_{\\mu\\lambda}-\\partial_\\lambda g_{\\mu\\nu})-\n \\frac{1}{2}(\\kappa_\\mu\\delta_\\nu^\\rho +\\kappa_\\nu\\delta_\\mu^\\rho-\\kappa^\\rho g_{\\mu\\nu})\\\/.\n\\label{210}\n\\end{equation}\nHere and in the sequel purely metric quantities pertaining to a\n$V_4$ are denoted by a bar, for example, \n$\\bar\\Gamma_{\\mu\\nu}{}^\\rho=\\{{\\rho\\atop\\mu\\nu}\\}$ are the Christoffel\nsymbols of the metric $g_{\\mu\\nu}$. We remark in\npassing that the connection coefficient $\\Gamma_{\\mu\\nu}{}^\\rho$,\ndefined with respect to a natural base in (\\ref{210}), is\nWeyl-invariant, obeying $\\Gamma_{\\mu\\nu}{}^\\rho =\\Gamma_{\\mu\\nu}{}^\\rho {}\\,'$, with\nthe change in the metric computed according to (2.3) being compensated\nby the change in the Weyl vector fields computed according to (2.4).\nThus the Weyl-covariant derivative $D\\phi^{(n,m)}$ of a quantity\n$\\phi^{(n,m)}$ is independent of the Weyl gauge chosen in the\nfamily (2.2), and, by definition, transforms again\nlike (\\ref{27}). Corresponding to Eq.(\\ref{29}) the Weyl- and $U(1)$-\ncovariant derivative of a spinor field with $w(\\psi)=-{3\\over 4}$ and\ncharge $e$ is given by\n\\begin{eqnarray}\n\\tilde D\\psi (x)&=&D\\psi (x)+{ie\\over\\hbar c} A\\cdot {\\bf 1}\\psi (x)\\nonumber\\\\\n&=&dx^\\mu\\biggl\\{\n\\biggl(\\partial_\\mu+i\\Gamma_\\mu (x)\\biggr)\\psi (x)+{3\\over 4}\\kappa\\cdot {\\bf 1}~\n\\psi (x)+{ie\\over\\hbar c} A_\\mu\\cdot {\\bf 1}~\\psi (x)\\biggr\\}\\,,\\label{211}\n\\end{eqnarray}\nwhere $\\Gamma_\\mu (x)$ is the spin connection\n\\begin{equation}\n\\Gamma_\\mu (x)=\\lambda^j_\\mu (x)\\frac{1}{2}\\Gamma_{jik}(x) S^{ik}~;\\qquad S^{ik}=\n{i\\over 4}[\\gamma^i,\\gamma^k]\\\/.\n\\label{212}\n\\end{equation}\nHere $\\gamma^i;~i=0,1,2,3$ are the constant Dirac matrices satisfying\n$\\{\\gamma^i,\\gamma^k\\}=\\gamma^i\\gamma^k+\\gamma^k\\gamma^i=2\\eta^{ik}\\cdot {\\bf 1}$ with\n$\\eta^{ik}=diag(1,-1,-1,-1)$ and $\\lambda_\\mu^j (x)$ are\nthe vierbein fields obeying\n\\begin{equation}\ng_{\\mu\\nu} (x)=\\lambda^i_\\mu (x)\\lambda^k_\\nu (x)\\eta_{ik}\\\/.\\label{213}\n\\end{equation}\nThe inverse vierbein fields used below are denoted by $\\lambda^\\mu_i(x)$.\nThe quantities $\\Gamma_{\\mu ik}=\\lambda^j_\\mu\\Gamma_{jik}$ appearing in\nthe first equation of (\\ref{212}) are the coefficients of the\nLorentz part of the connection mentioned in relation to (\\ref{25})\nand (\\ref{26}) with $\\Gamma_{jik}$ defined by\n\\begin{equation}\n\\omega_{ik}=\\bar\\omega_{ik}-\\frac{1}{2}(\\kappa_i\\theta_k-\\kappa_k\\theta_i)=\\theta^j\n\\Gamma_{jik}\n\\label{214}\n\\end{equation}\nwhere $\\bar\\omega_{ik}=\\theta^j\\bar\\Gamma_{jik}$ with\n$\\bar\\Gamma_{jik}=-\\bar\\Gamma_{jki}$ being the\nRicci rotation coefficients of a $V_4$ and \n$-\\frac{1}{2} (\\kappa_i\\eta_{kj}-\\kappa_k\\eta_{ij})\\theta^j$ denoting the\nWeyl addition in a $W_4$. In Eq. (\\ref{214}) $\\theta^j=\\lambda^j_\\mu (x)dx^\\mu$;\n$j=0,1,2,3$ are the fundamental one-forms representing a\nLorentzian basis in the dual tangent space, $T_x^*(W_4)$, at\n$x\\in W_4$. (Compare Appendix A of I.)\nThe form (\\ref{214}) for $\\omega_{ik}$ together with (\\ref{210})\nyields $D\\lambda^\\mu_i=0$ with $w(\\lambda^\\mu_i)=-\\frac{1}{2}$\nfor the Weyl-covariant derivative of the vierbein field and,\ncorrespondingly, $D_\\rho g_{\\mu\\nu}=0$ with the Weyl weights \n$w(g_{\\mu\\nu})=1$ [see Eq. (2.3)], $w(\\lambda^i_\\mu)=\\frac{1}{2}$ and\n$w(\\eta_{ik})=0$ [compare (\\ref{213})]. The relation $D_\\rho g_{\\mu\\nu}=0$,\nexpressing the fact that the metric is Weyl-covariant constant,\nreduces the connection on the general linear frame bundle [i.e.\\ the\n$Gl(4,R)$-bundle] in a Weyl space --- possessing a metric given\nonly modulo conformal transformations (2.3) --- to a Weyl frame\nbundle, called $P_W$ in I, possessing the structural group\n$SO(3,1)\\otimes D(1)$.\\\\\n\n\\noindent{\\bf B. Standard Model Extension: Weyl-Electroweak Theory (WEW Theory)}\n\nWe now extend the formalism developed in I to a unified electroweak\ntheory [neglecting as mentioned $SU(3)$ colour degrees of freedom]\nby extending the gauge group of the theory to the group \\cite{3a}\n\\begin{equation}\nG = SO(3,1)\\otimes D(1)\\otimes U(1)_Y\\times SU(2)_W\\\/,\n\\label{215}\n\\end{equation}\ni.e.\\ interpreting the $U(1)$ degree of freedom in I as\nweak hypercharge, $U(1)_Y$, and considering an additional\nweak isospin group $SU(2)_W$ (compare Weinberg's model of leptons \\cite{4}).\nThe underlying principal bundle over\n$W_4$ is now \n\\begin{equation}\nP=P(W_4,G)\\label{216}\n\\end{equation}\nwith $G$ given by (\\ref{215}). For the discussion of spinor fields\nof Dirac type one considers, as usual, the spin frame bundle\n$\\bar P=\\bar P (W_4,\\bar G)$\npossessing the universal covering group of $G$, i.e.\n\\begin{equation}\n\\bar G = Spin (3,1)\\otimes D(1)\\otimes U(1)_Y\\times SU(2)_W\n\\label{217}\n\\end{equation}\nas structural group (compare the discussion above and in I).\nA Dirac spinor field $\\psi$, with Weyl weight\n$w(\\psi)=-{3\\over 4}$ as before [see I], and hypercharge $Y$,\npossessing a definite representation character regarding\n$SU(2)_W$ i.e.\\ [we follow the standard model assignment]\n$I=\\frac{1}{2}$ (isodoublet), $Y=-\\frac{1}{2}$ for the left-handed\nchiral fields $\\psi_L(x)=\\frac{1}{2}(1-\\gamma_5)\\psi (x)$, and $I=0$\n(isosinglet), $Y=-1$, for the right-handed chiral fields\n$\\psi_R(x)=\\frac{1}{2} (1+\\gamma_5)\\psi (x)$, with\n\\begin{equation}\n\\gamma_5 =i\\gamma_0\\gamma_1\\gamma_2\\gamma_3\\,;\\qquad \\gamma_5{}^\\dagger =\\gamma_5\\,;\\qquad (\\gamma_5)^2=1\\\/.\n\\label{218}\n\\end{equation}\nThe chiral fields $\\psi_L (x)$ and $\\psi_R (x)$ will be regarded,\nrespectively, as a section on the spinor bundle $S$ associated\nto $\\bar P$ with fiber $F$ given by $\\tilde{{\\cal C}}={\\cal C}_4\\times {\\cal C}_2$\nfor $I=\\frac{1}{2}$; or given by $\\tilde {{\\cal C}}={\\cal C}_4\\times {\\bf {\\cal C}}$ for\n$I=0$, being thus defined by the bundle \n\\begin{equation}\nS=S (W_4, F=\\tilde{{\\cal C}},\\bar G)\\,.\n\\label{219}\n\\end{equation}\nHence the leptonic chiral fermion fields of Weyl weight $-{3\\over 4}$\nwill be\n\\begin{equation}\n\\psi_L={\\nu_L\\choose e_L}=\n {\\frac{1}{2} (1-\\gamma_5)\\psi_\\nu\\choose \\frac{1}{2}(1-\\gamma_5)\\psi_e}\\,, Y=-\\frac{1}{2}\\,;\n\\quad \\psi_R=e_R =\\frac{1}{2} (1+\\gamma_5)\\psi_e, Y=-1,\n\\label{220}\n\\end{equation}\nwith their adjoints $(\\bar\\psi =\\psi^\\dagger\\gamma_0)$:\n\\begin{equation}\n\\bar\\psi_L=(\\bar\\nu_L,\\bar e_L)=\\left(\\bar\\psi_\\nu\\frac{1}{2}\n(1+\\gamma_5),\\bar\\psi_e\\frac{1}{2} (1+\\gamma_5)\\right), \nY=\\frac{1}{2};~\\bar\\psi_R=\\bar e_R=\\bar\\psi_e\\frac{1}{2} (1-\\gamma_5); Y=1\\\/.\n\\label{221}\n\\end{equation}\n\nFor the scalar field we shall use as representation character\nwith respect to $SU(2)_W$ an isodoublet, $I=\\frac{1}{2}$, yielding thus\n\\begin{equation}\n\\phi = \\left({\\varphi_+\\atop\\varphi_0}\\right) \\quad {\\rm with} \\quad Y=\\frac{1}{2}; \n\\quad \\mbox{and}\\quad\n\\phi^\\dagger =(\\varphi^*_+,\\varphi^*_0)\\quad \\mbox{with} \\quad Y=-\\frac{1}{2}\n\\label{222}\n\\end{equation}\npossessing the Weyl weight $w(\\phi)=w(\\phi^\\dagger )=-\\frac{1}{2}$. Here $\\varphi_0$\nis a neutral complex field, and $\\varphi_+$\nis a complex field with positive charge, obeying $\\varphi^*_+=\\varphi_-$.\nThe relation between electric charge $Q$, isospin $(I_3)$, and weak\nhypercharge is, as usual,\n\\begin{equation}\nQ=I_3+Y\\\/.\n\\label{223}\n\\end{equation}\n\nThe field $\\phi$ may be regarded as a section on the bundle\n\\begin{equation}\nE=E(W_4, F={\\cal C}_2, G)\\label{224}\n\\end{equation}\nassociated to $P$. The square of the modulus of the scalar field is\nnow given by the $U(1)_Y$ and $SU(2)_W$ invariant of Weyl weight\n$w(\\Phi^2)=-1$:\n\\begin{equation}\n\\Phi^2=\\phi^\\dagger \\phi =\\varphi^*_+\\varphi_++\\varphi^*_0\\varphi_0=|\\varphi_+|^2+\n|\\varphi_0|^2.\n\\label{225}\n\\end{equation}\n\nThe invariant Yukawa coupling term of Weyl weight $-1$ for\nthe scalar and spinor fields --- reading\n$\\tilde\\gamma\\sqrt{\\varphi^*\\varphi} (\\bar\\psi\\psi)=\n\\tilde\\gamma\\Phi(\\bar\\psi\\psi)$ in I --- will now be written as\n\\begin{equation}\n\\tilde\\gamma\\{(\\bar\\psi_L\\phi)\\psi_R + \\bar\\psi_R (\\phi^\\dagger\\psi_L)\\}\\\/.\n\\label{226}\n\\end{equation}\n\nCalling the $U(1)_Y$ gauge potentials $B_\\mu$, i.e.\\ reserving as usual\nthe notation $A_\\mu$ for the electromagnetic gauge potentials,\nthe full $G$-covariant derivative of $\\phi$ is written as\n[compare (\\ref{215})]\n\\begin{equation}\n\\buildrel\\approx \\over D_\\mu\\!\\phi =D_\\mu\\phi +{i\\over 2}\\tilde g\nA^a_\\mu\\tau_a\\phi + i\\tilde g'Y B_\\mu\\cdot {\\bf 1}~\\phi \\\/.\n\\label{227}\n\\end{equation}\nHere the Weyl-covariant part is given by $D_\\mu\\phi=\\partial_\\mu\\phi +\\frac{1}{2}\n\\kappa_\\mu\\cdot {\\bf 1}\\phi$, and the $SU(2)_W$-gauge fields are denoted by\n$A^a_\\mu;~a=1,2,3;\\mu=0,1,2,3$. Moreover, $Y=\\frac{1}{2}$ in\n(\\ref{227}) according to (\\ref{222}). The Lie algebra\nof $SU(2)_W$ adapted to the choice (\\ref{222}) for\n$\\phi$ is given by $\\frac{1}{2}\\tau_a$ with $\\tau_a; a=1,2,3$ denoting the\nPauli matrices [summation over $a$ from 1 to 3 is understood in\n(\\ref{227})]. Finally, $\\tilde g$ and $\\tilde g'$ are dimensionless\ncoupling constants for the $SU(2)_W$ and $U(1)_Y$ coupling, respectively,\nwhich we write with a tilde in order not to confuse them with\nthe determinant of the metric tensor called $g$.\n\nA similar expression as (\\ref{227}) may be written down for\n$\\buildrel\\approx\\over{D}_\\mu\\!\\psi_L$ involving the Weyl-covariant\npart $D_\\mu\\psi_L =(\\partial_\\mu +i\\Gamma_\\mu)\n\\psi_L+{3\\over 4}\\kappa_\\mu\\cdot {\\bf 1} \\psi_L$, an $SU(2)_W$ part as in (\\ref{227}),\nand an $U(1)_Y$ part with $Y=-\\frac{1}{2}$. [Compare (\\ref{211}).]\nFor $\\buildrel\\approx\\over{D}_\\mu\\!\\psi_R$ the $A^a_\\mu$-contributions\nare absent due to the choice $I=0$ for the right-handed fermion field.\nWe take account of this in the notation by writing only one tilde\nfor the covariant differentiation of $\\psi_R$, i.e.\n$\\buildrel\\approx\\over{D}_\\mu\\!\\psi_R\\equiv\\buildrel\\sim\\over{D}_\\mu\\!\\psi_R=\nD_\\mu\\psi_R+i\\tilde g'Y B_\\mu\\cdot {\\bf 1}\\psi_R$ with $Y=-1$ according to\n(\\ref{220}).\n\nWe are now in a position to write down a $G$-gauge invariant\nLagrangean density of Weyl weight zero generalizing $\\tilde{\\cal L}_{W_4}$\nof $I$ [compare (I, 3.8)] to the case of an\nelectroweak theory including ``gravitation'', i.e.\\\ncontaining also a dynamics for the metric (determined modulo\nWeyl-transformations), in a scenario for the massless fields\n$\\phi,\\psi_L,\\psi_R,\\kappa_\\mu, B_\\mu$ and $A^a_\\mu$ possessing\nall a definit Weyl weight which is zero for the gauge fields\n$\\kappa_\\mu, B_\\mu, A^a_\\mu$, and is $w(\\phi)=-\\frac{1}{2}$ and $w(\\psi_L)=\nw(\\psi_R)=-{3\\over 4}$ as mentioned above. Again we use below the\nsame coefficient for the kinetic term of the scalar field\nand for the ${1\\over{12}}R$-term implying in I the validity of\nthe relation (I, 2.20), while (I, 2.21) is a consequence\nof the choice $w(\\psi)=-{3\\over 4}$. The Lagrangean for a massless\n$G$-invariant theory, called for short WEW theory (Weyl-electroweak theory),\nnow reads:\n\\begin{eqnarray}\t\n\\buildrel\\approx\\over{\\cal L}_{W_4}&=&\nK\\sqrt{-g}\\Biggl\\{\\frac{1}{2} g^{\\mu\\nu}(\\overrightarrow\\appD_\\mu\\phi)^\\dagger\n\\overrightarrow\\appD_\\nu\\phi -{1\\over {12}}\nR\\,\\phi^\\dagger\\phi -\\beta (\\phi^\\dagger\\phi)^2+\\tilde\\alpha R^2\n\\nonumber\\\\ \n& &\\qquad +{i\\over 2}\n\\left(\\bar\\psi_L\\gamma^\\mu{\\overrightarrow\\appD}_\\mu\\psi_L\n-\\bar\\psi_L {\\overleftarrow\\appD}_\\mu\n\\gamma^\\mu\\psi_L\\right)+{i\\over 2}\\left(\\bar\\psi_R\\gamma^\\mu\n\\overrightarrow{\\simD}_\\mu\\psi_R-\\bar\\psi_R\\overleftarrow{\\simD}_\\mu\\gamma^\\mu\\psi_R\\right)\n\\nonumber\\\\\n& &\n+\\tilde\\gamma[(\\bar\\psi_L\\phi)\\psi_R+\n\\bar\\psi_R(\\phi^\\dagger\\psi_L)]-\\tilde\\delta\n{1\\over 4}f_{\\mu\\nu}f^{\\mu\\nu} -\n\\buildrel\\approx\\over\\delta {1\\over 4}\\left(F_{\\mu\\nu}^a\nF^{\\mu\\nu}_a + B_{\\mu\\nu}B^{\\mu\\nu}\\right)\\Biggr\\}\\\/,\\label{228}\n\\end{eqnarray}\nHere $\\gamma^\\mu$ denote a set of $x$-dependent $\\gamma$-matrices with\nWeyl weight $w(\\gamma^\\mu)=-\\frac{1}{2}$ defined by\n\\begin{equation}\n\\gamma^\\mu=\\gamma^\\mu(x) =\\lambda^\\mu_i (x)\\gamma^i\\quad\\mbox{obeying}\\quad\n\\{\\gamma^\\mu,\\gamma^\\nu\\}=2 g^{\\mu\\nu}\\cdot {\\bf 1}\\\/.\n\\label{229}\n\\end{equation}\n\nThe meaning of the various terms in (\\ref{228}) is the same as in I and was\ndescribed there in detail. This is true except for the generalization\nof the covariant derivatives, as explained above, being denoted here\nby $\\buildrel\\approx\\over{D}_\\mu$ and $\\buildrel\\sim\\over{D}_\\mu$ with the arrows $\\rightarrow$ and $\\leftarrow$\nindicating, as usual, the action on $\\psi_L$, $\\psi_R$ and on $\\bar\\psi_L$, $\\bar\\psi_R$,\nrespectively, with a sign change involved according to the rule\n[compare (\\ref{211}) and (\\ref{227})],\n$\\overrightarrow\\appD_\\mu{}^\\dagger=\\gamma^0\\overleftarrow\\appD_\\mu\\gamma^0$ for the fermion fields and\nsimilarly for $\\phi$, i.e.\\ $({\\overrightarrow\\appD}_\\mu\\phi)^\\dagger\n=\\phi^\\dagger\\overleftarrow\\appD_\\mu$ with\n$\\overleftarrow\\appD_\\mu ={\\overrightarrow\\appD}_\\mu{}^\\dagger = \\buildrel\\approx\\over{D}_\\mu{}\\!^\\dagger$. Corresponding\nto this, i.e.\\ to the introduction of the new gauge fields $B_\\mu$ and\n$A^a_\\mu$, the last two terms in (\\ref{228}) replace the \nelectromagnetic term $-{1\\over K}{1\\over 4}F_{\\mu\\nu} F^{\\mu\\nu}$\nin (I, 3.8) with $F_{\\mu\\nu}=\\partial_\\mu A_\\nu-\\partial_\\nu A_\\mu$.\nSince the fields $B_\\mu$ and $A^a_\\mu$ have the dimension\n$[L^{-1}]$ ($L$=Length) as seen from (\\ref{227}), we replace the\nfactor ${1\\over K}$ (see below) appearing in front of the\nelectromagnetic term in (I, 3.8) by a constant $\\buildrel\\approx\\over\\delta$\nof dimension $[L^2]$. Moreover, the Lagrangean $\\buildrel\\approx\\over{\\cal L}_{W_4}$ is chiral\ninvariant, i.e. is invariant under {\\it global} $U(1)$ transformations\n($\\beta ' = const$):\n\\begin{equation}\n\\psi_L \\rightarrow e^{-i\\beta '}~\\psi_L \\\/;\\quad \\psi_R \\rightarrow e^{i\\beta '}~\\psi_R \\\/;\n\\quad \\phi \\rightarrow e^{-2i\\beta '}~\\phi \\\/,\n\\label{230a}\n\\end{equation}\nand analogously for $\\bar\\psi_L$, $\\bar\\psi_R$ and $\\phi^\\dagger$ with the\ncomplex conjugate phase factors.\n\nThe field strengths (curvatures) entering the expression (\\ref{228})\nin addition to $f_{\\mu\\nu}$ and the Weyl curvature scalar $R$ defined by\n[see (I, A31)]\n\\begin{equation}\nR=\\bar R-3\\bar\\nabla^\\rho \\kappa_\\rho +{3\\over 2} \\kappa^\\rho\\kappa_\\rho\\,,\n\\label{230}\n\\end{equation}\nwhere $\\bar R$ is the Riemannian part, are the $U(1)_Y$ gauge curvature\n\\begin{equation}\nB_{\\mu\\nu}=\\partial_\\mu B_\\nu -\\partial_\\nu B_\\mu~,\n\\label{231}\n\\end{equation}\nand the $SU(2)_W$ i.e.\\ Yang-Mills gauge curvature\n\\begin{equation}\nF^a_{\\mu\\nu} =\\partial_\\mu A^a_\\nu -\\partial_\\nu A^a_\\mu-\\tilde g f^a_{bc} A^b_\\mu A^c_\\nu\\\/,\n\\label{232}\n\\end{equation}\nwith $f^a_{bc}=\\varepsilon_{abc}$ denoting the structure constants of\n$SU(2)_W$ where $\\varepsilon_{abc}$ is the Levi-Civita symbol. The\noverall constant $K$ in (\\ref{228}) with dimension [Energy $\\cdot L^{-1}$]\nis a factor converting the length dimension in the curly brackets\n(which is $[L^{-2}]$) into [Energy $\\cdot L^{-3}$] in order to give\n$\\buildrel\\approx\\over{{\\cal L}}_{W_4}$ --- finally, after symmetry breaking ---\nthe correct dimension of an energy density. This factor $K$ drops\nout of the field equations in the Weyl-symmetric case discussed\nin this section and appears in (\\ref{228}) only for convenience.\nWe finally remark that the length dimension of the scalar field\n$\\phi$ is assumed to be $[L^0]$ and relative to this choice the\nfermion fields have length dimension $[L^{-\\frac{1}{2}}]$. With this\nconvention the Yukawa coupling constant $\\tilde\\gamma$ has length\ndimension $[L^{-1}]$.\n\nSo far we have not included a coupling $\\sim\\sqrt{-g} F_{\\mu\\nu} f^{\\mu\\nu}$\nin the Lagrangean (I, 3.8) or a coupling $\\sim\\sqrt{-g}B_{\\mu\\nu}\nf^{\\mu\\nu}$ in (\\ref{228}) which would also be of Weyl weight zero\nand hence would be allowed to occur. We intend to come back to\nan investigation of this point in a separate context and restrict the\ndiscussion here to the direct product structure of the abelian\ngauge groups involved, treating them thus as completely independent\nfrom each another.\n\nThe variation of the fields in the Lagrangean (\\ref{228}) now yields\nthe following set of $G$-covariant field equations:\n\\begin{eqnarray}\n&&\\delta\\phi^\\dagger :~~g^{\\mu\\nu}\\buildrel\\approx\\over{D}_\\mu\\buildrel\\approx\\over{D}_\\nu \\phi + {1\\over 6}R\\phi\n + 4\\beta(\\phi^\\dagger\\phi)\\phi - 2\\tilde\\gamma\\bar\\psi_R\\psi_L = 0\\\/,\\label{234}\\\\\n&&\\delta\\psi_L^\\dagger :~~-i\\gamma^\\mu\\buildrel\\approx\\over{D}_\\mu\\!\\psi_L - \\tilde\\gamma\\phi\\psi_R = 0\\\/,\\label{235}\\\\\n&&\\delta\\psi_R^\\dagger :~~-i\\gamma^\\mu\\buildrel\\sim\\over{D}_\\mu\\!\\psi_R - \\tilde\\gamma(\\phi^\\dagger\\psi_L) = 0\\\/,\\label{236}\\\\\n&&\\delta\\kappa_\\rho :~~\\tilde\\delta D_\\mu f^{\\mu\\rho} = -6\\tilde\\alpha D^\\rho R~,\n\\label{237}\\\\\n&&\\delta B_\\rho :~~\\buildrel\\approx\\over\\delta D_\\mu B^{\\mu\\rho} = \\tilde g'\\left[j^{(\\phi )}{}^\\rho+j^{(\\psi_L )}{}^\\rho\n +j^{(\\psi_R )}{}^\\rho\\right]\\\/,\\label{238}\\\\\n&&\\delta A^a_\\rho :~~\\buildrel\\approx\\over\\delta ~\\buildrel\\approx\\over{D}_\\mu\\! F_a^{\\mu\\rho}\\equiv ~\\buildrel\\approx\\over\\delta\\left[D_\\mu F_a^{\\mu\\rho}\n -\\tilde g f^b_{ac}A^c_\\mu F_b^{\\mu\\rho}\\right]=\n \\tilde g\\left[j^{(\\phi)}_a{}^\\rho + j^{(\\psi_L)}_a{}^\\rho\\right]\\\/,\n\\label{239}\\\\\n&&\\delta g^{\\mu\\nu} :~~{1\\over 6}\\Phi^2\\left[R_{(\\mu\\nu)}-\\frac{1}{2} g_{\\mu\\nu}R\\right]-\n 4\\tilde\\alpha R\\left[R_{(\\mu\\nu)}-{1\\over 4}g_{\\mu\\nu}R\\right]-\n 4\\tilde\\alpha\\left\\{D_{(\\mu}D_{\\nu )}R-g_{\\mu\\nu}D^\\rho D_\\rho R\\right\\}=\n \\nonumber\\\\\n&&\\qquad\\qquad =\\Theta^{(\\phi)}_{\\mu\\nu} + T^{(\\psi_L )}_{\\mu\\nu} + T^{(\\psi_R )}_{\\mu\\nu}\n + T^{(f)}_{\\mu\\nu} + T^{(B)}_{\\mu\\nu} + T^{(F_a)}_{\\mu\\nu}-g_{\\mu\\nu}\\tilde\\gamma\n \\left[(\\bar\\psi_L\\phi )\\psi_R + \\bar\\psi_R (\\phi^\\dagger\\psi_L)\\right]\\\/.\n\\label{240}\n\\end{eqnarray}\nHere we have used the following hermitean and $G$-gauge covariant\nexpressions for the weak hypercharge and isospin source currents:\n\\begin{equation}\nj^{(\\phi)}_\\rho = {i\\over 4}\\left(\\phi^\\dagger\\overrightarrow\\appD_\\rho\\phi -\n \\phi^\\dagger\\overleftarrow\\appD_\\rho\\phi\\right)\n\\label{241}\n\\end{equation}\nfor the $U(1)_Y$ $\\phi$-current (with $Y=\\frac{1}{2}$ for the $\\phi$-field),\n\\begin{equation}\nj^{(\\psi_L)}_\\rho = -\\frac{1}{2}(\\bar\\psi_L \\gamma_\\rho\\psi_L)~; \\qquad j^{(\\psi_R)}_\\rho = -(\\bar\\psi_R\\gamma_\\rho\\psi_R)\\\/,\n\\label{242}\n\\end{equation}\nfor the left- and right-handed $U(1)_Y$ fermion currents (with $Y=-\\frac{1}{2}$\nand $Y=-1$, respectively), and for the $SU(2)_W$ currents\n\\begin{equation}\nj^{(\\phi)}_{a\\rho} = {i\\over 2}\\left(\\phi^\\dagger\\frac{1}{2}\\tau_a\\overrightarrow\\appD_\\rho\\phi -\n \\phi^\\dagger\\overleftarrow\\appD_\\rho\\frac{1}{2}\\tau_a\\phi\\right)~,\n\\label{243}\n\\end{equation}\n\\begin{equation}\nj^{(\\psi_L)}_{a\\rho} = \\bar\\psi_L\\gamma_\\rho\\frac{1}{2}\\tau_a\\psi_L~; \\qquad j^{(\\psi_R)}_{a\\rho}\\equiv 0~.\n\\label{244}\n\\end{equation}\nThe $G$-gauge covariant expressions for the symmetric energy-momentum\ntensors appearing in (\\ref{240}) are:\n\\begin{eqnarray}\n\\Theta^{(\\phi)}_{\\mu\\nu} &=& \\frac{1}{2}\\left[(\\buildrel\\approx\\over{D}_\\mu\\!\\phi)^\\dagger(\\buildrel\\approx\\over{D}_\\nu\\!\\phi) +\n (\\buildrel\\approx\\over{D}_\\nu\\!\\phi)^\\dagger(\\buildrel\\approx\\over{D}_\\mu\\!\\phi)\\right]\n+{1\\over 6}\\left\\{D_{(\\mu}D_{\\nu)}\\Phi^2 - g_{\\mu\\nu} D^\\rho D_\\rho\\Phi^2 \\right\\}\n\\nonumber\\\\\n&\\hfil&\\qquad\\qquad\\qquad\\qquad\\qquad\n-g_{\\mu\\nu}\\left[\\frac{1}{2} g^{\\rho\\lambda}(\\buildrel\\approx\\over{D}_\\rho\\!\\phi)^\\dagger(\\buildrel\\approx\\over{D}_\\lambda\\!\\phi)\n -\\beta (\\phi^\\dagger\\phi)^2\\right]\\\/,\n\\label{245}\\\\\nT^{(\\psi_L)}_{\\mu\\nu} &=& {i\\over 2}\\left\\{\\bar\\psi_L \\gamma_{(\\mu}\\overrightarrow\\appD_{\\nu)}\\psi_L -\n\\bar\\psi_L\\overleftarrow\\appD_{(\\mu}\\gamma_{\\nu)}\\psi_L\\right\\} - g_{\\mu\\nu} {i\\over 2}\\left\\{\n\\bar\\psi_L\\gamma^\\rho\\overrightarrow\\appD_\\rho\\psi_L - \\bar\\psi_L\\overleftarrow\\appD_\\rho\\gamma^\\rho\\psi_L\\right\\}\\\/,\n\\label{246}\n\\end{eqnarray}\nand analogously for $T^{(\\psi_R)}_{\\mu\\nu}$, and\n\\begin{eqnarray}\nT^{(f)}_{\\mu\\nu} &=&~-\\,\\tilde\\delta\\left[f_{\\mu\\sigma}f_\\nu{}^\\sigma - {1\\over 4}\n g_{\\mu\\nu} f^{\\rho\\lambda}f_{\\rho\\lambda}\\right]\\\/,\n\\label{247}\\\\\nT^{(B)}_{\\mu\\nu} &=& -\\buildrel\\approx\\over\\delta\\left[B_{\\mu\\sigma}B_\\nu{}^\\sigma - {1\\over 4}\n g_{\\mu\\nu} B^{\\rho\\lambda}B_{\\rho\\lambda}\\right]\\\/,\n\\label{248}\\\\\nT^{(F_a)}_{\\mu\\nu} &=& -\\buildrel\\approx\\over\\delta\\left[F^a_{\\mu\\sigma}F_{a\\nu}{}^\\sigma - {1\\over 4}\n g_{\\mu\\nu} F_a^{\\rho\\lambda}F^a_{\\rho\\lambda}\\right]\\\/.\n\\label{249}\n\\end{eqnarray}\n\nWhile the last three energy-momentum tensors have vanishing trace\nthe traces of $\\Theta^{(\\phi)}_{\\mu\\nu}, T^{(\\psi_L)}_{\\mu\\nu}$ and $T^{(\\psi_R)}_{\\mu\\nu}$\nfor solutions of the field equations (\\ref{234}) -- (\\ref{236}) and their\nadjoints read:\n\\begin{eqnarray}\n\\Theta^{(\\phi)\\mu}_\\mu &=& -{1\\over 6}R\\phi^\\dagger\\phi + \\tilde\\gamma\n\\left[(\\bar\\psi_L\\phi)\\psi_R + \\bar\\psi_R(\\phi^\\dagger\\psi_L)\\right]\\\/,\n\\label{250}\\\\\nT^{(\\psi_L)\\mu}_\\mu &=& T^{(\\psi_R)\\mu}_\\mu = {3\\over 2}\\tilde\\gamma\n\\left[(\\bar\\psi_L\\phi)\\psi_R + \\bar\\psi_R(\\phi^\\dagger\\psi_L)\\right]\\\/.\n\\label{251}\n\\end{eqnarray}\nUsing these relations in computing the trace of Eq. (\\ref{240}),\ntogether with $D^\\rho D_\\rho R=0$ following\nfrom (\\ref{237}), one finds that the resulting\nequation is identically satisfied.\n\nIt is also interesting to compute the Weyl-covariant divergences of the\nenergy-momentum tensors (\\ref{245}) -- (\\ref{249}) for solutions of the\nfield equations, i.e. using Eqs. (\\ref{234}) -- (\\ref{239}) as well as the\nBianchi identities for the $F^a_{\\mu\\nu}$ reading, with $\\{\\rho\\mu\\nu\\}$ denoting\nthe cyclic sum of the indices $\\rho\\mu\\nu$,\n\\begin{equation}\n\\buildrel\\approx\\over{D}_{\\{\\rho}F_{\\mu\\nu\\}}^a = 0\\\/.\n\\label{252}\n\\end{equation}\nThe result is\n\\begin{eqnarray}\nD^\\mu\\Theta^{(\\phi)}_{\\mu\\nu} &=& {1\\over 6}\\left[R_{(\\nu\\rho)}-\\frac{1}{2} g_{\\nu\\rho}R\\right]\nD^\\rho\\Phi^2-{1\\over 12}\\Phi^2 D^\\rho f_{\\rho\\nu}+\n\\tilde\\gamma \\left[(\\bar\\psi_L\\psi_R\\overrightarrow\\appD_\\nu\\phi) + (\\phi^\\dagger\\overleftarrow\\appD_\\nu\\bar\\psi_R\\psi_L)\\right]\\\/,\n\\label{253}\\\\\nD^\\mu T^{(\\psi_L)}_{\\mu\\nu} &=& \\tilde\\gamma\\, Y_\\nu(\\phi,\\psi_L,\\psi_R)+\n\\tilde g' B_{\\nu\\rho}j^{(\\psi_L)\\rho}+\\tilde g F^a_{\\nu\\rho}j^{(\\psi_L)\\rho}_a\\\/,\n\\label{254}\\\\\nD^\\mu T^{(\\psi_R)}_{\\mu\\nu} &=& \\tilde\\gamma\\, Y_\\nu(\\phi,\\psi_L,\\psi_R)+\n\\tilde g' B_{\\nu\\rho}j^{(\\psi_R)\\rho}\\\/,\n\\label{255}\n\\end{eqnarray}\nwith the contribution $Y_\\nu (\\phi ,\\psi_L,\\psi_R)$ of the Yukawa coupling\nto Eqs. (\\ref{254}) and (\\ref{255}):\n\\begin{eqnarray}\n&&Y_\\nu (\\phi ,\\psi_L,\\psi_R) = {1\\over 4}\\left[\n\\left\\{\\bar\\psi_L(\\phi^\\dagger\\overrightarrow\\appD_\\nu\\psi_L) + (\\bar\\psi_L\\overleftarrow\\appD_\\nu\\phi)\\psi_R\\right\\}-\n\\left\\{(\\bar\\psi_L\\phi)\\overrightarrow{\\simD}_\\nu\\psi_R + \\bar\\psi_R\\overleftarrow{\\simD}_\\nu (\\phi^\\dagger\\psi_L)\\right\\}\\right.\n\\nonumber\\\\\n&&\\qquad\\qquad\\left.\n+\\left\\{(\\psi_L\\psi_R\\overrightarrow\\appD_\\nu\\phi) + (\\phi^\\dagger\\overleftarrow\\appD_\\nu\\bar\\psi_R\\psi_L)\\right\\}-\n\\left\\{(\\bar\\psi_L\\gamma_\\nu\\gamma_\\rho\\psi_R\\overrightarrow\\appD{}^\\rho\\phi) +\n (\\phi^\\dagger\\overleftarrow\\appD{}^\\rho\\bar\\psi_R\\gamma_\\rho\\gamma_\\nu\\psi_L)\\right\\}\\right]\\\/,\n\\label{256}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\nD^\\mu T^{(f)}_{\\mu\\nu} &=& 6\\tilde\\alpha f_{\\nu\\rho}D^\\rho R\\\/,\n\\label{257}\\\\\nD^\\mu T^{(B)}_{\\mu\\nu} &=& -\\tilde g' B_{\\nu\\rho}\\left[j^{(\\phi)\\rho}+j^{(\\psi_L)\\rho}+j^{(\\psi_R)\\rho}\\right]\\\/,\n\\label{258}\\\\\nD^\\mu T^{(F_a)}_{\\mu\\nu} &=& -\\tilde g F^a_{\\nu\\rho}\\left[j^{(\\phi)\\rho}_a + j^{(\\psi_L)\\rho}_a\\right]\\\/.\n\\label{259}\n\\end{eqnarray}\n\nThe last two equations as well as the field equations (\\ref{238})\nand (\\ref{239}) show that it is useful to introduce the following\ntotal $U(1)_Y$ and total $SU(2)_W$ current densities of Weyl weight zero\ncomposed of bosonic and fermionic contributions:\n\\begin{eqnarray}\nU(1)_Y~~:~~ J_\\rho &=& -{{\\partial\\buildrel\\approx\\over{\\cal L}_{W_4}}\\over {\\partial B^\\rho}} = K\\sqrt{-g}\\,\\tilde g'\n \\left[j^{(\\phi)}_\\rho + j^{(\\psi_L)}_\\rho + j^{(\\psi_R)}_\\rho\\right]\\\/,\n\\label{260}\\\\\nSU(2)_W~:~~ J_{a\\rho} &=& -{{\\partial\\buildrel\\approx\\over{\\cal L}_{W_4}}\\over {\\partial A^{a\\rho}}} = K\\sqrt{-g}\\,\\tilde g\n \\left[j^{(\\phi)}_{a\\rho} + j^{(\\psi_L)}_{a\\rho}\\right]\\\/.\n\\label{261}\n\\end{eqnarray}\nThese definitions allow the separation of the Lagrangean density $\\buildrel\\approx\\over{\\cal L}_{W_4}$\ninto a Weyl-invariant ``free'' part $\\buildrel\\approx\\over{\\cal L}{}\\!^{(0)}_{W_4}$\n(obtained from Eq. (\\ref{228})\nfor $B_\\rho\\equiv 0$ and $A^a_\\rho \\equiv 0$) and a Weyl-invariant interaction part\n$\\buildrel\\approx\\over{\\cal L}_{int}$ expressed in terms of the currents (\\ref{260}) and\n(\\ref{261}) and the corresponding gauge fields, i.e.\n\\begin{equation}\n\\buildrel\\approx\\over{\\cal L}_{int}~= - J^\\rho B_\\rho - J^\\rho_a A^a_\\rho \\\/,\n\\label{261a}\n\\end{equation}\nyielding thus the decomposition\n\\begin{equation}\n\\buildrel\\approx\\over{\\cal L}_{W_4} ~=~ \\buildrel\\approx\\over{\\cal L}{}\\!^{(0)}_{W_4} + \\buildrel\\approx\\over{\\cal L}_{int} + \\buildrel\\approx\\over{\\cal L}{}\\!^{(B,F_a)} \\\/,\n\\label{261b}\n\\end{equation}\nwhere $\\buildrel\\approx\\over{\\cal L}{}\\!^{(B,F_a)}$ is given by the last two terms in (\\ref{228})\nproportional to $\\buildrel\\approx\\over\\delta$ representing the Lagrangean density of the free\n$B_\\rho$ and $A^a_\\rho$ fields.\n\nThe energy-momentum balance for the interacting fields is represented\nby the Weyl-covariant divergence of Eq. (\\ref{240}). Using Eqs. (\\ref{253})\n--(\\ref{259}) together with the contracted Bianchi identities (I, A40)\nfor a $W_4$, it is seen that this is identically satisfied for solutions\nof the field equations for any value of $\\tilde\\gamma$. We shall find later \nin Sect. III that the analogous argument for the broken Weyl theory \nyields a constraint which is required to be satisfied for the\ndivergence relation following from the $g_{\\mu\\nu}$-equations --- representing\nthe over-all energy-momentum conservation in the broken case --- to\nvanish again.\n\nLet us now turn to the conservation relations for the currents defined\nin Eqs. (\\ref{241}) -- (\\ref{244}) as well as for the total\nelectromagnetic current. Introducing the charge operator\n\\begin{equation}\n\\hat q = \\frac{1}{2}(1 + \\tau_3)\n\\label{262}\n\\end{equation}\nfor the $\\phi$-field the total electromagnetic current is given by\n\\begin{eqnarray}\nj^{(e.m.)}_\\rho ={1 \\over {K\\sqrt{-g}}}\\left({1\\over\\tilde g'}J_\\rho + \n{1\\over\\tilde g}J_{3\\rho}\\right)\n&\\equiv& j^{(\\phi)}_\\rho+j^{(\\psi_L)}_\\rho+j^{(\\psi_R)}_\\rho +j^{(\\phi)}_{3\\rho}+j^{(\\psi_L)}_{3\\rho}\n\\nonumber\\\\\n&=&{i\\over 2}\\left(\\phi^\\dagger \\hat q\\overrightarrow\\appD_\\rho\\phi - \\phi^\\dagger\\overleftarrow\\appD_\\rho \\hat q\\phi\\right)\n-\\bar\\psi_e\\gamma_\\rho\\psi_e ~,\n\\label{263}\n\\end{eqnarray}\nHere $\\hat q$ projects out the $\\varphi_+$-component of $\\phi$ with charge\n$q=+1e$ in the $\\phi$-part of the electromagnetic current\nwhile the last term in (\\ref{263}) yields, with\n$-(\\bar e_L\\gamma_\\rho e_L)-(\\bar e_R\\gamma_\\rho e_R)=-\\bar\\psi_e\\gamma_\\rho\\psi_e$, the\nelectromagnetic current contribution of the charged lepton (electron) with\n$q=-1e$.\n\nIt is easy to show from the field equations (\\ref{235}) and (\\ref{236}) and\ntheir adjoints that the Weyl-covariant divergence of the currents (\\ref{242})\nyield\n\\begin{equation}\nD^\\rho j^{(\\psi_R)}_\\rho = -2D^\\rho j^{(\\psi_L)}_\\rho = i\\tilde\\gamma\n\\left[(\\bar\\psi_L \\phi )\\psi_R - \\bar\\psi_R(\\phi^\\dagger \\psi_R)\\right]\\\/.\n\\label{264}\n\\end{equation}\nThis implies for the fermion part of the electromagnetic current the relation\n\\begin{equation}\n\\buildrel\\approx\\over{D}\\!{}^\\rho\\left[j^{(\\psi_L)}_\\rho + j^{(\\psi_R)}_\\rho + j^{(\\psi_L)}_{3\\rho}\\right] =\ni\\tilde\\gamma\\left[(\\bar\\psi_L \\hat q\\phi)\\psi_R - \\bar\\psi_R(\\phi^\\dagger \\hat q\\psi_L)\\right]\\\/,\n\\label{265}\n\\end{equation}\nwhere we have, moreover, used (for $a=3$) the divergence relation for the\nfermionic isospin current\n\\begin{equation}\n\\buildrel\\approx\\over{D}\\!{}^\\rho j^{(\\psi_L)}_{a\\rho} \\equiv D^\\rho j^{(\\psi_L)}_{a\\rho}-\n\\tilde g \\varepsilon_{abc}A^b_\\rho j^{(\\psi_L)\\rho}_c\n= i\\tilde\\gamma\\left[(\\bar\\psi_L\\frac{1}{2}\\tau_a\\phi)\\psi_R -\n \\bar\\psi_R (\\phi^\\dagger \\frac{1}{2}\\tau_a\\psi_L)\\right]\n\\label{266}\n\\end{equation}\nfollowing from Eqs. (\\ref{244}) and (\\ref{235}).\nOn the other hand, one concludes from Eqs. (\\ref{238}) and (\\ref{239})\nthat the hypercharge and isospin currents appearing on the r.-h. sides\nof these equations are covariantly conserved.\n\nSimilarly, one concludes from the field equation (\\ref{234}) and\nits adjoint that the $\\phi$-part of the isospin current obeys\n\\begin{equation}\n\\buildrel\\approx\\over{D}\\!{}^\\rho j^{(\\phi)}_{a\\rho} = -i\\tilde\\gamma\\left[(\\bar\\psi_L\\frac{1}{2}\\tau_a\\phi)\\psi_R -\n \\bar\\psi_R(\\phi^\\dagger\\frac{1}{2}\\tau_a\\psi_L)\\right]\\\/,\n\\label{267}\n\\end{equation}\nyielding for the sum $j^{(\\phi)}_\\rho+j^{(\\phi)}_{3\\rho}$ the relation\n\\begin{equation}\n\\buildrel\\approx\\over{D}\\!{}^\\rho\\left[j^{(\\phi)}_\\rho + j^{(\\phi)}_{3\\rho}\\right] =\n-i\\tilde\\gamma\\left[(\\bar\\psi_L \\hat q\\phi)\\psi_R - \\bar\\psi_R(\\phi^\\dagger \\hat q\\psi_L )\\right]\\\/.\n\\label{268}\n\\end{equation}\nEqs. (\\ref{265}) and (\\ref{268}) together, finally, lead for the total\nelectromagnetic current (\\ref{263}) to the conservation relation\n\\begin{equation}\nD^\\rho j^{(e.m.)}_\\rho = 0\\\/,\n\\label{269}\n\\end{equation}\nwhere, for a fixed $3$-direction, we have replaced the $SU(2)_W$\nand Weyl-covariant derivative $\\buildrel\\approx\\over{D}_\\rho$ by $D_\\rho$.\n\nAs usual we now introduce the Weinberg angle $\\theta_W$ by the rotation\nrelating $A_{3\\rho}$ and $B_\\rho$ to the $q=0$ component\nof the the $SU(2)_W$ gauge fields $Z_\\rho$ and the electromagnetic fields\n$A_\\rho$:\n\\begin{eqnarray}\nA_{3\\rho} &=&~~ \\cos\\theta_W ~Z_\\rho + \\sin\\theta_W ~{e\\over \\hbar c}A_\\rho\n\\nonumber\\\\\nB_\\rho &=& -\\sin\\theta_W ~Z_\\rho + \\cos\\theta_W ~{e\\over \\hbar c}A_\\rho \\\/.\n\\label{270}\n\\end{eqnarray}\nThe factor $e\/\\hbar c$ in front of the photon field\nis introduced here for dimensional reasons\nwith the fields $A_{3\\rho}, B_\\rho , Z_\\rho ,$ and ${e\\over{\\hbar c}} A_\\rho$ all\nhaving length dimension $[L^{-1}]$.\nBelow we shall frequently abbreviate the electromagnetic potential\n${e\\over{\\hbar c}} A_\\mu$ with dimension $[L^{-1}]$ as $\\bar A_\\mu$. The\ndimensionless gauge coupling constants $\\tilde g$ and $\\tilde g'$ were introduced\nin Eq. (\\ref{227}). In addition we shall introduce below the\ndimensionless coupling constant $\\tilde g_0$ for the neutral fields $Z_\\rho$\nand a dimensionless strength for the electromagnetic coupling\ngiven by $\\tilde e =\\sqrt{4\\pi\\alpha_F}$ with $\\alpha_F =e^2\/\\hbar c = 1\/137.04$\ndenoting the fine-structure constant. As usual the elementary\nelectromagnetic charge --- a quantity with a dimension --- is denoted by \n$e$ with $-1|e|=-4.8032\\cdot 10^{-10}\\,esu$\nbeing the electron charge. [$1\\,esu = 1\\,dyn^{\\frac{1}{2}}\\,cm$; we are using\n$cgs$-units as conventional reference units. The intrinsic length\nunit obtained after Weyl-symmetry breaking will be introduced\nin Sect. III below.]\n\nUsing now a spherical basis for the isovector contributions\nin Eq. (\\ref{261a}) introducing the following charge changing current\ncomponents with $\\Delta q=\\pm 1$ [compare (\\ref{261})]\n\\begin{equation}\nJ^{(1)}_\\rho = {1 \\over \\sqrt{2}}\\left(J_{1\\rho} - iJ_{2\\rho}\\right)\\\/;\\qquad\nJ^{(1)}_\\rho {}^\\dagger = {1 \\over \\sqrt{2}}\\left(J_{1\\rho} + iJ_{2\\rho}\\right)\\\/,\n\\label{273}\n\\end{equation}\nand the corresponding gauge fields\n\\begin{equation}\nW_\\rho = {1 \\over \\sqrt{2}}\\left(A_{1\\rho} - iA_{2\\rho}\\right)\\\/;\\qquad\nW_\\rho{}^\\dagger = {1 \\over \\sqrt{2}}\\left(A_{1\\rho} + iA_{2\\rho}\\right)\\\/,\n\\label{274}\n\\end{equation}\nand expressing, furthermore, the fields $A_{3\\rho}$ and $B_\\rho$ in terms\nof the fields $Z_\\rho$ and $A_\\rho$ according to (\\ref{270}) the interaction\nLagrangean (\\ref{261a}) of Weyl weight zero may be written as\n\\begin{equation}\n\\buildrel\\approx\\over{\\cal L}_{int}~=-K\\sqrt{-g}\\left(\\tilde e\\, j^{(e.m.)}_\\rho \\bar A^\\rho +\n\\tilde g_0\\, j^{(0)}_\\rho Z^\\rho\\right)\n-\\left(J^{(1)}_\\rho{}^\\dagger W^\\rho + J^{(1)}_\\rho W^\\rho{}^\\dagger\\right)\\\/.\n\\label{275}\n\\end{equation}\nHere $j^{(e.m.)}_\\rho$ is the electromagnetic current defined in\n(\\ref{263}). In order to obtain the electromagnetic interaction\nin the conventional form given in (\\ref{275}) one has to demand\nthat the coupling constants $\\tilde g$ and $\\tilde g'$ and the Weinberg angle\n$\\theta_W$ are related by\n\\begin{equation}\n\\tilde g\\,\\sin \\theta_W = \\tilde g' \\cos \\theta_W = {\\tilde g\\tilde g'\\over{\\sqrt{\\tilde g^2+\\tilde g'^2}}} = \\tilde e \\\/,\n\\label{276}\n\\end{equation}\nwhere $\\tilde e$ is the dimensionless electromagnetic coupling strength\nintroduced above. The second term in the first bracket of\n(\\ref{275}) describes the coupling of the neutral\nweak current $j^{(0)}_\\rho$ to the neutral gauge fields $Z_\\rho$ with\na coupling strength given by \n\\begin{equation}\n\\tilde g_0 = {\\tilde g\\tilde g'\\over{\\tilde e}} = {\\tilde g\\over{\\cos\\theta_W}} = \\sqrt{\\tilde g^2 + \\tilde g'^2}\\\/,\n\\label{276a}\n\\end{equation}\nwhere $j^{(0)}_\\rho$ is defined by\n\\begin{equation}\nj^{(0)}_\\rho = j^{(\\phi)}_{3\\rho} + j^{(\\psi_L)}_{3\\rho} - \\sin^2\\theta_W ~j^{(e.m.)}_\\rho \\\/.\n\\label{277}\n\\end{equation}\nAll these definitions are the same as those appearing in the usual\nformulation of the standard electroweak model.\n\nAlong with $J^{(1)}_\\rho$ and $J^{(1)}_\\rho{}^\\dagger$ one may now also introduce\nthe current densities $J^{(e.m.)}_\\rho$ and $J^{(0)}_\\rho$ of\nWeyl weight zero by\n\\begin{equation}\nJ^{(e.m.)}_\\rho = K\\sqrt{-g}~\\tilde e j^{(e.m.)}_\\rho ~;\\qquad\nJ^{(0)}_\\rho = K\\sqrt{-g}~\\tilde g_0 j^{(0)}_\\rho\\\/.\n\\label{278}\n\\end{equation}\n\nThe last two terms in Eq. (\\ref{275}) represent the\ncoupling of the charged $SU(2)_W$ gauge fields to the charge\nchanging weak currents describing the weak decays of particles\nin phenomenological applications. For low momentum transfer\nprocesses, $p^2\\approx 0$, we have by identification with the\neffective current-current theory of weak interactions with the\nFermi constant $G_F$ the relation\n\\begin{equation}\n\\left(\\frac{1}{2}\\,{\\tilde g\\over{\\sqrt{2}}}\\right)^2\\,{1\\over{m_W^2}} =\n{G_F\\over{\\sqrt{2}}}\\\/,\n\\label{278a}\n\\end{equation}\nwhere $m_W$ is the mass of the $W$-field defined in Eq. (\\ref{347})\nbelow. Compare in this context also Eq. (\\ref{261}) as well\nas (\\ref{328}) below.\\\\\n\n\\noindent{\\bf C. Remarks Concerning Symmetry Breaking}\n\nWe, finally, compute the square of the Dirac operators\n\\begin{equation}\n\\buildrel\\approx\\over{\\D} = -i\\gamma^\\mu\\!\\buildrel\\approx\\over{D}_\\mu \\quad {\\rm and} \\quad\n\\buildrel\\sim\\over{\\D} = -i\\gamma^\\mu\\!\\buildrel\\sim\\over{D}_\\mu\n\\label{279}\n\\end{equation}\nin application to the fermion fields $\\psi_L$ and $\\psi_R$, respectively.\nFrom the field equations (\\ref{235}) and (\\ref{236}) one easily\nderives that\n\\begin{equation}\n\\buildrel\\approx\\over{\\D}\\Dtt\\psi_L = -i\\tilde\\gamma\\ga^\\mu\\psi_R \\buildrel\\approx\\over{D}_\\mu\\phi + \\tilde\\gamma^2\\phi(\\phi^\\dagger\\psi_L)\\\/,\n\\label{280}\n\\end{equation}\nand\\begin{equation}\n\\buildrel\\sim\\over{\\D}\\Dt\\psi_R = -i\\tilde\\gamma (\\phi^\\dagger\\overleftarrow\\appD_\\mu\\gamma^\\mu\\psi_L) +\n\\tilde\\gamma^2(\\phi^\\dagger\\phi)\\psi_R\\\/.\n\\label{281}\n\\end{equation}\nAs regards the $SU(2)_W$ degrees of freedom Eq. (\\ref{280}) is a\nmatrix equation for a two-component isospinor while (\\ref{281}) is\nan equation for an isoscalar with the round brackets denoting\n$SU(2)_W$ invariants. The left- and right-handed fields are coupled\nin these equations through the first terms on their r.-h. sides.\nWe observe that both equations {\\it decouple} and become eigenvalue\nequations for $\\psi_L$ and $\\psi_R$, respectively, which are moreover\n{\\it diagonal in spin space} provided the $\\phi$-field is covariant\nconstant, i.e. obeys\n\\begin{equation}\n\\buildrel\\approx\\over{D}_\\mu \\phi = 0\\\/.\n\\label{282}\n\\end{equation}\nA property of this type would annihilate the first term in the\nfield equations (\\ref{235}) for $\\phi$ and would yield an algebraic\nconstraint involving the curvature scalar $R$, the $\\phi$-field, and\nthe $\\psi$-fields. It is not immediately clear whether this is\nconsistent with the other field equations. Therefore we shall not\ndemand Eq. (\\ref{282}) to be satisfied. However, an equation of a\nsimilar nature will be used in Sect. III below when we investigate\nthe breaking of the Weyl symmetry to obtain, finally, a gauge\ntheory formulated in a Riemannian space-time in the limit.\n\nAfter these remarks concerning the standard electroweak theory \nand its formulation in a Weyl geometric framework we turn, as mentioned,\nto the breaking of the $G$-gauge symmetry. Let us, however, first\nconsider, at the end of this section, the so-called breaking of the\nelectroweak gauge symmetry $\\buildrel\\sim\\over{G} = SU(2)_W\\times U(1)_Y$ to the\nelectromagnetic gauge symmetry $U(1)_{e.m.}$ which is described in the\nstandard model as the result of a so-called ``spontanous symmetry breaking\ndue to a nonvanishing vacuum expectation value of the scalar field''.\n\nIn order to view the situation more clearly we investigate\nin Appendix A the coset representation for the field $\\phi$\nin terms of transformations $U(\\bar g_\\phi)$ or $U(\\phi)$ parametrizing\n$\\buildrel\\sim\\over{G}\\!\/H$ [see (A5) and (A11)] by\ngenerating the function $\\phi$ from the real function\n$\\hat\\phi =\\left({0\\atop\\hat\\varphi_0}\\right)$ with $\\hat\\phi$\nbeing invariant under the subgroup $H$ of $\\buildrel\\sim\\over{G}$, where $H$ is\nidentified with the {\\it electromagnetic} gauge group $U(1)_{e.m.} = U(1)_+$ generated\nby $\\hat q$ and obeying $\\hat q\\,\\hat\\phi =0$ [compare (A3) and (A7)]. The transition\nfrom $\\phi$ to $\\hat\\phi$ with the help of the transformation $U^{-1}(\\phi)$\nis to be regarded as a {\\it choice of coordinates} for the \nrepresentation of the scalar field in the theory and has,\nin the first place, nothing to do with a ``vacuum expectation\nvalue'' of this field. To adopt the origin $\\hat\\phi$ in $\\buildrel\\sim\\over{G}\\!\/H$\nas a parametrization for the scalar field is done for physical\nreasons establishing thereby the electromagnetic gauge group\n$H = U(1)_{e.m.}$ as stability group of the point $\\hat\\phi$ in the\nformalism and relate it to physical observations and experiments.\nThis choice is, actually, not a breaking of the original $\\buildrel\\sim\\over{G}$-gauge theory\nbut a different realization of it. It is thus better to say\nthat after transforming $\\phi$ to $\\hat\\phi$ with the help of $U^{-1}(\\phi)$\none has adopted an {\\it electromagnetic gauge} in the electroweak theory\nwith the residual gauge transformations being given by $U(h(\\alpha))\\inU(1)_{e.m.}$\n[see (A7), (A15) and the relation of these transformations to the so-called\n``Wigner rotations'']. A true symmetry reduction from $G$ to a\nsubgroup of $G$ is governed by a relation of the type (\\ref{282}) to\nwhich we turn in the next section \\cite{5}.\n\nConditions of the type of Eq. (\\ref{282}) where $\\phi$ is a section\non a bundle with a homogeneous space as fiber are well-known from\ndifferential geometry. They guarantee that in a reduction of the\nstructural group of a certain principal bundle $P(B,G)$ over the\nbase $B$ and with structural group $G$ to a bundle $P'(B,G')$\nover the same base and with a subgroup $G'$ of $G$ as structural\ngroup also the corresponding {\\it connection} reduces from a\n${\\bf g}$-valued to a ${\\bf g'}$-valued form, where ${\\bf g}$ and\n${\\bf g'}$ denote the respective Lie algebras of the groups $G$\nand $G'\\subset G$ (compare \\cite{6} for details). In concluding this\nsection let us, therefore, state the following theorem well-known from\nthe literature on differential geometry: {\\it As the condition for\na true symmetry reduction in the physical sense from a theory with\ngauge group $G$ to a theory with gauge group $G'\\subset G$, implying\nalso the reduction of the connection on $P(B,G)$ to the connection\non $P'(B,G')$, it is required that there exists a section $\\phi_E$ \non the bundle $E(B,G\/G',G)$, associated to $P(B,G)$, with fiber $F=G\/G'$ and\nstructural group $G$ which is covariant\nconstant, i.e. obeys $D\\phi_E = 0$, where $D$ is the covariant\nderivative on $E$.}\n\n\\section{Weyl-Symmetry Breaking}\nAs a term in the Lagrangean which breaks the Weyl-symmetry with the aim\nof introducing a scale of lengths with the help of the $\\phi$-field and\nestablish an electroweak theory of leptons in the presence\nof gravitation formulated in a $V_4$, we add the following expression\nof Weyl weight $+1$ to the Weyl-invariant Lagrangean density\n$\\buildrel\\approx\\over{\\cal L}_{W_4}$ given in (\\ref{228}):\n\\begin{equation}\n{\\cal L}_B = -{a\\over 2}K\\sqrt{-g}\\left\\{{1\\over 6}R + \\left[{mc\\over\\hbar}\\right]^2\n \\phi^\\dagger\\phi\\right\\}\\\/.\n\\label{31}\n\\end{equation}\nHere $a$ is a dimensionless constant, $R$ is the curvature scalar of\nthe ambient Weyl space $W_4$ [see (\\ref{230})] which is related here to the\nmass --- or rather Compton wave length --- of the universal scalar field $\\phi$\nby tying $R$ to the squared modulus $\\Phi^2 = \\phi^\\dagger\\phi$ of this field.\nThe expression (\\ref{31}) is independent of the standard model gauge fields\nassociated with the group $\\buildrel\\sim\\over{G}$ and thus leaves the $\\buildrel\\sim\\over{G}$-gauge invariance\nunaffected. However, the explicit breaking of the $D(1)$ symmetry caused\nby ${\\cal L}_B$ will lead, as we shall see, to nonzero masses not only for the\n$\\phi$-field but also for the fermion and the gauge boson fields in a manner\nsimilar to the situation realized in the standard electroweak theory.\n\nWe have mentioned at the end of Sect. II that the standard model is not\ncharacterized by a true symmetry reduction from a gauge group $\\buildrel\\sim\\over{G}$\nto a subgroup of $\\buildrel\\sim\\over{G}$. What is conventionally called a spontanous\nsymmetry breaking is a choice of an appropriate coordinatization\ntaking due recognition of the electromagnetic phase group $U(1)_{e.m.}$\nas a subgroup generated by $\\hat q$ in the formalism. On the contrary, \nadding (\\ref{31}) to the Lagrangean $\\buildrel\\approx\\over{\\cal L}_{W_4}$ will lead to a true\nbreaking of the $G$-gauge theory [see (\\ref{215})] to a theory\nwith the subgroup $G\\,'=SO(3,1)\\otimes\\buildrel\\sim\\over{G}$ of $G$ as gauge group.\nIt is the square of the modulus, $\\Phi^2$, of the $\\phi$-field which\nis the section on $E(W_4,G\/G\\,',G)$, required to be covariant constant\nin the sense of the theorem quoted at the end of Sect. II, which\ngoverns, as we shall see, the symmetry breaking by (\\ref{31})\nyielding, ultimately, a $V_4$ from a $W_4$ and the generation of\nnonvanishing masses. Hence the symmetry breaking relation will be\n\\begin{equation}\nD_\\mu \\Phi^2 \\equiv \\partial_\\mu\\Phi^2 + \\kappa_\\mu\\Phi^2 = 0\\\/,\n\\label{32}\n\\end{equation}\nwith $\\Phi^2\\in G\/G\\,'\\equiv D(1)$.\n\nHowever, before we come to this point, let us first derive the field\nequations following from a variational principle formulated with the\nLagrangean ${\\cal L}$ given by\n\\begin{equation}\n{\\cal L} = \\buildrel\\approx\\over{\\cal L}_{W_4} +~ {\\cal L}_B \\\/.\n\\label{33}\n\\end{equation}\nOne finds using the same notation as above:\n\\begin{eqnarray}\n&&\\delta\\phi^\\dagger :~~g^{\\mu\\nu}\\buildrel\\approx\\over{D}_\\mu\\buildrel\\approx\\over{D}_\\nu\\phi + {1\\over 6}R\\phi + 4\\beta\n (\\phi^\\dagger\\phi)\\phi - 2\\tilde\\gamma\\bar\\psi_R\\psi_L + a\\left[{mc\\over\\hbar}\\right]^2\\phi = 0\\\/,\n \\label{34}\\\\\n&&\\delta\\psi_L^\\dagger :~~-i\\gamma^\\mu\\buildrel\\approx\\over{D}_\\mu\\!\\psi_L - \\tilde\\gamma\\phi\\psi_R = 0\\\/,\\label{35}\\\\\n&&\\delta\\psi_R^\\dagger :~~-i\\gamma^\\mu\\buildrel\\sim\\over{D}_\\mu\\!\\psi_R - \\tilde\\gamma (\\phi^\\dagger\\psi_L) = 0\\\/,\\label{36}\\\\\n&&\\delta\\kappa_\\rho :~~\\tilde\\delta D_\\mu f^{\\mu\\rho} = -6\\tilde\\alpha D^\\rho R + {a\\over 4}\\kappa^\\rho\\\/,\n \\label{37}\\\\\n&&\\delta B_\\rho :~~\\buildrel\\approx\\over\\delta D_\\mu B^{\\mu\\rho} = \\tilde g'\\left[j^{(\\phi)\\rho} + j^{(\\psi_L)\\rho} +\n j^{(\\psi_R)\\rho}\\right]\\\/,\\label{38}\\\\\n&&\\delta A^a_\\rho :~~\\buildrel\\approx\\over\\delta \\buildrel\\approx\\over{D}_\\mu\\!F^{\\mu\\rho}_a = \\tilde g\\left[j^{(\\phi)\\rho}_a +\n j^{(\\psi_L)\\rho}_a\\right]\\\/, \\label{39}\\\\\n&&\\delta g_{\\mu\\nu} :~~{1\\over 6}(\\Phi^2+a)\\left[R_{(\\mu\\nu)}-\\frac{1}{2} g_{\\mu\\nu}R\\right]-\n 4\\tilde\\alpha R\\left[R_{(\\mu\\nu)}-{1\\over 4}g_{\\mu\\nu}R\\right]-4\\tilde\\alpha\\left\\{\n D_{(\\mu}D_{\\nu)}R-g_{\\mu\\nu}D^\\rho D_\\rho R\\right\\} = \n \\nonumber\\\\\n&&\\qquad\\qquad =\\Theta^{(\\phi)}_{\\mu\\nu} + T^{(\\psi_L)}_{\\mu\\nu} + T^{(\\psi_R)}_{\\mu\\nu} +\n T^{(f)}_{\\mu\\nu} + T^{(B)}_{\\mu\\nu} + T^{(F_a)}_{\\mu\\nu} - g_{\\mu\\nu}\\tilde\\gamma\n \\left[(\\bar\\psi_L\\phi)\\psi_R+(\\bar\\psi_R(\\phi^\\dagger\\psi_L)\\right] + \\nonumber\\\\\n&&\\qquad\\qquad\\qquad\\qquad +\\,g_{\\mu\\nu}{a\\over 2}\\left[{mc\\over\\hbar}\\right]^2\\Phi^2\\\/,\n \\label{310}\\\\\n&&\\delta a :~~{1\\over 6}R + \\left[{mc\\over\\hbar}\\right]^2\\Phi^2 = 0\\\/,\n\\label{311}\n\\end{eqnarray}\nwhere Eqs. (\\ref{35}), (\\ref{36}), (\\ref{38}) and (\\ref{39}) are unchanged;\ncompare Eqs. (\\ref{235}), (\\ref{236}), (\\ref{238}) and (\\ref{239}).\nThe energy-momentum tensors appearing in (\\ref{310}) are the same as those\ndefined in Eqs. (\\ref{245}) -- (\\ref{249}) of Sect. II. Only the\nenergy-momentum tensor for the scalar field is to be changed now, for\n$a \\neq 0$, to the expression given by the sum of the first and the last\nterm on the r.-h. side of (\\ref{310}), i.e. by\n\\begin{equation}\n\\Theta^{(\\phi)}_{\\mu\\nu}\\,' = \\Theta^{(\\phi)}_{\\mu\\nu} + \n g_{\\mu\\nu}\\,{a\\over 2}\\left[{mc\\over\\hbar}\\right]^2\\Phi^2\\\/.\n\\label{311a}\n\\end{equation}\n\nWe first turn to the trace condition following from (\\ref{310}). The trace\nof $\\Theta^{(\\phi)}_{\\mu\\nu}\\,'$ for solutions of the field equation (\\ref{34})\nand its adjoint is now given by\n\\begin{equation}\n\\Theta_\\mu^{(\\phi)}\\,'{}^\\mu = -{1\\over 6}R\\phi^\\dagger\\phi + \\tilde\\gamma\\left[(\\bar\\psi_L\\phi)\\psi_R +\n \\bar\\psi_R(\\phi^\\dagger\\psi_L)\\right] + a\\left[{mc\\over\\hbar}\\right]^2\\phi^\\dagger\\phi\\\/,\n\\label{312}\n\\end{equation}\nwhile the other traces of the energy-momentum tensors are the same as\nin Sect. II. With these and Eq. (\\ref{312}) one concludes from the trace\nof (\\ref{310}) that\n\\begin{equation}\n\\tilde\\alpha D^\\rho D_\\rho R = 0\\\/.\n\\label{313}\n\\end{equation}\nTaking the Weyl-covariant divergence of Eq. (\\ref{37}) one finds with \n(\\ref{313}) that the Weyl vector fields must satisfy the Lorentz-like\ncondition:\n\\begin{equation}\nD_\\rho \\kappa^\\rho \\equiv \\bar\\nabla_\\rho \\kappa^\\rho - \\kappa_\\rho\\kappa^\\rho = 0 \\qquad {\\rm for}\n \\qquad a\\neq 0\\\/,\n\\label{314}\n\\end{equation}\nwhere $\\bar\\nabla_\\rho$ denotes the metric covariant derivative\n[compare Appendix A of I]. Eq. (\\ref{314}) in turn implies that the\n$W_4$ curvature scalar (\\ref{230}) is now given by\n\\begin{equation}\nR = \\bar R - {3\\over 2}\\kappa_\\rho\\kappa^\\rho\\\/.\n\\label{315}\n\\end{equation}\n\nWe, finally, compute the divergence conditions for the solutions of the\nfield equations (\\ref{34}) -- (\\ref{311}) which follow from (\\ref{310})\nby taking the Weyl-covariant divergence of this equation and using the\ncontracted Bianchi identities (I, A40) for the $W_4$ as well as the\nequations\n\\begin{equation}\nD^\\mu T^{(f)}_{\\mu\\nu} = f_\\nu{}^\\rho \\left[\\,6\\tilde\\alpha D_\\rho R - {a\\over 4}\\kappa_\\rho\\right]\\\/.\n\\label{316}\n\\end{equation}\nBefore the symmetry breaking by ${\\cal L}_B$ these energy-momentum balance\nrelations for the set of interacting fields were identically satisfied in Sect. II.\nIn the broken case we now obtain that the following relations must hold\nfor the divergence relations, deduced from (\\ref{310}), to be fulfilled\nagain [compare (I, 4.16)]~:\n\\begin{equation}\nD^\\mu f_{\\mu\\nu} - 3f_{\\nu\\mu}\\kappa^\\mu = 0 \\qquad {\\rm for} \\qquad a\\neq 0\\\/.\n\\label{317}\n\\end{equation}\nThese relations are trivially satisfied for a Weyl vector field being\n``pure gauge'', i.e. implying $f_{\\mu\\nu} = 0$. This is identical with the\ncondition (\\ref{32}) being satisfied, which may be written as\n\\begin{equation}\n\\kappa_\\mu = -\\partial_\\mu\\log\\Phi^2\\\/.\n\\label{318}\n\\end{equation}\nThe Weyl vector field in this broken Weyl theory is thus derivable\nfrom a potential given by the modulus of the scalar field. This is\nin direct analogy to the case of the Christoffel connection,\n$\\bar\\Gamma_{\\mu\\nu}{}^\\rho=\\{{\\rho\\atop\\mu\\nu}\\}$, following from the\nrelation $\\bar\\nabla_\\rho g_{\\mu\\nu} = 0$ in (pseudo-)Riemannian geometry\nwith $g_{\\mu\\nu}\\in Gl(4,R)\/SO(3,1)$.\n\nThe field equations (\\ref{37}), finally, lead --- together with\n(\\ref{311}) which implies $D_\\rho R = 0$ for $D_\\rho \\Phi^2 = 0$ --- to\n\\begin{equation}\n\\kappa_\\mu = 0\\\/;\\qquad {\\rm i.e.}\\qquad \\Phi^2 = const \\\/.\n\\label{319}\n\\end{equation}\nThis shows that the Weyl space $W_4$ reduces completely to a\npseudo-Riemannian space $V_4$ with the scalar field possessing a\nconstant modulus. The value of this modulus can not be computed\nnumerically. On the other hand, the value for $\\Phi$ will determine\nthe fermion and gauge boson masses appearing in the broken Weyl theory\nas well as in the nonlinearity contained in the $\\phi$-equation\nyielding the ``Higgs dynamics'' in the standard model. A fixing of\nan ungauged $D(1)$ degree of freedom is, indeed, implicit in the\nstandard model. However, a relation of the type (\\ref{32}) for\nthe $D(1)$ gauge symmetry breaking does not appear in the standard \nmodel since this would require to go beyond the flat space formulation\nof the conventional description. In the present context we have to\ninvestigate, in the $V_4$ limit given by (\\ref{319}), the appearance\nof Einstein's equations for the metric coupled to the energy-momentum\ntensors of the now massive fermion and gauge boson fields and establish\nthe fact that gravitation, as we know it from general relativity, is\na natural part of the broken $G$-gauge dynamics described by ${\\cal L}$.\\\\\n\n\\noindent{\\bf A. Electromagnetism in the WEW Theory}\n\nIn this subsection we first turn to the field equations for the\nelectromagnetic fields $F_{\\mu\\nu} = \\partial_\\mu A_\\nu - \\partial_\\nu A_\\mu$\nand the fields $Z_{\\mu\\nu} = \\partial_\\mu Z_\\nu - \\partial_\\nu Z_\\mu$ following\nfrom Eqs. (\\ref{38}) and (\\ref{39}). The total $\\buildrel\\sim\\over{G}$-curvature,\nwritten in Lie algebra valued form with ${\\cal F}_{\\mu\\nu} = F^a_{\\mu\\nu}\\,\\frac{1}{2}\\tau_a$,\nis [compare Eqs. (\\ref{231}) and (\\ref{232})]\n\\begin{equation}\n{\\cal F}_{\\mu\\nu} + \\frac{1}{2}\\,{\\bf 1} B_{\\mu\\nu} = \\partial_\\mu\\!\\buildrel\\sim\\over{\\Gamma}_\\nu - \\partial_\\nu\\!\\buildrel\\sim\\over{\\Gamma}_\\mu\n +\\, i\\tilde g \\left[\\buildrel\\sim\\over{\\Gamma}_\\mu ,\\buildrel\\sim\\over{\\Gamma}_\\nu\\right]\\\/,\n\\label{320}\n\\end{equation}\nwith $\\buildrel\\sim\\over{\\Gamma}_\\mu$ as given by (\\ref{B9}). The definitions of the curvature\ncomponents $F^a_{\\mu\\nu}$ and $B_{\\mu\\nu}$ imply that in a spherical basis\nwe have, with Eqs. (\\ref{270}) and (\\ref{276}), the relations\n\\begin{eqnarray}\nF^3_{\\mu\\nu} &=& \\tilde g\\tilde g_0^{-1}\\,Z_{\\mu\\nu} + \\tilde g'\\tilde g_0^{-1}{e\\over{\\hbar c}}\\,F_{\\mu\\nu}\n - i\\tilde g \\left(W^\\dagger_\\mu W_\\nu - W^\\dagger_\\nu W_\\mu\\right)\\\/,\n\\label{321} \\\\\nB_{\\mu\\nu} &=& -\\tilde g'\\tilde g_0^{-1}\\,Z_{\\mu\\nu} + \\tilde g\\tilde g_0^{-1}{e\\over{\\hbar c}}\\,F_{\\mu\\nu}\\\/,\n\\label{322} \\\\\nF^-_{\\mu\\nu} &=& \\left(F^+_{\\mu\\nu}\\right)^\\dagger = \\partial_\\mu W_\\nu - \\partial_\\nu W_\\mu\n -i \\left\\{\\tilde g\\cos\\theta_W\\;[W_\\mu Z_\\nu - W_\\nu Z_\\mu]\n + \\tilde e\\;{e\\over{\\hbar c}}[W_\\mu A_\\nu - W_\\nu A_\\mu ]\\right\\}\\\/,\n\\label{323}\n\\end{eqnarray}\nwith $F^{\\pm}_{\\mu\\nu}={1\\over{\\sqrt 2}}(F^1_{\\mu\\nu} \\pm iF^2_{\\mu\\nu})$. In (\\ref{320})\n$F^3_{\\mu\\nu}\\,\\frac{1}{2}\\tau_3 +\\frac{1}{2}\\,{\\bf 1}B_{\\mu\\nu}$ is that part\nwhich commutes with $\\hat q$ while $F^{\\pm}_{\\mu\\nu}$, as defined by (\\ref{323}),\ndenote the off diagonal parts which do not commute with $\\hat q$.\n\nWe now rewrite the field equations (\\ref{38}) and (\\ref{39}) --- the latter\nat first for $a=3$ --- using (\\ref{321}), (\\ref{322}) and the definition \n(\\ref{263}) of the electromagnetic current and find , with $\\buildrel\\approx\\over{D}_\\mu\nF_3^{\\mu\\rho}\\equiv D_\\mu F_3^{\\mu\\rho}$ and Eqs. (\\ref{276}) and (\\ref{276a}),\n\\begin{equation}\n{e\\over{\\hbar c}}\\,D_\\mu F^{\\mu\\rho} = {\\buildrel\\approx\\over\\delta}{}^{-1}\\,\\tilde e\\, j^{(e.m.)\\rho} + \n i\\,\\tilde e\\,D_\\mu \\left(W^\\dagger{}^\\mu W^\\rho - W^\\dagger{}^\\rho W^\\mu\\right)\\\/,\n\\label{324}\n\\end{equation}\nand, with $j^{(0)}_\\rho$ as defined in (\\ref{277}),\n\\begin{equation}\nD_\\mu Z^{\\mu\\rho} = {\\buildrel\\approx\\over\\delta}{}^{-1}\\,\\tilde g_0\\,j^{(0)\\rho} + i\\,\\tilde e\\,\\tilde g\\tilde g'^{-1}\\,D_\\mu\n \\left(W^\\dagger{}^\\mu W^\\rho - W^\\dagger{}^\\rho W^\\mu\\right)\\\/.\n\\label{325}\n\\end{equation}\nWe continue to write here the covariant derivative as $D_\\mu$ disregarding for\nthe moment that $\\kappa_\\mu =0$ according to (\\ref{319}) in the broken case\nsince (\\ref{324}) and (\\ref{325}) are valid also in the Weyl symmetric\ntheory discussed in Sect. II. The l.-h. side of (\\ref{324}) could also be\nwritten $D_\\mu\\bar F^{\\mu\\rho}$ with the electromagnetic field strengths\n$\\bar F^{\\mu\\rho}=\\partial^\\mu\\bar A^\\rho - \\partial^\\rho\\bar A^\\mu$ of dimension $[L^{-2}]$.\nEqs. (\\ref{324}), moreover, show that besides the electromagnetic source\ncurrent $j^{(e.m.)}_\\rho$ there contributes also a term on the r.-h.\\,side\nwhich is bilinear in the $W_\\mu$-fields with $W_\\mu$ and $W^\\dagger_\\mu$ being\nrelated to the charge changing weak processes. The same remark applies to the\nlast term in (\\ref{325}) representing the contribution of the \n$W_\\mu$-fields to the neutral weak processes.\n\nFurthermore, Eqs. (\\ref{324}) and (\\ref{325}) imply current conservation\n[compare (\\ref{269})]\n\\begin{equation}\nD^\\rho j_\\rho^{(e.m.)}\\equiv \\bar\\nabla^\\rho j^{(e.m.)}_\\rho = {1\\over{\\sqrt{-g}}}\n \\partial_\\mu\\left(\\sqrt{-g}\\,g^{\\mu\\rho}\\,j^{(e.m.)}_\\rho\\right) = 0 \\\/,\n\\label{325c}\n\\end{equation}\nand, analogously, $D^\\rho j_\\rho^{(0)}=0$\nwith the Weyl-covariant divergence of the $W$-terms and of the \nl.-h. sides of these equations yielding zero. Writing\nEqs. (\\ref{324}) and (\\ref{325}) in the electromagnetic gauge (see Appendix B)\nit follows from (\\ref{B5}) -- (\\ref{B8}) and (\\ref{B16}) that these\nequations are $U(1)_{e.m.}$ gauge invariant.\n\nIt may be worth while in this context to write down explicitly the $U(1)_{e.m.}$ gauge\ninvariant source currents $\\hat j^{(e.m.)}_\\rho$ and $\\hat j^{(0)}_\\rho$ in the \nelectromagnetic gauge [compare Eqs. (\\ref{263}), (\\ref{277}), (\\ref{B3}),\n(\\ref{B14}) and (\\ref{B15})]\n\\begin{eqnarray}\n\\hat j^{(e.m.)}_\\rho &=& - \\hat{\\bar e}_L\\gamma_\\rho\\hat e_L - \\hat{\\bar e}_R\\gamma_\\rho\\hat e_R\n\\nonumber \\\\\n &=& - (\\hat\\vphi_0)^{-2}\\Biggl\\{|\\vphi_+|^2\\left(\\bar\\nu_L\\gamma_\\rho\\nu_L\\right) -\n |\\vphi_+|^2\\left(\\bar e_L\\gamma_\\rho e_L\\right) + \\vphi_+\\vphi^*_0\\left(\\bar\\nu_L\\gamma_\\rho e_L\\right)\n\\nonumber \\\\\n&\\hfil& \\qquad +\\vphi^*_+\\vphi_0\\left(\\bar e_L\\gamma_\\rho\\nu_L\\right)\\Biggr\\} -\n \\bar\\psi_e\\gamma_\\rho\\psi_e\\\/,\n\\label{324a}\n\\end{eqnarray}\nand\n\\begin{equation}\n\\hat j^{(0)}_\\rho = -{1\\over 8}\\tilde g\\tilde g' (\\hat\\vphi_0)^2\\,\\left(\\hat Z_\\rho \n + \\hat Z_\\rho^\\dagger\\right) + \\frac{1}{2}\\left(\\hat{\\bar\\nu}_L\n \\gamma_\\rho\\hat\\nu_L\\right) - \\frac{1}{2}\\left(\\hat{\\bar e}_L\\gamma_\\rho\\hat e_L\\right) -\n \\sin^2\\theta_W\\;\\hat j^{(e.m.)}_\\rho \\,.\n\\label{325a}\n\\end{equation}\n\nWe, finally, determine the field equations for $F^{\\pm}_{\\mu\\nu}$, i.e. for\n$a=1,2$ in (\\ref{39}). They read:\n\\begin{eqnarray}\n\\buildrel\\approx\\over{D}_\\mu (F^-){}^{\\mu\\rho} &\\equiv& D_\\mu(F^-){}^{\\mu\\rho} - i\\tilde g\\, W_\\mu F^{\\mu\\rho}_3\n +i\\tilde g\\, A^3_\\mu(F^-){}^{\\mu\\rho} =\\, {\\buildrel\\approx\\over\\delta}{}^{-1} \\tilde g\\; j^{(1)\\rho} \\\/,\n\\label{326} \\\\\n\\buildrel\\approx\\over{D}_\\mu (F^+){}^{\\mu\\rho} &\\equiv& D_\\mu (F^+){}^{\\mu\\rho} +i\\tilde g\\,W^\\dagger_\\mu F^{\\mu\\rho}_3\n -i\\tilde g\\, A^3_\\mu(F^+){}^{\\mu\\rho} =\\, {\\buildrel\\approx\\over\\delta}{}^{-1} \\tilde g\\; j^{(1)\\rho\\dagger}\\\/,\n\\label{327}\n\\end{eqnarray}\nwhere [compare (\\ref{261}) and (\\ref{273})]\n\\begin{equation}\nj^{(1)}_\\rho = {1\\over{\\sqrt 2}}\\left(j^{(\\phi)}_{1\\rho} + j^{(\\psi_L)}_{1\\rho} -\n i\\,j^{(\\phi)}_{2\\rho} - i\\,j^{(\\psi_L)}_{2\\rho}\\right)\n\\label{328}\n\\end{equation}\nis the charge changing current. In (\\ref{326}) and (\\ref{327}) $F_3^{\\mu\\rho}$ and\n$A^3_\\mu$ may be replaced according to Eqs. (\\ref{321}) and (\\ref{270}) in\norder to yield field equations involving only the fields $A_\\mu, Z_\\mu$ and\n$W_\\mu , W^\\dagger_\\mu$. Written in the electromagnetic gauge Eqs. (\\ref{326})\nand (\\ref{327}) are again $U(1)_{e.m.}$ gauge covariant. To establish this result\none needs the formula\n\\begin{equation}\n\\left(\\hat F^{\\mp}_{\\mu\\nu}\\right)' = e^{\\mp i{e\\over{\\hbar c}}\\alpha(\\phi ',\\phi)}\\;\n \\hat F^{\\mp}_{\\mu\\nu}\\\/,\n\\label{329}\n\\end{equation}\nwhich is easily derivable from (\\ref{323}) with the help of\n(\\ref{B5}) -- (\\ref{B8}). Moreover, one needs the following expression\nfor the the current $\\hat j^{(1)}_\\rho$ evaluated at the origin in $\\buildrel\\sim\\over{G}\\!\/H$.\nWith the help of Eqs. (\\ref{B3}) and (\\ref{B15}) one finds\n\\begin{equation}\n\\hat j^{(1)}_\\rho = - {1\\over 4}\\tilde g\\, (\\hat\\vphi_0)^2\\,\\hat W_\\rho + {1\\over{\\sqrt 2}}\\,\n \\hat{\\bar e}_L\\gamma_\\rho\\hat\\nu_L \\\/.\n\\label{330}\n\\end{equation}\nFrom the form of (\\ref{330}) the transformation rule\n\\begin{equation}\n\\left(\\hat j^{(1)}_\\rho\\right)' = e^{-i{e\\over{\\hbar c}}\\alpha(\\phi ',\\phi)}\\,\\hat j^{(1)}_\\rho\n\\label{331}\n\\end{equation}\nunder residual $U(1)_{e.m.}$ gauge transformations is at once apparent as a\nconsequence of (\\ref{B6}) and (\\ref{B16}), and correspondingly for\n$\\hat j_\\rho^{(1)}{}^\\dagger$ appearing in Eq. (\\ref{327}) after transformation\nto the electromagnetic gauge.\n\nIn order to gain information about the constant $\\buildrel\\approx\\over\\delta$ in\nEqs. (\\ref{324}) and (\\ref{325}) and bring (\\ref{324}) \n--- disregarding the $W_\\mu$-contributions for a moment --- into the\nform of Maxwell's equations in\nelectromagnetism, we first observe that each term in these\nequations has length dimension $[L^{-3}]$. Multiplying (\\ref{324}) by the\ncharge $e$ and introducing the fine-structure constant \n$\\alpha_F=\\tilde e^2\/4\\pi$ we can rewrite Eqs. (\\ref{324}) and (\\ref{325})\nas\n\\begin{eqnarray}\nD_\\mu F^{\\mu\\rho} &=& {4\\pi\\over{c}}\\,{\\bf j}^{(e.m.)\\rho} +i\\,{4\\pi\\over\\tilde e}\\,\n e\\,D_\\mu\\left(W^{\\dagger\\mu}W^\\rho - W^{\\dagger\\rho}W^\\mu\\right)\\\/,\n\\label{334} \\\\\nD_\\mu Z^{\\mu\\rho} &=& {1\\over{c}}\\,{\\bf j}^{(0)\\rho} + i\\,\\tilde e\\,\\tilde g\\tilde g'^{-1}\n D_\\mu\\left(W^{\\dagger\\mu}W^\\rho - W^{\\dagger\\rho}W^\\mu\\right)\\\/,\n\\label{335}\n\\end{eqnarray}\nby taking (compare (\\ref{346a}) and (\\ref{346b}) below)\n\\begin{equation}\n\\buildrel\\approx\\over\\delta = \\buildrel\\approx\\over\\delta{}\\!'\\,l_\\vphi^2 \\qquad \\mbox{with} \\qquad \\buildrel\\approx\\over\\delta{}\\!'= {1\\over\\tilde e} \\\/,\n\\label{336}\n\\end{equation}\nFurthermore, we have introduced here the following current densities:\n\\begin{equation}\n{\\bf j}^{(e.m.)\\rho} = e\\,c\\,l_\\vphi^{-2}\\,j^{(e.m.)\\rho}\\\/;\\qquad\n{\\bf j}^{(0)\\rho} = \\tilde e \\tilde g_0\\,c\\,l_\\vphi^{-2}\\,j^{(0)\\rho}\\\/.\n\\label{337}\n\\end{equation}\n\nWe have measured in (\\ref{336}) the constant $\\buildrel\\approx\\over\\delta$ of dimension\n$[L^2]$ in units of $l_\\vphi^2$ defined in Eq. (\\ref{346a}) and\ndetermined the numerical coefficient $\\buildrel\\approx\\over\\delta{}\\!'$ in such a way that\nthe first term on the r.-h. side of (\\ref{334}) has the\nconventional form known from electromagnetism. This fixed the\nnumerical constant $\\buildrel\\approx\\over\\delta{}\\!'$ to the value $\\tilde e^{-1}=1\/\\sqrt{4\\pi\\alpha_F}$.\\\\\n\n\\noindent{\\bf B. Gravitation in the Broken WEW Theory}\n\nEinstein's equations for the metric follow from (\\ref{310}), as we\nwill now show, with a total energy-momentum tensor $T_{\\mu\\nu}$ on\nthe r.-h. side for all the interacting massive and massless\nfields involved. Clearly, $T_{\\mu\\nu}^{(f)}=0$ as a consequence\nof (\\ref{319}). We first turn to the contributions of the\n$\\buildrel\\sim\\over{G}$-gauge fields contained in $T_{\\mu\\nu}^{(B)}+T_{\\mu\\nu}^{(Fa)}$\non the r.-h. side of (\\ref{310}) and split this expression\ninto the familiar electromagnetic contribution, a $Z_\\mu$-contribution,\nand $W_\\mu$-contributions in the following way:\n\\begin{equation}\nT_{\\mu\\nu}^{(B)} + T_{\\mu\\nu}^{(Fa)} = T_{\\mu\\nu}^{(F)} + T_{\\mu\\nu}^{(Z)} +\n T_{\\mu\\nu}^{(W)} + T_{\\mu\\nu}^{(WW)} \\\/.\n\\label{340}\n\\end{equation}\nUsing (\\ref{336}) we have here introduced the following\nenergy-momentum tensors possessing the dimension $[L^{-2}]$:\n\\begin{equation}\nT_{\\mu\\nu}^{(F)} = - {\\tilde e\\over{4\\pi}}\\,{1\\over{\\hbar c}}\\,l_\\vphi^2\\,\n \\left[F_{\\mu\\sigma}F_\\nu{}^\\sigma\n -{1\\over 4}\\,g_{\\mu\\nu}F^{\\rho\\lambda}F_{\\rho\\lambda}\\right]\n\\label{341}\n\\end{equation}\nfor the electromagnetic fields, and\n\\begin{eqnarray}\nT_{\\mu\\nu}^{(Z)} &=& -{1\\over\\tilde e}\\,l_\\vphi^2\\,\\left[Z_{\\mu\\sigma}Z_\\nu{}^\\sigma\n -{1\\over 4}\\,g_{\\mu\\nu}\\,Z^{\\rho\\lambda}Z_{\\rho\\lambda}\\right]\\\/,\n\\label{342} \\\\\nT_{\\mu\\nu}^{(W)} &=& -{1\\over\\tilde e}\\,l_\\vphi^2\\,\\left[F^+_{\\mu\\sigma}F_\\nu^{-\\sigma}\n + F^-_{\\mu\\sigma}F_\\nu^{+\\sigma} - \\frac{1}{2}\\,g_{\\mu\\nu}\\,F^{+\\rho\\lambda}F^-_{\\rho\\lambda}\\right]\\\/,\n\\label{343}\n\\end{eqnarray}\nfor the $Z$- and $W$-fields, respectively. The last term, $T_{\\mu\\nu}^{(WW)}$,\nin (\\ref{340}) is a lengthy expression constructed with \nthe fields $Z_{\\mu\\nu}, F_{\\mu\\nu}$\nand $W^\\dagger_\\mu ,W_\\nu$ containing terms of second and fourth order in the\n$W$-fields which we shall not write down explicitly. All terms in\n(\\ref{340}) are traceless, so that the contributions of the masses\nfor the $Z$- and $W$-fields cannot come from these expressions but\nmust be contained in the other contributions on the r.-h side of (\\ref{310}).\nIn fact, it is the energy-momentum tensor of the $\\phi$-field, (\\ref{311a}),\nwhich contains --- besides the mass $\\sqrt{a}\\,m$ of the $\\phi$-field\nitself --- the effects of the nonzero masses for the $Z$- and\n$W$-fields. In order to see this more clearly we consider the\nsymmetry breaking relation (\\ref{32}) which implies, by taking the\nWeyl-covariant divergence,\n\\begin{equation}\nD^\\mu D_\\mu \\Phi^2 = 0\\\/.\n\\label{344}\n\\end{equation}\nRelating this to the field equation (\\ref{34}) for $\\phi$ and\nits adjoint leads to the following result:\n\\begin{equation}\n\\left(\\buildrel\\approx\\over{D}_\\mu\\!\\phi\\right){}\\!^\\dagger\\left(\\buildrel\\approx\\over{D}{}\\!^\\mu\\phi\\right) =\n{1\\over 6}R\\phi^\\dagger\\phi\n+ 4\\beta\\left(\\phi^\\dagger\\phi\\right)^2 + a\\left[{mc\\over\\hbar}\\right]^2 \\phi^\\dagger\\phi\n - \\tilde\\gamma\\left[(\\bar\\psi_L\\phi)\\psi_R + \\bar\\psi_R(\\phi^\\dagger\\psi_L)\\right]\\\/.\n\\label{345}\n\\end{equation}\nEvaluating (\\ref{345}) at the origin $\\hat\\phi$ of $\\buildrel\\sim\\over{G}\\!\/H$ [compare Appendix A],\ni.e. considering (\\ref{345}) in the electromagnetic gauge, using moreover\n$D_\\mu\\hat\\vphi_0 =0$ for $\\hat\\vphi_0\\neq 0$, i.e. $\\kappa_\\mu =0,~ \\hat\\vphi_0 =const$ according\nto (\\ref{319}), one finds with the help of Eqs. (B3), (B18), (\\ref{311}) \nand (\\ref{315})\n\\begin{eqnarray}\n\\frac{1}{2}\\tilde g^2\\hat\\vphi_0^2\\,\\hat W^\\dagger_\\mu \\hat W^\\mu + {1\\over 4}\\tilde g_0^2\\,\\hat\\vphi_0^2\\,\n \\hat Z_\\mu \\hat Z^\\mu &=& {1\\over 6}\\bar R\\hat\\vphi_0^2 + 4\\beta\\hat\\vphi_0^4 +\n a\\left[{mc\\over\\hbar}\\right]^2\\hat\\vphi_0^2 -\n \\tilde\\gamma\\hat\\vphi_0\\hat{\\bar\\psi}_e\\hat\\psi_e \\nonumber \\\\\n&=& 4\\beta\\hat\\vphi_0^4 + \\left[{mc\\over\\hbar}\\right]^2 \\left(a-\\hat\\vphi_0^2\\right)\\hat\\vphi_0^2\n - \\tilde\\gamma\\,\\hat\\vphi_0\\,\\hat{\\bar\\psi}_e\\hat\\psi_e \\\/.\n\\label{346}\n\\end{eqnarray}\nThis is an interesting relation between the mass terms for the various\nfields involved in the theeory and the nonlinear self-coupling\nterm of the scalar field. \n\nWe first observe that each term in (\\ref{346}) has dimension $[L^{-2}]$\nand that the unit of lengths in which every quantity with a length\ndimension is to be measured is given by the length\n\\begin{equation}\nl_\\vphi = {\\hbar\\over{mc}}\n\\label{346a}\n\\end{equation}\nappearing on the r.-h. side of (\\ref{346}). This length was introduced\nby the Weyl-symmetry breaking Lagrangean ${\\cal L}_B$ defined in (\\ref{31}).\nIndeed, for $a = 1$ the mass of the scalar field is $m_\\phi \\equiv m$\nand the corresponding Compton wave length is $l_\\vphi$. This length will from\nnow on be adopted as the unit of lengths. This choice implies that\nall quantities with a length dimension have to be measured in units\nof $l_\\vphi$ as we already did for the constant $\\buildrel\\approx\\over\\delta$ in (\\ref{336}).\nFor the constants $\\beta$ and $\\tilde\\gamma$ this means that\n\\begin{equation}\n\\beta = \\beta '\\,l_\\vphi^{-2}\\\/,\\qquad \\tilde\\gamma = - \\tilde\\gamma '\\,l_\\vphi^{-1}\\\/,\n\\label{346b}\n\\end{equation}\nwith the primed quantities being numerical constants. The minus sign\nin the second equation is adopted here in order to obtain the correct\nsign for the electron mass in the last term of (\\ref{346}) which\nis thus given by\n\\begin{equation}\n- \\tilde\\gamma\\,\\hat\\vphi_0 = \\tilde\\gamma '\\,\\hat\\vphi_0\\,{mc\\over\\hbar} = {m_ec\\over\\hbar}\\\/.\n\\label{346c}\n\\end{equation}\nThe same argument applies to the $Z$- and $W$-boson fields\nof length dimension $[L^{-1}]$ and the corresponding masses.\nWe thus identify the masses of the charged ($q=\\mp 1e$) and neutral \n($q=0$) boson fields as well as the electron mass $m_e$ by the equations:\n\\begin{equation}\n2\\,m_W^2 = \\frac{1}{2}\\tilde g^2\\,\\hat\\vphi_0^2\\,m^2~;\\qquad m_Z^2 = {1\\over 4}\\tilde g_0^2\\,\\hat\\vphi_0^2\\,m^2~;\n\\qquad m_e = \\tilde\\gamma '\\,\\hat\\vphi_0\\,m~,\n\\label{347}\n\\end{equation}\nimplying the relation\n\\begin{equation}\nm_Z = {m_W\\over{\\cos\\theta_W}}\n\\label{348}\n\\end{equation}\nbetween the $Z$- and the $W$-masses, which is well-known from the\nstandard model. Below we shall sometimes denote by $\\tilde m_W$,\n$\\tilde m_Z$ and $\\tilde m_e$ the dimensionless quantities\n\\begin{equation}\n\\tilde m_W = {m_W\\over m} = \\frac{1}{2}\\tilde g\\,\\hat\\vphi_0~;\\qquad\n\\tilde m_Z = {m_Z\\over m} = \\frac{1}{2}\\tilde g_0\\,\\hat\\vphi_0~;\\qquad \\tilde m_e \n= {m_e\\over m} = \\tilde\\gamma '\\,\\hat\\vphi_0 \\\/.\n\\label{348a}\n\\end{equation}\n\nWe now focus the attention on the energy-momentum tensor\n$\\Theta_{\\mu\\nu}^{(\\phi)}{}'$ for the field $\\phi$ which was defined in\n(\\ref{311a}), reading with $a = 1$\n\\begin{eqnarray}\n\\Theta_{\\mu\\nu}^{(\\phi)}{}' &=& \\frac{1}{2}\\,\\left[\\left(\\buildrel\\approx\\over{D}_\\mu\\!\\phi\\right)^\\dagger\n \\left(\\buildrel\\approx\\over{D}_\\nu\\!\\phi\\right) + \\left(\\buildrel\\approx\\over{D}_\\nu\\!\\phi\\right)^\\dagger\n \\left(\\buildrel\\approx\\over{D}_\\mu\\!\\phi\\right)\\right]\n\\nonumber \\\\\n&\\hfil& \\qquad\\qquad\n- g_{\\mu\\nu}\\,\\Biggl\\{\\frac{1}{2}\\left(\n \\buildrel\\approx\\over{D}\\!{}^\\rho\\phi\\right)^\\dagger\\left(\\buildrel\\approx\\over{D}_\\rho\\!\\phi\\right)\n- \\beta\\,\\left(\\phi^\\dagger\\phi\\right)^2\n - {1\\over 2}\\,\\left[{mc\\over\\hbar}\\right]^2\\,\\phi^\\dagger\\phi\\Biggr\\}\\\/.\n\\label{349}\n\\end{eqnarray}\nUsing now the relation (\\ref{345}) for $a=1$ in order to to eliminate\nthe $(\\phi^\\dagger\\phi)^2$-coupling term proportional to $\\beta$ in (\\ref{349}) \none obtains, considering also (\\ref{310}) again,\n\\begin{eqnarray}\n\\Theta_{\\mu\\nu}^{(\\phi)}{}' &=& \\frac{1}{2}\\,\\left[\\left(\\buildrel\\approx\\over{D}_\\mu\\!\\phi\\right)^\\dagger\n\\left(\\buildrel\\approx\\over{D}_\\nu\\!\\phi\\right) + \\left(\\buildrel\\approx\\over{D}_\\nu\\!\\phi\\right)^\\dagger\n\\left(\\buildrel\\approx\\over{D}_\\mu\\!\\phi\\right)\\right]\n\\nonumber \\\\\n&\\hfil& \\quad\n- {1\\over 4} g_{\\mu\\nu}\\,\\Biggl\\{\n\\left(\\buildrel\\approx\\over{D}\\!{}^\\rho\\phi\\right)^\\dagger\\left(\\buildrel\\approx\\over{D}_\\rho\\phi\\right) -\n\\left[{mc\\over\\hbar}\\right]^2\\,\\left(\\Phi^2 + 1\\right)\\Phi^2\n-\\tilde\\gamma\\,\\left[(\\psi_L\\phi)\\psi_R + \\bar\\psi_R(\\phi^\\dagger\\psi_L)\\right]\\Biggr\\}\\\/.\n\\label{350}\n\\end{eqnarray}\nEvaluating this in the electromagnetic gauge yields with Eqs. (B3),\n(\\ref{346c}), (\\ref{347}), and (\\ref{348a}), and with\n$\\hat Z_\\mu^\\dagger = \\hat Z_\\mu$, the result\n\\begin{eqnarray}\n\\hat\\Theta_{\\mu\\nu}^{(\\phi)}{}' &=& \\tilde m_W^2\\left(\n\\hat W_\\mu^\\dagger \\hat W_\\nu + \\hat W_\\nu^\\dagger \\hat W_\\mu\\right) + \n\\tilde m_Z^2\\,\\hat Z_\\mu \\hat Z_\\nu\n\\nonumber \\\\\n&\\hfil&\\qquad\n- {1\\over 4} g_{\\mu\\nu}\\,\\Biggl\\{\n2 \\tilde m_W^2\\,\\hat W_\\rho^\\dagger \\hat W^\\rho + \\tilde m_Z^2\\,\n\\hat Z_\\rho \\hat Z^\\rho\n- \\left[{mc\\over\\hbar}\\right]^2\\,\\left(\\hat\\vphi_0^2 + 1\\right)\\hat\\vphi_0^2 +\n{m_ec\\over\\hbar}\\,\\hat{\\bar\\psi}_e\\hat\\psi_e\\Biggr\\}\\\/,\n\\label{351}\n\\end{eqnarray}\nshowing that the boson and fermion mass terms appear in the total\nenergy-momentum tensor through the tensor $\\hat\\Theta_{\\mu\\nu}^{(\\phi)}{}'$.\n\nThe r.-h. side of (\\ref{310}), i.e. the source term of the field\nequations for the metric, when evaluated in the electromagnetic gauge\nand for $\\kappa_\\mu =0,~ \\hat\\vphi_0=const$ now reads\n\\begin{equation}\n\\hat T_{\\mu\\nu} = \\Biggl\\{\\hat\\Theta_{\\mu\\nu}^{(\\phi)}{}' + \\hat T_{\\mu\\nu}^{(\\psi_L)}\n+ \\hat T_{\\mu\\nu}^{(\\psi_R)} + g_{\\mu\\nu}\\,{m_ec\\over\\hbar}\\,\n\\hat{\\bar\\psi}_e\\hat\\psi_e + \\hat T_{\\mu\\nu}^{(F)} + \n\\hat T_{\\mu\\nu}^{(Z)} + \\hat T_{\\mu\\nu}^{(W)} +\n\\hat T_{\\mu\\nu}^{(WW)} \\biggr\\}_{\\kappa_\\mu =0,~\\hat\\vphi_0 =const} \\\/,\n\\label{352}\n\\end{equation}\nwhere $\\hat\\Theta_{\\mu\\nu}^{(\\phi)}{}'$ is given by (\\ref{351}), and (B18)\nhas been used for the Yukawa term. After Weyl-symmetry breaking\naccording to (\\ref{31}), (\\ref{32}) yielding (\\ref{319})\nwe thus have to evaluate the r.-h. side of\n(\\ref{352}) for a vanishing Weyl vector field and for $\\hat\\vphi_0 = const$,\ni.e. in a $V_4$, which is indicated by the suffix on the curly\nbrackets. Moreover, we use the length $l_\\vphi$ as a universal unit.\nBy construction $\\hat T_{\\mu\\nu}$ satisfies the usual conservation\nrelations $\\bar\\nabla^\\mu\\hat T_{\\mu\\nu} = 0$ due to\nEqs. (\\ref{317}) -- (\\ref{319}).\n\nThe l.-h side of (\\ref{310}) for $R = \\bar R = const$ according to\nEqs. (\\ref{311}) and (\\ref{319}), and with $\\tilde\\alpha =0$ --- remembering\nthat $\\tilde\\alpha$ was introduced in $\\buildrel\\approx\\over{\\cal L}_{W_4}$, as discussed in I, to yield a\nnontrivial dynamics for the Weyl vector fields which now vanish --- is given\nin the electromagnetic gauge and in the $V_4$ limit, taking moreover $a=1$\n(see above), by\n\\begin{equation}\n{1\\over 6}\\,\\left(\\hat\\vphi_0^2 + 1 \\right)\\left[\\bar R_{\\mu\\nu}\n - \\frac{1}{2}\\,g_{\\mu\\nu}\\,\\bar R \\right] \\\/.\n\\label{353}\n\\end{equation}\nThis together with (\\ref{352}) yields, finally, a set of field equations\nfor the metric of the form\n\\begin{equation}\n\\bar R_{\\mu\\nu} - \\frac{1}{2}\\,g_{\\mu\\nu}\\,\\bar R = {1 \\over{{1\\over 6}[\\hat\\vphi_0^2 + 1]K}}~\n\\hat T_{\\mu\\nu}{}' \\\/.\n\\label{354}\n\\end{equation}\nHere appears the constant $K$ in the denominator on the\nr.-h. side since we want to measure the total energy-momentum\ntensor ultimately in the conventional units of $[Energy\/L^3]$ while\n$\\hat T_{\\mu\\nu}$ in (\\ref{352}) has dimension $[L^{-2}]$. This we indicate\nby a prime on the total source tensor which is given by\n$\\hat T_{\\mu\\nu}{}' = K\\,\\hat T_{\\mu\\nu}$. Eqs. (\\ref{354}) are\nidentical with Einstein's field equations in general relativity\nprovided we are entitled to make the dentification\n\\begin{equation}\n\\kappa_E = {1 \\over{{1\\over 6}[\\hat\\vphi_0^2 + 1]K}}~,\n\\label{355}\n\\end{equation}\nwhere $\\kappa_E$ is Einstein's gravitational constant, $\\kappa_E = 8\\pi N\/c^4 =\n2.076\\cdot 10^{-48} g^{-1} cm^{-1} sec^2$, and $N$ is Newton's\nconstant. Of course, in the framework adopted here also $\\kappa_E$\nis to be expressed in the proper units related to $l_\\vphi$ as the chosen\nintrinsic fundamental length unit replacing the $cgs$-units\nconventionally chosen for $\\kappa_E$ yielding the quoted numerical value\nof this constant. (For a general discussion on the transformation\nof the units for mass, length and time we refer to Nariai and Ueno\n\\cite{7} and Dicke \\cite{8}.) Eq. (\\ref{355}) implies that the\nover-all size of Einstein's gravitational constant \nand its dimension is determined by\nthe constant $K^{-1}$ with $\\hat\\vphi_0$, representing an elementary mass\nratio if the coupling constants $\\tilde g ,~\\tilde g_0$ or $\\tilde\\gamma '$\nin (\\ref{348a}) were known, leading to a correction in the relation\nbetween $\\kappa_E$ and $K$ as expressed by (\\ref{355}).\nThe squared modulus of the scalar field --- being a constant\nafter Weyl-symmetry breaking --- enters the gravitational constant\n$\\kappa_E$ in a manner reminiscent of the Brans-Dicke scalar-tensor\ntheory of gravitation \\cite{3} although, as mentioned, the strength\nof the gravitational coupling is determined in the presented broken\nWeyl theory essentially by $K^{-1}$ having the dimension $[L\/Energy]$.\nFor a more detailed investigation of this point see, however,\nSubsection D below.\nFurthermore, it is easy to show that taking the trace on both sides\nof (\\ref{354}) yields an identity after use of (\\ref{311}) has\nbeen made.\n\nIn the described situation where one considers the appearance of the\nunit of length as originating from a $D(1)$- or Weyl-symmetry breaking\nin a theory containing a universal scalar quantum field --- relating the\nestablished length unit to the mass of this field --- the quantities\n$\\hbar$ and $c$ as well as the fine-structure constant $\\alpha_F=\\tilde e^2\/4\\pi$\nare regarded as universal constants being by definition unrelated\nto the appearance of the length $l_\\vphi$. Summarizing we may say\nthat the scalar field in its $U(1)_{e.m.}$ gauge invariant form\n$\\hat\\phi = \\left({0\\atop{\\hat\\vphi_0(x)}}\\right)$, with $\\hat\\vphi_0(x)$ being a\nconstant, $\\hat\\vphi_0$, after $D(1)$-symmetry breaking, determines\naccording to Eqs. (\\ref{347}) and (\\ref{348a}) not only the $Z$- and\n$W$-boson masses as well as the electron mass in a way described as\nthe ``Higgs phenomenon'' in the standard electroweak model --- which, in\nfact, is just a choice of gauge called here the electromagnetic or nonlinear\ngauge --- but affects also the gravitational coupling constant in\na Brans-Dicke-like manner with $\\hat\\Phi^2 = \\hat\\vphi_0^2$ playing the role \nof the real scalar field.\\\\\n\n\\noindent{\\bf C. The Field Equations for the Scalar Field}\n\nIt is a surprising fact that the nonlinear $\\phi^4$-coupling term\nproportional to the constant $\\beta$ could be eliminated from the\nenergy-momentum tensor $\\Theta_{\\mu\\nu}^{(\\phi)}{}'$ due to Eq. (\\ref{345})\nfollowing from the Weyl-symmetry breaking relation $D_\\mu\\Phi^2 = 0$.\nIn the electromagnetic gauge Eq. (\\ref{345}) took the form of the\nfirst equation in (\\ref{346}) which then led, with $a=1$ and \nEqs. (\\ref{346b}) -- (\\ref{348a}), to $\\hat\\Theta_{\\mu\\nu}^{(\\phi)}{}'$\ngiven in (\\ref{351}).\n\nIn concluding this section let us now finally\nstudy the field equation (\\ref{34}) for $\\phi$ for the\ncase $a=1$, and $\\kappa_\\mu =0$, $\\hat\\vphi_0 = const$ according to (\\ref{319}),\nto see the influence of the $\\phi^4$-coupling on the dynamics of\nthe gauge and fermion fields in the nonlinear, i.e. electromagnetic,\ngauge. To this end one has to \ncompute $\\hat{\\buildrel\\approx\\over{D}}{}^\\mu\\hat{\\buildrel\\approx\\over{D}}_\\mu\\hat\\phi$ using (B1) and (B3). \nIt is then easy to show that (\\ref{34}) is, in the\nelectromagnetic gauge, equivalent to the following $U(1)_{e.m.}$ gauge\ncovariant equations:\n\\begin{equation}\ni{\\tilde g\\over{\\sqrt{2}}}\\,\\hat\\vphi_0\\,\\left[\\bar\\nabla^\\mu\\hat W_\\mu +\ni\\tilde e\\,\\hat{\\bar A}^\\mu\\hat W_\\mu - i\\tilde g_0\\,\\hat Z^\\mu\\hat W_\\mu\\right] +\n2\\tilde\\gamma '\\,{mc\\over\\hbar}\\,\\hat{\\bar e}_R \\hat\\nu_L = 0\\\/,\n\\label{359}\n\\end{equation}\n\\begin{equation}\ni{\\tilde g\\over{\\sqrt{2}}}\\,\\hat\\vphi_0\\,\\left[\\bar\\nabla^\\mu\\hat W^\\dagger_\\mu -\ni\\tilde e\\,\\hat{\\bar A}^\\mu\\hat W^\\dagger_\\mu + i\\tilde g_0\\,\\hat Z^\\mu\\hat W^\\dagger_\\mu\\right] -\n2\\tilde\\gamma '\\,{mc\\over\\hbar}\\,\\hat{\\bar\\nu}_L \\hat e_R = 0\\\/,\n\\label{360}\n\\end{equation}\n\\begin{equation}\n-\\left[ 2\\tilde m_W^2\\,\\hat W^\\dagger{}^\\mu\\hat W_\\mu + \n\\tilde m_Z^2\\,\\hat Z^\\mu\\hat Z_\\mu\\right] + {1\\over 6}\\bar R\\hat\\vphi_0^2 +\n4\\beta\\hat\\vphi_0^4 +\\left[{mc\\over\\hbar}\\right]^2 \\hat\\vphi_0^2 +\n{m_ec\\over\\hbar}\\,\\hat{\\bar\\psi}_e\\hat\\psi_e = 0\\\/,\n\\label{361}\n\\end{equation}\n\\begin{equation}\n\\bar\\nabla^\\mu\\hat Z_\\mu = 2\\tilde\\gamma '\\,{1\\over{\\tilde g_0\\,\\hat\\vphi_0}}\\,{mc\\over\\hbar}\\,\ni\\left(\\hat{\\bar e}_L\\hat e_R - \\hat{\\bar e}_R\\hat e_L\\right)\\\/.\n\\label{362}\n\\end{equation}\n\nHere (\\ref{359}) corresponds to the upper components in (\\ref{34})\nwhen evaluated for $\\hat\\phi =\\left({0\\atop\\hat\\vphi_0}\\right)$,\nEq. (\\ref{360}) is the adjoint equation of (\\ref{359}),\nwhile (\\ref{361}) and (\\ref{362}) are the real and imaginary parts\nof the lower components in (\\ref{34}), respectively. For comparison\nwith (\\ref{346}) Eq. (\\ref{361}) was multiplied by $\\hat\\vphi_0$. \nThe constant $\\beta$ only enters Eq. (\\ref{361}) \nwhich is seen to be identical to the first equation\nof (\\ref{346}) [prior to the use of (\\ref{311}) and (\\ref{348a})]\nwhich was derived above.\nHence the only equation containing the term proportional to\n$\\beta$ is, in fact, the algebraic equation Eq, (\\ref{361}), and this\nequation was used above to eliminate the $\\beta$-contribution\nfrom the source terms in Einstein's equations. The elimination of\nthe $\\beta$-term is a particular consequence of the Weyl-symmetry breaking\nby Eq. (\\ref{32}) in this Weyl-electroweak theory.\nStated physically one may say: The energy represented by\nthe term proportional to $\\beta (\\phi^\\dagger\\phi)^2$ in Eq. (\\ref{349})\nmay be reexpressed by the mass terms appearing in Eq. (\\ref{361}),\nusing also (\\ref{311}), so that the constant\n$\\beta$ disappears from the final equations. The field equation for $\\phi$\nis thus, finally, turned into the set of linear differential equations\n({\\ref{359}), (\\ref{360}) and (\\ref{362}). \\\\\n\n\\noindent{\\bf D. Determination of the Parameters of the Theory}\n\nThe following parameters appearing in the Lagrangean (\\ref{33})\nhave already been fixed so far: $a = 1$ in (\\ref{31}); $\\tilde\\alpha =0$\nin $\\buildrel\\approx\\over{\\cal L}_{W_4}$ [compare (\\ref{228})], and $\\buildrel\\approx\\over\\delta{}\\!'=1\/\\tilde e$ with\n$\\tilde e = \\sqrt{4\\pi\\alpha_F}=0.30282$ in (\\ref{336}). Moreover, $\\beta$\ndisappeared from the dynamics after the Weyl-symmetry breaking as\nwas shown in the last subsection. The remaining parameters to be\ndetermined are the following six quantities: The constants $\\tilde g$\nand $\\tilde g'$ together with the Weinberg angle $\\theta_W$ [compare (\\ref{276})];\nthe constants $\\hat\\vphi_0$ and $K$; the Yukawa coupling constant $\\tilde\\gamma '$\nand, last not least, the universal length unit $l_\\vphi =\\hbar\/mc$ or\nrather the reference mass $m = m_\\phi$. Besides the value for $\\tilde e$\nalready quoted (using $\\alpha_F^{-1}=137.04$) we have the following\nfive experimental data at our disposal: $m_e=0.510999\\, MeV\/c^2,\nm_Z=91.187\\, GeV\/c^2, m_W=80.41\\, \nGeV\/c^2, G_F=1.16639\\cdot 10^{-5}\\,GeV^{-2}$\nand $\\kappa_E=2.076\\cdot 10^{-48}g^{-1}cm^{-1}sec^2$.\n\nUnfortunately it is not possible to decide uniquely what the actual length\nscale $l_\\vphi$ is which is to be adopted as a universal unit in the theory.\nWe shall investigate two conceivable possibilities in somewhat greater detail:\n(a) the mass of the $\\phi$-field is identical with the $Z$-boson mass,\ni.e. $m=m^{(a)}=m_Z$, with\n$l_\\vphi =l_\\vphi^{(a)}=\\hbar\/m_Zc=0.2164\\cdot 10^{-15}\\,cm$, and (b) the\nmass of the $\\phi$-field is identical to the electron mass, i.e. $m=m^{(b)}=m_e$,\nwith $l_\\vphi =l_\\vphi^{(b)}=\\hbar\/m_ec=0.38610\\cdot 10^{-10}\\,cm$. As a\nthird possibility we only mention briefly the case when $l_\\vphi =l_\\vphi^{(c)}=1\\,cm$\ncorresponding to $m=m^{(c)}=0.1973\\cdot 10^{-4}\\,eV\/c^2$. This\npossibility could be of interest in connection with a very small but\nnonzero neutrino mass of order $m^{(c)}$ as the lower edge of the\nfermion mass spectrum. Of course, this last choice, $l_\\vphi = 1\\,cm$,\nis completely ad hoc being included here only as an orientation.\n\nFor case (a) we find the following numerical values: $\\cos\\theta_W = 0.8818,~\n|\\tilde g |=0.65316$ [from $G_F$], $\\tilde g' =0.34341$ and thus $|\\tilde g_0 |=0.73794$;\n$\\hat\\vphi_0^2 = 7.345$ and $|\\tilde\\gamma '|=0.2067\\cdot 10^{-5}$ [from\n$\\hat\\vphi_0 = (m_e\/m_Z)\\tilde\\gamma '^{-1}$]. For $\\hat\\vphi_0 >0$ the constants\n$\\tilde g$, $\\tilde g_0$, and $\\tilde\\gamma '$ must be positive, i.e. the absolute\nsigns in the quoted results are unnecessary. Finally one has \n$K=0.7190\/\\kappa_E$, i.e. $K$ is, for the case (a), essentially the \ninverse Einstein constant.\n\nFor the case (b) the constants $\\tilde g, \\tilde g', \\tilde g_0$ and $\\tilde\\gamma '$\nare the same as for the case (a) given above. However, now we have\n$\\hat\\vphi_0^2 = 2.339\\cdot 10^{11}$ due to $\\hat\\vphi_0 = \\tilde\\gamma '^{-1}$. This\nleads, finally, to $K=2.565\\cdot 10^{-11}\/\\kappa_E$ being a factor of\nthe order of $(m_e\/m_Z)^2$ smaller than in case (a). From this\nit is apparent that the relative\ncontributions of $\\hat\\vphi_0^2$ and $K$ in the relation (\\ref{355}) for\n$\\kappa_E$ depends strongly on the unit of lengths adopted.\n\nIn concluding this subsection we remark that the question of the size\nof the ``Higgs mass'' in the conventional formulation of the standard\nmodel has turned in the present broken Weyl-electroweak theory into\nthe question of the relative contributions of $\\hat\\vphi_0^2$ and $K$ to\nEinstein's gravitational constant and, correspondingly, into the\nquestion of the actual size of the true elementary length scale to be\nadopted in the theory. We, moreover, mention that in our determination of\nthe free parameters of the theory we used the observed electron, $Z_0$-\nand $W^{\\pm}$-boson masses disregarding radiative corrections.\n\nThe strength of the gravitational interaction is usually characterized\nby the Planck length $l_{Planck}=(N\\hbar \/c^3)^\\frac{1}{2} =1.616\\cdot 10^{-33}\\,cm$.\nWith this Einstein's gravitational constant $\\kappa_E$ may be written as\n$\\kappa_E\\cdot\\hbar c=8\\pi l_{Planck}^2=65.64\\cdot 10^{-66}\\,cm^2$.\nHowever, writing finally Einstein's field equations (\\ref{354}) relating\nthe contracted space-time curvature, i.e. the Einstein tensor $G_{\\mu\\nu}$,\nto the distribution of energy and momentum in a form independent of\na particular choice of a length unit yields\n\\begin{equation}\nG_{\\mu\\nu} = \\bar R_{\\mu\\nu} - \\frac{1}{2} g_{\\mu\\nu} \\bar R = {6\\over{[\\hat\\vphi_0^2 + 1]}}\\,T_{\\mu\\nu}\\,\n\\label{364}\n\\end{equation}\nwhere $G_{\\mu\\nu}$ and $T_{\\mu\\nu}$ are both to be measured in the same units of\nan inverse length squared. The gravitational coupling in dimensionless form\nis characterized in Eqs. (\\ref{364}) by the constant $6\/[\\hat\\vphi_0^2 + 1]$. For a\nmassless world, i.e. for $\\hat\\vphi_0 =0$ in Eqs. (\\ref{348a}), this dimensionless\ncoupling constant would at most be $6$; for the case (a) above it would be\n$0.719$ --- i.e. of the order of unity as mentioned --- and for the\ncase (b) it would be $2.565\\cdot 10^{-11}$.\n\n\n\\section{Discussion}\n\nWe investigated in this paper the semi-classical theory of a\nscalar-isospinor field $\\phi$ coupled to the chiral fermion fields\n$\\psi_L$ and $\\psi_R$ in the presence of the gauge fields $\\kappa_\\mu$ (Weyl\nvector fields) for the dilatation group $D(1)$, and the gauge fields\n$A_\\mu, Z_\\mu, W_\\mu$ and $W_\\mu^\\dagger$ for the electroweak gauge\ngroup $\\buildrel\\sim\\over{G} = SU(2)_W\\times U(1)_Y$.\nThe dynamics of this originally massless and scaleless theory\nwas formulated in a Weyl space $W_4$ characterized by a family of metrics\n$g_{\\mu\\nu}$ and associated Weyl vector fields $\\kappa_\\mu$, both determined\nonly up to Weyl transformations (2.3) and (2.4) corresponding\nto conformal rescalings of the metric and the related transformations of\nthe Weyl vector fields, respectively. The gauge structure of the original\nWeyl-electroweak theory (WEW theory) was given by the group\n$G = SO(3,1)\\otimes D(1)\\otimes\\buildrel\\sim\\over{G}$.\n\nIn order to investigate the appearance of nonzero masses and\nestablish a scale of lengths, $l_\\vphi$, in the theory which is associated\nwith the squared modulus, $\\Phi^2 = \\phi^\\dagger\\phi$, of Weyl weight $-1$\nof the scalar field and, furthermore, derive field equations of\nEinstein's type for the metric, we broke the Weyl-symmetry explicitly\nby a term in the Lagrangean involving the scalar curvature $R$ of\nthe $W_4$ and a mass term for the scalar field. The idea here is\nto establish an intrinsic length scale in an originally massless\nand scaleless Weyl-symmetric theory by attributing this nonzero mass\nand corresponding finite length unit to the scalar field $\\phi$. \nThen we studied how nonzero masses for\nthe various other interacting fields appear on the scene within the\nframework of a broken gauge theory containing as a subsymmetry\nthe electroweak gauge symmetry which is known to contain many\nfeatures in accord with observation. After the Weyl-symmetry\nbreaking we finally obtain a $U(1)_{e.m.}$ gauge covariant theory formulated\nin a Riemannian space $V_4$. The reduction of the Weyl geometry\nto a Riemannian geometry for the underlying space-time is\ngoverned by a true symmetry breaking relation, $D_\\mu\\Phi^2 =0$,\nimplying that the $D(1)$ gauge field $\\kappa_\\mu$ is ``pure gauge''\nwith the associated length curvature $f_{\\mu\\nu}$ being zero and\nthe gauge symmetry with group $G$ reducing to a gauge symmetry\nwith the subgroup $G\\,'= SO(3,1)\\otimes\\buildrel\\sim\\over{G}$. This is different\nfrom the so-called spontanous symmetry breaking in the electroweak\nsector of the theory which is better described as a choice of\ngauge by singling out a particular point $\\hat\\phi$ as origin in the\ncoset space $\\buildrel\\sim\\over{G}\\!\/H$, with $\\hat\\phi$ being invariant under\n$H\\equivU(1)_{e.m.}$, where $\\buildrel\\sim\\over{G}\\!\/H$ is isomorphic to the scalar field\n$\\phi$ (see Appendix A).\n\nThe transformation $\\phi\\longrightarrow\\hat\\phi$ is a gauge transformation\nwhich reshuffles the fields by putting the theory in a form\npossessing a residual $U(1)_{e.m.}$ gauge freedom and exhibiting the\nappearance of mass terms for the $\\hat Z_\\mu$-field, the $\\hat W_\\mu$-\nand $\\hat W_\\mu^\\dagger$-fields, and the electron field $\\hat\\psi_e$\nwithout, however, reducing the connection and covariant derivatives\nfrom a Lie\\,$\\buildrel\\sim\\over{G}$-valued form to a form characterized by a\ncorresponding expression associated with a subgroup of $\\buildrel\\sim\\over{G}$.\nThis is contrary to the situation for the $D(1)$ or\nWeyl-symmetry breaking described in this paper which, indeed, is a\ntrue symmetry reduction $G\\longrightarrow G\\,'$ in the sense of the\ntheorem quoted at the end of Sec. II ~leading to the appearance\nof the length scale $l_\\vphi$ in the theory freezing at the same\ntime the squared modulus $\\hat\\Phi^2$ of the scalar field to\nthe constant $\\hat\\vphi_0^2$. Now the question arises: What is the true\nnature of this scalar ``field'' $\\hat\\vphi_0$ which enters the $Z_0$- and\n$W^{\\pm}$-boson masses relating them and the electron mass to\nthe established length scale and, furthermore, enters the gravitational\nconstant in Einstein's field equations for the metric in the\n$V_4$ limit in a manner comparable to a Brans-Dicke field\n$\\hat\\vphi_0^2$. We try to answer this question in the following way: The\n$\\phi$-field, as it appears in the broken Weyl theory, is a\n{\\it vehicle for symmetry breaking}. $\\phi$ is not a matter field of\nthe usual type which would also possess a particle interpretation\nin a fully quantized theory. For this reason it is very unlikely\nthat this field, being in the $V_4$ limit reduced to a constant\nresponsible for the mass generation of the gauge boson and\ncharged fermion fields, would actually show up as a particle\nin high energy processes. It is a field necessary to establish\na scale in a theory. Now the next question arises: What is the\nactual size of this scale? Of course, here we have to rely on\nobservation. Since the measured masses $m_Z$ and $m_W$ are\nof the order of $100\\, GeV\/c^2$ it is suggestive to assume\nthat the mass scale established by the $\\phi$-field after\nWeyl-symmetry breaking is of this same order. The corresponding\nlength $l_\\vphi$ would thus be of the order of\n$l_\\vphi \\approx 0.2\\cdot 10^{-15}\\,cm$ [case (a) in Subsection III D].\nHowever, this identification, although reasonable, is\nnot compelling. Regardless of whether one fixes the mass $m$ at\nthe scale of $100\\, GeV\/c^2$ --- case (a) above --- or identifies\nthis $\\phi$-mass with the much smaller electron mass --- case\n(b) above --- the huge difference of the observed\nmasses for the $Z_0$- and $W^{\\pm}$-bosons, on the one hand, and the\nelectron mass, on the other hand, is mainly due to the Yukawa\ncoupling constant $\\tilde\\gamma '$ [compare Eqs. (\\ref{348a})].\nA point of particular interest in the presented broken Weyl theory,\nhowever, is that the special choice of the unit of lengths also\naffects the dimensionless coupling constant appearing in the field\nequations (\\ref{364}) for gravity.\n\n\n\n\\vspace{1cm}\n\\noindent{\\bf ACKNOWLEDGEMENT}\n\n\\noindent I thank Heinrich Saller for numerous discussions.\n\n\n\\newpage\n\\begin{appendix}\n\\section\n{Coset Representation of $\\phi$}\n\nThe coset representation of $\\phi$ is related to the proper disentanglement\nof the various $U(1)$ phase groups involved. The field $\\phi$ transforms\nunder $G=SO(3,1)\\otimes D(1)\\otimes \\buildrel\\sim\\over{G}$ corresponding\nto a representation with spin zero, Weyl weight $w(\\phi)=-\\frac{1}{2}$, isospin $I=\\frac{1}{2}$,\nand hypercharge $Y=\\frac{1}{2}$. The $D(1)$ factor of $G$ affects the modulus\n$\\Phi = \\sqrt{\\phi^\\dagger\\phi}$, while the electroweak gauge group\n\\begin{equation}\n\\buildrel\\sim\\over{G}\\;= SU(2)_W \\times U(1)_Y\n\\label{A1}\n\\end{equation}\ndetermines the orientation of the isospin degrees of freedom\nand the $U(1)_Y$ phase. We denote by $U(\\bar g)$, with $\\bar g\\in\\buildrel\\sim\\over{G}$,\nthe $2\\times 2$ representation of $\\buildrel\\sim\\over{G}$ operating on $\\phi$. We\nparametrize the elements of $\\buildrel\\sim\\over{G}$ by $\\bar g = \\bar g(b_a,\\beta)$\nwith $b_a;~a=1,2,3$, yielding a parametrization of $SU(2)_W$, and with\n$\\beta$ parametrizing the hypercharge transformations. (The angle $\\beta$ \nshould not be confused with the constant $\\beta$ used in Eq. (\\ref{228})\nand in the main text.) Thus the $U(1)_Y$\ntransformations are given by\n\\begin{equation}\nU\\Bigl(\\bar g(0,\\beta)\\Bigr) = \\left(\\begin{array}{cc} e^{i\\tilde g' Y\\beta}&0 \\\\\n 0&e^{i\\tilde g' Y\\beta} \\\\ \\end{array}\\right)\n\\label{A2}\n\\end{equation}\nwith $Y=\\frac{1}{2}$. Mathematically speaking, the transformations (\\ref{A2})\nare transformations of $U(2)$ which may be decomposed into the\ndirect product\n\\begin{equation}\nU(\\bar g(0,\\beta) = U(1)_+ \\otimes U(1)_-~,\n\\label{A3}\n\\end{equation}\nwhere the groups $U(1)_{\\pm}$ are generated by $\\frac{1}{2}(1\\pm\\tau_3)$,\nrespectively, i.e. by the electromagnetic charge $\\hat q=\\frac{1}{2}(1+\\tau_3)$\n[compare Eq. (\\ref{265})] and by\n\\begin{equation}\n\\hat q_o = \\frac{1}{2} (1 - \\tau_3)\\\/.\n\\label{A4}\n\\end{equation}\nThe decomposition of the original weak hypercharge and isospin transformations\ninto $\\hat q$ and $\\hat q_0$ contributions (of which, as shown below, only the\n$\\hat q$ contributions survive) corresponds to physically measurable\nsituations yielding the coupling to the $A_\\rho$-fields (electromagnetic\neffects) and the coupling to the $Z_\\rho$-fields (weak neutral effects).\n\nWe now like to introduce a representation of $\\phi$ which is characterized\nin terms of the cosets $\\buildrel\\sim\\over{G}\\!\/H$, where $H$ is the {\\it electromagnetic}\nsubgroup of $\\buildrel\\sim\\over{G}$, i.e. $H=U(1)_{e.m.} =U(1)_+\\subset\\buildrel\\sim\\over{G}$. We thus write\n\\begin{equation}\n\\phi = U(\\bar g_\\phi )~\\hat\\phi \\qquad {\\rm with} \\qquad \\hat q~\\hat\\phi =0\\\/.\n\\label{A5}\n\\end{equation}\nHere $U(\\bar g_\\phi)$ is an element of $\\buildrel\\sim\\over{G}\\!\/H$ parametrized by $\\phi$, and\n\\begin{equation}\n\\hat\\phi = \\left({0\\atop \\hat\\varphi_0}\\right)\\\/,\n\\label{A6}\n\\end{equation}\nwith $\\hat\\varphi_0$ being a real field, denotes the origin of the coset space.\n$\\hat\\phi$ is invariant under the electromagnetic gauge group $U(1)_{e.m.} =U(1)_+$\n[the stability group $H$] with transformations $h\\in H$ given by\n\\begin{equation}\nU(h(\\alpha)) = e^{-i{e\\over\\hbar c}\\hat q\\alpha} = \\left(\\begin{array}{cc}\n e^{-i{e\\over\\hbar c}\\alpha} & 0\\\\ 0 & 1 \\\\ \\end{array}\\right)\\\/,\n\\label{A7}\n\\end{equation}\nwhere the minus sign in the exponential is adopted for conventional reasons.\nDue to the splitting (\\ref{A3}) of the hypercharge transformations\nand the invariance of $\\hat\\phi$ by the contributions $U(1)_+$ generated\nby the charge $\\hat q$, the transformation $U(\\bar g_\\phi)$ --- which could be\ncalled a ``boost'' generating $\\phi$ from the fixed state $\\hat\\phi$ --- is\nseen to be given by the following element of $SU(2)_W\\otimes U(1)_-$ :\n\\begin{equation}\nU(\\bar g_\\phi ) = U\\Bigl(\\bar g(b_a(\\phi)\\Bigr)~e^{{i\\over 2}\\tilde g'\\hat q_o\\beta(\\phi)}\\\/,\n\\label{A8}\n\\end{equation}\nwith $\\hat q_o$ as given by (\\ref{A4}). Here the first factor on the r.-h. side\nis an element of $SU(2)_W$ parametrized by $b_a(\\phi)$, and the second\nfactor is the transformation\n\\begin{equation}\n\\left(\\begin{array}{cc} 1 & 0 \\\\ 0 & e^{{i\\over 2}\\tilde g'\\beta(\\phi)} \\\\\n\\end{array}\\right) \\in U(1)_-\n\\label{A9}\n\\end{equation}\nwith hypercharge phase angle $\\beta(\\phi)$. It is easy to show that one\ncan express the r.-h. side of (\\ref{A8}) in terms of the components\nof $\\phi$ in the following manner [compare (\\ref{222})]:\n\\begin{equation}\nU(\\bar g_\\phi ) = {1\\over {\\hat\\varphi_0}}\\left(\\begin{array}{cc}\n ~\\varphi^*_0~e^{{i\\over 2}\\tilde g'\\beta(\\phi)} & ~\\varphi_+ \\\\\n -\\varphi^*_+~e^{{i\\over 2}\\tilde g'\\beta(\\phi)} & ~\\varphi_0 \\\\ \\end{array}\n\\right)\\\/,\n\\label{A10}\n\\end{equation}\nwith $\\hat\\varphi_0 =\\sqrt{\\phi^\\dagger\\phi} = \\sqrt{|\\varphi_+|^2 + |\\varphi_0|^2}$\nexpressing the invariance of the modulus $\\Phi$ of $\\phi$ under\n``boosts'' parametrized in terms of $\\buildrel\\sim\\over{G}\\!\/H$.\nIdentifying $\\phi$ with the coset $U(\\phi)\\cdot H\\equiv U(\\bar g_\\phi)\\cdot H$\nwe may now take the simpler matrix $U(\\phi)$ defined by\n\\begin{equation}\nU(\\phi) = {1\\over {\\hat\\varphi_0}}\\left(\\begin{array}{cc}\n ~\\varphi^*_0 & ~\\varphi_+ \\\\\n -\\varphi^*_+ & ~\\varphi_0 \\\\ \\end{array}\n\\right)\\\/,\\quad {\\rm with} \\quad \\det U(\\phi)=1\n\\label{A11}\n\\end{equation}\nas a coset representative instead of $U(\\bar g_\\phi)$ and write Eq. (\\ref{A5}) as\n\\begin{equation}\n\\phi = U(\\phi)~\\hat\\phi\\\/.\n\\label{A12}\n\\end{equation}\n\nThe $\\buildrel\\sim\\over{G}$-transformation of $\\phi$ or, more exactly, the gauge\ntransformation $U(\\bar g(x))$ representing a change of section on the\nbundle $E$ [see (\\ref{224})] given by (the argument $x$ is suppressed)\n\\begin{equation}\n\\phi\\;' = U(\\bar g)~\\phi \\\/,\n\\label{A13}\n\\end{equation}\ntogether with the coset representation (\\ref{A12}) of $\\phi$ as well as of\n$\\phi '$ associated with the stability group $H=U(1)_{e.m.}$ now yields the\nfollowing decomposition of an arbitrary transformation\n$U(\\bar g)$ into boosts parametrized by $\\phi =\\phi (x)$ and $\\phi '=\\phi '(x)$,\nrespectively, and a stability group transformation\ncharacterized by an angle $\\alpha$ depending on $\\phi (x)$ and\non $\\bar g=\\bar g(x)$ which is written for short as $\\alpha (\\phi ',\\phi)$ :\n\\begin{equation}\nU(\\bar g) = U(\\phi ')~e^{-i{e\\over \\hbar c}\\hat q \\alpha (\\phi ',\\phi)}~\n U^{-1}(\\phi)\\\/.\n\\label{A14}\n\\end{equation}\nHere the subgroup transformation\n\\begin{equation}\nU(h(\\phi ',\\phi)) = e^{-i{e\\over \\hbar c}\\hat q \\alpha (\\phi ',\\phi)} =\n \\left(\\begin{array}{cc}\n e^{-i{e\\over{\\hbar c}}\\alpha(\\phi ',\\phi)} & 0 \\\\ 0 & 1 \\\\\n \\end{array}\\right)\n\\label{A15}\n\\end{equation}\ncould be called the ``Wigner rotation'' for the electroweak theory\nor the ``little group transformation'' at the origin $\\hat\\phi$ in the coset space\n$\\buildrel\\sim\\over{G}\\!\\!\/H$ which is associated with the transformation $U(\\bar g)=\nU(\\bar g(b_a,\\beta))$ sending $\\phi$ into $\\phi '$. The transformation\n(\\ref{A15}) with angle $\\alpha(\\phi ',\\phi)$ is in similar contexts \nusually called the {\\it nonlinear realization} of a gauge transformation\nof the group $\\buildrel\\sim\\over{G}$ on the stability subgroup $H$ of $G$ \\cite{5}.\nIn the standard model, however, this terminology is not\nused and one speaks instead of a symmetry breaking by the vacuum\nexpectation value of the scalar field $\\phi$ having the form (\\ref{A6}).\nIn the present case, with $\\buildrel\\sim\\over{G}$ possessing\nthe product structure (\\ref{A1}), one finds for the angle $\\alpha (\\phi ',\\phi)$\nby direct computation for the transformations $\\bar g=\\bar g(b_a,\\beta)$\nthe results\n\\begin{eqnarray}\n&&\\qquad {\\rm for} \\qquad \\bar g(0,0) ~~:~~~~~ \\alpha(\\phi ,\\phi) = 0\\\/, \\nonumber\\\\\n&&\\qquad {\\rm for} \\qquad \\bar g(0,\\beta)~~:~{e\\over {\\hbar c}}\\alpha(\\phi ',\\phi) =\n -\\tilde g'\\beta\\\/,\\nonumber \\\\\n&&\\qquad {\\rm for} \\qquad \\bar g(b_a,0)~~:~{e\\over {\\hbar c}}\\alpha(\\phi ',\\phi) = 0\\\/.\n\\label{A16}\n\\end{eqnarray}\nThe last line in (\\ref{A16}) implies that there are no residual $SU(2)_W$\ngauge transformations left on the stability\nsubgroup $H=U(1)_{e.m.}$. After transforming to the origin $\\hat\\phi$\nin $\\buildrel\\sim\\over{G}\\!\/H$ only one gauge degree of freedom remains which\nis of the form (\\ref{A15}) with $\\alpha(\\phi ',\\phi)$ given by (\\ref{A16})\ntogether with the corresponding transformation rule for the\nelectromagnetic potentials $A_\\mu$ [see Appendix B].\nThe result for the hypercharge transformation\n$\\bar g(0,\\beta)$ quoted in (\\ref{A16}) follows also directly from the\ndeterminant of the transformation (\\ref{A14}).\n\n\\section\n{The Electromagnetic Gauge}\n\nWe call the gauge obtained by realizing the transformations\n$U(\\bar g(b_a,\\beta)$ of $\\buildrel\\sim\\over{G}$ in terms of transformations $U(h(\\phi ',\\phi))$\nof the electromagnetic subgroup $H=U(1)_+$ of $\\buildrel\\sim\\over{G}$ the\nelectromagnetic or nonlinear gauge [compare Eqs. (\\ref{A14}) and (\\ref{A15})].\nTo characterize this gauge, which we shall denote by a hat,\nthe scalar field $\\phi$ of the theory is used: As described in Appendix A,\n$\\phi$ takes the form $\\hat\\phi =\\left({0\\atop\\hat\\vphi_0}\\right)$ in this gauge, and\nthe residual gauge transformations are the transformations of the\nstability subgroup $U(1)_+=U(1)_{e.m.}$ leaving $\\hat\\phi$ invariant.\n\nIt is of particular interest to determine the form the gauge potentials\n$A^a_\\mu$ and $B_\\mu$ take in the electromagnetic gauge, i.e. determine\n$\\hat A^a_\\mu$ and $\\hat B_\\mu$ and their residual gauge freedom given\nby the transformations $h(\\phi ',\\phi)$. Let us to this end rewrite the\n$G$-covariant derivative (\\ref{227}) of $\\phi$ in terms of the spherical\ncomponents $A^-_\\mu =W_\\mu ,~A^+_\\mu =W^\\dagger_\\mu$ and $A^3_\\mu$\n[compare (\\ref{274})] and express $A^3_\\mu$ and $B_\\mu$ in terms of\n$A_\\mu$ and $Z_\\mu$ with the help of (\\ref{270}). One finds with\n$\\tau_\\pm = {1\\over {2\\sqrt 2}}(\\tau_1\\pm i\\tau_2)$:\n\\begin{equation}\n\\buildrel\\approx\\over{D}_\\mu \\phi = \\left[D_\\mu + i\\tilde g \\left(W_\\mu\\tau_+ + W^\\dagger_\\mu\\tau_-\\right)\n +i\\,\\tilde e\\,{e\\over{\\hbar c}}\\hat q A_\\mu + \n i\\tilde g_0\\left(\\frac{1}{2}\\tau_3 - \\sin^2\\theta_W\\cdot\n \\hat q\\right)Z_\\mu\\right] \\phi \\\/.\n\\label{B1}\n\\end{equation}\n\nPerforming now a gauge transformation with $U^{-1}(\\phi)$, mapping\n$\\phi$ into $\\hat\\phi$, the covariant derivative (\\ref{B1}) of $\\phi$ is mapped into\n\\begin{equation}\n\\hat{\\buildrel\\approx\\over{D}}_\\mu \\hat\\phi = \\left[D_\\mu + i\\tilde g\\hat W_\\mu \\tau_+ - {i\\over 2}\n \\tilde g_0\\hat Z_\\mu\\right] \\hat\\phi \\\/,\n\\label{B2}\n\\end{equation}\nwhere we have used the fact that $\\hat q\\hat\\phi =0$, $\\tau_-\\hat\\phi =0$ and\n$\\frac{1}{2}\\tau_3\\hat\\phi = -\\frac{1}{2}\\hat\\phi$. We rewrite this for later use in terms of\nisospinor components, i.e.\n\\begin{equation}\n\\hat{\\buildrel\\approx\\over{D}}_\\mu\\hat\\phi = \\left({0\\atop{D_\\mu\\hat\\vphi_0}}\\right) + i \\left(\n{{{1\\over{\\sqrt 2}}\\,\\tilde g\\hat\\vphi_0\\,\\hat W_\\mu}\\atop{-{1\\over 2}\\,\\tilde g_0\\hat\\vphi_0\\,\\hat Z_\\mu}}\n\\right)\\\/,\n\\label{B3}\n\\end{equation}\nwhere $D_\\mu\\hat\\vphi_0 =\\partial_\\mu\\hat\\vphi_0 +\\frac{1}{2}\\kappa_\\mu\\hat\\vphi_0$ is the Weyl-covariant\nderivative of the real field $\\hat\\vphi_0$. From the residual transformation\nwith $U(h(\\phi ',\\phi))$ given by (\\ref{A15}) one now concludes that\n\\begin{equation}\n\\left(\\hat{\\buildrel\\approx\\over{D}}_\\mu\\hat\\phi\\right)' = \\left(\\begin{array}{cc}\n e^{-i{e\\over{\\hbar c}}\\alpha(\\phi ',\\phi)} & 0 \\\\ 0 & 1 \\\\\n \\end{array}\\right)~\\hat{\\buildrel\\approx\\over{D}}_\\mu\\hat\\phi\n\\label{B4}\n\\end{equation}\nis the nonlinear subgroup transformation rule for the $G$-covariant\nderivative of $\\hat\\phi$. One sees at once from the split form\n(\\ref{B3}) that (\\ref{B4}) implies for the residual (electromagnetic)\n$U(1)_+$ gauge transformations of the potentials $\\hat Z_\\mu$,\n$\\hat W_\\mu$ and $\\hat W_\\mu^\\dagger$ the behaviour\n\\begin{eqnarray}\n\\hat Z_\\mu ' &=& \\hat Z_\\mu \\\/, \\label{B5} \\\\\n\\hat W_\\mu ' &=& e^{-i{e\\over{\\hbar c}}\\alpha(\\phi ',\\phi)}~\\hat W_\\mu\\\/, \\label{B6} \\\\\n\\hat W_\\mu^\\dagger{}' &=& e^{+i{e\\over{\\hbar c}}\\alpha(\\phi ',\\phi)}~\\hat W_\\mu^\\dagger \\\/.\n \\label{B7}\n\\end{eqnarray}\nFor the electromagnetic potentials $\\hat A_\\mu$ applies the usual \nrule given by [see Eq. (\\ref{B13a}) below]\n\\begin{equation}\n\\hat A_\\mu ' = \\hat A_\\mu + \\partial_\\mu\\alpha(\\phi ',\\phi)\\\/.\n\\label{B8}\n\\end{equation}\nIt is clear from the definitions that $\\hat A_\\mu$ and $\\hat Z_\\mu$\nare real vector fields.\n\nIn concluding this appendix we observe with respect to the theorem\nquoted at the end of Sect. II that the connection in the present case\ndoes {\\it not} reduce from a $Lie \\buildrel\\sim\\over{G}$-valued to a\n$Lie H$-valued form. In fact, we have for the $Lie \\buildrel\\sim\\over{G}$-valued\ngauge potentials with Eqs. (\\ref{270}), (\\ref{274}) and (\\ref{276}):\n\\begin{eqnarray}\n\\buildrel\\sim\\over{\\Gamma}_\\mu &=& \\tilde g \\frac{1}{2}\\tau_a A^a_\\mu + \\tilde g' \\frac{1}{2} {\\bf 1} B_\\mu \\nonumber \\\\\n &=& \\left(\\begin{array}{cc}\n \\tilde e\\,\\bar A_\\mu + \\tilde g_0\\left[\\frac{1}{2} - \\sin^2\\theta_W\\right]Z_\\mu &\n \\tilde g{1\\over{\\sqrt 2}}W_\\mu \\\\ \\tilde g{1\\over{\\sqrt 2}}W^\\dagger_\\mu &\n -\\frac{1}{2}\\tilde g_0 Z_\\mu \\\\ \\end{array}\\right)\n\\label{B9}\n\\end{eqnarray}\nwith $\\bar A_\\mu = {e\\over{\\hbar c}} A_\\mu$.\nIn this notation the covariant derivative is written\n$\\buildrel\\approx\\over{D}_\\mu\\!\\phi = D_\\mu\\phi +i\\buildrel\\sim\\over{\\Gamma}_\\mu\\!\\phi$. The transformation to the\nelectromagnetic gauge with $U^{-1}(\\phi)$ yields\n\\begin{equation}\n\\hat{\\buildrel\\sim\\over{\\Gamma}}_\\mu = U^{-1}(\\phi)\\buildrel\\sim\\over{\\Gamma}_\\mu U(\\phi) - iU^{-1}(\\phi)~\\partial_\\mu U(\\phi)\n\\label{B10}\n\\end{equation}\nwhich is still $Lie \\buildrel\\sim\\over{G}$-valued and hence, according to the theorem of\nSect. II, the $\\buildrel\\sim\\over{G}$ gauge symmetry does not reduce but is nonlinearly\nrealized.\n\nFrom (\\ref{B10}) we compute for the fields $\\hat W_\\mu$ and $\\hat Z_\\mu$\nappearing in (\\ref{B2}) the expressions:\n\\begin{eqnarray}\n{\\tilde g\\over{\\sqrt 2}}\\hat W_\\mu &= \\Biggl[\\left(\\tilde e\\,\\bar A_\\mu +\n \\tilde g_0 [\\frac{1}{2} - \\sin^2\\theta_W ]Z_\\mu\\right)&\\vphi_0\\vphi_+ + {\\tilde g\\over{\\sqrt 2}}W_\\mu (\\vphi_0)^2\n -{\\tilde g\\over{\\sqrt 2}}W^\\dagger_\\mu (\\vphi_+)^2 \\nonumber \\\\\n &\\qquad\\qquad + \\frac{1}{2}\\tilde g_0 Z_\\mu\\vphi_0\\vphi_+\\Biggr] (\\hat\\vphi_0)^{-2}&\n -i\\left[\\vphi_0\\partial_\\mu\\!\\left({\\vphi_+\\over\\hat\\vphi_0}\\right)\n -\\vphi_+\\partial_\\mu\\!\\left(\\vphi_0\\over\\hat\\vphi_0\\right)\\right](\\hat\\vphi_0)^{-1} \\\/,\n\\label{B11} \\\\\n-\\frac{1}{2}\\tilde g_0\\hat Z_\\mu &= \\Biggl[\\left(\\tilde e\\,\\bar A_\\mu +\n \\tilde g_0 [\\frac{1}{2} -\\sin^2\\theta_W ]Z_\\mu\\right)&|\\vphi_+|^2\n + {\\tilde g\\over{\\sqrt 2}}W_\\mu \\vphi^*_+\\vphi_0\n +{\\tilde g\\over{\\sqrt 2}}W^\\dagger_\\mu\\vphi^*_0\\vphi_+ \\nonumber \\\\\n &\\qquad\\qquad - \\frac{1}{2}\\tilde g_0 Z_\\mu |\\vphi_0|^2\\Biggr] (\\hat\\vphi_0)^{-2}&\n -i\\left[\\vphi^*_+\\partial_\\mu\\!\\left({\\vphi_+\\over\\hat\\vphi_0}\\right)\n +\\vphi^*_0\\partial_\\mu\\!\\left({\\vphi_0\\over\\hat\\vphi_0}\\right)\\right](\\hat\\vphi_0)^{-1}\\\/.\n\\label{B12}\n\\end{eqnarray}\nFor $\\hat{\\bar A}_\\mu ={e\\over{\\hbar c}}\\hat A_\\mu $, by inserting also \n$\\hat Z_\\mu$ from (\\ref{B12}), one finds the relation:\n\\begin{eqnarray}\n\\tilde e\\,\\hat{\\bar A}_\\mu +& &\\tilde g_0[\\frac{1}{2} - \\sin^2\\theta_W]\\hat Z_\\mu =\n \\Biggl[\\left(\\tilde e\\,\\bar A_\\mu + \\tilde g_0 [\\frac{1}{2} - \\sin^2\\theta_W ]Z_\\mu\\right)|\\vphi_0|^2\n -{\\tilde g\\over{\\sqrt 2}}W_\\mu\\vphi_0\\vphi^*_+ \\nonumber \\\\\n& & -{\\tilde g\\over{\\sqrt 2}}W^\\dagger_\\mu\\vphi^*_0\\vphi_+\n - \\frac{1}{2}\\tilde g_0 Z_\\mu |\\vphi_+|^2\\Biggr] (\\hat\\vphi_0)^{-2}\n -i\\left[\\vphi_0\\partial_\\mu\\!\\left(\\vphi^*_0\\over\\hat\\vphi_0\\right)\n +\\vphi_+\\partial_\\mu\\!\\left(\\vphi^*_+\\over\\hat\\vphi_0\\right)\\right](\\hat\\vphi_0)^{-1}\\\/.\n\\label{B13}\n\\end{eqnarray}\n\nAfter the transformation (\\ref{B10}) has been carried out the residual\ngauge freedom is given by the transformations $U(h(\\phi ',\\phi))$ of the\nelectromagnetic subgroup $U(1)_+ = U(1)_{e.m.}$, i.e. by\n\\begin{equation}\n\\hat{\\buildrel\\sim\\over{\\Gamma}}_\\mu{}'= U(h(\\phi ',\\phi))\\,\\hat{\\buildrel\\sim\\over{\\Gamma}}_\\mu \\,U^{-1}(h(\\phi ',\\phi)) -\n i\\,U(h(\\phi ',\\phi))\\,\\partial_\\mu U^{-1}(h(\\phi ',\\phi))\\\/,\n\\label{B13a}\n\\end{equation}\nwith $U(h(\\phi ',\\phi))$ as defined in (\\ref{A15}) with phase angle (\\ref{A16}). For\nthe potentials $\\hat A_\\mu ', \\hat Z_\\mu ', \\hat W_\\mu '$ and\n$\\hat W_\\mu '^\\dagger$ this yields at once the relations (\\ref{B5}) -- (\\ref{B8}).\n\nWe finally quote the form of the fermion fields after transformation to\nthe origin in $\\buildrel\\sim\\over{G}\\!\/H$:\n\\begin{equation}\n\\hat\\psi_L = U^\\dagger(\\phi)\\,\\psi_L \\qquad {\\mbox {\\rm and}} \\qquad \\hat\\psi_R = \\psi_R \\\/,\n\\label{B14}\n\\end{equation}\nimplying that\n\\begin{equation}\n\\hat\\nu_L = (\\hat\\vphi_0)^{-1}[\\vphi_0\\,\\nu_L-\\vphi_+\\,e_L]\\\/,\\quad\n \\hat e_L = (\\hat\\vphi_0)^{-1}[\\vphi^*_+\\,\\nu_L+\\vphi^*_0\\,e_L]\\\/,\\quad \\hat e_R = e_R\\\/.\n\\label{B15}\n\\end{equation}\nThe residual $U(1)_{e.m.}$ gauge transformations of these fields are\n\\begin{equation}\n\\hat\\nu_L {}' = e^{-i{e\\over{\\hbar c}}\\alpha(\\phi ',\\phi)}~\\hat\\nu_L\\\/,\\quad\n \\hat e_L {}' = \\hat e_L\\\/,\\quad \\hat e_R {}' = \\hat e_R \\\/,\n\\label{B16}\n\\end{equation}\nand correspondingly for the adjoint fields $\\hat{\\bar\\nu}_L, \\hat{\\bar e}_L$\nand $\\hat{\\bar e}_R$. The Yukawa coupling (\\ref{226}) of $\\phi$ and the\nfermion fields may in this notation, together with $U^\\dagger(\\phi)\\,\\phi = \\hat\\phi$,\nbe written as\n\\begin{equation}\n\\tilde\\gamma\\left\\{(\\bar\\psi_L\\phi)\\psi_R + \\bar\\psi_R(\\phi^\\dagger\\psi_L)\\right\\}\\equiv\n\\tilde\\gamma\\left\\{(\\hat{\\bar\\psi_L}\\hat\\phi)\\hat\\psi_R + \\hat{\\bar\\psi_R}(\\hat\\phi^\\dagger\\hat\\psi_L)\\right\\}\n= \\tilde\\gamma\\,\\hat\\vphi_0 ~\\left\\{\\hat{\\bar e}_L \\hat e_R + \\hat{\\bar e}_R \\hat e_L\\right\\}\\\/.\n\\label{B17}\n\\end{equation}\nThe r.-h. side of (\\ref{B17}) may be written as \n$\\tilde\\gamma\\,\\hat\\vphi_0\\,\\hat{\\bar\\psi}_e\\hat\\psi_e$, with $\\hat\\psi_e$ denoting\nthe electron field transformed to the origin in $\\buildrel\\sim\\over{G}\\!\/H$, showing\nthat the Yukawa coupling represents, in effect, an electron mass term.\n\n\\end{appendix}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\chapter*{Abstract}\n\n\\begin{spacing}{1.5}\n\nAccretion disks are ubiquitous in the universe. Although difficult to observe directly,\ntheir presence is often inferred from the unique signature they imprint on the spectra \nof the systems in which they are observed. In addition, many properties of accretion-disk\nsystems that would be otherwise mysterious are easily accounted for by the presence of\nmatter accreting (accumulating) onto a central object. Since the angular momentum of the\ninfalling material is conserved, a disk naturally forms as a repository of angular momentum.\nDissipation removes energy and angular momentum from the system and allows the disk to accrete.\nIt is the energy lost in this process and ultimately converted to radiation that we observe.\n\nUnderstanding the mechanism that drives accretion has been the primary challenge in accretion\ndisk theory. Turbulence provides a natural means of dissipation and the removal of angular\nmomentum, but firmly establishing its presence in disks proved for many years to be difficult.\nThe realization in the 1990s that a weak magnetic field will destabilize a\ndisk and result in a vigorous turbulent transport of angular momentum has revolutionized\nthe field. Much of accretion disk research now focuses on understanding the implications\nof this mechanism for astrophysical observations. At the same time, the success of this\nmechanism depends upon a sufficient ionization level in the disk for the flow to be well-coupled\nto the magnetic field. Many disks, such as disks around young stars and disks in binary systems\nthat are in quiescence, are too cold to be sufficiently ionized, and so efforts to establish the\npresence of turbulence in these disks continues.\n\nThis dissertation focuses on several possible mechanisms for the turbulent transport of\nangular momentum in weakly-ionized accretion disks: gravitational instability, radial convection\nand vortices driving compressive motions. It appears that none of these mechanisms are very\nrobust in driving accretion. A discussion is given, based on these results, as to the most\npromising directions to take in the search for a turbulent transport mechanism that does not\nrequire magnetic fields. Also discussed are the implications of assuming that no turbulent\ntransport mechanism exists for weakly-ionized disks.\n\n\\end{spacing}\n\n\\chapter*{Acknowledgments}\n\n\\begin{spacing}{1.5}\n\nThere are many people without whose encouragement and assistance I would not have\nbegun, let alone completed, this dissertation. I am grateful to my parents, Larry\nand Joyce Johnson, for raising me in an environment of loving affection and\ndiscipline and for providing me with every opportunity to pursue a love for learning.\nI am grateful for the encouragement of Josh Coe and Jim Wolfe in making what seemed\nat the time like an uncertain switch from engineering to physics, and for the\nDepartment of Physics at the University of Illinois for giving me the chance to pursue\na doctorate even though I barely made the minimum required score on the Physics GRE.\nI am also grateful to my teachers, both at the University of Illinois and at the other\ninstitutions I have attended throughout my life, for their investment in my education.\n\nThe bulk of the research in this dissertation has been published in collaboration\nwith my adviser, Charles Gammie. I have thoroughly enjoyed working with him and\nhave greatly appreciated his unselfishness and accessibility. This dissertation\nand my own progress in astrophysics have benefited immensely from his insights and guidance.\nI am also grateful to Stu Shapiro, Alfred Hubler and Doug Beck for their willingness to serve on my defense committee and for their comments on improving the dissertation. Additional improvements have come through critical reviews of an earlier version of the\nmanuscript by Po Kin Leung and Ruben Krasnopolsky.\n\nThe completion of this dissertation would not have proceeded as smoothly as it\nhas were it not for the constant and loyal support of my wife, Amy Banner Dau Johnson.\nShe has helped me in countless ways, and has admirably accepted the sacrifices\nassociated with living on a graduate-student salary. The blessing of a family,\nincluding my three children Luke, Emily and Oliver (who could care less about the\nsalary), is one of the primary things that makes my work worthwhile.\n\nThe funding for this work has come from the National Science Foundation\n(grants AST 00-03091 and PHY 02-05155), the National Aeronautics and Space Administration\n(grant NAG 5-9180) and a Drickamer Research Fellowship from the Department of Physics at\nthe University of Illinois.\n\nAbove all, I wish to acknowledge the Lord Jesus Christ as the Giver of all these\ngood things, and as the Creator and Sustainer of the heavens that declare His glory,\nthe fascinating study of which He has enabled me to pursue.\n\n\\end{spacing}\n\n\\chapter{The Boussinesq Approximation}\\label{appendix}\n\nWe demonstrate here that the Boussinesq approximation to the linear\nperturbation equations is formally equivalent to a short-wavelength,\nlow-frequency limit of the full set of linear equations. We perform the\ndemonstration for the stratified shearing-sheet model since the standard\nshearing sheet is recovered in the limit of zero stratification.\n\nCombining equations (\\ref{LIN1a}) through (\\ref{LIN5a}) into a single\nequation for $\\delta v_x$ yields the following differential equation,\nfourth-order in time:\n\\begin{equation}\\label{DV4DT}\nF_4 \\delta v_x^{(4)} + F_3 \\delta v_x^{(3)} + F_2\\delta v_x^{(2)} + F_1\n\\delta v_x^{(1)} + F_0 \\delta v_x = 0,\n\\end{equation}\nwhere\n\\begin{equation}\\label{COEF4}\nF_4 = \\tilde{k}_x^2 \\left[(k_y^2 + k_z^2) \\left(1-\\frac{i}{\\tilde{k}_x L_P}\n\\right)^2 + k_y^2\\frac{2(\\qe+1)(\\qe+2)}{\\tilde{k}_x^2 H^2}\\right],\n\\end{equation}\n\\begin{equation}\nF_3 = -2 \\qe \\Omega \\tilde{k}_x k_y (k_y^2 + k_z^2) \\left(1+\\frac{i}\n{\\tilde{k}_x L_P}\\right),\n\\end{equation}\n\\begin{eqnarray}\nF_2 = c_s^2 \\tilde{k}_x^2 \\left[\\tilde{k}_x^2 \\left(1 + \\frac{1}{\\tilde{k}\n_x^2 L_P^2} + \\frac{N_x^2 + \\tilde{\\kappa}^2}{\\tilde{k}_x^2 H^2}\\right) \n\\left\\{ (k_y^2 + k_z^2)\\left(1-\\frac{i}{\\tilde{k}_x L_P}\\right)^2 + k_y^2\n\\frac{2(\\qe + 1)(\\qe + 2)}{\\tilde{k}_x^2 H^2} \\right\\} \\, + \\right. \\nonumber\n\\\\ \\left. (k_y^2 + k_z^2) \\left\\{(k_y^2 + k_z^2)\\left(1-\\frac{i}{\\tilde{k}_x L_P}\n\\right)^2 + k_y^2 \\frac{2(\\qe + 1)(3\\qe + 2)}{\\tilde{k}_x^2 H^2}\\right\\} \\right],\n\\end{eqnarray}\n\\begin{eqnarray}\nF_1 = 4\\qe \\Omega c_s^2 k_y \\tilde{k}_x^3\\left[(k_y^2 + k_z^2)\\left(1+\\frac{i}{\\tilde{k}_x L_P}\\right) \\left\\{1 + \\frac{i(3\\qe - 2)}\n{2\\qe \\tilde{k}_x L_P} + \\frac{\\tilde{\\kappa}^2}{4\\qe \\tilde{k}_x^2 L_P^2} \n- \\frac{N_x^2 + \\tilde{\\kappa}^2}{2\\tilde{k}_x^2 H^2}\\right\\} \\, + \\right.\n\\nonumber \\\\ \\left. 3k_y^2\\frac{(\\qe + 1)(\\qe + 2)}{\\tilde{k}_x^2 H^2}\n\\left(1 - \\frac{2i}{3\\qe \\tilde{k}_x L_P}\\right) \\right],\n\\end{eqnarray}\n\\begin{equation}\\label{COEF0}\nF_0 = c_s^2 \\tilde{k}_x^2 \\left[k_y^2 (N_x^2 + 2\\qe^2 \\Omega^2) +\nk_z^2(N_x^2 + \\tilde{\\kappa}^2)\\right] \\left[(k_y^2 + k_z^2)\\left(1-\\frac{i}\n{\\tilde{k}_x L_P}\\right)^2 + k_y^2 \\frac{2(\\qe+1)(3\\qe+2)}\n{\\tilde{k}_x^2 H^2}\\right].\n\\end{equation}\n\nThe above expressions have been written to make the short-wavelength\nlimit more apparent: all but the leading-order terms in brackets are\nproportional to factors of $(\\tilde{k}_xL_P)^{-1}$ or\n$(\\tilde{k}_xH)^{-1}$. Notice also that since one expects $H\/L_P \\ll 1$ for\nKeplerian disks with modest radial gradients,\n\\begin{equation}\n\\frac{1}{\\tilde{k}_x L_P} = \\frac{1}{k_y H\\tilde{\\tau}}\\frac{H}{L_P} \n\\ll \\frac{1}{k_y H\\tilde{\\tau}},\n\\end{equation}\n(for $\\tilde{\\tau} \\neq 0$) and therefore the short-wavelength limit is sufficient.\nOne needs to be careful in taking this limit, however, since $\\tilde{k}_x = k_y\n\\tilde{\\tau}$ goes through zero as a shwave goes from leading to trailing. The\napproximation is rigorously valid only for $\\tilde{\\tau} \\neq 0$, but we have \nnumerically integrated the full set of linear equations (equations (\\ref{LIN1a})\nthrough (\\ref{LIN5a}) with $k_z = 0$) and found good agreement with the \nBoussinesq solutions described in \\S 4.3 for all $\\tilde{\\tau}$ at sufficiently\nshort wavelengths.\\footnote{One must start with a set of initial conditions \nconsistent with equations (\\ref{LIN2}), (\\ref{LIN3}), (\\ref{LIN1}) and\n(\\ref{LIN5}) in order to accurately track the incompressive-shwave solutions. \nIn addition, suppression of the high-frequency compressive-shwave solutions\nnear $\\tilde{\\tau} = 0$ requires $k_y L_P \\gtrsim 200$, which for $H\/L_P = \n0.1$ implies $H k_y \\gtrsim 20$.}\n\nWith these assumptions in mind, to leading order in $(H k_y)^{-1}$ equation\n(\\ref{DV4DT}) becomes\n\\begin{eqnarray}\\label{EQVX}\n\\tilde{k}_x \\delta v_x^{(4)} - 2 \\qe \\Omega k_y \\delta v_x^{(3)} +\nc_s^2 \\tilde{k}_x \\tilde{k}^2 \\delta v_x^{(2)} + 4\\qe \\Omega c_s^2\n\\tilde{k}_x^2 k_y \\delta v_x^{(1)} \\, + \\nonumber \\\\ c_s^2 \\tilde{k}_x\n\\left[k_y^2(N_x^2 + 2\\qe^2 \\Omega^2) + k_z^2(N_x^2 + \\tilde{\\kappa}^2)\n\\right]\\delta v_x = 0.\n\\end{eqnarray}\nIf we assume $\\partial_t \\ll c_s k_y$, the two highest-order time derivatives\nare of lower order and can be neglected (thereby eliminating the compressive\nshwaves) and we have\n\\begin{equation}\\label{DV2DT}\n\\tilde{k}^2 \\ddot{\\delta v_x} + 4 \\qe \\Omega \\tilde{k}_x k_y\\dot{\\delta v_x}\n\\left[k_y^2(N_x^2 + 2\\qe^2 \\Omega^2) + k_z^2(N_x^2 +\\tilde{\\kappa}^2)\\right]\n\\delta v_x = 0.\n\\end{equation}\nThis is equivalent to equation (\\ref{BOUSSVX}).\n\nNotice also that the assumption $\\partial_t \\sim O(c_s k_y)$ applied to\nequation (\\ref{EQVX}) yields\n\\begin{equation}\n\\delta v_x^{(4)} + c_s^2 \\tilde{k}^2 \\delta v_x^{(2)} = 0\n\\end{equation}\nto leading order in $(H k_y)^{-1}$. This equation is of the same form\nas the short-wavelength limit of equation (\\ref{SOUND}) for the\ncompressive shwaves in the unstratified shearing sheet, confirming our\nclaim that short-wavelength compressive shwaves are unchanged at\nleading order by stratification.\n\n\n\\chapter{Summary and Outlook}\\label{conclusion}\n\n\\begin{spacing}{1.5}\n\nAngular momentum transport is key to the evolution of accretion disks. In ionized disks,\nmomentum transport is likely to be mediated internally by MHD turbulence generated by the\nMRI. Despite the success of this local shear instability in elucidating the accretion\nprocess in ionized disks, the complexity of its nonlinear outcome has raised a whole new\nset of questions regarding its effects upon disk evolution. The answers to most of these\nquestions will require the use of three-dimensional numerical simulations. I discuss some\nof the remaining open questions in Section~\\ref{saos1} and propose some simple numerical\nexperiments that will attempt to answer them.\n\nThe mechanism driving accretion in weakly-ionized disks remains unclear. I summarize the\nmain results of this dissertation in Section~\\ref{saos2}, results which mostly argue against\na turbulent transport of angular momentum in these disks. I also discuss some possible\ndirections to take in further pursuit of such a mechanism.\n\nThe fact that decades of research have not uncovered a robust turbulent transport mechanism\nfor weakly-ionized disks raises the possibility that at least some of these disks (or portions\nof them) are stable and do not accrete in a steady state as the standard disk model assumes.\nProposals for exploring the implications of this possibility are discussed in\nSection~\\ref{saos3}.\n\nFinally, while modeling turbulent shear stresses as an alpha viscosity has turned out to be\nuseful phenomenologically, representing disk turbulence as an alpha viscosity has its\nlimitations (see Section~\\ref{dmts}), and any model results that depend upon an accurate\nrepresentation of this aspect of disk physics are therefore suspect. The fundamental\nunderstanding of turbulent shear stresses in disks that has begun to emerge in recent years\nhas opened up an exciting opportunity for developing physically-motivated disk models based\nupon a first-principles treatment of disk turbulence. I conclude in Section~\\ref{saos4}\nwith a discussion of proposed research along these lines.\n\n\\section{Ionized Disks}\\label{saos1}\n\nThe phenomenological approach to modeling turbulent transport in accretion disks has been\nchallenged by the recent improved physical understanding of that transport in ionized disks.\nAt the same time, there are important issues with regard to MHD turbulence and its \nramifications for disk physics that must be understood before the standard disk model can be \nreplaced with models based upon a more accurate representation of turbulent transport. This \nsection discusses a few of these issues and proposes ways in which they can be investigated.\n\n\\subsection{Transition from 2D to 3D MHD Turbulence}\n\nAn important assumption underlying standard disk theory is that the global and local disk \nscales are well separated; small-scale turbulence is assumed to average to a smoothly-varying \nflow on large scales. The validity of this assumption can be tested by measuring the power \nspectrum of MHD turbulence in a series of three-dimensional shearing-box simulations with \nhorizontal scales much larger than the vertical scale, looking for a transition between \ntwo-dimensional and three-dimensional behavior. If there is a transition at some \ncharacteristic scale, presumably on the order of the disk scale height, this will confirm \nthe standard picture as well as provide a guide for the scales at which the assumptions of\nstandard disk theory can be appropriately applied in global models. If there is no transition \nto smoothly-varying two-dimensional flow, much of standard disk theory is invalid.\n\n\\subsection{Dynamics of MRI Turbulent Stresses}\\label{dmts}\n\nTurbulence is an extremely complex phenomenon, and the standard approach of modeling it as\na viscosity is clearly oversimplified. For example, there are key dynamical properties of\nMHD turbulence, such as the elastic properties that produce magnetic tension \\citep{op03},\nthat cannot be modeled with a viscosity \\citep{mg02}. In addition, the standard disk model\nassumes that the turbulent shear stress is isotropic, whereas MHD turbulence is inherently\nanisotropic. Any model results that depend upon an accurate representation of the turbulent\nshear stress are therefore suspect. \\cite{ogl03} has developed an analytic model for MHD\nturbulent stresses consisting of evolution equations for the turbulent stress tensor. Such\na model could provide the basis for more realistic analytic and numerical studies of accretion\ndisks, as well as be useful for global disk simulations since it would allow one to represent\nthe underlying turbulence accurately over long time scales without having to resolve the\nsmall-scale structure. I will attempt to test this model against local numerical\nsimulations of MHD turbulence and seek to constrain the values of its free parameters.\n\n\\subsection{Interaction of Waves with Turbulence}\\label{iwt}\n\nAnother key assumption underlying many disk studies is that waves will propagate through\nturbulence relatively unhindered. For example, models that have been developed to explain\nquasi-periodic oscillations (QPOs) in binary systems often invoke characteristic oscillations\nof the accretion disk as the source of the QPOs \\citep{wso01}. The implicit assumption in\nthese models is that the disk oscillations are negligibly affected by the disk turbulence.\nAs another example, analytic studies of warped accretion disks \\citep{pp83,pl95} have shown\nthat warps will propagate as non-dispersive waves if the turbulence in the disk is\nsufficiently small. The turbulence in these studies is modeled as a shear viscosity, but\nit is not clear how the turbulence will interact with warp propagation self-consistently. \nAs a final example, the standard disk model assumes that thermal energy is both generated \nand radiated locally, which results in a temperature dependence with radius that is not \nalways observed \\citep{rww99}. An alternative to this local dissipation process is a \nglobal process whereby waves are excited at one point in the disk and carry energy and \nangular momentum to another point in the disk before dissipating \\citep{asl88}. Again, \nwave propagation through the disk is assumed to proceed unhindered by the turbulence.\n\nThe general problem of the interaction of waves with turbulence has been well-studied\nanalytically, particularly in the context of solar oscillations (\\citealt{gk88,gk90}; see\nalso \\citealt{fg04}), but numerical studies along these lines are less advanced (see\n\\citealt{tea00} for one example). I will seek to further our understanding in this area by\ninvestigating, via numerical experiments, the effect of turbulence on waves propagating\nthrough a disk. A wave or superposition of wave modes calculated from linear theory will\nbe inserted into a magnetized turbulent numerical model, and the subsequent evolution\nwill be compared with the linear theory results. The outcome of these experiments will\ndetermine whether or not the neglect of wave-turbulence interactions is valid. If there\nis significant interaction, these experiments may provide a means for developing a\npredictive theory of wave propagation in turbulent disks.\n\n\\section{Weakly-Ionized Disks}\\label{saos2}\n\nThe prospects of discovering a turbulent transport mechanism in weakly-ionized accretion \ndisks appear to be dim. Since proving stability is much more difficult than demonstrating \ninstability, however, the search for such a transport mechanism continues. The most \npromising mechanism based upon the work discussed in this dissertation is the driving\nof compressive motions by vortices, but even this faces serious difficulties due to the \ndecaying angular momentum flux and the potential for the vortices to be unstable in three \ndimensions. It would be useful to redo in three dimensions the simulations discussed in \nChapter~\\ref{paper3}, to determine the rate at which the vortices become unstable and the \nnonlinear outcome of such an instability. Testing some of the vorticity-generation \nmechanisms discussed in \\S\\ref{pap3s4} is also a necessary next step in understanding the \nrelevance of this overall picture for driving accretion in weakly-ionized disks.\n\nThe possibility of a bypass transition to turbulence due to the transient amplification of\nlinear disturbances followed by a nonlinear feedback from trailing shwaves (shearing waves) \ninto leading shwaves\nhas not been fully explored. The effects of aliasing shown in Figure~\\ref{pap4f12} clearly\ndemonstrate that such a feedback mechanism can result in the overall growth of linear\ndisturbances into the nonlinear regime. While the feedback in Figure~\\ref{pap4f12} is\nentirely numerical, the high-resolution requirements for tracking these shwaves (as can\nbe seen in Figure~\\ref{pap4f1b}) implies that the nonlinear outcome of transient amplification\nhas not been tested at the resolutions we employ for the runs discussed in Chapter~\\ref{paper4}.\n\n\\cite{vmw98} calculated three-shwave interactions in the unstratified (incompressible) shearing\nsheet and found that there is feedback from trailing shwaves into leading shwaves for a small\nsubset of initial shwave vectors. It would be of interest to revisit this calculation in the\nstratified shearing sheet. One key difference between the stratified and unstratified\nshearing-sheet models is that in the latter case all linear perturbations decay after their transient\ngrowth, whereas in the former case the density perturbation does not decay. This implies that quasilinear interactions are more likely to take place. \nA comparison of Figures~\\ref{pap3f7} and \\ref{pap4f12} indicates that \nfeedback due to aliasing in the unstratified sheet does not result in\nany overall growth, whereas feedback in the stratified sheet does.\n\n\\section{Layered Disks}\\label{saos3}\n\nA more fruitful line of research may be to simply assume that a weakly-ionized flow is\nstable. Accretion in that case could proceed in surface layers that are ionized by\nnon-thermal radiative processes, with the mid-plane of the disk remaining inactive\n(see Figure~\\ref{conclf1} and \\citealt{gam96}). One of the few numerical studies of \nlayered accretion has shown that there may be some wave transport from the active, \nMHD-turbulent zone to the inactive zone \\citep{fs03}. This work could be expanded upon \nwith a more realistic treatment of the vertical structure and vertical boundary conditions, \nas well as a closer inspection of the stresses in the inactive zone, including an \ninvestigation of their dependence on simulation parameters. Understanding these processes\nwould be useful for developing more sophisticated models of layered disks.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=6.5in,clip]{conclf1.eps}\n\\caption[Ionization structure of YSO disks.]{\nIonization structure of YSO disks. The inner disk is coupled\nto the field via thermal ionization. At larger radii the surface layers\nare coupled but the midplane (hashed) is inactive. At still larger\nradii the density is lower and nonthermal ionization provides effective\ncoupling throughout the disk.}\n\\label{conclf1}\n\\end{figure}\n\nIn addition, since the dynamics of the dust layer in protoplanetary disks (disks which are\nlikely to be weakly-ionized) have important implications for understanding planet formation,\na useful and natural extension to the research proposed here is to incorporate gas-dust\ninteractions in some of my calculations. The effects of turbulence and wave transport on\nthe dust layer could be investigated by including dust particles of various sizes in numerical\nsimulations similar to those described in Section~\\ref{iwt}. In addition, interactions could be\ncalculated self-consistently by incorporating dust dynamics into layered disk models using\na two-fluid approach.\n\n\\section{Advanced Physical Disk Models}\\label{saos4}\n\nThe majority of analytic \\citep{ss73,lbp74,prin81,har98} and numerical \\citep{ia99,spb99,ia00,\nian00,mg02} accretion-disk studies incorporate the assumption of an alpha viscosity to model\ndisk turbulence. As discussed in \\S\\ref{dmts}, such an assumption has its shortcomings.\nA fundamental understanding of MHD turbulent stresses can be used to develop advanced disk\nmodels that will enhance our ability to explain observations of accretion systems.\n\nNumerical models that rely on an alpha viscosity could be improved upon by incorporating a more\nsophisticated model for the turbulent stresses based upon simulations and theoretical studies\nof turbulence, such as those described in Section~\\ref{dmts}. In addition to being useful\nfor modeling accretion systems, these models would provide a more reliable bridge between\nlocal and global MHD simulations and possibly be a means for understanding the complex\nresults of global MHD simulations at a more fundamental level.\n\nAnalytic models that incorporate more sophisticated turbulent-stress modeling could also be\ndeveloped in an effort to improve upon the standard disk model. In addition, there may\nbe key analytic results that need to be revisited with an enhanced understanding of turbulent\nstresses, such as the work on wave propagation in warped disks discussed in \\S\\ref{iwt}\n\\citep{pp83,pl95}. All of these proposals represent potential progress towards a\nfirst-principles understanding and modeling of accretion-disk systems.\n\n\\end{spacing}\n\n\\chapter{Curriculum Vitae}\n\n\\setlength{\\parskip}{0ex} \\setlength{\\parindent}{0ex}\n\n\\section*{\\centering Bryan Mark Johnson}\n\n\\subsection*{ADDRESS}\n\nLoomis Laboratory of Physics\n\nUniversity of Illinois at Urbana-Champaign\n\n1110 West Green Street\n\nUrbana, IL 61801\n\nPhone: (217) 333-2327\n\ne-mail: bmjohnso@uiuc.edu\n\n\\subsection*{EDUCATION}\n\nPh.D. in Physics, University of Illinois at Urbana-Champaign, 2005 (Advisor: Gammie).\n\nDissertation: {\\it Turbulent Angular Momentum Transport in Weakly-Ionized Accretion Disks}.\n\nB.S. Electrical Engineering, LeTourneau University, 1996 (Advisor: Knoop).\n\nSenior thesis: {\\it System Measurement with White Noise}.\n\n\\subsection*{EXPERIENCE}\n\nTeaching Assistant, University of Illinois at Urbana-Champaign, 2000-2002.\n\nSystems Integration Engineer, Northrop Grumman Corporation, 1997-2000.\n\n\\subsection*{HONORS AND AWARDS}\n\nDrickamer Research Fellowship, University of Illinois, 2004: {\\it to recognize a graduate student\nwho has demonstrated significant ability in research, the most prestigious prize given by\nthe Department of Physics to a graduate student}.\n\nR.G. LeTourneau Award for Outstanding Senior Engineering Student, LeTourneau University, 1996.\n\nGold Key Honor Society, LeTourneau University, 1996: {\\it represents the highest honor given\nby LeTourneau University to outstanding senior students}.\n\nDean's Scholarship, LeTourneau University, 1994.\n\nNorthrop Corporation Engineering Scholarship, Harper College, 1993.\n\nSquare D Engineering Scholarship, Harper College, 1992.\n\nPresident's Scholarship, Crown College, 1990.\n\n\\chapter{Introduction}\\label{intro}\n\n\\begin{spacing}{1.5}\n\nAccretion disks form around gravitating objects because the angular momentum\nof the infalling gas is conserved. In order for the gas to accrete, however,\nits angular momentum must be removed. Understanding the mechanism underlying this\nangular momentum transport is a long outstanding puzzle in accretion disk\ntheory. Much progress has been made in recent decades through the realization\nthat a weak magnetic field will destabilize the flow in an ionized disk and result in a\nturbulent transport of angular momentum. One of the key features of this mechanism,\nhowever, is that the gas in the disk must be sufficiently ionized to couple to the\nmagnetic field. It is likely that there are portions of disks, and perhaps entire\nclasses of disks, in which the ionization is too low for the gas to destabilize. The\nsearch for a turbulent transport mechanism in weakly-ionized disks continues, therefore,\nto this day, in an effort to place our understanding of the evolution of these disks on\nas firm a theoretical footing as that for ionized disks. My research, in collaboration\nwith Charles Gammie, has focused on several possible mechanisms: self-gravity, radial\nconvection and vortices driving compressive disturbances. This dissertation summarizes\nour main results and discusses their relevance to the question of what drives accretion\nin weakly-ionized disks.\n\nThe purpose of this chapter is to describe all of these ideas in detail and put them\nin their astrophysical context. I begin in \\S\\ref{intros1} with an overview of\naccretion disks: the systems in which they are observed and their general properties.\nThe importance of angular momentum transport for the evolution of disks is discussed in\n\\S\\ref{intros2}, followed by a more detailed discussion of angular momentum transport\nin both ionized and weakly-ionized disks (\\S\\S\\ref{intros3} and \\ref{intros4}). I\ngive a brief overview in \\S\\ref{intros5} of the model that is used throughout the\ndissertation for analytic and numerical studies. \\S\\ref{intros6} looks ahead to\nChapter~\\ref{conclusion} in which I summarize and chart a course for future \nwork.\\footnote{Portions of this chapter will be published in the proceedings of the\nWorkshop on Chondrites and the Protoplanetary Disk, November 8-11, 2004, Kauai, Hawaii.}\n\n\\section{Accretion Disks}\\label{intros1}\n\nAn accretion disk is a roughly cylindrical distribution of matter in orbit around a\ngravitating central object. It is supported against the gravitational pull of its\nhost object primarily by the centrifugal forces arising from the angular momentum of\nthe orbiting material. This support is slightly compromised, however, by the presence of\ndissipation or the application of external torques. As a result, angular momentum is\nredistributed through the disk and some of the disk material falls onto the central\nobject, i.e., it accretes. The gravitational potential energy lost during this process\nis typically converted into radiation, which is the basis for our observations of\naccretion disk systems. Although disks are rarely observed directly (i.e., by being\nresolved in a telescope image), they imprint a unique signature on the spectra of their\nhost systems. In addition, the accretion-disk paradigm easily accounts for many properties\nof astrophysical systems that would be difficult to explain otherwise.\n\nThe material in an accretion disk covers a wide range of density and temperature\nscales \\citep{bh98}. In most cases, collisional mean free paths are extremely short\ncompared to the length scales of interest, and mean times between collisions are short\ncompared to the time scales of interest. Disks are therefore usually modeled as a\ncontinuous fluid, using the macroscopic equations of gas dynamics rather than the\nmicroscopic equations of kinetic theory. If the fluid is ionized, it is referred to\nas a plasma.\n\nAccretion disks are found in a variety of astrophysical settings,\nincluding compact binary systems (with a white dwarf, black\nhole, or neutron star), active galactic nuclei (AGN), and young stars.\nAGN consist of a supermassive black hole ($M \\sim 10^8 \\msun$) surrounded\nby an accretion flow. The presence of a disk in AGN is inferred\nprimarily from the large luminosities of these systems, luminosities\nthat cannot be accounted for by stellar nuclear burning\nbut are easily provided by the large gravitational energy\nof the compact object. A spectacular exception to this indirect verification\nis the system NGC4258, an AGN in which the disk is directly observed via\nmaser emission \\citep{ww94}. Low Mass and High Mass X-Ray Binary (LMXRB and HMXRB)\nsystems consist of a neutron star (NS) or black hole (BH) accreting matter from its companion;\nCataclysmic Variables (CV) are binary systems in which the accreting object\nis a white dwarf (see Figure~\\ref{introf1}). The variability observed in these\nsystems can be accounted for by models in which instabilities in a disk surrounding\nthe compact object give rise to episodic accretion. Young Stellar Objects (YSO)\nare pre-main-sequence stars with a circumstellar disk, also known as protoplanetary\ndisks since they are thought to be the sites of planet formation. The presence of a\ndisk in these systems is confirmed in many cases by direct observation (e.g., \n\\citealt{burr96}).\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=6.0in,clip]{introf1.eps}\n\\caption{Schematic of a cataclysmic variable system.}\n\\label{introf1}\n\\end{figure}\n\n\\begin{deluxetable}{lccccccccc}\n\\tablecolumns{10}\n\\tablewidth{0pc}\n\\tablecaption{Example Accretion-Disk Systems \\label{introt1}}\n\\tablehead{System & Type & $\\frac{M}{\\msun}$ & $\\frac{\\mdot}{\\msun\n \\yr^{-1}}$ & $\\frac{L_{acc}}{\\lsun}$ & $R_c$ (cm) & $T$ (K)\n & $\\frac{H}{R_c}$ }\n\\startdata\n NGC 4258 & AGN & $4 \\times 10^7$ & $1 \\times 10^{-2}$ & $1 \\times 10^{10}$ &\n $1 \\times 10^{18}$ & $7 \\times 10^2$ & $2 \\times 10^{-3}$ \\\\\n Sco X-1 & LMXRB-NS & $1$ & $1 \\times 10^{-8}$ & $3 \\times 10^{4}$ & $1 \\times 10^{10}$\n & $3 \\times 10^{5}$ & $5 \\times 10^{-2}$ \\\\\n LMC X-3 & HMXRB-BH & $10$ & $1 \\times 10^{-8}$ & $1 \\times 10^{4}$ & $1 \\times 10^{11}$\n & $7 \\times 10^{4}$ & $3 \\times 10^{-2}$ \\\\\n UX Uma & CV & $0.5$ & $1 \\times 10^{-8}$ & $1 \\times 10^1$ & $1 \\times 10^{10}$ &\n $2 \\times 10^{5}$ & $6 \\times 10^{-2}$ \\\\\n TW Hydrae & YSO & $0.7$ & $5 \\times 10^{-10}$ & $1 \\times 10^{-2}$ & $1 \\times 10^{13}$\n & $1 \\times 10^2$ & $2 \\times 10^{-2}$ \\\\\n\\enddata\n\\end{deluxetable}\n\nTable~\\ref{introt1} lists typical values for various physical properties of these systems.\nThe mass of the accreting object is denoted by $M$ in Table~\\ref{introt1}, in\nunits of the solar mass ($\\msun = 2.0 \\times 10^{33}$ g), and the accretion rate\nby $\\mdot$. The accretion luminosity\n\\begin{equation}\nL_{acc} \\equiv \\frac{GM\\mdot}{R_{in}}\n\\end{equation}\n(where $R_{in}$\nis the inner radius of the disk and where $G = 6.673 \\times 10^{-8} \\cm^3 \\gm^{-1}\n\\s^{-2}$ is the gravitation constant) is given in units of the solar luminosity\n($\\lsun = 3.9 \\times 10^{33} \\erg \\s^{-1}$). $R_c$ is a characteristic radius for\nthe accretion disk, and $T$ is a characteristic temperature. The final column in\nTable~\\ref{introt1} contains the ratio of the vertical scale height $H$ to the local radius.\nHere\n\\begin{equation}\nH \\equiv \\frac{c_s}{\\Omega_K},\n\\end{equation}\nwhere $c_s$ is the isothermal sound speed and\n\\begin{equation}\n\\Omega_K = \\sqrt{\\frac{GM}{r^3}}\n\\end{equation}\nis the local Keplerian orbital frequency. The values for $T$ and $H$ in\nTable~\\ref{introt1} are obtained from the standard $\\alpha$-disk model, a\nderivation of which is outlined in the following section, using $\\alpha = 0.01$.\nValues for the other quantities in Table~\\ref{introt1} were obtained from the\nliterature (NGC4258: \\citealt{miy95}, \\citealt{gnb99}; Sco X-1: \\citealt{vea91};\nLMC X-3: \\citealt{pac83}, \\citealt{vanp96}; UX Uma: \\citealt{fea81}; TW Hydrae:\n\\citealt{mea00}, \\citealt{wea03}). The standard disk model assumes that the internal\nenergy of the disk material is efficiently radiated from the surfaces of the disk.\nOne implication of this assumption is that the disk is typically quite thin, as can\nbe seen from the last column of Table~\\ref{introt1}. In addition, if the mass of the\ndisk is much less than the mass of the central star, the orbital frequency of the gas\n$\\Omega = \\Omega_K + O(H\/r)^2$, so thin, low-mass disks have a nearly-Keplerian\nrotation profile.\n\n\\section{Angular Momentum Transport and Disk Evolution}\\label{intros2}\n\nThe accretion process consists of a net inward transport of matter and a\nnet outward transport of angular momentum; a small fraction of the matter\ncarries angular momentum outward, enabling the bulk of the matter to accrete.\nThis angular momentum transport can take place 1) internally via a local exchange of\nmomentum between fluid elements at adjacent radii or 2) externally via\na global mechanism such as the removal of angular momentum by a wind off\nthe surface of the disk (e.g. \\citealt{bp82}). The focus of this dissertation\nis on the former: the internal diffusion of angular momentum. While global\nmechanisms such as winds and jets are certain to play a role in many accretion\nsystems, their operation likely depends in a complicated manner upon the details\nof each particular system. Standard disk modeling typically ignores their effects \nand assumes that angular momentum is transported internally (e.g., \\citealt{prin81,rp91,\nsm94,nkh94,step97,gam99}). Whether or not it is possible to isolate this aspect of\ndisk evolution and get meaningful results is a question that can only be answered \nas more comprehensive disk models are developed.\n\nAs an example of the importance of global effects, as well as some of the\ndifficulties involved in modeling them, consider the torques from\nMHD winds (axisymmetric pressure-driven winds have zero torque).\nOutflows are widely observed from YSO, and it is likely that the\noutflows are magnetically driven. Outflows are more common in young,\nhigh-accretion-rate systems. Highly uncertain estimates for the\nmass loss rate suggest that about $10\\%$ of the accreted mass goes\ninto the jet and associated outflow. What is even less certain is the\namount of angular momentum in the outflows, and therefore the role that\nthey play in the evolution of disks on large scales (as opposed to the\ndisk at radii less than a few tenths of an AU). Wind models exist\n(e.g., \\citealt{bp82,shu94,shu00,wk93}), but there are large gaps in our\nunderstanding. We do not know what the strength of the mean vertical\nmagnetic field, which organizes the wind, ought to be, nor how that mean\nfield is transported radially through the disk, nor how the wind evolves\nin time. A nice summary of this situation is given in \\cite{kp00}.\n\nMolecular shear viscosity is a natural mechanism for coupling fluid elements locally\nand transporting angular momentum internally, but the large Reynolds\nnumbers of astrophysical flows (due to the large length scales involved) imply\nmolecular shear viscosities that are much too tiny to account for the observations.\nThe coupling due to molecular viscosity is simply too weak to explain the rapid\nvariability and accretion rates that are observed. For example, the outburst duration\nin CV ranges from $2-20$ days, with the interval between outbursts ranging from tens\nof days to tens of years \\citep{warn95}. The timescale for viscous diffusion over a\ndistance $l$ due to molecular viscosity is $l^2\/\\nu_m$, where $\\nu_m$ is the molecular\n(or kinematic) shear viscosity. Using a value $\\nu_m = 10^5 \\cm \\s^{-1}$ and a\ncharacteristic distance $l = 10^{10} \\cm$ (see \\citealt{bal03} and Table~\\ref{introt1})\nyields a viscous timescale of about $10^7$ years, which is orders of magnitude too large\nto account for the timescales of CV outbursts.\n\nStandard disk modeling circumvents this problem by assuming the presence of an\nenhanced ``anomalous viscosity'' due to turbulence. The large Reynolds numbers\nare used to advantage, since our experience with laboratory flows indicates\nthat the onset of turbulence typically occurs above a critical Reynolds number.\nThe {\\it assumption} of turbulent flow, along with the picture of turbulent eddies\nexchanging momentum with one another to drive accretion, underlies the majority of the\nphenomenological disk modeling that is currently used to explain observations.\n\nThe construction of a standard model for disk evolution proceeds as follows. We\nwill use a cylindrical coordinate system centered on the accreting object, with $r$, $\\phi$ and $z$ the radial,\nazimuthal and vertical coordinates, respectively. We start with mass conservation:\n\\begin{equation}\n\\frac{\\partial \\rho}{\\partial t}\n=\n-\\frac{1}{r} \\partial_r (r \\rho v_r)\n- \\frac{\\partial}{\\partial z} (\\rho v_z)\n\\end{equation}\nwhere $\\rho$ is the mass density and we\nassume axisymmetry ($\\partial\/\\partial\\phi = 0$), on average. Integrating this\nequation vertically through the disk gives\n\\begin{equation}\\label{MASSCON}\n\\frac{\\partial \\Sigma}{\\partial t}\n=\n-\\frac{1}{r} \\partial_r (r \\Sigma \\bar{v}_r) + \\dot{\\Sigma}_{ext}\n\\end{equation}\nwhere $\\Sigma = \\int dz \\rho$ is the surface density,\n$\\bar{v}_r$ is a vertical average of the radial velocity, and\n$\\dot{\\Sigma}_{ext}$ is the difference of $-\\rho v_z$ evaluated at the\nupper and lower surface of the disk. It includes infall onto the disk,\nmass loss in winds, and mass loss through photoevaporation. It is\npositive when mass flows into the disk, and negative when mass flows\nout.\n\nThe problem now is to find the radial velocity $\\bar{v}_r$, which we can do\nusing angular momentum conservation:\n\\begin{equation}\n\\frac{\\partial (\\rho l)}{\\partial t}\n=\n-\\frac{1}{r} \\frac{\\partial}{\\partial r} (r^2 \\Pi_{r\\phi})\n- \\frac{\\partial}{\\partial z} (r \\Pi_{z \\phi}).\n\\end{equation}\nHere $\\rho l$ is evidently the local density of angular momentum,\nand the right hand side of the equation is the divergence of an angular\nmomentum flux density. $\\Pi_{r\\phi}$ is a component of the stress tensor,\nsometimes referred to as the shear stress, with dimensions of pressure;\nit is the flux density of $\\phi$ momentum in the $r$ direction. Likewise\n$\\Pi_{z\\phi}$ is the flux density of $\\phi$ momentum in the $z$ direction.\nAgain integrating vertically,\n\\begin{equation}\\label{AMCON}\n\\frac{\\partial (\\Sigma l)}{\\partial t}\n=\n-\\frac{1}{r} \\frac{\\partial}{\\partial r} (r^2 W_{r\\phi} + r \\Sigma \\bar{v}_r l)\n+ \\tau + l \\dot{\\Sigma}_{ext}.\n\\end{equation}\nHere\n\\begin{equation}\nW_{r\\phi} \\equiv \\int dz \\Pi_{r\\phi} - r \\Sigma \\bar{v}_r l\n\\end{equation}\nis the integrated shear stress, but with one piece of it, proportional to\nthe radial mass flux, peeled off. In models which assume that angular momentum\ntransport is due to turbulence, $W_{r\\phi}$ is referred to as the ``turbulent\nshear stress.'' The external torque\n\\begin{equation}\n\\tau \\equiv -r \\Pi_{z\\phi}|_{\\rm lower}^{\\rm upper} - l \\dot{\\Sigma}_{ext}\n\\end{equation}\nis the angular momentum flux into the upper and lower surface of the\ndisk with one piece, proportional to the mass flux into the disk, peeled\noff. $\\tau$ includes the effects of, e.g., MHD winds; it is positive\nwhen angular momentum flows into the disk and negative when angular\nmomentum flows out. In a steady state ($\\partial\/\\partial t = 0$), the \ncondition $W_{r\\phi} > 0$ must be met for an outward transport of angular \nmomentum and inward accretion.\n\nThe mass and angular momentum conservation equations (\\ref{MASSCON}) and\n(\\ref{AMCON}) can be combined into a single equation governing the\nevolution of thin, Keplerian disks (multiply the continuity equation by\n$r v_\\phi$, subtract the angular momentum equation, solve for $\\bar{v}_r$,\nsubstitute back into the continuity equation):\n\\begin{equation}\\label{BASIC}\n\\frac{\\partial \\Sigma}{\\partial t}\n=\n\\frac{2}{r}\n\\frac{\\partial}{\\partial r} \\left(\n\\frac{1}{r\\Omega}\n\\frac{\\partial}{\\partial r} ( r^2 W_{r\\phi} )\n\\, - \\, \\frac{\\tau}{\\Omega} \\right)\n\\, + \\, \\dot{\\Sigma}_{ext}.\n\\end{equation}\n\nIf we assume that the shear stress $W_{r\\phi}$ is due to an\nanomalous viscosity $\\nu$, and that the external torques and mass\nloss\/infall are negligible, equation (\\ref{BASIC}) becomes the\nbasic equation for standard disk modeling:\n\\begin{equation}\\label{BASICPRIME}\n\\frac{\\partial \\Sigma}{\\partial t} =\n\\frac{3}{r}\n\\frac{\\partial}{\\partial r} \\left(\n\\frac{1}{r\\Omega}\n\\frac{\\partial}{\\partial r} ( r^2 \\Sigma \\nu \\Omega) \\right).\n\\end{equation}\nIn a steady state one can show that the accretion rate (inward mass flux\n$= -2 \\pi \\Sigma r \\bar{v}_r$) is given by\n\\begin{equation}\n\\dot{M} = 3 \\pi \\Sigma \\nu.\n\\end{equation}\n\nIn addition to setting $\\tau$ and $\\dot{\\Sigma}_{ext}$ to zero, \nstandard disk theory usually sets $\\nu = \\alpha c_s\nH$, which parameterizes our ignorance of $W_{r\\phi}$. If one\nreasonably assumes that the turbulent stress (an off-diagonal component\nof the stress tensor) must be associated with a pressure (an isotropic,\ndiagonal component of the stress tensor), then $\\alpha \\lesssim 1$.\nMost disk evolution models take $\\alpha = const.$, or allow it to assume\na few discrete values. For a disk around a solar-type star\nwith a temperature of $300{\\rm\\,K}$ at $1$ AU, this yields $\\dot{M} =\n9.9 \\times 10^{-9} \\alpha \\Sigma {\\rm\\,M_\\odot} {\\rm\\,yr}^{-1}$, or\n$\\sim 10^{-8} {\\rm\\,M_\\odot} {\\rm\\,yr}^{-1}$ for $\\alpha = 0.01$ and\n$\\Sigma = 10^2 {\\rm\\,g} {\\rm\\,cm}^{-2}$, roughly consistent with\nobserved accretion rates \\citep[e.g.][]{ghbc98}.\n\n\\section{Angular Momentum Transport in Ionized Disks}\\label{intros3}\n\nWhile phenomenological disk modeling can proceed along its merry way\nwithout a clear demonstration of the onset of turbulence in accretion\ndisk flows, recent progress towards a first-principles\nunderstanding of disk turbulence has raised the possibility that more\nphysically-motivated disk models can be developed. The primary\nbreakthrough in our understanding came with the realization that\nthe presence of a weak magnetic field destabilizes a disk on a dynamical\ntime scale, resulting in the onset of magnetohydrodynamic (MHD) turbulence and\na vigorous outward flux of angular momentum \\citep{bh91,bh98,bal03}.\n\nThis section gives a summary of the essential physics of this instability,\ngenerally termed the magneto-rotational instability, or MRI. The importance\nof ionization for the successful operation of the MRI in driving turbulence\nis also discussed, which leads in the following section to an overview of the\nmain question addressed by this dissertation: what drives accretion in disks\nthat are too weakly ionized to be unstable to the MRI?\n\n\\subsection{Magneto-Rotational Instability}\\label{MRI}\n\nThe MRI grows directly through exchange of angular momentum between\nradially-separated fluid elements. This can be understood with a simple\nmechanical analogy introduced by \\cite{bh92} and illustrated in\nFigure~\\ref{introf2}.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=6.5in,clip]{introf2.eps}\n\\caption[Mechanical analogy for the MRI.]{\nMechanical analogy for the MRI. Two masses, orbiting in the direction\nindicated by the arrow about a massive body (bottom of the frame), are\nconnected by a spring. The outer mass has higher angular momentum and\nlower orbital frequency. The lower mass is pulled back in its orbit by\nthe spring, reducing its angular momentum. It sinks to a lower orbit,\nwhere it orbits faster, stretching the string and increasing the torque.\nA runaway ensues. The masses may be thought of as fluid elements\nconnected by a magnetic field.}\n\\label{introf2}\n\\end{figure}\n\nImagine that two small masses orbit with frequency $\\Omega(r)$ about a\nthird, massive body. The masses are coupled by a spring; the natural\nfrequency of the spring-mass system is $\\gamma$. If $\\gamma \\gg\n\\Omega$, the bodies behave like a perturbed harmonic oscillator. But if\n$\\gamma$ is lowered until $\\gamma \\sim \\Omega$ the orbital motion of the\nbodies begins to influence the dynamics, and something interesting\nhappens. The outer mass has higher angular momentum but lower orbital\nfrequency. It is pulled forward in its orbit by the spring; its angular\nmomentum increases, moves to a higher orbit, and lowers its orbital\nfrequency further. This stretches the spring, increases the rate at\nwhich the outer body gains angular momentum, and a runaway ensues. The\nouter body heads outward, acquiring angular momentum from the inner\nbody. The inner body moves in a mirror image of the outer body as it\nloses angular momentum and falls inward.\n\nThere is an exact correspondence between the modes of the spring-mass\nmodel and the MRI. One can think of the masses as fluid elements and the\nspring as magnetic field. The field has characteristic frequency\n$\\gamma \\sim v_A\/\\lambda$, where $\\lambda$ is the separation of the\nmasses and $v_A = B\/\\sqrt{4\\pi\\rho}$ is the Alfv\\'en speed.\n$v_A\/\\lambda \\lesssim \\Omega$ implies instability.\n\nThe simplicity of the dynamics shown in Figure~\\ref{introf2} suggests that the MRI\nis robust. Instability requires the presence of a weak (subthermal,\ni.e. $B^2\/(8 \\pi \\rho) \\lesssim c_s^2$) magnetic field. A stronger\nmagnetic field seems unlikely (it would likely be ejected from the disk\nby magnetic buoyancy), but if it were present it would likely be\nassociated with other, even more powerful instabilities. The MRI also\nrequires an angular velocity that decreases outward ($d\\Omega^2\/d\\ln r <\n0$), which is always satisfied in Keplerian disks ($\\Omega \\propto\nr^{-3\/2}$). The maximum growth rate is $\\sim \\Omega$, independent of\nthe field strength (unless diffusive effects are present). This\nsurprising fact is easily understood once one realizes that the scale\n$\\lambda$ of the instability decreases with the field strength: $\\lambda\n\\sim v_A\/\\Omega$.\n\nWhat does the MRI tell us about $W_{r\\phi}$, the key quantity in the\nevolution equation? Numerical integration of the compressible MHD\nequations shows that the linear MRI initiates nonlinear turbulence.\nIn the turbulent state one can measure the average value of\n\\begin{equation}\nW_{r\\phi} = \\int dz \\left( \\rho v_r \\delta v_\\phi\n- \\frac{B_r B_\\phi}{4\\pi} \\right)\n\\end{equation}\nwhere $\\delta v_\\phi$ is the noncircular azimuthal component of the\nvelocity. There are two distinct contributors to the shear stress:\nhydrodynamic velocity fluctuations, and magnetic field fluctuations,\nsometimes referred to as the Reynolds and Maxwell stresses. Then $\\alpha\n= const. \\times W_{r\\phi}\/(\\Sigma c_s^2)$ (the constant is a matter of\nconvention and takes several values in the literature). Local\nnumerical models yield $\\alpha \\approx 0.01$ \\citep{hgb95,mt95,hgb96,\nmmts97}. Global numerical models yield similar but slightly larger\nvalues \\citep[e.g.][]{arm98,haw00,mhm00,ar01}.\n\nThe presence of turbulence does not guarantee the $W_{r\\phi} > 0$\nnecessary for outward angular momentum transport and accretion. For\nexample, the turbulence associated with vertical convection can produce\n$\\alpha < 0$ \\citep{sb96,cab96}. Also, vortices in nearly incompressible\ndisks produce $\\alpha \\approx 0$ (see Chapter~\\ref{paper3}). The fact\nthat MRI-driven turbulence has $W_{r\\phi} > 0$ is thus a nontrivial result,\nalthough one that might have been anticipated because of the central\nrole of angular momentum exchange in driving the linear MRI.\n\n\\subsection{MRI in Low-Ionization Disks}\\label{MRILowI}\n\nMRI-generated MHD turbulence is the likely angular momentum transport \nmechanism in AGN and in binary systems during outbursts. In portions of \nYSO disks, however, as well as in CV disks and X-Ray transients\nin quiescence \\citep{sgbh00,gm98,men00}, the plasma is cool\nand nearly neutral. The conductivity of the gas is small by\nastrophysical standards and the field is no longer frozen into the gas.\nIn some regions the field may be completely decoupled from the fluid,\njust as the Earth's lower atmosphere is decoupled from the Earth's\nmagnetic field.\n\nLow ionization levels change the field evolution through three separate\neffects: Ohmic diffusion, Hall drift, and ambipolar diffusion.\nFollowing the beautiful discussion of \\cite{bt01} \\citep[see also][]\n{war99,des04}, a measure of the correction to the field evolution\nequation (\\ref{INDUC}) due to Ohmic diffusion is given by the magnetic\nReynolds number $Re_M^{-1}\n\\equiv \\eta\/(v_A H)$, where $\\eta = c^2\/(4\\pi\\sigma_e)$ is the\nresistivity, $c$ is the speed of light and the conductivity $\\sigma_e$\nis proportional to the collision timescale for electrons with neutrals. Then\n\\begin{equation}\nRe_M \\simeq 2 \\times 10^{19} B \\left(\\frac{r}{{\\rm\\,AU}}\\right)^{3\/2}\n\\left(\\frac{n_e}{n}\\right) \\left(\\frac{M_*}{{\\rm\\,M_\\odot}}\\right)^{-1\/2}n^{-1\/2}\n\\end{equation}\nHere $n$ is the neutral number density and $n_e$ is the electron number\ndensity. Ohmic diffusion destroys flux (via reconnection) and converts\nmagnetic energy to thermal energy.\n\nHall drift can be thought of as arising from the relative mean motion of\nthe electrons and ions. The associated correction to the field\nevolution equation is, for conditions appropriate to circumstellar\ndisks, typically comparable to Ohmic diffusion. Hall drift does not\nchange the magnetic energy.\n\nAmbipolar diffusion arises from the relative mean motion of the\nions and neutrals. Unlike Hall drift, it converts magnetic energy\nto thermal energy. The ratio of the ambipolar to Ohmic term $\\sim 5\n\\times 10^{28} B^2 T^{-1\/2} n^{-2}$ \\citep[all cgs units; we have assumed\nthat the number densities of electrons and ions are equal, but see][]{des04}.\nIn low-density environments such as\nthe clouds from which young stars condense, ambipolar diffusion is the\ndominant nonideal effect; one can then think of the field as being\nlocked into the ion-electron fluid, which gradually diffuses through the\nneutrals. At higher densities ambipolar diffusion becomes less dominant,\nand at the highest densities found in disks Ohmic diffusion dominates.\nEvidently the precise variation of the relative importance of these effects\nwith location depends on the variation of ionization fraction, temperature,\nand field strength. In YSO disks, ambipolar diffusion tends to dominate in the\ndisk atmosphere and at $r \\gtrsim 10$ AU.\n\nThe linear theory of the MRI with Ohmic diffusion was first considered\nby \\cite{jin96}. Stability is recovered when the Ohmic diffusion rate\n$\\eta\/\\lambda^2$ exceeds the growth rate of the instability\n$v_A\/\\lambda$. Since $\\lambda \\lesssim H$, this occurs when $\\eta\/(v_A\nH) \\sim 1$. One can avoid expressing the stability condition in terms\nof the unknown field strength $B$, which likely arises through dynamo\naction induced by the MRI, by noting that for a subthermal field $c_s >\nv_A$. Then when $\\eta\/(c_s H) > 1$ the disk is stable, although the MRI\nmay be suppressed at even lower $\\eta$. Circumstellar disks have\n$\\eta\/(c_s H) > 1$, and are therefore stable to the MRI, over a large\nrange in radius \\citep[e.g.][]{gam96}.\n\nThe linear theory of the MRI with ambipolar diffusion was first treated\nby \\cite{bb94}, and reconsidered more recently by\n\\cite{des04,kb04,sw03,sal04}. The natural expectation is that the MRI\ndevelops even in the presence of ambipolar diffusion as long as a\nneutral particle manages to collide with an ion (which can see the\nmagnetic field) at least once per orbit. Assuming common values for the\ncollision strengths and ion mass \\citep[e.g.][]{bt01}, this condition\nbecomes\n\\begin{equation}\nA \\equiv 0.01 \\, n_e \\, (r\/AU)^{3\/2} (M\/{\\rm\\,M_\\odot})^{-1\/2}\n\\gtrsim 1.\n\\end{equation}\nThe more recent round of papers points out that even when $A < 1$ there\nare unstable perturbations within a band of wavevectors\n${\\mbox{\\boldmath $k$}}$ outside the purely axial wavevectors considered\nby \\cite{bb94}. These perturbations evade ambipolar damping by\norienting their magnetic field perturbations perpendicular to both\n${\\mbox{\\boldmath $B$}}$ and ${\\mbox{\\boldmath $k$}}$\n\\citep[see][]{des04}. Instability can thus survive when $A < 1$, albeit\nin a narrow band of wavevectors and at greatly reduced growth rates.\n\nThe linear theory of the MRI with Hall drift has been considered by\n\\cite{war99,bt01,des04}. There are always perturbations that become\nmore unstable as Hall drift is turned on. The maximum growth rate is\nnot affected. The combination of all these nonideal effects, together\nwith a best guess for the disk ionization structure is discussed in\n\\cite{sw03,sal04,des04}.\n\nEarly numerical experiments by \\cite{hgb95} using a scalar resistivity,\nno explicit viscosity, no Hall drift and no ambipolar diffusion\nsuggested that the MRI dynamo fails when $c_s H\/\\eta \\lesssim 10^4$. A\nsimilar but more thorough study by \\cite{fsh00} found similar results.\n\\cite{ss02a,ss02b} considered models that incorporated Ohmic diffusion\nand Hall drift, but not ambipolar diffusion (relevant in some regions of\nthe disk). The most relevant of Sano \\& Stone's models are probably\nthose with net toroidal field or zero net field. Their results\n\\citep[see Figs. 14 and 19 of][]{ss02b} suggest that Ohmic diffusion is\nthe governing nonideal effect; $\\alpha$ drops sharply when $c_s H\/\\eta <\n3 \\times 10^3$, and is only weakly dependent on the Hall parameter.\n\n\\section{Angular Momentum Transport in Weakly-Ionized Disks}\\label{intros4}\n\nAccretion rates inferred from observations of weakly-ionized disks indicate\nthat an enhanced viscosity or some other mechanism for angular momentum transport\nis operating in these disks. As discussed in the previous section, however, these same\ndisks are likely to be stable to the MRI. This leaves open the question of what\ngenerates turbulence in weakly-ionized disks. As long as there was no firm\ntheoretical understanding of the onset of turbulence in disks (ionized or not),\nit was reasonable to assume that all disks are turbulent due to their large\nReynolds numbers. Laboratory shear flows are turbulent above a critical\nReynolds number even though they are stable to infinitesimal perturbations\n(i.e., there is no linear instability to trigger the turbulence), and the\nextrapolation to astrophysical shear flows was a natural one to make. With the\nestablishment of a robust transport mechanism in MRI-induced MHD turbulence,\nthis assumption has come under critical scrutiny. If a mechanism can be\nestablished from first principles for ionized disks, it seems reasonable to\nmaintain the same standard for disks which are too weakly ionized to be\nMHD-turbulent. Although many attempts have been made, to date no robust transport\nmechanism akin to the MRI has been established for low-ionization disks. This\ndissertation investigates three possible mechanisms in detail: gravitational\ninstability, convection, and vortices driving compressive motions. Each of these\nmechanisms is summarized briefly here and discussed in detail in the remainder of\nthe dissertation. Since there are those who continue to argue for turbulence in\ndisks by way of analogy with laboratory shear flows, a brief overview is also given\nof the current state of this controversy.\n\n\\subsection{Gravitational Instability}\n\nGravitational instability arises when self-gravity in the disk overcomes\nthe stabilizing influences of pressure and rotation. The nonlinear\noutcome of this instability is either a gravito-turbulent state of\nmarginal stability or fragmentation of the fluid into bound clumps.\nIf a sustained gravito-turbulent state can be established, then steady\noutward angular momentum transport ensues. Instability tends to form temporary\nspiral enhancements in the density with a trailing orientation,\nand gravity then carries angular momentum along the spiral\n(there is a new term in $W_{r\\phi}$, a ``Newton'' stress given by $\\int\n\\, dz \\, g_r g_\\phi\/(8\\pi G)$, where $g_r$ and $g_\\phi$ are components of the gravitational field). Local numerical experiments exhibit a\ngravitational $\\alpha$ up to $\\sim 1$ \\citep{gam01}. If this state can\nbe maintained steadily throughout the disk, it would provide an effective\nturbulent transport mechanism in weakly-ionized disks \\citep{pac78}.\n\nA sustained gravito-turbulent state cannot be established, however, if\nthe cooling (due to radiation from the surfaces of the disk) is too strong.\nThen the clumps of matter formed by the\ninstability cool before they can collide and heat each other via shocks.\nFragmentation-- the formation of small, bound clumps-- results.\nThe mean cooling time can be used to distinguish a gravito-turbulent disk\nfrom a fragmenting disk:\n\\begin{equation}\\label{COOLT}\n\\tau_c \\equiv \\frac{\\langle U \\rangle}{\\langle \\Lambda \\rangle}.\n\\end{equation}\nwhere $U = \\int dz u$ and $u$ is the internal energy per unit volume,\nand $\\Lambda$ is the cooling function. The brackets $\\langle\\rangle$\nindicate an average over space and time, since the disk may have\nnonuniform density and temperature.\n\nChapter~\\ref{paper1} discusses in detail local numerical experiments\nwhich show that fragmentation occurs when $\\tau_c \\Omega \\lesssim 1$.\\footnote{\nThis result has been been demonstrated in other numerical experiments as well, both\nlocal and global \\citep{gam01,rabb03}.} These experiments also show\nthat fragmentation occurs for a wide range of parameters, indicating that a\ngravito-turbulent state is difficult to sustain. In addition, cooling\ntypically becomes more efficient with an increase in disk radius, making\nan extended, marginally-stable region unlikely. Gravitational instability thus\ndoes not appear to be a likely candidate for a turbulent transport mechanism\nin weakly-ionized disks.\n\n\\subsection{Convection}\n\nAs mentioned in \\S\\ref{MRI}, vertical convection appears to transport angular\nmomentum {\\it inward}, opposite to what is usually required for accretion.\nIndeed, arguments have been made that any incompressive disturbance or\nincompressible turbulence (of which convection is just one example) will\ndrive angular momentum in the wrong direction \\citep{bal00,bal03}.\n\nThe possible role of radial convection in driving angular momentum\ntransport has come to the fore recently with the work of \\cite{kb03} and\n\\cite{klr04} on the ``Global Baroclinic Instability''. One would expect\nthat the combination of weak radial gradients and strong Keplerian shear\nin circumstellar disks would preclude any instabilities due to radial\nconvection, yet \\cite{kb03} found turbulence and angular momentum\ntransport in global hydrodynamic simulations with a modest radial\nequilibrium entropy gradient. The claim in \\cite{klr04} is that this\nactivity, which grows on a dynamical time scale, is the result of a {\\it\nlocal} hydrodynamic instability due to the presence of the global\nentropy gradient.\n\nChapters~\\ref{paper2} and \\ref{paper4} describe analytic and numerical\nwork in a local model that attempts to confirm or refute these unexpected results.\nChapter~\\ref{paper2} describes a local stability analysis in radially-stratified\ndisks, an analysis which uncovers no exponentially-growing instabilities for\ndisks with a Keplerian rotation profile.\nChapter~\\ref{paper4} describes local numerical experiments which attempt\nto uncover any nonlinear instabilities that may be present. Disks with\nKeplerian shear are again found to be stable. It appears, therefore, that\nthe ``Global Baroclinic Instability'' claimed by \\cite{kb03} is either\nglobal or nonexistent.\n\n\\subsection{Vortices}\n\nThe absence of a robust instability mechanism for generating\nhydrodynamic turbulence does not necessarily imply the absence of\ninternal angular momentum transport. Chapter~\\ref{paper3} describes\nnumerical experiments which show significant\nshear stresses associated with finite-amplitude vortices that emit\ncompressive waves and shocks. In these experiments, an initial field of random\nvelocity perturbations with Mach number $\\sim 1$ forms anticyclonic\nvortices that provide an outward flux of angular momentum corresponding\nto an initial $\\alpha \\sim 0.001$ and decaying as $t^{-1\/2}$. These results were\nobtained in a two-dimensional local model, and are likely to be modified\nconsiderably by three-dimensional instabilities, which tend to destroy\ntwo-dimensional vortices \\citep{ker02,bm05}. In addition, these results\nleave open the key question of what generates the initial vorticity.\nBoth of these issues are discussed in more detail in Chapter~\\ref{paper3}.\n\n\\subsection{Nonlinear Hydrodynamic Instability}\n\nFor some, the lack of a well-established mechanism for generating turbulence in\nweakly-ionized disks does not necessarily imply the absence of turbulence (e.g.,\n\\citealt{rz99}). In the first place, our understanding of the onset of turbulence\nin simple laboratory shear flows is still incomplete, despite over a century of\ntheoretical effort. Even when linear theory predicts stability of these flows at\nall values of the Reynolds number, experiments consistently show the onset of\nturbulence above a critical Reynolds number. The failure of linear theory to predict\nthe outcome of experiments indicates that nonlinear instabilities (i.e., instabilities\ndue to finite-amplitude disturbances) are the likely source of turbulence in these\nflows. Perhaps an analogous mechanism operates in weakly-ionized disks. In addition,\nsince nonlinear stability is extremely difficult to {\\it prove} due to the complexity\nof nonlinear dynamics, the question of stability in weakly-ionized disks remains, in\nsome sense, an open question. As discussed in this section, however, no nonlinear\ninstability mechanism has yet been established for a Keplerian shear flow, despite its\napparent similarities with laboratory shear flows. In addition, the two main features \nthat distinguish an accretion disk flow from a laboratory shear flow-- rotational effects\nand the absence of rigid boundaries-- seem to argue for the nonlinear {\\it stability} of\nthe former.\n\nThe laboratory flow that most closely resembles an accretion disk flow is Couette-Taylor\nflow, which is flow between two concentric cylinders. Early theoretical and\nexperimental results for this flow were obtained by Rayleigh, Couette and Taylor\n(see \\citealt{dr81}). Its {\\it linear} stability is governed by the Rayleigh stability\ncriterion, which states that a necessary and sufficient criterion for stability\nto axisymmetric disturbances is that\n\\begin{equation}\\label{RAY}\n\\kappa^2 = \\frac{1}{r^3}\\frac{d}{dr}\\left(r^2\\Omega(r)\\right)^2 \\geq 0,\n\\end{equation}\nwhere $\\kappa^2$ is the square of the epicyclic frequency (also known as the Rayleigh \ndiscriminant). For a Rayleigh-stable flow, fluid elements displaced from circular orbits\nwill undergo epicycles about their equilibrium velocity at a frequency $\\kappa$.\nThe stability criterion (\\ref{RAY}) is equivalent to the requirement that the specific\nangular momentum of the mean flow decrease with radius.\n\nAnother laboratory flow that has been studied in depth is planar shear flow, also known\nas plane Couette flow \\citep{dr81}. This is flow between two parallel walls, the laminar\nstate of which is a streamwise (parallel to the walls) velocity that varies\nlinearly with distance from the walls. A necessary condition for the {\\it linear}\ninstability of a parallel shear flow with an arbitrary shear profile (i.e., an\nequilibrium velocity that is an arbitrary function of the coordinate perpendicular\nto the direction of flow) is that there be an inflection point in the equilibrium\nvelocity profile. Since a linear shear profile does not meet this condition, plane\nCouette flow is also predicted to be stable based upon linear theory.\n\nBoth Couette-Taylor flow and plane Couette flow show the onset of turbulence above a critical\nReynolds number, against the predictions of linear theory. Theoretical efforts to\nexplain this transition to turbulence have focused on 1) the transient amplification\nof linear disturbances coupled with a nonlinear feedback mechanism to close the amplifier\nloop (e.g., \\citealt{bt97}); 2) self-sustaining nonlinear processes that are triggered at\nfinite amplitude and are therefore not treatable by a linear analysis (e.g., \\citealt{wal97});\nor 3) some combination of nonlinear mechanisms and secondary linear instabilities (e.g.,\n\\citealt{fi93}). Reviews of these mechanisms can be found in \\cite{bo88}, \\cite{gross00}\nand \\cite{rem03}. All of them include some aspects of the nonlinear dynamics and are\ngenerically referred to as nonlinear instabilities. While a discussion of their detailed\noperation is not necessary for the purposes of this dissertation, it is important to note\nthat none of them has provided a complete understanding of the transition to turbulence\nin laboratory shear flows.\n\nThe application of these ideas to accretion disks has continued since the discovery of\nthe MRI \\citep{zahn91,dk92,dk93,dub93,ik01,rich01,rddz01,long02,cztl03,rich03,klr04,amn04,\nman04,rd04,yeck04,ur04,hdh05,man05,unrs05}. Much of this work has focused on the mechanism\nof transient amplification of linear disturbances coupled with nonlinear feedback, since\nthere are local nonaxisymmetric vortical perturbations which can experience an arbitrary\namount of transient growth at infinite Reynolds number (a result that was recognized as early as 1907 by Orr; see \\citealt{shep85}). These solutions are discussed in\ndetail in Chapter~\\ref{paper2}, where it is shown (\\S\\ref{pap2eis}) that an isotropic\nsuperposition of these perturbations has an energy that is constant with time. This seems\nto indicate that any potential mechanism for the onset of hydrodynamic turbulence in disks\nwould be an entirely nonlinear process. Only nonlinear simulations can fully answer this\nquestion, however, and a full investigation of the effects of transient amplification over\na wide range of initial perturbation amplitudes and spectra has not yet been made. To date,\nhowever, no numerical simulations have demonstrated a transition to turbulence from\ninfinitesimal perturbations, and the results of \\S\\ref{pap2eis} indicate that such a\ntransition may not occur for a physically-realistic set of low-amplitude perturbations.\n\nEarly simulations of MHD turbulence in MRI-unstable disks \\citep{hgb95,hgb96} found that\n1) when the magnetic fields were turned off the turbulence decayed away and 2) when\nrotational effects were removed, thereby converting the Keplerian flow into plane Couette\nflow, the turbulence increased and the magnetic field decayed away. While advocates of\nnonlinear instabilities in disk flows will often attribute the absence of turbulence in\nhydrodynamic simulations to numerical diffusion (e.g., \\citealt{long02}), this latter\nresult confirms the ability of these simulations to identify a nonlinear instability. In\naddition, \\cite{bal04} has argued that due to a nonlinear scale invariance of the equations\ngoverning the local disk flow, any local instabilities that are present should be present\nat all scales and therefore not require high resolutions for their manifestation in local\nnumerical simulations.\n\nTwo subsequent comprehensive studies of nonlinear instabilities in\nlocal numerical simulations \\citep{bhs96,hbw99} have confirmed these results. Both\nKeplerian and plane Couette flows were investigated, using codes with very different\ndiffusive properties, and rotational effects were cited as the key stabilizing factor in\ndisks. One of the mechanisms for nonlinear instability in plane Couette flow is the\ngeneration of streamwise vortices by the shear, resulting in a secondary instability due\nto inflections in the spanwise (across the mean flow) direction. The epicyclic motions\nof fluid elements in a rotating flow prevent these streamwise vortices from developing.\nWhen rotational effects are removed, the nonlinear instability of planar shear flow is\nreadily recovered.\n\nBefore the work of \\cite{hbw99}, the only Couette-Taylor (rotating-flow) experiments that\nshowed the onset of turbulence had shear profiles that were sufficiently non-Keplerian to\nmore closely resemble plane Couette flow than a rotationally-dominated flow (the shear\nprofiles were near to linear instability, expression [\\ref{RAY}]). More recent experiments,\nhowever, have shown the onset of turbulence in Couette-Taylor flow with a Keplerian\nshear profile \\citep{rich01,rddz01}, thus indicating the presence of a nonlinear instability.\nThese results, however, may simply highlight another key difference between laboratory\nshear flows and disk flows, namely the presence or absence of rigid boundaries. As noted in\n\\cite{go05}, early Couette-Taylor experiments revealed the importance of end effects in\ndisturbing the laminar flow. Torque measurements in a system with an aspect ratio of $23$\n(the ratio of the cylinder lengths to the width of the gap between the cylinders) and an end\nplate corotating with the outer cylinder were $100\\%$ larger than measurements with the end\nplate stationary. An aspect ratio $\\gtrsim 40$ was required to minimize the end effects.\nThe experimental setup described in \\cite{rddz01} has an aspect ratio of $25$.\n\n\\cite{go05} proposed a model for the nonlinear dynamics of turbulent shear flows and also used\ntheir model to predict the onset of linear and nonlinear instability in shear flows both with\nand without rotation. The model accounts for many aspects of laboratory shear flow experiments.\nFor reasonable model parameters, the model predicts nonlinear {\\it stability} for Keplerian\nshear flows in the absence of boundaries and nonlinear {\\it instability} for a wall-bounded\nexperiment with a Keplerian shear profile at sufficiently large Reynolds numbers. This is \nanother indication that the results observed by \\cite{rddz01} may be due to boundary effects.\n\n\\section{Local Model}\\label{intros5}\n\nSince the focus of this dissertation is on local mechanisms for angular momentum transport,\nall the analytic and numerical results are obtained in a local model of an accretion disk.\nSuch a model can be obtained by a rigorous expansion of the fluid equations in $|{\\bx}|\/r$,\nwhere ${\\bx} = (x,y,z) \\equiv (r-R_o, R_o(\\phi - \\phi_o - \\Omega(R_o) t),z) \\sim O(H)$ are\nthe local Cartesian coordinates of the fluid with respect to a fiducial radius $R_o$ and\nfiducial angle $\\phi_o$ (see Figure~\\ref{introf3}). Since the local coordinates are assumed\nto vary on the order of the disk scale height $H$, the local model expansion is only valid for\nthin disks with $H\/r \\ll 1$ (see Table~\\ref{introt1}). This local frame is corotating with\nthe fluid in the disk at a distance $R_o$ from the central object and at a frequency\n$\\Omega(R_o)$, the local rotation frequency of the disk. Local curvature is neglected, but\ncentrifugal and Coriolis forces are retained. The additional simplifying assumption of an\ninfinitesimally thin disk is made, which implies a vertical integration of the fluid variables.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=2.8in,clip]{introf3.eps}\n\\caption{Local coordinate system at $t = 0$.}\n\\label{introf3}\n\\end{figure}\n\nThe resulting equations of motion for a fluid in the local model (including self-gravity) are\n\\begin{equation}\\label{EQM}\n\\partial_t \\bv + (\\bv \\cdot \\bnabla) \\bv = -\\frac{1}{\\Sigma}\\bnabla \\left(P + \\frac{B^2}{8\\pi}\n\\right) + \\frac{({\\bB}\\cdot \\bnabla){\\bB}}{4\\pi \\rho} - \\bnabla\\phi - 2\\bO{\\bf \\times} \\bv +\n3 \\Omega^2 x \\ex,\n\\end{equation}\nwhere $\\bv$ is the fluid velocity with respect to the rotating frame, $B$ is the magnetic field,\n$P$ is the two-dimensional pressure, $\\Sigma$ is the column density and $\\phi$ is the disk\npotential with the time-steady axisymmetric component removed. The last two terms on the\nright-hand side of equation (\\ref{EQM}) incorporate the effects of Coriolis and centrifugal forces as well as the gravitational\nacceleration due to the central point mass and the time-steady axisymmetric component of the\ndisk. These equations of motions are valid for a disk system in which the gravitational\npotential is dominated by the central object; the fluid in such a disk follows a Keplerian\nrotation curve, $v_\\phi \\sim r^{-1\/2}$.\n\nThe continuity, internal energy and induction equations retain their usual form:\n\\begin{equation}\n\\partial_t \\Sigma + (\\bv \\cdot \\bnabla) \\Sigma + \\Sigma \\bnabla \\cdot \\bv = 0,\n\\end{equation}\n\\begin{equation}\\label{EQE}\n\\partial_t U + (\\bv \\cdot \\bnabla) U + (U + P) \\bnabla \\cdot \\bv + \\Lambda = 0,\n\\end{equation}\n(where $U$ is the internal energy per unit area and $\\Lambda$ is the cooling function) and\n\\begin{equation}\\label{INDUC}\n\\partial_t {\\bB} - \\bnabla {\\bf \\times} (\\bv {\\bf \\times} {\\bB}) = 0.\n\\end{equation}\nEquations (\\ref{EQM}) through (\\ref{INDUC}) are the equations of compressible ideal\nmagnetohydrodynamics (MHD) in the local model. Magnetic fields are assumed to be dynamically\nunimportant for most of research described in this dissertation, in which case equations\n(\\ref{EQM}) through (\\ref{EQE}) reduce to the equations of compressible hydrodynamics.\nThe disk self-gravity ($\\phi$ in equation [\\ref{EQM}]) and explicit cooling ($\\Lambda$ in\nequation [\\ref{EQE}]) are neglected except in the work discussed in Chapter~\\ref{paper1}.\nIn Chapters~\\ref{paper1} - \\ref{paper4}, the fluid is assumed to obey an ideal-gas equation of\nstate, $P = (\\gamma - 1) U$, where $\\gamma$ is the adiabatic index. Chapter~\\ref{paper3}\nassumes an isothermal equation of state $P = c_s^2 \\Sigma$.\n\nWith constant density and pressure in equilibrium, an exact steady-state solution to\nequations (\\ref{EQM}) through (\\ref{EQE}) is $\\bv_o = -\\frac{3}{2} \\Omega x \\ey$. This uniform shear\nvelocity is a manifestation of differential rotation of the fluid in the disk. As a result, \nthe (two-dimensional) local model is referred to as the ``shearing sheet''. The numerical \nimplementation of the shearing sheet requires a careful treatment of the boundary conditions\nin the radial direction. These boundary conditions are described in detail in \\cite{hgb95}.\nIn brief, one uses strictly periodic boundary conditions in $y$ and shearing-periodic boundary\nconditions in $x$. The latter is done by enforcing periodic boundary conditions in the radial\ndirection followed by an advection of the boundary fluid due to the shear. This assumes that\nthe shearing sheet is surrounded by identical boxes that are strictly periodic initially, with\na large-scale shear flow present across all the boxes.\n\nThe shearing-sheet equations are evolved using a ZEUS-based scheme \\citep{sn92}: a time-explicit,\noperator-split, finite-difference method on a staggered grid which uses an artificial viscosity\nto capture shocks. An important modification of the standard shearing sheet, introduced by\n\\cite{mass00}, is the splitting of the overall shear velocity from the rest of the flow.\nThis overcomes a practical limitation of the standard shearing sheet, which is the small\nCourant-limited time step imposed by the large shear velocities at the edges of the sheet;\nfor numerical stability of grid-based schemes the Courant condition requires time steps to\nbe lower than the grid spacing divided by the maximum velocity on the grid. The larger the\nbox, the more severe this limitation becomes. Separating out the shear removes this limitation\nand allows one to increase the size of the shearing sheet arbitrarily. This separation is done\nby replacing $v_y$ by $v_o + \\delta v_y$ in the fluid equations, and then evolving $\\delta v_y$;\nthis can be done because there is no evolution of $v_o$ directly ($\\partial_t v_o = 0$ and\n$\\bnabla v_o = {\\rm{constant}}$). Advection of other fluid variables by $v_o$ is done by\nsplitting the distance over which the fluid is sheared into an integral and fractional number\nof grid zones: the fluid variables are simply shifted an integral number of zones and then\nadvected in the usual manner for the remaining fractional part (which does not require a\nhigher effective velocity than any other part of the flow).\n\n\\section{Discussion}\\label{intros6}\n\nWhile a rigorous proof of the stability of weakly-ionized disks may well be impossible,\nthe results of this dissertation add to the already strong evidence against a turbulent\nangular momentum transport mechanism in weakly-ionized disks. Gravitational instability likely\nresults in fragmentation, radial convection is suppressed by differential rotation and\ntwo-dimensional vortices, which provide a decaying flux of angular momentum, are likely to be\nunstable in three dimensions. Evidence against a local, nonlinear, purely hydrodynamic\ninstability is mounting.\n\nAccretion may be driven globally by a magneto-centrifugal wind \\citep{bp82} or tidally-induced\nspiral waves \\citep{lars89,ls91}, or locally via spiral waves excited by planets embedded in\nthe disk \\citep{gr01,sg04}. There also exist global instabilities (e.g., \\citealt{pp84,pp85})\nthat result in a small amount of turbulence and angular momentum transport (e.g.,\n\\citealt{haw87}). In addition, there are instabilities associated with the dust layer in\nYSO disks (e.g., \\citealt{gl04}) that will generate some amount of turbulence in those\nsystems. While one or more of these mechanisms may play a role in transporting angular\nmomentum in certain systems, their dependence upon global structure or other special features\nin order to operate makes their broad application to weakly-ionized disks doubtful. \nAlternatively, weakly-ionized disks may simply be inactive except in ionized surface layers \n\\citep{gam96}.\n\nA detailed discussion of these possibilities is beyond the scope of this dissertation, but a\nbrief discussion of layered accretion is given in Chapter~\\ref{conclusion}, along with some \nproposals for future modeling based upon that idea. Chapter~\\ref{conclusion} also summarizes\nthe main results and implications of this work, and provides some direction as to where to go\nfrom here in the search for a turbulent angular momentum transport mechanism in weakly-ionized\naccretion disks. In addition, proposals are made for future investigations of the properties\nof turbulent stresses in ionized disks, with a view towards incorporating these properties in\nadvanced, physically-motivated disk models.\n\n\\end{spacing}\n\n\n\\chapter{Nonlinear Outcome of Gravitational Instability in\nDisks with Realistic Cooling}\\label{paper1}\n\n\\begin{spacing}{1.5}\n\n\\section{Chapter Overview}\n\nWe consider the nonlinear outcome of gravitational instability in\noptically-thick disks with a realistic cooling function. We use a\nnumerical model that is local, razor-thin, and unmagnetized. External\nillumination is ignored. Cooling is calculated from a one-zone model\nusing analytic fits to low temperature Rosseland mean opacities. The\nmodel has two parameters: the initial surface density $\\Sigma_o$ and\nthe rotation frequency $\\Omega$. We survey the parameter space and find: (1)\nThe disk fragments when $\\tce \\Omega \\sim 1$, where $\\tce$ is an\neffective cooling time defined as the average internal energy of the model\ndivided by the average cooling rate. This is consistent with earlier\nresults that used a simplified cooling function. (2) The initial\ncooling time $\\tco$ for a uniform disk with Toomre stability parameter $Q = 1$ can differ by orders\nof magnitude from $\\tce$ in the nonlinear outcome. The difference is\ncaused by sharp variations in the opacity with temperature. The\ncondition $\\tco \\Omega \\sim 1$ therefore does not necessarily indicate\nwhere fragmentation will occur. (3) The largest difference\nbetween $\\tce$ and $\\tco$ is near the opacity gap, where dust is absent\nand hydrogen is largely molecular. (4) In the limit of strong\nillumination the disk is isothermal; we find that an isothermal version\nof our model fragments for $Q \\lesssim 1.4$. Finally, we discuss some physical\nprocesses not included in our model, and find that most are likely to\nmake disks more susceptible to fragmentation. We conclude that disks\nwith $\\tce\\Omega \\lesssim 1$ do not exist.\\footnote{Published in ApJ\nVolume 597, Issue 1, pp. 131-141. Reproduction for this dissertation is \nauthorized by the copyright holder.}\n\n\\section{Introduction}\n\nThe outer regions of accretion disks in both active galactic nuclei\n(AGN) and young stellar objects (YSO) are close to gravitational\ninstability (for a review see, for AGN: \\citealt{sbf90}; YSOs:\n\\citealt{al93}). Gravitational instability can be of central importance in\ndisk evolution. In some disks, it leads to the efficient redistribution\nof mass and angular momentum (e.g. \\citealt{lar84,lr96,gam01}). In\nother disks, gravitational instability leads to fragmentation and the\nformation of bound objects. This may cause the truncation of\ncircumnuclear disks \\citep{good03}, or the formation of planets (e.g.\n\\citealt{boss97}, and references therein).\n\nWe will restrict attention to disks whose potential is dominated by the\ncentral object, and whose rotation curve is therefore approximately\nKeplerian. Gravitational instability to axisymmetric perturbations sets in when the sound\nspeed $c_s$, the rotation frequency $\\Omega$, and the surface density\n$\\Sigma$ satisfy\n\\begin{equation}\\label{QDEF}\nQ \\equiv {\\frac{c_s\\Omega}{\\pi G\\Sigma}} < Q_{crit} \\simeq 1\n\\end{equation}\n\\citep{toom64,glb65}. Here $Q_{crit} = 1$ for a ``razor-thin'' (two-\ndimensional) fluid disk model of the sort we will consider below, and\n$Q_{crit} = 0.676$ for a finite-thickness isothermal disk \\citep{glb65}. \\footnote{For global \nmodels with radial structure, nonaxisymmetric instabilities typically set in for\nslightly larger values of $Q$ (see \\citealt{boss98} and references therein).}\nThe instability condition (\\ref{QDEF}) can be rewritten, for a disk with\nscale height $H \\simeq c_s\/\\Omega$, around a central object of mass\n$M_*$,\n\\begin{equation}\\label{THICKCRIT}\nM_{disk} \\gtrsim \\frac{H}{r} M_*,\n\\end{equation}\nwhere $M_{disk} = \\pi r^2 \\Sigma$. For YSO disks $H\/r \\sim 0.1$ and\nthus a massive disk is required for instability. AGN disks are expected\nto be much thinner. The instability condition can be rewritten in a\nthird, useful form if we assume that the disk is in a steady state and its\nevolution is controlled by internal (``viscous'') transport of angular\nmomentum. Then the accretion rate $\\dot{M} = 3\\pi \\alpha c_s^2\n\\Sigma\/\\Omega$, where $\\alpha \\lesssim 1$ is the usual dimensionless\nviscosity of \\cite{ss73}, and\n\\begin{equation}\\label{INSTCRIT}\n\\dot{M} \\gtrsim \\frac{3 \\alpha c_s^3}{G}\n= 7.1 \\times 10^{-4}\\, \\alpha\n\\left( \\frac{c_s}{1\\kms}\\right)^3 \\,\\msun \\yr^{-1}\n\\end{equation}\nimplies gravitational instability (e.g. \\cite{sbf90}). Disks\ndominated by external torques (e.g. a magnetohydrodynamic [MHD] wind)\ncan have higher accretion rates (but not arbitrarily higher; see\n\\citealt{good03}) while avoiding gravitational instability.\n\nFor a young, solar-mass star accreting from a disk with $\\alpha =\n10^{-2} $ at $10^{-6} \\msun \\yr^{-1}$, equation (\\ref{INSTCRIT}) implies\nthat instability occurs where the temperature drops below $17 \\K$.\nDisks may not be this cold if the star is located in a warm molecular\ncloud where the ambient temperature is greater than $17 \\K$, or if the\ndisk is bathed in scattered infrared light from the central star\n(although there is some evidence for such low temperatures in the solar\nnebula, e.g. \\citealt{owen99}). If the vertically-averaged value of\n$\\alpha$ is small and internal dissipation is confined to surface\nlayers, as in the layered accretion model of \\cite{gam96}, then\ninstability can occur at higher temperatures, although equation\n(\\ref{THICKCRIT}) still requires that the disk be massive.\n\nAGN disk heating is typically dominated by illumination from a central\nsource. The temperature then depends on the shape of the disk. If the\ndisk is flat or shadowed, however, and transport is dominated by\ninternal torques, one can apply equation (\\ref{INSTCRIT}). For example,\nin the nucleus of NGC 4258 \\citep{miy95} the accretion rate may be as\nlarge as $10^{-2} \\msun \\yr^{-1}$ \\citep{las96,gnb99}. Equation\n(\\ref{INSTCRIT}) then implies that instability sets in where $T < 10^4\n(\\alpha\/10^{-2}) \\K$. If the disk is illumination-dominated then $Q$\nfluctuates with the luminosity of the central source.\n\nIn a previous paper \\citep{gam01}, one of us investigated the effect of\ngravitational instability in cooling, gaseous disks in a local model. A\nsimplified cooling function $\\Lambda$ was employed in these simulations,\nwith a fixed cooling time $\\tco$:\n\\begin{equation}\n\\Lambda = -\\frac{U}{\\tco},\n\\end{equation}\nwhere $U \\equiv$ the internal energy per unit area. Disk fragmentation\nwas observed for $\\tco\\Omega \\lesssim 3$. The purpose of this\npaper is to investigate gravitational instability in a local model with\nmore realistic cooling. \n\nSeveral recent numerical experiments have included cooling, as opposed\nto isothermal or adiabatic evolution, and we can ask whether these\nresults are consistent with \\cite{gam01}. \\cite{nbr00} studied a global\ntwo-dimensional (thin) SPH model in which the vertical density and\ntemperature structure is calculated self-consistently and each particle\nradiates as a blackbody at the surface of the disk. The initial\nconditions at a radius corresponding to the minimum initial value of Q\n($\\sim 1.5$) for these simulations were $\\Sigma_o \\approx 50\n{\\rm\\,g\\,cm^{-2}}, \\Omega \\approx 8 \\times 10^{-10} {\\rm \\,s^{-1}}$; the\ninitial cooling time under these circumstances is $\\tco \\approx 250 \\,\n\\Omega^{-1}$, so fragmentation is not expected and is not observed.\n\n\\cite{dmph01} consider a global three dimensional (3D) Eulerian\nhydrodynamics model in which the volumetric cooling rate varies with\nheight above the midplane so as to preserve an isentropic vertical\nstructure. The cooling time is fixed at each radius. Their cooling\ntime $\\gtrsim 10 \\Omega^{-1}$ at all radii, so fragmentation is not expected\nbased on the criterion of \\cite{gam01}. The simulations show structure\nformation due to gravitational instabilities but not fragmentation.\n\n\\cite{rabb03} consider a global 3D SPH model with a cooling time that is\na fixed multiple of $\\Omega^{-1}(r)$. They find that their disk\nfragments when $\\tco \\approx 3 \\Omega^{-1}$ and $M_{disk} = 0.1 M_*$.\nFor a more massive disk ($M_{disk} = 0.25 M_*$), fragmentation occurred\nat somewhat higher cooling times ($\\tco \\approx 10 \\Omega^{-1}$). This\nis effectively a global generalization of the local model problem\nconsidered by \\cite{gam01}. The fact that the results are so consistent\nsuggests that the local, thin approximation used in \\cite{gam01} and\nhere give a reasonable approximation to a global outcome.\n\n\\cite{mqws02} consider a global three dimensional SPH model of a\ncircumstellar disk. Explicit cooling is not included, but the equation\nof state switches from isothermal to adiabatic when gravitational\ninstability begins to set in. This is designed to account for the\ninefficient cooling of dense, optically thick regions. Fragmentation is\nobserved. Realistic cooling can have a complex influence on disk\nevolution, and it is not clear that switching between isothermal and\nadiabatic behavior ``brackets'' the outcomes that might be obtained when\nfull cooling is used.\n\nOther notable recent work, such as that by \\cite{boss02}, includes strong\nradiative heating in the sense that the effective temperature of the\nexternal radiation field $T_{irr}$ is comparable to or larger than the\ndisk midplane temperature $T_c$. In the limit that $T_{irr} \\ll T_c$ we\nrecover the limit considered here and in \\cite{gam01}; in the limit that\n$T_{irr} \\gg T_c$ the disk is effectively isothermal.\n\nThe plan of this paper is as follows. In \\S\\ref{pap1s2} we describe the model,\nwith a detailed description of the cooling function given in \\S\\ref{pap1s3}. The\nresults of numerical experiments are described in \\S\\ref{pap1s4}. Conclusions are\ngiven in \\S\\ref{pap1s5}.\n\n\\section{Model}\\label{pap1s2}\n\nThe model we use here is identical to that used in \\cite{gam01} in every\nrespect except that we use a more complicated cooling function. To make\nthe description more self-contained, we summarize the basic equations of\nthe model here. The model is local, in the sense that it considers a\nregion of size $L$ where $L\/r_o \\ll 1$ and $r_o$ is a fiducial radius. We\nuse a {\\it local Cartesian} coordinate system $x \\equiv r - r_o$ and $y\n\\equiv (\\phi - \\Omega t) r_o$, where $r,\\phi$ are the usual cylindrical\ncoordinates and $\\Omega$ is the orbital frequency at $r_o$. The model is also\nthin in the sense that matter is confined entirely to the plane of the disk.\n\nUsing the local approximation one can perform a formal expansion of the\nequations of motion in the small parameter $L\/r_o$. The resulting equations of\nmotion read, where $\\bv$ is the velocity, $P$ is the (two-dimensional)\npressure, and $\\phi$ is the gravitational potential with the time-steady\naxisymmetric component removed:\n\\begin{equation}\n\\frac{D\\bv}{D t} = -\\frac{\\bnabla P}{\\Sigma} - 2\\bO\\times\\bv\n + 3\\Omega^2 x \\ex - \\bnabla\\phi.\n\\end{equation}\nFor constant pressure and surface density, $\\bv = -\\frac{3}{2}\\Omega x\n\\ey$ is an equilibrium solution to the equations of motion. This linear\nshear flow is the manifestation of differential rotation in the local\nmodel.\n\nThe equation of state is\n\\begin{equation}\nP = (\\gamma - 1) U,\n\\end{equation}\nwhere $P$ is the two-dimensional pressure and $U$ the two-dimensional\ninternal energy. The two-dimensional (2D) adiabatic index $\\gamma$ can\nbe mapped to a 3D adiabatic index $\\Gamma$ in the low-frequency (static)\nlimit. For a non-self-gravitating disk $\\gamma = (3\\Gamma - 1)\/(\\Gamma +\n1)$ (e.g. \\citealt{ggn86,osa92}). For\na strongly self-gravitating disk, one can show that $\\gamma = 3 - 2\/\\Gamma$. We\nadopt $\\Gamma = 7\/5$ throughout, which yields $\\gamma = 11\/7$.\n\nThe internal energy equation is\n\\begin{equation}\n\\frac{\\del U}{\\del t} + \\nabla \\cdot (U \\bv) =\n\t-P\\bnabla\\cdot\\bv - \\Lambda,\n\\end{equation}\nwhere $\\Lambda = \\Lambda(\\Sigma,U,\\Omega)$ is the cooling function,\nfully described below. Notice that there is no heating term; heating is\ndue solely to shocks. Numerically, entropy is increased by artificial\nviscosity in shocks.\n\nThe gravitational potential is determined by the razor-thin disk Poisson\nequation:\n\\begin{equation}\n\\nabla^2\\phi = 4\\pi G \\Sigma \\, \\delta(z).\n\\end{equation}\nFor a single Fourier component of the surface density $\\Sigma_{\\bk}$\nthis has the solution\n\\begin{equation}\n\\phi = -\\frac{2\\pi G}{|\\bk|} \\Sigma_{\\bk} e^{i \\bk\\cdot{\\bf x}\n\t- |k z|}.\n\\end{equation}\nA finite-thickness disk has weaker self-gravity, but this does not\nqualitatively change the dynamics of the disk in linear theory\n\\citep{glb65}.\n\nWe integrate the governing equations using a self-gravitating\nhydrodynamics code based on ZEUS \\citep{sn92}. ZEUS is a\ntime-explicit, operator-split, finite-difference method on a staggered\nmesh. It uses an artificial viscosity to capture shocks. Our\nimplementation has been tested on standard linear and nonlinear\nproblems, such as sound waves and shock tubes. We use the ``shearing\nbox'' boundary conditions, described in detail by \\cite{hgb95}, and solve\nthe Poisson equation using the Fourier transform method, modified for\nthe shearing box boundary conditions. See \\cite{gam01} for further\ndetails on boundary conditions, numerical methods and tests.\n\nThe numerical model is always integrated in a region of size $L \\times\nL$ at a numerical resolution of $N \\times N$. In linear theory the disk\nis most responsive at the critical wavelength $2c_s^2\/G\\Sigma_o$.\\footnote{The\nwavelength corresponding to the minimum in the dispersion\nrelation for axisymmetric waves.}\nWe have checked the dependence of the outcome on $L$ and\nfound that as long as $L \\gtrsim 2c_s^2\/G\\Sigma_o$ the outcome does\nnot depend on $L$. We have also checked the dependence of the outcome\non $N$ and found that the outcome is\ninsensitive to $N$, at least for the models with $N \\geq 256$ that we use.\n\n\\section{Cooling Function}\\label{pap1s3}\n\nOur cooling function is determined from a one-zone model for the\nvertical structure of the disk. The disk cools at a rate per unit area\n\\begin{equation}\n\\Lambda \\equiv 2 \\sigma T_e^4,\n\\end{equation}\nwhich defines the effective temperature $T_e$. The cooling function\ndepends on the heat content of the disk and how that content is\ntransported from the disk interior to the surface: by radiation,\nconvection, or perhaps some more exotic form of turbulent transport such\nas MHD waves. Low temperature disks are expected to be convectively\nunstable (e.g. \\citealt{cam78,lp80}). \\cite{cass93} has argued,\nhowever, that the radiative heat flux in an adiabatically-stratified\ndisk is comparable to the heat dissipated by turbulence (in an\n$\\alpha$-disk model), suggesting that convection is incapable of\nradically altering the vertical structure of the disk. We will consider\nonly radiative transport.\n\nIf the disk is optically thick in the Rosseland mean sense, so that\nradiative transport can be treated in the diffusion approximation, then\n\\citep{hub90}\n\\begin{equation}\\label{TEFF}\nT_e^4 = \\frac{8}{3}\\frac{T_c^4}{\\tau} \n\\end{equation}\nwhere $\\tau$ is the Rosseland mean optical depth and $T_c$ is the\ncentral temperature. We will assume that $T_c \\approx T$, where\n\\begin{equation}\\label{TEMP}\nT = \\frac{\\mu m_p c_s^2}{\\gamma k_B},\n\\end{equation}\nand\n\\begin{equation}\\label{CS2}\nc_s^2 = \\gamma(\\gamma - 1)\\frac{U}{\\Sigma},\n\\end{equation}\nwhich follows from the equation of state and the assumption that the\nradiation pressure is small (we have verified that this is never\nseriously violated). Here $k_B$ is Boltzmann's constant, $m_p$ is the\nproton mass, and $\\mu$ is the mean mass per particle, which we have set\nto $2.4$ in models with initial temperature below the boundary between\nthe grain-evaporation opacity and molecular opacity and $\\mu = 0.6$ in\nmodels with initial temperature above the boundary.\n\nThe optical depth is\n\\begin{equation}\n\\tau \\equiv \\int_0^\\infty \\, dz \\, \\kappa(\\rho_z, T_z) \\rho_z\n\\end{equation}\nwhere $\\kappa$ is the Rosseland mean opacity, $\\rho_z$ and $T_z$ are\nlocal density and temperature, and $z$ is the height above the midplane.\nFollowing the usual one-zone approximation,\n\\begin{equation}\n\\int_0^\\infty \\, dz \\, \\kappa(\\rho_z, T_z) \\rho_z \\approx\nH \\kappa(\\bar{\\rho},\\bar{T}) \\bar{\\rho}\n\\end{equation}\nwhere the overbar indicates a suitable average and $H \\approx\nc_s(T)\/\\Omega$ is the disk scale height (we ignore the effects of\nself-gravity on the disk scale height, which is valid when locally $Q\n\\gtrsim 1$). Taking $\\bar{T} \\approx T$ and $\\bar{\\rho} \\approx\n\\Sigma\/(2 H)$ then gives a final, closed expression for $\\Lambda$.\n\nWe have adopted the analytic approximations to the opacities provided by\n\\cite{bl94}. These opacities are dominated by, in order of increasing\ntemperature: grains with ice mantles, grains without ice mantles,\nmolecules, H$^-$ scattering, bound-free\/free-free absorption and\nelectron scattering. The molecular opacity regime is commonly called the\n{\\it opacity gap}; it is too hot for dust, but too cold for H$^-$\nscattering to contribute much opacity. The opacity can be as much as\n$4$ orders of magnitude smaller than the $\\sim 5 \\gm \\cm^{-2}$ typical\nof the dust-dominated opacity regime. It turns out that this feature\nplays a significant role in the evolution of gravitationally-unstable\ndisks.\n\nTo sum up, the cooling function is\n\\begin{equation}\\label{COOL}\n\\Lambda(\\Sigma,U,\\Omega) = \\frac{16}{3} \\frac{\\sigma T^4}{\\tau}.\n\\end{equation}\nFor a power-law opacity of the form $\\kappa = \\kappa_0 \\rho^a T^b$, this\nimplies that\n\\begin{equation}\n\\Lambda \\sim \\Sigma^{-5 - 3 a\/2 + b} U^{4 + a\/2 - b}.\n\\end{equation}\nFrom this it follows that the cooling time $\\tc \\equiv U\/\\Lambda$\nscales as\n\\begin{equation}\n\\tc \\sim \\Sigma^{5 + 3 a\/2 - b} U^{-3 - a\/2 + b}.\n\\end{equation}\nIf the disk evolves quasi-adiabatically (as it does if the cooling time\nis long compared to the dynamical time) then $U \\sim \\Sigma^\\gamma$ and\n\\begin{equation}\n\\tc \\sim \\Sigma^{5 - 3 \\gamma + (a\/2) (3 - \\gamma) + b (\\gamma - 1)}.\n\\end{equation}\n\n\\begin{deluxetable}{lccc}\n\\tablecolumns{4}\n\\tablewidth{0pc}\n\\tabcolsep 0.5truecm\n\\tablecaption{Scaling Exponent for Cooling Time as a Function of Surface Density \\label{pap1t1}}\n\\tablehead{Opacity Regime & a & b & Exponent}\n\\startdata\nIce grains & 0 & 2 & 10\/7 \\\\\nEvaporation of ice grains & 0 & -7 & -26\/7 \\\\\nMetal grains & 0 & 1\/2 & 4\/7 \\\\\nEvaporation of metal grains & 1 & -24 & -89\/7 \\\\\nMolecules & 2\/3 & 3 & 52\/21 \\\\\nH$^-$ scattering & 1\/3 & 10 & 131\/21 \\\\\nBound-free and free-free & 1 & -5\/2 & -3\/7 \\\\\nElectron scattering & 0 & 0 & 2\/7 \\\\\n\\enddata\n\\end{deluxetable}\n\n\\noindent\nTable~\\ref{pap1t1} gives a list of values for this scaling exponent for our nominal\nvalue of $\\gamma = 11\/7$. Notice that, when ice grains or metal grains\nare evaporating, and in the bound-free\/free-free opacity regime, cooling\ntime {\\it decreases} as surface density {\\it increases}.\n\nOur cooling function is valid in the limit of large optical depth ($\\tau\n\\gg 1$). Since the disk becomes optically thin at some locations in the\ncourse of a typical run, we must modify this result so that the cooling\nrate does not diverge at small optical depth. A modification that\nproduces the correct asymptotic behavior is\n\\begin{equation}\\label{COOLNEW}\n\\Lambda = \\frac{16}{3} \\sigma T^4 \\frac{\\tau}{1 + \\tau^2}.\n\\end{equation}\nThis interpolates smoothly between the optically-thick and\noptically-thin regimes and is proportional to the (Rosseland mean)\noptical depth in the optically-thin limit. While it would be more\nphysically sensible to use a Planck mean opacity in the optically-thin\nlimit, usually the optically-thin regions contain little mass so their\ncooling is not energetically significant. An exception is in the\nopacity gap, where even high density regions become optically thin.\n\nOur simulations begin with $\\Sigma$ and $U$ constant. The velocity field is\nperturbed from the equilibrium solution to initiate the gravitational\ninstability. The initial velocities are $v_x = \\delta v_x$, $v_y = -\n\\frac{3}{2}\\Omega x + \\delta v_y$, where $\\delta \\bv$ is a Gaussian\nrandom field of amplitude $\\< \\delta v^2 \\>\/c_s^2 = 0.1$. The\npower spectrum of perturbations is white noise ($v_k^2 \\sim k^0$) in a band\nin wavenumber $k_{crit}\/4 < |k| < 4 k_{crit}$ surrounding the minimum\n$k_{crit} = 1\/(\\pi Q^2)$ (with $G = \\Sigma_o = \\Omega = 1$) in the\ndensity-wave dispersion relation. We have checked in particular cases\nthat for $10^{-3} < \\< \\delta v^2 \\>\/c_s^2 < 10$ the outcome is\nqualitatively unchanged. This is expected because disk\nperturbations (unlike cosmological perturbations) grow exponentially and\nthe initial conditions are soon forgotten.\n\nExcluding the initial velocity field, the initial conditions for a\nspatially-uniform disk consist of three parameters: $\\Sigma_o, U_o$, and $\\Omega$.\nWe fix $Q = 1$, leaving two degrees of freedom. In models with simple,\nscale-free cooling functions such as that considered by \\cite{gam01},\nthese degrees of freedom remain and can be scaled away by setting $G =\n\\Sigma_o = \\Omega = 1$. That is, there is a two-dimensional continuum\nof disks (with varying values of $\\Sigma_o$ and $\\Omega$, but the same\nvalue of $Q$) that are described by a single numerical\nmodel.\n\nThe opacity contains definite physical scales in density and\ntemperature. The realistic cooling function considered here therefore removes our freedom to\nrescale the disk surface density and rotation frequency. That is, there\nis now a one-to-one correspondence between disks with fixed $\\Sigma$ and\n$\\Omega$ and our numerical models.\n\nThe choice of $\\Sigma_o$ and $\\Omega$ as labels for the parameter space\nis not unique. Internally in the code we fix the initial\nvolume density (in $\\gcm$) and the initial temperature (in\nKelvins). These choices are difficult to interpret, however, since they\nare tied to quantities that change over the course of the simulation;\n$\\Omega$ and the mean value of $\\Sigma$ do not.\n\nThe cooling is integrated explicitly using a first-order scheme. The timestep is\nmodified to satisfy the Courant condition and to be less than a fixed fraction of\nthe shortest cooling time on the grid. We have varied this fraction and shown that\nthe results are insensitive to it, provided that it is sufficiently small.\n\n\\section{Nonlinear Outcome}\\label{pap1s4}\n\n\\subsection{Standard Run}\\label{STDRUN}\n\nConsider the evolution of a single ``standard'' run, with $\\Sigma_o =\n1.4 \\times 10^5 \\gm \\cm^{-2}$ and $\\Omega = 1.1 \\times 10^{-7}\n\\sec^{-1}$. This corresponds to $T_o = 1200$ and $\\tco = 9.0\n\\times 10^4 \\Omega^{-1}$. The model size is $L = 320\nG\\Sigma_o\/\\Omega^2$ and numerical resolution $1024^2$. The model\ninitially lies at the lower edge of the opacity gap.\n\nThe evolution of the kinetic, gravitational and thermal energy per unit\narea ($\\$, $\\$ and $\\$ respectively) normalized to\n$G^2\\Sigma_o^3\/\\Omega^2$,\\footnote{The natural unit that can be formed\nfrom $G$, $\\Sigma$ and $\\Omega$.} are shown in Figure~\\ref{pap1f1}. After\nthe initial phase of gravitational instability the model settles into a\nstatistically-steady, gravito-turbulent state. It does not fragment.\nCooling is balanced by shock heating. Energy for driving the shocks is\nextracted from the shear flow, and the mean shear flow is enforced by the\nboundary conditions.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=4.5in,clip]{pap1f1.eps}\n\\caption[Evolution of the kinetic, gravitational, and thermal energy per unit\narea in the standard run.]\n{Evolution of the kinetic, gravitational, and thermal energy per unit\narea, normalized to $G^2 \\Sigma_o^3\/\\Omega^2$, in the standard run, which\nhas $L = 320 G\\Sigma_o\/\\Omega^2$, resolution $1024^2$, and $\\tco = 9.0\n\\times 10^4 \\Omega^{-1}$.}\n\\label{pap1f1}\n\\end{figure}\n\nThe turbulent state transports angular momentum outward via hydrodynamic\nand gravitational shear stresses. The dimensionless gravitational shear stress\nis\n\\begin{equation}\n\\alpha_{grav} = \\frac{1}{\\<\\frac{3}{2} \\Sigma c_s^2\\>}\n\t\t\\int_{-\\infty}^{\\infty} dz \\frac{g_x g_y}{4 \\pi G}\n\\end{equation}\nwhere ${\\bf g}$ is the gravitational acceleration, and\nthe dimensionless hydrodynamic shear stress is\n\\begin{equation}\n\\alpha_{hyd} = \\frac{\\Sigma v_x \\delta v_y}{\\<\\frac{3}{2} \\Sigma c_s^2\\>}\n\\end{equation}\nwhere $\\<\\>$ denote a spatial average. Figure~\\ref{pap1f2} shows the evolution of $\\< \\alpha_{grav} \\>$\nand $\\< \\alpha_{hyd} \\>$ in the standard run. Averaged over the last $230 \\Omega^{-1}$\nof the run, $\\<\\< \\alpha_{hyd} \\>\\> = 0.0079$, $\\<\\< \\alpha_{grav} \\>\\> = 0.017$, and so the\ntotal dimensionless shear stress is $\\<\\< \\alpha \\>\\> = 0.025$, where $\\<\\<\\>\\>$ denote\na space and time average.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=5.in,clip]{pap1f2.eps}\n\\caption{Evolution of the gravitational and hydrodynamic \npieces of $\\< \\alpha \\>$ in the standard run.}\n\\label{pap1f2}\n\\end{figure}\n\nThe mean stability parameter $\\ \\equiv \\langle c_s\n\\rangle\\Omega\/\\pi G\\langle \\Sigma\\rangle$ averages $1.86$ over the\nlast $230 \\Omega^{-1}$ of the run. Because the temperature and\nsurface density vary strongly, other methods of averaging $Q$ will\ngive different results.\n\nFigure~\\ref{pap1f3} shows a snapshot of the surface density at $t = 50\n\\Omega^{-1}$. The structure is similar to that observed in\n\\cite{gam01}, with trailing density structures. The density\nstructures are stretched into a trailing configuration by the\nprevailing shear flow. Their scale is determined by the disk\ntemperature and surface density rather than the size of the box (see\n\\citealt{gam01}).\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=4.in,clip]{pap1f3.eps}\n\\caption[Map of surface density in the standard run.]\n{Map of surface density at $t = 50 \\Omega^{-1}$ in the standard run.\nDark shades (blue in color version) indicate low density ($0.2 \\Sigma_o$) \nand light shades (yellow in color version) indicate high density ($3\n\\Sigma_o$).}\n\\label{pap1f3}\n\\end{figure}\n\n\\subsection{Varying $\\Sigma_o$ and $\\Omega$}\n\nWe now turn to exploring the two-dimensional parameter space of\nmodels. First consider a series of models with the same initial\ncentral temperature, but with varying $\\tco$. As $\\tco$ is lowered\nthe time-averaged gravitational potential energy per unit area $\\<\\\\>$ increases\nmonotonically in magnitude. The gravito-turbulent state becomes more\nextreme, with larger $\\<\\< \\alpha \\>\\>$, larger perturbed velocities, and larger\ndensity contrasts. Eventually a threshold is crossed and the disk\nfragments.\n\nFragmentation is illustrated in Figure~\\ref{pap1f4}, which shows a snapshot from a\nrun with $\\Sigma_o = 6.6 \\times 10^3 \\gm \\cm^{-2}$, $\\Omega = 5.4 \\times\n10^{-9} \\sec^{-1}$. This corresponds to $T_o = 1200$, $\\tco =\n0.025\\Omega^{-1}$. The run has numerical resolution $256^2$ and $L = 80\nG\\Sigma_o\/\\Omega^2$. The largest bound object in the center of the\nfigure was formed from the collision and coalescence of several smaller\nbound objects. A snapshot of the optical depth at the same point in the\nsimulation is given in Figure~\\ref{pap1f5}. For each snapshot, red indicates high\nvalues of the mapped variable and blue indicates low values. Much of the disk is\noptically thick, but most of the low density regions are optically thin in the\nRosseland mean sense.\n\n\\begin{figure}[h]\n \\hfill\n \\begin{minipage}[t]{.45\\textwidth}\n \\begin{center}\n \\includegraphics[width=0.9\\textwidth,clip]{pap1f4.eps}\n \\caption[Map of surface density in a run with $\\tco = 0.025\\Omega^{-1}$.]\n {Map of surface density in a run with $\\tco = 0.025\\Omega^{-1}$.\n\tDark shades indicate both low density ($10^{-2} \\Sigma_0$, black in \n\tcolor version) and high density ($10^2 \\Sigma_0$, near the centers \n\tof bound objects, red in color version).}\n \\label{pap1f4}\n \\end{center}\n \\end{minipage}\n \\hfill\n \\begin{minipage}[t]{.45\\textwidth}\n \\begin{center}\n \\includegraphics[width=0.9\\textwidth,clip]{pap1f5.eps}\n \\caption[Map of optical depth in a run with $\\tco = 0.025 \\Omega^{-1}$.]\n {Map of optical depth $\\tau$ in a run with $\\tco = 0.025 \\Omega^{-1}$. \n Dark shades indicate both low $\\tau$ ($10^{-2}$, black in\n color version) and high $\\tau$ ($10^4$, near the centers\n of bound objects, red in color version).}\n \\label{pap1f5}\n \\end{center}\n \\end{minipage}\n \\hfill\n\\end{figure}\n\nLowering $\\tco$ sufficiently always leads to fragmentation. We have\nsurveyed the parameter space of $\\Omega$ and $\\Sigma_o$ to\ndetermine where the disk begins to fragment. Each model was run to $100\n\\Omega^{-1}$.\\footnote{In four cases we had to run the simulation longer\nto get converged results.} Figures~\\ref{pap1f6} and \\ref{pap1f7} summarize\nthe results. Two heavy solid lines are shown on each diagram. The upper line\nshows the most rapidly cooling simulations that show no signs of\ngravitational fragmentation ({\\it nonfragmentation point}). Quantitatively, we\ndefine this as the point at which the time-averaged gravitational potential energy\nper unit area is equal to $-3 G^2 \\Sigma_o^3\/\\Omega^2$.\\footnote{$-3$ is the\npotential energy per unit area of a wave at the critical wavelength in a $Q = 1$\ndisk with $\\delta \\Sigma\/\\Sigma = \\sqrt{3}\/\\pi$. No bound objects are\nobserved throughout the duration of these runs.} The lower line shows\nthe most slowly cooling simulations to show definite fragmentation ({\\it\nfragmentation point}). Quantitatively, we define this as the point at\nwhich the gravitational potential energy per unit area\n\n\\begin{figure}[hp]\n \\hfill\n \\begin{minipage}[t]{1.\\textwidth}\n \\begin{center}\n \\includegraphics[width=3.5in,clip]{pap1f6.eps}\n \\caption[Location of the critical curves as a function of initial volume\n density and temperature.]\n {Location of the critical curves as a function of initial volume density and\n temperature (in cgs units). Each contour line is an order of magnitude\n change in $\\tco$, solid\/dotted lines indicating positive\/negative\n integer values of log($\\tco$).}\n \\label{pap1f6}\n \\end{center}\n \\end{minipage}\n \\hfill\n \\begin{minipage}[b]{1.\\textwidth}\n \\begin{center}\n \\includegraphics[width=3.5in,clip]{pap1f7.eps}\n \\caption[Location of the critical curves as a function of initial surface\n density and rotation frequency.]\n {Location of the critical curves as a function of initial surface density\n and rotation frequency (in cgs units). Each contour line is an order of\n magnitude change in $\\tco$, solid\/dotted lines indicating\n positive\/negative integer values of log($\\tco$). The gap in the center\n of the plot is due to the discontinuous jump in the value of $\\mu$.}\n \\label{pap1f7}\n \\end{center}\n \\end{minipage}\n \\hfill\n\\end{figure}\n\\noindent\nis equal to $-300\nG^2 \\Sigma_o^3\/\\Omega^2$ {\\it at some point during the run}.\\footnote{These\nruns exhibit bound objects that persist for the duration of the run.}\nFigure~\\ref{pap1f6} shows the data in the $\\rho_o, T_o$ plane, while\nFigure~\\ref{pap1f7} shows the results in the $\\Sigma_o,\\Omega$ plane.\nLight contours are lines of constant $\\tco$.\n\nThe transition from persistent, gravito-turbulent outcomes to\nfragmentation is gradual and statistical in nature. Figure~\\ref{pap1f8} shows the\ngravitational potential energy per unit area in the transition region for a series\nof runs with $T_o = 1200\\K$. The abscissa is labeled with the initial\ncooling time $\\tco\\Omega$. There is a gradual, approximately\nlogarithmic increase in the magnitude of $\\<\\\\>$ as $\\tco$ decreases.\nRuns in this region exhibit the transient formation of small bound\nobjects which might collapse if additional physics (e.g. the effects of\nMHD turbulence) were included in the model. Eventually $-\\<\\\\>$ begins to\nincrease dramatically, and we define the {\\it transition point} as the\nbeginning of this steep increase in gravitational binding energy.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=4.75in,clip]{pap1f8.eps}\n\\caption{Mean gravitational potential energy as a function of initial cooling\ntime for a series of models with varying initial cooling time and $T_o = 1200$.}\n\\label{pap1f8}\n\\end{figure}\n\nFigure~\\ref{pap1f9} shows the run of $\\tco\\Omega$ for the fragmentation point,\ntransition point, and nonfragmentation point as a function of $T_o$.\nIt is surprising that a disk can begin to exhibit signs of\ngravitational collapse for $\\tco\\Omega$ as large as $10^6$, and evade\ncollapse for $\\tco\\Omega$ as small as $0.02$. A naive application of\nthe results of \\cite{gam01} would suggest that fragmentation should\noccur for $\\tco\\Omega \\lesssim 3$. Evidently this estimate can be off by\norders of magnitude, with the largest error for $T_o \\approx 10^3\\K$,\njust below the opacity gap.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=5.5in,clip]{pap1f9.eps}\n\\caption{\nInitial cooling times at the points of non-fragmentation, fragmentation\nand transition.\n}\n\\label{pap1f9}\n\\end{figure}\n\nThe physical argument for fragmentation at short cooling times is as\nfollows (e.g. \\citealt{sbf90}). Thermal energy is supplied to the\ndisk via shocks. Strong shocks occur when dense clumps collide with one\nanother; this occurs on a dynamical timescale $\\sim \\Omega^{-1}$. If\nthe disk cools itself more rapidly then shock heating cannot match\ncooling and fragmentation results. This argument is apparently\ncontradicted by Figure~\\ref{pap1f9}. The resolution lies in finding an appropriate\ndefinition of cooling time. The disk loses thermal energy on the\neffective cooling timescale\n\\begin{equation}\n\\tce^{-1} \\equiv \\frac{\\<\\< \\Lambda \\>\\>}{\\<\\< U \\>\\>}.\n\\end{equation}\nFigure~\\ref{pap1f10} shows the run of\n$\\tce$ at the fragmentation, transition, and non-fragmentation points.\nEvidently $\\tce$ at transition lies between $\\Omega^{-1}$ and $10\n\\Omega^{-1}$. Figure~\\ref{pap1f11} shows the run of $\\tco$ and $\\tce$ on the\ntransition line. Just below the opacity gap they differ by as much as\nfour orders of magnitude.\n\n\\begin{figure}[h]\n \\hfill\n \\begin{minipage}[t]{.45\\textwidth}\n \\begin{center}\n \\includegraphics[width=1.\\textwidth,clip]{pap1f10.eps}\n \\caption{Effective cooling times at the points of non-fragmentation,\n fragmentation and transition.}\n \\label{pap1f10}\n \\end{center}\n \\end{minipage}\n \\hfill\n \\begin{minipage}[t]{.45\\textwidth}\n \\begin{center}\n \\includegraphics[width=1.\\textwidth,clip]{pap1f11.eps}\n \\caption{Initial and effective cooling times at the transition between\n non-fragmentation and fragmentation.}\n \\label{pap1f11}\n \\end{center}\n \\end{minipage}\n \\hfill\n\\end{figure}\n\nWhy do $\\tco$ and $\\tce$ differ by such a large factor? The answer is\nrelated to the existence of sharp variations in opacity with\ntemperature. Consider a disk near the lower edge of the opacity gap.\nOnce gravitational instability sets in, fluctuations in temperature move\nparts of the disk into the opacity gap. There, the opacity is reduced\nby orders of magnitude. Since the cooling rate for an optically thick\ndisk is proportional to $\\kappa^{-1}$, the cooling time drops by a\nsimilar factor. Relatively small variations in temperature can thus\nproduce large variations in cooling rate.\n\nAs in \\cite{gam01}, the result $\\tce\\Omega \\gtrsim 1$ also implies a\nconstraint on $\\<\\< \\alpha \\>\\>$. Energy conservation implies that\n\\begin{equation}\\label{AVGSOL}\n\\qq \\Omega \\<\\< W_{xy} \\>\\> = \\<\\< \\Lambda \\>\\>,\n\\end{equation}\nwhere $W_{xy}$ is the total shear stress (hydrodynamic plus gravitational). Equivalently, stress by rate-of-strain\nis equal to the dissipation rate. Using the definition of $\\tce$, this implies\n\\begin{equation}\\label{ANALPHA}\n\\<\\< \\alpha \\>\\> = \\left(\\gamma (\\gamma - 1) \\frac{9}{4} \\Omega \\tce\n\t\\right)^{-1}.\n\\end{equation}\nHence $\\tce\\Omega \\gtrsim 1$ implies $\\<\\< \\alpha \\>\\> \\lesssim 1$. Figure~\\ref{pap1f12}\nshows $\\<\\< \\alpha \\>\\>$ vs $\\tce$ for a large number of runs plotted against\nequation (\\ref{ANALPHA}). For small values of $\\tce$ the numerical\nvalues lie below the line. These models are not in equilibrium (i.e., not in a\nstatistically-steady gravito-turbulent state), so the\ntime average used in equation (\\ref{AVGSOL}) is not well defined. For\nlarger values of $\\tce$ numerical results typically (there is noise in\nthe measurement of both $\\<\\< \\alpha \\>\\>$ and $\\tce$ because the time\naverage is taken over a finite time interval) lie slightly above the\nanalytic result.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=5.in,clip]{pap1f12.eps}\n\\caption[Time-averaged shear stress vs. effective cooling time for a series of runs.]\n{Time-averaged shear stress vs. effective cooling time for a series of\nruns. The solid line shows the analytic result, based on energy\nconservation, from equation (\\ref{ANALPHA}).}\n\\label{pap1f12}\n\\end{figure}\n\nThe bias toward points lying slightly above the line reflects the fact\nthat $\\<\\< \\alpha \\>\\>$ measures the rate of energy extraction from the shear\nwhile $\\tce$ measures the rate at which that energy is transformed into\nthermal energy. If energy is lost, perhaps to numerical averaging at\nthe grid scale, then more energy must be extracted from the shear flow\nto make up the difference. Overall, however, the agreement with the\nanalytic result is good and demonstrates good energy conservation\nin the code.\n\nThe relationship between $\\tce$ and $\\<\\< \\alpha \\>\\>$ is interesting but not\nparticularly useful because $\\tce$ is no more readily calculated than\n$\\<\\< \\alpha \\>\\>$; it depends on a complicated moment of the surface density and\ntemperature. Only for constant cooling time have we been able to evaluate this moment\nanalytically.\n\n\\subsection{Isothermal Disks}\n\nWe have assumed that external illumination of the disk is negligible.\nThis approximation is valid when the effective temperature $T_{irr}$ of\nthe external irradiation is small compared to the central temperature of\nthe disk. In the opposite limit, illumination controls the energetics\nof the disk and it is isothermal (if it is illuminated directly so that\nshadowing effects, such as those considered by \\cite{js03} are\nnegligible).\n\nIt is therefore worth studying the outcome of gravitational instability\nin an isothermal disk. The isothermal disk model has a single\nparameter: the initial value of $Q$. We ran models with varying values\nof $Q$ and with $\\<\\delta v^2\\>\/c_s^2 = 0.1$. We find that models with\n$Q \\lesssim 1.4$ fragment.\n\nIt is likely that the mass of the fragments, etc., depends on how an\nisothermal disk becomes unstable. Rapid fluctuation of the external\nradiation field is likely to produce a different outcome than dimming on a\ntimescale long compared to the dynamical time.\n\n\\section{Discussion}\\label{pap1s5}\n\nUsing numerical experiments, we have identified\nthose disks that are likely to fragment absent external\nheating. Disks with effective cooling times $\\tce \\lesssim\n\\Omega^{-1}$ are susceptible to fragmentation. This is what one might\nexpect based on the simple argument of \\cite{sbf90}: if the disk cools\nmore quickly than the self-gravitating condensations can collide with\none another, then those collisions (which occur on a timescale $\\sim\n\\Omega^{-1}$) cannot reheat the disk and fragmentation is inevitable. But our\nresults are at the same time surprising.\n\nThe effective cooling time depends on the nonlinear outcome of\ngravitational instability. It depends on the cooling function, which in\nturn depends sensitively on $\\Sigma$ and $U$. Since $\\Sigma$ and $U$\nvary strongly over the disk once gravitational instability has set in,\nit is difficult to estimate $\\tce$ directly. One might be tempted to\nestimate $\\tce(\\Sigma,\\Omega) \\simeq \\tco(\\Sigma_o,\\Omega,Q = 1)$, but\nour experiments show that this estimate can be off by as much as four\norders of magnitude. The effect is particularly pronounced near sharp\nfeatures in the opacity. For example, consider a model initially\nlocated just below the opacity gap with $\\tco\\Omega \\gg 1$.\nGravitational instability creates dense regions with higher\ntemperatures, where dust is destroyed. The result is rather like having\nto shed one's blanket on a cold winter morning: the disk loses its\nthermal energy suddenly. Pressure support is lost and gravitational\ncollapse ensues.\n\nThe difference between $\\tce$ and $\\tco (Q = 1)$ implies that a much\nlarger region of the disk is susceptible to fragmentation than naive\nestimates based on the approximation $\\tce \\approx \\tco$ might suggest.\nFor example, consider an equilibrium disk model with $Q \\gg 1$ at small\n$r$. As $r$ increases, $Q$ declines. Eventually $Q\\sim 1$ and\ngravitational instability sets in. There is then a range of radii where\n$Q \\sim 1$, $\\tce\\Omega \\gtrsim 1$ and recurrent gravitational\ninstability can transport angular momentum and prevent collapse.\nGenerally speaking, however, the cooling time decreases with increasing\nradius. Eventually $\\tce\\Omega \\sim 1$ and fragmentation cannot be\navoided. By lowering our estimate of $\\tce$, we narrow the range of\nradii over which recurrent gravitational instability can occur.\n\nThe general sense of our result is that it is extremely difficult to\nprevent a marginally-stable, $Q \\sim 1$, optically-thick disk from\nfragmenting and forming planets (in circumstellar disks) or stars (in\ncircumstellar and circumnuclear disks). This is particularly true for\ndisks with $T \\sim 10^3\\K$, whose opacity is dominated by dust grains,\ni.e. disks whose temperature lies within a factor of several of the\nopacity gap.\n\nOur numerical model uses a number of approximations. First, our\ntreatment is razor-thin, i.e. all the matter is in a thin slice at\n$z = 0$. The effect of finite thickness on linear stability has been\nunderstood since \\cite{glb65}: it is stabilizing because gravitational\nattraction of neighboring columns of disk is diluted by finite\nthickness. The size of the effect may be judged by the fact that $Q =\n0.676$ is required for marginal stability of a finite-thickness,\nisothermal disk. \n\nThe behavior of a finite-thickness disk in the nonlinear regime is more\ndifficult to predict. Shocks will evidently deposit some of their\nenergy away from the midplane, where it can be radiated away more\nquickly (because the energy is deposited at smaller optical depth - see \\citealt{pcdl00}).\nRadiative diffusion parallel to the disk plane (not included here) may\nenhance cooling of dense, hot regions. Both these effects are\ndestabilizing. Ultimately, however, a numerical study is required.\nThis is numerically expensive: one\nmust resolve the disk vertically, on the scale height $H$, and\nhorizontally, at the critical wavelength $2 \\pi Q H$.\n\nSecond, we have ignored magnetic fields. While there may be\nastrophysical situations where cool disks have such low ionization that\nthey are unmagnetized, most disks are likely to contain dynamically\nimportant magnetic fields that give rise to a dimensionless shear stress\n$\\<\\< \\alpha \\>\\> \\gtrsim 0.01$ (e.g. \\citealt{hgb95}). These fields are likely to\nremove spin angular momentum from partially collapsed objects,\ndestabilizing them. Numerical experiments including both gravitational\nfields and magnetohydrodynamics are necessarily three dimensional (the\ninstability of \\cite{bh91} requires $\\partial_z \\ne 0$), and are thus\nnumerically expensive.\n\nThird, we have fixed $\\gamma$ and $\\mu$ for the duration of each\nsimulation. This eliminates the soft spots in the equation of state\nassociated with ionization of atomic hydrogen and dissociation of\nmolecular hydrogen. In these locations the three dimensional $\\gamma$\ndips below $4\/3$, which is destabilizing.\n\nFourth, we have treated the physics of grain destruction and formation\nvery simply. In using the \\cite{bl94} opacities we implicitly assume\nthat grains reform in cooling gas on much less than a dynamical time.\nIt is likely that grain re-formation will take some time (e.g.\n\\citealt{hess91}) and this will further reduce the disk opacity and\nenhance fragmentation.\n\nFifth, we have neglected the effects of illumination. In the limit of\nstrong external illumination, i.e. when the effective temperature of the\nirradiation $T_{irr}$ is large compared to the disk central temperature\n$T_c$, the disk is isothermal (here $T_c$ is the temperature of a dense\ncondensation). We have carried out isothermal experiments and shown\nthat, for initial velocity perturbations with $\\< \\delta v^2 \\>\/c_s^2 = 0.1$,\ndisks with $Q \\lesssim 1.4$ fragment. Weaker illumination produces a more\ncomplicated situation that we have not explored here.\nIllumination-dominated disks that become unstable presumably do so\nbecause the external illumination declines, and the rate at which the\nexternal illumination changes may govern the nonlinear outcome.\n\nWe conclude that disks with $\\tce\\Omega \\lesssim 1$ do not exist.\nCooling in this case is so effective that fragmentation into condensed\nobjects-- stars, planets, or smaller accretion disks-- is inevitable.\n\nAs an example application of this result, consider the model for the\nnucleus of NGC 1068 recently proposed by \\cite{lb03}. Their model is an\nextended marginally-stable self-gravitating disk of the type\ninvestigated here and originally proposed by \\cite{glb65} for galactic\ndisks and \\cite{pac78} for accretion disks, although their disk is\nsufficiently massive that it modifies the rotation curve as well. Based\non their Figure 3, at a typical radius of $0.5\\pc$, $\\Sigma_o \\simeq\n10^4$ and $\\Omega \\simeq 10^{-9}$. According to our Figure 7 this disk is\nabout 2 orders of magnitude too dense to avoid fragmentation. While it\nmay be possible to avoid this conclusion by invoking strong external\nheating, the energy requirements are severe, as outlined in\n\\cite{good03}. The disk proposed by \\cite{lb03} would therefore fragment\ninto stars on a short timescale.\n\n\n\\end{spacing}\n\n\\chapter{Linear Theory of Thin, Radially-Stratified Disks}\\label{paper2}\n\n\\begin{spacing}{1.5}\n\n\\section{Chapter Overview}\n\nWe consider the nonaxisymmetric linear theory of radially-stratified\ndisks. We work in a shearing-sheet-like approximation, where the\nvertical structure of the disk is neglected, and develop equations for\nthe evolution of a plane-wave perturbation comoving with the shear flow\n(a shearing wave, or ``shwave''). We calculate a complete solution set\nfor compressive and incompressive short-wavelength perturbations in both\nthe stratified and unstratified shearing-sheet models. We develop\nexpressions for the late-time asymptotic evolution of an individual\nshwave as well as for the expectation value of the energy for an\nensemble of shwaves that are initially distributed isotropically in\n$k$-space. We find that: (i) incompressive, short-wavelength\nperturbations in the unstratified shearing sheet exhibit transient\ngrowth and asymptotic decay, but the energy of an ensemble of such\nshwaves is constant with time; (ii)\nshort-wavelength compressive shwaves grow asymptotically in the\nunstratified shearing sheet, as does the energy of an ensemble of such\nshwaves; (iii) incompressive shwaves in the stratified shearing sheet\nhave density and azimuthal velocity perturbations $\\delta \\Sigma$,\n$\\delta v_y \\sim t^{-{\\rm Ri}}$ (for $|{\\rm Ri}| \\ll 1$), where ${\\rm\nRi} \\equiv N_x^2\/ (\\qe \\Omega)^2$ is the Richardson number, $N_x^2$ is\nthe square of the radial\nBrunt-V$\\ddot{\\rm{a}}$is$\\ddot{\\rm{a}}$l$\\ddot{\\rm{a}}$ frequency and\n$\\qe \\Omega$ is the effective shear rate; (iv) the energy of an ensemble\nof incompressive shwaves in the stratified shearing sheet behaves\nasymptotically as ${\\rm Ri} \\, t^{1-4{\\rm Ri}}$ for $|{\\rm Ri}| \\ll 1$. For\nKeplerian disks with modest radial gradients, $|{\\rm Ri}|$ is expected\nto be $\\ll 1$, and there will therefore be weak growth in a single\nshwave for ${\\rm Ri} < 0$ and near-linear growth in the energy of an\nensemble of shwaves, independent of the sign of Ri.\\footnote{To be published in ApJ\nVolume 626, Issue 2. Reproduction for this dissertation is authorized by the\ncopyright holder.}\n\n\\section{Introduction}\n\nAngular momentum transport is central to the evolution of astrophysical\ndisks. In many disks angular momentum is likely redistributed\ninternally by magnetohydrodynamic (MHD) turbulence driven by the\nmagnetorotational instability (MRI; see \\citealt{bh98}). But in portions of\ndisks around young, low-mass stars, in cataclysmic-variable disks in\nquiescence, and in X-ray transients in quiescence \\citep{sgbh00,gm98,men00}, \ndisks may be composed of gas that is so neutral that the MRI fails. It is \ntherefore of interest to understand if there are purely hydrodynamic \nmechanisms for driving turbulence {\\it and} angular momentum transport \nin disks.\n\nThe case for hydrodynamic angular momentum transport is not promising.\nNumerical experiments carried out under conditions similar to those\nunder which the MRI produces ample angular momentum fluxes-- local\nshearing-box models-- show small or negative angular momentum fluxes\nwhen the magnetic field is turned off \\citep{hgb95,hgb96}. Unstratified \nshearing-sheet models show decaying angular momentum flux and kinetic \nenergy when nonlinearly perturbed, yet recover the well known, high \nReynolds number nonlinear instability of plane Couette flow when the \nparameters of the model are set appropriately (\\citealt{bhs96}; see, however, \nthe recent results by \\citealt{ur04}). Local models \nwith unstable vertical stratification show overturning and the development \nof convective turbulence, but the mean angular momentum flux is small and \nof the wrong sign \\citep{sb96}.\n\nLinear theory of global disk models has long indicated the presence of\ninstabilities associated with reflecting boundaries or features in the\nflow (see e.g., \\citealt{pp84,pp85,pp87,ggn86,gng87,ngg87,llcn99,lflc00}).\nNumerical simulations of the nonlinear outcome of these instabilities\nsuggest that they saturate at low levels and are turned off by modest \naccretion \\citep{bla87,haw91}. One might guess that in the nonlinear\noutcome these instabilities will attempt to smooth out the features that \ngive rise to them, much as convection tends to erase its parent entropy \ngradient. There are some suggestions, however, that such instabilities saturate\ninto long-lived vortices, which may serve as obstructions in the flow that\ngive rise to angular momentum transport \\citep{lcwl01}. We will consider\nthis possibility in a later publication.\n\nLinear theory has yet to uncover a {\\it local} instability of\nhydrodynamic disks that produces astrophysically-relevant angular\nmomentum fluxes. Because of the absence of a complete set of modes in\nthe shearing-sheet model, however, local linear stability is difficult\nto prove. Local nonlinear stability may be impossible to prove.\nComparison with laboratory Couette flow experiments is complicated by\nseveral factors, not least of which is the inevitable presence of solid\nradial boundaries in the laboratory that have no analogue in\nastrophysical disks.\n\nRecently, however, \\cite{kb03} (hereafter KB03) have claimed to find a local \nhydrodynamic instability in global numerical simulations: the ``Global \nBaroclinic Instability.'' The \ninstability arises in a model with scale-free initial conditions (an \nequilibrium entropy profile that varies as a power-law in \nradius) and thus does not depend on sharp features in the flow. \n\\cite{klr04} has performed a local linear stability analysis of a\nradially-stratified accretion disk in an effort to explain the numerical\nresults obtained by KB03. The instability mechanism invoked is the\nphenomenon of transient amplification as a shearing wave goes from\nleading to trailing. This is the mechanism that operates for\nnonaxisymmetric shearing waves in a disk that is nearly unstable to the\naxisymmetric gravitational instability \\citep{glb65,jt66,gt78}. It is the \npurpose of this work to clarify and extend the linear analysis of\n\\cite{klr04}. If this instability exists it could be important for the\nevolution of low-ionization disks.\n\nTo isolate the cause for instabilities originally observed in global 3D\nsimulations, KB03 perform both local and global 2D calculations\nin the ($R,\\phi$)-plane. The local simulations use a new set of boundary\nconditions termed the shearing-disk boundary conditions. The model is\ndesigned to simulate a local portion of the disk without neglecting\nglobal effects such as curvature and horizontal flow gradients. The\nboundary conditions, which are described in more detail in KB03,\nrequire the assumption of a power-law scaling for the mean values of\neach of the variables, as well as the assumption that the fluctuations\nin each variable are proportional to their mean values. The radial\nvelocity component in the inner and outer four grid cells is damped by\n$5\\%$ each time step in order to remove artificial radial oscillations\nproduced by the model.\\footnote{It is not surprising that shearing disk\nboundary conditions as implemented in KB03 produce features on the\nradial boundary, because the Coriolis parameter is discontinuous across\nthe radial boundary.}\n\nThe equilibrium profile for KB03's 2D runs was a constant surface density\n$\\Sigma$ with either a constant temperature $T$ or a temperature profile\n$T \\propto R^{-1}$. The constant-$T$ runs showed no instability while\nthose with varying $T$ (and thus varying entropy) sustained turbulence\nand positive Reynolds stresses.\\footnote{Notice that with a constant\n$\\Sigma$, the constant-$T$ runs have no variation in any of the\nequilibrium variables, so it is not clear that the effects being\nobserved in the 2D calculations are due to the presence of an entropy\ngradient rather than due simply to the presence of a pressure gradient.}\nThe fiducial local simulations were run at a resolution of $64^2$, with\na spatial domain of $R = 4$ to $6$ AU and $\\Delta \\phi = 30\\,^{\\circ}$.\nThe unstable run was repeated at a resolution of $128^2$, along with a\nrun at twice the physical size of the fiducial runs. One global model\n(with nonreflecting outflow boundary conditions) was run at a\nresolution of $128^2$ with a spatial domain of $R = 1$ to $10$ AU and\n$\\Delta \\phi = 360\\,^{\\circ}$. All the runs yielded similar results,\nwith the larger simulations producing vortices and power on large\nscales.\n\nKB03 have chosen the term ``baroclinic instability'' by way of\nanalogy with the baroclinic instability that gives rise to weather\npatterns in the atmosphere of the Earth and other planets (see e.g. \n\\citealt{ped87}).\\footnote{A baroclinic flow is one in which surfaces \nof constant density are inclined with respect to surfaces of constant\npressure. If these surfaces coincide, the flow is termed barotropic.} \nThe analogy is somewhat misleading, however, since the baroclinic \ninstability that arises in planetary contexts is due to a baroclinic \nequilibrium. In a planetary atmosphere, a baroclinically-unstable \nsituation requires stratification in both the vertical and latitudinal \ndirections.\\footnote{Contrary to the claim in \\cite{klr04}, the \ntwo-layer model \\citep{ped87} does not ignore the vertical structure; \nit simply considers the lowest-order vertical mode.} The stratification \nin KB03 is only in the radial direction, and as a result the\nequilibrium is barotropic. It is the perturbations that are baroclinic; \ni.e., the disk is only baroclinic at linear order in the amplitude of a \ndisturbance.\n\n\\cite{cab84} and \\cite{ks86} have analyzed a thin disk with a baroclinic\nequilibrium state (with both vertical and radial gradients). The latter\nfind that due to the dominant effect of the Keplerian shear, the\ninstability only occurs if the radial scale height is comparable to the\nvertical scale height, a condition which is unlikely to be\nastrophysically relevant. As pointed out in KB03, the salient feature\nthat is common to their analysis and the classical baroclinic\ninstability is an equilibrium entropy gradient in the horizontal\ndirection. As we show in \\S 2, however, an entropy gradient is not\nrequired in order for two-dimensional perturbations to be baroclinic;\nany horizontal stratification will do.\n\nThe ``Global Baroclinic Instability'' claimed by KB03 is thus\nanalogous to the classical baroclinic instability in the sense that both\nhave the potential to give rise to convection.\\footnote{The classical\nbaroclinic instability gives rise to a form of ``sloping convection''\n\\citep{hou02} since the latitudinal entropy gradient is inclined with\nrespect to the vertical buoyancy force.} When neglecting vertical\nstructure, however, the situation in an accretion disk is more closely\nanalogous to a shearing, stratified atmosphere, the stability of which\nis governed by the classical Richardson criterion \\citep{jwm61,chi70}. \nThe only additional physics in a disk is the presence of the Coriolis force.\nMost analyses of a shearing, stratified atmosphere, however, only \nconsider stratification profiles that are stable to convection. The primary\nquestion that \\cite{klr04} and this work are addressing, then, is\nwhether or not the presence of shear stabilizes a stratified equilibrium \nthat would be unstable in its absence.\n\nWe begin in \\S\\ref{pap2s2} by outlining the basic equations for a local model of a\nthin disk. \\S\\S\\ref{pap2s3} and \\ref{pap2s4} describe the local linear theory for\nnonaxisymmetric sinusoidal perturbations in unstratified and\nradially-stratified disks, respectively. We summarize and discuss the\nimplications of our findings in \\S\\ref{pap2s5}.\n\n\\section{Basic Equations}\\label{pap2s2}\n\nThe effect of radial gradients on the local stability of a thin disk can\nbe analyzed most simply in the two-dimensional shearing-sheet\napproximation\\footnote{See \\cite{rg92} for a discussion of why this\napproximation is appropriate for an analysis of local stability. See \nalso \\cite{mp77}, who use a similar approach to demonstrate the\nstability of unbounded viscous plane Couette flow.}.\nThis is obtained by a rigorous expansion of the equations\nof motion in the ratio of the vertical scale height $H$ to the local radius $R$,\nfollowed by a vertical integration of the fluid equations. The basic\nequations that one obtains (e.g., \\citealt{gt78}) are\n\\begin{equation}\\label{EQ1}\n\\dv{\\Sigma}{t} + \\Sigma \\bnabla \\cdot \\bv = 0,\n\\end{equation}\n\\begin{equation}\\label{EQ2}\n\\dv{\\bv}{t} + \\frac{\\bnabla P}{\\Sigma} + 2\\bO\\times\\bv - 2q\\Omega^2 x \\ex = 0,\n\\end{equation}\n\\begin{equation}\\label{EQ3}\n\\dv{\\,{\\rm{ln}} S}{t} = 0,\n\\end{equation}\nwhere $\\Sigma$ and $P$ are the two-dimensional density and pressure,\n$S \\equiv P \\Sigma^{-\\gamma}$ is monotonically related to the fluid\nentropy,\\footnote{With the assumptions of vertical hydrostatic equilibrium \nand negligible self-gravity, the effective two-dimensional adiabatic index\ncan be shown to be $\\gamma = (3\\gamma_{3D} - 1)\/(\\gamma_{3D} + 1)$ \n(e.g. \\citealt{ggn86}).} $\\bv$ is the fluid velocity and $d\/dt$ is the\nLagrangian derivative. The third and fourth terms in equation\n(\\ref{EQ2}) represent the Coriolis and centrifugal forces in the local\nmodel expansion, where $\\Omega$ is the local rotation frequency, $x$ is\nthe radial Cartesian coordinate and $q$ is the shear parameter (equal to\n$1.5$ for a disk with a Keplerian rotation profile). The gravitational\npotential of the central object is included as part of the centrifugal\nforce term in the local-model expansion, and we ignore the self-gravity\nof the disk.\n\nIt is worth emphasizing at this point that we have integrated out the\nvertical degrees of freedom in the model. We will later focus on\nperturbations with planar wavelengths that are small compared to a scale\nheight, and these perturbations will be strongly influenced by the\nvertical structure of the disk. \n\nEquations (\\ref{EQ1}) through (\\ref{EQ3}) can be combined into a single\nequation governing the evolution of the potential vorticity:\n\\begin{equation}\\label{PVEV}\n\\dv{}{t}\\left(\\frac{\\bnabla \\times \\bv + 2\\bO}{\\Sigma}\\right) \\equiv\n\\dv{\\bxi}{t} = \\frac{\\bnabla \\Sigma \\times \\bnabla P}{\\Sigma^3}.\n\\end{equation}\nIn two dimensions, $\\bxi$ has only one nonzero component and can\ntherefore be regarded as a scalar. Equation (\\ref{PVEV}) demonstrates \nthat for $P \\equiv P(\\Sigma)$ (as in the case of a strictly adiabatic evolution \nwith isentropic initial conditions), the potential vorticity of fluid elements \nis conserved. For $P \\neq P(\\Sigma)$, however, the potential vorticity \nevolves with time. A barotropic equilibrium stratification can result in \nbaroclinic perturbations that cause the potential vorticity to evolve at linear \norder. This can be seen by linearizing the scalar version of equation \n(\\ref{PVEV}):\n\\begin{equation}\\label{PVEVLIN}\n\\pdv{\\delta \\xi}{t} + \\bv_0 \\cdot \\bnabla \\delta \\xi + \\delta \\bv \\cdot \\bnabla \\xi_0 = \\frac{\\ez \\cdot(\\bnabla \n\\Sigma_0 \\times \\bnabla \\delta P - \\bnabla P_0 \\times \\bnabla \\delta \\Sigma)}{\\Sigma_0^3},\n\\end{equation}\nwhere we have dropped the term $\\propto \\bnabla \\Sigma_0 \\times \\bnabla\nP_0$. Notice that an entropy gradient is not required for the evolution of \nthe perturbed potential vorticity. For $S_0 = P_0 \\Sigma_0^\\gamma = \nconstant$, equation (\\ref{PVEVLIN}) reduces to\n\\begin{equation}\\label{PVEVLIN2}\n\\pdv{\\delta \\xi}{t} + \\bv_0 \\cdot \\bnabla \\delta \\xi + \\delta \\bv \\cdot \\bnabla \\xi_0 = \\frac{\\ez \\cdot(\\bnabla \nP_0 \\times \\bnabla \\delta S)}{\\gamma \\Sigma_0^2 S_0}.\n\\end{equation}\nPotential vorticity is conserved only in the limit of zero stratification ($P_0 \n= constant$) or adiabatic perturbations ($\\delta S = 0$).\n\n\\section{Unstratified Shearing Sheet}\\label{pap2s3}\n\nOur goal is to understand the effects of radial stratification, but we\nbegin by developing the linear theory of the standard (unstratified)\nshearing sheet, in which the equilibrium density and pressure are\nassumed to be spatially constant. This will serve to establish notation\nand method of analysis and to highlight the changes introduced by\nradial stratification in the next section.\n\nOur analysis follows that of \\cite{gt78} except for our neglect of\nself-gravity. The equilibrium consists of a uniform sheet with $\\Sigma =\n\\Sigma_0 = constant$, $P = P_0 = constant$, and $\\bv_0 = -q\\Omega x\n\\ey$. We consider nonaxisymmetric Eulerian perturbations about this\nequilibrium with space-time dependence $\\delta(t){\\rm exp} (ik_x(t) x +\nik_y y)$, where\n\\begin{equation}\\label{KX}\nk_x(t) \\equiv k_{x0} + q\\Omega k_y t\n\\end{equation}\n(with $k_{x0}$ and $k_y > 0$ constant) is required to allow for a spatial Fourier \ndecomposition of the perturbation. We will refer to these perturbations \nas shearing waves, or with some trepidation, but more compactly, as \n``shwaves''.\n\n\\subsection{Linearized Equations}\n\nTo linear order in the perturbation amplitudes, the dynamical equations\nreduce to\n\\begin{equation}\\label{LIN1s}\n\\frac{\\dot{\\delta \\Sigma}}{\\Sigma_0} + i k_x \\delta v_x \n\t+ i k_y \\delta v_y = 0,\n\\end{equation}\n\\begin{equation}\\label{LIN2s}\n\\dot{\\delta v}_x - 2\\Omega \\delta v_y \n\t+ ik_x \\frac{\\delta P}{\\Sigma_0} = 0,\n\\end{equation}\n\\begin{equation}\\label{LIN3s}\n\\dot{\\delta v}_y + (2- q)\\Omega \\delta v_x \n\t+ ik_y \\frac{\\delta P}{\\Sigma_0} = 0,\n\\end{equation}\n\\begin{equation}\\label{LIN4s}\n\\frac{\\dot{\\delta P}}{\\Sigma_0} + c_s^2 (i k_x \\delta v_x \n\t+ i k_y \\delta v_y) = 0,\n\\end{equation}\nwhere $c_s^2 = \\gamma P_0\/\\Sigma_0$ is the square of the equilibrium\nsound speed and an over-dot denotes a time derivative.\n\nThe above system of equations admits four linearly-independent\nsolutions. Two of these are the nonvortical shwaves (solutions for \nwhich the perturbed potential vorticity is zero), which in the absence \nof self-gravity can be solved for exactly. The remaining two solutions are \nthe vortical shwaves. When $k_y \\rightarrow 0$ the latter reduce to the \nzero-frequency modes of the axisymmetric version of equations \n(\\ref{LIN1s}) through (\\ref{LIN4s}). One of these (the entropy mode) \nremains unchanged in nonaxisymmetry (in a frame comoving with the \nshear). There is thus only one nontrivial vortical shwave in the \nunstratified shearing sheet.\n\nIn the limit of tightly-wound shwaves ($|k_x| \\gg k_y$), the nonvortical \nand vortical shwaves are compressive and incompressive, respectively. \nIn the short-wavelength limit ($H k_y \\gg 1$, where $H \\equiv c_s\/\\Omega$ \nis the vertical scale height), the compressive and incompressive solutions \nremain well separated at all times, but for $H k_y \\lesssim O(1)$ there is mixing \nbetween them near $k_x = 0$ as an incompressive shwave shears from leading \nto trailing.\\citep[][; also Goodman 2005, private communication]{chag97}\nWith the understanding that the distinction between compressive shwaves \nand incompressive shwaves as separate solutions is not valid for all time \nwhen $H k_y \\lesssim O(1)$, we generally choose to employ these terms over \nthe more general but less intuitive terms ``nonvortical'' and ``vortical.''\n\nBased upon the above considerations, it is convenient to study the \nvortical shwave in the short-wavelength, low-frequency ($\\partial_t \\ll \nc_s k_y$) limit. This is equivalent to working in the Boussinesq \napproximation,\\footnote{We demonstrate this equivalence in the Appendix.} \nwhich in the unstratified shearing sheet amounts to assuming incompressible \nflow. In this limit, equation (\\ref{LIN1s}) is replaced with\n\\begin{equation}\\label{LIN1b}\nk_x \\delta v_x + k_y \\delta v_y = 0.\n\\end{equation}\nThis demonstrates the incompressive nature of the vortical shwave in \nthe short-wavelength limit.\n\n\\subsection{Solutions}\n\nIn the unstratified shearing sheet, equation (\\ref{PVEVLIN}) for the \nperturbed potential vorticity can be integrated to give:\n\\begin{equation}\\label{POTVORT}\n\\delta \\xi_u = \\frac{i k_x \\delta v_y - i k_y \\delta v_x}{\\Sigma_0} - \n\\xi_0 \\frac{\\delta \\Sigma}{\\Sigma_0} = constant,\n\\end{equation}\nwhere $\\xi_0 = (2 - q)\\Omega\/\\Sigma_0$ is the equilibrium potential\nvorticity and we have employed the subscript $u$ to highlight the fact \nthat the perturbed potential vorticity is only constant in the unstratified \nshearing sheet. To obtain the compressive-shwave solutions, we set the \nconstant $\\delta \\xi_u$ to zero. Combining equations (\\ref{LIN3s}) and\n(\\ref{POTVORT}) with $\\delta \\xi_u = 0$, one obtains an expression for\n$\\delta v_{xc}$ in terms of $\\delta v_{yc}$ and its derivative:\n\\begin{equation}\\label{CVX}\n\\delta v_{xc} = \\frac{c_s^2 k_x k_y \\delta v_{yc} - \\xi_0 \\Sigma_0\n\\dot{\\delta v}_{yc}}{\\xi_0^2 \\Sigma_0^2 + c_s^2 k_y^2},\n\\end{equation}\nwhere the subscript $c$ indicates a compressive shwave. The associated \ndensity and pressure perturbations are\n\\begin{equation}\\label{CS}\n\\delta \\Sigma_c = \\frac{\\delta P_c}{c_s^2} = i \\frac{\\xi_0 \\Sigma_0 k_x \n\\delta v_{yc} + k_y \\dot{\\delta v}_{yc}}{\\xi_0^2 \\Sigma_0^2 + c_s^2 k_y^2}\n\\end{equation}\nvia equation (\\ref{POTVORT}). Reinserting equation (\\ref{CVX}) into\nequation (\\ref{LIN3s}), taking one time derivative and replacing\n$\\dot{\\delta P}$ via equation (\\ref{LIN4s}), we obtain the following\nremarkably simple equation:\n\\begin{equation}\\label{SOUND}\n\\ddot{\\delta v}_{yc} + \\left(c_s^2 k^2 + \\kappa^2\\right) \\delta v_{yc} = 0,\n\\end{equation}\nwhere $k^2 = k_x^2 + k_y^2$ and $\\kappa^2 = (2 - q)\\Omega^2$ is the\nepicyclic frequency. Changing to the dimensionless dependent variable\n\\begin{equation}\\label{TTAU}\nT \\equiv i \\sqrt{\\frac{2 c_s k_y}{q\\Omega}} \\left(q \\Omega t + \\frac{k_{x0}}\n{k_y}\\right) \\equiv i \\sqrt{\\frac{2 c_s k_y}{q\\Omega}} \\, \\tau\n\\end{equation}\nand defining\n\\begin{equation}\nC \\equiv \\frac{c_s^2 k_y^2 + \\kappa^2}{2 q\\Omega c_s k_y},\n\\end{equation}\nthe equation governing $\\delta v_{yc}$ becomes\n\\begin{equation}\\label{VYT}\n\\dv{^2\\delta v_{yc}}{T^2} + \\left(\\frac{1}{4}T^2 - C\\right) \\delta v_{yc} = 0.\n\\end{equation}\nThis is the parabolic cylinder equation (e.g. \\citealt{as72}), \nthe solutions of which are parabolic cylinder functions. One \nrepresentation of the general solution is\n\\begin{equation}\\label{CVY}\n\\delta v_{yc} = e^{-\\frac{i}{2}T^2}\\left[c_1 \\, M\\left(\\frac{1}{4} - \n\\frac{i}{2}C,\\frac{1}{2},\\frac{i}{2}T^2\\right) + c_2 \\, T \\, M\\left(\n\\frac{3}{4} - \\frac{i}{2}C,\\frac{3}{2},\\frac{i}{2}T^2\\right)\\right],\n\\end{equation}\nwhere $c_1$ and $c_2$ are constants of integration and $M$ is a \nconfluent hypergeometric function. This completely specifies the \ncompressive solutions for the unstratified shearing sheet, for any \nvalue of $k_y$.\n\nEquation (\\ref{VYT}) has been analyzed in detail by \\cite{ngg87}; \ntheir modal analysis yields the analogue of equation (\\ref{VYT}) in \nradial-position space rather than in the radial-wavenumber ($k_x = \nk_y \\tau$) space that forms the natural basis for our shwave \nanalysis. One way of seeing the correspondence between the \nmodes and shwaves is to take the Fourier transform of the asymptotic \nform of the solution. Appropriate linear combinations of the solutions \ngiven in equation (\\ref{CVY}) have the following asymptotic time \ndependence for $\\tau \\gg 1$:\n\\begin{equation}\n\\delta v_{yc} \\propto \\sqrt{\\frac{2}{T}}\\exp\\left(\\pm \\frac{i}{4}T^2 \n\\right) \\propto \\frac{1}{\\sqrt{k_x}}\\exp\\left(\\pm i \\int c_s k_x \n\\, dt \\right).\n\\end{equation}\nThe Fourier transform of the above expression, evaluated by the method of \nstationary phase for $H k_y \\gg 1$, yields\n\\begin{equation}\n\\delta v_{yc}(X) \\propto \\sqrt{\\frac{2}{X}}\\exp\\left(\\pm \\frac{i}{4}X^2\\right),\n\\end{equation}\nwhich is equivalent to the expressions given for the modes analyzed by \n\\cite{ngg87}, in which the dimensionless spatial variable (with zero frequency, \nso that corotation is at $x = 0$) is defined as\n\\begin{equation}\nX \\equiv \\sqrt{\\frac{2 q \\Omega k_y}{c_s}} x.\n\\end{equation}\n\nTo obtain the incompressive shwave, we use the condition of incompressibility\n(equation (\\ref{LIN1b})) to write $\\delta v_y$ in terms of $\\delta v_x$,\nand then combine the dynamical equations (\\ref{LIN2s}) and (\\ref{LIN3s})\nto eliminate $\\delta P$. The incompressive shwave is given by:\n\\begin{equation}\\label{IVX}\n\\delta v_{xi} = \\delta v_{xi0}\\frac{k_0^2}{k^2},\n\\end{equation}\n\\begin{equation}\\label{IVY}\n\\delta v_{yi} = -\\frac{k_x}{k_y} \\delta v_{xi},\n\\end{equation}\n\\begin{equation}\\label{IS}\n\\frac{\\delta \\Sigma_i}{\\Sigma_0} = \\frac{\\delta P_i}{\\gamma P_0} = \n\\frac{1}{i c_s k_y}\\left(\\frac{k_x}{k_y} \\frac{\\dot{\\delta v}_{xi}}{c_s} \n+ 2(q - 1) \\Omega \\frac{\\delta v_{xi}}{c_s}\\right),\n\\end{equation}\nwhere the subscript $i$ indicates an incompressive shwave, $k_0^2 =\nk_{x0}^2 + k_y^2$ and $\\delta v_{xi0}$ is the value of $\\delta v_{xi}$\nat $t=0$.\\footnote{\\cite{cztl03} obtained this solution by starting with\nthe assumption of incompressibility. In the incompressible limit, it is \nan exact nonlinear solution to the fluid equations.} This solution is uniformly valid \nfor all time to leading order in $(H k_y)^{-1} \\ll 1$.\n\n\\subsection{Energetics of the Incompressive Shwaves}\\label{pap2eis}\n\nWe define the kinetic energy in a single incompressive shwave as\n\\begin{equation}\nE_{ki} \\equiv \\frac{1}{2}\\Sigma_0 (\\delta v_{xi}^2 + \\delta v_{yi}^2) =\n\\frac{1}{2}\\Sigma_0 \\delta v_{xi}^2 \\frac{k^2}{k_y^2} =\n\\frac{1}{2}\\Sigma_0 \\delta v_{xi0}^2 \\frac{k_0^4}{k_y^2 k^2},\n\\end{equation}\nwhich peaks at $k_x = 0$. This is not the only possible\ndefinition for the energy associated with a shear-flow disturbance;\nsee Appendix A of \\cite{ngg87} for a discussion of the subtleties\ninvolved in defining a perturbation energy in a differentially-rotating\nsystem. The energy defined above can simply be regarded as a convenient\nscalar measure of the shwave amplitude.\n\nOne can also define an amplification factor for an individual shwave,\n\\begin{equation}\n{\\cal A } \\equiv \\frac{E_{ki}(k_x = 0)}{E_{ki}(t = 0)} =\n1 + \\frac{k_{x0}^2}{k_y^2},\n\\end{equation}\nwhich indicates that an arbitrary amount of transient amplification in \nkinetic energy can be obtained as one increases the amount of swing for \na leading shwave ($k_{x0} \\ll -k_y$). This is essentially the mechanism\ninvoked by \\cite{cztl03}, \\cite{ur04} and \\cite{amn04} to argue for the onset \nof turbulence in unmagnetized Keplerian disks.\n\nBecause only a small subset of all Fourier components achieve large\namplification (those with initial wavevector very nearly aligned with\nthe radius vector), one must ask what amplification is achieved for an\nastrophysically relevant set of initial conditions containing a\nsuperposition of Fourier components. It is natural to draw such a set\nof Fourier components from a distribution that is isotropic, or nearly\nso, when $k_0$ is large.\n\nConsider, then, perturbing a disk with a random set of incompressive\nperturbations (initial velocities perpendicular to $\\bk_0$) drawn from an\nisotropic, Gaussian random field and asking how the expectation value for the\nkinetic energy associated with the perturbations evolves with time. The\nevolution of the expected energy density is given by the following\nintegral:\n\\begin{equation}\n\\ = L^2 \\int d^2k_0 \\ \n= L^2 \\int d^2k_0 \\frac{1}{2}\\Sigma_0\n\\< \\delta v_{xi0}^2\\> \\frac{k_0^4}{k_y^2 k^2}.\n\\end{equation}\nwhere $\\<\\>$ indicates an average over an ensemble of initial\nconditions, the first equality follows from Parseval's theorem, the\nsecond equality follows from the incompressive shwave solution\n(\\ref{IVX})-(\\ref{IS}) and therefore applies only for $k_0 H \\gg 1$,\nand $L^2$ is a normalizing factor with units of length squared.\n\nFor initial conditions that are isotropic in $\\bk_0$ ($\\delta v_{xi0} =\n\\delta v_\\perp (k_0,\\theta) \\sin \\theta$, where $\\<\\delta v_\\perp^2\n(k_0)\\>$ is the expectation value for the initial incompressive\nperturbation as a function of $k_0$ and $\\tan\\theta=k_y\/k_{x0}$), the\nintegral becomes\n\\begin{equation}\\label{EKI}\n\\< E_i\\> = \\frac{1}{2}\\Sigma_0 L^2 \\int k_0 dk_0 \\< \\delta v_\\perp^2 (k_0)\\>\n\\int_0^{2\\pi} d\\theta \\, \\frac{1}{\\sin^2\\theta + (q\\Omega t \\sin\\theta \n+ \\cos\\theta)^2}.\n\\end{equation}\nChanging integration variables to $\\tau = q\\Omega t + \\cot\\theta$, the angular \nintegral becomes\n\\begin{equation}\n\\int_{-\\infty}^{\\infty} d\\tau \\, \\frac{2}{1 + \\tau^2} = 2\\pi,\n\\end{equation}\nwhich is independent of time; hence\n\\begin{equation}\n\\< E_i\\> = \\< E_i (t = 0)\\>\n\\end{equation}\nand we do not expect the total energy in incompressive shwaves to\nevolve.\\footnote{Notice that while the energy of each individual shwave\ndecays asymptotically, the energy of an ensemble does not. This is\ndue to the spread of amplification factors in the spectrum of shwaves;\nsome are amplified by very large factors while others are amplified\nvery little.} This same calculation has been performed in the context of \nplane Couette flow by \\cite{shep85}, who also points out that the \namplification factor due to a distribution of wavevectors in an angular \nwedge $\\Delta \\theta$ has an upper bound of $2\\pi\/(\\Delta \\theta)$. \nThis indicates that the amplification will be modest unless the initial \ndisturbance is narrowly concentrated around a single wavevector.\n\nAlthough this result may appear to depend in detail on the assumption of\nisotropy, one can show that it really only depends on $\\< E_{ki} (t =\n0)\\>$ being smooth near $\\sin \\theta = 0$, i.e. that there should not be\na concentration of power in nearly radial wavevectors. This can be seen\nfrom the following argument. If we relax the assumption of isotropy, the\nangular integral becomes\n\\begin{equation}\n\\int_0^{2\\pi} d\\theta \\, \\frac{\\< \\delta v_\\perp^2(k_0,\\theta) \\>}\n{\\sin^2\\theta + (q\\Omega t \\sin\\theta + \\cos\\theta)^2}.\n\\end{equation}\nFor $q\\Omega t \\gg 1$ the above integrand is sharply peaked in the\nnarrow regions around $\\tan \\theta = -1\/(q\\Omega t) \\ll 1$ (i.e., $\\sin\n\\theta \\simeq 0$). One can perform a Taylor-series expansion of $\\<\n\\delta v_\\perp^2 (k_0,\\theta) \\>$ in these regions, and as long as $\\<\n\\delta v_\\perp^2(k_0,\\theta) \\>$ itself is not sharply peaked it is well\napproximated as a constant. A modest relaxation of the assumption of\nisotropy, then, will result in an asymptotically constant value for the \nenergy integral.\n\nBased upon this analysis, large amplification in an individual shwave\ndoes not in itself argue for a transition to turbulence due to transient\ngrowth. One must also demonstrate that a ``natural'' set of\nperturbations can extract energy from the background shear flow. In the\ncase of the unstratified shearing sheet, the energy of a random set of\nincompressive perturbations remains constant with time. This is\nconsistent with the results of \\cite{ur04}, who see asymptotic decay in\nlinear theory, because they work with a finite set of wavevectors, each\nof which must decay asymptotically.\n\n\\subsection{Energetics of the Compressive Shwaves}\n\nHere we calculate the energy evolution of the compressive shwaves\nfor comparison purposes. We will consider the evolution of \nshort-wavelength compressive shwaves in which only the initial velocity is \nperturbed, both for simplicity and for consistency with our calculation \nof the short-wavelength incompressive shwaves. As before, we\nwill assume that the initial kinetic energy is distributed isotropically.\n\nWe use the WKB solutions to equation (\\ref{SOUND}) with $H k_y \\gg \n1$.\\footnote{These solutions are the short-wavelength, {\\it high}-frequency \n($\\partial_t \\sim O(c_s k_y)$) limit of the full set of linear equations in \nthe shearing sheet; see the Appendix.} With the initial density perturbation \nset to zero (consistent with our assumption of only initial velocity \nperturbations), the uniformly-valid asymptotic solution to leading order \nin $(H k_y)^{-1}$ is given by\n\\begin{equation}\\label{VYWKB}\n\\delta v_{yc} = \\delta v_{yc0} \\sqrt{\\frac{k_0}{k}} \\cos(W-W_0),\n\\end{equation}\n\\begin{equation}\\label{VXWKB}\n\\delta v_{xc} = \\frac{k_x}{k_y} \\delta v_{yc},\n\\end{equation}\n\\begin{equation}\\label{SWKB}\n\\delta \\Sigma_c = \\frac{i}{c_s^2 k_y}\\dot{\\delta v}_{yc},\n\\end{equation}\nwhere the WKB eikonal is given by\n\\begin{equation}\nW \\equiv \\int c_s k \\, dt = \\frac{H k_y}{q} \\int \\sqrt{1 + \\tau^2} \\, \nd\\tau = \\frac{H k_y}{2 q} \\left(\\tau \\sqrt{1 + \\tau^2} + \\ln\\left(\\tau + \n\\sqrt{1 + \\tau^2}\\right) \\right),\n\\end{equation}\nwith $W_0$ being the value of $W$ at $t=0$.\\footnote{This is\nnot the same WKB solution that is calculated in the tight-winding\napproximation by \\cite{gt78}; in that case $c_s k_y\/\\kappa \\ll 1$, the\nopposite limit to that which we are considering here. The two WKB \nsolutions match for $\\tau \\gg 1$ in the absence of self-gravity. We have \nverified the accuracy of this solution by comparing it to the exact \nsolution with acceptable results, and it is valid to leading order for all time.}\n\nUsing equation (\\ref{VXWKB}), the energy integral for the compressive \nshwaves in the short-wavelength limit is\n\\begin{equation}\n\\< E_{c}\\> = L^2 \\int d^2k_0 \\ = L^2 \n\\int d^2k_0 \\frac{1}{2}\\Sigma_0 \\< \\delta v_{yc}^2 \\>\\frac{k^2}{k_y^2}.\n\\end{equation}\nWith initial velocities now parallel to ${\\bk}_0$ (and again isotropic), \nthis becomes\n\\begin{equation}\n\\< E_{c}\\> = \n\\frac{1}{2}\\Sigma_0 L^2\n\\int k_0 dk_0 \\< \\delta v_\\|^2(k_0) \\>\n\\int_0^{2\\pi} d\\theta \\,\n\\sqrt{\\sin^2\\theta + (q\\Omega t \\sin\\theta + \\cos\\theta)^2}\\cos^2(W-W_0).\n\\end{equation}\nFor $q \\Omega t \\gg 1$, the angular integral is approximated by\n\\begin{equation}\n\\int_0^{2\\pi} d\\theta \\, |\\sin\\theta| \\left(1 + \\cos(2W-2W_0)\\right) \\simeq \n2 q \\Omega t + \\sqrt{\\frac{2\\pi q \\Omega}{c_s k_0}} \\cos(c_s k_0 q \n\\Omega t^2 - \\pi\/4),\n\\end{equation}\nwhere the second approximation comes from employing the method of\nstationary phase.\\footnote{The first approximation breaks down near\n$\\sin \\theta = 0$, but the contribution of these regions to the integral\nis negligible for $q \\Omega t \\gg 1$, in contrast to the situation for\nincompressive shwaves.} In the short-wavelength limit, then,\n\\begin{equation}\n\\< E_{c} (q \\Omega t \\gg 1) \\> = 2 q \\Omega t \\, \\< E_{c}(t = 0) \\>.\n\\end{equation}\nThus the kinetic energy of an initially isotropic distribution of compressive\nshwaves grows, presumably at the expense of the background shear flow. \n\nThe fate of a single compressive shwave is to steepen into a weak shock\ntrain and then decay. The fate of the field of weak shocks generated by\nan ensemble of compressive shwaves is less clear, but the mere presence\nof weak shocks does not indicate a transition to turbulence.\n\n\\section{Radially-Stratified Shearing Sheet}\\label{pap2s4}\n\nWe now generalize our analysis to include the possibility that the\nbackground density and pressure varies with $x$; this stratification is\nrequired for the manifestation of a convective instability. In order to\nuse the shwave formalism we must assume that the background\nvaries on a scale $L \\sim H \\ll R$ so that the local model expansion\n(e.g., the neglect of curvature terms in the equations of motion) is\nstill valid.\n\nWith this assumption the equilibrium condition becomes\n\\begin{equation}\\label{V0}\n\\bv_0 = \\left(-q\\Omega x + \\frac{P_0^\\prime(x)}{2\\Omega \n\\Sigma_0(x)}\\right)\\ey,\n\\end{equation}\nwhere a prime denotes an $x$-derivative. \nOne can regard the background flow as providing an effective shear rate\n\\begin{equation}\n\\qe \\Omega \\equiv -v_0^\\prime\n\\end{equation}\nthat varies with $x$, in which case $\\bv_0 = -\\int^x \\qe(s) ds \\, \\Omega\n\\ey$. \n\nLocalized on this background flow we will consider a shearing\nwave with $k_y L \\gg 1$. That is, we will consider nonaxisymmetric\nshort-wavelength Eulerian perturbations with spacetime dependence\n$\\delta(t) \\exp(i\\int^x \\tilde{k}_x(t,s) ds + ik_y y + ik_z z)$,\nwhere $k_y$ and $k_z$ are constants and\n\\begin{equation}\n\\tilde{k}_x(t,x) \\equiv k_{x0} + \\qe(x)\\Omega k_y t.\n\\end{equation} \n\nIt may not be immediately obvious that this is a valid expansion since \nthe shwaves sit on top of a radially-varying background (see \\citealt{toom69} \nfor a discussion of waves in a slowly-varying background). But\nthis is an ordinary WKB expansion in disguise. To see this, one\nneed only transform to ``comoving'' coordinates $x' = x$, $y'\n= y + \\int^x \\qe(s) ds \\Omega t$, $t' = t$ (this procedure may be more familiar\nin a cosmological context; as \\cite{bal88} has pointed out, this\nis possible for any flow in which the velocities depend linearly\non the spatial coordinates). In this frame the time-dependent\nwavevector given above is transformed to a time-independent wavevector.\nThe price paid for this is that $\\partial_x \\rightarrow\n\\partial_{x'} + \\qe \\Omega t \\partial_{y'}$, so new explicit time\ndependences appear on the right hand side of the perturbed equations\nof motion, and the perturbed variables no longer have time dependence\n$\\exp(i\\omega t')$. Instead, we must solve an ODE for $\\delta(t')$.\nThe $y'$ dependence can be decomposed as $\\exp(ik_y y')$. The \n$x'$ dependence can be treated via WKB, since the perturbation may \nbe assumed to have the form $W(\\epsilon x', \\epsilon t')\\, \\exp(i\\bk' \n\\cdot \\bx')$. This ``nearly diagonalizes'' the operator $\\partial_{x'}$.\nThus we are considering the evolution of a wavepacket in comoving\ncoordinates--- a ``shwavepacket''.\n\nFor this procedure to be valid two conditions must be met. First the\nusual WKB condition must apply, $k_y L \\gg 1$. Second, the parameters\nof the flow that are ``seen'' by the shwavepacket must change little on\nthe characteristic timescale for variation of $\\delta(t)$, which is\n$\\Omega^{-1}$ for the incompressive shwaves. For solid body rotation\n($\\tilde{q} = 0$) the group velocity (derivable from equation [\\ref{DRQ0}],\nbelow) is $|v_g| < N_x\/k$ (for positive squared Brunt-V\\\"ais\\\"al\\\"a\nfrequency $N_x^2$, defined below; for $N_x^2 < 0$ the waves grow in place),\nso the timescale for change of wave packet parameters in this case is\n$L\/|v_g| > k L\/N_x \\gg \\Omega^{-1}$. It seems reasonable to anticipate\nsimilarly long timescales when shear is present. As a final check, we\nhave verified directly, using a code based on the ZEUS code of\n\\cite{sn92}, that a vortical shwavepacket in the stratified shearing\nsheet remains localized as it swings from leading to trailing.\n\n\\subsection{Linearized Equations}\n\nTo linear order in the perturbation amplitudes, the dynamical equations\nreduce to\n\\begin{equation}\\label{LIN1a}\n\\frac{\\dot{\\delta \\Sigma}}{\\Sigma_0} + \\frac{\\delta v_x}{L_{\\Sigma}} \n+ i \\tilde{k}_x \\delta v_x + i k_y \\delta v_y + i k_z \\delta v_z = 0,\n\\end{equation}\n\\begin{equation}\\label{LIN2}\n\\dot{\\delta v}_x - 2\\Omega \\delta v_y + i\\tilde{k}_x \\frac{\\delta P}{\\Sigma_0} -\n\\frac{c_s^2}{L_P} \\frac{\\delta \\Sigma}{\\Sigma_0} = 0,\n\\end{equation}\n\\begin{equation}\\label{LIN3}\n\\dot{\\delta v}_y + (2- \\qe)\\Omega \\delta v_x + ik_y \\frac{\\delta P}{\\Sigma_0} = 0,\n\\end{equation}\n\\begin{equation}\\label{LIN4}\n\\dot{\\delta v}_z + ik_z \\frac{\\delta P}{\\Sigma_0} = 0,\n\\end{equation}\n\\begin{equation}\\label{LIN5a}\n\\frac{\\dot{\\delta P}}{\\Sigma_0} - c_s^2 \\frac{\\dot{\\delta \\Sigma}}\n{\\Sigma_0} + c_s^2 \\frac{\\delta v_x}{L_S} = 0,\n\\end{equation}\nwhere\n\\begin{equation}\n\\frac{1}{L_P} \\equiv \\frac{P_0^\\prime}{\\gamma P_0} = \n\\frac{1}{L_{\\Sigma}} + \\frac{1}{L_S} \\equiv\n\\frac{\\Sigma_0^\\prime}{\\Sigma_0} +\n\\frac{S_0^\\prime}{\\gamma S_0}\n\\end{equation}\ndefine the equilibrium pressure, density and entropy length scales in\nthe radial direction.\nWe have included the vertical component of the velocity in\norder to make contact with an axisymmetric convective instability that\nis present in two dimensions, after which we will set $k_z$ to zero. \n\nWe will be mainly interested in the incompressive shwaves because the\nshort-wavelength compressive shwaves are unchanged at leading order by\nstratification. We will therefore work solely in the Boussinesq\napproximation.\\footnote{We also drop the subscripts distinguishing\nbetween the compressive and incompressive shwaves.} In addition to the\nassumption of incompressibility, this approximation considers $\\delta P$\nto be negligible in the entropy equation; pressure changes are\ndetermined by whatever is required to maintain nearly incompressible\nflow. The original Boussinesq approximation applies only to\nincompressible fluids. It was extended to compressible fluids by\n\\cite{jeff30} and \\cite{sv60}. We show in the Appendix that it is\nformally equivalent to taking the short-wavelength, low-frequency limit\nof the full set of linear equations. From this viewpoint, assuming that\n$H k_y \\delta P\/P_0$ is of the same order as the other terms in the\ndynamical equations implies that $\\delta P\/P_0 \\sim (H k_y)^{-1} \\delta\n\\Sigma\/\\Sigma_0$, thus justifying its neglect in the entropy equation.\nWe therefore replace equations (\\ref{LIN1a}) and (\\ref{LIN5a}) with\n\\begin{equation}\\label{LIN1}\n\\tilde{k}_x \\delta v_x + k_y \\delta v_y + k_z \\delta v_z = 0\n\\end{equation}\nand\n\\begin{equation}\\label{LIN5}\n\\frac{\\dot{\\delta \\Sigma}}{\\Sigma_0} - \\frac{\\delta v_x}{L_S} = 0.\n\\end{equation}\n\nUsing equations (\\ref{LIN3}) and (\\ref{LIN5}) and the time derivative of\nequation (\\ref{LIN1}), one can express $\\dot{\\delta v}_y$ and $\\delta P$\nin terms of $\\delta v_x$ and $\\dot{\\delta v}_x$:\n\\begin{equation}\\label{DH}\n\\frac{\\delta P}{\\Sigma_0} = -i \\frac{\\tilde{k}_x \\dot{\\delta v}_x + 2(\\qe - 1) \n\\Omega k_y \\delta v_x}{k_y^2 + k_z^2},\n\\end{equation}\n\\begin{equation}\\label{DVY}\n\\dot{\\delta v}_y = \\frac{(-\\qe k_y^2 + (\\qe - 2)k_z^2) \\Omega \\delta v_x - \n\\tilde{k}_x k_y \\dot{\\delta v}_x}{k_y^2 + k_z^2}.\n\\end{equation}\nEliminating $\\delta P$ in equation (\\ref{LIN2}) via equation (\\ref{DH}) gives\n\\begin{equation}\n\\tilde{k}^2 \\dot{\\delta v}_x + 2(\\qe - 1) \\Omega \\tilde{k}_x k_y \\delta v_x = (k_y^2 + k_z^2) (2 \\Omega \\delta v_y + (c_s^2\/L_P) \\delta \\Sigma\/\\Sigma_0),\n\\end{equation}\nwhere $\\tilde{k}^2 = \\tilde{k}_x^2 + k_y^2 + k_z^2$. Taking the time\nderivative of this equation and eliminating $\\dot{\\delta \\Sigma}$ and\n$\\dot{\\delta v}_y$ via equations (\\ref{LIN5}) and (\\ref{DVY}), we obtain\nthe following differential equation for $\\delta v_x$:\n\\begin{equation}\\label{BOUSSVX}\n\\tilde{k}^2 \\ddot{\\delta v}_x + 4 \\qe \\Omega \\tilde{k}_x k_y \\dot{\\delta v}_x \n+ \\left[k_y^2\\left(N_x^2 + 2\\qe^2 \\Omega^2\\right) + k_z^2\\left(N_x^2 +\n \\tilde{\\kappa}^2\\right)\\right]\\delta v_x = 0,\n\\end{equation}\nwhere $\\tilde{\\kappa}^2 = 2(2-\\qe)\\Omega^2$ is the square of the\neffective epicyclic frequency and \n\\begin{equation}\nN_x^2 \\equiv -\\frac{c_s^2}{L_S L_P}\n\\end{equation}\nis the square of the\nBrunt-V$\\ddot{\\rm{a}}$is$\\ddot{\\rm{a}}$l$\\ddot{\\rm{a}}$ frequency in the\nradial direction.\\footnote{Notice that $N_x^2$, $\\qe$ and\n$\\tilde{\\kappa}^2$ are all functions of $x$ and vary on a scale $L \\sim\nH$.}\n\n\\subsection{Comparison with Known Results}\n\nSetting $k_y = 0$ in equation (\\ref{BOUSSVX}) yields the axisymmetric\nmodes with the following dispersion relation (for $\\delta(t) \\propto\ne^{-i\\omega t}$):\n\\begin{equation}\n\\omega^2 = \\frac{k_z^2}{k_{x0}^2 + k_z^2}\\left(N_x^2 + \\tilde{\\kappa}^2\\right).\n\\end{equation}\nThis is the origin of the H{\\o}iland stability criterion: the axisymmetric\nmodes are stable for $N_x^2 + \\tilde{\\kappa}^2 > 0$. In the absence of\nrotation this reduces to the Schwarzschild stability criterion: $N_x^2 >\n0$ is the necessary condition for stability. The effect of rotation is strongly \nstabilizing: if $N_x^2 < -\\tilde{\\kappa}^2$, as required for instability, then \n$L_S L_P \\sim H^2$; pressure and entropy must vary on radial scales of \norder the scale height for the disk to be H{\\o}iland unstable.\n\nNotice that effective epicyclic frequency $\\tilde{\\kappa}^2$ only stabilizes \nmodes with nonzero $k_z$. The stability of nonaxisymmetric\nshwaves with $k_z = 0$ (as in the mid-plane of a thin disk) is the open \nquestion that this work is addressing. In this limit and in\nthe absence of shear the Schwarzschild stability criterion is again\nrecovered: with $k_z = 0$ and $\\qe = 0$ in equation (\\ref{BOUSSVX}) the\ndispersion relation becomes\n\\begin{equation}\\label{DRQ0}\n\\omega^2 = \\frac{k_y^2}{k_{x0}^2 + k_y^2}N_x^2,\n\\end{equation}\nIf there is a region of the disk where the effective shear is zero, a\nWKB normal-mode analysis will yield the above dispersion relation and\nthere will be convective instability for $N_x^2 < 0$. It appears from\nequation (\\ref{BOUSSVX}) that differential rotation provides a\nstabilizing influence for nonaxisymmetric shwaves just as rotation\ndoes for the axisymmetric modes. Things are not as simple in\nnonaxisymmetry, however. The time dependence is no longer exponential,\nnor is it the same for all the perturbation variables. There is no clear\ncutoff between exponential and oscillatory behavior, so the question of\nflow stability becomes more subtle. \n\nAs discussed in the introduction, the Boussinesq system of equations in \nthe shearing-sheet model of a radially-stratified disk bear a close resemblance \nto the system of equations employed in analyses of a shearing, stratified \natmosphere. A sufficient condition for stability in the latter case is that \n\\begin{equation}\\label{RICH}\n{\\rm Ri} \\equiv \\frac{N_x^2}{(v_0^\\prime)^2} \\geq \\frac{1}{4}\n\\end{equation}\neverywhere in the flow, where Ri is the Richardson number, a measure of \nthe relative importance of buoyancy and shear. This stability criterion was \noriginally proved by \\cite{jwm61} and \\cite{how61} for incompressible \nfluids, and its extension to compressible fluids was demonstrated by \n\\cite{chi70}. The stability criterion is based on a normal-mode analysis \nwith rigid boundary conditions. Other than differences in notation (e.g., \nour radial coordinate corresponds to the vertical coordinate in a stratified \natmosphere), the key differences in our system are: (i) the equilibrium \npressure gradient in a disk is balanced by centrifugal forces rather than \nby gravity; (ii) the disk equations contain Coriolis force terms; (iii) most \natmospheric analyses only consider an equilibrium that is convectively \nstable, whereas we are interested in an unstable stratification; (iv) we\ndo not employ boundary conditions in our analytic model since we are \nonly interested in the possibility of a local instability. \n\nThe lack of boundary conditions in our model makes the applicability of \nthe standard Richardson stability criterion in determining local stability \nsomewhat dubious, since the lack of boundary conditions precludes \nthe decomposition of linear disturbances into normal modes. \nThe natural procedure for performing a local linear analysis in disks \nis to decompose the perturbations into shwaves, as we have done.\n\n\\cite{ehr53} consider both stable and unstable atmospheres and analyze \nan initial-value problem by decomposing the perturbations in time via\nLaplace transforms. For flow between \ntwo parallel walls, they find that an arbitrary initial disturbance behaves \nasymptotically as $t^{(\\alpha - 1)\/2}$ for $-3\/4 < {\\rm Ri} < 1\/4$, where\n\\begin{equation}\\label{ALPHA}\n\\alpha \\equiv \\sqrt{1 - 4 \\, {\\rm{Ri}}},\n\\end{equation}\nwhich grows algebraically for ${\\rm Ri} < 0$. The disturbance grows \nexponentially only for ${\\rm Ri} < -3\/4$. For a semi-infinite flow, the \npower-law behavior in time holds for $-2 < {\\rm Ri} < 1\/4$, with \nexponential growth for ${\\rm Ri} < -2$. These results illustrate the \nimportance of boundary conditions in determining stability.\n\nIn the $k_z = 0$ limit that we are concerned with here, the correspondence \nbetween the disk and atmospheric models turns out to be exact in the \nshwave formalism. This is because the Coriolis force only appears in\nequation (\\ref{BOUSSVX}) via $\\tilde{\\kappa}^2$, which disappears\nwhen $k_z = 0$. The equation describing the time evolution of shwaves\nin both a radially-stratified disk and a shearing, stratified atmosphere \nis thus\n\\begin{equation}\\label{BOUSSVX2D}\n\\tilde{k}^2 \\ddot{\\delta v}_x + 4 \\qe \\Omega \\tilde{k}_x k_y \\dot{\\delta v}_x\n+ k_y^2\\left(N_x^2 + 2\\qe^2 \\Omega^2\\right)\\delta v_x = 0.\n\\end{equation}\nWe analyze the solutions to this equation in the following \nsection.\\footnote{This equation is also obtained in a shwave analysis\nof interchange instability in a disk with a poloidal magnetic field\n\\citep{ssp95}, with $N_x^2$ replaced by a magnetic buoyancy frequency.}\n\n\\subsection{Solutions}\n\nChanging time variables in equation (\\ref{BOUSSVX2D}) to \n$\\tilde{\\tau} \\equiv \\tilde{k}_x\/k_y$, the\ndifferential equation governing $\\delta v_x$ becomes\n\\begin{equation}\\label{ODE}\n(1 + \\tilde{\\tau}^2)\\dv{^2\\delta v_x}{\\tilde{\\tau}^2} + 4 \\tilde{\\tau} \n\\dv{\\delta v_x}{\\tilde{\\tau}} + ({\\rm{Ri}} + 2)\\delta v_x = 0.\n\\end{equation}\nThe solutions to equation (\\ref{ODE}) are\nhypergeometric functions. With the change of variables $z \\equiv\n-\\tilde{\\tau}^2$, equation (\\ref{ODE}) becomes\n\\begin{equation}\\label{ODEZ}\nz(1-z)\\dv{^2\\delta v_x}{z^2} + \\frac{1- 5z}{2}\\dv{\\delta v_x}{z} - \n\\frac{{\\rm{Ri}} + 2}{4}\\delta v_x = 0.\n\\end{equation}\nThe hypergeometric equation \\citep{as72}\n\\begin{equation}\\label{HGEQ}\nz(1-z)\\dv{^2\\delta v_x}{z^2} + \\left[c - (a + b + 1)z\\right]\\dv{\\delta v_x}{z} \n- ab\\delta v_x = 0\n\\end{equation}\nhas as its two linearly independent solutions $F(a,b;c;z)$ and $z^{1-c}\nF(a-c+1,b-c+1;2-c;z)$. Comparison of equations (\\ref{ODEZ}) and \n(\\ref{HGEQ}) shows that $a = (3 - \\alpha)\/4$, $b = (3 + \\alpha)\/4$ and \n$c = 1\/2$, where $\\alpha$ is defined in equation (\\ref{ALPHA}).\n\nThe general solution for $\\delta v_x$ is thus given by\n\\begin{equation}\\label{SOLVX}\n\\delta v_x = C_1 \\,F\\left(\\frac{3 - \\alpha}{4},\\frac{3 + \\alpha}{4};\n\\frac{1}{2};-\\tilde{\\tau}^2\\right) + C_2 \\, \\tilde{\\tau}\\,F\\left(\\frac{\n5 - \\alpha}{4},\\frac{5 + \\alpha}{4};\\frac{3}{2};-\\tilde{\\tau}^2\\right),\n\\end{equation} \nwhere $C_1$ and $C_2$ are constants of integration representing\nthe two degrees of freedom in our reduced system. These two \ndegrees of freedom can be represented physically by the initial velocity \nand displacement of a perturbed fluid particle in the radial direction. \nThe radial Lagrangian displacement $\\xi_x$ is obtained from \nequation (\\ref{SOLVX}) by direct integration,\\footnote{In our notation, \na subscript $x$ or $y$ on the symbol $\\xi$ indicates a Lagrangian displacement, not \na component of the potential vorticity, which is a scalar.}\n\\begin{equation}\\label{SOLX}\n\\xi_x = \\int \\delta v_x \\, dt = -\\frac{C_2}\n{\\qe \\Omega {\\rm Ri}} \\,F\\left(\\frac{1 - \\alpha}{4},\\frac{1 + \\alpha}{4};\n\\frac{1}{2};-\\tilde{\\tau}^2\\right) + \\frac{C_1}{\\qe \\Omega}\\,\\tilde{\\tau}\\,\nF\\left(\\frac{3 - \\alpha}{4},\\frac{3 + \\alpha}{4};\\frac{3}{2};\n-\\tilde{\\tau}^2\\right).\n\\end{equation}\nThe solutions for the other perturbation variables can be obtained from \nequations (\\ref{LIN1}), (\\ref{LIN5}) and (\\ref{DH}) with $k_z = 0$:\n\\begin{equation}\n\\delta v_y = -\\tilde{\\tau} \\delta v_x,\n\\end{equation}\n\\begin{equation}\\label{SOX}\n\\frac{\\delta \\Sigma}{\\Sigma_0} = \\frac{\\xi_x}{L_S}\n\\end{equation}\nand\n\\begin{equation}\\label{SOLDH}\n\\frac{\\delta P}{P_0} = \\frac{\\gamma \\Omega}{i c_s k_y} \\left[\\qe \n\\tilde{\\tau} \\dv{}{\\tilde{\\tau}}\\left(\\frac{\\delta v_x}{c_s}\\right) + \n2(\\qe - 1) \\frac{\\delta v_x}{c_s}\\right].\n\\end{equation}\nIt can be seen from the latter equation and the solution for $\\delta\nv_x$ that $\\delta P\/P_0$ remains small compared to $\\delta v_x\/c_s$ in the\nshort-wavelength limit. This demonstrates the consistency of the Boussinesq approximation. \n\nThe hypergeometric functions can be transformed to a form valid for\nlarge $\\tilde{\\tau}$ (see \\citealt{as72} equations 15.3.7 and 15.1.1).\nAn equivalent form of the solution for $|\\tilde{\\tau}| \\gg 1$ is\n\\begin{eqnarray}\n\\delta v_x = (C_1 V_1 + {\\rm{sgn}}(\\tilde{\\tau}) C_2 V_2) \\, |\\tilde{\\tau}|^\n{\\frac{\\alpha - 3}{2}} F\\left(\\frac{3 - \\alpha}{4}, \\frac{5 - \\alpha}{4}; 1 -\n \\frac{\\alpha}{2}; -\\frac{1}{\\tilde{\\tau}^2}\\right) + \\nonumber \\\\ (C_1 V_3 +\n {\\rm{sgn}}(\\tilde{\\tau}) C_2 V_4) \\, |\\tilde{\\tau}|^{-\\frac{\\alpha+3}{2}} \nF\\left(\\frac{3 + \\alpha}{4}, \\frac{5 + \\alpha}{4}; 1 + \\frac{\\alpha}{2}; \n-\\frac{1}{\\tilde{\\tau}^2}\\right), \\; \\; \\;\n\\end{eqnarray}\nwhere ${\\rm{sgn}}(\\tilde{\\tau})$ is the arithmetic sign of\n$\\tilde{\\tau}$ and the constants $V_i$ are given by\n\\begin{eqnarray}\nV_1 \\equiv \\frac{\\Gamma\\left(\\frac{1}{2}\\right) \\Gamma\\left(\\frac{\\alpha}\n{2}\\right)} {\\Gamma\\left(\\frac{3 + \\alpha}{4}\\right)\\Gamma\\left(-\\frac{1 -\n \\alpha}{4}\\right)} \\;\\; , \\;\\; V_2 \\equiv \\frac{\\Gamma\\left(\\frac{3}{2}\\right)\n \\Gamma\\left(\\frac{\\alpha}{2}\\right)} {\\Gamma\\left(\\frac{5 + \\alpha\n }{4}\\right)\\Gamma\\left(\\frac{1 + \\alpha}{4}\\right)} , \\;\\; \\nonumber \\\\\nV_3 \\equiv \\frac{\\Gamma\\left(\\frac{1}{2}\\right) \\Gamma\\left(-\\frac{\\alpha}\n{2}\\right)} {\\Gamma\\left(\\frac{3 - \\alpha}{4}\\right)\\Gamma\\left(-\\frac{1 +\n \\alpha}{4}\\right)} \\;\\; , \\;\\; V_4 \\equiv \\frac{\\Gamma\\left(\\frac{3}{2}\\right)\n \\Gamma\\left(-\\frac{\\alpha}{2}\\right)} {\\Gamma\\left(\\frac{5 - \\alpha}\n{4}\\right)\\Gamma\\left(\\frac{1 - \\alpha}{4}\\right)}.\n\\end{eqnarray}\nExpanding the above form of the solution for $|\\tilde{\\tau}| \\gg 1$, we obtain\n\\begin{equation}\\label{DVAS}\n\\delta v_x = \\left(C_1 V_1 + {\\rm{sgn}}(\\tilde{\\tau}) C_2 V_2\\right)\\,\n |\\tilde{\\tau}|^{\\frac{\\alpha - 3}{2}} + \\left(C_1 V_3 + {\\rm{sgn}}\n(\\tilde{\\tau}) C_2 V_4\\right) \\, |\\tilde{\\tau}|^{-\\frac{\\alpha + 3}{2}} + \nO(\\tilde{\\tau}^{-2}).\n\\end{equation}\n\nAn equivalent form of $\\xi_x$ for $|\\tilde{\\tau}| \\gg 1$ is\n\\begin{eqnarray}\n\\xi_x = \\left(-\\frac{C_2 X_1}{\\qe \\Omega {\\rm Ri}} + {\\rm{sgn}}\n(\\tilde{\\tau}) \\frac{C_1 X_2}{\\qe \\Omega}\\right) \\, |\\tilde{\\tau}|^{\\frac{\n\\alpha - 1}{2}} F\\left(\\frac{3 - \\alpha}{4}, \\frac{1 - \\alpha}{4}; 1 - \n\\frac{\\alpha}{2}; -\\frac{1}{\\tilde{\\tau}^2}\\right) + \\nonumber \\\\ \\left(\n-\\frac{C_2 X_3}{\\qe \\Omega {\\rm Ri}} + {\\rm{sgn}}(\\tilde{\\tau}) \n\\frac{C_1 X_4}{\\qe \\Omega}\\right) \\, |\\tilde{\\tau}|^{-\\frac{\\alpha+1}{2}} \nF\\left(\\frac{3 + \\alpha}{4}, \\frac{1 + \\alpha}{4}; 1 + \\frac{\\alpha}{2};\n-\\frac{1}{\\tilde{\\tau}^2}\\right), \\; \\; \\;\n\\end{eqnarray}\nwhere the constants $X_i$ are given by\n\\begin{eqnarray}\nX_1 \\equiv \\frac{\\Gamma\\left(\\frac{1}{2}\\right) \\Gamma\\left(\\frac{\\alpha}\n{2}\\right)} {\\Gamma\\left(\\frac{1 + \\alpha}{4}\\right)\\Gamma\\left(\\frac{1 + \n\\alpha}{4}\\right)} \\;\\; , \\;\\; X_2 \\equiv \\frac{\\Gamma\\left(\\frac{3}{2}\\right)\n \\Gamma\\left(\\frac{\\alpha}{2}\\right)} {\\Gamma\\left(\\frac{3 + \\alpha}\n{4}\\right)\\Gamma\\left(\\frac{3 + \\alpha}{4}\\right)} , \\;\\; \\nonumber \\\\\nX_3 \\equiv \\frac{\\Gamma\\left(\\frac{1}{2}\\right) \\Gamma\\left(-\\frac{\\alpha}\n{2}\\right)} {\\Gamma\\left(\\frac{1 - \\alpha}{4}\\right)\\Gamma\\left(\\frac{1 - \n\\alpha}{4}\\right)} \\;\\; , \\;\\; X_4 \\equiv \\frac{\\Gamma\\left(\\frac{3}{2}\\right)\n \\Gamma\\left(-\\frac{\\alpha}{2}\\right)} {\\Gamma\\left(\\frac{3 - \\alpha}\n{4}\\right)\\Gamma\\left(\\frac{3 - \\alpha}{4}\\right)}.\n\\end{eqnarray}\nExpanding $\\xi_x$ for $|\\tilde{\\tau}| \\gg 1$ yields\n\\begin{equation}\\label{DSAS}\n\\xi_x = \\left(-\\frac{C_2 X_1}{\\qe \\Omega {\\rm Ri}} + {\\rm{sgn}}\n(\\tilde{\\tau}) \\frac{C_1 X_2}{\\qe \\Omega}\\right) \\, |\\tilde{\\tau}|^\n{\\frac{\\alpha - 1}{2}} + \\left(-\\frac{C_2 X_3}{\\qe \\Omega {\\rm Ri}} + \n{\\rm{sgn}}(\\tilde{\\tau}) \\frac{C_1 X_4}{\\qe \\Omega}\\right) \\, |\\tilde{\\tau}|^\n{-\\frac{\\alpha+1}{2}} + O(\\tilde{\\tau}^{-2}). \n\\end{equation}\n\nThe dominant contribution for each perturbation variable at late times is thus\n\\begin{equation}\n\\delta P \\propto \\delta v_x \\sim t^{\\frac{\\alpha - 3}{2}},\n\\end{equation}\n\\begin{equation}\n\\delta \\Sigma \\propto \\xi_x \\sim t^{\\frac{\\alpha - 1}{2}},\n\\end{equation}\nand\n\\begin{equation}\n\\delta v_y \\propto t \\delta v_x \\sim t^{\\frac{\\alpha - 1}{2}}.\n\\end{equation}\nThis leads to one of our main conclusions: the density and $y$-velocity\nperturbations will grow asymptotically for $\\alpha > 1$, i.e.\n${\\rm{Ri}} \\propto N_x^2 < 0$.\\footnote{Notice that this is the same time \ndependence obtained by \\cite{ehr53} in a modal analysis; see the discussion \nsurrounding equation (\\ref{ALPHA}). These power law time-dependences \ncan be obtained more efficiently by solving the large-$\\tilde{\\tau}$ limit \nof equation (\\ref{SOLVX}).} For small Richardson number, \nhowever (as is expected for a Keplerian disk with modest radial gradients), \n$\\alpha \\sim 1 - 2{\\rm Ri}$ and the asymptotic growth is extremely slow:\n\\begin{equation}\n\\delta \\Sigma \\sim \\delta v_y \\sim t^{-{\\rm Ri}}.\n\\end{equation}\n\nIn the stratified shearing sheet, the right-hand side of equation \n(\\ref{PVEVLIN}) governing the evolution of the perturbed potential \nvorticity is no longer zero. The form of this equation for the incompressive \nshwaves is\n\\begin{equation}\\label{DPVEV}\n\\dot{\\delta \\xi} = \\frac{d}{d t}\\left(\\frac{i\\tilde{k}_x \\delta v_y - i k_y \\delta v_x}\n{\\Sigma_0}\\right) = \\frac{c_s^2 k_y}{i L_P \\Sigma_0^2}\\delta \\Sigma.\n\\end{equation}\nThe asymptotic time dependence of the perturbed potential vorticity can\nbe obtained by integrating equation (\\ref{DPVEV}):\n\\begin{equation}\n\\delta \\xi \\sim t^{\\frac{\\alpha + 1}{2}} \\sim t^{1 - {\\rm Ri}}\n\\end{equation}\nfor $\\tilde{\\tau} \\gg 1$ and $|{\\rm Ri}| \\ll 1$. As noted in \\S 2, an entropy \ngradient is not required to generate vorticity. For $N_x^2 = 0$, $\\alpha = 1$ \nand the perturbed potential vorticity grows linearly with time. The unstratified \nshearing sheet is recovered in the limit of zero stratification ($1\/L_P \\rightarrow \n0$), since in this limit equation (\\ref{DPVEV}) reduces to $\\xi = constant$.\n\n\\subsection{Energetics of the Incompressive Shwaves}\n\nFor a physical interpretation of the incompressive shwaves in the stratified \nshearing sheet, we repeat the analysis of section 3.3 for the solution \ngiven in the previous section. For a complete description of the energy in\nthis case, however, we must include the potential energy of a fluid element \ndisplaced in the radial direction. Following \\cite{jwm61}, an \nexpression for the energy in the Boussinesq approximation is obtained by\nsumming equation (\\ref{LIN2}) multiplied by $\\delta v_x$ \nand equation (\\ref{LIN3}) multiplied by $\\delta v_y$. Replacing \n$\\delta \\Sigma\/\\Sigma_0$ by $\\xi_x\/L_S$ via equation (\\ref{SOX}) \nresults in the following expression for the energy evolution:\n\\begin{equation}\\label{ENERGY}\n\\dv{E_k}{\\tilde{\\tau}} \\equiv \\dv{}{\\tilde{\\tau}} \\left(\\frac{1}{2}\\Sigma_0 \n\\delta v^2 + \\frac{1}{2} \\Sigma_0 N_x^2 \\xi_x^2 \\right) = \\Sigma_0 \\delta \nv_x \\delta v_y,\n\\end{equation}\nwhere $\\delta v^2 = \\delta v_x^2 + \\delta v_y^2$. The three terms in equation \n(\\ref{ENERGY}) can be identified as the kinetic energy, potential energy \nand Reynolds stress associated with an individual shwave. One may readily \nverify that the vortical shwaves (see equations (\\ref{IVX})-(\\ref{IS})) in the \nunstratified shearing sheet ($N_x^2 = 0$) satisfy equation (\\ref{ENERGY}).\n\nThe right hand side of equation (\\ref{ENERGY}) can be rewritten $-\\tilde{\\tau}\n\\delta v_x^2$ and individual trailing shwaves ($\\tilde{\\tau} > 0$) are therefore\nassociated with a negative angular momentum flux. If the energy were\npositive definite this would require that individual shwaves always\ndecay. But when $N_x^2 < 0$ (${\\rm Ri} < 0$) the potential energy\nassociated with a displacement is negative, so the energy $E_k$ can be\nnegative and a negative angular momentum flux is not enough to halt\nshwave growth. \n\nOur next step is to write the constants of integration $C_1$ and $C_2$ \nin terms of the initial radial velocity and displacement of the shearing \nwave, $\\delta v_{x0}$ and $\\xi_{x0}$:\n\\begin{equation}\nC_1 = \\frac{\\qe \\Omega {\\rm Ri}\\, \\delta v_{x2}(\\tilde{\\tau}_0) \\, \n\\xi_{x0} + \\xi_{x1}(\\tilde{\\tau}_0) \\, \\delta v_{x0}}{\\delta v_{x1}\n(\\tilde{\\tau}_0) \\, \\xi_{x1}(\\tilde{\\tau}_0) + {\\rm Ri}\\,\\delta v_{x2}\n(\\tilde{\\tau}_0) \\, \\xi_{x2}(\\tilde{\\tau}_0)} \\;\\; , \\;\\;\nC_2 = \\frac{-\\qe \\Omega {\\rm Ri}\\, \\delta v_{x1}(\\tilde{\\tau}_0) \\,\n\\xi_{x0} + {\\rm Ri}\\,\\xi_{x2}(\\tilde{\\tau}_0) \\, \\delta v_{x0}}{\\delta \nv_{x1}(\\tilde{\\tau}_0) \\, \\xi_{x1}(\\tilde{\\tau}_0) + {\\rm Ri}\\,\\delta \nv_{x2}(\\tilde{\\tau}_0) \\, \\xi_{x2}(\\tilde{\\tau}_0)},\n\\end{equation}\nwhere $\\tilde{\\tau}_0 = k_{x0}\/k_y$, $\\delta v_{x1}$ is the \nhypergeometric function given by equation (\\ref{SOLVX}) with $C_1 \n= 1$ and $C_2 = 0$, and the other functions are similarly defined. These \nexpressions can be simplified by noticing that the denominator of $C_1$ \nand $C_2$ is the Wronskian of the differential equation for $\\xi_x$:\\footnote{\nBased upon the relationship between a hypergeometric function \nand its derivatives, $\\delta v_{x1} = d(\\xi_{x2})\/d\\tilde{\\tau}$ and \n${\\rm Ri} \\delta v_{x2} = -d(\\xi_{x1})\/d\\tilde{\\tau}$.}\n\\begin{equation}\\label{ODEX}\n(1 + \\tilde{\\tau}^2)\\dv{^2\\xi_x}{\\tilde{\\tau}^2} + 2 \\tilde{\\tau} \n\\dv{\\xi _x}{\\tilde{\\tau}} + {\\rm Ri}\\xi_x = 0.\n\\end{equation}\nThe Wronskian of this equation is\n\\begin{equation}\n{\\cal W} \\equiv \\dv{\\xi_{x2}}{\\tilde{\\tau}} \\, \\xi_{x1}\n- \\dv{\\xi_{x1}}{\\tilde{\\tau}} \\, \\xi_{x2}\n= \\exp \\left(-\\int^{\\tilde{\\tau}} \\frac{2\\tau^2}\n{1 + \\tau^2} \\, d\\tau \\right) = \\frac{1}{1 + \\tilde{\\tau}^2}.\n\\end{equation}\nWe further simplify the analysis by setting the initial displacement $\\xi_{x0}$\nto zero.\n\nWith these simplifications, the solution given by equations \n(\\ref{SOLVX}) and (\\ref{SOLX}) becomes\n\\begin{eqnarray}\n\\frac{\\delta v_x}{\\delta v_{x0}} = \\left(1 + \\tilde{\\tau}_0^2 \\right)\n\\left[F\\left(\\frac{1 - \\alpha}{4},\\frac{1 + \\alpha}{4};\\frac{1}{2};\n-\\tilde{\\tau}_0^2\\right) \\,F\\left(\\frac{3 - \\alpha}{4},\\frac{3 + \\alpha}{4};\n\\frac{1}{2};-\\tilde{\\tau}^2\\right) + \\right. \\nonumber \\\\ \\left. {\\rm Ri}\\,\n\\tilde{\\tau}_0\\,F\\left(\\frac{3 - \\alpha}{4},\\frac{3 + \\alpha}{4};\n\\frac{3}{2};-\\tilde{\\tau}_0^2\\right) \\, \\tilde{\\tau}\\,F\\left(\\frac{5 - \\alpha}\n{4},\\frac{5 + \\alpha}{4};\\frac{3}{2};-\\tilde{\\tau}^2\\right)\\right],\n\\end{eqnarray}\n\\begin{eqnarray}\n\\frac{\\xi_x}{\\delta v_{x0}} = \\left(1 + \\tilde{\\tau}_0^2\\right)\\left[\n-\\frac{1}{\\qe \\Omega} \\tilde{\\tau}_0\\,F\\left(\\frac{3 - \\alpha}{4},\n\\frac{3 + \\alpha}{4};\\frac{3}{2};-\\tilde{\\tau}_0^2\\right) \\,F\\left(\n\\frac{1 - \\alpha}{4},\\frac{1 + \\alpha}{4};\\frac{1}{2};-\\tilde{\\tau}^2\n\\right) + \\right. \\nonumber \\\\ \\left. \\frac{1}{\\qe \\Omega} F\\left(\\frac{1 \n- \\alpha}{4},\\frac{1 + \\alpha}{4};\\frac{1}{2};-\\tilde{\\tau}_0^2\\right) \n\\,\\tilde{\\tau}\\, F\\left(\\frac{3 - \\alpha}{4},\\frac{3 + \\alpha}{4};\n\\frac{3}{2};-\\tilde{\\tau}^2\\right)\\right].\n\\end{eqnarray}\n\nAs in section 3.3, the energy integral for the incompressive perturbations \nis given by\n\\begin{equation}\n\\< E_i \\> = \\frac{1}{2}\\Sigma_0 L^2 \\int k_0 dk_0 \\< \\delta v_\\perp^2(k_0)\\>\n\\int_0^{2\\pi} d\\theta \\sin^2\\theta \\left[ \\left(1 + \\tilde{\\tau}^2\\right) \n\\left(\\frac{\\delta v_x}{\\delta v_{x0}}\\right)^2 + N_x^2 \\left(\n\\frac{\\xi_x}{\\delta v_{x0}}\\right)^2\\right],\n\\end{equation}\nfor initial perturbations perpendicular to and isotropic in $\\bk_0$. \nChanging integration variables to $\\tilde{\\tau} = \\qe\\Omega t + \n\\cot\\theta$, the angular integral becomes\n\\begin{eqnarray}\\label{EINT}\n2 \\int_{-\\infty}^{\\infty} d\\tilde{\\tau} \\, \n\\left[ \\left(1 + \\tilde{\\tau}^2\\right) \\left\\{\\xi_{x1}\n(\\tilde{\\tau} - \\qe \\Omega t)\\delta v_{x1}(\\tilde{\\tau}) + {\\rm{Ri}}\n \\, \\xi_{x2}(\\tilde{\\tau} - \\qe \\Omega t) \\delta v_{x2}(\\tilde{\\tau})\n\\right\\}^2 + \\right. \\nonumber \\\\ \\left. {\\rm Ri} \\, \\left\\{\\xi_{x2}\n(\\tilde{\\tau} - \\qe \\Omega t)\\xi_{x1} (\\tilde{\\tau}) - \\xi_{x1}\n(\\tilde{\\tau} - \\qe \\Omega t) \\xi_{x2}(\\tilde{\\tau})\\right\\}^2\\right],\n\\end{eqnarray}\nwhere we have used the relation $\\sin\\theta = (1 + \\tilde{\\tau}_0^2)\n^{-1}$. In the limit of large $\\qe \\Omega t$, the dominant \ncontribution to the angular integral comes from the region $0 \\lesssim \n\\tilde{\\tau} \\lesssim \\qe \\Omega t$. This can be seen from the \nfollowing argument. Using the expansions given by equations \n(\\ref{DVAS}) and (\\ref{DSAS}), we find the angular integrand is\n\\begin{equation}\n2 |\\tilde{\\tau}(\\tilde{\\tau} - \\qe \\Omega t)|^{\\alpha-1}\\left[(V_1 \nX_1 + {\\rm sgn}(\\tilde{\\tau}){\\rm sgn}(\\tilde{\\tau} - \\qe \\Omega \nt){\\rm{Ri}} \\, V_2 X_2)^2 + {\\rm Ri} X_1^2 X_2^2 ({\\rm sgn}\n(\\tilde{\\tau}) - {\\rm sgn}(\\tilde{\\tau} - \\qe \\Omega t))^2\\right]\n\\end{equation}\nfor $|\\tilde{\\tau}| \\gg 1$ and $|\\tilde{\\tau} - \\qe \\Omega t| \\gg 1$. \nUsing the relation $\\Gamma(n+1) = n\\Gamma(n)$, one can easily\nshow that\n\\begin{equation}\nX_2 = \\frac{2}{\\alpha - 1}V_1 \\;\\;\\;\\; {\\rm and} \\;\\;\\;\\; V_2 = \\frac{2}\n{\\alpha + 1}X_1.\n\\end{equation}\nThe integrand therefore simplifies to\n\\begin{equation}\n|\\tilde{\\tau}(\\tilde{\\tau} - \\qe \\Omega t)|^{\\alpha-1}V_1^2 \nX_1^2\\frac{2}{1 - \\alpha}\\left[{\\rm sgn}\n(\\tilde{\\tau}) - {\\rm sgn}(\\tilde{\\tau} - \\qe \\Omega t)\\right]^2,\n\\end{equation}\nwhich is zero unless $0 < \\tilde{\\tau} < \\qe \\Omega t$ (for $t > 0$). \nFor large $\\qe \\Omega t$, therefore, the angular integral is \napproximately given by\n\\begin{equation}\n\\frac{16 V_1^2 X_1^2}{1 - \\alpha} \\int_\\nu^{\\qe \\Omega \nt - \\nu} d\\tilde{\\tau} \\, \\left[\\tilde{\\tau}\\left(\\tilde{\\tau}-\\qe \n\\Omega t\\right)\\right]^{\\alpha-1} = \\frac{16 V_1^2 X_1^2}{\\alpha \n(1 - \\alpha)} \\left. \\frac{(\\tilde{\\tau} \\qe \\Omega t)^{\\alpha}}\n{\\qe \\Omega t} F\\left(\\alpha,1-\\alpha;1+\\alpha;\\frac{\\tilde\n{\\tau}}{\\qe \\Omega t}\\right) \\right | ^{\\qe \\Omega t - \\nu}_{\\nu},\n\\end{equation}\nwhere $1 \\ll \\nu \\ll \\qe \\Omega t$. For $\\qe \\Omega t \\gg \\nu$, the \nabove expression can be approximated by evaluating it at \n$\\tilde{\\tau} = \\qe \\Omega t$, giving\n\\begin{equation}\n\\< E_i (\\qe \\Omega t \\gg 1) \\> \\simeq \\, 16 V_1^2 X_1^2\n\\frac{\\Gamma(1+\\alpha) \\Gamma(\\alpha)}{\\alpha (1 - \\alpha) \n\\Gamma(2\\alpha)} (\\qe \\Omega t)^{2\\alpha-1} \\, \\< E_i(t = 0) \\>,\n\\end{equation}\nwhere we have used equation 15.3.7 in \\cite{as72} to evaluate \n$F(a,b;c;1)$.\\footnote{We have numerically integrated the angular \nintegral (\\ref{EINT}) and found this to be an excellent approximation \nat late times.}\n\nNotice that there is no power-law growth in the perturbation energy for\n${\\rm{Ri}} > 1\/4$,\\footnote{For ${\\rm{Ri}} > 1\/4$, $\\alpha$ is \nimaginary and ${\\rm{Re}}[t^{2\\alpha-1}] = t^{-1}\\cos(2|\\alpha| \\ln t)$.}\nconsistent with the classical Richardson criterion (\\ref{RICH}). In \nour analysis the energy decays with time for $2\\alpha-1<0$, or \n${\\rm{Ri}} > 3\/16$. Thus the energy of an initial isotropic set of \nincompressive perturbations in a radially-stratified shearing sheet-model \ngrows asymptotically (for ${\\rm{Ri}} < 3\/16$), just like the compressive \nshwaves and {\\it unlike} the incompressive shwaves in an unstratified \nshearing sheet, for which the energy is constant in time.\n\nThe growth of an ensemble of incompressive shwaves in a stratified disk is {\\it not}\ndue to a Rayleigh-Taylor or convective type instability. There is \nasymptotic growth for $0 < {\\rm{Ri}} < 3\/16$, and convective \ninstability requires ${\\rm{Ri}} < 0$. One can also see \nthis by examining the asymptotic energy for small values of\n$|{\\rm{Ri}}|$, such as would be expected for a Keplerian disk with\nmodest radial gradients:\n\\begin{equation}\n\\< E_i (\\qe \\Omega t \\gg 1) \\> \\simeq \\left[2 \\pi^2 {\\rm Ri} + O({\\rm Ri}^2)\\right]\\qe \n\\Omega t^{1-4{\\rm Ri}+O({\\rm Ri}^2)} \\, \\< E_i(t = 0) \\>.\n\\end{equation}\nEvidently for small values of Ri the near-linear growth in time of the\nenergy is independent of the sign of Ri and therefore $N_x^2$.\\footnote{\nThis asymptotic expression assumes ${\\rm Ri} \\neq 0$. Notice that the energy \nat late times can have the opposite sign to the initial energy because the \npotential energy is negative for $N_x^2 < 0$.}\n\n\\section{Implications}\\label{pap2s5}\n\nWe have studied the nonaxisymmetric linear theory of a thin, \nradially-stratified disk. Our findings are: (i) incompressive,\nshort-wavelength perturbations in the unstratified shearing sheet\nexhibit transient growth and asymptotic decay, but the energy of an\nensemble of such shwaves is constant with time; \n(ii) short-wavelength compressive shwaves grow asymptotically\nin the unstratified shearing sheet, as does the energy of an ensemble of\nsuch shwaves, which in the absence of any other dissipative effects\n(e.g., radiative damping) will result in a compressive shwave steepening\ninto a train of weak shocks; (iii) incompressive shwaves in the\nstratified shearing sheet have density and azimuthal velocity\nperturbations $\\delta \\Sigma$, $\\delta v_y \\sim t^{-{\\rm Ri}}$ (for\n$|{\\rm Ri}| \\ll 1$); (iv) incompressive shwaves in the stratified\nshearing sheet are associated with an angular momentum flux proportional\nto $-\\tilde{k}_x\/k_y$; leading shwaves therefore have positive angular\nmomentum flux and trailing shwaves have negative angular momentum\nflux\\footnote{This is consistent with the asymptotic result one obtains\nfrom a WKB analysis of incompressive waves \\citep{bal03}.}; (v) the\nenergy of an ensemble of incompressive shwaves in the stratified\nshearing sheet behaves asymptotically as $t^{1-4{\\rm Ri}}$ for $|{\\rm\nRi}| \\ll 1$. For Keplerian disks with modest radial gradients, $|{\\rm\nRi}|$ is expected to be $\\ll 1$, and there will therefore be weak growth\nin a single shwave for ${\\rm Ri} < 0$ and near-linear growth in the\nenergy of an ensemble of shwaves, independent of the sign of Ri.\n\nAlong the way we have found the following solutions: (i) an exact solution \nfor nonvortical shwaves in the unstratified shearing sheet, equations \n(\\ref{CVX}), (\\ref{CS}) and (\\ref{CVY}); (ii) a WKB-solution for the \nnonvortical,\ncompressive shwaves in the short-wavelength, high-frequency limit,\nequations (\\ref{VYWKB})-(\\ref{SWKB}); (iii)\na solution for incompressive shwaves in the unstratified shearing sheet\nvalid in the short-wavelength, low-frequency limit, equations\n(\\ref{IVX})-(\\ref{IS}); (iv) a solution for incompressive shwaves in\nthe radially-stratified shearing sheet (also valid in the short-wavelength,\nlow-frequency limit), equations (\\ref{SOLVX})-(\\ref{SOLDH}).\n\nOur results are summarized in Figure~\\ref{pap2f1}, which shows the regions of\namplification and decay for shwaves in a stratified disk in the\n$N_x^2\/\\Omega, \\tilde{q}$ plane.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=6.in,clip]{pap2f1.eps}\n\\caption[A summary of analytic results for shwaves (shearing waves) in a\nstratified disk.]\n{A summary of analytic results for shwaves (shearing waves) in a\nstratified disk. The relevant parameters are the local dimensionless\nshear rate\n$\\tilde{q} = -\\frac{1}{2}d\\ln\\Omega^2\/d\\ln r$ and the dimensionless\nBrunt-V\\\"ais\\\"al\\\"a frequency $N_x^2\/\\Omega^2$. The expected location\nof a thin, smooth disk is shown as a vertically extended ellipse near\n$\\tilde{q} = 1.5$, $N_x^2\/\\Omega^2 = 0$. The far right region (shaded in\nthe figure) is forbidden by the H{\\o}iland criterion. When $\\tilde{q} =\n0$ shear is absent and a modal analysis is possible; instability is\npresent for $N_x^2 < 0$. Solitary shwaves with ${\\rm Ri} =\nN_x^2\/(\\tilde{q}^2\\Omega^2) < 0$ experience asymptotic power-law growth\n($\\propto t^{-{\\rm Ri}}$ for small ${\\rm Ri}$); since each shwave grows the energy\nof an ensemble of shwaves does as well. For $0 < {\\rm Ri} < 3\/16$ solitary\nshwaves decay but the energy of an ensemble of shwaves grows as a\npower-law in time. For ${\\rm Ri} > 3\/16$ both solitary shwaves and the energy\nof an ensemble of shwaves asymptotically decay.}\n\\label{pap2f1}\n\\end{figure}\n\nThe presence of power-law growth of incompressive shwaves in stratified\ndisks opens the possibility of a transition to turbulence as amplified\nshwaves enter the nonlinear regime. Any such transition would depend,\nhowever, on the nonlinear behavior of the disk after the shwaves break.\nIt is far from clear that they would continue to grow. We will evaluate\nthe nonlinear behavior of the disk in subsequent work.\n\nOur results are essentially in agreement with the numerical results\npresented by \\cite{klr04},\nthat is, we find that arbitrarily large amplification factors can be\nobtained by starting with appropriate initial conditions. Our results,\nhowever, clarify the nature and asymptotic time dependence of the\ngrowth. Our results on the unstratified shearing sheet are also\nconsistent with the results of \\cite{shep85} and \\cite{amn04}, who find that an isotropic\nensemble of incompressive shwaves have fixed energy.\n\n\n\\end{spacing}\n\n\n\\chapter{Vortices in Thin, Compressible, Unmagnetized Disks}\\label{paper3}\n\n\\begin{spacing}{1.5}\n\n\\section{Chapter Overview}\n\nWe consider the formation and evolution of vortices in a hydrodynamic\nshearing-sheet model. The evolution is done numerically using a version\nof the ZEUS code. Consistent with earlier results, an injected vorticity\nfield evolves into a set of long-lived vortices each of which has radial\nextent comparable to the local scale height. But we also find that the\nresulting velocity field has positive shear stress $\\<\\Sigma \\delta v_r\n\\delta v_\\phi\\>$. This effect appears only at high resolution. The\ntransport, which decays with time as $t^{-1\/2}$, arises primarily because the vortices drive compressive\nmotions. This result suggests a possible mechanism for angular momentum\ntransport in low-ionization disks, with two important caveats: a\nmechanism must be found to inject vorticity into the disk, and the\nvortices must not decay rapidly due to three-dimensional \ninstabilities.\\footnote{Submitted to ApJ. Reproduction for this dissertation\nis authorized by the copyright holder.}\n\n\\section{Introduction}\n\nAstrophysical disks are common because the specific angular momentum of\nthe matter inside them is well-conserved. They evolve because angular\nmomentum conservation is weakly compromised, either because of diffusion\nof angular momentum within the disk or because of direct application of\nexternal torques.\n\nIn astrophysical disks composed of a well-ionized plasma it is likely\nthat some, perhaps most, of the evolution is driven by diffusion of\nangular momentum within the disk. This view is certainly consistent\nwith observations of steadily accreting cataclysmic variable systems\nlike UX Ursa Majoris \\citep{bap98,bap04}, whose radial\nsurface-brightness profile is consistent with steady accretion-flow\nmodels in which the bulk of the accretion energy is dissipated within\nthe disk.\n\nAngular momentum diffusion in well-ionized disks is likely driven by\nmagnetohydrodynamic (MHD) turbulence. Analytic analyses, numerical\nexperiments, and laboratory evidence strongly suggest that\nwell-coupled plasmas in differentially-rotating flows are subject to\nthe magnetorotational instability (MRI; \\citealt{bh91,bh98,bal03}).\nBut MHD turbulence is initiated by the MRI only so long as the\nplasma is sufficiently ionized to couple to the magnetic field\n\\citep{kb04,des04}. In disks around young stars, cataclysmic-variable \nand X-ray binary disks in quiescence, and possibly the outer parts of AGN \ndisks, the plasma may be too neutral to support magnetic activity \n\\citep{gm98,men00,sgbh00,mq01,ftb02}. This motivates interest in\nnon-MHD angular momentum transport mechanisms.\n\nWithin the last few years, a body of work has been developed suggesting\nthat vortices can be generated as a result of global hydrodynamic\ninstability \\citep{haw87,bh88,haw90,llcn99,lflc00} or local hydrodynamic\ninstability \\citep{kb03}, that vortices in disks may be long-lived\n\\citep{gl99,gl00,ur04,bm05}, and that these vortices may be\nrelated to an outward flux of angular momentum \\citep{lcwl01,bm05}.\nIf these claims can be verified then the consequences for low-ionization\ndisks would be profound.\n\nHere we investigate the evolution of a disk that is given a large\ninitial vortical velocity perturbation. Our study is done in the\ncontext of a (two-dimensional) shearing-sheet model, which permits us to\nresolve the dynamics to a degree that is not currently possible in a\nglobal disk model. Our model is also fully compressible, unlike previous\nwork using a local model \\citep{ur04,bm05}. The former assume incompressible\nflow and the latter use the anelastic approximation (e.g., \\citealt{gou69}),\nwhich filters out the high-frequency acoustic waves. We will show that\ncompressibility and acoustic waves play an essential part in the angular\nmomentum transport.\n\nOur paper is organized as follows. In \\S2 we describe the model. In\n\\S3 we describe the evolution of a fiducial, high-resolution model.\nIn \\S4 we investigate the dependence of the results on model\nparameters. And in \\S5 we describe implications, with an emphasis on\nkey open questions: are the vortices destroyed by three-dimensional\ninstabilities?; and do mechanisms exist that can inject vorticity into the\ndisk?\n\n\\section{Model}\\label{pap3s2}\n\nThe shearing-sheet model is obtained via a rigorous expansion of the\ntwo-dimensional hydrodynamic equations of motion to lowest order in\n$H\/R$, where $H = c_s\/\\Omega$ is the disk scale height ($c_s$ is the\nisothermal sound speed and $\\Omega$ is the local rotation frequency)\nand $R$ is the local radius. See \\cite{ngg87} for a description. Adopting\na local Cartesian coordinate system where the $x$ axis is oriented\nparallel to the radius vector and the $y$ axis points forward in\nazimuth, the equations of motion become\n\\begin{equation}\\label{EQUA1}\n\\dv{\\Sigma}{t} + \\Sigma \\bld{\\nabla} \\cdot \\bld{v} = 0,\n\\end{equation}\n\\begin{equation}\\label{EQUA2}\n\\dv{\\bld{v}}{t}\n+ \\frac{\\bld{\\nabla} P}{\\Sigma} + 2\\bld{\\Omega}\\times\\bld{v} - 2q\\Omega^2 x\n\\ex = 0,\n\\end{equation}\nwhere $\\Sigma$ and $P$ are the two-dimensional density and pressure,\n$\\bld{v}$ is the fluid velocity and $d\/dt$ is the\nLagrangian derivative. The third and fourth terms in equation\n(\\ref{EQUA2}) represent the Coriolis and centrifugal forces in the local\nmodel expansion, where $q = -(1\/2) \\, d\\ln\\Omega^2\/d\\ln r$ is\nthe shear parameter. We will assume throughout that $q = 3\/2$,\ncorresponding to a Keplerian shear profile. We\nclose the above equations with an isothermal equation of state\n\\begin{equation}\\label{EQUA3}\nP = c_s^2 \\Sigma,\n\\end{equation}\nwhere $c_s$ is constant in time and space.\n\nEquations (\\ref{EQUA1}) through (\\ref{EQUA3}) can be combined to show\nthat the vertical component of potential vorticity\n\\begin{equation}\n\\xi \\equiv \\frac{(\\bld{\\nabla} \\times \\bld{v} + 2\\bld{\\Omega})\\cdot \\ez}{\\Sigma}\n\\end{equation}\nis a constant of the motion; i.e., the potential vorticity of fluid elements\nin two dimensions is conserved.\n\nAn equilibrium solution to the equations of motion is\n\\begin{equation}\n\\Sigma = \\Sigma_0 = const.\n\\end{equation}\n\\begin{equation}\nP = c_s^2 \\Sigma_0 = const.\n\\end{equation}\n\\begin{equation}\nv_x = 0\n\\end{equation}\n\\begin{equation}\nv_y = -q \\Omega x\n\\end{equation}\nThus the differential rotation of the disk makes an appearance in the\nform of a linear shear.\n\nWe integrate the above equations using a version of the ZEUS code\n\\citep{sn92}. ZEUS is a time-explicit, operator-split scheme on a\nstaggered mesh. It uses artificial viscosity to capture shocks. Our\ncomputational domain is a rectangle of size $L_x \\times L_y$ containing\n$N_x \\times N_y$ grid cells. The numerical resolution is therefore $\\Delta x\n\\times \\Delta y = L_x\/N_x \\times L_y\/N_y$.\n\nOur code differs from the standard ZEUS algorithm in two respects.\nFirst, we have implemented a version of the shearing-box boundary\nconditions. The model is then periodic in the $y$ direction; the $x$\nboundaries are initial joined in a periodic fashion, but they are\nallowed to shear with respect to each other, becoming periodic again\nwhen $t = n L_y\/(q\\Omega L_x)$, $n = 1,2,\\ldots$. A detailed\ndescription of the boundary conditions is given in \\cite{hgb95}.\n\nSecond, we treat advection by the mean flow $\\bld{v}_0 = -q\\Omega x \\ey$\nseparately from advection by the perturbed flow $\\delta \\bld{v} \\equiv\n\\bld{v} - \\bld{v}_0$. Mean-flow advection can be done by interpolation, using\nthe algorithm described in \\cite{gam01}, which is similar to the FARGO\nscheme \\citep{mass00}. This has the advantage that the timestep is not\nlimited by the mean flow velocity (it is $|\\delta \\bld{v}|$ rather than\n$|\\bld{v}|$ that enters the Courant condition). This permits the use of a\ntimestep that is larger than the usual timestep by $\\sim L_x\/H$ if $L_x\n\\gg H$. The shear-interpolation scheme also makes the algorithm more\nnearly translation-invariant in the $x-y$ plane, thereby more\nnearly embodying an important symmetry of the underlying equations.\n\n\\subsection{Initial Conditions}\n\nWithout a specific model for the process that is injecting the\nvorticity, it is difficult to settle on a particular set of initial\nconditions, or to know how these initial conditions ought to vary\nwhen the size of the box is allowed to vary. Our choice of initial\nconditions is therefore somewhat arbitrary. We use a set of initial\n(incompressive) velocity perturbations drawn from a Gaussian random\nfield. The amplitude of the perturbations is characterized by $\\sigma =\n\\<|\\delta \\bld{v}\/c_s|^2\\>^{1\/2}$. The power spectrum is $|\\delta \\bld{v}|^2\n\\sim k^{-8\/3}$, corresponding to the energy spectrum ($E_k \\sim k^{-5\/3}$)\nof a two-dimensional Kolmogorov inverse turbulent cascade, with\ncutoffs at $k_{min} = (1\/2) (2\\pi\/H)$ and $k_{max} = 32 k_{min}$\\footnote{We\nhave compared our fiducial run to runs with a different range in $k$,\ncorresponding to vorticity injection either at scales $\\sim H$ or scales\n$\\sim 0.1 H$. The results are qualitatively the same.}.\nThe surface density is not perturbed. These initial conditions correspond\nto a set of purely vortical perturbations. The parameters for our fiducial\nrun are $L_x = L_y = 4H$ and $\\sigma = 0.4$.\n\n\\subsection{Code Verification}\n\nAlthough our basic algorithm has already been tested (see \\citealt{gam01}),\nwe test the current version of our code by making a comparison with linear\ntheory. Due to the underlying shear, small-amplitude perturbations in the\nshearing sheet are naturally decomposed in terms of shearing waves\nor {\\it shwaves} (see Chapter~\\ref{paper2}), Fourier\ncomponents in the ``co-shearing'' frame. These have time-dependent\nwavenumber $\\bld{k}(t) = k_x(t)\\ex + k_y\\ey$, where $k_x(t) = k_{x0}\n+ q\\Omega k_y t$ and $k_{x0}$ and $k_y$ are constant. The evolution of\na single Fourier component can be calculated by integrating an ordinary\ndifferential equation for the amplitude of the shwave. For purely vortical\n(nonzero {\\it potential} vorticity)\nor non-vortical perturbations, the evolution can be obtained analytically. The\nexplicit expression for the amplitude of a vortical (incompressive) shwave is\n\\begin{equation}\n\\delta v_{xi} = \\delta v_{x0} \\frac{k_0^2}{k^2 } =\n\\delta v_{x0} \\frac{1 + \\tau_0^2}{1+\\tau^2},\n\\end{equation}\nwhere $k^2 = k_x^2 + k_y^2$, $\\tau = q\\Omega t + k_{x0}\/k_y$ and a\nsubscript $0$ on a quantity indicates its value at $t = 0$.\\footnote{This\nsolution is valid at all times only for short-wavelength vortical perturbations\n($kH \\gg 1$).} The amplitude of a non-vortical (compressive) shwave\nsatisfies the differential equation\n\\begin{equation}\n\\dv{^2 \\delta v_{yc}}{t^2} + \\left(c_s^2 k^2 + \\Omega^2\\right) \\delta v_{yc} = 0,\n\\end{equation}\nthe solutions of which are parabolic cylinder functions. See Chapter~\\ref{paper2}\nfor further details on the shwave solutions.\n\nFigures~\\ref{pap3f1a} and \\ref{pap3f1b} compare the numerical evolution of both vortical and\ncompressive shwave amplitudes with their analytic solutions. The initial\nshwave vector ($k_{x0}, k_y$) is ($-16\\pi\/L_x, 4\\pi\/L_y$) for the\nvortical shwave and ($-8\\pi\/L_x, 2\\pi\/L_y$) for the compressive shwave.\nThe other model parameters\n\n\\begin{figure}[hp]\n \\hfill\n \\begin{minipage}[t]{1.\\textwidth}\n \\begin{center}\n \\includegraphics[width=3.5in,clip]{pap3f1a.eps}\n \\caption[Evolution of the radial velocity amplitude for a vortical shwave.]\n {Evolution of the radial velocity amplitude for a vortical shwave.\n The heavy line is the analytic result, and the light lines\n are numerical results with (in order of increasing accuracy) $N_x = N_y\n = 32, 64, 128$ and $256$.}\n \\label{pap3f1a}\n \\end{center}\n \\end{minipage}\n \\hfill\n \\begin{minipage}[b]{1.\\textwidth}\n \\begin{center}\n \\includegraphics[width=3.5in,clip]{pap3f1b.eps}\n \\caption[Evolution of the azimuthal velocity amplitude for a nonvortical shwave.]\n {Evolution of the azimuthal velocity amplitude for a nonvortical shwave.\n The heavy line is the analytic result, and the light lines\n are numerical results with (in order of increasing accuracy) $N_x = N_y\n = 32, 64, 128$ and $256$.}\n \\label{pap3f1b}\n \\end{center}\n \\end{minipage}\n \\hfill\n\\end{figure}\n\\noindent\nare the same as those in the fiducial run, except\nthat $L = 0.5H$ for the vortical-shwave evolution since $k_y H \\gg 1$ is\nrequired to prevent mixing between vortical and non-vortical shwaves\nnear $\\tau = 0$. The shwaves are well resolved until the radial\nwavelength $\\lambda_x = 4 \\times \\Delta x$, and the code is capable of\ntracking both potential-vorticity and compressive perturbations with\nhigh accuracy.\n\nWithout a specific model for the process that is injecting the\nvorticity, it is difficult to settle on a particular set of initial\nconditions, or to know how these initial conditions ought to vary\nwhen the size of the box is allowed to vary. Our choice of initial\nconditions is therefore somewhat arbitrary. We use a set of initial\n(incompressive) velocity perturbations drawn from a Gaussian random\nfield. The amplitude of the perturbations is characterized by $\\sigma =\n\\<|\\delta \\bld{v}|^2\\>^{1\/2}$. The power spectrum is appropriate for two\ndimensional Kolmogorov turbulence, $|\\delta \\bld{v}|^2 \\sim k^{-8\/3}$, with\ncutoffs at $k_{min} = (1\/2) (2\\pi\/H)$ and $k_{max} = 32 k_{min}$.\nThe surface density is not perturbed. These initial conditions correspond\nto a set of purely vortical perturbations.\n\n\\section{Results}\\label{pap3s3}\n\nThe evolution of the potential vorticity in our fiducial run is shown in\nFigure~\\ref{pap3f2}. The snapshots are shown in lexicographic order beginning with\nthe initial conditions, which have equal positive and negative $\\delta \\xi$.\n\nOne of the most remarkable features of the fiducial run evolution is the\nappearance of comparatively stable, long-lived vortices. These vortices\nhave negative $\\delta \\xi$ and are therefore dark in Figure~\\ref{pap3f2}. Similar\nvortices have been seen by \\cite{gl99,gl00}, \\cite{lcwl01} and \\cite{ur04}. Cross\nsections of one of the vortices at the end of the run are shown in \nFigures~\\ref{pap3f3a} and \\ref{pap3f3b}.\nIn our models the vortices are not associated with easily identifiable\nfeatures in the surface density, since the perturbed vorticity is not large\nenough to require, through the equilibrium condition, an order unity\nincrease in the local pressure.\n\nWhile the vortices are long-lived, they do decay. Figure~\\ref{pap3f4} shows the\nevolution of the perturbed (noncircular) kinetic energy\n\\begin{equation}\nE_K \\equiv \\frac{1}{2} \\Sigma (\\delta v_x^2 + \\delta v_y^2)\n\\end{equation}\nin the fiducial run. Evidently the kinetic energy decays approximately\nas $t^{-1\/2}$ (which is remarkable in that, if the vortices would\ncorrespond to features in {\\it luminosity} that decay as $t^{-1\/2}$,\nthey could produce flicker noise; see \\citealt{press78}). Runs with half and\ntwice the resolution decay in the same fashion, but if the resolution is\nreduced to $64^2$ the kinetic energy decays exponentially. Resolution\n\n\\begin{figure}[hp]\n\\centering\n\\includegraphics[width=6.in,clip]{pap3f2.eps}\n\\caption[Evolution of the potential vorticity in the fiducial run.]\n{Panels show the evolution of the potential vorticity in the fiducial\nrun. The size is $4 H \\times 4 H$ and the numerical resolution is\n$1024^2$. The initial conditions are shown in the upper left corner,\nand the other frames follow in lexicographic order at intervals of $22.2\n\\Omega^{-1}$. Dark shardes (blue and black in the color version) indicate potential vorticity smaller than $\\Omega\/(2\n\\Sigma_0)$; light shades (yellow and red in the color version) show positive potential vorticity\nperturbations. Evidently only the ``anticyclonic'' (negative potential\nvorticity perturbation) vortices survive. Each vortex sheds sound\nwaves, which steepen into trailing shocks.}\n\\label{pap3f2}\n\\end{figure}\n\n\\begin{figure}[hp]\n \\hfill\n \\begin{minipage}[t]{1.\\textwidth}\n \\begin{center}\n \\includegraphics[width=3.5in,clip]{pap3f3a.eps}\n \\caption[Radial slice of a vortex at the end of the fiducial run.]\n {Radial slice of a vortex at the end of the fiducial run. The heavy line\n shows the potential vorticity, the light line shows the magnitude of the\n velocity and the dotted line shows the surface density.}\n \\label{pap3f3a}\n \\end{center}\n \\end{minipage}\n \\hfill\n \\begin{minipage}[b]{1.\\textwidth}\n \\begin{center}\n \\includegraphics[width=3.5in,clip]{pap3f3b.eps}\n \\caption[Azimuthal slice of a vortex at the end of the fiducial run.]\n {Azimuthal slice of a vortex at the end of the fiducial run. The heavy line\n shows the potential vorticity, the light line shows the magnitude of the\n velocity and the dotted line shows the surface density.}\n \\label{pap3f3b}\n \\end{center}\n \\end{minipage}\n \\hfill\n\\end{figure}\n\n\\begin{figure}[hp]\n\\centering\n\\includegraphics[width=6.in,clip]{pap3f4.eps}\n\\caption[Evolution of kinetic energy in time for the fiducial run.]\n{Evolution of kinetic energy in time for the fiducial run, on a log-log\nscale. The solid line shows a $t^{-1\/2}$ decay for comparison purposes.}\n\\label{pap3f4}\n\\end{figure}\n\n\\noindent\nof at least 128 zones per scale height appears to be\nrequired.\n\nWhat is even more remarkable is that the vortices are associated with an\noutward angular momentum flux, due to the driving of compressive\nmotions by the vortices. Figure~\\ref{pap3f5} shows the evolution\nof the dimensionless angular momentum flux\n\\begin{equation}\n\\alpha \\equiv \\frac{1}{L_x L_y\\Sigma_0 c_s^21} \\int \\Sigma \\delta v_x \\delta v_y dx dy\n\\end{equation}\nfor models with a variety of resolutions. The data has been boxcar\nsmoothed over an interval $\\Delta t = 10 \\Omega^{-1}$ to make the plot\nreadable. Again, a resolution of at least $512^2$ appears to be\nrequired for a converged measurement of the shear stress. For the most\nhighly resolved models $\\alpha$ evolves like the kinetic energy,\n$\\propto t^{-1\/2}$.\n\n\\begin{figure}[hp]\n\\centering\n\\includegraphics[width=5.25in,clip]{pap3f5.eps}\n\\caption{Evolution of the shear stress $\\alpha$ in the fiducial run and a set of\nruns at lower resolutions.}\n\\label{pap3f5}\n\\end{figure}\n\nCompressibility is crucial for development of the anglar momentum flux.\nWe have demonstrated this in two ways. First, we have taken the\nfiducial run and decomposed the velocity field into a compressive and an\nincompressive part (i.e., into potential and solenoidal pieces in Fourier \nspace) and measured the stress associated with each. For a\nset of snapshots taken from the last half of the fiducial run, the\naverage total $\\alpha = 0.0036$; the incompressive component is\n$\\alpha_i = -0.0006$; the compressive component is $\\alpha_c = 0.0032$.\nThe remaining alpha $\\alpha_x = 0.00099$ is in cross-correlations\nbetween the incompressive and compressive pieces of the velocity field.\nAs argued in \\cite{bal00} and \\cite{bal03}, both incompressive trailing\nshwaves and incompressive turbulence tend to transport angular\nmomentum inward, whereas trailing compressive disturbances transport\nangular momentum outward. Our negative (positive) value for $\\alpha_i$\n($\\alpha_c$) is consistent with this.\n\nSecond, we have reduced the size of the model and reduced the amplitude\nof the initial perturbation so that it scales with the shear velocity at\nthe edge of the model (constant ``intensity'' of the turbulence, in\nUmurhan and Regev's parlance). Thus the Mach number of the turbulence\nis reduced in proportion to the size of the box. We have compared four\nmodels, with $L = (4, 2, 1, 0.5) H$ and $\\sigma = (0.8, 0.4, 0.2, 0.1)\nc_s$. We would expect the lower Mach number models to have\nsmaller-amplitude compressive velocity fields and therefore, consistent\nwith the above results, smaller angular momentum flux $\\alpha$.\nAveraging over the second half of the simulation, we find $\\alpha =\n(0.0031, 0.0018, 7.2 \\times 10^{-5}, -9.5 \\times 10^{-7})$.\n\nAn additional confirmation of our overall picture can be seen in\nFigure~\\ref{pap3f6}, in which we show a snapshot of the velocity divergence\nsuperimposed on the potential vorticity for a medium-resolution\n($256^2$) version of the fiducial run.\\footnote{At higher resolutions,\nshocks are generated earlier in the simulation from smaller vortices,\nand it is more difficult to see the effect we are describing due to the\nrandom nature of the vortices at this early stage.} The position of\nthe shocks with respect to the vortices in this figure is consistent\nwith our interpretation that the former are generated by the latter.\n\nThe smallest of our simulations ($L = 0.5H$) is nearly incompressible,\nbut we continue to observe $t^{-1\/2}$ decay (least squares fit power law\nis $-0.49$) at late times. The reason that we see decay while\n\\cite{ur04} do not may be that: (1) the remaining compressibility in our\nmodel causes added dissipation; (2) the numerical dissipation in our\ncode is larger than that of \\cite{ur04}; (3) the code used by\n\\cite{ur04} could somehow be aliasing power from trailing shwaves to\nleading shwaves (although they do explicitly discuss, and dismiss, this\npossibility).\n\n\\begin{figure}[hp]\n \\hfill\n \\begin{minipage}[t]{1.\\textwidth}\n \\begin{center}\n \\includegraphics[width=3.5in,clip]{pap3f6.eps}\n \\caption[Snapshot of the velocity divergence superimposed on the potential\nvorticity in a medium-resolution version of the fiducial run.]\n {Snapshot of the velocity divergence superimposed on the potential\nvorticity in a medium-resolution ($256^2$) version of the fiducial run.\nThe thin (red in the color version) contours indicate negative divergence and are\nassociated with shocks. The thick (blue in the color version) contours indicate negative\npotential vorticity.}\n \\label{pap3f6}\n \\end{center}\n \\end{minipage}\n \\hfill\n \\begin{minipage}[b]{1.\\textwidth}\n \\begin{center}\n \\includegraphics[width=3.5in,clip]{pap3f7.eps}\n \\caption[Evolution of a vortical shwave amplitude in a low-resolution run]\n {Evolution of a vortical shwave amplitude in a low-resolution ($64^2$)\nrun, in units of $\\tau$. The initial shwave vector ($k_{x0},k_y$) is\n($-16\\pi\/L_x, 4\\pi\/L_y$), corresponding to $\\tau_0 = -4$. The interval\nbetween successive peaks (a numerical effect due to aliasing) is $\\tau =\nN_x\/n_y$, where $n_y = 2$ is the azimuthal wavenumber.}\n \\label{pap3f7}\n \\end{center}\n \\end{minipage}\n \\hfill\n\\end{figure}\n\nTo highlight the dangers of aliasing for our finite-difference code, in\nFigure~\\ref{pap3f7} we show the evolution of a vortical shwave amplitude at low\nresolution ($64^2$), in units of $\\tau$. We use the same parameters as\nthose in our linear-theory test (Figures~\\ref{pap3f1a} and \\ref{pap3f1b}), \nfor which the initial shwave\nvector corresponds to $\\tau_0 = -4$. The initially-leading shwave swings\ninto a trailing shwave, the radial wavelength is eventually lost near the\ngrid scale, and due to aliasing the code picks up the evolution of the\nshwave again as a leading shwave. Repeating this test at higher\nresolutions indicates that successive swings from leading to trailing\noccur at an interval of $\\tau = N_x\/n_y$, where $n_y = 2$ is the\nazimuthal wavenumber of the shwave. This is equivalent to $k_x(t) =\n2\\pi\/\\Delta x$. The decay of the successive linear solutions with time\nis due to numerical diffusion.\n\nFigure~\\ref{pap3f7} suggests that it is easier to inject power into the simulation\ndue to aliasing rather than to remove power due to numerical diffusion.\nWe do not believe, however, that aliasing is affecting our\nhigh-resolution results. In addition, if we assume that the flow in our\nsimulations can be modeled as two-dimensional Kolmogorov turbulence,\nthen $\\delta v_{rms} \\sim \\lambda^ {1\/3}$, where $\\delta v_{rms}$ is\nthe rms velocity variation across a scale $\\lambda$. The velocity due to the\nmean shear at these scales is $\\delta v_{shear} \\sim q \\Omega \\lambda$,\nand $\\delta v_{rms}\/\\delta v_{shear} \\sim \\lambda^{-2\/3}$. The velocities at\nthe smallest scales are thus dominated by turbulence rather than by the\nmean shear. This conclusion is supported by the convergence of our\nnumerical results at high resolution.\n\nOur model contains two additional numerical parameters: the size $L$ and\nthe initial turbulence amplitude $\\sigma$. Figure~\\ref{pap3f8} shows the evolution\nof $\\alpha$ for several values of $\\sigma$. Evidently for small enough\nvalues of $\\sigma$ the $\\alpha$ amplitude is reduced, but for near-sonic\ninitial Mach numbers the $\\alpha$ amplitude saturates (or at least the\ndependence on $\\sigma$ is greatly weakened). Figure~\\ref{pap3f9} shows the\nevolution for several values of $L$ but the same initial $\\sigma$ and\nthe identical initial power spectrum. For large enough $L$ the shear\nstress appears to be independent of $L$.\n\n\\begin{figure}[hp]\n \\hfill\n \\begin{minipage}[t]{1.\\textwidth}\n \\begin{center}\n \\includegraphics[width=3.5in,clip]{pap3f8.eps}\n \\caption[Evolution of the shear stress $\\alpha$ in a set of runs at with varying\ninitial $\\sigma$.]\n {Evolution of the shear stress $\\alpha$ in a set of runs at with varying\ninitial $\\sigma$. Apparently for low values of $\\sigma$ the shear\nstress is reduced, but for initial Mach number near $1$ the stress\nsaturates. All runs have $L = 4 H$.}\n \\label{pap3f8}\n \\end{center}\n \\end{minipage}\n \\hfill\n \\begin{minipage}[b]{1.\\textwidth}\n \\begin{center}\n \\includegraphics[width=3.5in,clip]{pap3f9.eps}\n \\caption[Evolution of a vortical shwave amplitude in a low-resolution run]\n {Evolution of the shear stress $\\alpha$ in a set of runs at with varying\ninitial $L$, but the same initial Mach number $\\sigma$.}\n \\label{pap3f9}\n \\end{center}\n \\end{minipage}\n \\hfill\n\\end{figure}\n\nFinally, we have studied the autocorrelation function of the potential\nvorticity as a means of characterizing structure inside the flow.\nFigures~\\ref{pap3f10a} and \\ref{pap3f10b} shows the autocorrelation function\nmeasured in the fiducial\nmodel and in an otherwise identical model with $L = 8H$. Evidently\nthe potential vorticity is correlated over about one-half a scale height\nin radius, independent of the size of the model. This supports the idea\nthat compressive effects limit the size of the vortices, since the shear\nflow becomes supersonic across a vortex of size $\\sim H$ \\citep{bs95,lcwl01}.\n\n\\begin{figure}[hp]\n \\hfill\n \\begin{minipage}[t]{1.\\textwidth}\n \\begin{center}\n \\includegraphics[width=3.5in,clip]{pap3f10a.eps}\n\\caption{Autocorrelation function of the potential vorticity $\\xi$ for the fiducial\nmodel with $L = 4H$.}\n \\label{pap3f10a}\n \\end{center}\n \\end{minipage}\n \\hfill\n \\begin{minipage}[b]{1.\\textwidth}\n \\begin{center}\n \\includegraphics[width=3.5in,clip]{pap3f10b.eps}\n\\caption{Autocorrelation function of the potential vorticity $\\xi$ for a model\nwith $L = 8H$.}\n \\label{pap3f10b}\n \\end{center}\n \\end{minipage}\n \\hfill\n\\end{figure}\n\n\\section{Conclusion}\\label{pap3s4}\n\nThe presence of long-lived vortices in weakly-ionized disks may be an\nintegral part of the angular momentum transport mechanism in these\nsystems. The key result we have shown here is that compressibility of\nthe flow is an extremely important factor in providing a significant,\npositively-correlated average shear stress with its associated outward\ntransport of angular momentum. Previous results using a local model\nhave assumed incompressible flow and either report no angular momentum\ntransport \\citep{ur04} or report a value ($\\alpha \\sim 10^{-5}$, \\citealt{\nbm05}) that is two orders of magnitude lower than what we find when we\ninclude the effects of compressibility. Global simulations\n\\citep{gl99,gl00,lcwl01} have a difficult time accessing the high\nresolution that we have shown is required for a significant shear\nstress due to compressibility.\n\nOur work leaves open the key question of what happens in three\ndimensions. Our vortices, which have radial and azimuthal extent\n$\\lesssim H$, are inherently three-dimensional. Three-dimensional\nvortices are susceptible to the elliptical instability \\citep{ker02} and\nare likely to be destroyed on a dynamical timescale. The fact that \nvortices persist in our two-dimensional simulations and not in the local \n(three-dimensional) shearing-box calculations of \\cite{bhs96} is likely \ndue to dimensionality. The recent numerical results of \\cite{bm05} indicate \nthat vortices near the disk midplane are quickly destroyed, whereas vortices \nsurvive if they are a couple of scale heights away from the midplane. \nStrong vertical stratification away from the midplane may enforce \ntwo-dimensional flow and allow the vortices that we consider here to survive.\n\nThe initial conditions in \\cite{bm05} are analytic solutions for\ntwo-dimensional vortices that are stacked into a three-dimensional\ncolumn. The stable, off-midplane vortices apparently arise due to the\nbreaking of internal gravity waves generated by the midplane vortices\nbefore they become unstable. There is also an unidentified instability\nthat breaks a single off-midplane vortex into several vortices. These\nsimulations leave open the question of whether stable off-midplane\nvortices can be generated from a random set of initial vorticity \nperturbations rather than the special vortex solutions that are imposed.\n\nOur work also leaves open the key question of what generates the initial\nvorticity. One possibility is that material builds up at particular\nradii in the disk, resulting in a global instability (e.g.\n\\citealt{pp84,pp85}) and a breakdown of the flow into vortices\n\\citep{lcwl01}. Another possibility for vortex generation in variable\nsystems is that the MHD turbulence, which likely operates during an\noutburst but decays as the disk cools \\citep{gm98}, leaves behind some\nresidual vorticity. The viability of such a mechanism could be tested\nwith non-ideal MHD simulations such as those of \\cite{fs03} and\n\\cite{ss03}. Yet another possibility is that differential illumination of the disk\nsomehow produces vorticity. Since the temperature of most circumstellar\ndisks is controlled by stellar illumination, small variations in\nillumination could produce hot and cold spots in the disk that interact\nto produce vortices. The final possibility that we consider is the generation\nof vorticity via baroclinic instability, which is likely to operate in disks\nwhose vertical stratification is close to adiabatic \\citep{ks86}. The\nnonlinear outcome of this instability in planetary atmospheres is the\nformation of vortices, although it is far from clear that the same outcome\nwill occur in disks. Finally, we note that a residual amount of vorticity \ncan be generated from finite-amplitude compressive perturbations. We have \nperformed a series of runs with zero initial vorticity and perturbation \nwavelengths on the order of the scale height, and the results are qualitatively similar to Figure~\\ref{pap3f5} with the shear stress reduced by nearly two \norders of magnitude.\n\n\\end{spacing}\n\n\\chapter{Nonlinear Stability of Thin, Radially-Stratified Disks}\\label{paper4}\n\n\\begin{spacing}{1.5}\n\n\\section{Chapter Overview}\n\nWe perform local numerical experiments to investigate the nonlinear\nstability of thin, radially-stratified disks. We demonstrate the\npresence of radial convective instability when the disk is nearly in\nuniform rotation, and show that the net angular momentum transport\nis slightly inwards, consistent with previous investigations of\nvertical convection. We then show that a convectively-unstable\nequilibrium is stabilized by differential rotation. Convective instability\n(corresponding to ${\\rm Ri} \\rightarrow -\\infty$, where Ri is the radial\nRichardson number) is suppressed when ${\\rm Ri} \\gtrsim -1$, i.e. when\nthe shear rate becomes greater than the growth rate. Disks\nwith a nearly-Keplerian rotation profile and radial gradients on the\norder of the disk radius have ${\\rm Ri} \\gtrsim -0.01$ and are\ntherefore stable to local nonaxisymmetric disturbances. One\nimplication of our results is that the ``Global Baroclinic\nInstability'' claimed by \\cite{kb03} is either global or nonexistent.\n\n\\section{Introduction}\n\nIn order for astrophysical disks to accrete, angular momentum must be\nremoved from the disk material and transported outwards. In many disks,\nthis outward angular momentum transport is likely mediated internally by\nmagnetohydrodynamic (MHD) turbulence driven by the magnetorotational\ninstability (MRI; see \\citealt{bh98}). A key feature of this transport\nmechanism is that it arises from a local shear instability and is therefore \nvery robust. In addition,\nMHD turbulence transports angular momentum {\\it outwards}; some other forms\nof turbulence, such as convective turbulence, appear to transport angular\nmomentum inwards \\citep{sb96}. The mechanism is only effective, however, if\nthe plasma in the disk is sufficiently ionized to be well-coupled to the\nmagnetic field (see \\S\\ref{MRILowI}). In portions of disks around young,\nlow-mass stars, in cataclysmic-variable disks in quiescence, and in X-ray\ntransients in quiescence \\citep{sgbh00,gm98,men00}, the plasma may be too\nneutral for the MRI to operate. This presents some difficulties for\nunderstanding the evolution of these systems, since no robust transport mechanism\nakin to MRI-induced turbulence has been established for purely-hydrodynamic\nKeplerian shear flows.\n\nSuch a mechanism has been claimed recently by \\cite{kb03},\nwho find vortices and an outward transport of angular momentum in the\nnonlinear outcome of their global simulations. The claim is that this\nnonlinear outcome is due to a {\\it local} instability (the ``Global\nBaroclinic Instability'') resulting from the presence of an equilibrium\nentropy gradient in the radial direction. The instability mechanism\ninvoked \\citep{klr04} is an interplay between transient amplification of\nlinear disturbances and nonlinear effects. The existence of such a\nmechanism would have profound implications for understanding the\nevolution of weakly-ionized disks.\n\nIn Chapter~\\ref{paper2}, we have\nperformed a linear stability analysis for local nonaxisymmetric\ndisturbances in the shearing-wave formalism. While the linear theory\nuncovers no exponentially-growing instability (except for convective\ninstability in the absence of shear), interpretation of the results\nis somewhat difficult due to the nonnormal nature of the linear\ndifferential operators\\footnote{A nonnormal operator is one that is\nnot self-adjoint, i.e. it does not have orthogonal eigenfunctions.}:\none has a coupled set of differential\nequations in time rather than a dispersion relation, which results\nin a nontrivial time dependence for the perturbation amplitudes\n$\\delta(t)$. In addition, transient amplification does occur for\na subset of initial perturbations, and linear theory cannot tell us\nwhat effect this will have on the nonlinear outcome. For these\nreasons, and in order to test for the presence of local nonlinear\ninstabilities, we here supplement our linear analysis with local\nnumerical experiments.\n\nWe begin in \\S2 by outlining the basic equations for a local model of\na thin disk. In \\S3 we summarize the linear theory results from Chapter~\\ref{paper2}.\nWe describe our numerical model and nonlinear results in \\S\\S4 and 5, and\ndiscuss the implications of our findings in \\S6.\n\n\\section{Basic Equations}\n\nThe simulations of \\cite{kb03} are two-dimensional (without vertical\nstructure), since the salient feature supposedly giving rise to the\ninstability is a radial entropy gradient. The simplest model to use\nfor a local verification of their global results is the two-dimensional\nshearing sheet (see, e.g., \\citealt{gt78}). This local approximation\nis made by expanding the equations of motion in the ratio of the disk\nscale height $H$ to the local radius $R$, and is therefore only valid\nfor thin disks ($H\/R \\ll 1$). The vertical structure is removed by\nusing vertically-integrated quantities for the fluid variables\\footnote{This\nvertical integration is not rigorous; we are assuming that important\nvertical structure does not develop to affect our results.}. The\nbasic equations that one obtains are\n\\begin{equation}\\label{EQN1}\n\\dv{\\Sigma}{t} + \\Sigma \\bnabla \\cdot \\bv = 0,\n\\end{equation}\n\\begin{equation}\\label{EQN2}\n\\dv{\\bv}{t} + \\frac{\\bnabla P}{\\Sigma} + 2\\bO\\times\\bv - 2q\\Omega^2 x \\ex = 0,\n\\end{equation}\n\\begin{equation}\\label{EQN3}\n\\dv{\\,{\\rm{ln}} S}{t} = 0,\n\\end{equation}\nwhere $\\Sigma$ and $P$ are the two-dimensional density and pressure, $S\n\\equiv P \\Sigma^{-\\gamma}$ is the fluid entropy,\\footnote{For a\nnon-self-gravitating disk the two-dimensional adiabatic index $\\gamma =\n(3\\gamma_{3D} - 1)\/(\\gamma_{3D} + 1)$ (e.g. \\citealt{ggn86}).} $\\bv$ is\nthe fluid velocity and $d\/dt$ is the Lagrangian derivative. The third and\nfourth terms in equation (\\ref{EQN2}) represent the Coriolis and centrifugal\nforces in the local model expansion, where $\\Omega$ is the local rotation\nfrequency, $x$ is the radial Cartesian coordinate and $q$ is the shear\nparameter (equal to $1.5$ for a disk with a Keplerian rotation profile).\nThe gravitational potential of the central object is included as part of\nthe centrifugal force term in the local-model expansion, and we ignore the\nself-gravity of the disk.\n\n\\section{Summary of Linear Theory Results}\n\nAn equilibrium solution to equations (\\ref{EQN1}) through (\\ref{EQN3}) is\n\\begin{equation}\\label{P0}\nP = P_0(x),\n\\end{equation}\n\\begin{equation}\n\\Sigma = \\Sigma_0(x),\n\\end{equation}\n\\begin{equation}\\label{VEQ}\n\\bv \\equiv \\bv_0 = \\left(-q\\Omega x + \\frac{P_0^\\prime}{2\\Omega\n\\Sigma_0}\\right)\\ey,\n\\end{equation}\nwhere a prime denotes an $x$ derivative.\nOne can regard the background flow as providing an effective shear rate\n\\begin{equation}\n\\qe \\Omega \\equiv -v_0^\\prime\n\\end{equation}\nthat varies with $x$, in which case $\\bv_0 = -\\int^x \\qe(s) ds \\, \\Omega\n\\ey$. Due to this background shear, localized disturbances can be\ndecomposed in terms of ``shwaves'', Fourier modes in a frame comoving with\nthe shear. These have a time-dependent radial wavenumber given by\n\\begin{equation}\\label{KXEFF}\n\\tilde{k}_x(t,x) \\equiv k_{x0} + \\qe(x)\\Omega k_y t.\n\\end{equation}\nwhere $k_{x0}$ and $k_y$ are constants. Here $k_y$ is the azimuthal\nwave number of the shwave.\n\nIn the limit of zero stratification,\n\\begin{equation}\nP_0(x) \\rightarrow constant,\n\\end{equation}\n\\begin{equation}\n\\Sigma_0(x) \\rightarrow constant,\n\\end{equation}\n\\begin{equation}\\label{V0U}\n\\bv_0 \\rightarrow -q\\Omega x \\ey,\n\\end{equation}\n\\begin{equation}\n\\qe \\rightarrow q,\n\\end{equation}\nand\n\\begin{equation}\n\\tilde{k}_x \\rightarrow k_x \\equiv k_{x0} + q\\Omega k_y t.\n\\end{equation}\nIn Chapter~\\ref{paper2}, we analyze the time dependence of the shwave amplitudes for both\nan unstratified equilibrium and a radially-stratified equilibrium. As\ndiscussed in more detail in Chapter~\\ref{paper2}, applying the shwave formalism to a\nradially-stratified shearing sheet effectively uses a short-wavelength\nWKB approximation, and is therefore only valid in the limit $k_y L \\gg 1$,\nwhere the background varies on a scale $L \\sim H \\ll R$. The disk scale\nheight $H \\equiv c_s \\Omega$, where $c_s = \\sqrt{\\gamma P_0\/\\Sigma_0}$.\n\nThere are three nontrivial shwave solutions in the unstratified shearing\nsheet, two nonvortical and one vortical. The radial stratification\ngives rise to an additional vortical shwave. In the limit of \ntightly-wound shwaves ($|k_x| \\gg k_y$), the nonvortical and vortical\nshwaves are compressive and incompressive, respectively. The former \nare the extension of acoustic modes to nonaxisymmetry, and to leading \norder in $(k_y L)^{-1}$ they are the same both with and without \nstratification. Since the focus of our investigation is on convective \ninstability and the generation of vorticity, we repeat here only the\nsolutions for the incompressive vortical shwaves and refer the\nreader to Chapter~\\ref{paper2} for further details on the nonvortical shwaves.\n\nIn the unstratified shearing sheet, the solution for the incompressive shwave is given\nby:\n\\begin{equation}\n\\delta v_x = \\delta v_{x0}\\frac{k_0^2}{k^2},\n\\end{equation}\n\\begin{equation}\n\\delta v_y = -\\frac{k_x}{k_y} \\delta v_x\n\\end{equation}\nand\n\\begin{equation}\\label{IS4}\n\\frac{\\delta \\Sigma}{\\Sigma_0} = \\frac{\\delta P}{\\gamma P_0} =\n\\frac{1}{i c_s k_y}\\left(\\frac{k_x}{k_y} \\frac{\\dot{\\delta v}_x}{c_s}\n+ 2(q - 1) \\Omega \\frac{\\delta v_x}{c_s}\\right),\n\\end{equation}\nwhere $k^2 = k_x^2 + k_y^2$, ($k_0,\\delta v_{x0}$) are the values of\n($k, \\delta v_x$) at $t=0$ and an overdot denotes a time \nderivative.\\footnote{As\ndiscussed in Chapter~\\ref{paper2}, this solution is valid for all time\nonly in the short-wavelength limit ($k_y H \\gg 1$); for $H k_y \\lesssim \nO(1)$, an initially-leading incompressive shwave will turn into a \ncompressive shwave near $k_x = 0$.}\n\nThe kinetic energy for a single incompressive shwave can be defined as\n\\begin{equation}\nE_k \\equiv \\frac{1}{2}\\Sigma_0 (\\delta v_x^2 + \\delta v_y^2) =\n\\frac{1}{2}\\Sigma_0 \\delta v_{x0}^2 \\frac{k_0^4}{k_y^2 k^2},\n\\end{equation}\nan expression which varies with time and peaks at $k_x = 0$.\nIf one defines an amplification factor for an individual shwave,\n\\begin{equation}\\label{AMP}\n{\\cal A } \\equiv \\frac{E_k(k_x = 0)}{E_k(t = 0)} =\n1 + \\frac{k_{x0}^2}{k_y^2},\n\\end{equation}\nit is apparent that an arbitrary amount of transient amplification in\nthe kinetic energy of an individual shwave can be obtained as one \nincreases the amount of swing for a leading shwave ($k_{x0} \\ll -k_y$).\n\nThis transient amplification of local nonaxisymmetric disturbances is\nreminiscent of the ``swing amplification'' mechanism that occurs in \ndisks that are marginally-stable to the axisymmetric gravitational \ninstability \\citep{glb65,jt66,gt78}. In that context, nonaxisymmetric \nshwaves experience a short period of exponential growth near $k_x = 0$ \nas they swing from leading to trailing. In order for this mechanism \nto be effective in destabilizing a disk, however, a feedback mechanism \nis required to convert trailing shwaves into leading shwaves\n\\citep{bt87}. The arbitrarily-large amplification implied by equation\n(\\ref{AMP}) has led some authors to argue for a bypass transition to\nturbulence in hydrodynamic Keplerian shear flows \\citep{cztl03,ur04,amn04}.\nThe reasoning is that nonlinear effects somehow provide the necessary\nfeedback. We show in Chapter~\\ref{paper2} that a ensemble of incompressive shwaves drawn\nfrom an isotropic, Gaussian random field has a kinetic energy that is a\nconstant, independent of time. This indicates that a random set\nof vortical perturbations will not extract energy from the mean shear. It\nis clear, however, that the validity of this mechanism as a transition to\nturbulence can only be fully explored via numerical experiments. No\nnumerical experiments to date have demonstrated a {\\it transition} to\nturbulence in Keplerian shear flows.\n\nIn the presence of radial stratification, there are two linearly-independent\nincompressive shwaves. The radial-velocity perturbation satisfies the\nfollowing equation (we use a subscript $s$ to distinguish the stratified\nfrom the unstratified case):\n\\begin{equation}\\label{BOUSS2D}\n(1 + \\tilde{\\tau}^2)\\dv{^2\\delta v_{xs}}{\\tilde{\\tau}^2} + 4 \\tilde{\\tau}\n\\dv{\\delta v_{xs}}{\\tilde{\\tau}} + ({\\rm{Ri}} + 2)\\delta v_{xs} = 0,\n\\end{equation}\nwhere\n\\begin{equation}\n\\tilde{\\tau} \\equiv \\tilde{k}_x\/k_y = \\qe\\Omega t + k_{x0}\/ky\n\\end{equation}\nis the time variable and\n\\begin{equation}\n{\\rm Ri} \\equiv \\frac{N_x^2}{(\\qe\\Omega)^2}\n\\end{equation}\nis the Richardson number, a measure of the relative importance of buoyancy\nand shear \\citep{jwm61,how61,chi70}\\footnote{As discussed in Chapter~\\ref{paper2}, \nequation (\\ref{BOUSS2D}) is the same equation that one obtains for the\nincompressive shwaves in a shearing, stratified atmosphere.}. Here\n\\begin{equation}\nN_x^2 \\equiv -\\frac{c_s^2}{L_S L_P}\n\\end{equation}\nis the square of the Brunt-V$\\ddot{\\rm{a}}$is$\\ddot{\\rm{a}}$l$\\ddot{\\rm{a}}$\nfrequency in the radial direction, where $L_P \\equiv \\gamma P_0\/P_0^\\prime$\nand $L_S \\equiv \\gamma S_0\/S_0^\\prime$ are the equilibrium pressure and \nentropy length scales in the radial direction. The solutions for the other\nperturbation variables are related to $\\delta v_{xs}$ by\n\\begin{equation}\\label{XIY}\n\\delta v_{ys} = -\\tilde{\\tau} \\delta v_{xs},\n\\end{equation}\n\\begin{equation}\n\\frac{\\delta \\Sigma_s}{\\Sigma_0} = \\frac{1}{L_S} \\int \\delta v_{xs}\\,dt\n\\end{equation}\nand\n\\begin{equation}\\label{SOLH}\n\\frac{\\delta P_s}{P_0} = \\frac{\\gamma \\Omega}{i c_s k_y} \\left[\\qe\n\\tilde{\\tau} \\dv{}{\\tilde{\\tau}}\\left(\\frac{\\delta v_{xs}}{c_s}\\right) +\n2(\\qe - 1) \\frac{\\delta v_{xs}}{c_s}\\right].\n\\end{equation}\n\nSince the solutions to equation (\\ref{BOUSS2D}) are hypergeometric\nfunctions, which have a power-law time dependence, it cannot in general be\naccurately treated with a WKB analysis; there is no asymptotic region in\ntime where equation (\\ref{BOUSS2D}) can be reduced to a dispersion\nrelation. If, however, there is a region of the disk where the effective\nshear is zero, $\\tilde{\\tau} \\rightarrow constant$ and equation\n(\\ref{BOUSS2D}) can be expressed as a WKB dispersion relation:\n\\begin{equation}\\label{DRQ}\n\\omega^2 = \\frac{k_y^2}{k_{x0}^2 + k_y^2}N_x^2,\n\\end{equation}\nwith $\\delta(t) \\propto \\exp(-i\\omega t)$.\nFor $\\qe \\simeq 0$ and $N_x^2 < 0$, then, there is convective instability.\nFor disks with nearly-Keplerian rotation profiles and modest radial\ngradients, $\\qe \\simeq 1.5$ and one would expect that the instability is\nsuppressed by the strong shear. Due to the lack of a dispersion relation, \nhowever, there is no clear cutoff between exponential and oscillatory time \ndependence, and establishing a rigorous analytic stability criterion is difficult.\n\nFor $\\qe \\neq 0$, the asymptotic time dependence for each perturbation variable\nat late times is\n\\begin{equation}\n\\delta P_s \\propto \\delta v_{xs} \\sim t^{\\frac{\\alpha - 3}{2}},\n\\end{equation}\n\\begin{equation}\n\\delta \\Sigma_s \\propto \\delta v_{ys} \\sim t^{\\frac{\\alpha - 1}{2}},\n\\end{equation}\nwhere\n\\begin{equation}\n\\alpha \\equiv \\sqrt{1 - 4 \\, {\\rm{Ri}}}.\n\\end{equation}\nThe density and $y$-velocity perturbations therefore grow asymptotically for\n$\\alpha > 1$, i.e. ${\\rm{Ri}} < 0$. For small Richardson number, as is\nexpected for a nearly-Keplerian disk with modest radial gradients,\n$\\alpha \\simeq 1 - 2{\\rm Ri}$ and the asymptotic growth is extremely slow:\n\\begin{equation}\\label{ASYMP}\n\\delta \\Sigma_s \\sim \\delta v_{ys} \\sim t^{-{\\rm Ri}}.\n\\end{equation}\nThe energy of an ensemble of shwaves grows asymptotically as\n$t^{2\\alpha-1}$, or $t^{1-4{\\rm Ri}}$ for small Ri. The ensemble energy\ngrowth is thus nearly linear in time for small Ri, independent of the sign\nof Ri.\\footnote{We show in Chapter~\\ref{paper2} that there is also linear growth in the\nenergy of an ensemble of compressive shwaves.}\n\nThe velocity perturbations are changed very little by a weak radial\ngradient. One would therefore expect that, at least in the linear\nregime, transient amplification of the kinetic energy for an individual\nshwave is relatively unaffected by the presence of stratification.\nThere is, however, an associated density perturbation in the stratified \nshearing sheet that is not present in the unstratifed sheet.\\footnote{\nThe amplitude of the density perturbation in the unstratified sheet is \nan order-of-magnitude lower than the velocity perturbations in the \nshort-wavelength limit; see equation(\\ref{IS4}).} This results in \ntransient amplification of the {\\it potential} energy of an individual \nshwave. We do not derive in Chapter~\\ref{paper2} I a general closed-form expression\nfor the energy of an ensemble of incompressive shwaves in the stratified \nshearing sheet, so it is not entirely clear what effect this \nqualitatively new piece of the energy will have on an ensemble of \nshwaves in the linear regime.\n\nIn any case, the question of whether or not radial stratification can play \na role in generating turbulence by interacting with the transient \namplification of linear disturbances or by some other nonlinear mechanism \ncan only be fully answered with a nonlinear study. For this reason, and \ndue to the subtleties involved in the linear analysis, we now turn to the\nmain focus of this paper, which is a series of local numerical experiments\nin a radially-stratified shearing sheet.\n\n\\section{Numerical Model}\n\nTo investigate local nonlinear effects in a radially-stratified thin\ndisk, we integrate the governing equations (\\ref{EQN1}) through (\\ref{EQN3})\nwith a hydrodynamics code based on ZEUS \\citep{sn92}. This is a\ntime-explicit, operator-split, finite-difference method on a staggered\nmesh. It uses an artificial viscosity to capture shocks. The\ncomputational grid is $L_x \\times L_y$ in physical size with $N_x \\times\nN_y$ grid cells, where $x$ is the radial coordinate and $y$ is the\nazimuthal coordinate. The boundary conditions are periodic in the\n$y$-direction and shearing-periodic in the $x$-direction. The shearing-box\nboundary conditions are described in detail in \\cite{hgb95}. As described \nin \\cite{mass00} and \\cite{gam01}, advection by the linear shear flow can be\ndone by interpolation. Rather than using a linear interpolation scheme as\nin \\cite{gam01}, we now do the shear transport with the same upwind \nadvection algorithm used in the rest of the code. This is less diffusive \nthan linear interpolation, and the separation of the shear from the bulk \nfluid velocity means that one is not Courant-limited by large shear \nvelocities at the edges of the computational domain.\n\nWe use the following equilibrium profile, which in general gives rise to \nan entropy that varies with radius:\n\\begin{equation}\nh_0(x) = h_a\\left[1 - \\epsilon \\cos\\left(\\frac{2\\pi x}{L_x}\\right)\\right] \\;\n\\; , \\; \\; \\Sigma_0(x) =\n\\left[\\frac{h_0(\\Gamma-1)}{\\Gamma K}\\right]^{\\frac{1}{\\Gamma-1}} \\; \\; , \\;\n\\; P_0(x) = K \\Sigma_0^\\Gamma,\n\\end{equation}\nwhere $h_a$, $\\epsilon$,\n$\\Gamma$ and $K$ are model parameters. The flow is maintained in\nequilibrium by setting the initial velocity according to equation\n(\\ref{VEQ}). This equilibrium yields a\nBrunt-V$\\ddot{\\rm{a}}$is$\\ddot{\\rm{a}}$l$\\ddot{\\rm{a}}$ frequency\n\\begin{equation}\\label{N2}\nN_x^2(x) = \\frac{(\\gamma-\\Gamma) {h_0^\\prime}^2}{\\gamma (\\Gamma-1) h_0},\n\\end{equation}\nwhich can be made positive, negative or zero by varying $\\gamma-\\Gamma$.\n\nWe fix some of the model parameters to yield an equilibrium profile that\nis appropriate for a thin disk. In particular, we want $H\/L_P \\sim H\/R \\ll\n1$ in order to be consistent with our use of a razor-thin (two-dimensional)\ndisk model. In addition, we want the equilibrium values for each fluid\nvariable to be of the same order to ensure the applicability of our linear\nanalysis. These requirements can be met by choosing $K = 1$, $\\epsilon =\n0.1$, $L_x = 12$ and $h_a = \\bar{c}_s^2 \\Gamma\/(\\Gamma-1)$, where\n$\\bar{c}_s \\equiv \\sqrt{\\< P_0\/\\Sigma_0 \\>} \\equiv 1$ is (to within a factor of\n$\\sqrt{\\gamma}$) the $x$ average of the sound speed. Since the equilibrium\nprofile changes with $\\Gamma$, we choose a fixed value of $\\Gamma = 4\/3$,\nwhich for $\\gamma = \\Gamma$ corresponds to a three-dimensional adiabatic\nindex of $7\/5$. These numbers yield $|H\/L_P| \\leq 0.2$. Our unfixed model \nparameters are thus $L_y$, $q$ and $\\gamma$.\n\nThe sinusoidal equilibrium profile we are using generates radial\noscillations in the shearing sheet due to truncation error. We apply an\nexponential-damping term to the governing equations in order to reduce the\nspurious oscillations and therefore get cleaner growth-rate measurements.\nWe damp the oscillations until their amplitude is equal to that of\nmachine-level noise, and subsequently apply low-level random perturbations\nto trigger any instabilities that may be present.\n\nAs a test for our code, we evolve a particular solution for the\nincompressive shwaves in the radially-stratified shearing sheet (equations\n[\\ref{BOUSS2D}] and [\\ref{XIY}]-[\\ref{SOLH}]). The initial conditions\nare $\\delta v_x\/\\bar{c}_s = \\delta \\Sigma\/\\Sigma_0 = 1 \\times 10^{-4}$ and\n$k_{x0} = -128\\pi\/L_x$. We set $L_y = 0.375$ and $k_y = 2\\pi\/L_y$ in\norder to operate in the short-wavelength regime, and the other model\nparameters are $q = 1.5$ and $\\gamma - \\Gamma = -0.3102$. The latter\nvalue yields a minimum value for $N_x^2(x)$ of $-0.01$. The results of\nthe linear theory test are shown in Figures~\\ref{pap4f1a} and \\ref{pap4f1b}.\n\n\\begin{figure}[hp]\n \\hfill\n \\begin{minipage}[t]{1.\\textwidth}\n \\begin{center}\n \\includegraphics[width=3.5in,clip]{pap4f1a.eps}\n \\caption[Evolution of the radial velocity amplitude for a vortical shwave in the\n radially-stratified shearing sheet.]\n {Evolution of the radial velocity amplitude for a vortical shwave in the\n radially-stratified shearing sheet (Run 1). The heavy line is the analytic result,\n and the light lines are runs with a numerical resolution of (in order of increasing\n accuracy) $N_x \\times N_y = 1024 \\times 16$, $2048 \\times 32$ and $4096 \\times 64$.\n The number of grid cells are chosen so that the shwave initially has the same number\n of grid cells per wavelength in both the $x$ and $y$ directions. The results are\n shown for a test point at the minimum in $N_x^2$.}\n \\label{pap4f1a}\n \\end{center}\n \\end{minipage}\n \\hfill\n \\begin{minipage}[b]{1.\\textwidth}\n \\begin{center}\n \\includegraphics[width=3.5in,clip]{pap4f1b.eps}\n \\caption[Evolution of the density amplitude for a vortical shwave in the\n radially-stratified shearing sheet.]\n {Evolution of the density amplitude for a vortical shwave in the\n radially-stratified shearing sheet (Run 1). The heavy line is the analytic result,\n and the light lines are runs with a numerical resolution of (in order of increasing\n accuracy) $N_x \\times N_y = 1024 \\times 16$, $2048 \\times 32$ and $4096 \\times 64$.}\n \\label{pap4f1b}\n \\end{center}\n \\end{minipage}\n \\hfill\n\\end{figure}\n\n\\section{Nonlinear Results}\n\n\\begin{deluxetable}{lll}\n\\tablecolumns{3}\n\\tablewidth{0pc}\n\\tabcolsep 0.5truecm\n\\tablecaption{Summary of Code Runs \\label{pap4t1}}\n\\tablehead{Run & Description & Figure(s)}\n\\startdata\n1 & Linear theory test & \\ref{pap4f1a}-\\ref{pap4f1b} \\\\\n2 & External potential, $\\Omega = 0$ & \\ref{pap4f2}-\\ref{pap4f4} \\\\\n3 & External potential, $\\Omega = 1$ & \\ref{pap4f5}-\\ref{pap4f6} \\\\\n4 & Uniform rotation ($q = 0$) & \\ref{pap4f7}-\\ref{alpharun4} \\\\\n5 & External potential, $\\Omega = 1$, boost & \\ref{pap4f9} \\\\\n6 & Small shear ($-\\infty \\lesssim {\\rm Ri} \\lesssim -1$) & \\ref{pap4f10} \\\\\n7 & Small shear ($q = 0.2, {\\rm Ri} \\gtrsim -1$) & \\ref{pap4f11} \\\\\n8 & Aliasing ($q = 0.2, {\\rm Ri} \\gtrsim -1$) & \\ref{pap4f12} \\\\\n9 & Parameter survey & \\ref{pap4f13} \\\\\n10 & Keplerian disk ($q = 1.5, {\\rm Ri} \\simeq -0.004$) & \\ref{pap4f14} \\\\\n\\enddata\n\\end{deluxetable}\n\nTable~\\ref{pap4t1} gives a summary of the runs that we have performed. A\ndetailed description of the setup and results for each is given in the\nfollowing subsections. Our primary diagnostic is a measurement of growth\nrates, and the probe that we use for these measurements is an average over\nazimuth of the absolute value of $v_x = \\delta v_x$ at the minimum in\n$N_x^2$. Measuring $v_x$ allows us to demonstrate the damping of the\ninitial radial oscillations, and the average over azimuth masks the\ninteractions between multiple WKB modes with different growth rates in our\nmeasurements. We will reference this probe with the following definition:\n\\begin{equation}\nv_t \\equiv \\langle |v_x(x_{min},y)| \\rangle,\n\\end{equation}\nwhere here angle brackets denote an average over $y$ and $x_{min}$ is the\n$x$-value at which $N_x^2(x)$ is a minimum.\n\n\\subsection{External Potential in Non-Rotating Frame}\n\nAs a starting problem, we investigate a stratified flow with $\\bv_0 = 0$.\nSuch a flow can be maintained in equilibrium by replacing the tidal force\nin equation (\\ref{EQN2}) with an external potential $\\Phi = -h_0$. This can\nbe done in either a rotating or non-rotating frame. It is a particularly\nsimple way of validating our study of convective instability in the shearing\nsheet. The condition $\\bv_0 = 0$ implies $\\qe = 0$, and therefore equation\n(\\ref{DRQ}) should apply in the WKB limit, with the expected growth rate\nobtained by evaluating equation (\\ref{N2}) locally.\\footnote{The fastest growing\nWKB modes will be the ones with a growth rate corresponding to the minimum\nin $N_x^2$.} We have performed a fiducial run with an imposed external\npotential in a non-rotating frame ($\\Omega = 0$ in equation [\\ref{EQN2}]) to\ncompare with the outcome expected from the Schwarzschild stability criterion\nimplied by equation (\\ref{DRQ}). We set $\\gamma - \\Gamma = -0.3102$,\ncorresponding to $N_{x,min}^2 = -0.01$, and $L_y = L_x$. The expected growth\nrate for this Schwarzschild-unstable equilibrium is $0.1$ (in units of the\naverage radial sound-crossing time). The numerical resolution for the fiducial\nrun is $512 \\times 512$, and all the variables are randomly perturbed at an\namplitude of $1.0 \\times 10^{-12}$.\n\nA plot of $v_t$ as a function of time is given in Figure~\\ref{pap4f2}, showing\nthe initial damping followed by exponential growth in the linear regime. The\nanalytic growth rate is shown on the plot for comparison. A least-squares\nfit of the data in the range $100 \\leq t \\leq 250$ yields a measured growth\nrate of $0.0978$.\\footnote{Measurements of the growth rate earlier in the\nlinear regime or over a larger range of data yield results that differ from\nthis value by at most $5\\%$.} Figure~\\ref{pap4f3a} shows a cross section of $N_x^2$\nas a function of $x$ after the instability has begun to set in, and Figure~\\ref{pap4f3b}\nshows cross\nsections of the entropy early and late in the nonlinear regime. The growth is\ninitially concentrated near the minimum points in $N_x^2(x)$. Eventually the entropy\nturns over completely and settles to a nearly constant value. Figure~\\ref{pap4f4}\nshows two-dimensional snapshots of the entropy in the nonlinear regime. Runs with\nthe same equilibrium profile except $\\gamma - \\Gamma \\geq 0$ are stable. There\nis also a long-wavelength axisymmetric instability that is present for\n$\\gamma - \\Gamma < 0$ even in the absence of the small-scale nonaxisymmetric\nmodes. We measure its growth rate to be $0.07$. Due to the long-wavelength\nnature of these modes, they are not treatable by a local linear analysis.\n\n\\begin{figure}[hp]\n\\centering\n\\includegraphics[width=3.5in,clip]{pap4f2.eps}\n\\caption{Evolution of $v_t$ as a function of time for Run 2 (external potential,\nnon-rotating frame).}\n\\label{pap4f2}\n\\end{figure}\n\n\\begin{figure}[hp]\n \\hfill\n \\begin{minipage}[t]{1.\\textwidth}\n \\begin{center}\n \\includegraphics[width=3.5in,clip]{pap4f3a.eps}\n \\caption[Plot of $N_x^2$ as a function of $x$ for Run 2.]\n {Plot of $N_x^2$ (averaged over $y$) as a function of $x$ for Run 2. The dotted line\n shows the equilibrium profile, and the solid line shows a snapshot during the nonlinear\n regime. Growth initially occurs at the minimum in $N_x^2$.}\n \\label{pap4f3a}\n \\end{center}\n \\end{minipage}\n \\hfill\n \\begin{minipage}[b]{1.\\textwidth}\n \\begin{center}\n \\includegraphics[width=3.5in,clip]{pap4f3b.eps}\n \\caption[Plot of the entropy as a function of $x$ for Run 2.]\n {Plot of the entropy (averaged over $y$) as a function of $x$ for Run 2. The dotted line\n shows the equilibrium profile, and the solid lines show snapshots during the nonlinear\n regime. The entropy eventually settles to a nearly-constant value.}\n \\label{pap4f3b}\n \\end{center}\n \\end{minipage}\n \\hfill\n\\end{figure}\n\n\\begin{figure}[hp]\n\\centering\n\\includegraphics[width=6.in,clip]{pap4f4.eps}\n\\caption[Snapshots of the entropy in the nonlinear regime for Run 2.]\n{Snapshots of the entropy in the nonlinear regime for Run 2, indicating\nmaximum growth for modes near the grid scale and the eventual turnover of\nthe equilibrium entropy profile to its average value. Dark shades indicate \nvalues above (red in the color version) and below (blue in the color version) \nthe average value (yellow in the color version).}\n\\label{pap4f4}\n\\end{figure}\n\n\\subsection{External Potential in Rotating Frame}\n\nWe have performed the same test as described in \\S5.1 in a rotating frame\n($\\Omega = 1$ in equation [\\ref{EQN2}]). Figure~\\ref{pap4f5} shows the\nexponential growth in the linear regime for this run, with a measured growth\nrate of $0.0977$. Figure~\\ref{pap4f6} shows snapshots of the entropy in the\nnonlinear regime. The results are similar to the nonrotating case, except that\n1) rotation suppresses the long-wavelength axisymmetric instability; 2) the\nnonlinear outcome exhibits more coherent structures in the rotating case\nincluding transient vortices; and 3) these coherent structures eventually\nbecome unstable to a Kelvin-Helmholz-type instability.\n\n\\begin{figure}[hp]\n\\centering\n\\includegraphics[width=5.5in,clip]{pap4f5.eps}\n\\caption[Evolution of $v_t$ as a function of time for Run 3 (external potential,\nrotating frame).]\n{Evolution of $v_t$ as a function of time for Run 3 (external potential,\nrotating frame). The dotted line shows the expected growth rate.}\n\\label{pap4f5}\n\\end{figure}\n\n\\begin{figure}[hp]\n\\centering\n\\includegraphics[width=6.in,clip]{pap4f6.eps}\n\\caption[Snapshots of the entropy in the nonlinear regime for Run 3.]\n{Snapshots of the entropy in the nonlinear regime for Run 3. Dark shades \nindicate values above (red in the color version) and below (blue in the \ncolor version) the average value (yellow in the color version).}\n\\label{pap4f6}\n\\end{figure}\n\n\\subsection{Uniform Rotation}\n\nHaving demonstrated the viability of simulating convective instability in\nthe local model, we now turn to the physically-realistic equilibrium\ndescribed in \\S4. We begin by setting the shear parameter $q$ to zero\nin order to make contact with the results of \\S\\S5.1 and 5.2. This is\nanalogous to a disk in uniform rotation. The other model parameters are\nthe same as for the previous runs. While there is still an effective\nshear $-0.05 \\lesssim \\qe \\lesssim 0.05$, near $\\qe = 0$ one expects the\nmodes to obey equation (\\ref{DRQ}) in the WKB limit. Figures~\\ref{pap4f7}\nand \\ref{pap4f8} give the linear and nonlinear results for this run. The\nmeasured growth rate in the linear regime is $0.0809$.\n\n\\begin{figure}[hp]\n\\centering\n\\includegraphics[width=5.5in,clip]{pap4f7.eps}\n\\caption[Evolution of $v_t$ as a function of time for Run 4 ($q = 0$,\nrotating frame).]\n{Evolution of $v_t$ as a function of time for Run 4 ($q = 0$). The\ndotted line shows the expected growth rate, and the solid lines are runs\nwith (in order of increasing growth) $L_y = 12$, $6$, $3$ and $1.5$.}\n\\label{pap4f7}\n\\end{figure}\n\n\\begin{figure}[hp]\n\\centering\n\\includegraphics[width=6.in,clip]{pap4f8.eps}\n\\caption[Snapshots of the entropy in the nonlinear regime for Run 4.]\n{Snapshots of the entropy in the nonlinear regime for Run 4. Notice that\nthe maximum growth does not occur for modes at the grid scale.}\n\\label{pap4f8}\n\\end{figure}\n\nConsistent with results from numerical simulations of vertical convection\n\\citep{sb96,cab96}, the angular momentum transport associated with radial \nconvection is {\\it inwards}. Figure~\\ref{alpharun4} shows the evolution of \nthe dimensionless angular momentum flux\n\\begin{equation}\n\\alpha \\equiv \\frac{1}{L_x L_y \\} \\int \\Sigma \\delta v_x \\delta v_y dx dy,\n\\end{equation}\nwhere $\\$ is the radial average of the equilibrium pressure, for an \nextended version of Run 4. Averaging over the last $1200 \\Omega^{-1}$ yields \n$\\alpha \\sim -10^{-5}$.\n\n\\begin{figure}[hp]\n\\centering\n\\includegraphics[width=6.in,clip]{alpharun4.eps}\n\\caption\n{Evolution of the dimensionless angular momentum flux due to radial convection.}\n\\label{alpharun4}\n\\end{figure}\n\nThere are two reasons for the larger error in the measured growth rate for\nthis run: 1) the equilibrium velocity gives rise to numerical diffusion due to\nthe motion of the fluid variables with respect to the grid; and 2) since the\ngrowing modes are being advected in the azimuthal direction, the maximum growth\ndoes not occur at the grid scale. The latter effect can be seen in Figure~\\ref{pap4f8};\nseveral grid cells are required for a well-resolved wavelength. In order to resolve\nsmaller wavelengths, we have repeated this run with $L_y = 6$, $3$ and $1.5$. The\nresults are plotted in Figure~\\ref{pap4f7} along with the results from the $L_y = 12$\nrun. The measured growth rate for the $L_y = 1.5$ run is $0.0924$.\n\nTo quantify the effects of numerical diffusion, we have performed a series\nof tests similar to Run 2 (external potential in a rotating frame) but with\nan overall boost in the azimuthal direction. Figure~\\ref{pap4f9} shows measured growth\nrates as a function of boost at three different numerical resolutions. The\nlargest boost magnitude in this plot corresponds to the velocity at the\nminimum in $N_x^2$ for a run with $q = 1.5$.\n\n\\begin{figure}[hp]\n\\centering\n\\includegraphics[width=4.in,clip]{pap4f9.eps}\n\\caption[Growth rates as a function of azimuthal boost.]\n{Growth rates as a function of azimuthal boost in a series of runs with\nan external potential and $N_{x,min}^2 = -0.01$. The dotted line shows\nthe analytic growth rate from linear theory. The open circle denotes the\ngrowth rate that was measured in Run 4, with the boost corresponding to\nthe magnitude of the velocity at the minimum in $N_x^2$ for Run 3 ($q = 0$).\nThe largest boost magnitude corresponds to the velocity at the minimum in\n$N_x^2$ for Run 10 ($q = 1.5$).}\n\\label{pap4f9}\n\\end{figure}\n\n\\subsection{Shearing Sheet}\n\nTo investigate the effect of differential rotation upon the growth of this\ninstability, we have performed a series of simulations with nonzero $q$.\nIntuitively, one expects the instability to be suppressed when the shear\nrate is greater than the growth rate, i.e. for ${\\rm Ri} \\gtrsim -1$.\nFigure~\\ref{pap4f10} shows growth rates from a series of runs with $N_{x,min}^2\n= -0.01$ and small, nonzero values of $q$ at three numerical resolutions. This\nfigure clearly demonstrates our main result: convective instability is suppressed by\ndifferential rotation. The expected growth rate from linear theory ($\\sqrt{|N_x^2|}$\nat $\\qe = 0$) is shown in Figure~\\ref{pap4f10} as a dotted line. If there is a\nradial position where $\\qe(x) = 0$ (i.e., ${\\rm Ri} = -\\infty$), $v_t$ at that\nposition looks similar to that of the previous runs (very little deviation from a\nstraight line); these measurements are indicated on the plot with solid points.\nFor $q \\gtrsim 0.055$ there is no longer any point where $\\qe(x) = 0$; in that case\n$v_t$ was measured at the radial average between the minimum in $N_x^2(x)$ and the\nminimum in $\\qe(x)$, since this is where the maximum growth occured. The data for\nthese measurements, which are indicated in Figure~\\ref{pap4f10} with open points, is\nnot as clean as it is for the runs with ${\\rm Ri} = -\\infty$ (see Figure~\\ref{pap4f11}).\nAll of the growth rate measurements in Figure~\\ref{pap4f10} were obtained by a least-squares\nfit of the data in the range $1 \\times 10^{-9} < v_t\/\\bar{c}_s < 1 \\times 10^{-5}$.\nThe dashed line in Figure~\\ref{pap4f10} indicates the value of $q$ for which\n${\\rm Ri}_{min} = -1$.\n\n\\begin{figure}[hp]\n\\centering\n\\includegraphics[width=3.5in,clip]{pap4f10.eps}\n\\caption[Growth rates as a function of $q$.]\n{Growth rates as a function of $q$ with $N_{x,min}^2 = -0.01$.\nSee the text for a discussion.}\n\\label{pap4f10}\n\\end{figure}\n\nSome of the growth in Figure~\\ref{pap4f10} appears to be due to aliasing. This is a\nnumerical effect in finite-difference codes that results in an artificial transfer of\npower from trailing shwaves into leading shwaves as the shwave is lost at the grid scale.\nOne expects aliasing to occur approximately at intervals of\n\\begin{equation}\\label{ALIAS}\n\\Delta \\tilde{\\tau} = \\frac{N_x}{n_y}\\frac{L_y}{L_x},\n\\end{equation}\nwhere $n_y$ is the azimuthal shwave number. This interval corresponds to $\\Delta\n\\tilde{k}_x(t) = 2\\pi\/dx$, where $dx = L_x\/N_x$ is the radial grid scale. Based upon\nexpression (\\ref{ALIAS}), aliasing effects should be more pronounced at lower numerical\nresolution because the code has less time to evolve a shwave before the wavelength of the\nshwave becomes smaller than the grid scale. It can be seen from the far-right data point\nin Figure~\\ref{pap4f10} (Run 7 in Table~\\ref{pap4t1}) that the measured growth rate\n{\\it decreases} with increasing resolution. The evolution of $v_t$ for this run is\nshown in Figure~\\ref{pap4f11}.\n\n\\begin{figure}[hp]\n\\centering\n\\includegraphics[width=5.in,clip]{pap4f11.eps}\n\\caption[Evolution of $v_t$ as a function of time for Run 7.]\n{Evolution of $v_t$ as a function of time for Run 7 ($q = 0.2$ and $N_{x,min}^2 = -0.01$).}\n\\label{pap4f11}\n\\end{figure}\n\nThe effects of aliasing can be seen explicitly by evolving a single shwave, as was done for\nour linear theory test (Figures~\\ref{pap4f1a} and \\ref{pap4f1b}). Figure~\\ref{pap4f12}\nshows the evolution of the density perturbation for a single shwave using the same\nparameters that were used for Run 7: $L_x = L_y = 12$, $N_{x,min}^2 = -0.01$ and $q = 0.2$.\nThe initial shwave vector used was $(k_{x0}, k_y) = (-8\\pi\/L_x, 8\\pi\/L_y)$. This\ncorresponds to $n_y = 4$, and the expected aliasing interval (\\ref{ALIAS}) is therefore\n$\\Delta \\tilde{\\tau} = N_x\/4$. Runs at three numerical resolutions are plotted in\nFigure~\\ref{pap4f12}, and the aliasing interval at each resolution is consistent with\nexpression (\\ref{ALIAS}). It is clear from Figure~\\ref{pap4f12} that a lower resolution\nresults in a larger overall growth at the end of the run. It also appears that the growth\nseen in Figure~\\ref{pap4f12} requires a negative entropy gradient. We have performed this\nsame test with $N_x^2 > 0$, and while aliasing occurs at the same interval, there is no\noverall growth in the perturbations. This is likely due to the fact that the perturbations\ndecay asymptotically for $N_x^2 > 0$ (see expression [\\ref{ASYMP}]).\n\n\\begin{figure}[hp]\n\\centering\n\\includegraphics[width=4.5in,clip]{pap4f12.eps}\n\\caption[Evolution of the density perturbation for Run 8.]\n{Evolution of the density perturbation for a single shwave with $q = 0.2$, $N_{x,min}^2 =\n-0.01$ and $L_y = L_x$ (Run 8). The linear theory result is shown as a dotted line, along\nwith results at three numerical resolutions.\nAliasing occurs when $\\tilde{k}_x(t) = 2\\pi\/dx$. The overall growth, which is greater at\nlower numerical resolution, requires $N_x^2 < 0$.}\n\\label{pap4f12}\n\\end{figure}\n\nFigure~\\ref{pap4f13} summarizes the parameter space we have surveyed, indicating that\nthere is instability only for $\\qe \\simeq 0$ and $N_x^2 < 0$. The numerical\nresolution in all of these runs is $512 \\times 512$. Figure~\\ref{pap4f14} shows the\nevolution of the radial velocity in Run 10, a run with realistic\nparameters for a disk with a nearly-Keplerian rotation profile and radial\ngradients on the order of the disk radius: $q = 1.5$ and $N_{x,min}^2 =\n-0.01$ (corresponding to ${\\rm Ri} \\simeq -0.004$). Clearly no instability\nis occurring on a dynamical timescale. This plot is typical of all runs\nfor which the evolution was stable. To give a sense for the minimum growth\nrate that we are able to measure, we have also plotted in Figure~\\ref{pap4f14}\nthe results from several unstable runs with $q = 0$ and a boost equivalent to the\nvelocity at the minimum in $N_x^2$ for Run 10. It is difficult to measure a growth rate for\nthe smallest value of $N_{x,min}^2$, but it is clear that there is activity present in this\nrun which does not occur in the stable run. Based upon Figure~\\ref{pap4f14}, a\nconservative estimate for the minimum growth rate that should be detectable in our\nsimulations is $0.0025 \\Omega$.\n\n\\begin{figure}[hp]\n\\centering\n\\includegraphics[width=4.5in,clip]{pap4f13.eps}\n\\caption[Parameter space surveyed in a search for nonlinear instabilities.]\n{Parameter space surveyed in a search for nonlinear instabilities. Closed\n(open) circles denote runs that were unstable (stable). The\nonly instability found was convective instability for $\\qe \\simeq 0$\nand $N_x^2 < 0$ (${\\rm Ri} \\rightarrow -\\infty$). (We do not include on\nthis plot the runs shown in Figure~\\ref{pap4f10}.)}\n\\label{pap4f13}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=4.in,clip]{pap4f14.eps}\n\\caption[Evolution of $v_t$ as a function of time for Run 10.]\n{Evolution of $v_t$ as a function of time for Run 10 ($q = 1.5$,\n$N_{x,min}^2 = -0.01$). Also shown are runs with $q = 0$ and an overall\nboost equivalent to the velocity at the minimum in $N_x^2$ for Run 10, for\n$N_{x,min}^2 = -0.01$ (measured growth rate of $0.058$), $N_{x,min}^2 = -0.003$\n(measured growth rate of $0.021$) and $N_{x,min}^2 = -0.001$ (measured growth\nrate of $0.0025$).}\n\\label{pap4f14}\n\\end{figure}\n\n\\section{Implications}\n\nOur results seem to indicate that nearly-Keplerian disks with weak radial\ngradients are stable to local nonaxisymmetric disturbances, although we cannot\nexclude instability at very high Reynolds number. Figure~\\ref{pap4f10}\ndemonstrates that convective instability, present when the shear is nearly zero,\nis stabilized by differential rotation. Perturbations simply do not have time to grow\nbefore they are pulled apart by the shear.\n\nAn important implication of our results is that the instability claimed by \\cite{kb03}\nis {\\it not} a linear or nonlinear local nonaxisymmetric instability.\nFigure~\\ref{pap4f12} suggests that the results of \\cite{kb03} may be due,\nat least in part, to aliasing. They use a finite difference code at fairly\nlow numerical resolution ($\\leq 128^2$), and growth is only observed in\nruns with a negative entropy gradient. Curvature effects and the effects of\nboundary conditions, which may also play a role in their global results,\ncannot be tested in our local model.\n\n\\end{spacing}\n\n\\chapter{List of Figures}\\newpage\n\\addtocounter{page}{-1}\n\\listoffigures\n\n\\chapter{List of Tables}\\newpage\n\\addtocounter{page}{-1}\n\\listoftables\n\n\\mainmatter\n\n\\include{introduction}\n\\include{paper1}\n\\include{paper2}\n\\include{paper4}\n\\include{paper3}\n\\include{conclusion}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{s:intro}\nThe (Maker-Breaker) perfect matching game on a graph $G$ with an even number of vertices is played by two players who alternately select edges from $G$, with Breaker choosing first.\nMaker wins when she has chosen the edges of a perfect matching. If she never does so, Breaker wins. The (Maker-Breaker) Hamilton cycle game is defined \nanalogously. We restrict our attention, and the definition of these games, to graphs $G$ that are cliques. \n\nChv\\'{a}tal and Erd\\H{o}s~\\cite{CE78} proved that for sufficiently large $n$ and $G$ a clique on $n$ vertices, \nif both players play optimally then Maker can ensure she wins the Hamilton cycle game in her first $2n$ moves. Since the edges of every cycle with an even number \nof vertices can be partitioned into two matchings this means that Maker will also win the perfect matching game for even $n$.\n\nChv\\'{a}tal and Erd\\H{o}s~\\cite{CE78} also introduced the biased version of such games, where for some integer $b$, Breaker selects $b$ edges in each turn. They showed that for any positive $\\epsilon$, for $n$ sufficiently large, if $b>\\frac{(1+\\epsilon)n}{\\log{n}}$ then, \nBreaker can ensure that he selects all edges incident to one of the vertices. For such values of $b$, Breaker wins both the biased Hamilton cycle game and the biased perfect matching game. Krivelevich~\\cite{Kri11} obtained an essentially matching lower bound, showing that for $b<\\frac{(1-\\epsilon)n}{\\log{n}}$ \nMaker wins the biased Hamilton cycle game, and hence also the biased perfect matching game\\footnote{A slightly stronger result is stated in \\cite{Kri11}, but a careful reading of the proof \nshows that the result stated here is actually what is proven. In particular the use of Lemma 3 stated therein means the approach given there can do no better.}. \n\nFor Maker to win the (unbiased) perfect matching game she must make at least $\\frac{n}{2}$ moves as to obtain the edges of the matching. Indeed, Breaker can prevent Maker from just choosing \nthe edges of a matching by stealing the last edge, so Maker cannot ensure she wins in fewer than $\\frac{n}{2}+1$ moves. Hefetz et al.~\\cite{HKS09} showed that this lower bound is tight; i.e., they proved that when $n$ is sufficiently large Maker can win in $\\frac{n}{2}+1$ moves. \n\nIn the same vein, Maker will need at least n+1 moves to win the unbiased Hamilton cycle game, and Hefetz and Stich~\\cite{HS09} proved that she can always do so. \n\nIn this paper we focus on the number of moves Maker needs to win the biased perfect matching game. This question has been previously studied by \nMikala\\v{c}ki and Stojakovi\\'{c}~\\cite{MS17}. They showed that there is some $\\delta>0$ such that if $b<\\frac{\\delta n}{\\log{n}}$, then Maker can win in $(1+o(1))\\frac{n}{2}$ moves. \nThey asked whether this remains true for larger values of $b$. We answer this question in the affirmative by showing:\n\n\\begin{theorem}\n\\label{themaintheorem}\nFor every $f(n)$ which is $\\omega(1)$ and every sufficiently large $n$, if $b<\\frac{n}{\\log{n}}-\\frac{f(n) n}{(\\log{n})^{5\/4}}$ then Maker can win the biased perfect matching game in $\\frac{(1+o(1))n}{2}$ steps. \n\\end{theorem}\n\nWe note that this improves the upper bound from \\cite{Kri11} on the bias which ensures that Maker can win this game. \n\nMaker's strategy is a 2-stage approach. In the next section we discuss the goal of each stage and sketch the proof that she can successfully complete them.\nIn Section \\ref{s:details} we fill in the details. \n\nWe close this section with four definitions. At any point in the game, we use $d_B(v)$ to denote the degree of the vertex $v$ in the graph chosen by Breaker and $d_M(v)$ to denote its degree in the graph chosen by Maker. We let $E_B$ and $E_M$ be the edges picked by Breaker and Maker.\n\n\\section{A Proof Sketch}\\label{s:sketch}\n\nSince Maker can always pretend Breaker has chosen more edges than he actually has, we can and do assume that $f(n) =o(\\log{\\log{n}})$. As noted above, Maker's approach has two stages. \n\nAt the end of the first stage, Maker has chosen a subgraph which contains a matching $M$ such that $|V(M)|=(1-o(1))\\frac{n}{2}$. \nAt this point every vertex $v$ satisfies $d_M(v) \\le 3$ and if $v$ is in $M$, $d_M(v)=1$. Thus \nMaker chooses at most $\\frac{n}{2} +o(n)$ edges in the first stage. In the second stage Maker chooses only edges within $V(G)-V(M)$. \nThe edges chosen in the first stage ensure that Maker can make these choices so that after $14|V(G)-V(M)|$ turns, her graph will contain a matching with vertex set $V(G)-V(M)$ and hence she has won the game in $\\frac{n}{2} +o(n)$ turns. \n\nNow, if Maker chose no edges within $V(G)-V(M)$ during the first stage, then the second stage simply consists of playing the perfect matching game on a clique with $n'=o(n)$ vertices. But $b=\\omega(\\frac{n'}{\\log{n'}})$, hence Breaker wins this game and Maker will not be able to successfully complete the second stage. Thus,\nduring the first stage, as well as choosing the edges of $M$, Maker must choose edges between the elements of $V(G)-V(M)$ so that the graph $F$ formed by these non-matching edges has a structure that allows her to carry out the second stage. \nTo specify this structure precisely we need some definitions.\n\nWe say that a tree is {\\it matchable} if it has a perfect matching. \n\nWe say a rooted tree $T$ with root $r$ is {\\it augmenting} if every node at an odd depth has exactly one child. Thus every leaf is at even depth and there is a unique matching $N_T$\nin T, with vertex set T-r containing, for each vertex at odd depth, the edge from this vertex to its unique child. We note that for every leaf $x$ of $T$, letting $P_x$ be the path of $T$ from $x$ to $r$, we have that $N_x=(N_T-E(P_x)) \\cup (E(P_x)-N_T)$ is a matching of $T$ with vertex set $V(T)-x$.\n\nFor some even $p$, $F$ will have $p$ components that are not matchable trees and each of these will be augmenting. This means that in the second stage Maker need only construct a matching $N$, with $\\frac{p}{2}$ edges \ncontaining exactly one leaf in each component that is an augmenting tree. Since $T-x$ has a matching for every component $T$ of $F$ and leaf $x$ of $T$, we can extend $N \\cup M$ to the desired perfect matching. \n\nWe\nneed to impose some additional structure on the augmenting trees in order to ensure we can construct $N$. Specifically,\nthat each of these trees has a large number of leaves, and all of these leaves have low degree in Breaker's graph. We say a node v is {\\it troublesome} if $d_B(v)>\\frac{n}{\\sqrt{\\log{n}}}$. We let $\\ell=\\lceil \\sqrt{f(n)}(\\log{n})^{1\/4} \\rceil $. \nWe say that a tree $T$ is {\\it nice} if it is augmenting, has $\\ell$ leaves; none of which is troublesome, has maximum degree 3, and all of its nontroublesome nonleaf nodes at even depth have two children.\n\nModifying the approach of Krivelevich, we shall show: \n\n \\begin{lemma}\n\\label{stage2lemma}\nGiven that Maker has chosen a graph each of whose components is a nice tree, matchable, or an edge and such that the number of components that are nice trees is an even $p$ lying between $\\frac{n}{\\sqrt{\\log{n}}}$ and $\\frac{4n}{\\sqrt{\\log{n}}}+1$, she has a strategy that allows her to construct a matching with $p\/2$ edges whose vertex set contains exactly one leaf from each nice tree in $14p$ moves. \n\n\n\\end{lemma}\n\nSo, to complete the proof we need only show that Maker has a strategy that ensures she can construct a graph as in the hypothesis of Lemma \\ref{stage2lemma} in $\\frac{n}{2}+o(n)$ turns. \n\n If no vertex ever became troublesome, then as long as there are at least $\\frac{n}{\\sqrt{\\log{n}}}+2$ unmatched vertices we could simply pair two unmatched vertices to increase the size of \n the matching. Then the remaining unmatched vertices could be the roots of the nice trees and we could build these trees, each with $\\ell$ leaves, by repeatedly choosing two edges (in two consecutive turns) \n from a leaf of such a tree to two vertices in different edges of the matching, and also add the two matching edges these vertices are in to the tree. This increases the number of leaves of the tree by one (see Figure \\ref{fig:strategy}). Since there are no troublesome vertices, there will always be plenty of choices \n for the two matching edges. We note that each tree would have exactly $4\\ell-3$ vertices so the total size of our trees would be $O(p\\ell)=o(n)$. \n \\begin{figure}[ht]\n \\centering\n \\begin{tikzpicture}[scale=0.5]\n \n \\begin{scope}[every node\/.style={circle, fill=black, draw, inner sep=0pt,\n minimum size = 0.15cm\n }]\n \n \n \n \\node[] (t01) at (2,10) {};\n \\node[] (t02) at (1,9) {};\n \\node[] (t03) at (3,9) {};\n \\node[] (t04) at (1,8) {};\n \\node[] (t05) at (3,8) {};\n \\node[] (t06) at (0,7) {};\n \\node[] (t07) at (2,7) {};\n \\node[] (t08) at (0,6) {};\n \\node[] (t09) at (2,6) {};\n \n \\node[] (t11) at (3.5,5) {};\n \\node[] (t12) at (2.5,4) {};\n \\node[] (t13) at (4.5,4) {};\n \\node[] (t14) at (2.5,3) {};\n \\node[] (t15) at (4.5,3) {};\n \n \\node[] (t21) at (7,9) {};\n \\node[] (t22) at (6,8) {};\n \\node[] (t23) at (8,8) {};\n \\node[] (t24) at (6,7) {};\n \\node[label={[label distance=4]245:$v$}] (t25) at (8,7) {};\n \n \\node[] (m01) at (12,10) {};\n \\node[] (m02) at (14,10) {};\n \\node[] (m11) at (12,9) {};\n \\node[] (m12) at (14,9) {};\n \\node[label={[label distance=2]165:$u$}] (m21) at (12,8) {};\n \\node[label={[label distance=2]15:$w$}] (m22) at (14,8) {};\n \\node[] (m31) at (12,7) {};\n \\node[] (m32) at (14,7) {};\n \\node[] (m41) at (12,6) {};\n \\node[] (m42) at (14,6) {};\n \\node[] (m51) at (12,5) {};\n \\node[] (m52) at (14,5) {};\n \\node[] (m61) at (12,4) {};\n \\node[] (m62) at (14,4) {};\n \\node[label={[label distance=2]195:$y$}] (m71) at (12,3) {};\n \\node[label={[label distance=2]345:$z$}] (m72) at (14,3) {};\n \\node[] (m81) at (12,2) {};\n \\node[] (m82) at (14,2) {};\n \n \\end{scope}\n\n \n \\begin{scope}[every edge\/.style={draw=black}]\n \n \n \n \\path[thick] (t01) edge node {} (t02);\n \\path[thick] (t01) edge node {} (t03);\n \\path[ultra thick, dotted] (t02) edge[draw=blue] node {} (t04);\n \\path[ultra thick, dotted] (t03) edge[draw=blue] node {} (t05);\n \\path[thick] (t04) edge node {} (t06);\n \\path[thick] (t04) edge node {} (t07);\n \\path[ultra thick, dotted] (t06) edge[draw=blue] node {} (t08);\n \\path[ultra thick, dotted] (t07) edge[draw=blue] node {} (t09);\n \n \\path[thick] (t11) edge node {} (t12);\n \\path[thick] (t11) edge node {} (t13);\n \\path[ultra thick, dotted] (t12) edge[draw=blue] node {} (t14);\n \\path[ultra thick, dotted] (t13) edge[draw=blue] node {} (t15);\n \n \\path[thick] (t21) edge node {} (t22);\n \\path[thick] (t21) edge node {} (t23);\n \\path[ultra thick, dotted] (t22) edge[draw=blue] node {} (t24);\n \\path[ultra thick, dotted] (t23) edge[draw=blue] node {} (t25);\n \n \\path[ultra thick, dotted] (m01) edge[draw=blue] node {} (m02);\n \\path[ultra thick, dotted] (m11) edge[draw=blue] node {} (m12);\n \\path[ultra thick, dotted] (m21) edge[draw=blue] node {} (m22);\n \\path[ultra thick, dotted] (m31) edge[draw=blue] node {} (m32);\n \\path[ultra thick, dotted] (m41) edge[draw=blue] node {} (m42);\n \\path[ultra thick, dotted] (m51) edge[draw=blue] node {} (m52);\n \\path[ultra thick, dotted] (m61) edge[draw=blue] node {} (m62);\n \\path[ultra thick, dotted] (m71) edge[draw=blue] node {} (m72);\n \\path[ultra thick, dotted] (m81) edge[draw=blue] node {} (m82);\n \n \\path[very thick, dashed] (t25) edge node {} (m21);\n \\path[very thick, dashed] (t25) edge node {} (m71);\n \n \\end{scope}\n\n \n \\begin{scope}[every node\/.style={draw=none,rectangle}]\n \n \n \\node (Tlabel) at (7,7.5) {$T$};\n \\node (Mlabel) at (15.5,6) {$M$};\n \n \\end{scope}\n \\end{tikzpicture}\n \\caption{If no vertex becomes troublesome, Maker's strategy first selects a matching $M$ and then builds a collection of trees by choosing edges from a leaf to two matching edges. In the figure, matching edges are represented by dotted lines. Maker chooses edges from the leaf $v$ of tree $T$ to the edges $uw$ and $yz$. Thus leaf $v$ of $T$ is replaced by leaves $w$ and $z$.}\n \\label{fig:strategy}\n\\end{figure}\n \n Troublesome vertices complicate our choices in two ways. There are fewer choices for the edges out of them, and we want to deal with the troublesome vertices before Breaker picks all the \n edges incident to them. This may mean we cannot pick an edge we otherwise would have picked because we have to choose an edge with a troublesome endpoint instead.\n Thus, we may have to root a tree of size exceeding two at a troublesome vertex even though there are many more than $\\frac{n}{\\sqrt{\\log{n}}}$ unmatched vertices because Breaker has picked \n all the edges between the troublesome vertex and the unmatched vertices. \n This will increase the number of trees of size exceeding two we create.\n Furthermore, Breaker may have chosen all the edges \n from a troublesome leaf of one of the trees we want to make nice to vertices in our matching. This will force us to pick an edge \n from this leaf to a singleton tree to construct a matchable tree with four or more vertices. \n This explains our need for both matchable and nice trees of size exceeding two. \n \n The reason we can handle the trouble caused by troublesome vertices is that there are not very many of them. We carry out fewer than $n$ turns and Breaker chooses fewer than $\\frac{n}{\\log{n}}$ edges \n in each turn. Hence, the total degree in Breaker's graph is at most $\\frac{2n^2}{\\log{n}}$ and there are at most $\\frac{2n}{\\sqrt{\\log{n}}}$ troublesome vertices. We exploit this fact repeatedly. \n\nFor any set S we let $\\tau(S)$ be the number of troublesome\nvertices\nin $S$. We say a tree $T$ is {\\it small} if $|V(T)|<4(\\tau(V(T))+\\ell)$. \n\nWe shall show: \n\\begin{lemma}\n\\label{stage1lemma}\nMaker has a strategy to choose, in an initial set of turns, a graph F which, for some even $p$ with $\\frac{n}{\\sqrt{\\log{n}}} \\le p \\le \\frac{4n}{\\sqrt{\\log{n}}}+1$, has $p$ components that are small nice trees and whose remaining components are small matchable trees each of which either contains a troublesome vertex or is an edge.\n\\end{lemma}\n\nWe note that $F$ has at most $p$ components that are not an edge and contain no troublesome vertex.\nSo, $F$ has at most $\\frac{2n}{\\sqrt{\\log{n}}}+p \\le \\frac{6n}{\\sqrt{\\log{n}}}+1$ components that are not edges. Each such component $T$ is small, i.e. $|V(T)| \\le 4(\\tau(V(T)+\\ell)$. Since there are at most $\\frac{2n}{\\sqrt{\\log{n}}}$ troublesome vertices, the total number of vertices in components of $F$ that are not edges is at most \n$\\frac{24n\\ell}{\\sqrt{\\log{n}}}+4\\ell+\\frac{8n}{\\sqrt{\\log{n}}}$.\n\nSince this is $o(n)$, Maker constructs $F$ in $\\frac{n}{2}+o(n)$ turns and so combining\nLemmas \\ref{stage2lemma} and \\ref{stage1lemma} proves Theorem \\ref{themaintheorem}. \n\nThis completes our intuitive sketch of the approach. It remains to set out the precise strategy used in each stage \n and prove that both can be successfully applied. We do so in the next section.\n \n\\section{The Details} \\label{s:details}\n\n\\subsection{Stage 1 algorithm formalization}\n\n To begin we set out the invariants that Maker will ensure hold throughout Stage 1 and her strategy for doing so.\n We remark that Maker may decide during\n one turn what to do in both that turn and the next. Because of this, each step of the algorithm will take either 1 turn or 2 turns. \n\nAfter $i$ turns in Stage 1, Maker will ensure that the graph she has chosen consists of a matching $M_i$, a set $q_i$ \nof small matchable trees, each with a troublesome root, \nand for some $p_i > \\frac{n}{\\sqrt{\\log{n}}}$, a family ${\\cal T}_i=\\{T_1,...,T_{p_i}\\}$ of augmenting trees, each of maximum degree three. Furthermore, \nShe will attempt to also ensure that every non-leaf non-troublesome node at even depth of one of the augmenting trees has degree 2.\nHowever, when adding children underneath a nontroublesome node she will add two, in two consecutive turns, and between these turns this will not be true. \n\n\nWe have ${\\cal T}_0=\\set{T_1,...,T_n}$,$q_i=0$, $p_i=n$ where each $T_i$ is a singleton, and $E_M=E_B=\\emptyset$.\n\n\\begin{enumerate}\n \\item \\textbf{If no tree in ${\\cal T}_i$ has a leaf that is troublesome:}\n \\begin{enumerate}\n \\item \\textbf{If $p_i>2\\ceil{\\frac{n}{\\sqrt{\\log{n}}}}+1$ and there are $u,v$ such that $\\deg_M(u)=\\deg_M(v)=0$ and $(u,v)\\notin E_B$}\n \n Maker chooses $e=(u,v)$. She has removed 2 trees and added an edge to the matching, so $p_{i+1}=p_i-2$, ${\\cal T}_{i+1}={\\cal T}\\setminus \\set{\\set{u},\\set{v}}$ and $M_{i+1}=M_i\\cup \\set{e}$.\n \n \\item \\textbf{Otherwise if there is $T_j\\in {\\cal T}_i$ with fewer than $\\ell$ leaves}\n \n Maker picks a leaf $v$ of $T_j$, and finds an edge $(w,z)\\in M_i$ such that neither $w$ nor $z$ is troublesome and $(v,w)\\notin E_B$. Maker chooses $(v,w)$ and adds this edge and $e$ to $T_j$. \n \n Then, after Breaker's turn, Maker finds another edge $(x,y)\\in M_i$ such that neither $x$ nor $y$ is troublesome and $(v,x)\\notin E_B$, chooses $(v,x)$ and adds this edge and $(x,y)$ to $T_j$. \n \n The number of trees in ${\\cal T}_i$ hasn't changed in either step so $p_{i+2}=p_{i+1}=p_i$ however,$M_{i+1}=M_i\\setminus\\set{w,z}$ and $M_{i+2}=M_{i+1}\\setminus\\set{(x,y)}$. \n \n \n \n \n \\item \\textbf{Otherwise}\n \n We end the algorithm.\n \\end{enumerate}\n \\item \\textbf{Otherwise, consider a tree $T_j \\in {\\cal T}_i$ and troublesome leaf $v$ of $T_j$ where the pair $(j,v)$ is chosen so as to maximize $d_B(v)$}\n \\begin{enumerate}\n \\item[(a)] \n If possible, Maker chooses an edge $vw$ such that $w$ is contained in an edge $(w,z)$ of $M_i$ such that neither $w$ nor $z$ is troublesome and $(v,w)\\notin E_B$. She adds $vw$ and $wz$ to $T_j$. Set $M_{i+1}=M_i\\setminus{(w,z)}$ and $p_{i+1}=p_i$.\n \\item[(b)] Otherwise, Maker chooses an edge $vw$ such that $d_M(w)=0$, $w$ is not troublesome, and $vw \\not\\in E_B$ and adds it to $T_j$. \n She then adds $T_j$ to the set of matchable trees containing a troublesome vertex and deletes $T_j$ and $z$ from ${\\cal T}_i$. So $p_{i+2}=p_i-2$, ${\\cal T}_i ={\\cal T}\\backslash \\{T_j,w\\}$, $M_{i+2}=M_i$, $q_{i+2}=q_i+1$.\n \\end{enumerate}\n \n\n\\end{enumerate}\nWe need to show that this strategy shows that Lemma \\ref{stage1lemma} holds. \n\n\\subsection{Proof of Lemma \\ref{stage1lemma}}\n\n\n\n\n\nWe will show that Maker can always choose an edge with the desired properties; hence, that\nthe first stage ends. We then demonstrate that when the stage terminates, the properties\nrequired by Lemma \\ref{stage1lemma} hold. Before doing either we make some useful observations \nabout properties that hold throughout the process. \n\nWe note first that as discussed above, we have fewer than $\\frac{2n}{\\sqrt{\\log{n}}}$ troublesome vertices.\n\nEach non-singleton tree of any ${\\cal T}_i$ is composed of some edges that were in the matching \nat some point and others that were not. The first edge chosen from the root of such a tree \nwas never in the matching and was chosen in Case 2 a) or Case 1 b). If this edge was chosen in Case 2 a) then the \nroot is troublesome so there are never more than $\\frac{2n}{\\sqrt{\\log{n}}}$ such trees.\nIf this edge was chosen in 1 b), then in the iteration it was chosen Maker did not carry out 1 a). \nThis means that there were at most $ \\lceil \\frac{2n}{\\sqrt{\\log{n}}} \\rceil$ singleton trees at this point\nas otherwise for every singleton tree consisting of a (nontroublesome) vertex $u$ there would be at least $\\lceil \\frac{n}{\\sqrt{\\log{n}}} \\rceil$ \nchoices for a $v$ such that we could apply 1 a) for $u$ and $v$.\n\n\nIn every iteration, the set of non-troublesome roots of the nonsingleton trees that are not in the matching, are contained in the set of the last \n$\\lceil \\frac{2n}{\\sqrt{\\log{n}}} \\rceil$ vertices to be in singleton trees. \n\nHence, we always have that the total number of trees of size at least three in Maker's graph \nis at most $\\frac{4n}{\\sqrt{\\log{n}}}+1$. \n\nThe size of a tree of ${\\cal T}_i$ can be increase in Case 1 b) or Case 2. Each iteration of 1 b) increases the number of leaves \nin the tree by 1, thus, we may only have $\\ell-1$ such turns per tree. Now, \nin Case 1 b) 4 vertices are added to a tree; while in Case 2, at most two vertices are added.\nFurthermore, every time we carry out Case 2 a), we add a troublesome internal vertex to the tree, and we carry out Case 2 b) for each tree at most once. We have that at every point in the algorithm each of the nonsingleton trees not in the matching is small. \n\n Combining this with the last paragraph, and the fact that there are at most $\\frac{2n}{\\sqrt{\\log{n}}}$ troublesome \n vertices, we have that the union of these trees has size at most\n$\\frac{16\\ell n}{\\sqrt{\\log{n}}}+4l+\\frac{8n}{\\sqrt{\\log{n}}}$.\n\n\n\n\n\n\n\n\\begin{lemma}\nMaker is always able to choose an edge as the strategy 1 algorithm specifies she should. \n\\end{lemma}\n\n\\begin{proof}\n$~$\n\n\n\n Case 1: \n \n At most $2b$ vertices can become troublesome in any turn. We need only show if we cannot carry out 1a) but there is some tree $T_j$ of ${\\cal T}_i$ with fewer than $2\\ell$ leaves, then for any leaf $v$ of $T_j$ there are at least $3b+2$ edges of $M_i$ that contain no troublesome vertices and Breaker has not isolated from $v$. This implies the two turn process is possible, as Breaker can only choose $b$ edges from $v$ and make $2b$ vertices troublesome in her intervening turn. \n \n By our bound on the size of the non-singleton trees and number of troublesome vertices, there are at least $n \n -\\frac{16\\ell n} {\\sqrt{\\log{n}}}-4\\ell-\\frac{12n} {\\sqrt{\\log{n}}}$ vertices that are either singletons or in matching edges not containing a troublesome vertex. As shown above, since we did not carry out 1a), there are at most $\\lceil \\frac{2n}{\\sqrt{\\log{n}}}\\rceil $ singletons. Since $v$ is not troublesome, there are most $\\frac{n}{\\sqrt{\\log{n}}}~ w$ such that $vw \\in E_B$. So there are indeed the desired $3b+2$ matching edges. \n \n \n Case 2:\n \n We first show that if for every $v$ from which we choose an edge in the first turn of a Case 2 step, we have $d_B(v)< n-\\frac{16\\ell n}{\\sqrt{\\log{n}}}-4\\ell-\\frac{20n}{\\sqrt{\\log{n}}}$, then Maker will always be able to find an appropriate edge in each turn of a Case 2\n step. We then prove that we can ensure this upper bound on $d_B(v)$ holds. \n \n The first step is similar to the analysis for Case 1. We know that there are at least $\\frac{16\\ell n}{\\sqrt{\\log{n}}}+4l+\\frac{20n}{\\sqrt{\\log{n}}}$\n vertices of $G$ that are not joined to $v$ by an edge of Breaker's graph. At least $\\frac{12n}{\\sqrt{\\log{n}}}$ of these are singletons or in a matching edge. Since there are at most $\\frac{2n}{\\sqrt{\\log{n}}}$ troublesome vertices, it follows that we can carry out one of Step 2 a) or Step 2 b). \n \n It remains to show that whenever Maker attempts to choose an edge from a vertex $v$ in the first turn of a Caser 2 step, we have $d_B(v)< n - \\frac{16\\ell n}{\\sqrt{\\log{n}}}-4l-\\frac{20n}{\\sqrt{\\log{n}}}$. We assume for a contradiction this is not the case and consider \n the sequence of her (one or two turn) steps culminating with the first turn in which this bound does not hold. \n \n \n We look at the suffix of this sequence starting immediately after the last step where a choice was made via an application of Case 1. We let $k$ be the number of steps in this suffix. Since, we are in Case 2 for each of these $k$ steps, we have $k \\le \\frac{2n}{\\sqrt{\\log{n}}}$ steps. Furthermore, each of these steps consists of only one turn. We let $a_i$ be the vertex that Maker makes into a non-leaf in the $i^{th}$ of these $k$ turns\/steps and note that the $a_i$ are distinct. For each $i$, we let \n $d^j_i = d_B(a_i) - |\\{a_l| l \\ge j,a_ia_l \\in E_B\\}|$, after $j$ steps of the sequence.\n \n Now, Maker carried out a step before this sequence of $k$ steps, as at the start of the game there are no troublesome vertices. Before this step there were no troublesome vertices and there were at most two turns in the step, each of which can increase a Breaker degree by at most 2b. So we end up with $ d^0_i \\le \\frac{n}{\\sqrt{\\log n}}+\\frac{2n}{\\log n}$. \n\n We define the potential function $p(j)$, as follows:\n\n\\begin{align*}\np(j) := \\frac{1}{k-j+1}\\sum_{i=j}^{k}d^{j-1}_i.\n\\end{align*}\n\nNote that, we are assuming that $p(k) = d^{k-1}_k \\ge n -\\frac{16\\ell n}{\\sqrt{\\log{n}}}-4l-\\frac{20n}{\\sqrt{\\log{n}}}$. On the other hand\n\\begin{align*}\np(1) = \\frac{1}{k}\\sum_{i=1}^{k}d^0_i \\le \\frac{1}{k} \\times k \\times (\\frac{n}{\\sqrt{\\log n}}+\\frac{2n}{\\log n})=\\frac{n}{\\sqrt{\\log n}}+\\frac{2n}{\\log n}.\n\\end{align*}\n\nWe derive a contradiction by bounding the increase in the potential function in each step. \nBy definition $\\sum_{i=j+1}^{k}d^j_i-\\sum_{i=j+1}^{k}d^{j-1}_i$ is the sum of the number of \nedges Breaker chose in the $j^{th}$ turn with exactly one end in $\\{a_{j+1},...,a_k\\}$ \nand the number of edges he has chosen between $a_j$ and $\\{a_{j+1},...,a_k\\}$ by the end of the \n$j^{th}$ turn. The first of these is at most $b$ and the second is at most $k-j$. \n\nWe obtain: \n\n\\begin{align*}\n(*) \\sum_{i=j+1}^{k}d^j_i \\le \\sum_{i=j+1}^{k}d^{j-1}_i + b+k-j.\n\\end{align*}\n\nBy our definition of $d^j_i$ and choice of $a_i$ to maximize the degree in the Breaker's graph,\nat the start of the $j^{th}$ turn,\n\\begin{align*}\n\\forall j\\le i \\le k, d^{j-1}_i \\le d_B(a_i) \\le d_B(a_j) \\le d^{j-1}_j+(k-j).\n\\end{align*}\n \nSumming up and dividing by $k-j+1$, we obtain\n\\begin{align*}\nd^{j-1}_j \\ge \\frac{\\sum_{i=j}^{k}d^{j-1}_i}{k-j+1} -(k-j).\n\\end{align*}\n \nThus, noting $\\sum_{i=j+1}^{k}d^{j-1}_i = \\sum_{i=j}^{k}d^{j-1}_i + d^{j-1}_j$:\n\\begin{align*}\n\\sum_{i=j+1}^{k}d^{j-1}_i\\le \\frac{k-j}{k-j+1}\\sum_{i=j}^{k}d^{j-1}_i+(k-j).\n\\end{align*}\n \nCombining this with $(*)$ yields: \n\\begin{align*}\n\\frac{\\sum_{i=j+1}^{k}d^{j}_i}{k-j} \\le \\frac{\\sum_{i=j}^{k}d^{j-1}_i}{k-j+1}+\\frac{b}{k-j}+2.\n\\end{align*}\nHence \n\\begin{align*}\np(k) \\le p(1)+ 2k+b\\sum_{r=1}^{k} \\frac{1}{k} \\le O(\\frac{n}{\\sqrt{\\log{n}}})+b(\\log{n}).\n\\end{align*}\n\nSince $b<\\frac{n}{\\log{n}}-\\frac{f(n)n}{(\\log{n})^{5\/4}}$ and $\\ell=\\lceil \\sqrt{f(n)} \\rceil =o(f(n))$,\nthe desired result follows. \n\\end{proof}\n\nIt remains to show that the set of trees created by the end of Stage 1 have the properties required \nby Lemma \\ref{stage1lemma}. \n\nNote that $p$ is necessarily even upon completion of the phase - as $n$ is even, the set of vertices in the matchable trees is even, and by construction all trees in $t$ have an odd number of vertices,\nso we must have an even number of trees.\n\nClearly there are no leaves of any tree in ${\\cal T}_i$ that are troublesome when the stage terminates, \nas termination only occurs in Case 1. Furthermore, \nsince Case 1 b) was not carried out in the last iteration every tree in ${\\cal T}_i$ has \n$\\ell>1$ leaves.\n\nOur construction also ensures:\n\\begin{enumerate}\n \\item all our leaves are at even depth (except in the middle of a 2-turn step),\n\\item whenever we add a node at odd depth to the tree we always add exactly one child along with it, \n\\item whenever we make a node at even depth into a nonleaf we add exactly two children underneath it if it is nontroublesome(Case 1b)) and one vertex if it is troublesome (Case 2), and \n\\item we never add children under a nonleaf except in the step in which it becomes a nonleaf. \n\\end{enumerate}\n\nThus, at the end of the Stage, ${\\cal T}_i$ consists of an even number \n$p$ of nice trees. It remains to bound $p$. \n\n\n We have shown above, that there are always at most $\\frac{4n}{\\sqrt{\\log n}}+1$ non-singleton trees. Thus, at termination, $|{\\cal T}_i| \\le \\frac{4n}{\\sqrt{\\log n}}+1$. \n \n It remains to show that $p_i$ is always at least $\\frac{n}{\\sqrt{\\log{n}}}$. We note that \n $p_i$ can only be reduced in Cases 1a) or 2b) and, never by more than 2. Now, Case 1a) only applies if $p_i > \\frac{2n}{\\sqrt{\\log n}}$\n so $p_i$ cannot be reduced below $\\frac{n}{\\sqrt{\\log{n}}}$.\nFurthermore, as shown above, when we carry out Case 2b) by choosing an edge from some $v$, there are at least $\\frac{16\\ell n}{\\sqrt{\\log{n}}}+4l+\\frac{20n}{\\sqrt{\\log{n}}}$ vertices $w$ such that $vw$ is not an edge of Breaker's graph. \nAs noted above, at least $\\frac{8n}{\\sqrt{\\log{n}}}$ of these are in singleton trees or matching edges. \nFurther, by our bound on the number of troublesome vertices,\nfewer than $ \\frac{4n}{\\sqrt{\\log{n}}}$ of these are in matching edges that contain a troublesome vertex.\nFinally no other matching edge contains one of these vertices as otherwise we would have carried out Case 2a).\nIt follows that there are at least $ (\\frac{4n}{\\sqrt{\\log n}})$ singleton trees whenever we carry out Case 2b). \nThus, we obtain $ p_i > \\frac{4n}{\\sqrt{\\log n}}-1$ upon completion of the phase. So Lemma \\ref{stage1lemma} has been proved. \n\n\n\\subsection{Proof of Lemma \\ref{stage2lemma}}\n\nWe now present and analyze the strategy that Maker will use in the second phase: Strategy 2. \n\n In doing so we consider an auxiliary multigraph $F'$, whose structure depends on a\n specific graph $F$ satisfying the hypotheses of Lemma \\ref{stage2lemma} that Maker has constructed. For each component $T_i$ of $F$ that is a nice tree, we let $S_i$ be the set $\\ell$ leaves of $T_i$. $F'$ has vertices $v_1,...,v_p$. The number of edges \n between $v_i$ and $v_j$ is the number of unchosen edges between $S_i$ and $S_j$. We note that there are at most $\\ell^2$ edges between \n two vertices of $F'$ and since no vertex in any $S_i$ is troublesome, the minimum degree of $F'$ is at least:\n \n $\\ell^2(p-1)-\\frac{\\ell n}{\\sqrt{\\log{n}}} > f(n){\\sqrt{\\log{n}}}(\\frac{n}{\\sqrt{\\log{n}}}-1) - \\frac{\\ell n}{\\sqrt{\\log{n}}} >\n \\frac{f(n) n}{4}$ which is $\\omega(n)$.\n Thus, although $b=\\omega( \\frac{|V(F')|}{\\log{|V(F')|}} )$ we have that $b=o(\\frac{\\delta(F')}{\\log{\\delta(F')}})$. This allows us to adapt the strategy of \\cite{Kri11} to show that Maker can choose a Hamilton cycle in $F'$ in $14p$ moves.\n This Hamilton cycle contains a perfect matching corresponding to a matching containing exactly one leaf from each component of $F$ which is a nice tree, so we have proven Lemma \\ref{stage2lemma}. \n \n As noted in \\cite{Kri11}, it is enough to present a random strategy that creates a Hamilton cycle with positive probability in \n $14p$ steps. For if Maker cannot ensure he always creates a Hamilton cycle in $14p$ turns, then Breaker can ensure he never \n does so. \n \nWe consider applying the 2 phase strategy given below to $F'$. Note that at the start neither Maker nor Breaker\nhas chosen any edges of $F'$. We define $d_B$ and $d_M$ as before. When Maker chooses an edge, she assigns it a direction, we \nlet $d^+_M(v)$ be the outdegree of $v$ in the resultant digraph and define the {\\it danger} of $v$ to be $d_B(v)-2bd^+_M(v)$. \n\\vskip0.2cm\n\nStrategy 2: \n\\vskip0.2cm\nPhase 1: While there is a vertex whose outdegree in the Maker digraph is less than $10$, Maker chooses such a vertex $v$ so as to maximize its danger.\nIf possible Maker chooses an edge out of $v$ uniformly at random from all the unchosen edges incident to $v$, to add to Maker's digraph. \nOtherwise, Maker concedes defeat. \n\\vskip0.2cm\nPhase 2: We let $P$ be a longest path in Maker's graph chosen if possible to induce a cycle. If there is a cycle through $V(P)$ then Maker stops, \nwinning if this is a Hamilton cycle and losing otherwise. Otherwise, Maker looks for a path $Q$ with the same vertex set as $P$ such \nthat some edge between the endpoints of $Q$ has not been chosen. If she finds such a path she chooses the edge between its endpoints, otherwise she \nconcedes defeat. \n\\vskip0.4cm\n\nWe let $M^*$ be the graph chosen by Maker at the end of Phase 1. We will show: \n\n\\begin{lemma}\n\\label{anotherlemma1}\nWith positive probability $M^*$ is connected and for any longest path $P$ of any supergraph $M'$ of $M^*$, \neither $P$ is Hamiltonian and $M'$ has a Hamiltonian cycle or \nthe set $\\{e|e \\in E(F'),~ s.t.~ M'+e ~contains~a~cycle ~on ~V(P)\\}$ has size at least $\\frac{11pn}{\\log{n}}+1$. \n\\end{lemma}\n\nNow, there are at most $10p$ turns in the first phase. Furthermore, \nfor every turn in the second phase in which Maker chooses an edge $e$, if $V(P)$ \nis not the vertex set of a component, then the length of the longest path \nin Maker's graph increases as in the new Maker's graph there is a path using \nthe vertices of $P$ and any vertex joined to $P$ by an edge. So, if $M^*$ is \nconnected, then the second phase has at most $p$ turns and if it stops because \n$V(P)$ is a cycle, then this is a Hamiltonian cycle. \n\nSo, if $M^*$ is connected, Breaker chooses at most $\\frac{11pn}{\\log{n}}$ edges in total.\nHence, if in addition for any longest path $P$ of any supergraph $M'$ of $M^*$ \neither $P$ is Hamiltonian and $M'$ has a Hamiltonian cycle or \nthe set $\\{e|e \\in E(F'),~ s.t.~ M'+e ~contains~a~cycle ~on ~V(P)\\}$ has size at least $\\frac{11pn}{\\log{n}}+1$\nthen Phase 2 must terminate with the construction of a Hamiltonian cycle. \n\nThus Lemma \\ref{anotherlemma1} implies Lemma \\ref{stage2lemma}, and it remains to prove it. \n\n\n\n \n\\subsection{The Proof of Lemma \\ref{anotherlemma1}}\n\\label{thedetails}\n\n\nThe first key result is similar to one given by Gebauer and Szabo in \\cite{GS09}. The only differences are \nthat we consider a multigraph, we use $d^+_M$ instead of $d_M$, we have a slightly weaker bound on $b$, and we \nobtain a weaker bound on $d_B$. This latter fact significantly simplifies the proof which uses \na potential function argument, similar to, but slightly more complicated than \nthat used in the proof of Lemma \\ref{stage1lemma}.\n\n\\begin{lemma}\n\\label{lowdegreelemma} \nFor sufficiently large C, If Maker applies Strategy 2 to an input satisfying the hypotheses of Lemma \\ref{stage2lemma}, then throughout the algorithm every vertex of $F'$ with \n$d^+_M(v) <10$ satisfies $d_B(v) < 3n$.\n\\end{lemma}\n\n\\begin{proof} \n We assume for a contradiction that after some number $k$ of moves in the first phase, there is a vertex $v$ \n with $d_M^+(v)<10$ and $d_B(v)>3n$ this implies that the danger of $v$ exceeds $\\frac{5n}{2}$.\n \n We let $a_i$ be the vertex from which Maker chooses an edge in the $i^{th}$ turn and let \n $A_i$ be the set consisting of the vertices appearing in $\\{a_i,...,a_k\\}$. We note this is a set \n even though $\\{a_i,...,a_k\\}$ may be a multiset. \n \n We let the potential after $i-1$ turns be the average danger of the elements in $A_i$.\n We note that Breaker's $i^{th}$ turn increases this potential by at most $\\frac{2b}{|A_{i+1}|} $\n We note that if $A_i$ and $A_{i+1}$ are the same set (i.e. if $a_i=a_j$ for some $j>i$),\n then Maker's $i^{th}$ turn decreases this potential by exactly $\\frac{2b}{|A_i|}$.\n Moreover, since $a_i$ has maximum danger in $A_i$, Maker's turn never increases \n the potential. \n \n \n Thus, unless $|A_i|>|A_{i+1}|$, the potential does not increase in the $i^{th}$ turn.\n Furthermore, the total increase in the potential in turns such that $|A_i|>|A_{i+1}|$ is at most\n \\begin{align*}\n \\sum_{k=1}^{|A_1|} \\frac{2b}{k} \\le \\sum_{k=1}^n \\frac{2b}{k} \\le b(\\log{n}) \\le 2n.\n \\end{align*}\n The desired result follows.\n \\end{proof}\n\nNow, $F'$ has minimum degree $\\omega(n)$, so this lemma implies that every random choice of a directed edge that Maker makes \nis not that different from simply choosing a random vertex of $F'$ as an out-neighbour. This us allows us to use standard techniques to \nshow the following two lemmas.\n\n\\begin{definition}\nFor any $S \\subseteq V(F')$, we define $N(S)=N_{M^*}(S)$ to be those vertices outside of S joined to some vertex of S by an edge \nof $M^*$. \n\\end{definition}\n\n \\begin{lemma}\n \\label{expansion}\n If Maker adopts Strategy 2, then with probability $1-o(1)$, \n for every $S \\subseteq V(F')$, if $|S| \\le \\frac{p}{2}$ then $N(S)$ is nonempty\n while if $|S| \\le \\frac{p}{100}$,\n then $|N(S)| >2|S|$. \n \\end{lemma}\n\n\\begin{proof}\nAgain we modify an argument of Krivelevich. We will give an upper bound to the probability of having $S$ and $N(S)=A$ such that $S\\cup A \\subseteq V(F')$, $A\\cap S=\\emptyset$, and $A$ is small enough for the conclusion of the lemma to be false. In order to do so, we will use the union bound on that probability for fixed $S$ and $N(S)$. \n\nFix $j$ and $k$. Then, by using the Stirling's approximation to bound $\\binom{p}{j+k}$, the number of choices for $S$ and $A$ such that $|S|=j$ and $|A|=k$ and $|S\\cap A|=0$ is:\n\\[\\binom{p}{j+k}\\binom{j+k}{j}\n \\le \\binom{p}{j+k}2^{j+k}\n \\le \\power{\\frac{p}{j+k}}{j+k} (2e)^{j+k}\\]\n\nNoting that $\\forall v\\in S$ Maker chooses 10 edges in the first phase from $v$.\nSince none of the leaves corresponding to $v$ in the first phase were troublesome, by Lemma \\ref{lowdegreelemma} the number of unchosen edges out of $v$ when we choose those edges is at least $p\\ell^2-\\frac{\\ell n}{\\log{n}}-3n$. Hence, since $\\ell=\\ceil{\\sqrt{f(n)}\\power{\\log n}{\\frac{1}{4}}}=\\omega(1)\\cdot \\power{\\log n}{\\frac{1}{4}}$ and $p=\\Theta\\left(\\frac{n}{\\sqrt{\\log{n}}}\\right)$ the probability that an edge we pick is from $v$ to a vertex in $S\\cup A$ is at most \n\\begin{align*}\n \\frac{\\ell^2(|S|+|A|)}{pl^2-\\frac{\\ell n}{\\log{n}}-3n}= \\frac{j+k}{p-\\frac{n}{\\omega(1)\\cdot \\power{\\log n}{\\frac{1}{4}}\\log{n}}-3\\frac{n}{\\omega(1)\\cdot \\sqrt{\\log n}}}=\\frac{j+k}{p-o(p)}\\le \\frac{j+k}{\\frac{9p}{10}}=\\frac{j+k}{p}\\frac{10}{9}.\n\\end{align*}\n \n Noting that we are only interested in the cases that violate the conclusion of the lemma, so $\\frac{j+k}{p}<\\frac{9}{10}$. \n \n Given that we are choosing $10j$ out edges out of our set $S$ during Strategy 2, the probability that we make choices so that $N(S) \\subseteq A$ is less than\n \\begin{align*}\n \\power{\\frac{j+k}{p}}{10j}\n \\power{\\frac{10}{9}}{10j}\n \\leq\n \\power{\\frac{j+k}{p}}{8j} \\power{\\frac{10}{9}}{8j}.\n \\end{align*}\n \n Thus, by the union bound, the probability that there exists a set $S$ of $j$ nontroublesome vertices of $V(F')$ such that $|N(S)|=k$ is bounded above by: \n \n \\begin{align*}\n \\power{\\frac{j+k}{p}}{7j-k} \\power{\\frac{10}{9}}{8j}(2e)^{j+k}.\n \\end{align*}\n \n Recall that the sum of the $t$ first terms of a geometric sequence defined by $a_0$ and $a_{i+1}=\\lambda a_i$ with $\\lambda\\in (0,1)$ is: $a_0\\cdot \\frac{1-\\lambda^{t+1}}{1-\\lambda}=a_0\\cdot O(1)$. We will now study 2 cases, when $j\\leq \\frac{p}{100}$ and when $j>\\frac{p}{100}$. \n \n If we have a set $S$ of size $j \\le \\frac{p}{100}$ that violates the conclusion of the lemma, then the size of $N(S)$ must be $k<2j$, so the probability is bounded above by: \n\n \\begin{align*}\n \\power{\\frac{j+k}{p}}{j} \\power{\\frac{1}{25}}{3j} \\power{\\frac{200e}{81}}{4j} \\le \\power{\\frac{j+k}{p}}{j} \\power{\\frac{1}{5}}{j}\n \\end{align*}\n Assume that $i=(j+k)$ is fixed, then since $k\\leq 2j$, we can conclude that $i\\geq j\\geq \\frac{i}{3}$. Considering just one vertex of $S$ we know that $i=|S \\cup N(S)|\\geq 11$ so we must also have $j\\geq 4$. So by summing over $j$ we get:\n \n \\[\\sum_{j=\\max\\left\\{4,\\ceil{\\frac{i}{3}}\\right\\}}^{i} \\power{\\frac{i}{5p}}{j}=\\power{\\frac{i}{5p}}{\\max\\left\\{4,\\ceil{\\frac{i}{3}}\\right\\}}\\cdot O(1)\\]\n \n Noting that $\\power{\\frac{i}{5p}}{\\max\\left\\{4,\\ceil{\\frac{i}{3}}\\right\\}}$\n is less than $\\frac{1}{25p^2}$ when $i\\leq \\sqrt{p}$ and less than $\\power{\\frac{1}{25}}{\\sqrt{p}}$ otherwise since $j+k\\leq 4j\\leq \\frac{p}{25}$ by considering whether $i$ is greater or less than $\\sqrt{p}$, by summing over $i$ we get:\n \n \\[\\sum_{i=11}^{\\frac{p}{25}} \\power{\\frac{i}{5p}}{\\ceil{\\frac{i}{3}}}\\cdot O(1)\\leq \\frac{p}{25}\\left(\\frac{1}{25p^2} +\\power{\\frac{1}{25}}{\\sqrt{p}}\\right)\\cdot O(1)=o(1)\\]\n \nSo, the probability of the conclusion failing for some $S$ such that $|S|=j$, $j\\leq \\frac{p}{100}$ is $o(1)$. \n\nOtherwise, $\\frac{p}{100} \\le j \\le \\frac{p}{2}$ and the only case for $k$ that violates the conclusion of the lemma is $k =0$. In this case, the probability that there is some set $S$ of $j$ nontroublesome vertices of $V(F')$ such that $N(S)=\\emptyset$ can be bounded above by: \n\\begin{align*}\n &\\power{\\frac{j+k}{p}}{7j-k} \\power{\\frac{10}{9}}{8j} (2e)^{j+k}\\\\\n = &\\power{\\power{\\frac{j}{p}}{7} \\power{\\frac{10}{9}}{8} 2e}{j}\\\\\n \\le &\\power{\\frac{20e}{9}\\cdot \\power{\\frac{5}{9}}{7}}{j}\\\\\n \\le &\\power{\\frac{1}{100}}{j}\n\\end{align*}\nGiven our bounds on $j$, by summing over $j$ and using the sum of the first terms of a geometric sequence, we get at most $\\power{\\frac{1}{100}}{\\frac{n}{100\\sqrt{\\log n}+100}}\\cdot O(1)$ which is $o(1)$.\n\n\\end{proof}\n\n \\begin{lemma}\n \\label{usingexpansion}\n If\n every $S \\subseteq V(F')$ with $|S| \\le \\frac{p}{100}$ satisfies $|N(S)| >2|S|$\n then the following holds: \n \n (*) for any endpoint $u$ of any longest path $P$ of any supergraph $M'$ of $M^*$ there are at least $\\frac{p}{100}$\n vertices $w$ such that $u$ and $w$ are the endpoints of a path on $V(P)$. \n \\end{lemma}\n \n Lemma \\ref{usingexpansion} is an immediate corollary of Lemma 6.3.3 of \\cite{HKSSbook}, which follows from a well-known lemma of P\\'{o}sa \\cite{Pos76}.\n \n Now, if for every $S \\subseteq V(F')$, with $|S| \\le \\frac{p}{2}$, $N(S)$ is nonempty, then every component of $M^*$\n contains more than half its vertices so $M^*$ is connected. Furthermore, if (*) holds, for any longest \n path $P$ of any supergraph $M'$ of $M^*$, there are at least $\\frac{p}{100}$ vertices that are endpoints of paths on $V(P)$.\n Applying (*) to each of these paths we see that furthermore, if (*) holds, there are $\\frac{p^2}{20000}$ pairs of vertices that form the \n endpoints of paths on $V(P)$. By our lower bound on the degrees in $F'$,\n there must be $\\frac{\\ell^2p^2}{20000} -\\frac{p\\ell n}{\\sqrt{\\log{n}}}>\\frac{11pn}{\\log{n}}+1$ edges of $F'$ that join such a pair of vertices. \n So, Lemmas \\ref{expansion} and \\ref{usingexpansion} imply Lemma \\ref{anotherlemma1}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nAll rings in the paper are commutative with unity. The space $\\mathcal{R}_n$ of finitely generated rings marked on $n$ generators is by definition the set of pairs $(A,(x_1,\\dots,x_n))$ where $A$ is a ring endowed with a family of ring generators $(x_1,\\dots,x_n)$, up to marked ring isomorphism. This is a topological space, where a prebasis of neighbourhoods of the marked ring $(A,(x_1,\\dots,x_n))$ is given by the sets $V_P=V_P(A,(x_1,\\dots,x_n))$, for $P\\in\\mathbf{Z}[x_1,\\dots,x_n]$, defined as follows: if $P=0$ (respectively $P\\neq 0$) in $A$, $V_P$ is the set of marked rings on $n$ generators in which $P=0$ (resp. $P\\neq 0$), where $P$ is evaluated on the $n$-tuple of marked generators. The space $\\mathcal{R}_n$ is a compact metrizable, totally disconnected topological space, which is moreover countable by noetherianity. Hence every element of this space has a well-defined Cantor-Bendixson rank. Informally, this is the necessary (ordinal) number of times we have to remove isolated points so that the point itself becomes isolated. It is easy to check that the Cantor-Bendixson rank of a given marked ring does not depend on the choice of generators, and only on the isomorphism class within rings. See the beginning of Section \\ref{sec:cb} for more details.\n\nTo every noetherian ring, we can associate an ordinal-valued length, characterized by the formula\n$$\\ell(A)=\\sup\\{\\ell(B)+1|B\\text{ proper quotient of }A\\}\\quad\\text{(Agreeing }\\ell(\\{0\\})=0).$$\nThis length was introduced in \\cite{Bass,Gull} and further studied in \\cite{Krau,Bro1,Bro2}. It can be computed in a more explicit way (see Section \\ref{lengthbis}). In particular, if the Krull dimension of $A$ is an ordinal $\\alpha$, then $\\omega^\\alpha\\le\\ell(A)<\\omega^{\\alpha+1}$, where the left-hand inequality is an equality if and only if $A$ is a domain.\n\nThis length provides an obvious upper bound for the Cantor-Bendixson rank, but this is not optimal as every finite nonzero ring is isolated and therefore has zero Cantor-Bendixson rank, while it has finite but nonzero length. Accordingly we introduce the reduced length, an ordinal-valued function characterized by the formula\n$$\\ell'(A)=\\sup\\{\\ell'(B)+1|B\\text{ quotient of }A\\text{ with non-artinian kernel}\\}.$$\nIn the context of finitely generated rings, non-artinian just means infinite. A precise formula is given in Section \\ref{reducedo}. In particular, if the Krull dimension of $A$ is a finite number $d$, then $\\omega^{d-1}\\le\\ell'(A)<\\omega^{d}$, where the left-hand inequality is an equality if $A$ is a domain, and agreeing $\\omega^{-1}=0$.\n\n\\begin{thm}\nLet $A$ be a finitely generated ring. Then its Cantor-Bendixson rank\ncoincides with its reduced length.\\label{main}\n\\end{thm}\n\n\\begin{cor}\nIf $A$ is a finitely generated domain of Krull dimension $d$, then its Cantor-Bendixson rank is $\\omega^{d-1}$.\\label{cor:cbrkdom} \n\\end{cor}\n\nActually, every ring can be viewed as a module over itself generated by one\nelement, and ideals and submodules coincide; therefore it is a more general point of view if we consider the space of modules generated by $k$ marked elements over $\\mathbf{Z}[x_1,\\dots,x_n]$ (or even any commutative ring), the case $k=1$ corresponding to the space of rings marked by $n$ elements. Actually the point of view and language of modules (extensions, etc.) are very natural and useful in this context and it would have been awkward to restrict to rings. The definition of length and reduced length is extended to modules in the next sections, and in Section \\ref{sec:cb} we will obtain Theorem \\ref{main} as a particular case of Theorem \\ref{cb}, which holds in the general context of modules over finitely generated rings. Comments on the case of modules over some infinitely generated rings are included in Section \\ref{sec:other}.\n\n\\begin{rem}\nIt also makes sense to talk about the set of {\\it integral domains} generated by $n$ given elements; this can be viewed as a closed subset of $\\mathcal{R}_n$ and corresponds bijectively to $\\text{Spec}(\\mathbf{Z}[X_1,\\dots,X_n])$. It is proved in \\cite{CDN} that, {\\it inside this space}, the Cantor-Bendixson rank of any $d$-dimensional domain is $d$ and in particular is finite (this contrasts with Corollary \\ref{cor:cbrkdom}).\n\\end{rem}\n\n\n\\section{Length}\\label{lengthbis}\n\nLet $A$ be a ring (always assumed commutative). If $M$ is an $A$-module, define $\\Lambda(M)$ as the set of elements $m\\in M$ such that $Am$ has finite length. This is a submodule of $M$; if moreover $M$ is noetherian, then $\\Lambda(M)$ itself has finite length.\n\nLet $\\mathcal{P}$ be a prime ideal in $A$ and $M$ a noetherian $A$-module. As usual, $A_\\mathcal{P}$ denotes the local ring of $A$ at $\\mathcal{P}$ ($A_\\mathcal{P}=S^{-1}A$, where $S=A-\\mathcal{P}$), and $\\ell_{A_\\mathcal{P}}$ denotes the length function for finite length $A_\\mathcal{P}$-modules.\n\nDefine \n$$\\ell_\\mathcal{P}(M)=\\ell_{A_\\mathcal{P}}(\\Lambda(M\\otimes_AA_\\mathcal{P})).$$\n\nRecall that $\\textnormal{Asso}_A(M)$ is defined as the set of prime ideals $\\mathcal{P}$ of $A$ such that $A\/\\mathcal{P}$ embeds as a submodule of $M$. It is known to be finite if $M$ is noetherian, and non-empty if moreover $M\\neq 0$.\n\n\\begin{lem}\nWe have $\\ell_\\mathcal{P}(M)>0$ if and only if $\\mathcal{P}\\in\\textnormal{Asso}_A M$.\\label{lengthass}\n\\end{lem}\n\\begin{proof}\nIf $\\mathcal{P}\\in\\textnormal{Asso}_A M$, then $A\/\\mathcal{P}$ embeds into $M$; by flatness of $A_\\mathcal{P}$, this implies that $A_\\mathcal{P}\/\\mathcal{P} A_\\mathcal{P}$ embeds into $M\\otimes A_\\mathcal{P}$; so that $\\Lambda(M\\otimes A_\\mathcal{P})$ is non-zero and therefore has non-zero length.\n\nConversely suppose that $\\Lambda(M\\otimes A_\\mathcal{P})$ is non-zero. Then there exists an associated ideal $\\mathcal{Q}'\\in\\textnormal{Asso}_{A_\\mathcal{P}}(\\Lambda(M\\otimes A_\\mathcal{P}))$. We can write $\\mathcal{Q}'$ as $\\mathcal{Q} A_\\mathcal{P}$, for some prime ideal $\\mathcal{Q}$ of $A$ contained in $\\mathcal{P}$. Now $A_\\mathcal{P}\/\\mathcal{Q} A_\\mathcal{P}$ embeds into $\\Lambda(M\\otimes A_\\mathcal{P})$ and therefore has finite length; this forces $\\mathcal{Q}=\\mathcal{P}$. In other words, we have just proved that $$\\textnormal{Asso}_{A_\\mathcal{P}}(\\Lambda(M\\otimes A_\\mathcal{P}))=\\{\\mathcal{P} A_\\mathcal{P}\\}.$$ On the other hand, $$\\textnormal{Asso}_{A_\\mathcal{P}}(\\Lambda(M\\otimes A_\\mathcal{P}))\\subset \\textnormal{Asso}_{A_\\mathcal{P}}(M\\otimes A_\\mathcal{P})$$ $$=\\{\\mathcal{Q} A_\\mathcal{P}|\\mathcal{Q}\\in\\textnormal{Asso}_A(M),\\mathcal{Q}\\subset\\mathcal{P}\\}.$$ Therefore $\\mathcal{P}\\in\\textnormal{Asso}_A(M)$. \n\\end{proof}\n\n\\begin{lem}\nConsider a short exact sequence $0\\to K\\to M\\to N\\to 0$ of noetherian $A$-modules. Let $\\mathcal{P}$ be a prime ideal in $A$. Suppose that the $A_\\mathcal{P}$-module $K\\otimes A_\\mathcal{P}$ has finite length. Then $\\ell_\\mathcal{P}(M)=\\ell_\\mathcal{P}(N)+\\ell_\\mathcal{P}(K)$.\\label{lengthext}\n\\end{lem}\n\\begin{proof}\nBy flatness of $A_\\mathcal{P}$, we get an exact sequence\n$$0\\to K\\otimes A_\\mathcal{P}\\to M\\otimes A_\\mathcal{P}\\to N\\otimes A_\\mathcal{P}\\to 0,$$\nwhich by the assumption on $K$ induces an exact sequence\n$$0\\to K\\otimes A_\\mathcal{P}\\to \\Lambda(M\\otimes A_\\mathcal{P})\\to \\Lambda(N\\otimes A_\\mathcal{P})\\to 0.$$ \n\\end{proof}\n\n\\begin{lem}\nLet $\\mathcal{P}$ be a prime ideal in a noetherian ring $A$. Equivalences:\n\\begin{itemize}\n\\item[(i)] The sequence $(\\ell_\\mathcal{P}(A\/\\mathcal{P}^n))$ is bounded; \n\\item[(ii)] $\\mathcal{P}$ is a minimal prime ideal.\n\\end{itemize}\\label{lengthinfinity}\n\\end{lem}\n\\begin{proof}\n$\\ell_\\mathcal{P}(A\/\\mathcal{P}^n)=\\ell_{A_\\mathcal{P}}(\\Lambda(A_\\mathcal{P}\/\\mathcal{P}^n A_\\mathcal{P}))=\\ell_{A_\\mathcal{P}}(A_\\mathcal{P}\/\\mathcal{P}^n A_\\mathcal{P})$.\n\nIf $\\mathcal{P}$ is minimal, then $\\mathcal{P} A_\\mathcal{P}$ is the radical of the artinian ring $A_\\mathcal{P}$, so that $\\mathcal{P}^n A_\\mathcal{P}$ is eventually zero and therefore the sequence $A_\\mathcal{P}\/\\mathcal{P}^n A_\\mathcal{P}$ eventually stabilizes.\n\nConversely, if the sequence above is bounded, then it is stationary, so that in the local ring $A_\\mathcal{P}$, $(\\mathcal{P} A_\\mathcal{P})^n=(\\mathcal{P} A_\\mathcal{P})^{n+1}$ for some $n$. By Nakayama's Lemma, this forces $(\\mathcal{P} A_\\mathcal{P})^n=0$, so that $A_\\mathcal{P}$ is actually artinian, i.e. $\\mathcal{P}$ is a minimal prime ideal.\n\\end{proof}\n\nLet now $A$ be a ring. Say that a prime ideal $\\mathcal{P}$ in $A$ is conoetherian if $A\/\\mathcal{P}$ is noetherian. Define, for every conoetherian prime ideal $\\mathcal{P}$, its coheight as the ordinal\n$$\\text{coht}(\\mathcal{P})=\\sup\\{\\text{coht}(\\mathcal{Q})+1|\\mathcal{Q}\\text{ prime ideal properly containing }\\mathcal{P}\\}.$$\nThe noetherianity assumption makes this definition valid.\n\nIf $M$ is a noetherian $A$-module, then every $\\mathcal{P}\\in\\textnormal{Asso}_A(M)$ is conoetherian. Define, for every ordinal $\\alpha$,\n$$\\ell_\\alpha(M)=\\sum\\ell_\\mathcal{P}(M),$$\nwhere $\\mathcal{P}$ ranges over conoetherian prime ideals of coheight $\\alpha$ in $A$.\nThis is a finite sum as $\\ell_\\mathcal{P}(M)\\neq 0$ only when $\\mathcal{P}\\in\\textnormal{Asso}_A(M)$. Besides, the (ordinal-valued) Krull dimension of $M$ is defined as $\\sup\\text{coht}(\\mathcal{P})$, where $\\mathcal{P}$ ranges over prime ideals of $A$ containing the annihilator of $M$ (or, equivalently, over all associated primes of $M$).\n\n\\begin{lem}\nConsider a short exact sequence $0\\to K\\to M\\to N\\to 0$ of noetherian $A$-modules. Let $\\alpha$ be an ordinal. Suppose that $K$ has Krull dimension $\\le\\alpha$. Then $\\ell_\\alpha(M)=\\ell_\\alpha(N)+\\ell_\\alpha(K)$.\\label{lengthexto}\n\\end{lem}\n\\begin{proof}\nIt suffices to prove that for every prime ideal $\\mathcal{P}$ of coheight $\\alpha$, we have $\\ell_\\mathcal{P}(M)=\\ell_\\mathcal{P}(N)+\\ell_\\mathcal{P}(K)$. In view of Lemma \\ref{lengthext}, it is enough to obtain that the $A_\\mathcal{P}$-module $K\\otimes A_\\mathcal{P}$ has finite length. Indeed, $K$ can be written as a composite extension of modules $A\/\\mathcal{Q}_i$, where $\\mathcal{Q}_i$ are prime ideals of $A$. Then all $\\mathcal{Q}_i$ have coheight $\\le\\alpha$. Therefore either $\\mathcal{Q}_i=\\mathcal{P}$ or $\\mathcal{Q}_i$ is not contained in $\\mathcal{P}$. In the latter case, we have $A\/\\mathcal{Q}_i\\otimes A_\\mathcal{P}=0$, while $A\/\\mathcal{P}\\otimes A_\\mathcal{P}$ is the residual field of $A_\\mathcal{P}$ and therefore has length one. By flatness of $A_\\mathcal{P}$, we can thus write $K\\otimes A_\\mathcal{P}$ as a composite extension of modules of length $\\le 1$, so that $K\\otimes A_\\mathcal{P}$ has finite length. \n\\end{proof}\n\n\nLet $M$ be a noetherian $A$-module. Define the ordinal-valued length function of $M$ as\n$$\\ell(M)=\\sum_\\alpha\\omega^\\alpha\\cdot\\ell_\\alpha(M),$$\nwhen the sum ranges over the ordinals $\\alpha$ in \\textit{reverse order}. \n\nNote that if $M$ has finite Krull dimension (as most usual noetherian modules) then the exponents in the above ``polynomial\" are finite, i.e. $\\ell(M)<\\omega^\\omega$.\n\nThe following proposition gives a characterization of the ordinal-valued length function $\\ell$.\n\n\\begin{prop}\nLet $A$ be a ring and $M$ a noetherian $A$-module. Then\n$$\\ell(M)=\\sup\\{\\ell(N)+1|N\\textnormal{ proper quotient of }M\\}.$$\\label{caractl}\n\\end{prop}\n\nIn other words, $\\ell$ coincides with the ``Krull ordinal\" of the Noetherian ordered set of submodules of $M$, introduced in \\cite{Gull}, as well as (with a slight variant) in \\cite{Bass}. This inductive definition makes sense more generally for any noetherian module over any ring (commutative or not); it can be viewed as a quantitative gauge of noetherianity.\n\n\\begin{proof}\nSuppose that we have an exact sequence\n$$0\\to K\\to M\\to N\\to 0,$$\nwith $K\\neq 0$. Let $\\alpha$ be the Krull dimension of $K$, and pick $\\beta\\ge\\alpha$. Then, by Lemma \\ref{lengthexto}, $\\ell_\\beta(M)=\\ell_\\beta(N)+\\ell_\\beta(K)$. In particular, if $\\beta>\\alpha$, then $\\ell_\\beta(M)=\\ell_\\beta(N)$, and $\\ell_\\alpha(M)>\\ell_\\alpha(N)$. Therefore $\\ell(M)>\\ell(N)$.\n\nNow let us prove the other inequality, namely\n$$\\ell(M)\\le\\sup\\{\\ell(N)+1|N\\textnormal{ proper quotient of }M\\}.$$\n\n\\begin{itemize}\n\\item Zeroth case: $M=0$. Then we just get $0=\\sup\\emptyset$.\n\n\\item First case: $\\ell(M)$ is a successor ordinal. This occurs if and only if $\\ell_\\mathcal{P}(M)>0$ for some \\textit{maximal} ideal $\\mathcal{P}$, in which case $\\mathcal{P}\\in\\textnormal{Asso}_A(M)$ by Lemma \\ref{lengthass}, i.e. $M$ has a submodule $K$ isomorphic to $A\/\\mathcal{P}$. It is then straightforward that \n$\\ell(M)=\\ell(M\/K)+1$.\n\n\\item Second case: the least $\\alpha$ such that $\\ell_\\alpha(M)\\neq 0$ is a limit ordinal. Pick $\\mathcal{P}\\in\\textnormal{Asso}_A(M)$ with $\\text{coht}(\\mathcal{P})=\\alpha$. Find an exact sequence $0\\to K\\to M\\to N\\to 0$ with $K\\simeq A\/\\mathcal{P}$. For every $\\beta<\\alpha$, there exists a prime ideal $\\mathcal{P}_\\beta$ containing $\\mathcal{P}$ with $\\text{coht}(\\mathcal{P}_\\beta)=\\beta$. Find a submodule $V_\\beta$ of $K$ such that $K\/V_\\beta$ is isomorphic to $A\/\\mathcal{P}_\\beta$. From the exact sequences $0\\to K\\to M\\to N\\to 0$, $0\\to K\/V_\\beta\\to M\/V_\\beta\\to N\\to 0$ and Lemma \\ref{lengthexto}, we get:\n$$\\ell_\\gamma(M\/V_\\beta)=\\ell_\\gamma(M)\\text{ if }\\gamma>\\alpha;$$\n$$\\ell_\\alpha(M\/V_\\beta)=\\ell_\\alpha(M)-1;$$\n$$\\ell_\\beta(M\/V_\\beta)\\ge 1.$$\nNow write $\\ell(M)=P+\\omega^\\alpha$, where\n$$P=\\sum_{\\gamma>\\alpha}\\omega^\\gamma\\cdot\\ell_\\gamma(M)+\\omega^\\alpha\\cdot(\\ell_\\alpha(M)-1).$$\nThen we get\n$$\\ell(M\/V_\\beta)\\ge P+\\omega^\\beta,$$and thus\n$$\\sup_{\\beta<\\alpha}\\ell(M\/V_\\beta)\\ge P+\\sup_{\\beta<\\alpha}\\omega^\\beta=P+\\omega^\\alpha=\\ell(M).$$\n\\item Third case: the least $\\alpha$ such that $\\ell_\\alpha\\neq 0$ is a successor ordinal $\\alpha=\\beta+1$. Pick $\\mathcal{P}\\in\\textnormal{Asso}_A(M)$ with $\\text{coht}(\\mathcal{P})=\\alpha$ and choose a prime ideal $\\mathcal{Q}$ of coheight $\\beta$ containing $\\mathcal{P}$. Find an exact sequence $0\\to K\\to M\\to N\\to 0$ with $K\\simeq A\/\\mathcal{P}$. For every $n$, there exists a submodule $V_n$ of $K$ such that $K\/V_n$ is isomorphic to $A\/(\\mathcal{Q}^n+\\mathcal{P})$. By Lemma\n\\ref{lengthinfinity}, $(\\ell_\\beta(K\/V_n))$ is unbounded when $n\\to\\infty$. From the exact sequences $0\\to K\\to M\\to N\\to 0$, $0\\to K\/V_n\\to M\/V_n\\to N\\to 0$ and Lemma \\ref{lengthexto}, we get:\n$$\\ell_\\gamma(M\/V_n)=\\ell_\\gamma(M)\\text{ if }\\gamma>\\alpha;$$\n$$\\ell_\\alpha(M\/V_n)=\\ell_\\alpha(M)-1;$$\n$$\\ell_\\beta(M\/V_n)\\to\\infty\\text{ when }n\\to\\infty,$$\nand therefore $\\sup_n\\ell(M\/V_n)\\ge\\ell(M)$.\n\\end{itemize}\n\\end{proof}\n\n\n\\section{Reduced length}\\label{reducedo}\n\nDefine, for every ordinal $\\alpha$, the ordinal $\\alpha'$ as $\\alpha'=\\alpha+1$ if $\\alpha<\\omega$ and $\\alpha'=\\alpha$ otherwise.\nIf $M$ is a noetherian $A$-module, define its reduced length as follows\n\n$$\\ell'(M)=\\sum_\\alpha\\omega^\\alpha\\cdot\\ell_{\\alpha'}(M),$$\nwhere as usual the sum ranges over ordinal in reverse order. Observe that the reduced length is characterized by the length, as a consequence of the formula\n$$\\ell(M)=\\omega\\cdot\\ell'(M)+\\ell_0(M).$$\n\n\\begin{prop}\nIf $M$ is any noetherian $A$-module, then $\\ell'(M)=\\sup_N(\\ell'(N)+1)$, where $N$ ranges over all quotients of $M$ with non-artinian kernel. Moreover, $\\ell'(N)=\\ell'(M)$ if $N$ is a quotient of $M$ with artinian kernel.\\label{sigmaprime}\n\\end{prop}\n\\begin{proof}\nThe proof is similar to that of Proposition \\ref{caractl}, so let us just sketch it, stressing on the differences.\n\nFirst we have to prove that $\\ell(N)<\\ell(M)$ for every quotient $N$ of $M$ with non-artinian kernel $K$. The proof is the same, just noticing that then the Krull dimension of $K$ is at least one.\n\nIt remains to prove the reverse inequality\n$$\\ell'(M)\\ge\\sup\\{\\ell'(N)+1|N\\textnormal{ quotient of }M\\textnormal{ with non-artinian kernel}\\}.$$\n\n\\begin{itemize}\n\\item Zeroeth case: $M$ is artinian. Then we just get $0=\\sup\\emptyset$.\n\n\\item First case: $\\ell_1(M)\\neq 0$. Then $M$ has a submodule $K$ isomorphic to $A\/\\mathcal{P}$ for some prime ideal $\\mathcal{P}$ of coheight one. Then $\\ell'(M)=\\ell'(M\/K)+1$.\n\n\\item Second case (respectively third case): the least $\\alpha\\ge 1$ such that $\\ell_\\alpha(M)\\neq 0$ is a limit ordinal (resp. is a successor ordinal $\\ge 2$). Go on exactly like in the case of non-reduced length.\n\\end{itemize}\n\nFinally, if $N$ is a quotient of $M$ with artinian kernel, it follows from Lemma \\ref{lengthexto} that $\\ell_i(N)=\\ell_i(M)$ for all $i\\ge 1$, and therefore $\\ell'(N)=\\ell'(M)$.\n\\end{proof}\n\n\n\\section{Cantor-Bendixson rank}\\label{sec:cb}\n\nLet $M$ be a module over a ring $A$.\nLet $\\textnormal{Sub}_A(M)$ be the set of\nsubmodules of $M$. This is a closed subset of $2^M$, endowed with the product topology which makes it a Hausdorff compact space; if $M$ has countable cardinality it is moreover metrizable. Let $\\textnormal{Quo}_A(M)$ the set of quotients of $M$. It can be defined as a topological space that coincides with $\\textnormal{Sub}_A(M)$, in which we view its elements as quotients of $M$ through the correspondence $K\\leftrightarrow M\/K$. In particular, $M\\in\\textnormal{Quo}_A(M)$ corresponds to $\\{0\\}\\in\\textnormal{Sub}_A(M)$. \n\nIf $N$ is a quotient of $M$, then there is a natural embedding of $\\textnormal{Quo}_A(N)$ into $\\textnormal{Quo}_A(M)$. It is continuous so has closed image; moreover its image is open if and only if the $\\text{Ker}(M\\to N)$ is finitely generated (the easy argument is given in a similar context in \\cite[Lemma~1.3]{CGP}); this is fulfilled if $M$ is Noetherian. \n\nIn any topological space $X$ define by transfinite induction $I_0(X)$ as the set of isolated points of $X$, and $I_\\alpha(X)=I_0(X-\\bigcup_{\\beta<\\alpha}I_\\beta(X))$. For $x\\in X$, set\n$\\textnormal{CB-rk}(x,X)=\\infty$ if $x\\notin\\bigcup I_\\alpha(X)$ and $\\textnormal{CB-rk}(x,X)=\\alpha$ if $x\\in I_\\alpha$.\nAs all $I_\\alpha(X)$ are pairwise disjoint, this is well-defined. Agree that $\\infty>\\alpha$ for every ordinal $\\alpha$.\n\nNow for every $A$-module $M$, set $\\textnormal{CB-rk}(M)=\\textnormal{CB-rk}(M,\\textnormal{Quo}_A(M))$. Observe that if $N$ is a quotient of $M$, then $\\textnormal{CB-rk}(N)=\\textnormal{CB-rk}(N,\\textnormal{Quo}_A(M))$, as the natural embedding $\\textnormal{Quo}_A(N)\\to\\textnormal{Quo}_A(M)$ is open (because the kernel of $M\\to N$ is finitely generated by noetherianity). In particular, if $N$ is generated by $k$ elements, then it can be viewed as a quotient of $A^k$, the free module of rank $k$, and $\\textnormal{CB-rk}(N)$ coincides with the Cantor-Bendixson rank of $N$ inside the space $\\textnormal{Quo}_A(A^k)$ of finitely generated $A$-modules over $k$ marked generators.\n\n\\begin{lem}\nEvery noetherian $A$-module $M$ satisfying $\\textnormal{CB-rk}(M)=0$ (i.e. $M$ is isolated) has finite length. More precisely, every noetherian $A$-module of infinite length contains a decreasing sequence of non-zero submodules $(N_n)$ with trivial intersection.\\label{rkzero}\n\\end{lem}\n\n\\begin{rem}\nThe converse is not true in general: for instance if $A$ is an infinite field and $M$ is a 2-dimensional vector space. However every module of finite cardinality is obviously isolated.\n\\end{rem}\n\n\\begin{proof}[Proof of Lemma \\ref{rkzero}]\nOtherwise, $M$ has a non-maximal associated ideal $\\mathcal{P}$. As it is clear that being isolated is inherited by submodules, we can suppose that $M=A\/\\mathcal{P}$. We can even suppose that $\\mathcal{P}=\\{0\\}$, so that $A=M$ is a noetherian domain which is not a field. Then if $\\mathcal{M}$ is a maximal ideal in $A$, then $\\{0\\}\\neq\\mathcal{M}^n\\to \\{0\\}$ in $\\textnormal{Sub}_A(A)$ and we get a contradiction.\n\\end{proof}\n\n\n\\begin{thm}\nLet $M$ be a noetherian $A$-module. Suppose that every artinian subquotient of $M$ has finite cardinality. Then $\\textnormal{CB-rk}(M)=\\ell'(M)$.\\label{cb}\n\\end{thm}\n\\begin{proof}\nSuppose that the statement is proved for every proper quotient of $M$.\n\nSuppose that $\\textnormal{CB-rk}(M)>\\ell'(M)$. Then there exists a sequence of proper quotients $M\/W_n$, converging to $M$, such that $\\textnormal{CB-rk}(M\/W_n)\\ge\\ell'(M)$. By induction, $\\ell'(M\/W_n)\\ge\\ell'(M)$. By Proposition \\ref{sigmaprime}, this forces $W_n$ to be artinian, i.e. contained in the maximal artinian submodule $S$ of $M$. By assumption, $S$ is finite. As $W_n\\to\\{0\\}$ in the space of submodules of $M$, this forces that eventually $W_n=0$, a contradiction.\n\nSuppose that $\\textnormal{CB-rk}(M)<\\ell'(M)$. Then, in view of Proposition \\ref{sigmaprime}, there exists a quotient $M\/W$, with $W$ non-artinian, such that $\\ell'(M\/W)\\ge\\textnormal{CB-rk}(M)$. By induction we get $\\textnormal{CB-rk}(M\/W)\\ge\\textnormal{CB-rk}(M)$. As $W$ is non-artinian, it contains by Lemma \\ref{rkzero} a properly decreasing sequence $(W_n)$ of non-zero submodules with trivial\nintersection. Then $$\\ell'(M\/W_n)\\ge\\ell'(M\/W)\\ge\\textnormal{CB-rk}(M),$$ and $$\\textnormal{CB-rk}(M)\\ge\\sup(\\textnormal{CB-rk}(M\/W_n)+1)$$ $$=\\sup(\\ell'(M\/W_n)+1)\\ge\\textnormal{CB-rk}(M)+1,$$\na contradiction.\n\\end{proof}\n\nThe following lemma is well-known, and implies that Theorem \\ref{main} is a corollary of Theorem \\ref{cb}.\n\n\\begin{lem}\nLet $A$ be finitely generated ring. Then every finitely generated simple $A$-module has finite cardinality.\n\\end{lem}\n\\begin{proof}\nEvery such $A$-module $M$ can be viewed as a finitely generated $\\mathbf{Z}$-algebra which is a field. Let $F$ be its prime subfield. Then by the Nullstellensatz, $M$ is a finite extension of $F$. If $F$ is a finite field we are done. If $F=\\mathbf{Q}$, then $M=\\mathbf{Q}[X]\/P(X)$, where $P$ is a monic polynomial with coefficients in $\\mathbf{Z}[1\/k]$ for some integer $k>1$. Hence $M$ can be written as the increasing union of proper subrings\n$\\mathbf{Z}[1\/n!k][X]\/P(X)$, hence is not a finitely generated ring, a contradiction.\n\\end{proof}\n\n\n\\section{Comments on some other rings}\\label{sec:other}\n\nTheorem \\ref{cb} fails when $A$ possesses an infinite simple $A$-module $M$ (i.e. $A$ has an infinite index maximal ideal). Indeed, we have $\\ell'(M\\times M)=0$, while $\\textnormal{CB-rk}(M\\times M)=1$.\n\nIn all cases, we have:\n\n\\begin{prop}\n$$\\ell'(M)\\le\\textnormal{CB-rk}(M)\\le\\ell(M).$$\n\\end{prop}\nHere we set $(\\alpha+1)-1=\\alpha$, and $\\alpha-1=\\alpha$ if $\\alpha$ is not a successor ordinal.\n\nThe left-hand inequality has already been settled in the proof of Theorem \\ref{cb}, where we did not make use of the assumption on maximal ideals. The right-hand inequality is obtained by a straightforward induction. It is not optimal: for instance if $\\ell(M)$ is a successor ordinal, it is easy to check that $\\textnormal{CB-rk}(M)\\le\\ell(M)-1$. I do not know to which extent this can be improved. However, the following example provides a quite unexpected behaviour.\n\nLet $A$ be a local principal domain, with maximal ideal $I$ of infinite index. Denote the cyclic indecomposable $A$-modules $M_n=A\/I^n$.\n\nLet $T\\simeq {M_1}^{n_1}\\oplus\\dots\\oplus{M_k}^{n_k}$ be an $A$-module of finite length $\\ell(T)=\\sum in_i$, where $n_k\\neq 0$. Set $\\ell^*(T)=\\ell(T)-k$. (Agree that $\\ell^*(0)=0$.)\n\n\\begin{lem}\nLet $M$ be a finitely generated $A$-module, and $T$ its torsion submodule. Then, for $n\\ge 0$,\n\\begin{itemize}\n\\item if $M\\simeq A^{2n}\\oplus T$, then $\\textnormal{CB-rk}(M)=\\omega\\cdot n+\\ell^*(T)$;\n\\item if $M\\simeq A^{2n+1}\\oplus T$, then $\\textnormal{CB-rk}(M)=\\omega\\cdot n+\\ell(T)+1$.\n\\end{itemize}\n\\end{lem}\n\nObserve that on the other hand, $\\ell(A^k\\oplus T)=\\omega\\cdot k+\\ell(T)$, and $\\ell'(A^k\\oplus T)=\\ell(T)$.\n\n\\begin{proof}\nLet us argue by induction on $\\ell(M)$.\n\nSuppose that $M\\simeq A^{2n}\\oplus T$ with $\\ell^*(T)\\neq 0$. Then $T\\simeq {M_1}^{n_1}\\oplus\\dots\\oplus{M_k}^{n_k}$, where $n_k\\neq 0$ and $\\sum n_i\\ge 2$. Inside the socle of $T$, one can find infinitely many cyclic submodules $D_j$, with $D_j\\cap D_m=\\{0\\}$ for $n\\neq m$. Thus \n$(M\/D_j)$ is a sequence of distinct modules tending to $M$. Now by induction $\\textnormal{CB-rk}(M\/D_j)=\\omega\\cdot n+\\ell^*(T\/D_j)=\\omega\\cdot n+\\ell^*(T)-1$ (except maybe for one single value of $j$, if $n_k=1$),\nso $\\textnormal{CB-rk}(M)\\ge\\omega\\cdot n+\\ell^*(T)$.\n\nSuppose now that $M\\simeq A^{2n}\\oplus T$ (with in mind $\\ell^*(T)=0$ although we do not need it). If $n\\ge 1$, then $M$ is a limit for $d\\to\\infty$ of modules isomorphic to $A^{2n-1}\\oplus M_d\\oplus T$, which by induction have $\\textnormal{CB-rk}\\ge\\omega\\cdot (n-1)+d$. So $\\textnormal{CB-rk}(M)\\ge\\omega\\cdot n$. This also obviously holds if $n=0$.\n\nSuppose that $M\\simeq A^{2n+1}\\oplus T$. Then $M$ is a limit of modules isomorphic to $A^{2n}\\oplus M_d\\oplus T$, for $d\\to\\infty$, which by induction have $\\textnormal{CB-rk}\\ge\\omega\\cdot n+\\ell^*(M_d\\oplus T)=\\omega\\cdot n+\\ell(T)$ for $d$ large enough. Thus $\\textnormal{CB-rk}(M)\\ge\\omega\\cdot n+\\ell(T)+1$.\n\nSo we have established, in all cases, $\\textnormal{CB-rk}(M)\\ge r(M)$, where $r(M)$ is the right-hand term given by the proposition. Let us now prove the upper bound $\\textnormal{CB-rk}(M)\\le r(M)$, again by induction.\n\nSuppose that $M=A^{2n}\\oplus T$, and let $M\/N$ be a quotient of $M$. If $N$ is not contained in $T$, then the rank drops, so that by induction $\\textnormal{CB-rk}(M\/N)