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+{"text":"\\section{Introduction}\n\\label{sec:intr}\n\nIn our previous paper \\cite{Inan_Kisselev:2021} we probed the\nanomalous quartic gauge couplings (QGCs) in the $\\gamma\\gamma\n\\rightarrow \\gamma\\gamma$ process at the Compact Linear Collider\n(CLIC) \\cite{Braun:2008,Boland:2016}. Both the unpolarized and\npolarized light-by-light scatterings were considered, and the bounds\non QGCs were obtained. The neutral anomalous quartic couplings are\nof particular interest. The anomaly interactions $\\gamma ZZZ$,\n$\\gamma\\gamma ZZ$, and $\\gamma\\gamma\\gamma Z$ at the LHC were\nanalyzed in \\cite{Chapon:2009}-\\cite{Eboli:2004}. The LHC\nexperimental bounds on QGCs were presented by the CMS\n\\cite{CMS:QGCs} and ATLAS \\cite{ATLAS:QGCs} Collaborations (see also\n\\cite{Schoeffel:2021}). The bounds on the anomalous\n$\\gamma\\gamma\\gamma Z$ vertex can be also derived from the\nconstraints on the $\\mathcal{B}(Z\\rightarrow\\gamma\\gamma\\gamma)$\nbranching ratio obtained at the LEP \\cite{L3:Z_decay} and LHC\n\\cite{ATLAS:Z_decay}. As for $e^+e^-$ colliders, they may operate in\n$e\\gamma$ and $\\gamma\\gamma$ modes~\\cite{Ginzburg:1981}. The bounds\non QGCs in $e^+e^-$, $e\\gamma$ and $\\gamma\\gamma$ collisions were\ngiven in \\cite{Eboli:1994}-\\cite{Koksal:2014}. In particular, the\nlimits on the quartic couplings for the vertex $\\gamma\\gamma\\gamma\nZ$ were derived in \\cite{Gutierrez:2014} using LEP~2 data for the\nreactions $e^+e^- \\rightarrow \\gamma\\gamma\\gamma, \\gamma\\gamma Z$. A\nsimilar analysis for the exclusive $\\gamma Z$ production with intact\nprotons at the LHC was done in \\cite{Baldenegro:2017}. The search\nfor virtual SUSY effects in the process $\\gamma\\gamma \\rightarrow\n\\gamma Z$ at high energies was presented in \\cite{Gounaris:1999_1}.\n\nAs one can see, the anomalous $\\gamma\\gamma\\gamma Z$ vertex urgently\nneeds to be examined in high energy $e^+e^-$ collisions. That is\nwhy, in the present paper we study the process (see\nFig.~\\ref{fig:3gammasZ})\n\\begin{equation}\\label{process}\n\\gamma(p_1, \\mu)+ \\gamma(p_2, \\nu) \\rightarrow \\gamma(p_3, \\rho) +\nZ(p_4, \\alpha) \\;,\n\\end{equation}\nwhere $p_1, p_2, p_3, p_4$ are boson momenta, $\\mu, \\nu, \\rho,\n\\alpha$ are boson Lorentz indices, and ingoing particles are real\npolarized photons generated at the CLIC by the laser Compton\nbackscattering~\\cite{Kramer:1994}-\\cite{Telnov:1998}.\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[scale=0.6]{2_gammas_gamma_Z.eps}\n\\caption{The process $\\gamma + \\gamma \\rightarrow \\gamma + Z$.}\n\\label{fig:3gammasZ}\n\\end{center}\n\\end{figure}\nOur main goal is to derive bounds on anomaly couplings for the\nvertex $\\gamma\\gamma\\gamma Z$ which can be reached at the CLIC using\nboth polarized and unpolarized photon beams. The great potential of\nthe CLIC in probing new physics is well-known\n\\cite{Dannheim:2012}-\\cite{Franceschini:2020}. Let us underline that\na physical potential of a linear high energy $e^+e^-$ collider may\nbe significantly enhanced, provided the polarized beams are\nused~\\cite{polarized_beams,CLIC_lum}.\n\nLet $\\lambda_e$ be the helicity of the initial electron beam, while\n$\\lambda_0$ be the helicity of the ingoing laser photon beam. In our\ncalculations, we will consider two sets of these helicities, with\nopposite sign of $\\lambda_e$,\n\\begin{align}\\label{helicities}\n(\\lambda_e^{(1)}, \\lambda_0^{(1)}; \\lambda_e^{(2)}, \\lambda_0^{(2)})\n&= (0.8, 1; 0.8, 1) \\;, \\nonumber \\\\\n(\\lambda_e^{(1)}, \\lambda_0^{(1)}; \\lambda_e^{(2)}, \\lambda_0^{(2)})\n&= (-0.8, 1; -0.8, 1) \\;,\n\\end{align}\nwhere the superscripts 1 and 2 enumerate the beams. We will work in\nthe effective field theory framework. Previously effective\nLagrangians were used in \\cite{Stohr:1994}-\\cite{Novotny:1995} for\nexamining the $\\gamma\\gamma\\gamma Z$ interaction in the $Z\n\\rightarrow \\gamma\\gamma\\gamma$ decay, as well as in\n\\cite{Baillargeon:1996}, \\cite{Gutierrez:2014}, and\n\\cite{Baldenegro:2017}. Anomalous quartic gauge couplings (QGCs) are\ninduced at the dimension-six level already. However, they are not\nindependent of anomalous trilinear gauge couplings. That is why, in\nour paper, we study anomalous QGCs which enter the effective\nLagrangian at dimension-eight without contributing to anomalous\ntrilinear gauge interactions.\n\nThe paper is organized as follows. In the next section, the effective\nLagrangian is described, and Feynman rules for the anomalous\n$\\gamma\\gamma\\gamma Z$ vertex are presented. The helicity amplitudes\nare studied in Sec.~\\ref{sec:amplitudes}. In\nSec.~\\ref{sec:numerical_results}, both differential and total cross\nsections for the process \\eqref{process} are calculated, and bounds\non the QGCs are given. In Sec.~\\ref{sec:unit_const}, unitarity\nconstraints on anomalous quartic couplings are obtained. In\nAppendix~A, polarization tensors for the vertex $\\gamma\\gamma\\gamma\nZ$ are listed. The explicit expressions for the anomalous\ncontributions to the helicity amplitudes are given in Appendix~B.\nSome formulas for Wigner's $d$-function are collected in Appendix~C.\nFinally, in Sec.~\\ref{sec:concl}, we summarize our results and give\nconclusions.\n\n\n\\section{Effective Lagrangian}\n\\label{sec:Lagrangian}\n\nIt is appropriate to describe the anomalous $\\gamma\\gamma\\gamma Z$\ninteraction by means of an effective Lagrangian. Given parity is\nconserved and gauge invariance is valid, there are only two\nindependent operators with dimension 8. Following\n\\cite{Stohr:1994,Horejsi:1994}, we take the Lagrangian\n\\begin{equation}\\label{Lagrangian}\n\\mathcal{L}_{\\gamma\\gamma\\gamma Z} = g_1 O_1 + g_2 O_2 \\;,\n\\end{equation}\nwith the operators\n\\begin{equation}\\label{operators_G-R}\nO_1 = F^{\\rho\\mu} F^{\\alpha\\nu} \\partial_\\rho F_{\\mu\\nu} Z_\\alpha\n\\;, \\quad O_2 = F^{\\rho\\mu} F^\\nu_\\mu \\partial_\\rho F_{\\alpha\\nu}\nZ^\\alpha \\;,\n\\end{equation}\nwhere  $F_{\\mu\\nu} = \\partial_\\mu A_\\nu - \\partial_\\nu A_\\mu$. The\noperators $O_{1,2}$ arise from a $SU(2)\\times U(1)_Y$ effective\nLagrangian with two operators like\n$B_{\\mu\\nu}B^{\\nu\\nu}B_{\\rho\\sigma}B^{\\rho\\varrho}$, four operators\nlike $W_{\\mu\\nu}W^{\\nu\\nu}W_{\\rho\\sigma}W^{\\rho\\varrho}$, and four\noperators like $B_{\\mu\\nu}B^{\\nu\\nu}W_{\\rho\\sigma}W^{\\rho\\varrho}$,\nwhere $W_\\mu$ and $B_\\mu$ are the $SU(2)$ and hypercharge gauge\nfields, respectively \\cite{Fichet:2014}. We consider only CP\nconserving operators hence the dual field strength tensors\n$\\tilde{W}_{\\mu\\nu}$ and $\\tilde{B}_{\\mu\\nu}$ are not used. The\ncoupling $g_{1,2}$ are linear combinations of ten coefficients of\ndimension-eight operators mentioned above. Note that certain\ncombinations of these coefficients must obey so-called positivity\nconstraints \\cite{Remmen:2019}-\\cite{Bi:2019}.\n\nAs one can see, this Lagrangian contains no derivatives of the $Z$\nboson field (correspondingly, no $p_4$ in the momentum space), that\nsimplifies a derivation of Feynman rules for the $\\gamma\\gamma\\gamma\nZ$ vertex.\n\nSome authors use the Lagrangian \\cite{Novotny:1995}\n\\begin{equation}\\label{Lagrangian_N}\n\\mathcal{L}^{(\\mathrm{N})}_{\\gamma\\gamma\\gamma Z} = G_1 \\bar{O}_1 +\nG_2 \\bar{O}_2 \\;,\n\\end{equation}\nwith the operators\n\\begin{equation}\\label{operators_B}\n\\bar{O}_1 =  F^{\\mu\\nu}F_{\\mu\\nu} F^{\\rho\\sigma}Z_{\\rho\\sigma} \\;,\n\\quad \\bar{O}_2 = F^{\\mu\\nu} F_{\\nu\\rho}F^{\\rho\\sigma}Z_{\\sigma\\mu}\n\\;,\n\\end{equation}\nwhere $Z_{\\mu\\nu} = \\partial_\\mu Z_\\nu - \\partial_\\nu Z_\\mu$, or the\nLagrangian \\cite{Baldenegro:2017}\n\\begin{equation}\\label{Lagrangian_B}\n\\mathcal{L}^{(\\mathrm{B})}_{\\gamma\\gamma\\gamma Z} = \\zeta O +\n\\tilde{\\zeta} \\tilde{O} \\;,\n\\end{equation}\nwith the operators\n\\begin{equation}\\label{operators_B}\nO = F^{\\mu\\nu}F_{\\mu\\nu} F^{\\rho\\sigma}Z_{\\rho\\sigma} \\;, \\quad\n\\tilde{O} = F^{\\mu\\nu}\n\\tilde{F}_{\\mu\\nu}F^{\\rho\\sigma}\\tilde{Z}_{\\rho\\sigma} \\;,\n\\end{equation}\nwhere $\\tilde{F}_{\\mu\\nu} =\n(1\/2)\\,\\varepsilon^{\\mu\\nu\\rho\\sigma}F_{\\varrho\\sigma}$, and\n$\\tilde{Z}_{\\mu\\nu} =\n(1\/2)\\,\\varepsilon^{\\mu\\nu\\rho\\sigma}Z_{\\varrho\\sigma}$.\n\nUsing integration by parts and equations of motion, one can easily\nobtain the following relations between two bases for the effective\nLagrangian \\cite{Novotny:1995},\n\\begin{equation}\\label{operators_N_vs_G-R}\n\\bar{O}_1 = -8O_1 \\;, \\quad \\bar{O}_2 = 2(O_2 - O_1) \\;.\n\\end{equation}\nWe also have the relations \\cite{Baldenegro:2017}\n\\begin{equation}\\label{operators_G-R_vs_B}\nO = \\bar{O}_1 = -8 O_1 \\;, \\quad \\tilde{O} = 4\\bar{O}_2 - 2\\bar{O}_1\n= 8(O_2 + O_1) \\;.\n\\end{equation}\nThe above listed equations enable us to relate anomalous coupling in\neqs.~\\eqref{Lagrangian}, \\eqref{Lagrangian_N}, and\n\\eqref{Lagrangian_B}. In particular, we find\n\\begin{equation}\\label{couplings_G-R_B}\ng_1 = 8(\\tilde{\\zeta} - \\zeta) \\;, \\quad g_2 = 8 \\tilde{\\zeta} \\;.\n\\end{equation}\n\nThe Feynman rules for the effective anomalous vertex, resulting from\nthe Lagrangian \\eqref{Lagrangian}, are given by \\cite{Horejsi:1994}\n\\begin{align}\\label{Feynman_rules}\nP^{\\mu\\nu\\rho\\alpha} = \\mathcal{P}&\\{ g_1[(p_1\\cdot p_2)(p_2\\cdot\np_3) g^{\\mu\\nu}g^{\\rho\\alpha} - (p_1\\cdot p_3) p_2^\\mu p_1^\\nu\ng^{\\rho\\alpha}\n\\nonumber \\\\\n&- (p_1\\cdot p_3)p_1^\\nu p_2^\\alpha g^{\\mu\\rho} + p_2^\\mu p_1^\\nu\np_1^ \\rho p_3^\\alpha ]\n\\nonumber \\\\\n&+ g_2[ -(p_1\\cdot p_2)(p_1\\cdot p_3) g^{\\mu\\alpha}g^{\\nu\\rho} +\n(p_2\\cdot p_3) p_1^\\nu p_1^\\alpha g^{\\mu\\rho}\n\\nonumber \\\\\n&- (p_2\\cdot p_3) p_1^\\nu p_1^\\rho g^{\\mu\\alpha} + (p_2\\cdot p_3)\np_1^\\nu p_2^\\alpha g^{\\mu\\rho} + 2(p_2\\cdot p_3) p_2^\\mu p_1^\\rho\ng^{\\nu\\alpha}\n\\nonumber \\\\\n&- (p_1\\cdot p_3) p_2^\\rho p_1^\\alpha g^{\\mu\\nu} + p_3^\\mu p_1^\\nu\np_2^\\rho p_1^\\alpha ] \\} \\;,\n\\end{align}\nwhere $\\mathcal{P}$ denotes possible permutations $(p_1,\\mu)\n\\leftrightarrow (p_2,\\nu) \\leftrightarrow (p_3, \\rho)$, and all\nmomenta in the $\\gamma\\gamma\\gamma Z$ vertex are assumed to be\nincoming ones. Correspondingly, the polarization tensor is equal to\n\\begin{equation}\\label{polarization_tensor}\nP_{\\mu\\nu\\rho\\alpha}(p_1, p_2, p_3) = g_1 \\sum_{i=1}^4\nP_{\\mu\\nu\\rho\\alpha}^{(1,i)}(p_1, p_2, p_3) + g_2 \\sum_{i=1}^7\nP_{\\mu\\nu\\rho\\alpha}^{(2,i)}(p_1, p_2, p_3) \\;.\n\\end{equation}\nElectromagnetic gauge invariance results in equations $p_1^\\mu\nP_{\\mu\\nu\\rho\\alpha} = p_2^\\nu P_{\\mu\\nu\\rho\\alpha} = p_3^\\rho\nP_{\\mu\\nu\\rho\\alpha} = 0$. Note that terms proportional to $p_1^\\mu,\np_2^\\nu, p_3^\\rho$ are omitted in \\eqref{Feynman_rules}, since they\ndo not contribute to the matrix element, see\neq.~\\eqref{matrix_element} below. Explicit expressions for the\ntensors $P_{\\mu\\nu\\rho\\alpha}^{(1,i)}$ and\n$P_{\\mu\\nu\\rho\\alpha}^{(2,i)}$ are presented in Appendix~A. To\ncalculate helicity amplitudes for the process \\eqref{process}, one\nhas to make the replacement $p_3 \\rightarrow -p_3$ in the Feynman\nrules for the $\\gamma\\gamma\\gamma Z$ vertex given by\neqs.~\\eqref{Feynman_rules}, \\eqref{polarization_tensor}, and\n\\eqref{P1.1}-\\eqref{P2.7}.\n\n\n\\section{Helicity amplitudes}\n\\label{sec:amplitudes}\n\nWe work in the c.m.s. of the colliding real photons, $\\vec{p}_1 +\n\\vec{p}_2 = 0$, where the momenta are given by\n\\begin{align}\\label{momenta}\np_1^\\mu &=(p,0,0,p) \\;,\n\\nonumber \\\\\np_2^\\mu &=(p,0,0,-p) \\;,\n\\nonumber \\\\\np_3^\\mu &=(k, 0, k\\sin\\theta, k\\cos\\theta) \\;,\n\\nonumber \\\\\np_4^\\mu &=(E, 0, -k\\sin\\theta, -k\\cos\\theta) \\;.\n\\end{align}\nHere $E = \\sqrt{k^2 + m_Z^2}$, with $m_Z$ being the mass of the $Z$\nboson. The Mandelstam variables of the process \\eqref{process} are\n\\begin{align}\\label{Mandelstam_var}\ns &= (p_1 + p_2)^2 = 4p^2 \\;,\n\\nonumber  \\\\\nt &= (p_1 - p_3)^2 = -2pk (1 - \\cos\\theta) \\;,\n\\nonumber  \\\\\nu &= (p_2 - p_3)^2 = -2pk (1 + \\cos\\theta) \\;,\n\\end{align}\nwhere $\\theta$ is a scattering angle in the c.m.s. Note that $s + t\n+ u = m_Z^2$.\n\nIn the chosen system the polarization vectors are equal to\n\\begin{align}\\label{pol_vectors_initial}\n\\varepsilon_\\mu^+(p_1) &= \\varepsilon_\\mu^-(p_2) =\n\\frac{1}{\\sqrt{2}}(0, 1, i, 0) \\;,\n\\nonumber \\\\\n\\varepsilon_\\mu^-(p_1) &= \\varepsilon_\\mu^+(p_2) =\n\\frac{1}{\\sqrt{2}}(0, 1, -i, 0) \\;,\n\\nonumber \\\\\n\\varepsilon_\\mu^+(p_3) &= \\varepsilon_\\mu^-(p_4) =\n\\frac{1}{\\sqrt{2}}(0, 1, i\\cos \\theta, -i\\sin\\theta) \\;,\n\\nonumber \\\\\n\\varepsilon_\\mu^-(p_3) &= \\varepsilon_\\mu^+(p_4) =\n\\frac{1}{\\sqrt{2}}(0, 1, -i\\cos\\theta,\ni\\sin\\theta) \\;, \\nonumber \\\\\n\\varepsilon_\\mu^0(p_4) &= \\frac{1}{m_Z}(k, 0, -E\\sin\\theta, -\nE\\cos\\theta) \\;.\n\\end{align}\nThey obey the orthogonality condition $\\varepsilon_\\mu^\\lambda (k)\nk^\\mu = 0$. Correspondingly, we get the helicity vectors of the\nfinal photon and $Z$ boson,\n\\begin{align}\\label{pol_vectors_final}\n\\varepsilon_\\mu^{*+}(p_3) &= \\varepsilon_\\mu^{*-}(p_4) =\n\\frac{1}{\\sqrt{2}}(0, 1, -i\\cos \\theta, i\\sin\\theta) \\;,\n\\nonumber \\\\\n\\varepsilon_\\mu^{*-}(p_3) &= \\varepsilon_\\mu^{*+}(p_4) =\n\\frac{1}{\\sqrt{2}}(0, 1, i\\cos\\theta, -i\\sin\\theta) \\;,\n\\nonumber \\\\\n\\varepsilon_\\mu^{*0}(p_4) &= \\varepsilon_\\mu^0(p_4) =\n\\frac{1}{m_Z}(k, 0, -E\\sin\\theta, - E\\cos\\theta)  \\;.\n\\end{align}\n\nThe matrix element of the process \\eqref{process} with the definite\nhelicities of the incoming and outgoing bosons can be written as\n\\begin{equation}\\label{matrix_element}\nM_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}(p_1, p_2, p_3) =\nP_{\\mu\\nu\\rho\\alpha}(p_1, p_2, p_3)\n\\,\\varepsilon_\\mu^{\\lambda_1}(p_1)\\varepsilon_\\nu^{\\lambda_2}(p_2)\n\\varepsilon_\\rho^{*\\lambda_3}(p_3)\\varepsilon_\\alpha^{*\\lambda_4}(p_4)\n\\;,\n\\end{equation}\nwhere the polarization tensor $P_{\\mu\\nu\\rho\\alpha}$ is given by\neq.~\\eqref{polarization_tensor}. We have calculated the anomalous\nhelicity amplitudes, and present their explicit expressions in\nAppendix~B. Using these expressions, we obtain the unpolarized\namplitude squared\n\\begin{equation}\\label{M2}\n\\sum_{\\lambda_1\\ldots\\lambda_4}|M_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}|^2\n= \\frac{1}{4} [g_1^2(3A + 2B) - 4g_1g_2(A + B) + 4g_2^2(A + B)] \\;,\n\\end{equation}\nwhere\n\\begin{equation}\\label{A_B}\nA = s^2t^2 + t^2u^2 + u^2s^2 \\;, \\quad B = stu\\,m_Z^2 \\;.\n\\end{equation}\nWith a help of relations \\eqref{couplings_G-R_B}, we get from\n\\eqref{M2} the differential cross section\n\\begin{align}\\label{M2_B}\n\\frac{d\\sigma_{\\gamma\\gamma\\rightarrow\\gamma Z}}{d\\Omega} &=\n\\frac{\\beta}{64\\pi^2 s} \\frac{1}{4}\n\\sum_{\\lambda_1\\ldots\\lambda_4}|M_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}|^2\n\\nonumber \\\\\n&= \\frac{\\beta}{16\\pi^2 s}\\,[(3\\zeta^2 + 3\\tilde{\\zeta}^2 - 2\\zeta\n\\tilde{\\zeta})A + 2(\\zeta^2 + \\tilde{\\zeta}^2)B] \\;,\n\\end{align}\nwhere $\\beta = 1 - m_Z^2\/s$, in a full agreement with eq.~(2.3) in\n\\cite{Baldenegro:2017}.\n\nTo estimate a SM background, we take analytical expressions for the\nSM helicity amplitudes from Appendix~A in \\cite{Gounaris:1999_1}.\nBoth $W$ boson loops \\cite{Bailargeon:1991,Jikia:1994} and charged\nfermion loops \\cite{Bailargeon:1991,Bij:1989} contribute to these\namplitudes. As shown in \\cite{Gounaris:1999_1}, for $s > (250 \\\n\\mathrm{GeV})^2$ the dominant SM amplitudes\n$A_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}$ are the $W$-loop non-flip\namplitudes $A^W_{++++}(s,t,u)$ and $A^W_{+-+-}(s,t,u) =\nA^W_{+--+}(s,u,t)$. Almost negligible are $A^W_{+++0}(s,t,u)$ and\n$A^W_{+-+0}(s,t,u) = A^W_{+--0}(s,u,t)$. The rest are even smaller.\nThe fermion-loop amplitudes are comparable only to very small\n$W$-loop amplitudes \\cite{Gounaris:1999_1}. Similar properties of\nthe SM helicity amplitudes are also valid for the process\n$\\gamma\\gamma\\rightarrow\\gamma\\gamma$ \\cite{Gounaris:1999_2}.\n\nAnother possible background comes from the SM process $\\gamma\\gamma\n\\rightarrow \\gamma l^+l^-$ where the invariant mass of the lepton\npair, $m_{l^+l^-}$, is close to the Z boson mass $m_Z$. We have\nobtained the cross section of the process to be of order 10$^{-3}$\nfb for $|m_{l^+l^-} - m_Z| < 10$ GeV. So, this background can be\nsafely ignored.\n\n\n\\section{Numerical results}\n\\label{sec:numerical_results}\n\nThe differential cross section of the process\n$\\gamma\\gamma\\rightarrow\\gamma Z$ depends on spectra of the Compton\nbackscattered (CB) photons $f_{\\gamma\/e}(x_i)$, their helicities\n$\\xi(E_\\gamma^{(i)}, \\lambda_0)$ ($i = 1,2$), and helicity\namplitudes \\cite{Inan_Kisselev:2021,Sahin:2009},\n\\begin{align}\\label{diff_cs}\n\\frac{d\\sigma}{d\\cos \\theta} &= \\frac{\\beta}{128\\pi s}\n\\int\\limits_{x_{1 \\min}}^{x_{\\max}} \\!\\!\\frac{dx_1}{x_1}\n\\,f_{\\gamma\/e}(x_1) \\int\\limits_{x_{2 \\min}}^{x_{\\max}}\n\\!\\!\\frac{dx_2}{x_2} \\,f_{\\gamma\/e}(x_2)\n\\nonumber \\\\\n&\\times \\bigg\\{ \\left[ 1 + \\xi \\left( E_\\gamma^{(1)},\\lambda_0^{(1)}\n\\right) \\right] \\left[ 1 + \\xi \\left( E_\\gamma^{(2)},\\lambda_0^{(2)}\n\\right) \\right] \\sum_{\\lambda_3 \\lambda_4}|M_{++\\lambda_3\n\\lambda_4}|^2\n\\nonumber \\\\\n&\\quad + \\left[ 1 + \\xi \\left( E_\\gamma^{(1)},\\lambda_0^{(1)}\n\\right) \\right] \\left[ 1 - \\xi \\left( E_\\gamma^{(2)},\\lambda_0^{(2)}\n\\right) \\right] \\sum_{\\lambda_3 \\lambda_4} |M_{+-\\lambda_3\n\\lambda_4}|^2\n\\nonumber \\\\\n&\\quad + \\left[ 1 - \\xi \\left( E_\\gamma^{(1)},\\lambda_0^{(1)}\n\\right) \\right] \\left[ 1 + \\xi \\left( E_\\gamma^{(2)},\\lambda_0^{(2)}\n\\right) \\right] \\sum_{\\lambda_3 \\lambda_4}|M_{-+\\lambda_3\n\\lambda_4}|^2\n\\nonumber \\\\\n&\\quad + \\left[ 1 - \\xi \\left( E_\\gamma^{(1)},\\lambda_0^{(1)}\n\\right) \\right] \\left[ 1 - \\xi \\left( E_\\gamma^{(2)},\\lambda_0^{(2)}\n\\right) \\right] \\sum_{\\lambda_3 \\lambda_4} |M_{--\\lambda_3\n\\lambda_4}|^2 \\bigg\\} ,\n\\end{align}\nwhere $\\lambda_3 = +, -$, $\\lambda_4 = +, -, 0$, $x_1 =\nE_{\\gamma}^{(1)}\/E_e$ and $x_2 = E_{\\gamma}^{(2)}\/E_e$ are the\nenergy fractions of the CB photon beams, $x_{1 \\min} =\np_\\bot^2\/E_e^2$, $x_{2 \\min} = p_\\bot^2\/(x_{1} E_e^2)$, and\n$p_{\\bot}$ is the transverse momentum of the outgoing particles.\nNote that $\\sqrt{s x_1 x_2}$ is the invariant energy of the\nbackscattered photons. The explicit expressions for\n$f_{\\gamma\/e}(x_i)$ and $\\xi(E_\\gamma^{(i)}, \\lambda_0)$ can be\nfound in \\cite{Inan_Kisselev:2021}.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\hspace*{-0.4cm}\n\\includegraphics[scale=0.6]{WTDE750.eps}\n\\caption{The differential cross sections for the process\n$\\gamma\\gamma\\rightarrow\\gamma Z$ as functions of the invariant mass\nof the outgoing bosons for the CLIC energy $\\sqrt{s} = 1500$ GeV.\nThe left, middle and right panels correspond to the electron beam\nhelicities $\\lambda_e = 0.8, -0.8$, and 0, respectively. On each\nplot the curves denote (from the top downwards) the differential\ncross sections for the couplings $g_1 = 10^{-12} \\mathrm{\\\nGeV}^{-4}$, $g_2 = 0$, and $g_1 = 0$, $g_2 = 10^{-12} \\mathrm{\\\nGeV}^{-4}$, the anomalous contributions for the same values of\ncouplings, the SM cross section.}\n\\label{fig:WTDE750}\n\\end{center}\n\\end{figure}\n\\begin{figure}[htb]\n\\begin{center}\n\\hspace*{-0.4cm}\n\\includegraphics[scale=0.6]{WTDE1500F.eps}\n\\caption{The same as in Fig.~\\ref{fig:WTDE750}, but for the $e^+e^-$\ncollider energy $\\sqrt{s} = 3000$ GeV, and coupling sets $g_1 =\n10^{-13} \\mathrm{\\ GeV}^{-4}$, $g_2 = 0$, and $g_1 = 0$, $g_2 =\n10^{-13} \\mathrm{\\ GeV}^{-4}$.} \\label{fig:WTDE1500F}\n\\end{center}\n\\end{figure}\n\nThe differential cross sections are shown in\nFigs.~\\ref{fig:WTDE750}, \\ref{fig:WTDE1500F} as functions of the\ninvariant mass of the $\\gamma Z$ system. We have imposed the cut on\nthe rapidity of the final bosons, $|\\eta| < 2.5$, and considered the\nregion $m_{\\gamma Z} > 250$ GeV. As one can see, the anomalous cross\nsections dominate the SM one for $m_{\\gamma Z} > 600$ GeV. The\neffect is more pronounced for the collision energy $\\sqrt{s} = 3000$\nGeV, especially as $m_{\\gamma Z}$ grows. Note that for $\\sqrt{s} =\n3000$ GeV the differential cross sections depend weakly on electron\nbeam helicity $\\lambda_e$. In Figs.~\\ref{fig:WCUTE750},\n\\ref{fig:WCUTE1500F} the total cross sections are presented\ndepending on $m_{\\gamma Z, \\mathrm{min}}$, minimal invariant mass of\ntwo outgoing bosons. The anomalous contribution dominates both the\ninterference one and SM cross section. The ratio of the total cross\nsection to the SM one grows with an increase of $m_{\\gamma Z}$,\nbeing more than one order of magnitude at large $m_{\\gamma Z}$.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[scale=0.6]{WCUTE750.eps}\n\\caption{The total cross sections for the process\n$\\gamma\\gamma\\rightarrow\\gamma Z$ as functions of the minimal\ninvariant mass of the outgoing bosons for the $e^+e^-$ collider\nenergy $\\sqrt{s} = 1500$ GeV. The left, middle and right panels\ncorrespond to the electron beam helicities $\\lambda_e = 0.8, -0.8$,\nand 0, respectively. On each plot the curves denote (from the top\ndownwards) the total cross sections for the couplings $g_1 =\n10^{-12} \\mathrm{\\ GeV}^{-4}$, $g_2 = 0$, and $g_1 = 0$, $g_2 =\n10^{-12} \\mathrm{\\ GeV}^{-4}$, the anomalous contributions for the\nsame values of couplings, the SM cross section.}\n\\label{fig:WCUTE750}\n\\end{center}\n\\end{figure}\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[scale=0.6]{WCUTE1500F.eps}\n\\caption{The same as in Fig.~\\ref{fig:WCUTE750}, but for the\n$e^+e^-$ collider energy $\\sqrt{s} = 3000$ GeV, and coupling sets\n$g_1 = 10^{-13} \\mathrm{\\ GeV}^{-4}$, $g_2 = 0$, and $g_1 = 0$, $g_2\n= 10^{-13} \\mathrm{\\ GeV}^{-4}$.}\n\\label{fig:WCUTE1500F}\n\\end{center}\n\\end{figure}\n\nThe knowledge of the total cross sections and planned CLIC\nintegrated luminosities \\cite{CLIC_lum} enables us to calculate the\nexclusion regions for the QGCs. In our study we consider leptonic\n(electrons and muons) decays of the $Z$ boson. Let $s(b)$ be the\ntotal number of signal (background) events, and $\\delta$ the\npercentage systematic error. The number of events is defined as\n$\\sigma \\times L \\times \\mathcal{B}(Z \\rightarrow e, \\mu)$. The\nexclusion significance is given by \\cite{Zhang:2020}\n\\begin{equation}\\label{S_excl}\nS_{\\mathrm{excl}} = \\sqrt{ 2\\left[ s - b \\ln \\left( \\frac{b + s +\nx}{2b} \\right) - \\frac{1}{\\delta^2}\\ln \\left( \\frac{b - s + x}{2b}\n\\right) - (b + s -x) \\left( 1 + \\frac{1}{\\delta^2 b} \\right) \\right]\n} \\;,\n\\end{equation}\nwhere\n\\begin{equation}\\label{x}\nx = \\sqrt{(s+b)^2 - 4\\delta^2 s b^2\/(1 + \\delta^2 b)} \\;.\n\\end{equation}\nWe define the regions $S_{\\mathrm{excl}} \\leqslant 1.645$ as a\nregions that can be excluded at the 95\\% C.L. in the process\n$\\gamma\\gamma \\rightarrow \\gamma Z$ at the CLIC. To reduce the SM\nbackground, we impose the cut $m_{\\gamma Z} > 1000$ GeV, in addition\nto the bound $|\\eta| < 2.5$. The expected integrated luminosity at\nthe CLIC can be found, for instance, in \\cite{CLIC_lum}.\n\nIt is worth considering the unpolarized case first. One can obtain\nfrom eq.~\\eqref{M2} that the anomalous contribution to the\nunpolarized total cross section is proportional to the coupling\ncombination $3g_1^2 - 4g_1 g_2 + 4g_2^2$, provided terms\nproportional to $m_Z^2\/s \\ll 1$ are neglected in it. In such a case,\nthe exclusion regions are ellipses in the plane $(g_1 - g_2)$\nrotated clockwise through the angle $0.5 \\arctan 8 \\simeq\n41.4^\\circ$ around the origin. It is clear that our process is\nslightly more sensitive to the coupling $g_2$ rather than to $g_1$.\nOur 95\\% C.L. exclusion regions for anomalous QGCs for the\nunpolarized process $\\gamma\\gamma\\rightarrow\\gamma Z$ at the CLIC\nare shown in Figs.~\\ref{fig:excl_750}, \\ref{fig:excl_1500}. The\nresults are presented for $\\delta = 0$, $\\delta = 5\\%$, and $\\delta\n= 10\\%$.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[scale=0.6]{exclusion_E750.eps}\n\\caption{The 95\\% C.L. exclusion regions for the couplings $g_1,\ng_2$ in the unpolarized reaction $\\gamma\\gamma \\rightarrow \\gamma Z$\nat the CLIC with the systematic errors $\\delta = 0\\%$ (black\nellipse), $\\delta = 5\\%$ (blue ellipse), and $\\delta = 10\\%$ (red\nellipse). The inner regions of the ellipses are inaccessible. The\ncollision energy is $\\sqrt{s} = 1500$ GeV, the integrated luminosity\nis $L = 2500$ fb$^{-1}$. The cut on the outgoing photon invariant\nmass $m_{\\gamma\\gamma} > 1000$ GeV was imposed.}\n\\label{fig:excl_750}\n\\end{center}\n\\end{figure}\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[scale=0.6]{exclusion_E1500.eps}\n\\caption{The same as in Fig.~\\ref{fig:excl_750}, but for $\\sqrt{s} =\n3000$ GeV and $L=5000$ fb$^{-1}$.}\n\\label{fig:excl_1500}\n\\end{center}\n\\end{figure}\n\nIn Tabs.~\\ref{tab:excl_750}, \\ref{tab:excl_1500} we show the\nexclusion bounds on the couplings $g_1$ and $g_2$ for three values\nof the electron beam helicity $\\lambda_e$ and corresponding\nintegrated luminosity $L$. Let us underline that this time we did\nnot neglect the terms proportional to $m_Z^2$, both for unpolarized\nand for polarized reactions. As one can see, the best bound on the\ncouplings $g_{1,2}$ is approximately $5 \\times 10^{-15}$ GeV$^{-4}$\nfor the $e^+e^-$ energy $\\sqrt{s} = 3000$ GeV and electron beam\nhelicity $\\lambda_e = 0.8$.\n\n\\begin{table}[h]\n    \\centering \\caption{The 95\\% C.L. exclusion limits on the anomalous\n        quartic couplings $g_1$ and $g_2$  for the collision energy\n        $\\sqrt{s} = 1500$ GeV, and the cut $m_{\\gamma Z} > 1000$ GeV.\n        \\bigskip} \\label{tab:excl_750}\n    \\begin{tabular}{||c||c|c|c|c||}\n        \\hline\n        $\\lambda_e$ & & 0 & $-0.8$ & 0.8 \\\\\n        \\hline\n        \\quad $L$, fb$^{-1}$ & & 2500 & 2000 & 500 \\\\\n        \\hline\n        \\makecell{$|g_1|$, GeV$^{-4}$ \\\\ $(g_2=0)$} & \\makecell{$\\delta=0\\%$ \\\\  $\\delta=5\\%$ \\\\ $\\ \\, \\delta=10\\%$ } &\n        $\\makecell{ 4.19\\times10^{-14} \\\\ 5.32\\times10^{-14} \\\\ 6.81\\times10^{-14}}$ &\n        $\\makecell{ 6.25\\times10^{-14} \\\\ 7.91\\times10^{-14} \\\\ 1.02\\times10^{-13}}$ &\n        $\\makecell{ 4.42\\times10^{-14} \\\\ 5.38\\times10^{-14} \\\\ 6.78\\times10^{-14}}$ \\\\\n        \\hline\n        \\makecell{$|g_2|$, GeV$^{-4}$ \\\\  $(g_1=0)$} & \\makecell{$\\delta=0\\%$ \\\\ $\\delta=5\\%$ \\\\ $\\ \\, \\delta=10\\%$ } &\n        $\\makecell{ 3.61\\times10^{-14} \\\\ 4.63\\times10^{-14} \\\\ 5.91\\times10^{-14}}$ &\n        $\\makecell{ 5.47\\times10^{-14} \\\\ 6.94\\times10^{-14} \\\\ 8.87\\times10^{-14}}$ &\n        $\\makecell{ 4.53\\times10^{-14} \\\\ 5.51\\times10^{-14} \\\\ 6.94\\times10^{-14}}$ \\\\\n        \\hline\n    \\end{tabular}\n\\end{table}\n\\begin{table}[h]\n    \\centering \\caption{The same as in Tab.~\\ref{tab:excl_750}, but for\n        the energy $\\sqrt{s} = 3000$ GeV and different values of the\n        integrated luminosities. \\bigskip} \\label{tab:excl_1500}\n    \\begin{tabular}{||c||c|c|c|c||}\n        \\hline\n        $\\lambda_e$ & & 0 & $-0.8$ & 0.8 \\\\\n        \\hline\n        \\quad $L$, fb$^{-1}$ & & 5000 & 4000 & 1000 \\\\\n        \\hline\n        \\makecell{$|g_1|$, GeV$^{-4}$ \\\\ $(g_2=0)$ } & \\makecell{$\\delta=0\\%$ \\\\ $\\delta=5\\%$ \\\\ $\\ \\, \\delta=10\\%$ } &\n        $\\makecell{ 5.98\\times10^{-15} \\\\ 1.33\\times10^{-14} \\\\ 1.85\\times10^{-14}}$ &\n        $\\makecell{ 7.14\\times10^{-15} \\\\ 1.73\\times10^{-14} \\\\ 2.39\\times10^{-14}}$ &\n        $\\makecell{ 5.13\\times10^{-15} \\\\ 7.79\\times10^{-15} \\\\ 1.04\\times10^{-14}}$ \\\\\n        \\hline\n        \\makecell{$|g_2|$, GeV$^{-4}$ \\\\ $(g_1=0)$} & \\makecell{$\\delta=0\\%$ \\\\ $\\delta=5\\%$ \\\\ $\\ \\, \\delta=10\\%$ } &\n        $\\makecell{ 5.18\\times10^{-15} \\\\ 1.16\\times10^{-14} \\\\ 1.62\\times10^{-14}}$ &\n        $\\makecell{ 6.62\\times10^{-15} \\\\ 1.60\\times10^{-14} \\\\ 2.21\\times10^{-14}}$ &\n        $\\makecell{ 5.19\\times10^{-15} \\\\ 7.87\\times10^{-15} \\\\ 1.05\\times10^{-14}}$ \\\\\n        \\hline\n    \\end{tabular}\n\\end{table}\n\nRecently, the bounds on the anomalous quartic couplings for the\nvertex $\\gamma\\gamma\\gamma Z$ were obtained via $\\gamma Z$\nproduction with intact protons in the forward region at the LHC\n\\cite{Baldenegro:2017}. To examine this process, the effective\nLagrangian \\eqref{Lagrangian_B} was used with the anomalous\ncouplings $\\zeta, \\tilde{\\zeta}$. Both for integrated luminosity 300\nfb$^{-1}$ and high luminosity 3000 fb$^{-1}$ sensitivities were\nfound to be similar, $\\zeta, \\tilde{\\zeta} \\sim 1 \\times 10^{-13}$\nat the 95\\% C.L. Taking into account the relations between couplings\n$\\zeta, \\tilde{\\zeta}$ and our couplings $g_1, g_2$,\n\\eqref{couplings_G-R_B}, we expect that the sensitivities of $g_1,\ng_2 \\sim 8\\times 10^{-13}$ can be reached at the LHC (HL-LHC). These\nvalues should be compared with the CLIC bounds in\nTabs.~\\ref{tab:excl_750} and \\ref{tab:excl_1500}. Note that the\nexpected sensitivity from the $Z\\rightarrow\\gamma\\gamma\\gamma$ decay\nsearch at the LHC \\cite{ATLAS:Z_decay} is approximately three orders\nof magnitude smaller than that obtained in \\cite{Baldenegro:2017}.\n\n\n\\section{Unitarity constraints on anomalous quartic couplings}\n\\label{sec:unit_const}\n\nThe anomalous contribution to the total cross section rises as\n$s^3$. Thus, the contribution of the effective operators in\n\\eqref{Lagrangian} may lead to unitarity violation at high energies.\nThat is why we need to study bounds imposed by partial-wave\nunitarity. The partial-wave expansion of the helicity amplitude in\nthe center-of-mass system was derived in \\cite{Jacob:2000} and used\nin a number of papers \\cite{Gounaris:1994}. It looks like\n\\begin{align}\\label{helicity_ampl_expansion}\nM_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}(s, \\theta, \\varphi) &=\n16\\pi \\sum_J (2J + 1) \\sqrt{(1 + \\delta_{\\lambda_1\\lambda_2})(1 +\n\\delta_{\\lambda_3\\lambda_4})}\n\\nonumber \\\\\n&\\times \\,e^{i(\\lambda - \\mu)\\phi} \\,d^J_{\\lambda\\mu}(\\theta)\n\\,T^J_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}(s) \\;,\n\\end{align}\nwhere $\\lambda = \\lambda_1 - \\lambda_2$, $\\mu = \\lambda_3 -\n\\lambda_4$, $\\theta(\\phi)$ is the polar (azimuth) scattering angle,\nand $d^J_{\\lambda\\mu}(\\theta)$ is the Wigner (small) $d$-function\n\\cite{Wigner}. Relevant formulas for the $d$-functions are given in\nAppendix~C. In our case $\\lambda, \\mu$ are even numbers, $\\lambda,\n\\,\\mu = 0, \\pm 2$ (see below). If we choose the plane $(x - z)$ as a\nscattering plane, then $\\phi = 0$ in\n\\eqref{helicity_ampl_expansion}. Parity conservation means that\n\\begin{equation}\\label{parity_conservation}\nT^J_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}(s) = (-1)^{\\lambda_1 -\n\\lambda_2 - \\lambda_3 + \\lambda_4}\n\\,T^J_{-\\lambda_1-\\lambda_2-\\lambda_3-\\lambda_4}(s) \\;.\n\\end{equation}\nPartial-wave unitarity in the limit $s \\gg (m_1 + m_2)^2$ requires\nthat\n\\begin{equation}\\label{parity_wave_unitarity}\n\\left| T^J_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}(s) \\right| \\leq 1\n\\;.\n\\end{equation}\nUsing orthogonality of the $d$-functions\n\\eqref{Wigner_func_orthogonality}, we find the partial-wave\namplitude\n\\begin{align}\\label{parity wave_func}\nT^J_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}(s) &= \\frac{1}{32\\pi}\n\\frac{1}{\\sqrt{(1 + \\delta_{\\lambda_1\\lambda_2})(1 +\n\\delta_{\\lambda_3\\lambda_4})}} \\int\\limits_{-1}^1 \\!\\!\nM_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}(s, z)\n\\,d^{J}_{\\lambda\\mu}(z) \\,dz \\;.\n\\end{align}\nHere and in what follows, $z = \\cos\\theta$. Note that\n$M_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4} = g_1\nM^{(1)}_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4} + g_2\nM^{(2)}_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}$, and the helicity\namplitudes $M^{(1,2)}_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}$ are\ngiven in Appendix~B.\n\n\n\\subsection{Unitarity bounds on coupling $\\bs{g_1}$ ($\\bs{g_2 = 0}$)}\n\nTo obtain a unitarity bound on the coupling $g_1$, we put $g_2 = 0$.\nLet us note that due to eq.~\\eqref{parity_conservation}, it is\nsufficient to examine the helicity amplitudes with $\\lambda_1 = +1$\nonly. Moreover, it is enough to consider four amplitudes,\n$M^{(1)}_{++++}$, $M^{(1)}_{+-+-}$, $M^{(1)}_{+--+}$, and\n$M^{(1)}_{++--}$, since the rest are suppressed by small factor\n$m_Z\/\\sqrt{s}$ or zero. \\\\\n\n\\textbf{1.} $\\lambda_1 = \\lambda_2 = \\lambda_3 = \\lambda_4 = 1$,\nthen $\\lambda = \\mu = 0$. The helicity amplitude is given by the\nfirst of equations \\eqref{independent_ampl_M1},\n\\begin{equation}\\label{M1_++++}\nM_{++++}(s,z) = g_1 M^{(1)}_{++++}(s) = -\\frac{g_1 }{4} \\,s(s -\nm_Z^2) \\;.\n\\end{equation}\nUsing eqs.~\\eqref{d-function_00}-\\eqref{integral_Legendre}, we find\nthat the partial-wave amplitude with $J=0$ is the only non-zero\namplitude,\n\\begin{equation}\\label{T1_++++}\nT^0_{++++}(s) = - \\frac{g_1}{128\\pi} \\,s(s - m_Z^2)\n\\int\\limits_{-1}^1 d^{\\,0}_{00}(z) \\,dz = - \\frac{g_1}{64\\pi} \\,s(s\n- m_Z^2) \\;.\n\\end{equation}\nCorrespondingly, we obtain from \\eqref{parity_wave_unitarity},\n\\eqref{T1_++++}\n\\begin{equation}\\label{T1++++_unitarity}\n|g_1| \\leq 64\\pi [s(s - m_Z^2)]^{-1} \\;.\n\\end{equation}\n\n\\textbf{2.} $\\lambda_1 = -\\lambda_2 = \\lambda_3 = -\\lambda_4 = 1$,\nthen $\\lambda = \\mu = 2$. According to \\eqref{independent_ampl_M1},\n\\begin{equation}\\label{M1_+-+-}\nM_{+-+-}(s,z) = g_1 M^{(1)}_{+-+-}(s,z) = -\\frac{g_1}{4}\n\\,\\frac{s^3}{s - m_Z^2} \\left( \\frac{1 + z}{2}\\right)^{\\!2} .\n\\end{equation}\nIt follows from \\eqref{parity wave_func}, \\eqref{d-function_22} that\n\\begin{align}\\label{T1+-+-_J}\nT^J_{+-+-}(s) &= -\\frac{g_1}{128\\pi} \\,\\frac{s^3}{s - m_Z^2}\n\\int\\limits_{-1}^1 \\left( \\frac{1 + z}{2}\\right)^{\\!2}\n\\,d^{J}_{22}(z) \\,dz = -\\frac{g_1}{128\\pi} \\,\\frac{s^3}{s - m_Z^2}\n\\nonumber \\\\\n&\\times \\int\\limits_{-1}^1 \\left( \\frac{1 + z}{2}\\right)^{\\!4}\n{}_2F_1 \\left( 2-J, J+3; 1; \\frac{1-z}{2} \\right) dz \\;.\n\\end{align}\nLet $(1 - z)\/2 = x$, then $(1 + z)\/2 = 1 - x$, and we find\n\\begin{align}\\label{}\nT^J_{+-+-}(s) &= -\\frac{1}{64\\pi} \\,\\frac{g_1 s^3}{s - m_Z^2}\n\\int\\limits_0^1 (1 - x)^4 \\,{}_2F_1( 2-J, J+3; 1; x) \\,dx\n\\nonumber \\\\\n&= -\\frac{3g_1}{8\\pi\\Gamma(4+J)\\Gamma(3-J)} \\frac{s^3}{s - m_Z^2}\n\\;,\n\\end{align}\nwhere we used formulas 2.21.1.5 and 2.21.1.6 in \\cite{Prudnikov_v3}.\nThus, only three partial-waves amplitudes, $T^0_{+-+-}(s)$,\n$T^1_{+-+-}(s)$, and $T^2_{+-+-}(s)$, are non-zero. The most\nimportant for us is $T^0_{+-+-}(s)$, since it results in the\nstrongest constraint on the coupling $g_1$,\n\\begin{equation}\\label{T1+-+-_unitarity}\n|g_1| \\leq 32\\,\\pi (s - m_Z^2) s^{-3}\\;.\n\\end{equation}\n\n\\textbf{3.} $\\lambda_1 = -\\lambda_2 = -\\lambda_3 = \\lambda_4 = 1$,\nthen $\\lambda = 2, \\,\\mu = -2$, and we have\n\\begin{equation}\\label{M1_+--+}\nM_{+--+}(s,z) = g_1 M^{(1)}_{+--+}(s,z) = -\\frac{g_1}{4}\n\\,\\frac{s^3}{s - m_Z^2} \\left( \\frac{1 - z}{2}\\right)^{\\!2} .\n\\end{equation}\nUsing eq.~\\eqref{d-function_2-2} and first relation in\n\\eqref{d-function_symmetry}, after substitutions $(1 + z)\/2 = x$,\n$(1 - z)\/2 = 1 - x$, we reduce this case to the previous one. As a\nresult, we come again to the upper bound \\eqref{T1+-+-_unitarity}.\n\n\\textbf{4.} $\\lambda_1 = \\lambda_2 = -\\lambda_3 = -\\lambda_4 = 1$,\nthen $\\lambda = \\mu = 0$, and\n\\begin{equation}\\label{M1_++--}\nM_{++--}(s,z) = g_1 M^{(1)}_{++--}(s,z) = -\\frac{g_1}{8}\n\\,\\frac{s^3}{s - m_Z^2} (3 + z^2) \\;.\n\\end{equation}\nOnly two partial-waves amplitudes, $T^0_{++--}(s)$ and\n$T^2_{++--}(s)$, are non-zero,\n\\begin{equation}\\label{T1_++--}\nT^0_{++--}(s) =  -\\frac{5\\,g_1}{192\\pi} \\,\\frac{s^3}{s - m_Z^2} \\;,\n\\quad T^2_{++--}(s) =  -\\frac{g_1}{960\\pi} \\,\\frac{s^3}{s - m_Z^2}\n\\;.\n\\end{equation}\nThe strongest bound on $g_1$ comes from unitarity constraint on\n$T^0_{++--}(s)$,\n\\begin{equation}\\label{T1++--_unitarity}\n|g_1| \\leq \\frac{192\\,\\pi}{5} (s - m_Z^2) s^{-3} \\;.\n\\end{equation}\n\n\n\\subsection{Unitarity bounds on coupling $\\bs{g_2}$ ($\\bs{g_1 = 0}$)}\n\nTo derive a unitarity bound on the coupling $g_2$, we take $g_1 =\n0$. It is sufficient to consider three amplitudes, $M^{(2)}_{++++}$,\n$M^{(2)}_{+-+-}$, and $M^{(2)}_{+--+}$. The rest are suppressed by\nsmall factor $m_Z\/\\sqrt{s}$ or zero. \\\\\n\n\\textbf{1.} $\\lambda_1 = \\lambda_2 = \\lambda_3 = \\lambda_4 = 1$,\nthen $\\lambda = \\mu = 0$. The helicity amplitude is given by the\nfirst of equations \\eqref{independent_ampl_M2}\n\\begin{equation}\\label{M2_++++}\nM_{++++}(s,z) = g_2 M^{(2)}_{++++}(s) = \\frac{g_2}{2} \\,s(s - m_Z^2)\n\\;.\n\\end{equation}\nAs a result, we get\n\\begin{equation}\\label{T2++++_unitarity}\n|g_2| \\leq 32\\pi [s(s - m_Z^2)]^{-1} \\;.\n\\end{equation}\n\n\\textbf{2.} $\\lambda_1 = -\\lambda_2 = \\lambda_3 = -\\lambda_4 = 1$,\nthen $\\lambda = \\mu = 2$, and we find from\n\\eqref{independent_ampl_M2}\n\\begin{equation}\\label{M2_+-+-}\nM_{+-+-}(s,z) = g_2 M^{(2)}_{+-+-}(s,z) = \\frac{g_2}{2}\n\\,\\frac{s^3}{s - m_Z^2} \\left( \\frac{1 + z}{2}\\right)^{\\!2} .\n\\end{equation}\nWe follow the derivation of eq.~\\eqref{T1+-+-_unitarity} and come to\nthe inequality\n\\begin{equation}\\label{T2+-+-_unitarity}\n|g_2|  \\leq 16\\pi (s - m_Z^2) s^{-3} \\;.\n\\end{equation}\n\n\\textbf{3.} $\\lambda_1 = -\\lambda_2 = -\\lambda_3 = \\lambda_4 = 1$,\nthen $\\lambda = 2, \\,\\mu = -2$, and we obtain\n\\begin{equation}\\label{M2_+--+}\nM_{+--+}(s,z) = g_2 M^{(2)}_{+--+}(s,z) = \\frac{g_2}{2}\n\\,\\frac{s^3}{s - m_Z^2} \\left( \\frac{1 - z}{2}\\right)^{\\!2} .\n\\end{equation}\nUsing the first relation in \\eqref{d-function_symmetry} and\neq.~\\eqref{d-function_2-2}, we can reduce this case  to the previous\none to get eq.~\\eqref{T2+-+-_unitarity}.\n\n\\textbf{4.} $\\lambda_1 = \\lambda_2 = -\\lambda_3 = -\\lambda_4 = 1$.\nThe helicity amplitude $M^{(2)}_{++--}(s,z) = 0$.\n\n\n\\subsection{Unitarity bounds on couplings $\\bs{g_1}$ and $\\bs{g_2}$}\n\nNow we consider a general case with $g_1, g_2 \\neq 0$. Note that\n$M^{(2)}_{++++} = - 2M^{(1)}_{++++}$, $M^{(2)}_{+-+-} =\n-2M^{(1)}_{+-+-}$, and $M^{(2)}_{+--+} = - 2M^{(1)}_{+--+}$.\nCorrespondingly, $M_{++++} = (g_1 - 2g_2)M^{(1)}_{++++}$, etc. From\nformulas derived in two previous subsections we immediately get the\nfollowing bound on a linear combination of $g_1$ and $g_2$,\n\\begin{equation}\\label{g1_g2_unitarity}\n|g_1 - 2g_2| \\leq 32\\,\\pi (s - m_Z^2) s^{-3} \\;.\n\\end{equation}\nLet us underline that $M_{++--}(s,z) = g_1 M^{(1)}_{++--}(s,z)$. It\nmeans that inequality \\eqref{T1++--_unitarity} holds for a general\ncase ($g_1, g_2 \\neq 0$). It enables us to obtain constraints\nseparately on each coupling. If the couplings $g_1$, $g_2$ have the\nsame sign, then\n\\begin{equation}\\label{g1_g2_same_signs}\n|g_1| \\leq  \\frac{192\\,\\pi}{5} (s - m_Z^2) s^{-3}  \\;, \\quad |g_2|\n\\leq \\frac{176\\,\\pi}{5}(s - m_Z^2) s^{-3}  \\;.\n\\end{equation}\nIf the signs of the couplings $g_1$, $g_2$ are opposite, we obtain\n\\begin{equation}\\label{g1_g2_opposite_signs}\n|g_1| \\leq  32\\,\\pi \\,(s - m_Z^2) s^{-3} \\;, \\quad |g_2| \\leq\n16\\,\\pi \\,(s - m_Z^2) s^{-3}  \\;.\n\\end{equation}\nThe bounds on the couplings $g_1, g_2$ along with their numerical\nvalues are collected in Tab.~\\ref{tab:unit_lim}. We have taken into\naccount that $m_Z^2\/s \\ll 1$ for the CLIC energies.\n\n{\\setlength{\\extrarowheight}{4pt}\n\\begin{table}[h]\n\\centering \\caption{Unitarity constraints on the anomalous couplings\nwhen just one coupling is non-zero (second and third columns), and\nwhen both couplings are non-vanishing (fourth and fifth columns for\nthe couplings of the same sign, sixth and seventh columns for the\ncouplings of opposite signs). The numerical values of the bounds are\ngiven for the collision energy $\\sqrt{s} = 1500(3000)$ GeV.\n\\bigskip} \\label{tab:unit_lim}\n\\begin{tabular}{||c||c|c||c|c||c|c||}\n  \\hline\n\\multicolumn{1}{||c||} {} & \\multicolumn{2}{c||} {1 operator ($g_2 =\n0$ or $g_1 = 0$)} & \\multicolumn{2}{c||} {2 operators ($g_1 g_2 >\n0$)} & \\multicolumn{2}{c||} {2 operators ($g_1 g_2 < 0$)}  \\\\\n  \\hline\n  $g_1$ & $32\\pi s^{-2}$ & 20(1.2) TeV$^{-4}$ & $\\frac{192}{5}s^{-2}$\n  & 24(1.5) TeV$^{-4}$ & $32\\pi s^{-2}$ & 20(1.2) TeV$^{-4}$ \\\\ [2pt]\n  \\hline\n  $g_2$  & $16\\pi s^{-2}$ & 10(0.6) TeV$^{-4}$ & $\\frac{176}{5}s^{-2}$\n  & 22(1.4) TeV$^{-4}$ & $16\\pi s^{-2}$ & 10(0.6) TeV$^{-4}$ \\\\ [2pt]\n  \\hline\n\\end{tabular}\n\\end{table}\n}\n\nTo summarize, in spite of the fact that the anomalous contribution\nto the total cross section is proportional to $s^3$, the unitarity\nis not violated in the region of the anomalous QGCs presented in\nTabs.~\\ref{tab:excl_750}, \\ref{tab:excl_1500}.\n\n\n\\section{Conclusions}\n\\label{sec:concl}\n\nIn the present paper, the CLIC discovery potential for exclusive\n$\\gamma Z$ production in the scattering of the Compton backscattered\nphotons at the $e^+e^-$ collision energies 1500 GeV and 3000 GeV is\nstudied. We have shown that such a process provides an opportunity\nof searching for the anomalous quartic neutral gauge couplings for\nthe $\\gamma\\gamma\\gamma Z$ vertex at the CLIC. Both unpolarized and\npolarized initial electron beams are examined. To describe the\nanomalous quartic gauge couplings we used the effective Lagrangian\nwhich conserves gauge invariance. Although quartic gauge couplings\nare already induced at the dimension-six level, we considered the\neffective Lagrangian with CP conserving dimension-eight operators\nwithout contributing to anomalous trilinear gauge interactions.\n\nWe have derived the explicit expressions for the anomalous\ncontributions to the helicity amplitudes of the process\n$\\gamma\\gamma \\rightarrow \\gamma Z$. After that the differential and\ntotal cross sections are calculated depending on $m_{Z\\gamma}$, the\ninvariant mass of the $\\gamma Z$ system. It is shown that the\nanomalous contribution dominates both the interference and SM cross\nsections. Moreover, the ratio of the total cross section to the SM\none grows with the increase of $m_{Z\\gamma}$, being more\napproximately one order of magnitude at large $m_{\\gamma Z}$.\n\nIt enabled us to obtain the exclusion regions for the anomalous\ncouplings with the systematic errors of 0\\%, 5\\%, and 10\\%. We have\nconsidered the $Z$ boson decay into leptons (electron and muons).\nFor both couplings, $g_{1,2}$, the best bounds are equal to\napproximately $4.4 \\times 10^{-14}$ GeV$^{-4}$ and $5.1 \\times\n10^{-15}$ GeV$^{-4}$, for the $e^+e^-$ energies 1500 GeV and 3000\nGeV, respectively. They are achieved when electron beam helicity is\nequal to 0.8. We have checked that the unitarity is not violated in\nthe region of the couplings considered in the paper. Our best bound\non the anomalous couplings for the collision energy 3000 GeV is\nroughly two orders of magnitude stronger than the limits which can\nbe reached at the LHC and HL-LHC. This points to a great potential\nof the CLIC and other future leptonic colliders to probe the\nanomalous $\\gamma\\gamma\\gamma Z$ couplings.\n\n\n\n\\setcounter{equation}{0}\n\\renewcommand{\\theequation}{A.\\arabic{equation}}\n\n\\section*{Appendix A}\n\\label{app:A}\n\nHere we present explicit expressions for components of the\npolarization tensor \\eqref{polarization_tensor}. They are the\nfollowing\n\\begin{align}\\label{P1.1}\nP_{\\mu\\nu\\rho\\alpha}^{(1.1)} &= (p_1\\cdot p_2)[ (p_1\\cdot p_3) +\n(p_2\\cdot p_3)] g_{\\mu\\nu}g_{\\rho\\alpha} + (p_1\\cdot p_3)[ (p_1\\cdot\np_2) + (p_2\\cdot p_3) ] g_{\\mu\\rho}g_{\\nu\\alpha}\n\\nonumber \\\\\n&+ (p_2\\cdot p_3)[ (p_1\\cdot p_2) + (p_1\\cdot p_3) ]\ng_{\\nu\\rho}g_{\\mu\\alpha} \\;,\n\\end{align}\n\\begin{align}\\label{P1.2}\nP_{\\mu\\nu\\rho\\alpha}^{(1.2)} &= - \\{ [ (p_1\\cdot p_2) + (p_1\\cdot\np_3)] p_{3\\nu} p_{2\\rho}g_{\\mu\\alpha} + [ (p_1\\cdot p_2) + (p_2\\cdot\np_3) ] p_{3\\mu}p_{1\\rho} g_{\\nu\\alpha}\n\\nonumber \\\\\n&\\quad + [ (p_1\\cdot p_3) + (p_2\\cdot p_3) ]\np_{2\\mu}p_{1\\nu}g_{\\rho\\alpha} \\} \\;,\n\\end{align}\n\\begin{align}\\label{P1.3}\nP_{\\mu\\nu\\rho\\alpha}^{(1.3)} &= -[ (p_1\\cdot p_2)( p_{1\\rho} +\np_{2\\rho} )p_{3\\alpha}g_{\\mu\\nu} + (p_1\\cdot p_3)( p_{1\\nu} +\np_{3\\nu} )p_{2\\alpha} g_{\\mu\\rho}\n\\nonumber \\\\\n&\\quad + (p_2\\cdot p_3)( p_{2\\mu} + p_{3\\mu} )p_{1\\alpha}g_{\\nu\\rho}\n] \\;,\n\\end{align}\n\\begin{align}\\label{P1.4}\nP_{\\mu\\nu\\rho\\alpha}^{(1.4)} &= p_{2\\mu}p_{1\\nu} ( p_{1\\rho} +\np_{2\\rho} ) p_{3\\alpha} + p_{3\\mu} ( p_{1\\nu} + p_{3\\nu} ) p_{1\\rho}\np_{2\\alpha}\n\\nonumber \\\\\n&+ p_{3\\nu} p_{2\\rho}( p_{2\\mu} + p_{3\\mu} ) p_{1\\alpha} \\;,\n\\end{align}\nand\n\\begin{align}\\label{P2.1}\nP_{\\mu\\nu\\rho\\alpha}^{(2.1)} = &- 2[(p_1\\cdot p_3)(p_2\\cdot p_3)\ng_{\\mu\\nu}g_{\\rho\\alpha} + (p_1\\cdot p_2)(p_2\\cdot p_3)\ng_{\\mu\\rho}g_{\\nu\\alpha}\n\\nonumber \\\\\n&+ (p_1\\cdot p_2)(p_1\\cdot p_3) g_{\\nu\\rho} g_{\\mu\\alpha}] \\;,\n\\end{align}\n\\begin{align}\\label{P2.2}\nP_{\\mu\\nu\\rho\\alpha}^{(2.2)} &= (p_1\\cdot\np_2)[p_{3\\mu}p_{3\\alpha}g_{\\nu\\rho} +\np_{3\\nu}p_{3\\alpha}g_{\\mu\\rho}] + (p_1\\cdot\np_3)[p_{2\\mu}p_{2\\alpha}g_{\\nu\\rho} +\np_{2\\rho}p_{2\\alpha}g_{\\mu\\nu}]\n\\nonumber \\\\\n&+ (p_2\\cdot p_3)[p_{1\\nu}p_{1\\alpha}g_{\\mu\\rho} +\np_{1\\rho}p_{1\\alpha}g_{\\mu\\nu}] \\;,\n\\end{align}\n\\begin{equation}\\label{P2.3}\nP_{\\mu\\nu\\rho\\alpha}^{(2.3)} = - 2[(p_1\\cdot p_2)p_{3\\mu}p_{3\\nu}\ng_{\\rho\\alpha} + (p_1\\cdot p_3)p_{2\\mu}p_{2\\rho} g_{\\nu\\alpha} +\n(p_2\\cdot p_3)p_{1\\nu}p_{1\\rho} g_{\\mu\\alpha}] \\;,\n\\end{equation}\n\\begin{align}\\label{P2.4}\nP_{\\mu\\nu\\rho\\alpha}^{(2.4)} &= (p_1\\cdot\np_2)[p_{3\\mu}p_{1\\alpha}g_{\\nu\\rho} +\np_{3\\nu}p_{2\\alpha}g_{\\mu\\rho}] + (p_1\\cdot\np_3)[p_{2\\mu}p_{1\\alpha}g_{\\nu\\rho} +\np_{2\\rho}p_{3\\alpha}g_{\\mu\\nu}]\n\\nonumber \\\\\n&+ (p_2\\cdot p_3)[p_{1\\nu}p_{2\\alpha}g_{\\mu\\rho} +\np_{1\\rho}p_{3\\alpha}g_{\\mu\\nu}] \\;,\n\\end{align}\n\\begin{align}\\label{P2.5}\nP_{\\mu\\nu\\rho\\alpha}^{(2.5)} &= 2\\{ (p_1\\cdot\np_2)[p_{3\\mu}p_{2\\rho}g_{\\nu\\alpha} +\np_{3\\nu}p_{1\\rho}g_{\\mu\\alpha}] + (p_1\\cdot\np_3)[p_{2\\mu}p_{3\\nu}g_{\\rho\\alpha} +\np_{2\\rho}p_{1\\nu}g_{\\mu\\alpha}]\n\\nonumber \\\\\n&+ (p_2\\cdot p_3)[p_{3\\mu}p_{1\\nu}g_{\\rho\\alpha} +\np_{2\\mu}p_{1\\rho}g_{\\nu\\alpha}] \\} \\;,\n\\end{align}\n\\begin{align}\\label{P2.6}\nP_{\\mu\\nu\\rho\\alpha}^{(2.6)} = &-\\{ (p_1\\cdot\np_2)[p_{3\\mu}p_{2\\alpha}g_{\\nu\\rho} +\np_{3\\nu}p_{1\\alpha}g_{\\mu\\rho}] + (p_1\\cdot\np_3)[p_{2\\mu}p_{3\\alpha}g_{\\nu\\rho} +\np_{2\\rho}p_{1\\alpha}g_{\\mu\\nu}]\n\\nonumber \\\\\n&+ (p_2\\cdot p_3)[p_{1\\nu}p_{3\\alpha}g_{\\mu\\rho} +\np_{1\\rho}p_{2\\alpha}g_{\\mu\\nu}] \\} \\;,\n\\end{align}\n\\begin{align}\\label{P2.7}\nP_{\\mu\\nu\\rho\\alpha}^{(2.7)} = - ( p_{2\\mu}p_{3\\nu}p_{1\\rho} +\np_{3\\mu}p_{1\\nu}p_{2\\rho} ) (p_{1\\alpha} + p_{2\\alpha} +\np_{3\\alpha}) \\;.\n\\end{align}\nNote that the last tensor does not contribute to the matrix element\n\\eqref{matrix_element}, since it is proportional to $p_{4\\alpha}$.\nOne can directly check that\n\\begin{align}\\label{polarization_tensor_gauge_inv}\np_1^\\mu \\sum_{i=1}^4 P_{\\mu\\nu\\rho\\alpha}^{(1.i)} &= p_2^\\nu\n\\sum_{i=1}^4 P_{\\mu\\nu\\rho\\alpha}^{(1.i)} = p_3^\\rho \\sum_{i=1}^4\nP_{\\mu\\nu\\rho\\alpha}^{(1.i)} = 0 \\;,\n\\nonumber \\\\\np_1^\\mu \\sum_{i=1}^7 P_{\\mu\\nu\\rho\\alpha}^{(2.i)} &= p_2^\\nu\n\\sum_{i=1}^7 P_{\\mu\\nu\\rho\\alpha}^{(2.i)} = p_3^\\rho \\sum_{i=1}^7\nP_{\\mu\\nu\\rho\\alpha}^{(2.i)} = 0 \\;.\n\\end{align}\n\n\n\n\\setcounter{equation}{0}\n\\renewcommand{\\theequation}{B.\\arabic{equation}}\n\n\\section*{Appendix B}\n\\label{app:B}\n\nIn accordance with eq.~\\eqref{polarization_tensor}, any anomalous\nhelicity amplitude is the sum of two terms,\n\\begin{equation}\\label{helicity_ampl_sum}\nM_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4} = g_1\nM_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}^{(1)} + g_2\nM_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}^{(2)} \\;.\n\\end{equation}\nThere are $2^3\\times3 = 24$ helicity amplitudes\n$M^{(1)}_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}$ and,\ncorrespondingly, 24 amplitudes\n$M^{(2)}_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}$ for the process\n\\eqref{process}. Bose-Einstein statistics and parity invariance\ndemand that there exist nine independent helicity amplitudes\n$M^{(1)}_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}$ with $\\lambda_1 =\n+1$, six for transverse $Z$ and three for longitudinal $Z$. Our\ncalculations resulted in the following helicity amplitudes\n$M^{(1)}_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}$ with $\\lambda_1 =\n+1$\n\\begin{align}\\label{independent_ampl_M1}\nM^{(1)}_{++++}(s,t,u) &= \\frac{1}{4} s(t + u) \\;,\n\\nonumber \\\\\nM^{(1)}_{+++-}(s,t,u) &= 0 \\;,\n\\nonumber \\\\\nM^{(1)}_{++-+}(s,t,u) &= \\frac{1}{2} \\frac{tu}{t + u} m_Z^2 \\;,\n\\nonumber \\\\\nM^{(1)}_{++--}(s,t,u) &= \\frac{1}{2}\\frac{s(t^2 + tu + u^2)}{t + u}\n\\;,\n\\nonumber \\\\\nM^{(1)}_{+-++}(s,t,u) &= \\frac{1}{4} \\frac{tu}{t + u} m_Z^2 \\;,\n\\nonumber \\\\\nM^{(1)}_{+-+-}(s,t,u) &= \\frac{1}{4}\\frac{su^2}{t + u} \\;,\n\\nonumber \\\\\nM^{(1)}_{+++0}(s,t,u) &= 0 \\;,\n\\nonumber \\\\\nM^{(1)}_{++-0}(s,t,u) &= \\frac{i}{2\\sqrt{2}}\\sqrt{stu} \\,\\frac{t -\nu}{t + u} \\,m_Z \\;,\n\\nonumber \\\\\nM^{(1)}_{+-+0}(s,t,u) &=  -\\frac{i}{2\\sqrt{2}}\\frac{u\\sqrt{stu}}{t +\nu} \\,m_Z \\;.\n\\end{align}\nThree more amplitudes $M^{(1)}_{+\\lambda_2\\lambda_3\\lambda_4}$ can\nbe obtained by exchanging Mandelstam variables $t$ and\n$u$~\\cite{Gounaris:1999_1,Glover:1993},\n\\begin{align}\\label{dependent_ampl_M1}\nM^{(1)}_{+--+}(s,t,u) &= M^{(1)}_{+-+-}(s,u,t) =\n\\frac{1}{4}\\frac{st^2}{t + u}\\;,\n\\nonumber \\\\\nM^{(1)}_{+---}(s,t,u) &= M^{(1)}_{+-++}(s,u,t) = \\frac{1}{4}\n\\frac{tu}{(t + u)} m_Z^2 \\;,\n\\nonumber \\\\\nM^{(1)}_{+--0}(s,t,u) &= M^{(1)}_{+-+0}(s,u,t) =\n-\\frac{i}{2\\sqrt{2}}\\frac{t\\sqrt{stu}}{t + u}\\,m_Z \\;.\n\\end{align}\n\nNine independent helicity amplitudes\n$M^{(2)}_{\\lambda_1\\lambda_2\\lambda_3\\lambda_4}$ with $\\lambda_1 =\n+1$ are\n\\begin{align}\\label{independent_ampl_M2}\nM^{(2)}_{++++}(s,t,u) &= -\\frac{1}{2} s(t + u) \\;,\n\\nonumber \\\\\nM^{(2)}_{+++-}(s,t,u) &= 0 \\;,\n\\nonumber \\\\\nM^{(2)}_{++-+}(s,t,u) &= 0 \\;,\n\\nonumber \\\\\nM^{(2)}_{++--}(s,t,u) &= 0 \\;,\n\\nonumber \\\\\nM^{(2)}_{+-++}(s,t,u) &= - \\frac{1}{2} \\frac{tu}{t + u} m_Z^2 \\;,\n\\nonumber \\\\\nM^{(2)}_{+-+-}(s,t,u) &= -\\frac{1}{2}\\frac{su^2}{t + u} \\;,\n\\nonumber \\\\\nM^{(2)}_{+++0}(s,t,u) &= 0 \\;,\n\\nonumber \\\\\nM^{(2)}_{++-0}(s,t,u) &= 0 \\;,\n\\nonumber \\\\\nM^{(2)}_{+-+0}(s,t,u) &= \\frac{i}{\\sqrt{2}}\\frac{u\\sqrt{stu}}{t +\nu}m_Z \\;.\n\\end{align}\nThe other three  helicity amplitudes\n$M^{(2)}_{+\\lambda_2\\lambda_3\\lambda_4}$ are given by\n\\begin{align}\\label{dependent_ampl_M2}\nM^{(2)}_{+--+}(s,t,u) &= M^{(2)}_{+-+-}(s,u,t) =\n-\\frac{1}{4}\\frac{st^2}{t + u} \\;,\n\\nonumber \\\\\nM^{(2)}_{+---}(s,t,u) &= M^{(2)}_{+-++}(s,u,t) = - \\frac{1}{2}\n\\frac{tu}{t + u} m_Z^2 \\;,\n\\nonumber \\\\\nM^{(2)}_{+--0}(s,t,u) &= M^{(2)}_{+-+0}(s,u,t) =\n\\frac{i}{\\sqrt{2}}\\frac{t\\sqrt{stu}}{t + u}m_Z \\;.\n\\end{align}\nNote that all amplitudes $M^{(1,2)}_{\\lambda_1\\lambda_2\\lambda_30}$\nare equal to zero in the limit $m_Z = 0$.\n\nThe amplitudes with $\\lambda_1 = -1$ can be obtained from\nconstraints imposed by parity\ninvariance~\\cite{Gounaris:1999_1,Glover:1993},\n\\begin{equation}\\label{parity_relations}\nM^{(1,2)}_{-\\lambda_2\\lambda_3\\lambda_4}(s,t,u) = (-1)^{1 -\n\\lambda_4} M^{(1,2)}_{+-\\lambda_2-\\lambda_3-\\lambda_4}(s,t,u) \\;.\n\\end{equation}\nNote that we have directly calculated all 48 helicity amplitudes\nusing eq.~\\eqref{matrix_element}. Our calculations show that\nrelations \\eqref{dependent_ampl_M1}, \\eqref{dependent_ampl_M2}, and\n\\eqref{parity_relations} really hold.\n\n\n\n\\setcounter{equation}{0}\n\\renewcommand{\\theequation}{C.\\arabic{equation}}\n\n\\section*{Appendix C}\n\\label{app:C}\n\nWigner's $d$-functions \\cite{Wigner} are related to the Jacobi\npolynomials $P^{(\\alpha, \\,\\beta)}_n(z)$ with nonnegative $\\alpha,\n\\beta$ \\cite{Varshalovich},\n\\begin{align}\\label{d-function}\nd^J_{\\lambda\\mu}(z) &= \\left[ \\frac{(J + \\lambda)!(J - \\lambda)!}{(J\n+ \\mu)!(J - \\mu)!}\\right]^{1\/2} \\left( \\frac{1 -\nz}{2}\\right)^{(\\lambda - \\mu)\/2} \\left( \\frac{1 +\nz}{2}\\right)^{(\\lambda + \\mu)\/2}\n\\nonumber \\\\\n&\\times P^{(\\lambda-\\mu, \\,\\lambda+\\mu)}_{J - \\lambda}(z) \\;,\n\\end{align}\nwhere $z = \\cos\\theta$. The $d$-functions obey the orthogonality\ncondition \\cite{Varshalovich}\n\\begin{equation}\\label{Wigner_func_orthogonality}\n\\int\\limits_{-1}^1 d^{J}_{\\lambda\\lambda'}(z)\n\\,d^{J'}_{\\lambda\\lambda'}(z) \\,dz = \\frac{2}{2J + 1} \\,\\delta_{JJ'}\n\\;.\n\\end{equation}\nIn its turn, the Jacobi polynomial is related to the hypergeometric\nfunction \\cite{Varshalovich},\n\\begin{equation}\\label{Jacobi_polynomial}\nP^{(\\rho, \\,\\sigma)}_n(z) = \\frac{\\Gamma(\\rho + 1 + n)}{\\Gamma(\\rho\n+ 1) \\,n!} \\,{}_2F_1 \\!\\left( -n, \\rho+\\sigma + n + 1; \\rho +1;\n\\frac{1-z}{2} \\right) .\n\\end{equation}\nNote that $P^{(\\alpha, \\,\\beta)}_n(-z) = (-1)^n P^{(\\beta,\n\\,\\alpha)}_n(z)$, and, correspondingly,\n\\begin{equation}\\label{d-function_symmetry}\nd^J_{\\lambda\\mu}(-z) = (-1)^{J - \\lambda}d^J_{\\mu -\\lambda}(z) \\;,\n\\quad d^J_{\\lambda\\mu}(z) = (-1)^{\\lambda - \\mu}d^J_{-\\lambda\n-\\mu}(z) \\;.\n\\end{equation}\nIn particular, we get\n\\begin{align}\nd^J_{22}(z) &= \\left( \\frac{1 + z}{2}\\right)^{\\!2} {}_2F_1 \\!\\left(\n2-J, J+3; 1; \\frac{1-z}{2} \\right) , \\label{d-function_22} \\\\\nd^J_{2-2}(z) &= (-1)^J \\left( \\frac{1 - z}{2}\\right)^{\\!2} {}_2F_1\n\\!\\left( 2-J, J+3; 1; \\frac{1+z}{2} \\right) . \\label{d-function_2-2}\n\\end{align}\nIn the simplest case, $\\lambda = \\mu = 0$, we find\n\\begin{align}\\label{d-function_00}\nd^J_{00}(z) &= P_J(z) \\;,\n\\end{align}\nwhere $P_J(z)$ being the Legendre polynomial. Using table integral\n7.231.1 in \\cite{Gradshteyn}, we derive the following formula\n\\begin{equation}\\label{integral_Legendre}\n\\int\\limits_{-1}^1 z^m P_J(z) \\,dz = \\frac{1}{2} \\left[ 1 + (-1)^J\n\\right] (-1)^{J\/2} \\frac{\\Gamma \\!\\left( \\frac{J-m}{2} \\right)\n\\Gamma \\!\\left( \\frac{1+m}{2}\\right)}{{\\Gamma \\!\\left(\n-\\frac{m}{2}\\right) \\Gamma \\!\\left( \\frac{J+m+3}{2}\\right)}} \\;,\n\\end{equation}\nwith integer $J$ and even number $m \\geq 0$. To obtain unitarity\nconstraints on the anomalous couplings, we need integral\n\\eqref{integral_Legendre} with $m=0, \\,2$.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Selected topics on $A$-Connections}\nLet $(A,\\sharp_A,[\\,,\\,]_A)$ be a Lie algebroid and $V$ a vector\nbundle with the same base manifold $M^m$ ($m=dim\\,M$). By an\n$A$-connection we shall understand an $A$-covariant derivative\n$\\nabla:\\Gamma A\\times\\Gamma V\\rightarrow\\Gamma V$ ($\\Gamma$\ndenotes the space of cross sections of a vector bundle), written\nas $(a,v)\\mapsto\\nabla_av$, which is $\\mathds{R}$-bilinear and has\nthe properties\n\\begin{equation}\\label{propconex}\n\\nabla_{fa}v=f\\nabla_av,\\;\\nabla_a(fv)=f\\nabla_av+\\sharp_Aa(f)v\n\\hspace{2mm}(f\\in C^\\infty(M)).\\end{equation}\n\nAccordingly, the value $\\nabla_av(x)$ depends only on $a(x)$ and\non $v|_{U_x}$ where $U_x$ is a neighborhood of $x\\in M$. In order\nto write down the local expression of $\\nabla$, we take a local\nbasis $(b_i)_{i=1}^s$ $(s=rank\\,A)$ of $\\Gamma A$, with the dual\nbasis $(b^{*i})$ of $\\Gamma A^*$ and a local basis $(w_u)_{u=1}^r$\n$(r=rank\\,V)$ of $\\Gamma V$. Then, with the notation\n$\\Omega^k(A)=\\Gamma\\wedge^kA^*$ for the space of $A$-{\\it forms}\nof degree $k$ and using the Einstein summation convention, we get\n\\begin{equation}\\label{conexloc} \\nabla_{b_i}w_u=\\omega_u^t(b_i)w_t,\\;\n\\omega_u^t=\\gamma_{iu}^tb^{*i}\\in\\Omega^1(A),\\end{equation} and we\nsay that $(\\omega_u^t)$ is the {\\it local connection matrix}.\nCorrespondingly, the curvature\n$$R_\\nabla(a_1,a_2)w=\\nabla_{a_1}\\nabla_{a_2}w-\\nabla_{a_2}\\nabla_{a_1}w-\n\\nabla_{[a_1,a_2]_A}w$$\ngets the local expression\n\\begin{equation}\\label{curbloc}\nR_\\nabla(b_i,b_j)w_u=\\Omega_u^t(b_i,b_j)w_t,\\;\n\\Omega_u^t=d_A\\omega_u^t-\\omega_u^s\\wedge\\omega_s^t\\in\\Omega^2(A),\\end{equation}\nwhere $d_A$ denotes the $A$-exterior differential \\cite{Mk}. We\nwill say that $(\\Omega_u^t)$ is the {\\it local curvature matrix}\nand a change of the basis $(w_u)$ implies an\n$ad(Gl(r,\\mathds{R}))$-transformation of $(\\Omega_u^t)$. Like in\nclassical differential geometry, one has the covariant derivative\nmachinery of $V$-tensors and tensor valued $A$-forms and the\ncomputation of the $d_A\\Omega_u^t$ produces the Bianchi identity\nthat may be written under the form $\\nabla\\Omega_u^t=0$.\n\nIn the study of characteristic classes we shall need the direct\nproduct of two Lie algebroids $p_c:A_c\\rightarrow M_c$ $(c=1,2)$\nand we recall its definition given in \\cite{Mk}. Consider the\npullback bundles $\\pi_c^{-1}A_c$, where $\\pi_c$ is the projection\nof $M_1\\times M_2$ on $M_c$. Identify\n\\begin{equation}\\label{pullsections}\n\\Gamma(\\pi_c^{-1}A_c)\\equiv\\{\\sigma:M_1\\times M_2\\rightarrow\nA_c\\,\/\\,p_c\\circ\\sigma=\\pi_c\\}\\end{equation} and notice that local\nbases $(b_i^{(c)})$ of $\\Gamma A_c$ have natural lifts to local\nbases of $\\Gamma(\\pi_c^{-1}A_c)$, which will also be denoted by\n$(b_i^{(c)})$. Take local cross sections\n$$\\sigma_{(c)}=\\sigma_{(c)}^ib^{(c)}_i,\\,\n\\kappa_{(c)}=\\kappa_{(c)}^ib^{(c)}_i\\hspace{2mm}(\\sigma_{(c)}^i,\n\\kappa_{(c)}^i\\in C^\\infty(M_1\\times M_2))$$ (there is no summation\non indices in parentheses) and define the following anchors and\nbrackets\n$$\\sharp_{(c)}\\,\\sigma_{(c)}=\\sigma_{(c)}^i\\sharp_{A_c}\nb^{(c)}_i,[\\sigma_{(c)},\\kappa_{(c)}]_{(c)}=\n\\sigma_{(c)}^i\\kappa_{(c)}^j[b^{(c)}_i,b^{(c)}_j]_{A_c}$$\n$$+\\{\\sigma_{(c)}^j[(\\iota_{*c}\\sharp_{A_c}b^{(c)}_j)\\kappa_{(c)}^i]\n-\\kappa_{(c)}^j[(\\iota_{*c}\\sharp_{A_c}b^{(c)}_j)\\sigma_{(c)}^i]\\}b^{(c)}_i,$$\nwhere $\\iota_{*c}$ is the natural injection of $TM_c$ in\n$T(M_1\\times M_2)$. In these operations, the $M_{(c-1\\,{\\rm\nmod.}\\,2)}$-variable is just a passive parameter and, since the\nanchor and bracket of each $A_c$ are invariant, the results are\nindependent of the choice of the bases. Thus, the vector bundles\n$\\pi_c^{-1}A_c$ are Lie algebroids over $M_1\\times M_2$ and the\ndirect product of the Lie algebroids $A_c$ is the Whitney sum\n$\\mathcal{A}=\\pi_1^{-1}A_1\\oplus\n\\pi_2^{-1}A_2$ endowed with the direct sum of the anchors and brackets of\nthe two pullbacks (in particular, $[b_i^{(1)},b_j^{(2)}]=0$).\n\\begin{prop}\\label{conectlinks} Let $q:V\\rightarrow M_1$ be a vector\nbundle on $M_1$. Then, any $\\mathcal{A}$-connection\n$\\tilde{\\nabla}$ on the pullback $\\pi_1^{-1}(V)$ defines a\ndifferentiable family $\\nabla^{(x_2)}$ $(x_2\\in M_2)$ of\n$A_1$-connections on $V$. Conversely, any $x_2$-parameterized,\ndifferentiable family of $A_1$-connections on $V$ is induced by an\n$\\mathcal{A}$-connection on $\\pi_1^{-1}(V)$.\\end{prop}\n\\begin{proof} Assume that we have the\ncovariant derivatives $\\tilde{\\nabla}_{\\sigma^i_{(1)}b_i^{(1)}+\n\\sigma^j_{(2)}b_j^{(2)}}(\\nu^uw_u)$, where $\\sigma^i_{(c)},\\nu^u$\nare local, differentiable functions on $M_1\\times M_2$. Then, the\nrequired family of connections on $V$ is given by the covariant\nderivatives\n$$(\\nabla^{(x_2)}_{\\xi^ib_i^{(1)}}(\\eta^uw_u))(x_1)=\n(\\tilde{\\nabla}_{\\xi^ib_i^{(1)}}(\\eta^uw_u))(x_1,x_2),$$ where\n$\\xi^i,\\eta^u$ are local, differentiable functions on $M_1$,\n$x_1\\in M_1,x_2\\in M_2$ and we use an identification like\n(\\ref{pullsections}) for $V$. Notice that, if the local connection\nmatrices of $\\tilde{\\nabla}$ are\n\\begin{equation}\\label{eqfamilie}\n\\tilde\\omega_u^v=\\gamma_{(1)iu}^v(x_1,x_2)b_{(1)}^{*i}\n+\\gamma_{(2)ju}^v(x_1,x_2)b_{(2)}^{*j},\\end{equation} the\nconnection $\\nabla^{(x_2)}$ has the matrices\n$\\gamma_{(1)iu}^v(x_1,x_2)b_{(1)}^{*i}$ with the fixed value of\n$x_2$. Conversely, if the family $\\nabla^{(x_2)}$ is given, we get\nan $\\mathcal{A}$-connection $\\tilde{\\nabla}$ by adding the local\nequations $\\tilde{\\nabla}_{b_j^{(2)}}w_u=0$. The local matrices of\nthis connection $\\tilde{\\nabla}$ are the same as the matrices of\n$\\nabla^{(x_2)}$ where $x_2$ is allowed to vary in $M_2$.\n\\end{proof}\n\nIn particular, we may apply Proposition \\ref{conectlinks} for\n$M_1=M,M_2=I=\\{0\\leq \\tau\\leq1\\},A_1=A,A_2=TI$. Then, an $\n\\mathcal{A}$-connection $\\tilde{\\nabla}$ on $\\pi_1^{-1}(V)$ is called a\n{\\it link} between the $A$-connections $\\nabla^0,\\nabla^1$ on $V$.\nFormula (\\ref{eqfamilie}) shows that the local connection forms of\n$\\tilde{\\nabla}$ are given by\n\\begin{equation}\\label{eqlink}\n\\tilde{\\omega}_u^v=\\omega_{(\\tau)u}^v+\\lambda_u^v(x,\\tau)d\\tau,\\end{equation}\nwhere $\\omega_{(\\tau)}$ is the local connection matrix at the\nfixed value $\\tau$ and $\\lambda_u^v\\in C^\\infty(M\\times I)$. A\nsimple calculation gives the corresponding local curvature forms\n\\begin{equation}\\label{curblink}\n\\tilde{\\Omega}_u^v=\\Omega_{(\\tau)u}^v+\\Lambda_u^v\\wedge d\\tau,\n\\end{equation} where $$\n\\Lambda_u^v=d_A\\lambda_u^v+\n\\lambda_u^w\\omega_{(\\tau)w}^v-\\lambda_w^v\\omega_{(\\tau)u}^w+\n\\frac{\\partial\\omega_{(\\tau)u}^v}{\\partial \\tau}$$\n(the partial derivative with respect to $\\tau$ is applied to the\ncoefficients of the form).\n\nNow, we present another ``selected topic\". Let $V\\rightarrow M$ be\na vector bundle of rank $r$ endowed with either a positive,\nsymmetric tensor $g_+\\in\\Gamma\\odot^2V^*$ or a $2$-form\n$g_-\\in\\Gamma\\wedge^2V^*$. We shall say that $(V,g_{\\pm})$ is a\n{\\it quasi-(skew)-metric vector bundle}. Notice that we do not ask\n$rank\\,g_\\pm$ to be constant on $M$. An $A$-connection $\\nabla$ on\n$V$ such that $\\nabla g_\\pm=0$ will be called a {\\it\nquasi-(skew)-metric connection}. If $g_{\\pm}$ is non degenerate,\nthe particle ``quasi\" will not be used and the connection is\ncalled {\\it orthogonal} for $g_+$ and {\\it symplectic} for $g_-$.\nFor a Lie algebroid $A$ over $M$ we shall denote by $L$ a generic,\nintegral leaf of the distribution $im\\,\\sharp_A$ and by $L_x$ the\nleaf through the point $x\\in M$. In what follows we establish\nproperties of a quasi-(skew)-metric connection that are relevant\nto the construction of characteristic classes.\n\\begin{prop}\\label{theorem1} Assume that there exists a\nquasi-(skew)-metric connection $\\nabla$ on $(V,g_\\pm)$. Then, the\nfollowing properties hold. 1. If $x\\in M$ and $k\\in\\Gamma V$ is\nsuch that $k|_{L(x)}\\in K|_{L(x)}$ $(K=ann\\,g_\\pm)$, then\n$\\nabla_ak(x)\\in K_x$, $\\forall a\\in\\Gamma A$. 2.\n$q=rank\\,g_\\pm|_L$ is constant along each leaf $L$ and $\\forall\nx\\in M$ there exists an open neighborhood $U_x$ where $V$ has a\nlocal basis of cross sections of the form $(s_h,t_l)$\n$(h=1,...,q,\\,l=1,...,r-q)$ such that $t_l|_{U_x\\cap L_x}\\in\nK|_{U_x\\cap L_x}$ and the projections $[s_h]=s_h\\,({\\rm mod.}\\,K)$\ndefine a canonical basis of the (skew)-metric vector bundle\n$((V\/K)|_{U_x\\cap L_x},g'_\\pm)$, where $g'_\\pm$ is non-degenerate\nand induced by $g_\\pm$. 3. With respect to this basis, the\n$A$-connection $\\nabla$ has local equations\n\\begin{equation}\\label{eclocaleconex} \\nabla s_h=\\varpi_{(1)h}^ks_k\n+\\varpi_{(2)h}^pt_p,\\; \\nabla\nt_l=\\varpi_{(3)l}^ks_k+\\varpi_{(4)l}^pt_p,\\end{equation} where the\ncoefficients are local $1$-$A$-forms, $\\varpi_{(3)l}^k(x)=0$ and\n$(\\varpi_{(1)h}^k(x))\\in o(q)$, the orthogonal Lie algebra, in the\n$g_+$-case, $(\\varpi_{(1)h}^k(x))\\in sp(q,\\mathds{R})$, the\nsymplectic Lie algebra, in the $g_-$-case. 4. The curvature of\n$\\nabla$ has the local expression\n\\begin{equation}\\label{eclocalecurb} R_\\nabla s_h=\\Phi_{(1)h}^ks_k\n+\\Phi_{(2)h}^pt_p,\\; R_\\nabla\nt_l=\\Phi_{(3)l}^ks_k+\\Phi_{(4)l}^pt_p,\\end{equation} where the\ncoefficients are local $2$-$A$-forms and $\\Phi_{(3)l}^k(x)=0$,\n$(\\Phi_{(1)h}^k(x))\\in o(q)$ in the $g_+$-case,\n$(\\Phi_{(1)h}^k(x))\\in sp(q,\\mathds{R})$ in the\n$g_-$-case.\\end{prop}\n\\begin{proof} 1. For any $a\\in\\Gamma A,v\\in\\Gamma V$ one has\n\\begin{equation}\\label{conexsympl}\n(\\nabla_ag_\\pm)_x(v(x),k(x))=(\\sharp_Aa)_x(g_\\pm(v,k))-\ng_{\\pm,x}(\\nabla_av(x),k(x))\\end{equation}\n$$-g_{\\pm,x}(v(x),\\nabla_ak(x))=0.$$ Since\n$(\\sharp_Aa)_x(g_\\pm(v,k))$ depends only on $k|_{L(x)}\\in\nann\\,g_\\pm$, it vanishes, and we get the required result.\n\n2. $\\nabla g_\\pm=0$ is equivalent with the fact that $g_\\pm$ is\npreserved by parallel translations along paths in a leaf $L$\n\\cite{F}, therefore, $g_{\\pm}$ has a constant rank $q$ along\n$L_x$. This implies the existence of bases with the required\nproperties on a neighborhood $U_x\\cap L$ of $x$. (In the metric\ncase canonical means orthonormal and in the skew-metric case\ncanonical means symplectic.) Then, take any extension of such a\nbasis to $U_x$ and shrink the neighborhood $U_x$ as needed to\nensure the linear independence of the extended cross sections.\n\n3. The equality $\\varpi_{(3)l}^k(x)=0$ is an immediate consequence\nof part 1. Then, in (\\ref{conexsympl}), replace $v,k$ by\n$s_h,s_k$. Since the canonical character of the basis $(s_{h}|_L)$\nimplies $g_{\\pm}(s_{h}|_L,s_{k}|_L)= const.$, we get\n$$g_{\\pm,x}(\\nabla_as_h(x),s_k(x))\n+g_{\\pm,x}(s_h(x),\\nabla_as_k(x))=0,$$ whence,\n$(\\varpi_{(1)i}^j(x))\\in o(q),\\, sp(q,\\mathds{R})$, respectively.\n\n4. The (skew)-metric condition (\\ref{conexsympl}) also implies\n$$\\sharp_A\n[a_1,a_2]_A(g_\\pm(v_1,v_2))=g_\\pm(\\nabla_{[a_1,a_2]_A}v_1,v_2)+\ng_\\pm(v_1,\\nabla_{[a_1,a_2]_A}v_2),$$ where $a_1,a_2\\in\\Gamma A,\nv_1,v_2\\in\\Gamma V$, whence, after some obvious cancellations we\nget\n\\begin{equation}\\label{auxcurv}\\sharp_A[a_1,a_2]_A(\\omega(v_1,v_2)) =\n-\\omega(R_\\nabla(a_1,a_2)v_1,v_2)-\\omega(v_1,R_\\nabla(a_1,a_2)v_2).\\end{equation}\nLike in the proof of 3, (\\ref{auxcurv}) for $s_h,s_k$ implies\n$(\\Phi_{(1)i}^j(x))\\in o(q),\\, sp(q,\\mathds{R})$, respectively.\nThen, (\\ref{auxcurv}) for $s_h,t_l$ together with part 1 of the\nproposition implies $\\Phi_{(3)l}^k(x)=0$.\n\\end{proof}\n\nIn the theory of characteristic classes we need the Weil algebra\n$I(Gl(r,\\mathds{R}))$ $=\\oplus_{k\\geq0}I^k(Gl(r,\\mathds{R}))$,\nwhere $I^k(Gl(r,\\mathds{R}))$ is the space of real, ad-invariant,\nsymmetric, $k$-multilinear functions (equivalently, invariant,\nhomogeneous polynomials of degree $k$) on the Lie algebra of the\ngeneral, linear group $(r=rank\\,V)$. Using the exterior product\n$\\wedge$, such functions may be evaluated on arguments that are\nlocal matrices of $\\wedge$-commuting $A$-forms on $M$ with\ntransition functions of the adjoint type and the result is a\nglobal $A$-form on $M$ (e.g.,\n\\cite{V}). Secondary characteristic classes appear as a\nconsequence of vanishing phenomena encountered in the evaluation\nprocess described above. We shall need the following vanishing\nphenomenon (see \\cite{F}):\n\\begin{prop}\\label{theorem2} If the bundle $V$ endowed with the\nform $g_\\pm$ has a connection $\\nabla$ such that $\\nabla g_\\pm=0$\nand if $R_\\nabla(a_1,a_2)k_x=0$ for $x\\in M$, $a_1,a_2\\in\\Gamma\nA$, $k\\in ker\\,g_{\\pm,x}$, then, $\\forall\\phi\\in\nI^{2k-1}(Gl(r,\\mathds{R}))$, one has $\\phi(\\Phi)=0$, where $\\Phi$\nis the local curvature matrix of the connection $\\nabla$.\n\\end{prop}\n\\begin{proof} By $\\phi(\\Phi)$ we understand the\nevaluation of $\\phi$ where all the arguments are equal to $\\Phi$.\nIt is known that (with a harmless abuse of terminology and\nnotation) the required functions $\\phi$ are spanned by the Chern\npolynomials\n\\begin{equation}\\label{Chern}\nc_h(F)=\\frac{1}{h!}\\delta^{v_1...v_h}_{u_1...u_h}f_{v_1}^{u_1}\n...f_{v_h}^{u_h}\\end{equation} ($\\delta^{...}_{...}$ is the\nmulti-Kronecker index), which are the sums of the principal minors\nof order $h$ in $det(F-\\lambda Id)$ $(F\\in gl(r,\\mathds{R}))$.\nWith the notation of Proposition \\ref{theorem1} and since\n$R_\\nabla(a_1,a_2)k_x=0$, we have to take\n$$F=\\left(\\begin{array}{cc}\n\\Phi_{(1)}&0\\vspace{2mm}\\\\\\Phi_{(2)}&0\\end{array}\\right).$$ Therefore,\n$\\forall x\\in M$, we have $c_h(F)=c_h(\\Phi_{(1)x})$. It is known\nthat the polynomials $c_{2l-1}$ vanish on $o(q)$ and on\n$sp(q,\\mathds{R})$ (in the first case $\\Phi_{(1)}$ is\nskew-symmetric; for the second case see Remark 2.1.10 in \\cite{V},\nfor instance).\\end{proof}\n\\section{Secondary characteristic classes}\nA brief exposition of the classical theory of real characteristic\nclasses may be found in \\cite{V}. In this section, we present a\nLie algebroid version of the basic facts of the theory.\n\nConsider the direct product Lie algebroid $\\mathcal{A}=A\\times\nT\\Delta^k\\rightarrow M\\times\\Delta^k$, where\n$$\\Delta^k=\\{(t_0,t_1,...,t_k)\\in\\mathds{R}^{k+1}\\,\/\\,t_h\\geq0,\\,\n\\sum_{h=0}^k t_h=1\\}$$ is the standard $k$-simplex, $A$ is a Lie\nalgebroid over $M$ and $T\\Delta^k$ is the tangent bundle of\n$\\Delta^k$ endowed with the standard orientation\n$\\kappa=dt^1\\wedge...\\wedge dt^k$. Then,\n$\\forall\\Phi\\in\\Omega^*(\\mathcal{A})$, the fiber-integral\n$\\int_{\\Delta^k}\\Phi$ is defined as zero except for the case\n$$\\Phi=\\alpha\\wedge\\kappa,\\hspace{3mm}\\alpha=\n\\frac{1}{p!}\\alpha_{i_1...i_p}(x,t)b^{*i_1}\\wedge...\\wedge b^{*i_p}\n\\;\\;(x\\in M,t\\in\\Delta^k)$$ when\n$$\\int_{\\Delta^k}\\Phi=\\frac{1}{p!}\n\\left(\\int_{\\Delta^k}\\alpha_{i_1...i_p}(x,t)\\kappa\\right)\nb^{*i_1}\\wedge...\\wedge b^{*i_p}\\in\\Omega^p(A)$$ ($b_i$ is a local\nbasis of cross sections of $A$). The same proof as in the\nclassical case (e.g., \\cite{V}, Theorem 4.1.6) yields the Stokes\nformula:\n\\begin{equation}\\label{Stokes}\\int_{\\Delta^k}d_\\mathcal{A}\n\\Phi-d_{A}\\int_{\\Delta^k}\\Phi=\n(-1)^{deg\\,\\Phi-k}\\int_{\\partial{\\Delta^k}}\\iota^*\\Phi,\\hspace{5mm}\\iota:\n\\partial{\\Delta^k}\\subseteq{\\Delta^k}.\\end{equation}\n\nAssume that we have $k+1$ $A$-connections $\\nabla^{(s)}$ on the\nvector bundle $V\\rightarrow M$ that have the local connection\nmatrices $\\omega_{(\\alpha)}$ $(\\alpha=0,...,k)$ with respect to\nthe local basis $(w_u)$ of $V$. Then, the convex combination\n\\begin{equation}\\label{averagecon}\n\\nabla^{(t)}=\\sum_{\\alpha=0}^kt^\\alpha\\nabla^{\\alpha},\\hspace{3mm}\nt=(t^0,...,t^k)\\in\\Delta^k,\\end{equation} defines a family of\n$A$-connections parameterized by $\\Delta^k$ with the corresponding\n$\\mathcal{A}$-connection $\\tilde{\\nabla}$ on\n$\\pi_1^{-1}(V)\\rightarrow M\\times\\Delta^k$\n$(\\pi_1:M\\times\\Delta^k\\rightarrow M)$. The connection and\ncurvature matrices of $\\tilde{\\nabla}$ will be denoted by\n$\\tilde{\\omega},\\tilde{\\Omega}$; generally, the curvature matrix\nof a connection will be denoted by the upper case of the letter\nthat denotes the connection matrix. There exists a homomorphism\n$$\\Delta(\\nabla^0,...,\\nabla^k):I^h(Gl(r,\\mathds{R}))\n\\rightarrow\\Omega^{2h-k}(A),$$ defined by R. Bott in the classical\ncase, given by\n\\begin{equation}\\label{Bottmare}\n\\Delta(\\nabla^0,...,\\nabla^k)\\phi=(-1)^{\\left[\\frac{k+1}{2}\\right]}\n\\int_{\\Delta^k}\\phi(\\tilde{\\Omega}),\\hspace{2mm}\\phi\\in\nI^h(Gl(r,\\mathds{R})).\\end{equation} Moreover, Bott's proof in the\nclassical case (\\cite{V}, Proposition 4.2.3) also holds in the Lie\nalgebroid version and yields the following formula\n\\begin{equation}\\label{diffBott}\nd_A(\\Delta(\\nabla^0,...,\\nabla^k)\\phi) =\n\\sum_{\\alpha=0}^k(-1)^\\alpha\\Delta(\\nabla^0,...,\\nabla^{\\alpha-1},\n\\nabla^{\\alpha+1},...,\\nabla^k)\\phi.\\end{equation}\n\nLet $\\nabla$ be an $A$-connection on the  vector bundle\n$V\\rightarrow M$. As a consequence of the Bianchi identity,\n$\\forall\\phi\\in I^h(Gl(r,\\mathds{R}))$,\n$\\Delta(\\nabla)\\phi\\in\\Omega^{2h}(A)$ is a $d_A$-closed $A$-form\nand the $A$-cohomology classes defined by the $A$-forms\n$\\Delta(\\nabla)\\phi$ are called the $A$-{\\it principal\ncharacteristic classes of $V$}\n\\cite{F}. If $\\nabla^0,\\nabla^1$ are two $A$-connections, formula\n(\\ref{diffBott}) yields\n\\begin{equation}\\label{Bott2conex}\\Delta(\\nabla^1)\\phi-\\Delta(\\nabla^0)\\phi=\nd_A\\Delta(\\nabla^0,\\nabla^1)\\phi.\\end{equation} Therefore, the\nprincipal characteristic classes do not depend on the choice of\nthe connection.\n\nThe $\\mathcal{A}$-connection $\\tilde{\\nabla}$ to be used in\ndefinition (\\ref{Bottmare}) of $\\Delta(\\nabla^0,\\nabla^1)\\phi$ is\nthe link between $\\nabla^0,\\nabla^1$ given by the family of\n$A$-connections\n$$\\nabla^{(\\tau)}=(1-\\tau)\\nabla^0+\\tau\\nabla^1=\\nabla^0+\\tau D,\\hspace{3mm}\nD=\\nabla^1-\\nabla^0,\\;\\;(\\tau\\in I).$$ For this link, we have\n(\\ref{eqlink}) and (\\ref{curblink}) where\n$\\lambda_u^v=0,\\partial\\omega_{(\\tau)u}^v\/\\partial\\tau=\\alpha$,\nthe local matrix of the connection difference $D$, and formula\n(\\ref{Bottmare}) yields\n\\begin{equation}\\label{defDelta} \\Delta(\\nabla^0,\\nabla^1)\\phi\n=h\\int_0^1\\phi(\\alpha,\\underbrace{\\Omega_{(\\tau)},...,\\Omega_{(\\tau)}}_{(h-1)-{\\rm\ntimes}})d\\tau,\\end{equation} where\n\\begin{equation}\\label{Omegat}\n\\Omega_{(\\tau)}=(1-\\tau)\\Omega_{(0)}+\\tau\\Omega_{(1)}\n+\\tau(1-\\tau)\\alpha\\wedge\\alpha\\end{equation} is the local\ncurvature matrix of the connection $\\nabla^{(\\tau)}$.\n\nWe shall use the Lehmann version of the theory of secondary\ncharacteristic classes \\cite{{L},{V}}. Let $(J_0,J_1)$ be two\n(proper) homogeneous ideals of $I=I(Gl(r,\\mathds{R}))$. Define the\nalgebra\n\\begin{equation}\\label{Weil}W(J_0,J_1)=(I\/J_0)\\otimes(I\/J_1)\\otimes(\\wedge(I^+))\n\\hspace{3mm}(I^+=\\oplus_{k>0}I^k),\\end{equation} with the graduation\n$$deg\\,[\\phi]_{J_0}\n=deg\\,[\\phi]_{J_1}=2h,\\,deg\\,\\hat\\phi=2h-1,$$ and the differential\n$$d[\\phi]_{J_0}=d[\\phi]_{J_1}=0,\\,d\\hat\\phi=[\\phi]_{J_1}-[\\phi]_{J_0},$$\nwhere we refer to the three elements defined by $\\phi\\in I^h$ in\nthe factors of $W$.\n\nNow, take a vector bundle $V\\rightarrow M$ and two $A$-connections\n$\\nabla^0,\\nabla^1$ on $V$ such that $J_c\\subseteq\nker\\,\\Delta(\\nabla^c)$, $c=0,1$. By putting\n\\begin{equation}\\label{defrho}\\rho[\\phi]_{J_0}=\\Delta(\\nabla^0)\\phi,\\,\n\\rho[\\phi]_{J_1}=\\Delta(\\nabla^1)\\phi,\\,\n\\rho\\hat\\phi=\\Delta(\\nabla^0,\\nabla^1)\\phi,\\end{equation} we get a\nhomomorphism of differential graded algebras\n$$\\rho(\\nabla_0,\\nabla_1):W(J_0,J_1)\\rightarrow\\Omega(A)$$\nwith an induced cohomology homomorphism\n$$\\rho^{*}(\\nabla^0,\\nabla^1):H^*(W(J_0,J_1))\\rightarrow\nH^*(A).$$ The cohomology classes in $im\\,\\rho^{*}$ that are not\nprincipal characteristic classes are called $A$-{\\it secondary\ncharacteristic classes}.\n\nIf $J$ is a homogeneous ideal of $I(Gl(r,\\mathds{R}))$, two\n$A$-connections $\\nabla,\\nabla'$ on $V$ are called $J$-{\\it\nhomotopic connections} if there exists a finite chain of links\n$\\tilde{\\nabla}^0,...,\\tilde{\\nabla}^n$ that starts with $\\nabla$,\nends with $\\nabla'$ and is such that $J\\subseteq\\cap_{l=0}^n\nker\\,\\Delta(\\tilde{\\nabla}^l)$. By replacing the usual Stokes'\nformula by formula (\\ref{Stokes}) in the proof of Theorem 4.2.28\nof \\cite{V}, one gets\n\\begin{prop}\\label{invhomot} {\\rm \\cite{L}} The cohomology homomorphism\n$\\rho^{*}(\\nabla^0,\\nabla^1)$ remains unchanged if\n$\\nabla^0,\\nabla^1$ are replaced by $J_0,J_1$-homotopic\nconnections $\\nabla^{'0},\\nabla^{'1}$, respectively ($J_c\\subseteq\nker\\,\\Delta(\\nabla^c)$, $J_c\\subseteq ker\\,\\Delta(\\nabla^{'c})$,\n$c=0,1$).\\end{prop}\n\\begin{corol}\\label{corolhomotopie} The secondary characteristic\nclasses are invariant by any $J_0J_1$-homotopy of the\nconnections.\\end{corol}\n\nDenote by $J_{\\rm odd}\\subseteq I(Gl(r,\\mathds{R}))$ the ideal\nspanned by $\\{\\phi\\in I^{2h-1}(Gl(r,\\mathds{R})),\\\\h=1,2,...\\}$.\nAs explained in Proposition \\ref{theorem2}, if $\\nabla$ is an\northogonal connection for some metric $g$ on the vector bundle\n$V$, then $J_{\\rm odd}\\subseteq ker\\,\\Delta(\\nabla)$. Notice that\nthere always exist positive definite metrics $g$ on $V$ and\ncorresponding metric $A$-connections $\\nabla$, $\\nabla g=0$ (e.g.,\ntake $\\nabla_a=\\nabla'_{\\sharp_Aa}$, where $\\nabla'$ is a usual\northogonal connection on $(V,g)$). Furthermore, any two orthogonal\n$A$-connections on $V$ are $J_{\\rm odd}$-homotopic. Indeed, if\n$\\nabla,\\nabla'$ are orthogonal for the same metric $g$, then\n$(1-\\tau)\\nabla+\\tau\\nabla'$ $(0\\leq\\tau\\leq1)$ defines an\northogonal link. If orthogonality is with respect to different\nmetrics $g,g'$, then $(1-\\tau)g+\\tau g'$ is a metric on the\npullback of $V$ to $M\\times[0,1]$ and a corresponding metric\nconnection provides an orthogonal link between two orthogonal\nconnections $\\bar{\\nabla},\\bar{\\nabla}'$ with the metrics $g,g'$,\nrespectively. Thus, there exists a chain of three orthogonal links\nleading from $\\nabla$ to $\\bar{\\nabla}$, from $\\bar{\\nabla}$ to\n$\\bar{\\nabla}'$ and from $\\bar{\\nabla}'$ to $\\nabla'$, which\nproves the $J_{\\rm odd}$-homotopy of $\\nabla,\\nabla'$.\n\nNow, let $(V,g_\\pm)$ be a quasi-(skew)-metric vector bundle that\nhas a $K$-flat quasi-(skew)-metric connection $\\nabla^1$\n$(K=ann\\,g_\\pm)$. Then, Proposition \\ref{theorem2} tells us that\n$J_{\\rm odd}\\subseteq ker\\,\\Delta(\\nabla^1)$. Accordingly (like in\nthe case of the Maslov classes \\cite{V}), if we also take an\northogonal $A$-connection $\\nabla^0$ on $V$, we shall obtain\nsecondary characteristic classes corresponding to the ideals\n$J_0=J_1=J_{\\rm odd}$.\n\nFollowing \\cite{V}, Theorem 4.2.26, we may replace the algebra\n$W(J_0,J_1)$ by the algebra\n\\begin{equation}\\label{Maslov} \\tilde{W}=\n\\mathds{R}[c_2,c_4,...]\\otimes\\mathds{R}[c'_2,c'_4,...]\n\\otimes\\wedge(\\hat{c}_1,\\hat{c}_3,..),\\end{equation} where\n$c_\\centerdot$ are the Chern polynomials and the accent and hat\nindicate the place in the three factors of (\\ref{Maslov}); the\nhomomorphism $\\rho(\\nabla^0,\\nabla^1)$ is defined like on\n$W(J_0,J_1)$, while using orthogonal and quasi-(skew)-metric\n$A$-connections, respectively, and we get the same set of\ncharacteristic classes. Then, by the same argument like for\n\\cite{V}, Theorem 4.4.37  we get\n\\begin{prop}\\label{clsecFergen} The $A$-secondary characteristic\nclasses of $(V,g_\\pm)$ are the real linear combinations of\ncup-products of $A$-Pontrjagin classes of $V$ {\\rm\n\\cite{F}} and classes of the form\n\\begin{equation}\\label{clMV}\n\\mu_{2h-1}=[\\Delta(\\nabla^0,\\nabla^1)c_{2h-1}]\\in\nH^{4h-3}(A).\\end{equation}\\end{prop}\n\nThe classes $\\mu_{2h-1}$ will be called {\\it simple $A$-secondary\ncharacteristic classes}.\n\\begin{rem}\\label{obsclFern} {\\rm\nIf we start with an arbitrary vector bundle $(V,g_\\pm)$, a\n$K$-flat, quasi-(skew)-metric $A$-connection $\\nabla^1$ may not\nexist. Furthermore, if $\\nabla^1$ exists, it may happen that all\nthe secondary characteristic classes vanish. For instance, if we\nhave a non-degenerate form $g_-$, a usual connection on the bundle\nof $g_-$-canonical frames produces an $A$-connection $\\nabla^1$\nsuch that $\\nabla^1 g_-=0$ and, since $K=0$, we get $A$-secondary\ncharacteristic classes. Because of the $J_{\\rm odd}$-homotopy of\northogonal connections, these classes do not depend on the choice\nof the orthogonal connection $\\nabla^0$. Moreover, these classes\nare independent of the skew-metric connection $\\nabla^1$ because\nof the existence of the link $(1-\\tau)\\nabla^1+\\tau\\nabla^{'1}$\nbetween two such connections. But, the structure group of $V$ may\nbe reduced from the symplectic to the unitary group\n\\cite{V} and a unitary connection $\\bar\\nabla$ on $V$ will be skew-metric\nand orthogonal simultaneously. From (\\ref{defDelta}), and taking\n$\\nabla^0=\\nabla^1=\\bar\\nabla$, we see that the secondary\ncharacteristic classes above vanish.}\\end{rem}\n\\section{Characteristic classes of morphisms}\nLet $A$ be an arbitrary Lie algebroid on $M$, $V,W$ vector bundles\nwith the same basis $M$ and $\\varphi:V\\rightarrow W$ a morphism\nover the identity on $M$. The $A$-connections $\\nabla^V,\\nabla^W$\non $V,W$, respectively, will be called $\\varphi$-compatible if\n$\\nabla^W\\circ\\varphi=\\varphi\\circ\\nabla^V$. An equivalent way to\ncharacterize compatibility is obtained by considering the vector\nbundle $S=V\\oplus W^*$ , which is endowed with the $2$-forms\n\\begin{equation}\\label{omegainS} g_\\pm((v_1,\\nu_1),(v_2,\\nu_2))=\n<\\nu_2,\\varphi(v_1)>\\pm<\\nu_1,\\varphi(v_2)>,\\end{equation}\n$v_1,v_2\\in V,\\, \\nu_1,\\nu_2\\in W^*$. It suffices to work with one\nof these forms, but it is nice to mention that both may be used with\nthe same effect. The pair of $A$-connections $\\nabla^V,\\nabla^W$\nproduces an $A$-connection $\\nabla^S=\\nabla^V\\oplus\\nabla^{W^*}$ on\n$S$, where $\\nabla^{W^*}$ is defined by\n$$<\\nabla^{W^*}_a\\nu,v>=(\\sharp_Aa)<\\nu,v>-<\\nu,\\nabla^W_av>,\\hspace{3mm}\\nu\\in\nW^*,v\\in V.$$ A straightforward calculation shows that\n$\\nabla^V,\\nabla^W$ are $\\varphi$-compatible iff either\n$\\nabla^Sg_+=0$ or $\\nabla^Sg_-=0$. We also notice that the forms\n$g_\\pm$ have the same annihilator\n\\begin{equation}\\label{KinS} K=ker\\,\\varphi\\times\nker\\,^t\\varphi\\end{equation} where the index $t$ denotes\ntransposition.\n\\begin{prop}\\label{FerinS}\nIf $V=A$, if $W=A'$ is a second Lie algebroid and if $\\varphi$ is\na base-preserving Lie algebroid morphism, then there exist\n$K$-flat, $\\varphi$-compatible $A$-connections\n$(\\nabla,\\nabla')$.\\end{prop}\n\\begin{proof} We may proceed like in \\cite{F}. Take a neighborhood\nof $M$ where $\\Gamma A,\\Gamma A'$ have the fixed local bases\n$(b_i),(b'_u)$. Define local $A$-connections\n$\\nabla^U,\\nabla^{'U}$ by asking that\n\\begin{equation}\\label{conexpeU} \\nabla^U_{b_i}b_j=[b_i,b_j]_A,\\;\n\\nabla^{'U}_{b_i}b'_u=[\\varphi b_i,b'_u]_{A'},\\end{equation}\nthen, extending the operators to arbitrary local cross sections in\naccordance with the properties of a connection. Using the local\nexpression $\\varphi b_i=\\varphi_i^ub'_u$, it is easy to check that\n$\\varphi\\circ\\nabla^U=\\nabla^{'U}\\circ\\varphi$. If we consider a\nlocally finite covering $\\{U_\\sigma\\}$ of $M$ by such\nneighborhoods $U$ and glue up the local connections by a\nsubordinated partition of unity $\\{\\theta_\\sigma\\in\nC^\\infty(M)\\}$, we get $\\varphi$-compatible, global\n$A$-connections $\\nabla,\\nabla'$ defined by\n\\begin{equation}\\label{lipireconex}\\nabla_va(x)=\\sum_{x\\in\nU_\\sigma}\\theta_\\sigma(x)\\nabla^{U_\\sigma}_va(x),\\;\n\\nabla'_va'(x)=\\sum_{x\\in\nU_\\sigma}\\theta_\\sigma(x)\\nabla^{'U_\\sigma}_va'(x),\\end{equation}\nwhere $x\\in M,v\\in A_x,a\\in\\Gamma A,a'\\in\\Gamma A'$.\n\nNow, we notice that the local connections (\\ref{conexpeU}) satisfy\nthe following properties\n\\begin{equation}\\label{propcloc} \\nabla^U_{b_i}a=[b_i,a]_A,\\;\n\\nabla^{'U}_{b_i}a'=[\\varphi b_i,a']_{A'}.\\end{equation} Indeed, if\nwe put $a=f^jb_j,a'=h^ub'_u$, (\\ref{conexpeU}) and the properties of\nthe Lie algebroid bracket imply (\\ref{propcloc}). Furthermore, using\n(\\ref{propcloc}), it is easy to check the following properties of\nthe global compatible connections (\\ref{lipireconex})\n\\begin{equation}\\label{propluinabla} \\nabla_vk(x)=[\\tilde{v},k]_A(x),\n\\end{equation}\n\\begin{equation}\\label{propluinabla'} <\\nabla^{'*}_v\\alpha'(x),a'(x)>\n=(\\sharp_Av)<\\alpha',a'>-<\\alpha'(x),[\\varphi\\tilde{v},a']_{A'}(x)>,\\end{equation}\n$\\forall x\\in M,k\\in\\Gamma(ker\\,\\varphi),a'\\in\\Gamma\nA',\\alpha'\\in\\Gamma(ker\\,^t\\varphi)$ and $\\tilde{v}=\\nu^ib_i$ is a\ncross section of $\\Gamma A$ that extends $v\\in A_x$. The\nrestrictions put on $k,\\alpha'$ ensure the correctness of the\npassage from the covariant derivative to the Lie algebroid bracket\nand the independence of the result on the choice of $\\tilde{v}$.\nFormulas (\\ref{propluinabla}), (\\ref{propluinabla'}) imply\n$\\varphi(\\nabla_vk)=0$, $\\nabla^{'*}_v\\alpha'\\circ\\varphi=0$, which\nmeans that $ker\\,\\varphi$ and $ker\\,^t\\varphi$ are preserved by the\nconnections $\\nabla,\\nabla'$, respectively.\n\nFinally, if we denote $S=A\\oplus A^{'*}$ and\n$\\nabla^S=\\nabla\\oplus\\nabla^{'*}$, we can compute the curvature\n$[R_{\\nabla^S}(a_1,a_2)(k,\\alpha')](x)$, which has components on $A$\nand $A^{'*}$. The component on $A$ is\n$$(\\nabla_{a_1}\\nabla_{a_2}-\\nabla_{a_2}\\nabla_{a_1}\n-\\nabla_{[a_1,a_2]_A})\\tilde{k}(x)\\stackrel{(\\ref{propluinabla})}{=}\n([\\tilde{a}_1,[\\tilde{a}_2,\\tilde{k}]_A]_A$$ $$-\n[\\tilde{a}_2,[\\tilde{a}_1,\\tilde{k}]_A]_A-\n[[\\tilde{a}_1,\\tilde{a}_2]_A,\\tilde{k}]_A)(x)=0,$$ where tilde\ndenotes extensions to cross sections and the final result holds\nbecause of the Jacobi identity. For the component on $A^{'*}$ we\nget the following evaluation on any $a'\\in\\Gamma A'$:\n$$<(\\nabla^{'*}_{a_1}\\nabla^{'*}_{a_2}-\\nabla^{'*}_{a_2}\\nabla^{'*}_{a_1}\n-\\nabla^{'*}_{[a_1,a_2]_A})\\tilde{\\alpha}',a'>(x)\n\\stackrel{(\\ref{propluinabla'})}{=}<\\tilde{\\alpha}',\n[\\varphi\\tilde{a}_2,[\\varphi\\tilde{a}_1,\\tilde{k}]_{A'}]_{A'}$$ $$-\n[\\varphi\\tilde{a}_1,[\\varphi\\tilde{a}_2,\\tilde{k}]_{A'}]_{A'}-\n[[\\varphi\\tilde{a}_1,\\varphi\\tilde{a}_2]_{A'},\\tilde{k}]_{A'}>(x)=0,$$\nwhere the annulation is justified by the Jacobi identity again.\nTherefore, $$[R_{\\nabla^S}(a_1,a_2)(k,\\alpha')](x)=0,$$ which is the\nmeaning of $K$-flatness.\n\\end{proof}\n\\begin{rem}\\label{obsrankconst} {\\rm During the proof of Proposition \\ref{FerinS} we\nsaw that $ker\\,\\varphi$ is preserved by $\\nabla$, hence, it is\npreserved by the parallel translation along the paths in the leaves\n$L$ of $A$. This shows that $rank\\,\\varphi$ is constant along the\nleaves $L$.}\\end{rem}\n\\begin{rem}\\label{nouindistins} {\\rm If we use the\ndefinition of $\\nabla^{'*}$ in the left hand side of\n(\\ref{propluinabla'}) and take into account the relation\n$ann\\,ker\\,^t\\varphi=im\\,\\varphi$ we obtain the following equivalent\nform of (\\ref{propluinabla'}):\n\\begin{equation}\\label{eqpropluinabla'}\n\\nabla'_va'(x)=[\\varphi\\tilde v,a']_{A'}(x)\\;({\\rm\nmod.}\\,im\\,\\varphi_x)\\hspace{2mm}\\forall x\\in M,v\\in A_x.\n\\end{equation}}\\end{rem}\n\\begin{defin}\\label{defdistins} {\\rm\nA pair of $\\varphi$-compatible $A$-connections that satisfy the\nproperties (\\ref{propluinabla}), (\\ref{eqpropluinabla'}) will be\ncalled a {\\it distinguished pair} (in \\cite{F} one uses the term\nbasic connections).}\\end{defin}\n\nNow, we see that we may use Proposition \\ref{clsecFergen} in order\nto get secondary characteristic classes for the bundle $S=A\\oplus\nA^{'*}$ endowed with the quasi-(skew)-metrics (\\ref{omegainS}), with\na connection $\\nabla^1=\\nabla\\oplus\\nabla^{'*}$, where\n$(\\nabla,\\nabla')$ is a distinguished pair of $A$-connections, and\nwith an orthogonal connection\n$\\nabla^0=\\nabla^{g_A}\\oplus\\nabla^{g_{A'}*}$, where $g_A,g_{A'}$\nare metrics on the bundles $A,A'$ and $\\nabla^{g_A},\\nabla^{g_{A'}}$\nare corresponding orthogonal connections on $A,A'$.\n\\begin{defin}\\label{defclmor} {\\rm The above constructed\nsecondary characteristic classes of $A\\oplus A^{'*}$ will be called\nthe {\\it characteristic classes} of the base-preserving morphism\n$\\varphi$. In particular, one has the {\\it simple characteristic\nclasses} $\\mu_{2h-1}(\\varphi)\\in H^{4h-3}(A)$.}\\end{defin}\n\nThe secondary characteristic classes of the Lie algebroid $A$\ndefined in \\cite{F} are the simple characteristic classes of the\nmorphism $\\varphi=\\sharp_A:A\\rightarrow TM$.\n\\begin{prop}\\label{isomorfism} All the characteristic classes\nof a base-preserving isomorphism $\\varphi:A\\rightarrow A'$ are\nzero.\\end{prop} \\begin{proof} If $\\varphi$ is an isomorphism, then\n$g_-$ is non degenerate and we are in the situation discussed in\nRemark \\ref{obsclFern}.\\end{proof}\n\nThus, the characteristic classes of a morphism may be seen as a\nmeasure of its non-isomorphic character.\n\\begin{prop}\\label{invarlaconex} The characteristic\nclasses of a base preserving morphism $\\varphi:A\\rightarrow A'$ of\nLie algebroids do not depend on the choice of the orthogonal\nconnection and of the distinguished pair of compatible connections\nrequired by their definition.\\end{prop}\n\\begin{proof} The proposition is a consequence of Corollary\n\\ref{corolhomotopie}. In the previous section we have seen that two\northogonal $A$-connections are $J_{\\rm odd}$-homotopic. On the other\nhand, take two $\\varphi$-distinguished pairs of $A$-connections\n$\\nabla,\\nabla';\\tilde{\\nabla},\\tilde{\\nabla}'$. Then, it is easy to\ncheck that, $\\forall t\\in[0,1]$,\n$(1-t)\\nabla+t\\tilde{\\nabla},(1-t)\\nabla'+t\\tilde{\\nabla}'$ is a\n$\\varphi$-distinguished pair again. Therefore, $J_{\\rm\nodd}$-homotopy also holds for the corresponding quasi-(skew)-metric\nconnections on $S$ and we are done.\\end{proof}\n\nWe also have another consequence of Corollary \\ref{corolhomotopie}:\n\\begin{prop}\\label{morfismehomotope} Two homotopic, base-preserving\nmorphisms $\\varphi_0,\\varphi_1:A\\rightarrow A'$ of Lie algebroids\nhave the same secondary characteristic classes.\\end{prop}\n\\begin{proof} By homotopic morphisms we understand  morphisms\n$\\varphi_0,\\varphi_1$ that are linked by a differentiable family\nof morphisms $\\varphi_\\tau:A\\rightarrow A'$ $(0\\leq\\tau\\leq1)$.\nThe corresponding forms $g_{+,\\tau}$ on $S=A\\oplus A^{'*}$ are\ndifferent, but, still, all the connections $\\nabla^{1,\\tau}$\nrequired in the construction of the secondary classes have\nskew-symmetric local connection and curvature matrices. Therefore,\nthe $J_{\\rm odd}$-homotopy holds and we are done.\\end{proof}\n\\begin{rem}\\label{clasebicaract} {\\rm\nIn the case of an arbitrary pair of morphisms\n$\\varphi_0,\\varphi_1:A\\rightarrow A'$ we can measure the\ndifference between the secondary characteristic classes as\nfollows. Notice the existence of the {\\it bi-characteristic\nclasses}\n$\\bar{\\mu}_{2h-1}(\\varphi_1,\\varphi_2)=[\\Delta(\\nabla^1,\\nabla^2)c_{2h-1}]\\in\nH^{4h-3}(A)$ where $\\nabla^1,\\nabla^2$ are $A$-connections defined\non $S=A\\oplus A^{'*}$ by distinguished, $\\varphi_{1,2}$-compatible\nconnections respectively. Then, formula (\\ref{diffBott}) yields\n$$d_A\\Delta(\\nabla^0,\\nabla^1,\\nabla^2)c_{2h-1}=\\Delta(\\nabla^0,\\nabla^1)c_{2h-1}\n+\\Delta(\\nabla^1,\\nabla^2)c_{2h-1}+\\Delta(\\nabla^2,\\nabla^0)c_{2h-1},$$\nwhere $\\nabla^0$ is an orthogonal connection on $S$. Accordingly,\nwe get\n\\begin{equation}\\label{claseperechi} {\\mu}_{2h-1}(\\varphi_1)-\n{\\mu}_{2h-1}(\\varphi_2)=\n\\bar{\\mu}_{2h-1}(\\varphi_1,\\varphi_2).\\end{equation}}\\end{rem}\n\nIn what follows we give explicit local expressions of $A$-forms\nthat represent the characteristic classes $\\mu_{2h-1}(\\varphi)$.\nTake a point $x\\in M$ and an open neighborhood $U$ of $x$\ndiffeomorphic to a ball. Assume that $(\\nabla^U,\\nabla^{'U})$ and\n$(\\nabla,\\nabla')$ are pairs of local, respectively global,\ndistinguished, $\\varphi$-compatible $A$-connections on $A,A'$.\nThen, if $0\\leq\\chi\\in C^\\infty(M)$ is equal to $1$ on the compact\nclosure $\\bar V$ of the open neighborhood $V\\subseteq U$ of $x$\nand equal to $0$ on $M\\backslash U$, then the convex combinations\n$$\\bar{\\nabla}=\\chi\\nabla^U+(1-\\chi)\\nabla,\\,\n\\bar{\\nabla}'=\\chi\\nabla^{'U}+(1-\\chi)\\nabla'$$\ndefine a global pair of distinguished $A$-connections that\ncoincides with $(\\nabla^U,\\nabla^{'U})$ on $V$.\n\nAccordingly, in formula (\\ref{clMV}) for $S=A\\oplus A^{'*}$ we may\nalways use a connection $\\nabla^1$ such that the expressions\n(\\ref{conexpeU}) hold on the neighborhood $V$. Then, if we denote\n\\begin{equation}\\label{expresiiloc1} \\begin{array}{c}\n[b_i,b_j]_A=\\gamma_{ij}^kb_k,\\,\n[b'_u,b'_v]_{A'}=\\gamma_{uv}^{'w}b'_w,\\vspace{2mm}\\\\\n\\sharp_Ab_i=\\rho_i^j\\frac{\\partial}{\\partial x^j},\\,\n\\sharp_{A'}b'_u=\\rho_u^{'j}\\frac{\\partial}{\\partial x^j}\n\\,\\varphi(b_i)=\\varphi_i^sb'_s\\end{array}\\end{equation}\n(remember that we use the Einstein summation convention), where\n$x^i$ are local coordinates on $M$ and $(b_i)(,b'_u)$ are the\nbases used in (\\ref{conexpeU}), we get the following connection\nmatrix of $\\nabla^1$ on the neighborhood $V$\n\\begin{equation}\\label{expresiiloc2} \\left( \\begin{array}{cc}\n\\gamma_{ij}^kb^{*i}&0\\vspace{2mm}\\\\\n0&(-\\varphi_i^t\\gamma_{tu}^{'s}+\\rho^{'j}_u\n\\frac{\\partial\\varphi_i^s}{\\partial x^j})b^{*i}\\end{array}\\right)\\end{equation}\n(in (\\ref{expresiiloc2}), $b^{*i}$ is the dual basis of $b_i$).\n\nFurthermore, let $g^U,g^{'U}$ be local metrics on $A,A'$ such that\n$(b_i),(b'_u)$ are orthonormal bases and $g,g'$ arbitrary, global\nmetrics on $A,A'$. Then, define the metrics\n$$\\chi g^U+(1-\\chi)g,\\,\\chi g^{'U}+(1-\\chi)g'$$ and take\nan orthogonal connection $\\nabla^0$ whose components are\ncorresponding orthogonal connections. The connection matrix of\n$\\nabla^0$ on the neighborhood $V$, with respect to the same local\nbases like in (\\ref{expresiiloc2}), will be of the form\n\\begin{equation}\\label{expresiiloc3} \\left(\n\\begin{array}{cc}\\varpi_i^j&0\\vspace{2mm}\\\\ 0&-\\varpi_s^{'t}\n\\end{array}\\right),\\end{equation} where\n$(\\varpi_i^j),(\\varpi_s^{'t})$ are skew-symmetric matrices of\nlocal $1$-$A$-forms.\n\nIf these connections $\\nabla^0,\\nabla^1$ are used, then, along\n$V$, the difference matrix $\\alpha$ of formula (\\ref{defDelta}) is\nthe difference between the matrices (\\ref{expresiiloc2}) and\n(\\ref{expresiiloc3}). Furthermore, we can compute the matrix\n$\\Omega_{(\\tau)}$ by using formula (\\ref{Omegat}), where\n$\\Omega_{(0)}$ is a skew-symmetric matrix. The final result may be\nformulated as follows\n\\begin{prop}\\label{propexpresmu} If a point $x\\in M$ is fixed,\nthere exist global representative $A$-forms $\\Xi_{2h-1}\\in\n\\Omega^{4h-3}(A)$ of the characteristic classes $\\mu_{2h-1}$ such\nthat \\begin{equation}\\label{localmu} \\Xi_{2h-1}|_V=\n\\frac{1}{(2h-2)!}\\int_0^1\\left(\n\\delta^{\\sigma_1...\\sigma_{2h-1}}_{\\kappa_1...\\kappa_{2h-1}}\n\\alpha_{\\sigma_1}^{\\kappa_1}\\wedge\\Omega_{(\\tau),\\sigma_2}^{\\kappa_2}\n\\wedge...\\wedge\\Omega_{(\\tau),\\sigma_{2h-1}}^{\\kappa_{2h-1}}\\right)d\\tau,\n\\end{equation} for some neighborhood $V$ of $x$. In\n(\\ref{localmu}), the factors are the entries of the matrices\n$\\alpha,\\Omega_{(\\tau)}$ given by formulas (\\ref{expresiiloc2}),\n(\\ref{expresiiloc3}) and Greek indices run from $1$ to\n$dim\\,A+dim\\,A'$.\\end{prop}\n\\begin{proof} Use the expression (\\ref{Chern}) of the Chern\npolynomials and the connections $\\nabla^0,\\nabla^1$ constructed\nabove. \\end{proof}\n\nThe difficulty in using Proposition \\ref{propexpresmu}, besides\nits complexity in the case $h>1$, consists in the fact that\nformula (\\ref{localmu}) does not define global $A$-forms; for\nneighborhoods of different points $x_1\\neq x_2$ we have different\npairs of distinguished connections $\\bar{\\nabla},\\bar{\\nabla}'$.\nHowever, we can use Proposition \\ref{propexpresmu} in order to\nextend a result proven for a Lie algebroid $A$\n($\\varphi=\\sharp_A$) in\n\\cite{F}:\n\\begin{prop}\\label{comparmodul}\nThe secondary class $\\mu_1(\\varphi)$ is equal to the modular class\nof the morphism $\\varphi$.\\end{prop}\n\\begin{proof} Recall that the modular\nclass of a morphism is defined by\n$\\mu(\\varphi)=\\mu(A)-\\varphi^*\\mu(A')\\in H^1(A)$, where\n$\\mu(A),\\mu(A')$ are the modular classes of the Lie algebroids\n$A,A'$, respectively, \\cite{{GMM},{KW},{KGW}}. Furthermore, the\nmodular class $\\mu(A)$ is defined as follows\n\\cite{{ELW},{F},{H},{KGW}}. The line bundle\n$\\wedge^sA\\otimes\\wedge^mT^*M$ $(s=rank\\,A)$ has a flat\n$A$-connection defined, by means of local bases, as follows\n\\begin{equation}\\label{flattop}\n\\nabla_{b_i}((\\wedge_{j=1}^sb_j)\\otimes(\\wedge_{h=1}^mdx^h))\n=\\sum_{j=1}^sb_1\\wedge...\\wedge [b_i,b_j]_A\\wedge...\\wedge\nb_s\\otimes(\\wedge_{h=1}^mdx^h)\\end{equation}\n$$+(\\wedge_{j=1}^sb_j)\\otimes\nL_{\\sharp_Ab_i}(\\wedge_{h=1}^mdx^h),$$ where $L$ is the Lie\nderivative. Then, for\n$\\sigma\\in\\Gamma(\\wedge^sA\\otimes\\wedge^mT^*M)$ (which exists if\nthe line bundle is trivial; otherwise we go to its double\ncovering), one has $\\nabla_a\\sigma=\\lambda(a)\\sigma$ where\n$\\lambda$ is a $d_A$-closed $1$-$A$-form and defines the\ncohomology class $\\mu(A)$, which is independent on the choice of\n$\\sigma$.\n\nFrom (\\ref{flattop}) it follows easily that $\\mu(A),\\mu(A')$ are\nrepresented by the $A$-forms\n\\begin{equation}\\label{formemodulare}\n\\lambda=\\sum_{i,k,j}(\\gamma_{ik}^k+\\frac{\\partial\\rho_i^j}{\\partial\nx^j})b^{*i},\\,\n\\lambda'=\\sum_{s,t,h}(\\gamma_{st}^t+\\frac{\\partial\\rho_s^{'j}}{\\partial\nx^j})b^{'*s}\\end{equation} where the notation is that of\n(\\ref{expresiiloc1}). Notice that, even though the expressions\n(\\ref{formemodulare}) are local, the forms $\\lambda,\\lambda'$ are\nglobal $A$-forms because the connection that was used in their\ndefinition is global.\n\nOn the other hand, using formulas (\\ref{expresiiloc2}),\n(\\ref{expresiiloc3}) and since the trace of a skew-symmetric\nmatrix is zero, we may see that the $A$-form $\\Xi_1$ defined in\nProposition \\ref{propexpresmu} is such that\n$\\Xi|_{1V}=(\\lambda-\\varphi^*\\lambda')|_{V}$, where $V$ is a\nneighborhood of a fixed point $x\\in M$. Accordingly, there exists\na locally finite, open covering $\\{V_\\alpha\\}$ of $M$ and there\nexists a family of pairs of $A$-connections\n$(\\nabla^{0\\alpha},\\nabla^{1\\alpha})$ that provide representative\n$1$-$A$-forms $\\Xi_{1\\alpha}$ of the characteristic class\n$\\mu_1(\\varphi)$ such that\n\\begin{equation}\\label{auxmodular}\n\\Xi_{1\\alpha}|_{V_\\alpha}=(\\lambda-\\varphi^*\\lambda')|_{V_\\alpha}.\\end{equation}\nThen, if we take a partition of unity $\\{\\theta_\\alpha\\in\nC^\\infty(M)\\}$ subordinated to $\\{V_\\alpha\\}$ and glue up the\nfamilies $\\nabla^{0\\alpha},\\nabla^{1\\alpha}$, like in\n(\\ref{lipireconex}), we get connections $\\nabla^0,\\nabla^1$ that\ndefine the representative $A$-form\n$$\\Xi_1(x)=\\sum_{x\\in\nV_\\alpha}\\theta_\\alpha(x)\\Xi_{1\\alpha}(x)=(\\lambda-\\varphi^*\\lambda')(x),\n\\hspace{2mm}x\\in M$$ of $\\mu_1(\\varphi)$.\nThis justifies the required conclusion.\\end{proof}\n\\begin{example}\\label{Ps-Nij} {\\rm  An\ninteresting example appears on a Poisson-Nijenhuis manifold\n$(M,P,N)$, where $P$ is a Poisson bivector field and $N$ is a\nNijenhuis tensor. Then $^tN:(T^*M,N\\circ\\sharp_P)\n\\rightarrow (T^*M,\\sharp_P)$ is a morphism of cotangent Lie\nalgebroids. The modular class of the morphism $^tN$ was studied in\n\\cite{DF} and it would be interesting to get information about\nother characteristic classes of this morphism.}\\end{example}\n\nThe calculation of the classes $\\mu_{2h-1}$ for $h>1$ is much more\ncomplicated. One of the difficulties is the absence of a global\nconstruction of a distinguished pair of connections.\n\\begin{example}\\label{exdistins} {\\rm Let $\\varphi:A\\rightarrow A$ be\nan endomorphism of the Lie algebroid $A$ and assume that there\nexists an $A$-connection $\\nabla$ on $A$ that satisfies condition\n(\\ref{propluinabla}) and whose torsion\n$$T_\\nabla(a_1,a_2)=\\nabla_{a_1}a_2-\\nabla_{a_2}a_1-[a_1,a_2]_A,\\hspace{2mm}\na_1,a_2\\in\\Gamma A,$$ takes values in $K=ker\\,\\varphi$. Then, it\nis easy to check that the formula\n$$\\nabla'_{a_1}a_2=[\\varphi a_1,a_2]_A + \\varphi\\nabla_{a_2}a_1$$\ndefines a second $A$-connection that is $\\varphi$-compatible with\n$\\nabla$ and satisfies condition (\\ref{eqpropluinabla'}).\nTherefore, $(\\nabla,\\nabla')$ is a distinguished\npair.}\\end{example}\n\nAnother difficulty is produced by the complicated character of the\nexpression (\\ref{localmu}). A simple example follows.\n\\begin{example}\\label{exhmare} {\\rm If the Lie algebroids\n$A,A'$ have anchors zero, the $A$-connections are tensors and\nformula (\\ref{expresiiloc2}) gives the local connection matrices\nof a global, flat $A$-connection $\\nabla^1$ as required in the\ndefinition of the characteristic classes (flatness is just Jacobi\nidentity). In the simplest case\n$A=M\\times\\mathcal{G},A'=M\\times\\mathcal{G}'$ where\n$\\mathcal{G},\\mathcal{G}'$ are Lie algebras, we may take\n$\\varpi_i^j=0$ in (\\ref{expresiiloc3}), which gives a flat metric\nconnection $\\nabla^0$. Then, formula (\\ref{Omegat}) reduces to\n$$\\Omega_{(\\tau)}=\\tau(1-\\tau)\\alpha\\wedge\\alpha$$ where $\\alpha$\nis the matrix (\\ref{expresiiloc2}). Accordingly, like in \\cite{V},\nTheorem 4.5.11, we get the representative $A$-forms\n$$\\Xi_{2h-1}=\\frac{1}{(2h-2)!}\\nu_h\n\\delta^{\\sigma_1...\\sigma_{2h-1}}_{\\kappa_1...\\kappa_{2h-1}}\n\\alpha_{\\sigma_1}^{\\kappa_1}\\wedge\\alpha_{\\sigma_2}^{\\lambda_2}\n\\wedge\\alpha_{\\lambda_2}^{\\kappa_2}\\wedge...\\wedge\n\\alpha_{\\sigma_2h-1}^{\\lambda_{2h-1}}\n\\wedge\\alpha_{\\lambda_{2h-1}}^{\\kappa_{2h-1}}$$ of the classes\n$\\mu_{2h-1}$, where $\\alpha^{\\cdot}_{.}$ are the entries of the\nmatrix (\\ref{expresiiloc2}) and\n$$\\nu_h=\\int_0^1\\tau(1-\\tau)d\\tau=\\sum_{i=1}^{2h-2}(-1)^{h+i+1}\\frac{2^i}{4h-i-3}\n\\left(\\begin{array}{c}2h-2\\vspace{2mm}\\\\ i\\end{array}\\right).$$\n}\\end{example}\n\\begin{rem}\\label{bazediferite} {\\rm So far, we do not have\na good definition of characteristic classes of a morphism between\nLie algebroids over different bases. Using the terminology and\nnotation of\n\\cite{KGW}, let us\nconsider a morphism \\begin{equation}\\label{genmorf}\n\\begin{array}{ccc} A&\n\\stackrel{\\varphi}{\\rightarrow}&B\\vspace{2mm}\\\\ \\downarrow&\n&\\downarrow\\vspace{2mm}\\\\\nM&\\stackrel{f}{\\rightarrow}&N\\end{array}\\end{equation} between the\nLie algebroids $A,B$ and assume that the mapping $f$ is\ntransversal to the Lie algebroid $B$. Then, Proposition 3.11 of\n\\cite{KGW} tells us that $\\varphi=f_B^{!!}\\circ\\varphi'$, where\n$f_B^{!!}:f^{!!}B\\rightarrow B$, $\\varphi':A\\rightarrow f^{!!}B$\nare the natural projections of the pullback Lie algebroid\n$f^{!!}B$. Furthermore, Proposition 3.12 of \\cite{KGW}  tells that\nthe modular class of the non base preserving morphism $\\varphi$ is\nequal to the modular class of the base preserving morphism\n$\\varphi'$. This equality may be extended by definition to all the\ncharacteristic classes of $\\varphi$, but it is not clear whether\nthis definition is good (it does not loose information about\n$\\varphi$) even in the indicated particular case.}\\end{rem}\n\\section{Relative characteristic classes}\nFrom Proposition \\ref{comparmodul} and a known result on modular\nclasses (\\cite{KGW}, formula (2.5)) we see that the first class\n$\\mu_1(\\varphi)$ has a nice behavior with respect to the\ncomposition of morphisms namely, for the morphisms\n$\\varphi:A\\rightarrow A',\\psi:A'\\rightarrow A''$ one has\n\\begin{equation}\\label{modcompus} \\mu_1(\\psi\\circ\\varphi)=\n\\mu_1(\\varphi)+\\varphi^*(\\mu_1(\\psi)).\\end{equation} In this section\nwe give a proof of (\\ref{modcompus}) by means of the definition of\nthe characteristic classes of a morphism and we shall see why the\nresult does not extend to the higher classes $\\mu_{2h-1}$, $h>1$.\nThe proof will use a kind of relative characteristic classes that\nare interesting in their own right; in particular, we will show\nthat the relative classes defined by the jet Lie algebroid $J^1A$\n\\cite{CF} are cohomological images of the absolute characteristic\nclasses of a morphism $\\varphi:A\\rightarrow A'$.\n\nLike in the definition of the characteristic classes of $\\varphi$\nwe can produce characteristic classes of $\\psi:A'\\rightarrow A''$\nmodulo $\\varphi:A\\rightarrow A'$ as follows. Take the Lehmann\nmorphism $\\rho^*(D^0,D^1)$ for an orthogonal $A$-connection $D^0$\non the vector bundle $A'\\oplus A^{''*}$ associated with a sum of\nEuclidean metrics $g_{A'},g_{A''}$ and an $A$-connection $D^1$ on\n$A'\\oplus A^{''*}$, which is the sum of {\\it distinguished\n$A$-connections} $\\nabla',\\nabla''$ on $A',A''$, respectively.\nHere by a distinguished pair we mean a pair of $A$-connections\n$(\\nabla',\\nabla'')$ that satisfies the following properties\n\\begin{equation}\\label{distins2} \\begin{array}{l}\n\\psi\\nabla'_aa'=\\nabla''_a(\\psi a'),\\;\\;a\\in A_x\\,(x\\in M),\\,\na'\\in\\Gamma A',\\vspace{2mm}\\\\\n\\nabla'_ak(x)=[\\varphi\\tilde{a},k]_{A'}(x),\\;\\;k\\in\\Gamma ker\\,\\psi,\n\\vspace{2mm}\\\\\n\\nabla''_aa''(x)=[\\psi\\varphi\\tilde{a},a'']_{A''}(x)\\;\\;({\\rm\nmod.\\,}im\\,\\psi),\\end{array}\\end{equation} where the sign tilde\ndenotes the extension to a cross section. One can construct a\n$\\psi$-distinguished pair of $A$-connections $\\nabla',\\nabla''$ by\nreplacing the local formulas (\\ref{conexpeU}) by\n\\begin{equation}\\label{conexpeU1} \\nabla^{'U}_{b_i}b'_{j'}\n=[\\varphi b_i,b'_{j'}]_{A'},\\;\n\\nabla^{''U}_{b_i}b''_{j''}=[\\psi\\varphi b_i,b''_{j''}]_{A''},\n\\end{equation}\nthen gluing the local connections via a partition of unity. (In\n(\\ref{conexpeU1}) $(b_i),(b'_{i'}),(b''_{i''})$ are local bases of\n$\\Gamma A,\\Gamma A',\\Gamma A''$, respectively.)\n\\begin{defin}\\label{defclrel} {\\rm The\ncharacteristic $A$-cohomology classes in $im\\,\\rho^*(D^0,D^1)$\nwill be called {\\it relative characteristic classes} of $\\psi$\nmodulo $\\varphi$. In particular, $$\\mu_{2h-1}(\\psi\\,{\\rm\nmod.}\\,\\varphi) =[\\Delta(D^0,D^1)]\\in H^{4h-3}(A)$$ are the {\\it\nsimple relative characteristic classes}.}\\end{defin}\n\\begin{prop}\\label{1relativ} For $h=1$, the relative and absolute\ncharacteristic class $\\mu_1$ of the morphism $\\psi$ are related by\nthe equality \\begin{equation}\\label{pullbackmu1}\n\\mu_{1}(\\psi\\,{\\rm mod.}\\,\\varphi)\n=\\varphi^*\\mu_{1}(\\psi).\\end{equation}\n\\end{prop}\n\\begin{proof} By absolute classes we understand characteristic\nclasses $\\mu_{2h-1}(\\psi) \\in H^{4h-3}(A')$. The partition of\nunity argument given for (\\ref{expresiiloc2}) shows that we may\nassume the following local expressions of distinguished\n$A'$-connections on $A',A''$\n\\begin{equation}\\label{conexpeU2} \\bar{\\nabla}^{'U}_{b'_{i'}}b'_{j'}\n=[b'_{i'},b'_{j'}]_{A'},\\;\n\\bar{\\nabla}^{''U}_{b'_{i'}}b''_{j''}=[\\psi b'_{i'},b''_{j''}]_{A''}.\n\\end{equation}\nConnections (\\ref{conexpeU2}) induce $A$-connections\n$\\tilde{\\nabla}',\\tilde{\\nabla}''$ and we shall compute the local\nmatrices of the induced connections. By definition, we have\n$$\\tilde{\\nabla}^{'U}_{b_i}b'_{j'}=\n\\bar{\\nabla}^{'U}_{\\varphi b_i}b'_{j'},\\;\n\\tilde{\\nabla}^{''U}_{b_i}b''_{j''}=\n\\bar{\\nabla}^{''U}_{\\varphi b_i}b''_{j''}$$ and it is easy to\ncheck that the $A$-connections\n$\\tilde{\\nabla}^{'U},\\tilde{\\nabla}^{''U}$ satisfy conditions\n(\\ref{distins2}). Therefore,\n$\\tilde{\\nabla}^{'U},\\tilde{\\nabla}^{''U}$ may be used in the\ncalculation of the relative characteristic classes of $\\psi$ mod.\n$\\varphi$. If we denote $\\varphi b_i=\\varphi_i^{j'}b'_{j'}$ and\nuse expressions (\\ref{conexpeU2}) and the properties of the Lie\nalgebroid brackets we obtain the local connection matrices\n\\begin{equation}\\label{conexinduse} \\begin{array}{l}\n\\tilde{\\omega}^{'k'}_{j'}=\n\\varphi^*\\bar{\\omega}^{'k'}_{j'}-<d_{A'}\\varphi_i^{k'},b_{j'}>b^{*i},\\vspace{2mm}\\\\\n\\tilde{\\omega}^{''k''}_{j''}=\n\\varphi^*\\bar{\\omega}^{''k''}_{j''}-<d_{A''}\\varphi_i^{k'},\\psi b_{j'}>b^{*i}.\n\\end{array}\\end{equation}\nFormula (\\ref{conexinduse}) allows us to write down the local\nconnection matrix of the connection\n$D^1=\\tilde{\\nabla}'+\\tilde{\\nabla}^{''*}$ required by the\ndefinition of the relative classes. Furthermore, we may assume\nthat the local matrix of the orthogonal connection $D^0$ that we\nuse is skew-symmetric. Accordingly, and since $\\psi$ is a Lie\nalgebroid morphism, (\\ref{conexinduse}) yields\n$$\n\\Delta(D^0,D^1)c_1=tr\\,\\left(\\begin{array}{cc}\n\\tilde{\\omega}^{'k'}_{j'}&0\\vspace{2mm}\\\\0&\n-\\tilde{\\omega}^{''j''}_{k''}\\end{array}\\right) =\\varphi^*\ntr\\,\\left(\\begin{array}{cc}\n\\bar{\\omega}^{'k'}_{j'}&0\\vspace{2mm}\\\\0&\n-\\bar{\\omega}^{''j''}_{k''}\\end{array}\\right)=\n\\varphi^*\\Delta(\\bar{\\nabla}^0,\\bar{\\nabla}^1)c_1,$$\nwhere $\\bar{\\nabla}^1=\\bar{\\nabla}'+\\bar{\\nabla}^{''*}$ and\n$\\bar{\\nabla}^0$ is an orthogonal $A'$-connection on $A'\\oplus\nA^{''*}$. This result justifies (\\ref{pullbackmu1}).\\end{proof}\n\\begin{prop}\\label{2relativ}\nFor $h=1$, the relative and absolute characteristic class $\\mu_1$\nof the morphisms $\\varphi,\\psi$ are related by the equality\n\\begin{equation}\\label{eqKGW} \\mu_1(\\psi\\circ\\varphi)=\n\\mu_1(\\varphi)+\\mu_1(\\psi\\,{\\rm mod.}\\,\\varphi).\\end{equation}\\end{prop}\n\\begin{proof} In\nthe computation of $\\mu_1(\\psi\\circ\\varphi)$ we may use an\n$A$-connection $\\nabla+\\nabla^{''*}$ on $A\\oplus A^{''*}$ where,\non the specified neighborhood $U$, $\\nabla$ is given by\n(\\ref{conexpeU}) and $\\nabla''$ is given by (\\ref{conexpeU1}),\nwhile in the computation of $\\mu_{1}(\\psi\\,{\\rm mod.}\\,\\varphi)$\nwe shall use the connections $\\nabla',\\nabla''$ of\n(\\ref{conexpeU1}). Thus, the non-zero blocks of the local\ndifference matrix $\\alpha$ that enters into the expression of the\nrepresentative $1-A$-form of $\\mu_1(\\psi\\circ\\varphi)$ are given\nby the local matrix of\n\\begin{equation}\\label{blocuri}\n\\nabla''-\\nabla=\\nabla''-\\nabla'+\\nabla'-\\nabla\\end{equation} and\nthe opposite of its transposed matrix (in spite of the notation,\ncalculation (\\ref{blocuri}) is for the connection matrices not for\nthe connections). Then, if we use orthogonal connections of\nmetrics where the bases used in (\\ref{conexpeU1}) are orthonormal\nbases (therefore, with trace zero), formula (\\ref{blocuri})\njustifies (\\ref{eqKGW}).\\end{proof}\n\\begin{corol}\\label{corolarKGW} The characteristic class $\\mu_1$\nof a composed morphism $\\psi\\circ\\varphi,\\psi$ is given by formula\n(\\ref{modcompus}).\\end{corol} \\begin{proof} The result is an\nobvious consequence of formulas (\\ref{pullbackmu1}) and\n(\\ref{eqKGW}).\\end{proof}\n\\begin{rem}\\label{obsfinala} {\\rm Formulas (\\ref{modcompus}),\n(\\ref{pullbackmu1}), do not hold for $h>1$ because of the more\ncomplicated expression of the polynomials $c_{2h-1}$ (there is no\nnice formula for the determinant of a sum of matrices).}\\end{rem}\n\nWe finish by showing the relation between the characteristic\nclasses of the base-preserving Lie algebroid morphism\n$\\varphi:A\\rightarrow A'$ and the relative classes defined by the\nfirst jet Lie algebroid $J^1A$; for $\\varphi=\\sharp_A:A\\rightarrow\nTM$ this relation was established in \\cite{CF}.\n\nThe first jet bundle $J^1A$ may be defined as follows. Let $D$ be\na $TM$-connection on $A$ and let $Da$ denote the covariant\ndifferential of a cross section $a\\in\\Gamma A$ (i.e.,\n$Da(X)=D_Xa$, $X\\in\\Gamma TM$). The properties of a connection\ntell us that $Da\\in Hom(TM,A)$ and, if $a(x_0)=0$ for some point\n$x_0\\in M$, then $Da(x_0):T_{x_0}M\\rightarrow A_{x_0}$ is a linear\nmapping that is independent of the choice of the connection $D$.\n(This is not true if $a(x_0)\\neq0$.) If $(x^h)$ are local\ncoordinates of $M$ around $x_0$ and $(b_i)$ is a local basis of\n$\\Gamma A$, and if $a=\\xi^i(x^h)b_i$, the local matrix of\n$Da(x_0)$ is $(D_h\\xi^i(x_0))$ (the covariant derivative tensor),\nwhich reduces to $(\\partial\\xi^i\/\\partial x^h(x_0))$ if\n$\\xi^i(x_0)=0$.\n\nNow, for any point $x_0\\in M$, the space of $1$-jets of cross\nsections of $A$ at $x_0$ is\n\\begin{equation}\\label{spjetx}\nJ^1_{x_0}A=\\Gamma A\/\\{a\\in\\Gamma\nA\\,\/\\,a(x_0)=0,Da(x_0)=0\\}\\end{equation} and each $a\\in\\Gamma A$\ndefines an element $j_x^1a\\in J^1_{x_0}A$ called the $1$-jet of\n$a$ at $x_0$. With the local coordinates and basis considered\nabove, we may write\n$$a=\\xi^i(x^h)b_i=(\\xi^i(x_0)+\\frac{\\partial\\xi^i}{\\partial\nx^h}(x_0)(x^h-x^h(x_0))+o((x^h-x^h(x_0))^2))b_i.$$ Hence,\n$$j_{x_0}^1a=\\xi^i(x_0)j_{x_0}^1b_i+\n\\frac{\\partial\\xi^i}{\\partial x^h}(x_0)j_{x_0}^1((x^h-x^h(x_0))b_i)$$ and\n\\begin{equation}\\label{bazejet}\nj_{x_0}^1b_i,\\,j_{x_0}^1((x^h-x^h(x_0))b_i)=j_{x_0}^1\n(x^hb_i)-x^h(x_0)j_{x_0}^1b_i\\end{equation} is a basis of the\nvector space $J^1_{x_0}A$ such that\n$(\\xi^i(x_0),\\partial\\xi^i\/\\partial x^h(x)(x_0))$ are coordinates\nwith respect to this basis.\n\nA change of the local coordinates and basis of $A$ gives the\ntransition formulas\n\\begin{equation}\\label{schimbcoordjet}\n\\tilde{x}^h=\\tilde{x}^h(x^k),\\,\\tilde{\\xi}^i=\\lambda^i_j(x^k)\\xi^j,\\,\n\\frac{\\partial\\tilde{\\xi}^i}{\\partial\\tilde{x}^h}=\n\\frac{\\partial{x}^k}{\\partial\\tilde{x}^h}\\left(\n\\frac{\\partial\\lambda_j^i}{\\partial{x}^k}\\xi^j+\\lambda_j^i\n\\frac{\\partial{\\xi}^j}{\\partial{x}^k}\\right)\\end{equation}\nand may be seen as the composition of the change of the\ncoordinates $(x^h)$ with the change of the basis $(b_i)$, while\nthe order of the two changes is irrelevant. This remark allows for\nan easy verification of the fact that the change of the\ncoordinates discovered above in $J^1_{x_0}A$ has the cocycle\nproperty. Accordingly, (\\ref{schimbcoordjet}) shows that\n$J^1A=\\cup_{x\\in M}J^1_xA$ has a natural structure of a\ndifferentiable manifold and vector bundle $\\pi:J^1A\\rightarrow M$\nover $M$ called the first jet bundle of $A$.\n\nFrom (\\ref{bazejet}), we see that $(j^1b_i,j^1(x^hb_i))$ is a\nlocal basis of cross sections of $J^1A$ at each point of the\ncoordinate neighborhood where $x^h$ are defined. This basis\nconsists of $1$-jets of local cross sections of $A$, therefore,\nthe cross sections of $J^1A$ are locally spanned by $1$-jets of\ncross sections of $A$ over $C^\\infty(M)$. In the case of a Lie\nalgebroid $A$, the previous remark allows us to define a Lie\nalgebroid structure on $J^1A$ by putting\n\\begin{equation}\\label{strluiJ1A} \\sharp_{J^1A}(j^1a)=\\sharp_Aa,\\;\n[j^1a_1,j^1a_2]_{J^1A}=j^1[a_1,a_2]_A\\end{equation} and by\nextending the bracket to general cross sections via the axioms of\na Lie algebroid. We refer the reader to Crainic and Fernandes\n\\cite{CF} for details. A general, global expression of the Lie\nalgebroid bracket of $J^1A$ was given by Blaom \\cite{B}.\n\nMoreover, (\\ref{strluiJ1A}) shows that the natural projection\n$\\pi^1:J^1A\\rightarrow A$, $\\pi^1(j^1a)=a$ is a base-preserving\nmorphism of Lie algebroids and, if $\\varphi:A\\rightarrow A'$ is a\nmorphism of Lie algebroids, we may define relative characteristic\nclasses of $\\varphi$ modulo $\\pi^1$. Following \\cite{CF}, there\nexist flat $J^1A$-connections $\\nabla^{j^1},\\nabla^{'j^1}$ on\n$A,A'$, respectively, given by\n\\begin{equation}\\label{Jconex}\n\\nabla^{j^1}_{fj^1a_1}a_2=f[a_1,a_2]_A,\\, \\nabla^{'j^1}_{fj^1a}a'=f[\\varphi\na,a']_{A'},\\end{equation} where $a,a_1,a_2\\in\\Gamma A, a'\\in\\Gamma\nA',f\\in C^\\infty(M)$. These connections obviously satisfy\nconditions (\\ref{distins2}), hence,\n$D^1=\\nabla^{j^1}+\\nabla^{'*j^1}$ is a $J^1A$-connection on\n$A\\oplus A^{'*}$ that may be used in Definition \\ref{defclrel} for\nthe present case. We shall prove the following result\n\\begin{prop}\\label{propCF} The relative characteristic classes of\n$\\varphi$ modulo $\\pi^1$ are the images of the corresponding\nabsolute characteristic classes of $\\varphi$ by the homomorphism\n$\\pi^{1*}:H^*(A)\\rightarrow H^{*}(J^1A)$.\\end{prop}\n\\begin{proof} Here, we have the particular case of the situation\nthat existed in Proposition \\ref{1relativ} where $\\pi^1$ comes\ninstead of $\\varphi$ and $\\varphi$ comes instead of $\\psi$.\nTherefore, we may construct connections that correspond to\n(\\ref{conexpeU2}) and the induced $J^1A$-connections and get the\ncorresponding formulas (\\ref{conexinduse}). If we use the local\nbases (\\ref{bazejet}) of $\\Gamma J^1A$, the components\n$\\varphi_i^{k'}$ that appear in (\\ref{conexinduse}) are constant\nand (\\ref{conexinduse}) simply tell us that the local connection\nforms of the induced connections are the pullback of the\nconnection forms of the connections (\\ref{conexpeU2}) by $\\pi^1$.\nOf course, the same holds for the curvature forms, and, if we also\nuse a $J^1A$-orthogonal connection of $A\\oplus A^{'*}$ that is\ninduced by an $A$-orthogonal connection, we see that $\\pi^{1*}$\ncommutes with the Lehmann morphism, which implies the required\nresult.\\end{proof}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nConsider a sample ${\\mathbf x}_1{,}\\ldots{,} {\\mathbf x}_n$, where ${\\mathbf x}_i\\in\\RR^d$. We have a\nfeedforward neural network with a given architecture (but the weights\nare unknown). Each sample point ${\\mathbf x}_i$ has binary labels, either +1 or\n-1. Sauer's lemma provides an upper bound on the number of possible\nlabelings that could be generated by a hypothesis class (the\n\\emph{growth function}) in terms of the VC dimension of the hypothesis\nclass.\n\nWe are interested in hypothesis classes corresponding to neural\nnetworks with a fixed architecture but unspecified weights. While it\nis hard to exactly specify the VC dimension of this class, upper\nbounds on the VC dimension and the growth function are easily derived,\nsee for example~\\cite[Section 6.2]{anthony2009neural}. The growth\nfunction for a feedforward, linear threshold network is upper bounded\nby $(enk\/W)^W$, where $k$ is the number of neurons in the network, and\n$W$, the number of weights.\n\nOur goal in this paper is to generate labels of the sample uniformly\nat random from the set of all possible labelings that a given\nfeedforward architecture can provide. We obtain a polynomial time (in\nboth the number of samples and the size of the network), near uniform\nsampling from arbitrary feedforward networks.  In the special case of\na single neuron, we also provide a random walk based algorithm for\nperfectly uniform sampling, and with polynomial mixing time for the\nrandom walk.\n\nAside from the theoretical interest in generating labelings, we are\nalso motivated by questions in property testing. Namely, we want to\nestimate the statistics of all labelings generated by a given\narchitecture.  As an example, we may want to find out the the\nprobability that a subset of samples are all labeled the same if all\nlabels were generated at random from the given architecture.  In\nfuture work, we intend to leverage these insights into better\ninitializations of neural networks while training.\n\nWe obtain these results by developing insights on random walks between\nchambers of intersecting hyperplanes in high dimensions.  This is a\nwell studied area, see for example~\\cite{stanley2004introduction}.\nGeneral arrangements of these hyperplanes intersect in complicated\nways, as in our problem, and random walks between these chambers is\nnontrivial. It is common to visualize the geometry of these\narrangments by means of a \\emph{chamber graph}, see Chapter 7 of\n\\cite{ovchinnikov2011graphs} for a synopsis of such chamber graphs.\nRandom walks over hyperplane arrangements appears in contexts quite\ndifferent from ours. For example, Bidigere, Hanlon and Rockmore\nmodeled card shuffling in \\cite{bidigare1999combinatorial}, with such\nrandom walks. Some other applications are\nin~\\textit{e.g.,}\\xspace~\\cite{brown1998random,athanasiadis2010functions,pike2013eigenfunctions,bjorner2008random}.\n\nThe statistics of the random walks considered in the references\nabove is different from ours. Typically, these authors provide an\nexplicit expression to estimate the eigenvalues of the random walk to\nbound the mixing time. In our paper, we use conductance to\nunderstand the mixing properties of our random walk as\nin~\\cite{levin2017markov} and~\\cite{berestycki2016mixing}.\n\n\\ignore{The more general problem of uniformly sampling geometric objects is\nextensively studied in Markov Chain Monte Carlo (MCMC) literature,\ne.g. Dyer, Frieze and Kannan's work~\\cite{dyer1991random} on\nestimating the volume of high dimensional convex bodies.  }\n\n\\section{Setup and Notations}\n\\newcommand{\\W}{{\\textbf W}} We consider a feed-forward linear\nthreshold neural network with $L$ layers.  The input to the network is\n$d-$dimensional and there is a single binary output label. Namely,\ni.e. any neuron with parameters $\\textbf{w},b$, (${\\textbf w}\\in \\RR^d$,\n$b\\in\\RR$) outputs $\\sigma(\\textbf{x}^T\\textbf{w}+b)$ on an input\n${\\mathbf x}\\in\\RR^d$, where $\\sigma(u)=1$ if $u\\ge 0$ and $\\sigma(u)=0$\notherwise. In subsequent work, we extend our results to more general\nactivation functions.\n\nLet $N$ be the graph of the feedforward neural network with a fixed\narchitecture and $W$ different parameters (the weights and thresholds\nput together). Let $\\W\\in \\RR^W$, and let $N_{\\W}$ be the neural\nnetwork which assigns the parameters of $N$ to be $\\textbf{W}$. For\nany given architecture $N$, let\n$f_{\\W}:\\textbf{x}\\in\\mathbb{R}^d\\to\\{0,1\\}$ be the function expressed\nby $N_{\\W}$.\n\nThe vectors $\\textbf{x}\\in\\mathbb{R}^d$ are the input and\n$f_{\\W}(\\textbf{x})$ are the labels assigned to $\\textbf{x}$.\nFor a length $n$ sample\n$X=\\{\\textbf{x}_1,\\cdots,\\textbf{x}_n\\in \\mathbb{R}^d\\}$, let\n\\[\n  S_X=\\{(f_{\\W}(\\textbf{x}_1),\\cdots,f_{\\W}(\\textbf{x}_n))\\mid\n  \\textbf{W}\\in \\mathbb{R}^W\\}\n\\]\nbe the set of all labelings that can be generated on $X$ by the\narchitecture $N$. Note that the set $S_X\\subset \\sets{0,1}^n$ and for\n$W<n$,~\\cite[Section 6.2]{anthony2009neural}\n(or~\\cite{shalev2014understanding})\n\\[\n|S_X|\\le \\Paren{\\frac{enk}W}^W.\n\\]\nWhen $W\\ge n$, $|S_X|\\le 2^n$, or $X$ is potentially shattered.\n\n\\noindent\n\\textbf{Problem} For a given architecture $N$ and data $X$, how\ncan we randomly sample from $S_X$, in time polynomial in both $n$ and\n$W$, such that any labeling $v\\in S_X$ appears with\nprobability at least $\\Omega(1\/|S_X|)$?\n\n\\paragraph{Background}\nA hyperplane in $\\RR^d$ (or a hyperplane\nin $d$ dimensions) is the set of all points ${\\textbf w}\\in\\RR^d$ \nsatisfying ${\\mathbf x}^T {\\textbf w}=0$ for some fixed vector ${\\mathbf x}\\in\\RR^d$. \nLet $N$ be a single neuron with input dimension $d$. As before,\n$X=\\Sets{{\\mathbf x}_1{,}\\ldots{,} {\\mathbf x}_n\\in \\RR^d}$ is a length $n$ sample.\n\nLet ${\\textbf w}\\in \\RR^d$ and $b\\in\\RR$.  Physically, the vector in\n$d+1$ dimensions, $({\\textbf w}, b)\\in\\RR^{d+1}$ defines the parameters of\nthe single neuron $N$.  For each sample point\n$\\textbf{x}_i\\in \\mathbb{R}^d$, define $P_i$ to be the hyperplane in\nthe parameter space $\\RR^{d+1}$:\n\\[\n\\textbf{x}_i^T\\textbf{w}+b=0.\n\\] \nWe start with a visualization from~\\cite{anthony2009neural}.\n\\begin{theorem} All parameter vectors that belong to the same connected\n  component of $\\mathbb{R}^{d+1}\\backslash \\bigcup_i P_i$ label $X$ in\n  the same way. Conversely, different components have different\n  labelings on $X$.\n\\end{theorem}\n\nWe recall a few standard terms regarding hyperplane arrangements\nformed by $P_1{,}\\ldots{,} P_n$.\n\\begin{list}{$\\bullet$}{\\setlength{\\listparindent}{0pt}\\setlength{\\leftmargin}{10pt}}\n\n\\item The connected components in\n  $\\mathbb{R}^{d}\\backslash \\bigcup_{i=1}^nP_i$ are called\n  \\emph{chambers} (or regions).\n\n\\item The \\emph{chamber graph} is constructed as follows: assign a\nvertex to every chamber. Two vertices are connected if their associated \nchambers share a common face. \n\n\\item Any hyperplane arrangement is \\emph{centered} if the\n  intersection of the component hyperplanes contains the origin. \n  In our case, $\\bigcap_{i=1}^nP_i$ always contains the origin,\n  \\ie the samples generate a \\emph{centered} arrangement\n  in the parameter space.\n\n\\item A collection of $n$ centered hyperplanes in $\\RR^{d+1}$ is\n  in \\emph{general} position, if for all $k\\le d+1$, every\n  intersection of $k$ distinct hyperplanes forms a $d+1-k$-dimension\n  linear space, and any intersection of more than $d+1$ hyperplanes is\n  contains only the origin. Randomly chosen\n  planes are in general position almost surely.\n\\end{list}\n\n\\paragraph{Psuedo polynomial optimal training algorithm}\nA theoretically useful framework was introduced in Theorem 4.1\nof~\\cite{arora2016understanding} for ReLU networks, where the network\nsize $W$ is treated as a constant, and we look at the dependency\npurely on the sample size $n$ (thereby treating $n^W$ as a\npolynomial).\n\nWe note that our near-uniform polynomial time sampling procedure\nimplies a probabilistic, psuedo-polynomial training algorithm that\nattains the global minimum for any feedforward linear threshold neural\nnetwork. This implication is immediate from the coupon collector\nproblem---since given any confidence, generating at most\n$O(n^W\\log n)$ samples guarantees that we have seen every possible\nlabeling that can be produced.\n\n\\section{Properties of hyperplane arrangements}\nWe summarize a few useful properties of hyperplane arrangments that\nwe will use in our arguments in the paper.\n\\begin{proposition}[{\\cite[Theorem 3.1]{anthony2009neural}}]\n  The number of chambers in a centered hyperplane arrangement formed\n  by $n$ hyperplanes in $d$ dimensions in the general position is\n$$2\\sum_{i=0}^{d-1}\\binom{n-1}{i}.$$\n\\end{proposition}\nIn fact, even sampling all labels of a sample of size $n$, even when\nthe network consists of a single neuron, in time polynomial in both\n$n$ and dimension $d$ of the data points, is non-trivial. The number\nof chambers by Theorem 1 is the number of labels on a size-$n$ sample,\nwhich from the above Proposition is roughly $O(n^d)$.  Clearly,\ntrivial enumeration of labels is out of question. As we will see later\nin Section~\\ref{s:rcr}, this is not the only difficulty even for a\nsingle neuron.\n\n\\begin{proposition}\n  Let $Q_1{,}\\ldots{,} Q_n$ form a centered hyperplane arrangement in $d$\n  dimensions. Let $\\v_i\\in\\RR^d$ be any vector normal to the \n  hyperplane $Q_i$. If $\\v_1{,}\\ldots{,} \\v_n$ have rank $r$,\n  then any chamber in the hyperplane arrangment has at least $r$\n  faces.\n\\end{proposition}\n\\begin{proof}\n  Let $V=\\sets{\\v_1{,}\\ldots{,} \\v_n, -\\v_1{,}\\ldots{,} -\\v_n}$.  Suppose the\n  proposition is false. Then there exists a chamber with exactly $b<r$ faces.\n  Without loss of generality, let $\\textbf{u}_1\\in V,\\cdots,\\textbf{u}_b\\in V$\n  be the normal vectors of the $b$ different faces of this chamber respectively,\n  such that for any point $\\textbf{x}$ within the chamber\n  \\[\n    \\u_i^T{\\mathbf x}>0.\n  \\] \n  Since the rank of $\\v_1{,}\\ldots{,} \\v_n$ is $r>b$, we can choose a vector\n  $\\u_{b+1}\\in V$ such that $\\u_{b+1}$ is linearly independent from\n  $\\u_1,\\cdots,\\u_b$. \n\n  We now show that the hyperplane that determined by $\\u_{b+1}$ is\n  also a face of the chamber by proving that there is a point\n  $\\textbf{x}'$ in the chamber satisfying\n  \\begin{equation}\n    \\label{eq:const}\n    \\u_i^T\\textbf{x}'>0\n    \\;\\;\\;\\; 1\\le i\\le b\n    \\quad\n    \\text{ and }\n    \\quad\n    \\u_{b+1}^T\\textbf{x}'=0. \n  \\end{equation}\n  Since $\\u_{b+1}$ is linearly independent of $\\u_1{,}\\ldots{,} \\u_b$, we\n  can choose a vector $\\textbf{y}$ such that $\\textbf{y}^T\\u_i=0$ for\n  $1\\le i\\le b$ but $\\textbf{y}^T\\u_{b+1}\\not=0$.\n  Now let $\\textbf{x}$ be any point in the chamber and set\n  $\\textbf{x}'=\\textbf{x}+t\\textbf{y}$ where\n   $t=-\\u_{b+1}^T{\\mathbf x}\/\\u_{b+1}^T\\textbf{y}$.\n   It is easy to verify now that ${\\mathbf x}'$ satisfies~\\eqref{eq:const}.\n   This contradicts the assumption that the chamber contained\n   $b$ faces, where $b<r$.\n\\end{proof}\n\\begin{proposition}\nThe chamber graph of any hyperplane arrangement $Q_1{,}\\ldots{,} Q_n$ in $\\RR^d$ in general position satisfies (i) the degree of any vertex is at least $d$ and at most $n$, and (ii) any pair of vertices has graph distance at most $n$.\n\\end{proposition}\n\\begin{proof}\n(i) from Proposition $2$, (ii) from \\cite[Lemma 7.15]{ovchinnikov2011graphs}.\n\\end{proof}\n\n\\section{Sampling labelings }\nFor the sample $X=\\sets{{\\mathbf x}_1{,}\\ldots{,} {\\mathbf x}_n}$, where ${\\mathbf x}_i\\in\\RR^d$, \n$S_X$ is the set of all possible labels generated on $X$\nby the network $N$. We would like to sample from $S_X$ uniformly.\n\nIn Section~\\ref{s:rcr}, we let $N$ be a single neuron and even this\nturns out to be non-trivial. Inspired by the inductive approach for\ncomputing hyperplane partition number \\cite[Chapter\n2]{stanley2004introduction}, we derive Algorithm RS (for Recursive\nSampling) in Section~\\ref{s:rcr} that generates a label from $S_X$\nalmost uniformly.\n\nIn Sections~\\ref{s:rw} and~\\ref{s:an} we expand in two directions. In\nSection~\\ref{s:rw}, we provide means to perfectly sample from all\nlabelings of a single neuron using a random walk on $S_X$ with a perfectly\nuniform stationary distribution. This allows us to sample from $S_X$\nperfectly uniformly.  The mixing time of this random walk is as yet\nunproven, but we provide partial evidence (empirical as well as proofs\nfor small dimensions) that this random walk is fast mixing, with\nmixing time at most linear in the number of dimensions and at most\nquadratic in the number of samples.\n\nIn Section~\\ref{s:an}, we build on our RS approach to sample from\narbitrary feedforward networks in time (true) polynomial in the sample\nsize $n$, network size $W$ and input dimension ($d$), showing that\neven for arbitrary networks, we get near-uniform sampling of the\npossible labels that could be produced by the network.\n\n\\subsection{The recursive approach}\n\\label{s:rcr}\nFrom Theorem 1, to sample uniformly from $S_X$ we only need to sample the\nweights uniformly from the connected components of $\\mathbb{R}^{d+1}\\backslash \\bigcup_iP_i$.\n\n\nHowever, even for a single neuron, this is not trivial. As already\nnoted from Proposition 2, the basic combinatorial difficulty comes\nfrom the fact that there are roughly $n^d$ labelings for almost all\nsamples $X$---therefore the number of chambers is exponential in the\ndimension $d$. Clearly one can not simply enumerate all the possible\ncomponents.\n\nBut a bigger difficulty comes from the fact that the arrangement of\nthe hyperplanes can be very heterogeneous. The volume of some of the\nchambers can be arbitrarily small and therefore such chambers may be\ndifficult to find. We settle this problem by using a recursive\nsampling approach that is inspired by the inductive approach for\ncomputing hyperplane partition number.\n\nOur recursive algorithm RS($\\v_1{,}\\ldots{,} \\v_k$) (see Algorithm $1$) takes\nas its inputs $k$ unit vectors $\\v_1{,}\\ldots{,} \\v_k$, all from, say\n$\\RR^m$. The vectors $\\v_1{,}\\ldots{,} \\v_k$ are interpreted as normal\nvectors of $k$ distinct centered hyperplanes in $\\RR^m$. For\nsimplicity, the reader can assume that these hyperplanes are in the\ngeneral position, but they do not have to be. To sample from $S_X$, we\nwould therefore simply call RS($\\tilde{{\\mathbf x}}_1{,}\\ldots{,} \\tilde{{\\mathbf x}}_n$),\nwhere $\\tilde{{\\mathbf x}}_i= \\frac{({\\mathbf x}_i,-1)}{||({\\mathbf x}_i,-1)||_2}\\in\\RR^{d+1}$\nand ${\\mathbf x}_i\\in X$.\n\nThe call RS($\\v_1{,}\\ldots{,} \\v_k$) works recursively on the\ndimension $m$ of the vectors $\\v_i$ and $k$, by calling RS with a new set of\nvectors $\\u_1{,}\\ldots{,} \\u_{k'}$ in $\\RR^{m-1}$ with $k'\\le k-1$.\nThe base case is when RS is called with vectors in 1 dimension or when\n$k=1$. When RS is called with vectors in 1 dimension, the problem is\ntrivial since there is only one centered hyperplane arrangement in\n$1-$dimension, the origin. When RS is called with $k=1$ (no matter the\ndimension of the single input vector), the problem is also trivial\nsince there are only two chambers for one hyperplane.\n\nTo generate the vectors $\\u_i$ in $\\RR^{m-1}$, we choose a\nhyperplane at random from $\\v_1{,}\\ldots{,} \\v_k$, say $\\v_i$, and compute\nthe intersection of $\\v_i$ with all the remaining hyperplanes. These\nintersections are at most $k-1$  hyperplanes in $\\mathbb{R}^{m-1}$ \nand let $\\u_1{,}\\ldots{,} \\u_{k'}$ be the unit normal vectors of\nthese hyperplanes (written in the specific orthonormal basis indicated).\n\n\n\\begin{algorithm}[t]\n\\caption{RS($\\v_1{,}\\ldots{,} \\v_k$)}\\label{alg:1}\n\\textbf{Input:} $\\textbf{v}_1,\\cdots,\\textbf{v}_k\\in \\mathbb{R}^m$, interpreted\nas unit normal vectors of $k$ (distinct) centered hyperplanes in an $m-$dimensional space. \\\\\n\\textbf{Output:} point $\\textbf{y}\\in \\mathbb{R}^m$ representing a chamber\nin the hyperplane arrangement formed by $\\v_1{,}\\ldots{,} \\v_k$, \\\\\n\nLet $P_i$ be the hyperplane in $\\RR^m$ orthogonal to $\\v_i$.\n\\begin{itemize}\n\\item[1.]If $m=1$ output -1 or 1 with equal probability. If $m>1$ but $k=1$, output $\\v_1$ or $-\\v_1$ with equal probability.\n\\item[2.] Uniformly choose an index $I$ from $\\sets{1{,}\\ldots{,} k}$. \n\\item[3.] For hyperplane $P_I$, choose an arbitrary orthonormal basis\n  $B\\in\\RR^{(m-1)\\times m}$. Note that $P_I$ is a $(m-1)$-dimensional\n  linear space in $\\RR^m$, and the $m-1$ rows of $B$ contain the \n  orthonormal basis vectors, each being a vector in $\\RR^m$.\n\\item[4.] Compute the intersection of $P_I$ with $P_j$,\n  $j\\in \\sets{1{,}\\ldots{,} k}\\backslash \\sets{I}$. \n\\item[5.] Set $\\v_j'$ to be the unit vector in $P_I$ normal to $P_I\\cap P_j$\n  (written using the basis $B$),\n  $j\\in \\sets{1{,}\\ldots{,} k}\\backslash \\sets{I}$. Note $\\v_j'\\in\\RR^{m-1}$.\n\\item[6.]$\\textbf{x}=RS(\\u_1,\\cdots,\\u_{k'})$, where $\\u_1,\\cdots,\\u_{k'}$ are the distinct vectors among $\\{\\v_j'\\mid j\\not=I\\}$. Note $k'\\le k-1$.\n\\item[7.] Compute the smallest distance $\\delta$ of $\\textbf{x}^TB$ to\n  the planes $P_j$ with $j\\not=I$.\n\\item[8.] Let $t$ be -1 or 1 with equal probability, output\n  $\\textbf{y}=\\textbf{x}^TB+t\\delta\\textbf{v}_I$.\n\\end{itemize}\n\\end{algorithm}\n\n\\begin{theorem}\n  Let $V=\\sets{\\v_1{,}\\ldots{,} \\v_k}$ where $\\v_i\\in\\RR^m$ and rank of\n  $V$ is $m$. Let $C_V$ be the set of non-empty chambers induced by the $k$\n  centered hyperplanes orthogonal to the vectors in $V$. Algorithm\n  RS($\\v_1{,}\\ldots{,} \\v_k$) runs in $O(km^3)$ time and any chamber in the\n  hyperplane arrangment induced by $V$ is sampled with probability at\n  least\n  $$\n  \\frac{1}{2^m\\binom{k}{m}}\\ge \\left(\\frac{m}{2ek}\\right)^m,\\text{ where }e\\text{ is the base of nature logorithm}.\n  $$\n\\end{theorem}\n\\begin{proof} (Outline only)\n  The algorithm will run at most $m$ recursive iterations. For each\n  iteration, we need $O(m^2)$ to compute the base of the null space\n  (Step 3) and $O(km^2)$ time in Step 7 to compute the projection of\n  each input vectors to the plane chosen in Step 2. This yields the\n  total complexity to be $O(km^3)$.\n\n  To see the probability lower bound, define\n  $$p(m, k)=\\min_{V,c\\in C_V}\\mathrm{Pr}[RS(V)=c],\\text{ with }\\text{rank}(V)=m\\text{ and }|V|\\le k.$$\n  We now claim that\n  $$p(m, k)\\ge \\frac{m}{2k}p(m-1, k-1).$$\n  This is because any chamber $c\\in C_V$ has at least $m$ faces by\n  Proposition 2. For any chamber $c\\in C_V$, we therefore have\n  probability at least $\\frac{m}{2k}$ of choosing both a hyperplane\n  that forms the face of $c$ and the direction of the hyperplane that\n  faces the chamber $c$. Conditioned on this choice of hyperplane and\n  direction, we need to obtain the probability that the recursive call\n  in step in Step 6 returns a point in the face of $c$.\n\n  Observe that the face of $c$ is a $m-1$-dimensional linear space. In\n  Step 6, note that the rank of $\\sets{ \\u_1{,}\\ldots{,} \\u_{k'}}$ is exactly\n  $m-1$, but $k'$ can be less than $k-1$. The theorem follows by solving the recursive\n  inequality, standard approximations on binomial coefficient and by\n  noting that when $m=1$, there are two chambers, thus yielding\n  $p(1,k)=1\/2$ for all $k$.\n\\end{proof}\nNote that when $\\text{rank}(X)=d$ the above probability is\n${\\cal O}\\Paren{\\frac{d!}{2^dn^d}}$, a factor $\\frac{1}{2^d}$ off the\nhyperplane slicing bound $2\\sum_{i=0}^{d-1}\\binom{n-1}{i}$ in\nProposition 1. Note also that if the input vectors in $\\RR^d$ have\nrank $m<d$, the above approach still works. We can effectively project\ndown the inputs into $\\RR^m$ by choosing a basis for $\\RR^d$\nthat contains $d-m$ vectors that are orthogonal to the span of the\ninput vectors.\n\n\\subsection{A random walk approach}\n\\label{s:rw}\nTo mitigate the fact that the recursive approach above only yields\napproximately uniform sampling, We introduce a random walk based\nalgorithm that samples arbitrarily close to uniform. Specifically, we\nrun Algorithm NRW on a \\emph{lazy chamber graph}, both outlined\nbelow. One component of Algorithm NRW is Algorithm Chamber, that\ndetermines which chamber an input point belongs to.\n\\begin{algorithm}[t]\n\\caption{NRW}\\label{alg:2}\n\\textbf{Input:} walk length $T$ and hyperplanes $P_1,\\cdots,P_n$ in $\\mathbb{R}^m$\\\\\n\\textbf{Output:} point $\\textbf{w}\\in \\mathbb{R}^m$ and chamber $c$\n\\begin{itemize}\n\\item[1.] Initialize $\\textbf{w}_0=RS(\\textbf{v}_1,\\cdots,\\textbf{v}_n)$, where $\\textbf{v}_i$ is a normal vector of $P_i$\n\\item[2.] Set $c_0= $ Chamber$({\\textbf w}_0)$. $c_0$ will be the chamber\n  in the arrangement $\\sets{P_i}$ that contains\n  $\\textbf{w}_0$.\n\\item[3.] For $t=1$ through $T$, do\n  \\begin{itemize}\n  \\item[a.] Uniformly choose a face of $c_{t-1}$\n  \\item[b.] Set $c_t$ to be the chamber adjacent to $c_{t-1}$ and across the\n    face chosen in step (a.)\n  \\item[c.] Set $\\textbf{w}_t$ to any point in the chamber $c_t$\n    \\end{itemize}\n \\item[4.] Output ${\\textbf w}_T$ and $c_T$\n \\end{itemize}\n\\end{algorithm} \n\n\\begin{algorithm}[b]\n\\caption{Chamber}\\label{alg:3}\n\\textbf{Input:} point $\\textbf{w}\\in \\mathbb{R}^m$ and hyperplanes $P_1,\\cdots, P_n$\\\\\n\\textbf{Output:} The faces $P_{i_1},\\cdots,P_{i_k}$ \nof the chamber containing $\\textbf{w}$.\n\\begin{itemize}\n\\item[1.]Compute $\\sigma_i=\\text{sign}(\\textbf{w}^T\\textbf{v}_i)$.\n\\item[2.]For $1\\le i\\le n$ do:\n\\begin{itemize}\n\\item[3.]Define a linear program with $\\textbf{w}^T\\textbf{v}_i=0$ and $\\sigma_j(\\textbf{w}^T\\textbf{v}_j)>0$ for $j\\not =i$.\n\\item[4.]If the linear programming in step $3$ has a solution, add $P_i$ to the collection.\n\\end{itemize}\n\\end{itemize}\n\\end{algorithm} \n\n\\begin{theorem}\nAlgorithm Chamber runs in polynomial time both on $d$ and $n$.\n\\end{theorem}\n\\begin{proof}\nThe theorem follows since linear programming can be solved in polynomial time \\cite{megiddo1986complexity}.\n\\end{proof}\n\n\\paragraph{Analysis} We first analyze random walk defined by\nAlgorithm NRW over the simple chamber graph, assuming the hyperplanes are in general position.\nWith this assumption any vertex in the \\emph{chamber graph} has degree\nat least $d$ and at most $n$ from Proposition 3. Furthermore, from\nProposition 3 the graph is connected and the distance between any two\nvertexes is at most $n$.\n\nSince the random walk is a reversible Markov chain, the stationary\ndistribution $\\pi$ of the random walk will be proportional to the\ndegree of the vertices~\\cite[Chapter 1.6]{levin2017markov}. From our\nobservation on the bounds of degrees in Proposition 3,\nwe will therefore have for any two vertices $u$ and $v$\n\\[\n  \\frac{d}{n}\\le\\frac{\\pi(u)}{\\pi(v)}\\le \\frac{n}{d}.\n\\]\nThe more fundamental question is the mixing time of the\nrandom walk, or how quickly the walk generates stationary\nsamples. While there are several approaches to analyze the mixing time, we\nfocus on Cheeger's inequality \\cite[Theorem 13.14]{levin2017markov}\nthat bounds the spectral gap of the random walk's transition matrix \nusing the \\emph{conductance} of the graph. Recall that the\nconductance of a graph is\n$$\\min_{A\\subset V, \\text{vol}|A|\\le \\frac{1}{2}\\text{vol}|V|}\\frac{|\\partial A|}{\\text{vol}|A|},$$\nwhere $V$ is the vertex set, $\\partial A$ is size of the cut between\n$A$ and $V\\backslash A$, $\\text{vol}|A|$ is the sum of degrees of\nvertexes in $A$. The following theorem gives a lower bound on the\nconductance of chamber graph when dimension $d=2$.\n\\begin{theorem}\nThe chamber graph of $2$-dimensional hyperplane arrangement with size $n$ that is in the general position has conductance lower bounded by $\\frac{1}{2n}$.\n\\end{theorem}\n\\newcommand{\\text{vol}}{\\text{vol}}\n\\begin{proof}\n  For any set $A$ of vertices in the chamber graph with size no\n  greater than $\\frac{1}{2}|V|$, we will show that the conductance of $A$, \n  $\\frac{|\\partial A|}{\\text{vol}|A|}$, is lower bounded as follows\n  \\[\n    \\frac{|\\partial A|}{\\text{vol}|A|}\\ge \\frac{1}{2n}.\n  \\] \n  Let $X$ be the set with smallest volume satisfying\n  \\[ \n    X = \\arg \\min_{\\substack{A\\\\ \\text{vol}(A)\\le {\\frac12} \\text{vol}(V)}} \\frac{|\\partial A|}{\\text{vol}|A|}.\n  \\]\n  We first claim that $X$ must be connected. If not, we can write $X$\n  as the union of (maximally) connected components, \\ie\n  $X=\\bigcup_{i=1}^r X_i$, where $X_i$ are the maximally connected\n  components within $X$ (in particular, note that there are no edges\n  between distinct $X_i$). \n  Then, if $a_i=\\partial X_i$ and $b_i=\\text{vol}(X_i)$, then\n  \\[\n    \\frac{|\\partial X|}{\\text{vol}|X|} = \n    \\frac{a_1+a_2+\\cdots+a_r}{b_1+b_2+\\cdots+b_r}\\ge\n    \\min_{i\\in[r]}\\frac{a_i}{b_i},\n  \\]\n  implying that $X_i$ has lower conductance than $X$ and is smaller in\n  size than $X$, a contradiction.\n\n  Let $S$ be the boundary surface of the chambers corresponding to\n  vertexes in $X$. Since $X$ is connected, we must have $S$ to be\n  piece-wise line segments. \n\n  We now claim that $S$ will partition the chamber graph into two\n  connected components. Since $X$ is connected, we just have to show\n  that $V\\backslash X$ is also connected.\n\n  Suppose not, and let $V\\backslash X= \\bigcup_{i=1}^{m} Y_i$, where\n  $Y_i$ are maximally connected, and $V\\backslash X$ is the union of\n  $m$ different connected components.\n  Let $c_i=\\partial Y_i$ and $d_i=\\text{vol}(Y_i)$. Then we have\n  \\[\n    \\sum_{i=1}^m c_i=|\\partial X|,\n  \\]\n  and since $\\text{vol}(V\\backslash X)= \\text{vol}(V)-\\text{vol}(X)$ and\n  $\\text{vol}(X)\\le {\\frac12} \\text{vol}(V)$, we have\n  \\[\n    \\sum_{i=1}^m d_i\\ge {\\frac12} \\text{vol}(V)\\ge  \\text{vol}(X).\n  \\]\n  Therefore, there must be some component $i$ such that\n  \\[\n    \\frac{c_i}{d_i}\\le \\frac{\\sum c_i}{\\sum d_i} \\le \\frac{|\\partial\n      X|}{\\text{vol}|X|}.\n  \\] \n  If $Y_i$ satisfies $\\text{vol}(Y_i)\\le \\frac{1}{2}\\text{vol}|V|$, \n  then again we have a contradiction because of the following. If\n  $c_i\/d_i <  \\frac{|\\partial X|}{\\text{vol}|X|}$, we are done. \n  If   $c_i\/d_i = \\frac{|\\partial X|}{\\text{vol}|X|}$, it means\n  that every component in $V\\backslash X$ has conductance\n  $\\frac{|\\partial X|}{\\text{vol}|X|}$. But if there are more than\n  two components in $V\\backslash X$, then $X$ has a larger \n  cut $\\partial X$ than each of the components, and therefore\n  must have a larger volume as well, contradicting the assumption\n  on $X$.\n\n  If $\\text{vol}(Y_i)\\ge \\frac{1}{2}\\text{vol}|V|$, then consider the \n  set $Z= V \\backslash Y_i$. Note that\n  \\[\n    Z = X \\bigcup \\bigl( \\cup_{j\\ne i} Y_j \\bigr).\n  \\]\n  Now $|\\partial Z|=|\\partial Y_i| \\le |\\partial X|$. This follows\n  since there is no boundary between $Y_i$ and any of the other $Y_j$,\n  and the only boundary $Y_i$ has is with $X$. Furthermore,\n  $\\text{vol}(Z) > \\text{vol}(X)$, implying that $Z$ has lower conductance \n  than $X$, again a contradiction.\n\n\n  Now, we know that the boundary $S$ between the chambers in $X$ and\n  the rest of the hyperplane arrangement is exactly a piece-wise line\n  segment that separates $\\mathbb{R}^2$ into two connected\n  components. There are only 3 possibilities, as shown in Figure 1. We\n  now observe $\\text{vol}|X|$ is exactly the sum of the\n  $1$-dimensional faces in the arrangement that intersect with\n  $X$. Since there are at most $n$ lines in the arrangement, there\n  exist a line $P$ that intersect with $X$ (or $V\\backslash X$) by at\n  least $\\frac{\\text{vol}|X|}{n}$ many faces, see figure 1.  The\n  number of faces in $S$ is no less than the number of faces in $P$,\n  because any line that intersects with $P$ in $X$ must also intersect\n  with $S$, and at most two lines can intersect at the same point on\n  $S$ by our general position assumption. The theorem now follows.\n\\begin{figure}\n  \\centering \\vspace{-.5in}\n\\includegraphics[width=0.6\\textwidth]{part1.pdf}\n\\vspace{-1.5in}\n\\caption{Possibility of piece-wise linear partition}\n\\label{fig:slowmixing}\n\\end{figure}\\end{proof}\n\nFor the general dimension case, we have the following conjecture.\nSee Appendix for justification and partial proofs.\n\\begin{conj}\n  The conductance of any $d$-dimensional general position hyperplane\n  arrangement of size $n$ is lower bounded by\n$\\frac{1}{\\text{poly}(n,d)}$.\n\\end{conj}\n\n\\begin{remark}\n  Note that the requirement for general position of the hyperplanes is\n  necessary for fast mixing given by the Conjecture above. Else it is\n  easy to construct a hyperplane arrangement with mixing time lower\n  bounded by $O(\\frac{n^d}{2^d})$. As shown in Figure 2, the cut made\n  by the gray shaded top plane has only $4$ boundary chamber but the\n  total number of chambers below the plane is roughly $n^2$ (in two\n  dimensions, while in $d+1$ dimensions, we will have the cut and\n  volumne to be $2^{d+1}$ and $O(n^d)$ respectively).\n\\begin{figure}[!b]\n\\includegraphics[width=0.5\\textwidth]{hyperplane.pdf}\n\\vspace{-.7in}\n\\caption{Hyperplane arrangement with small conductance}\n\\label{fig:slowmixing}\n\\end{figure}\n\\end{remark}\n\n\\paragraph{Lazy Chamber graph}\nAlgorithm NRW on the regular chamber graph will not give an\n\\emph{exact} uniform sampling, but is off by a factor of $d\/n$ as\nmentioned above. This is easily fixed by adding dummy vertices and\ndummy edges to each vertex in the chamber graph raising the degree of\nevery vertex in the original chamber graph to $4n$. Call such a\ngraph to be \\emph{lazy chamber graph}.\n\nWe will call the vertex in the original chamber graph to be\n\\emph{chamber vertex} and the dummy vertices added to be\n\\emph{augmentation} vertices. The stationary probability of the new\nrandom walk, restricted on the chamber vertices, is exactly uniform.\nIf the Algorithm NRW on the chamber graph is fast mixing, we can\nshow that Algorithm NRW on the lazy chamber graph is also fast mixing:\n\n\\begin{lemma}\n  If the conductance of the chamber graph is $g$, the lazy chamber\n  graph has conductance $\\ge \\frac{g}{8n^2}$.\n\\end{lemma}\n\\begin{proof}\nWe only need to show that any subset $A$ of vertex in the lazy chamber\ngraph we have\n$\\frac{|\\partial A|}{|A|}\\ge \\frac{g}{8n}$.\nWe observe that if an augmentation vertex is in $A$, then the chamber\nvertex attached to it must also be included in $A$.  We denote\n$A'\\subset A$ to be the set of all chamber vertexes in $A$. \n\nThe vertexes in $A'$ can be partitioned into two classes,\n$A'=B'\\cup C'$ where $B'$ is the set of all chamber vertices that have\nall their attached augmentation vertices in $A$ and $C'$ is the\ncomplement of $B'$ in $A'$.\nSimilarly, $B$, $C$ to be the sets that contains\nalso the attached new vertex of $B'$ and $C'$ in $A$). We have\n$$\\frac{|\\partial{A'}|}{|A'|}=\\frac{|\\partial B'|+|\\partial C'|}{|B'|+|C'|}\\ge \\frac{|\\partial B'|}{|B'|},$$\nsince all vertexes in $C'$ are boundary vertexes. Note that\n$3n*|B'|\\le |A| \\le \\frac{4n*|V|}{2}$ since any vertex in $B'$ will\nattach at least $3n$ new vertex in order to make degree $4n$, we have\n$|B'|\\le \\frac{2}{3}|V|$. Now, by the definition of conductance we\nhave $\\frac{|\\partial B'|}{|B'|}\\ge g\/2$. This is because, if\n$|B'|\\le \\frac{|V|}{2}$ then $\\frac{|\\partial B'|}{|B'|}\\ge\ng$. Otherwise, we have\n$\\frac{1}{3}|V|\\le A'\\backslash B'|\\le \\frac{1}{2}|V|$, thus\n$|\\partial B'|\\ge g|A'\\backslash B'|\\ge \\frac{1}{3}g|V|$ and\n$|B'|\\le \\frac{2}{3}|V|$, we have\n$\\frac{|\\partial B'|}{|B'|}\\ge \\frac{g}{2}$. Therefore, we have\n$$\\frac{|\\partial A|}{|A|}\\ge \\frac{|\\partial B'|+|\\partial C|}{|B|+|C|}\\ge \\frac{|\\partial B'|}{|B|}\\ge \\frac{|\\partial B'|}{4n*|B'|}\\ge \\frac{g}{8n}.$$\nNow since $\\text{vol}|A|\\le n|A|$, the theorem follows.\n\\end{proof}\n\nCombining all the results, we have\n\\begin{theorem}\n  Assuming conjecture 1. For an given parameter $\\epsilon>0$ and $X$ in the general position,\n  Algorithm NRW run on the lazy chamber graph generated by $S_X$ can\n  generate labels from $S_X$ with distribution $\\epsilon$ close (in\n  variational distance) to uniform, and runs in time\n  $\\text{poly}(d,n,\\log(1\/\\epsilon))$.\n\\end{theorem}\n\\begin{proof}\nBy the relationship between mixing time and spectral gap \\cite[Theorem 2.2]{berestycki2016mixing}, we have\n$$t_{\\text{mix}}(\\epsilon)\\le \\frac{1}{g}\\log\\left(\\frac{1}{2\\epsilon n^d}\\right).$$\nThe theorem follows since the spectral gap is lower bounded by square\nof conductance by Cheeger's inequality \\cite[Theorem 13.14]{levin2017markov}.\n\\end{proof}\n\n\\subsection{Sampling for arbitrary neural networks}\n\\label{s:an}\nWe now consider the sampling for arbitrary neural networks. Let\n$X=\\{\\textbf{x}_1,\\cdots,\\textbf{x}_n\\}$ be the samples, we choose the\nweights of the network layer by layer. At layer $\\ell$ we use the\nprevious sampled weights in layers $1,\\cdots,\\ell-1$ to generate\noutputs $\\textbf{x}_1^{\\ell},\\cdots,\\textbf{x}_n^{\\ell}$, where\n$\\textbf{x}_i^{\\ell}$ is output of layer $\\ell-1$ with input\n$\\textbf{x}_i$, a binary vector. For each neuron in layer $\\ell$ we\n\\emph{independently} sample weights using Algorithm RS with input\n$\\{(1,\\textbf{x}_{1}^{\\ell}),\\cdots,(1,\\textbf{x}_{n}^{\\ell})\\}$.\n\nTo illustrate the idea more concretely, consider neural networks with\none hidden layer. Let $X$ to be the input samples of dimension $d$,\nfor each neuron in the hidden layer, we use Algorithm RS to generate\nthe weights \\emph{independently}. We now fix the weights we sampled\nfor the neuron in the hidden layer and view the function that\nexpressed by the hidden layer to be some function\n$h:=\\mathbb{R}^d\\rightarrow \\{0,1\\}^{u_2}$, where $u_2$ is the number\nof neurons in the hidden layer. We now define\n$\\textbf{x}_i'=h(\\textbf{x}_i)$ to be the new input sample for the\noutput layer, and again use Algorithm RS to sample the weights for the\noutput neuron with input $X'$.\n\\begin{theorem}\n  For a neural network with fixed architecture, $k$ neurons and $W$\n  parameters, the above sampling procedure runs in $O(nW^3)$\n  time. Given a sample $X$, each labeling in $S_X$ produced by this\n  architecture appears with probability at least\n$\\left(\\frac{W}{2enk}\\right)^W.$\n\\end{theorem}\n\\begin{proof}\nWe use induction on the layers. For any given labeling produced by\nweights $\\textbf{w}$, let $p(\\ell)$ to be the probability that the\noutput of layer $\\ell$ is consistent with the output on weight\n$\\textbf{w}$. We have\n$$p(\\ell)\\ge p(\\ell-1)\\prod_{i=1}^{u_{\\ell}}\\left(\\frac{d_i}{2e n}\\right)^{d_i},$$\nwhere $d_i$ is the input dimension of the $i$th neuron in layer\n$\\ell$, and the product term comes from Theorem 2 and\nindependence. Note that the rank of the outputs $X^{\\ell}$ may reduced\nafter passing the previous layers, however, this will only make the\nprobability larger than $\\left(\\frac{d_i}{2e n}\\right)^{d_i}$ by\nTheorem 2.  Now, the theorem follows with the same argument as\nin~\\cite[Theorem 6.1]{anthony2009neural} for bounding VC dimension of\nlinear threshold neural networks.\n\\end{proof}\n\n\\ignore{\\section{Simulations}\nWe run our recursive algorithm for randomly choosing samples with different dimension $d\\in \\{3,4,5,6,7,8,9,10\\}$ and size $n\\in\\{10,11,12,13,14,15\\}$, for each pair of $d,n$ we run the sampling procedure $30000$ times to count the empirical distribution on the different labeling. We then compute the ratio of the maximum and minimal probability that appears in the balling that we sampled, and rounding the ratio to be integers. One can see from figure 2 that, for each  sample size $n$ there is a peak for the probability ratio when the dimension of sample increases. For given dimension $d$, one can see that for $d$ is small the ratio will increase according the increasing of $n$, for $d$ is large the ratio will decrease when the $n$ is increasing. We also runs our sampling procedure on MNIST data set with $1000$ sample, the run time is around $5$ mins.\n\\begin{figure}[b]\n\\centering\n\\includegraphics[width=.6\\textwidth]{ratio.pdf}\n\\caption{Ratio of maximum and minimal empirical probability}\n\\label{fig:slowmixing}\n\\end{figure}}\n\n\n\n\n\\medskip\n\n\\small\n\n\\bibliographystyle{IEEEtran}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{The Great White Hope}\n\\label{sec:feynman}\n\n\nThe origins of John Wheeler's long and winding path to general relativity are hard to trace; all we have to go by are Wheeler's later recollections of the pipe dreams of a physicist in his mid-20s, or, as Wheeler liked to put it, his ``great white hopes.'' Of these there were two, which Wheeler recounted in two interviews, conducted by Charles Weiner and Gloria Lubkin (WL) in 1967 and by Finn Aaserud (FA) in 1988 (Session I),\\footnote{\\url{https:\/\/www.aip.org\/history-programs\/niels-bohr-library\/oral-histories\/5063-1}} respectively.\n\nThe first great white hope was electrons. After the discovery of the positron in 1932, the theory of electrons and positrons based on the Dirac equation, ``pair theory'', had been worked out by Dirac, Heisenberg, Oppenheimer and Furry. Wheeler felt that pair theory offered ``mechanisms for binding electrons in very small regions of space that never got a thorough discussion'' (WL), and that electrons might well be present in the nucleus after all (an assumption that had been dropped after the discovery of the neutron); in fact that electrons and positrons may form its fundamental constituents.\n\nThe second great white hope was scattering, which was to be viewed as the fundamental process from which all other characteristics of (primarily nuclear) interactions were to be derived. Both great white hopes were (at least in hindsight) also imbued with snappy and parallel slogans \\citep{wheeler_1989_the-young}, ``Everything as Electrons'' and ``Everything as Scattering,'' and even if these precise titles are not actually contemporary, they do show an essential characteristic of Wheeler's thinking: An extreme reductionism, a reduction to simple, catchy thoughts and a very small number of fundamental building blocks, a radical Ockhamism if you will. In particular, we see here what Wheeler would later call ``daring conservatism'': Taking (an element of) a well-established theory, and trying to use it beyond its usual domain of applicability, i.e., electrons in the nucleus or scattering theory to describe stationary states. But this unique approach did not show at the time. Feynman would later remark (perhaps apocryphally) that: \n\n\\begin{quote}\nSome people think Wheeler's gotten crazy in his later years, but he's always been crazy.\\footnote{Cited in \\citep{overbye_2002_peering} and in \\citep{wong_2008_remembering}. In the former, Kip Thorne is given as the reference for Feynman's supposed remark.}\n\\end{quote}\n\nWhile the later Wheeler would happily have publicized catchy slogans for crazy ideas, such as ``everything as electrons,'' there are no outward indications of his grand vision at the time. As he would remark in the interview with Weiner and Lubkin:\n\n\\begin{quote}\nNobody was as crazy as I was, to think that you could explain everything in terms of electrons. And this I think illustrates a weakness of my approach at that time, to have this secret hope nursed internally and talk about it occasionally with close friends but not feeling particularly at ease about bringing it out on a public platform...\n\\end{quote}\n\nSimilarly, a talk \\citep{wheeler_1934_interaction} he gave on (alpha particle) scattering at the APS meeting in Washington, DC, gave no indication (at least from the extant abstract) of the central role he was envisioning for scattering in fundamental physics, nor did his central paper on the subject, in which he famously introduced the S-Matrix \\citep{wheeler_1937_on-the-mathematical}.\n\nIt would take the meeting with a fellow eccentric to tickle Wheeler into presenting hints of his crazy ideas on ``everything as electrons.''\\footnote{We shall have no more to say about ``Everything as Scattering,'' which was here mentioned only to illustrate the early traces of daring conservatism in Wheeler's thinking. This notion was very influential in Feynman's later diagrammatic formulation of renormalized QED, which did effectively become a pure scattering theory, see \\citep{blum_2017_the-state}.} We refer, of course, to his PhD student, Richard Feynman, with whom he worked out what would later be known as Wheeler-Feynman electrodynamics \\citep{wheeler_1945_interaction}. In this theory, the field-mediated electromagnetic interaction of Maxwell's theory is replaced by a direct interaction at a (spatial) distance between charged point-like particles. This interaction is not instantaneous (in time), but is rather the sum of a retarded and an advanced component, corresponding to the two possible solutions of Maxwell's field equations and ensuring compliance with the special theory of relativity. When a given charged point particle (electron) exerts an advanced force on the other electrons in the universe, it will experience a retarded back-reaction, which will in fact be instantaneous. Imposing the ``absorber boundary condition'' (which in the corresponding field theory would imply that there are enough electrons in the universe to absorb all outgoing radiation, so that there is no radiation ``escaping to infinity'') then ensures that this instantaneous back-reaction is equal to the radiation reaction of the usual field theory. This then not only implies empirical equivalence with field theory, but also eliminates the possible difficulties with causality an advanced interaction might otherwise suggest.\n\nWheeler and Feynman give conflicting stories concerning the origins of their joint work on action-at-a-distance electrodynamics. Feynman in his 1965 Nobel lecture relates how, as an undergraduate, he had hit upon the idea of replacing the electromagnetic field with action-at-a-distance in order to eliminate the divergences of quantum electrodynamics (QED), how he had then learned as a graduate student that one cannot explain radiation reaction in this way, how he had tried to get radiation reaction into his action-at-a-distance framework as the backreaction of electrons, and how he had then presented this idea to Wheeler. Upon which Wheeler ``then went on to give a lecture as though he had worked this all out before and was completely prepared, but he had not, he worked it out as he went along,'' a lecture that ended in the conclusion that one would have to take into account advanced solutions of Maxwell's equations, in order to get an immediate radiation reaction. From this conversation then grew, as Feynman recalls it, their joint work on action-at-a-distance electrodynamics.\n\nWheeler's version of events is considerably younger, first published only a year after Feynman's death \\citep{wheeler_1989_the-young}. He recalls how Feynman had expressed some interest in Wheeler's idea of everything as electrons, how Wheeler had ``animated by the concept of everything as electrons' worked out one Sunday (up to a factor of 2) how one might get radiation reaction as a backreaction from the absorber even in a theory without fields, and how he had then presented his caculation to Feynman, who was able to sort out the missing factor of 2. But priority issues are not our concern here. In fact, in this case not only is the issue undecidable, the two stories are not even entirely incompatible: If we assume that both Feynman and Wheeler omitted substantial parts of the story, the two accounts might actually be merged together to form a coherent narrative.\n\nFor our story, another aspect evidenced by the two different accounts is much more important: It clearly shows that Wheeler and Feynman came to action-at-a-distance electrodynamics with very different motivations. For Feynman it was the divergence difficulties of QED, an issue which had somewhat dropped out of fashion in the late 1930s due to the great interest in meson physics, but an issue that would soon resurface in the late 1940s and ultimately earn Feynman the Nobel Prize. For Wheeler, it was his great white hope, his ``everything as electrons.''\n\nAccording to Feynman's recollections, it was he who got to write the first draft of their joint paper:\n\n\\begin{quote}\n[Wheeler] asked me to write the paper - I wrote this thing up in 27 pages, which we could have sent to a journal, but he began to think, ``No, it's too great a business, we'll write it good.'' And that of course made delays, and got interrupted with the war, and he got it so big that it was five parts - the whole reorientation of physics from a different point of view. I never went along with him on that. I mean, you know, with the idea that it's so marvelous, it's a reorientation of physics, you have to write five papers, and all of physics is turned upside down. But I felt that 27 pages were what it deserved. This was written mostly by him. See, it was a rewrite of the 27 pages, so to speak. I wouldn't say a rewrite because he didn't use the 27 pages as a basis, but the same ideas are developed, which I tried to write much more briefly, and which he tried to write in an historical context, about the arguments of Tetrode and Einstein - you see, it's a relatively long thing, and I didn't really write it, you understand.\n\\end{quote}\n\nThe manuscript that Feynman is referring to is probably identical with an untitled manuscript from 1941 in the Feynman Papers at Caltech (Box 6, Folder 1).\\footnote{The number of pages that this manuscript has depends on how exactly one counts, e.g., if one includes handwritten inserts, typed pages containing just one extraneous paragraph, and figure captions. A case can certainly be made for 27.} To assess the difference in tone, compare the first sentence of Feynman's draft:\n\n\\begin{quote}\nThe attempts to develop a satisfactory scheme of quantum electrodynamics have met with several difficulties, some of which are found not to be a result of the process of quantization, but to be contained in the classical electron theory itself.\n\\end{quote}\n\nwith the opening of the final, printed version:\n\n\\begin{quote}\nIt was the 19th of March in 1845 when Gauss described the conception of an action at a distance propagated with a finite velocity, the natural generalization to electrodynamics of the view of force so fruitfully applied by Newton and his followers. In the century between then and now what obstacle has discouraged the general use of this conception in the study of nature?\n\\end{quote}\n\nWheeler's grand vision clearly came out in the historical pathos. But as far as the motivation provided for reformulating classical electron theory in terms of action-at-a-distance, Wheeler fully adopted Feynman's motivation of solving the difficulties of QED. There is no hint of Wheeler's great white hope and so we need to ask the question, what did Wheeler mean when decades later he referred to his work as being ``animated by the concept of `everything as electrons'? Now, one might simply take this to mean that he was looking for a purely particulate description of nature, only electrons, no fields, and that action-at-a-distance was the implementation of this program. There is certainly something to this, and in his later recollections Wheeler certainly emphasized the ``everything as particles'' aspect of his early period, contrasting it with his later pure field approach. \n\nBut there is another aspect that appears to have been equally important, if not more so. Wheeler's program was specifically focused on electrons, and especially on their continuing role in the nucleus. It was thus not simply concerned with the electrons as generic particles, but with electrons as a very specific kind of particles, distinct from nucleons or mesons. Now, he was well aware that there were significant indications that electromagnetically interacting electrons were not the primary constituents of the nucleus, or even present there at all: The short range of the nuclear interactions and the huge kinetic energies obtained by electrons thus confined were the indications most typically cited. It seems, now, that Wheeler hoped that the action-at-a-distance formulation might help to put aside these objections to ``everything as electrons,'' as he also later stated in his autobiography \\citep[p. 164-165]{wheeler_1998_geons}:\n\n\\begin{quote}\nI had another motivation as well for pursuing action at a distance, for I clung to my hope that all of the matter in the world could be reduced to electrons and positrons. Yet I knew that if an electron and a positron were to be crowded together in subnuclear dimensions, some way would have to be found to get around the prediction of conventional theory that they would quickly radiate away their energy in the form of electromagnetic fields. Perhaps, I thought, an action-at-a-distance version of electromagnetic theory - one without fields - might explain the suppression of such radiation and permit the particles to live happily in such a confined space.\n\\end{quote}\n\nThere are only vague hints at this in the published paper with Feynman, which contains a section on ``Advanced effects associated with incomplete absorption,'' which discusses the physics of electrons in an incompletely absorbing cavity. This may be interpreted as Wheeler thinking about electrons in the nucleus, but this possibility is not mentioned explicitly. Stronger hints can be found in a lecture he gave, several months after the publication of the first Wheeler-Feynman paper, at a symposium of the American Philosophical Society on ``Atomic Energy and Its Implications.`` This symposium, conducted in November 1945, only three months after the use of atomic bombs on Hiroshima and Nagasaki, was a rather serious affair ``devoted to the atomic bomb'' (as per the opening words of Henry DeWolf Smyth), featuring talks by J. Robert Oppenheimer on ``Atomic Weapons'', Joseph H. Willits on ``Social Adjustments to Atomic Energy'' and Irving Langmuir on ``World Control of Atomic Energy'' (Proceedings of the American Philosophical Society, Vol. 90, No. 1, January 1946). Wheeler recalled that he ``didn't really want to talk about atomic energy'' but rather ``about what lay beyond it.'' \\citep[p. 168]{wheeler_1998_geons} So, inspired by his collaboration with Feynman and eager to do fundamental physics after the long war work, he chose this unusual venue to provide first hints on his grander schemes in a talk entitled ``Problems and Prospects in Elementary Particle Research''.\n\nThe hints towards his hope that action-at-a-distance electrodynamics might revolutionize nuclear physics are still very vague, merely by association \\citep[p. 45-46]{wheeler_1946_problems}:\n\n\\begin{quote}\n[T]he theory of [nuclear] mesons, may be said to be at present in a state of free experimentation with ideas and great uncertainty as to principle, both because of the incompleteness of our present experimental picture and because of difficulty in tying the proposed hypotheses to already existing theories. [...] The difficulties of inventing a new theory on the basis of  incomplete experimental evidence suggest that one possibility acceptable at this time is the conservative one of extending the range of applicability of already existing and well-established theories.\\\\\nThe second theory whose problems we consider is therefore the formalism of electron-positron pairs.\n\\end{quote}\n\nWe see here the notion of daring conservatism (without that name yet) appear in print for the first time, but as to how it is to be applied, there is nothing to go by but a lone and easily overlooked ``therefore.'' Wheeler would focus on cosmic ray physics in the following years, setting aside once again his great white hope (with the exception of some work with his student Gilbert Plass on counting degrees of freedom and thus doing thermodynamics - Planck's law - in an action-at-a-distance theory). But in June 1946, he submitted an application to the Guggenheim Foundation to pursue a project on an ``Analysis of the Problem of Measurement in Electron Theory.'' Through this scholarship, he hoped to obtain a leave of absence from Princeton to work on his foundational ideas. And he could think of no one better suited to help him in this endeavor than his old mentor Niels Bohr. As he wrote in his application:\n\n\\begin{quote}\nOnly a start could be made in the [...] program by the applicant himself and in collaboration with his student, R.P. Feynman; the war stopped further work. [...]\nThe primary reason for this proposal is the feeling that the stage has now come in the theoretical work where new concepts and points of view are essential. To develop and test such points of view it appears that by far the most effective course is to take up a close association with Niels Bohr. The writer's association with this scientist has convinced him that if there is hope of making advancement in the fundamental problems outlined above, this hope is best justified by Bohr's ability to see into the future, his courage and judgment in considering and testing new concepts. [...] [T]he Applicant therefore is proposing to go to work with Professor Bohr in Denmark.\n\\end{quote}\n\nEven in this proposal, Wheeler was still rather guarded about what he wanted to do exactly. The proposed title recalled Bohr and Rosenfeld's paper on the Problem of Measurement in QED \\citep{bohr_1933_zur-frage}, which to Wheeler was the model for an incisive theoretical analysis of current theory \\citep{hartz_2015_uses}, and which he referred to as ``a classic paper'' in his application. And when listing the tasks for his project, Wheeler remained excessively general:\n\n\\begin{quote}\n(1) The consistent formulation of the mathematical formalism of the theory of electrons and positrons.\\\\\n(2) A definition in terms of idealized experiments of the possibilities of measurement in the theory of the electrons and positrons.\\\\\n(3) The interpretation of these idealized experiments in terms of the formalism - and of the formalism in terms of the idealized experiments.\n\\end{quote}\n\nConsequently, Eugene Wigner, who acted as one of the references for Wheeler's application, was also rather non-committal about his views on Wheeler's project (received 24 June 1946):\n\n\\begin{quote}\nAs to his proposed study, I must admit that I find it quite impossible to make any predictions. [...] I know from personal contacts that Professor Wheeler is most deeply interested in the project which he outlined.\n\\end{quote}\n\nIn any case, while Wheeler's application was granted only five days after it was received (and five days before even receiving Wigner's letter of reference), Wheeler could not use the grant as soon as he had planned. As he wrote to Henry Allan Moe of the Guggenheim Foundation on 2 July 1946:\n\n\\begin{quote}\nI have now learned on my return to Princeton that there is a distinct possibility that the six-month leave of absence granted by the University to Professor Wigner may have to be extended through the second term of 1946-47 to allow him to accomplish most effectively his task to give a new direction to the work of the Oak Ridge Laboratory. I am afraid after examination of the situation with the Department here that it would create embarrassing difficulties for the work now underway to have me gone for the whole of the Spring term. However, it appears that I may be able to count on a period May 15 - October 1, 1947. Even this range of time can, however, not now be made entirely definite. [...] I am sorry to have to report to you that the situation is in this uncertain state. I should like to have your advice as to how I can best take into account these difficulties in a manner acceptable to the Foundation.\n\\end{quote}\n\nThere was no problem. Moe replied on 5 July 1946:\n\n\\begin{quote}\nIt is a pleasure to welcome you to the company of Guggenheim Fellows -- the distinguished company as I think.\\\\\nAs to the date of starting your Fellowship, there's no need to settle that now. [...] You may count on us to ``play ball.''\n\\end{quote}\n\nAfter some back and forth, Wheeler finally received his scholarship for the period from 1 July 1949 to 30 June 1950 (i.e., also for an extended time period). For family reasons \\citep[p. 183]{wheeler_1998_geons}, he also opted to make Paris his home during his stay in Europe, rather than Copenhagen. And even though he did visit Copenhagen on several occasions during his time in Europe, much of his work with Bohr was focused not on electron-positron theory, but rather on completing a paper with David Hill on the collective model of the nucleus, which Bohr ended up not co-signing (see Acknowledgments in \\citep{hill_1953_nuclear}). \n\nOne reason for this appears to have been a major advance that occurred in the years from 1946 to 1949: The problem of the electromagnetic self-energy of the electron had been solved, among others by Feynman, without the need to eliminate the self-interaction of the electron. Wheeler had always presented the elimination of the self-interaction as the prime motivation for action-at-a-distance. Consequently, when writing an updated version of his 1945 talk in 1949, he duly acknowledged the recent advances in quantum electrodynamics and used a much more cautious language when talking about Wheeler-Feynman electrodynamics. In 1945, he had concluded his brief elaborations on Wheeler-Feynman electrodynamics with the words:\n\n\\begin{quote}\nIt is too soon to say whether the translation of the revised classical theory into quantum mechanics will remove the outstanding divergences. To test this point is an important problem for the future.\n\\end{quote}\n\nIn 1949 he wrote:\n\n\\begin{quote}\nNaturally the ultimate complete equivalence of this approach to the usual field theoretical treatment makes it clear that nothing new can result so long as this equivalence is strictly maintained. What may come out by changes and reinterpretations of the existing theory of action at a distance is uncertain \\citep{wheeler_1949_elementary}.\n\\end{quote}\n\nThat he did not yet give up action at a distance entirely at this time (as Feynman did, see \\citep{blum_2017_the-state}) is clearly due to his great white hope of constructing the atomic nucleus from electrons, a conjecture that was largely unaffected by the advances in QED. Indeed, we have one bit of evidence that Wheeler did in fact discuss this idea with Bohr during his stay in Europe, a letter sent to Bohr from Paris on 21 January 1950,\\footnote{Archives for the History of Quantum Physics, Bohr Scientific Correspondence, Microfilm 34.} announcing his arrival in Copenhagen for the 27th. This letter is primarily concerned with their joint work on the collective nucleus, but then, on page two, Wheeler brings in ``everything as electrons'':\n\n\\begin{quote}\nThe other problem, about which I am very anxious to get your opinion, is the question: is it possible to exclude a picture of elementary particle constitution based entirely on positive and negative electrons?\n\\end{quote}\n\nWheeler discussed his hope how half-retarded, half-advanced action-at-a-distance might solve the problems usually brought forth:\n\n\\begin{quote}\n(1) \\emph{Localizability}. To localize an electron in a distance of $\\approx e^2\/mc^2$ it is sufficient to have a potential well of radius $(e^2\/mc^2)$ and depth $(137)^2 mc^2$ [...]\\\\\n(4) \\emph{Source of potential}. Acceleration and velocity of an electron bound in such a potential are so great that electrostatic forces are negligible in comparison with radiative forces. Radiational transfer of energy to the outer world is itself negligible owing to the symmetry of the charge-current distribution. The radiational forces within the system can therefore be considered in a good approximation as half-advanced, half-retarded [...]. [I]t does not seem impossible to suppose that the electronic system generates a self-consistent potential, in which however the correlations between movements of interacting electrons must have an altogether dominating importance, in contrast to the atomic case.\n\\end{quote}\n\nAll this was very qualitative. And even if Wheeler's hopes concerning the nuclear binding potential could be realized, he was aware of the fact that there was another difficulty, which he had only very vague ideas on how to address: That of spin and statistics. If a neutron, e.g., was really to be considered as consisting of electrons and positrons only, it would have to contain an equal number of electrons and positrons to ensure its neutrality, and thus would have to be an integer-spin boson. Wheeler was clearly interested in Bohr's judgment as to whether the project was worth pursuing despite this apparently insurmountable difficulty:\n\n\\begin{quote}\nThough I joke with you about my heresies, I am trying to be just as honest and as open as I can about the elementary particle problem. I know that there is no one who has your insight. So it will be a great privilege to talk with you over these and other problems...\n\\end{quote}\n\nAs this letter was discussed during Wheeler's visit to Copenhagen, we have no evidence as to Bohr's take on the matter. Soon after his visit to Copenhagen, Wheeler cut his sabbatical short to work on the hydrogen bomb. When Wheeler returned to foundational research after the hydrogen bomb interlude, his focus was now on a different, but related problem: action-at-a-distance gravity.\n\n\\section{Action-at-a-distance Gravity}\n\\label{sec:AAD}\n\nThe idea of action-at-a-distance gravity can be traced back to 1941, when Wheeler and Feynman had worked out the basics of their absorber theory and Feynman first presented their joint work at a seminar in Princeton. Feynman later recalled that Einstein had remarked \\citep[p. 80]{feynman_1985_surely}:\n\n\\begin{quote}\nI find [...] that it would be very difficult to make a corresponding theory for gravitational interaction...\n\\end{quote}\n\nWheeler appears to have been immediately intrigued by this challenge. Although soon mainly occupied by war work, he wrote a letter to Einstein on 3 November 1943 (Einstein Papers), requesting a meeting with Einstein to discuss ``where the force of gravitation fits into the point of view'' of action-at-a-distance theory. The meeting took place on 14 November in Wilmington, Delaware  (Letter from Einstein to Wheeler, 6 November 1943, Einstein Papers) and we can surmise some of its content from a letter which Wheeler wrote to Einstein after the meeting, on 2 December. Wheeler had developed a general framework for action-at-a-distance theories, which he referred to as ``the theory of world lines.'' In this framework not only was there no more talk of fields, there was even ``no reference to the concept of a space-time continuum.'' All that was left were the world lines of individual particles $a$, $b$, etc., the points on which were identified by  some parameter, $\\alpha$, $\\beta$, etc. The exact parameterization of the world lines was to be considered arbitrary, and all statements on physics were supposed to be independent of it.\n\nThe dynamics were now determined by functions that connect points on two worldlines. In his letter to Einstein, Wheeler referred to these functions as light cones, which most clearly reveals their physical interpretation. In his later notes, he would mainly use the term liaison. And when he finally wrote a paper on AAD gravity some 50 years later, together with Daniel Wesley, he called them associators \\citep{wesley_2003_towards-an-action-at-a-distance}.\\footnote{One reason for Wheeler's return to action-at-a-distance gravity after so many years appears to have been that it had by then become abundantly clear that quantum gravity would suffer from similar divergence difficulties as QED, difficulties that in electrodynamics the Wheeler-Feynman absorber theory had aimed to solve. Since these difficulties could not be solved by renormalization in the case of gravity, it appeared attractive to revisit the old AAD formulation. This was pointed out to us Daniel Wesley, who co-authored the AAD gravity paper with Wheeler as an undergraduate student, in an email to one of the authors (AB), 11 April 2019.} We will be referring to them as liaisons throughout this paper, as this is the term that Wheeler used for most of the period under study.\\footnote{We will also be using a notation that Wheeler introduced only after his letter to Einstein. This is additionally motivated by the fact that reproducing Wheeler's notation in the Einstein letter presents somewhat of a typesetting challenge.}\n\nThe liaison function $\\alpha_+ (\\beta) $ then returns the point $\\alpha$ that is on the forward lightcone of $\\beta$.\\footnote{\\label{fn:conjugate} In general, the forward lightcone might intersect the other world line more than once. The liaison was supposed to be a single-valued function that singles out one of those points and thus constituted an object somewhat more restricted than a lightcone.} Similarly, $\\beta_+ (\\alpha) $ returns the point $\\beta$ that is on the forward light cone of $\\alpha$. Wheeler also introduced liaisons $\\alpha_-$, $\\beta_-$ for the backward lightcone, which are the inverses of $\\beta_+$ and $\\alpha_+$ respectively. Wheeler now proposed to Einstein that one could construct physics from these functions by setting up non-trivial, parameterization-independent relations. He pitched the expression\n \n \\begin{equation}\n \\alpha' = \\alpha_- (\\gamma_+ (\\beta_+ (\\alpha))),\n \\end{equation}\n \nwhere $\\alpha'$ is the  point on world line $a$ that one reaches by moving along the forward light cone from a point $\\alpha$ to world line $b$, then further on the forward light cone to world line $c$ and then on the backward light cone back to world line $a$. As can easily be seen by imagining all three particles as being at rest in the (not yet constructed) three-dimensional background space, if $\\alpha'=\\alpha$ the three particles are on one line in three-space. If $\\alpha' \\neq \\alpha$ they are not. One could thus, merely from three one-dimensional world lines and the liaisons between them, distinguish between a line and a triangle through the (indeed parameterization-independent) statement that $\\alpha$ and $\\alpha'$ are equal or unequal, respectively. Wheeler further conjectured:\n \n \\begin{quote}\n With a number of particles greater than three, one can build up more complex geometrical concepts.\n \\end{quote}\n \nApparently, Wheeler had hoped that in this manner one would be able to construct a theory that would be fully equivalent to general relativity, just as Wheeler-Feynman absorber theory was equivalent to Maxwell. In their discussion, Wheeler and Einstein appear to have focused on a surprisingly specific difficulty with such an equivalence: A universe with two particles. In the world line theory, Wheeler asserted, ``no physics at all is possible'' in such a setup, because the only expression one can study, namely $\\alpha_- (\\beta_+(\\alpha))$, is trivially the identity, since $\\alpha_-$ is \\emph{defined} to be the inverse of $\\beta_+$.\\footnote{In general, it is of course possible to first move along the forward lightcone from worldline $a$ to worldline $b$ and then back, along the backward lightcone, to worldline $a$ and arrive at a point different from the point one started out from, when conjugate points are involved. In the liaison framework this issue is avoided: As mentioned in footnote \\ref{fn:conjugate}, the liaison has to be a single-valued function and thus it was natural on many levels for Wheeler to simply define the backward liaisons as the inverses of the forward ones.} In general relativity, on the other hand, there were, as Einstein pointed out, two-body solutions with a rich, non-trivial four-dimensional geometry.\\footnote{The two-body solutions by Weyl and Levi-Civit\\`a that Einstein was referring to are discussed in detail in an editorial footnote of the Collected Papers of Albert Einstein \\citep[p. 437]{buchwald_2018_the-collected}.} There was thus, as Wheeler remarked, ``an apparent discrepancy between the general theory of relativity and the general theory of action at a distance.''\n\nIn his letter to Einstein, Wheeler began to develop an understanding, which he would further develop over the course of the following years, that his world line theory was in fact right in implying that there should be ``no physics'' in a two-body universe. Wheeler would present first hints of his search for an AAD formulation of gravity in several talks given in the first postwar years, such as in his programmatic 1945 talk at the American Philosophical Society, already mentioned earlier, where he remarked:\n\n\\begin{quote}\nJust as the proper recognition of [...] atomicity requires in the electromagnetic theory a modification in the use of the field concept equivalent to the introduction of the concept of action at a distance, so it would appear that in the gravitational theory we should be able in principle to dispense with the concepts of space and time and take as the basis of our description of nature the elementary concepts of world line and light cone.\n\\end{quote}\n\nBut his stance on the two-body problem was only spelled out in the revised 1949 version of the talk. In defense of world line theory, Wheeler had adopted a relationalist view of space and time, in the tradition of Ernst Mach who had formulated such ideas in the late 19th century in opposition to the Newtonian notions of absolute space and time. Mach's ideas, in the shape of the more or less formalized ``Mach's Principle'' of the relativity of inertia, had played an important role in Einstein's application of general relativity to cosmology \\citep{smeenk_2014_einsteins}. But the existence of two-body solutions, Wheeler now argued, called into question the validity of Mach's principle in general relativity. In a strictly relationalist theory, an idealized two-body problem would be non-dynamical: There was only one length scale to relate things to, the distance between the two point masses, and hence that distance should not be representable as changing, since there was nothing that its change could be measured against. Any statement that general relativity claimed to be able to make about the time evolution of the distance between the two bodies was thus empty.\n\nWhile the centrality of the two-body problem was not to persist (already in the 1949 lecture, Wheeler voiced his doubts whether the paradox would necessarily be resolvable by action-at-a-distance gravity), the years 1943-49 see gravitation moving to the center of Wheeler's attention, with new interesting paths of inquiry popping up, such as the unresolved problem of gravitational radiation reaction. While his talks of the period, as mentioned earlier, contained somewhat defeatist language concerning the prospect of Wheeler-Feynman electrodynamics, this is not the case for action-at-a-distance gravity. As his hopes for explaining the nuclear forces through electrodynamics were waning, gravity was increasingly presenting itself as a worthwhile field of study. Wheeler was beginning to realize the untapped potential of general relativity, though it must be admitted that he does not yet appear to have had a definite program for using this potential. The most specific part of his new interest in gravitational theory was the focus on the role of Mach's principle (beyond the particular case of the two-body universe) in establishing the relation between the world line theory and general relativity. This became, as we shall see, a central theme in Wheeler's thinking about gravitation.\n\nWheeler appears to have worked on AAD gravity quite a bit during his 1949\/50 stay in Europe. In a letter to Gregory Breit of 28 December 1949 (Gregory Breit Papers, Yale University Archives), he wrote:\n\n\\begin{quote}\nI am working quietly, sometimes on the reconciliation of the individual particle model of the nucleus and the liquid drop model [i.e., collective models of the nucleus], sometimes constructing a description of nature which makes no use of the concepts of space and time (analogue in gravitation theory of electromagnetic action-at-a-distance).\n\\end{quote}\n\nSimilarly in a letter to Feynman of 10 November 1949 (Feynman Papers, Caltech). And also in the letter to Bohr of 21 January 1950, already cited earlier, Wheeler talked about action-at-a-distance gravity, explicitly connecting it with Machian ideas of making ``force dependent upon the number of particles in the universe'' and mentioning another letter (not extant) to Wilhelm Magnus, mathematician at the Courant Institute, who had provided Wheeler ``with some information about one of the group theoretical aspects of the problem.'' In later recollections, Wheeler even misremembered that he had proposed ``doing similar ideas [to Wheeler-Feynman] for gravitation theory'' (FA) already in his application to the Guggenheim Foundation in 1946. But at the time, his main focus was still on electromagnetism. With the success of renormalized QED, electrodynamics was on its way out for Wheeler, and Bohr appears to have disabused him of the last vestiges of ``everything as electrons.'' But gravitation, in Wheeler's mind, was up and coming! In April 1951, he returned to Princeton from Los Alamos, and while still chiefly concerned with the work on the hydrogen bomb \\citep[p. 218]{wheeler_1998_geons}, he did find some time to ponder these foundational questions. In a notebook entitled ``Action at a distance I'', we find an entry dated 10 November 1951, in which Wheeler considers two different pathways to AAD gravity:\n\n\\begin{quote}\nCan thus work towards desired [action] principle from either one of two directions---(1) math. convenience + naturalness; (2) correspondence.\n\\end{quote}\n\nAnd at this point the second pathway, establishing the theory through correspondence with the field theory of general relativity, clearly seemed the less favored, especially since the correspondence could not be exact for small numbers of particles, as he had established in his discussions with Einstein several years earlier. A few lines above the remark just quoted, Wheeler had written\n\n\\begin{quote}\nBut should satisfy the principle of correspondence to ordinary general relativity in the limit of infinitely many infinitely small masses (continuous mass distribution).\n\\end{quote}\n\nonly to then qualify this remark by a ``probably'' inserted after ``should.'' So how did Wheeler's pursuit of AAD gravity along the lines of mathematical convenience and naturalness look? Still using the liaisons as his central dynamical variables, Wheeler's idea was now to set up an action principle (similar to the Fokker action in Wheeler-Feynman electrodynamics), written as an integral over world line parameters, the integrand being some function of the liaisons. When varied, this action would return differential equations for determining the liaison functions. As to how this action should look, Wheeler thought that it might involve counting closed cycles of liaisons, as this could provide a notion of local world line density without having to invoke an underlying space-time. \n\nWhile these general ideas appear to have been present already in late 1951, Wheeler did not elaborate on it any further in his AAD notebook for quite a while. The next entry dealing with liaison theory dates from 17 March 1953. It would appear that Wheeler's purely mathematical approach turned out to be inadequate. For in 1952, he switched gears and started to engage general relativity head-on.\n\n\\section{Teaching Relativity}\n\\label{sec:course}\n\nEmbarking on his study of general relativity and the corresponding action-at-a-distance formulation, Wheeler asked to teach a course on relativity at Princeton in the academic year 1952\/53. His request was granted on 6 May 1952, the day on which Wheeler began his first in a long series of notebooks on Relativity.\\footnote{We will be citing frequently from Wheeler's first two relativity notebooks, which we will be abbreviating as WR1 and WR2, respectively. These notebooks are to be found in the John Wheeler Papers, held at the American Philosophical Society in Philadelphia, Section V, Volumes 39 and 40. } It is somewhat surprising that Wheeler was the first to teach a dedicated relativity course at the Princeton physics department. After all, Wheeler himself had clearly profited from the fact that Princeton was  \\emph{the} center for relativity in the US at the time. We have already mentioned his personal discussions with Einstein. And in his first writing on relativity, the 1945 talk at the American Philosophical Society, every author cited in the section on gravitation, aside from Ernst Mach, worked in Princeton. However, they all worked at the Institute of Advanced Study, resulting in a great divide between the accumulated expertise on relativity at Princeton and the lack of relativity teaching. There was expertise at the university and Wheeler tapped into that as well, as recalled by Churchill Eisenhart, son of Princeton professor Luther Eisenhart:\n\n\\begin{quote}\nAs I understand it, after [Luther Eisenhart] retired he and John Wheeler were working together at writing a book called Mathematics Essential for the Theory of Relativity. [...] Dad and Wheeler, as I understand it, were bringing together in their book the mathematics, from here and there in the various branches of mathematics, you need for the general field theory.\\footnote{Interview on 10 July 1984 with Churchill Eisenhart conducted by William Aspray, available at \\url{https:\/\/www.princeton.edu\/mudd\/finding_aids\/mathoral\/pmc09.htm}. In this interview, Churchill Eisenhart also recalls that the manuscript for Wheeler and Eisenhart's book disappeared under mysterious circumstances after Luther Eisenhart's death.}\n\\end{quote}\n\nEisenhart retired in 1945, around the time that Wheeler began thinking about AAD gravitation.\\footnote{For biographical information on Eisenhart, see \\citep{lefschetz_1969_luther}.} But Eisenhart (like Valentine Bargmann, another Princeton University expert on differential geomertry) was a mathematician, and indeed up until Wheeler's initiative general relativity was only taught in the mathematics department at Princeton \\citep{kaiser_1998_a-psi}.\\footnote{Kaiser erroneously gives the year of Wheeler's first course as 1954\/55. The 52\/53 course, which is well documented by Wheeler's notebook, indeed did not yet show up in the Princeton course catalogue. A course on relativity by Wheeler is listed for 1953\/54. This course catalogue had not been available to Kaiser at the time he wrote his paper.} So, despite the immense tradition and expertise that Wheeler could draw on in his exploration of general relativity, he was indeed the first one to teach it to Princeton physics graduate students.\n\nWheeler's notebook opens with his thoughts on his upcoming course:\n\n\\begin{quote}\n5:55 pm. Learned from [Allen] Shenstone [then head of the Princeton Physics Department] 1\/2 hour ago the great news that I can teach relativity next year. I wish to give the best possible course. To make the most of the opportunity, would be good to plan for a book on the subject. Points to be considered:\\\\\n(1) a short introductory outline of the whole\\\\\n(2) Emphasis on the Mach point of view \\\\\n(3) Many tie-ups with other fields of physics. Mention these in class; in the book put them in the ends of chapters as examples\n\\end{quote}\n\nThe last two remarks are especially noteworthy. In remark (3), we can already see a recurring theme in Wheeler's later work on and in relativity, both intellectually and institutionally,\\footnote{On Wheeler's role in ensuring that the institutionalization of research in general relativity would take place in the disciplinary context of physics, see \\citep{lalli_2017_building}.} namely to establish general relativity as a physical theory, rather than a mathematical or philosophico-cosmological one. Here we see the decidedly pedagogical aspect of this theme, as only in this manner could it legitimately be taught to and applied by physics students. This emphasis should of course also be viewed in light of the predominantly mathematical tradition in relativity at Princeton University.\n\nWheeler's personal intellectual perspective on general relativity shows in remark (2), which hints at how strongly Wheeler's interest in General Relativity was at this point tied up with the prospect of an action-at-a-distance formulation, in which space-time disappears as an independent entity. Indeed, action-at-a-distance was a defining element in Wheeler's subsequent course as documented by his notes. The first term, which dealt primarily with special relativity, saw frequent references to the Wheeler-Feynman papers, including a long discussion of Wheeler-Feynman electrodynamics itself, stretching from December 1952 to January 1953.\n\nWheeler's general relativity class began in February 1953 with a first class meeting on 5 February in which topics for seminar reports were discussed. Here Wheeler had already honed in on some key physical problems, problems that would be defining elements of the upcoming renaissance of relativity: Gravitationally collapsing stars, gravitational radiation, and empirical cosmology. These three problems were joined, in a list of topics for seminar reports discussed in the first class meeting (WR1, p. 47), by an `Assessment of Unified Theories.'' Here, Wheeler was clearly attempting to make contact with the general relativity scene as it presented itself to him at Princeton, as witnessed by the list of references for this report, which included not only the obvious Einstein (specifically his latest paper, which had just appeared in the January issue of the physical review \\citep{einstein_1953_a-comment}) but also work by Eisenhart, who had now, in retirement, turned to the study of non-symmetric metrics as they appeared in Einstein's Unified Field Theory \\citep{eisenhart_1951_generalized}. The suggested topics for seminar reports are followed by an unsorted list of further topics Wheeler wanted to cover, which included both the ``Mach point of view'' and, immediately afterward, ``our particulate point of view.'' The list also contains the entry ``Variational principle and connection with quantum theory,'' a clear reminiscence to the least-action (in modern parlance: path integral) formulation of quantum mechanics that Feynman had developed precisely in the attempt to quantize Wheeler-Feynman electrodynamics. So, also in his list of topics for the second half of the course, we see the two central foci of physical problems (where Wheeler's identification of the central ones was clearly very influential) and of Wheeler-Feynman-Mach action-at-a-distance gravity, now joined by a rising interest in the idea of a unified field theory stimulated by the Princeton milieu.\n\nOf course these questions were interrelated. The question of gravitational radiation, for example, was connected with the construction of an action-at-a-distance theory. The empirical adequacy of Wheeler-Feynman electrodynamics required imposing the so-called absorber condition that any electromagnetic radiation ultimately be absorbed, with nothing ever escaping to infinity. Is the gravitational world of GR ``non-absorptive'', Wheeler asks on the following page (WR1, p. 49), labeling it a ``\\emph{very} vital question to look at.'' Wheeler was thus following an intellectual trajectory typical for the renaissance of GR: In pursuing a speculative extension of GR (action-at-a-distance in this case), he was forced to reflect on fundamental questions of GR proper (gravitational radiation). The question of absorber boundary conditions was really more of a side issue in this study of GR for ulterior purposes, however. As we saw in the last section, the central challenge for Wheeler was to construct a least action formalism for AAD gravity using liaisons. Wheeler pursued this program in parallel to teaching the course, and his lecture notes are consequently interspersed with research notes, initially focussing on the construction of liaison theory. In the following, we will focus almost exclusively on the research notes, leaving the exact reconstruction of the curriculum of Wheeler's course aside.\n\nAfter his purely mathematical approach to this problem appears to have led nowhere, Wheeler's aim was now to construct liaison theory by studying its correspondence to regular GR. The first challenge here was to establish the locus of the correspondence, i.e., to identify the correct field quantity in GR that was to be reconstructed from the liaison formulation, the actual  ``gravitational field.'' Some two weeks into the course, Wheeler began to focus his attention on the Riemann tensor (WR1, p. 57). In an AAD theory, this would ultimately (via the liaisons) have to be reconstructed solely from the matter content (possibly merely in the form of singular worldlines), along with some sort of boundary conditions. Wheeler was thus led back to Einstein's original question concerning the realization of Mach's principle in GR: Is this sufficient to uniquely determine the Riemann tensor (up to coordinate transformations)? Wheeler ultimately reached a conclusion similar to that of \\citet{einstein_1917_kosmologische}, namely ``that there is a one to one correspondence between mass distribution and metric only when space closes up on itself.'' (WR1, p. 103)\n\nIt should be noted that Wheeler was aware that this statement was merely a plausible conjecture: ``Any proof of uniqueness of case where metric is made to close up on itself? Very important question of principle.'' (WR1, p. 105) His main source for this conjecture was  \\emph{The Meaning of Relativity} \\citep{einstein_1953_the-meaning}. He continued to discuss this matter at Princeton with Weyl (WR1, p. 111), Wigner and von Neumann (WR1, p. 120), all of whom disagreed with Wheeler's assessment. Wheeler took this aversion to Mach's principle to also be a result of unfortunate formulation of the principle and gave his class the task of coming up with a better ``presentation of Mach's principle in 2 pages for an elementary physics student.'' (WR1, p. 135). Despite these difficulties, Mach's principle remained central to Wheeler's research program as it provided an analog of the Wheeler-Feynman absorber boundary conditions in general relativity. For some time, closure of the universe became an unquestionable fact to Wheeler, as he explained to his students:\n\n\\begin{quote}\nQuestion raised in class whether mass density enough to permit open or closed universe, in view of expansion rate. Answer: [...] closure comes first, density knowledge too poor to permit proof of contradiction; closure so fundamental to whole Mach idea that in present state of knowledge think of density value having to yield precedence to Mach principle. (WR1, p. 104)\n\\end{quote}  \n\nWith the Riemann tensor identified as ``the field'' (WR1, p. 96), the possibility was now established to construct the action for liaison theory through correspondence with the usual field (Hilbert) action:\n \n \\begin{quote}\n Set up an experimental procedure to get $R_{ijkl}$ locally by liaisons between a number of particles. In this way tie up $R_{ijkl}$ with liaison picture. Hence express $R$ in terms of local liaisons. Hence get variation principle in terms of local liaisons. (WR1, p. 89)\n \\end{quote}\n \nThe ``experimental procedure'' was supposed to involve some sort of ``batting back and forth'' of light signals (WR1, p. 90) which would be a physical realization of the connection between two points established by a liaison. But Wheeler's attempts to tie up the Riemann tensor with the liaison picture ended inconclusively: He attempted to find the liaison function between two world lines from general relativity, where the light signals $\\kappa$ travel from one particle world line to another on light-like geodesics, soon focusing on the limiting flat space case, where the two world lines $x$ and $\\overline{x}$ are straight (WR1, p. 97), i.e. the system of equations:\n\n\\begin{eqnarray*}\nx^i (s) & = & x^i (0) + s \\left(\\frac{dx^i}{ds} \\right)_{s=0} \\\\\n\\overline{x}^i (\\overline{s}) & = & \\overline{x}^i (0) + \\overline{s} \\left(\\frac{d\\overline{x}^i}{d\\overline{s}} \\right)_{\\overline{s}=0} \\\\\n\\kappa^i &=& \\overline{x}^i - x^i \\\\\n\\kappa^{\\alpha} \\kappa_{\\alpha} &=& 0\n\\end{eqnarray*}\n\nwhich was supposed to give a relation between the parameters $s$ and $\\overline{s}$, i.e., the liaison function giving for any point on one world line the point on the other one that lies on the first point's light cone. But even this simplified, non-gravitational trial calculation (27 March 1953; WR1, p. 99) ended inconclusively. His simple idea of obtaining the liaison action merely by translating the Hilbert action into liaison language faltered. Although now fully immersed in general relativity and tensor calculus, he returned to his original mathematical approach and began to pursue (8 April) a new approach to the liaison action, no longer based on counting cycles, but rather on counting the number of (forward) liaisons entering and exiting a given volume element, a setup inspired (as he himself remarked) by the neutron balance in a nuclear chain reaction (WR1, p. 114).\n\nIn all this searching, Wheeler was well aware that he was pursuing an entirely new style of doing physics. On 18 March 1953, in the margins of notes on liaison theory (AAD notebook), he remarked:\n\n\\begin{quote}\nThis mushy thinking may in end be much better, if less attractive, to present than the usual 1,2, 3 type of argument with which one at the end so often presents his special conclusions.\n\\end{quote}\n\nWhat was driving him down this road of ``mushy thinking'' appears to have been the feeling of pursuing something grand, the ``great white hope'' feeling for which we here have the first contemporary archival evidence. Framing to himself his attempt to eliminate space and time, he wrote \n\n\\begin{quote} \nUndoing work of early man, that theoretical physicist who left no records. (26 March 1953; WR1, p. 97)\n\\end{quote}\n\nand also, for the first time in extant writing, coined one of his snappy slogans to describe his project, a ``universe of particles'' (WR1, p. 108). Indeed, though still bogged down in the attempt at formulating a liaison theory of gravitation alone, Wheeler always had in the back of his mind the further goal of combining this with electromagnetism and thereby achieving Einstein's goal of a unified theory (though without fields), and ultimately push on to include also the intrinsic properties of particles, such as spin:\n\n\\begin{quote}\nDon't feel discouraged about how much will still remain to do after expressing mere gravitation theory in liaison form. Should serve as guide in trying to put combined gravitation-electromagnetic theory in liaison form, and in later trying to put everything in neutrino language... (WR1, p. 113; 8 April 1953)\n \\end{quote}\n \nThe term ``neutrino'' appears here for the first time prominently in Wheeler's relativity notebook. Its significance for Wheeler is somewhat hard to grasp, as it can imply two distinct things: It appears as the barest possible point particle, carrying no charge or mass (only spin, possibly), or it can appear as a spinor field, the elementary carrier of spin and associated with the weak nuclear interaction, a reading that goes back to Wheeler's 1945 American Philosophical Society talk, where he referred to the neutrino as a ``field of interaction.'' This should be kept in mind in the following. What both notions have in common is that the neutrino is associated with the introduction of spin into the theory, which also appears to be the role in which it is invoked here. As the hope of recasting general relativity in liaison form faded, the fleshing out of the world line picture, i.e., the construction of a more sophisticated model of matter that would also include intrinsic properties such as spin, moved to the center of Wheeler's thinking.\n \n \\section{Particles as Singularities in the Field}\n \\label{sec:singularities}\n \n At some time in the spring of 1953, a shift began to occur in Wheeler's research agenda. Despite the day-to-day evidence we have from his notebooks, it is hard to date it exactly. It was rather a gradual shift, even though Wheeler's later use of religious metaphors to describe this tradition might rather imply an instantaneous conversion:\n \n\\begin{quote}\nThe idea of action at a distance I gave up, not because the action and the distance was complicated, but because the particle was complicated. It was just the wrong basic starting point for the description of physics, to think of a particle. Pair theory made clear, and renormalization theory, that what one thought was an electron was really an infinite number of pairs of positive and negative electrons indeterminate in number and that the whole of space is filled with pairs. [...] \\emph{And of course nobody gets religion like a reformed drunkard.} As I've often said about this subject, the fanaticism, if you would like to call it that, with which I pursued the opposite approach---that it's a pure field theory explanation of nature that one ought to work at---comes from having worked so hard at a pure particle explanation of what one sees. (LW, emphasis by us)\n\\end{quote}\n\nInterestingly the reasons that Wheeler gives for abandoning the particle approach (in particular the rise of renormalization theory) may well have been essential for his abandoning of the ``everything as electrons'' program, but played no role for his assessment of action-at-a-distance gravity, which, as we have seen, he was pursuing well into the 1950s. And his shift to field theory did not initially involve thinking of the particle as something ``complicated.'' Rather, he merely shifted from thinking of the particle world lines as the primary elements of the theory to thinking of them as secondary, derived objects, as singular lines in the field, whose equations of motion could be derived from the (vacuum) field equations simply by requiring consistent boundary conditions. This program goes back to the 1920s \\citep{einstein_1927_allgemeine}\\footnote{See \\citep{havas_1989_the-early} and \\citep{lehmkuhl_2017_general}.}. Wheeler focused primarily on the approach by Leopold Infeld, which was first formulated in a paper by Einstein, Infeld (then Einstein's assistant) and Banesh Hoffmann \\citep{einstein_1938_the-gravitational} and consequently goes by the name of EIH. It remained a major focus of Infeld's research all through the 1940s.\n\nWheeler had been interested in EIH early on and, in \\citep{infeld_1949_on-the-motion}, he is in fact credited with pointing out the fact that the EIH program has only a trivial (Minkowski) zero-mass limit, and that consequently a separate proof is needed in order to show that test particles follow geodesics in a non-trivial background field. The first reference to a paper by Infeld in Wheeler's notebook, however, appears only on 14 April 1953 (WR1, p. 125), several days after his last attempt to construct a liaison action (using the divergence of liaison lines in a small volume element) had ended inconclusively. Already in that attempt he had had to assume a pre-exisiting (though not necessarily metric) space in which to place the volume element. Wheeler was thus setting aside his ambitious goal of reconstructing space and time entirely from the world lines and liaisons, hoping that ``that deduction will come later'' (WR1, p. 113). Turning to the Einstein-Infeld-Hoffmann approach was a further step in this direction. After an intense study of Lichnerowicz's formulation of general relativity as initial value problem,\\footnote{He had been pointed to these mathematical works by Arthur Wightman; WR1, p.121.} which he hoped to combine with the EIH approach (the notes carry the header ``Geodesics from Field Eqns \\emph{or} Initial Conditions on Field Eqns''), he formulated, on the last pages of his first relativity notebook, a new research program on 1 May (WR1, p. 150).\n\nBefore we turn to this research program, we should briefly discuss the attraction of the EIH approach. For it is quite striking that only a few years earlier the EIH approach had been adopted as the basis for another attempt at a theory of everything, Peter Bergmann's construction of a theory of quantum gravity.\\footnote{For more details, see \\citep{blum_2016_quantum}.} Bergmann's hope had been that by transferring the EIH approach to quantum theory, the equations of the quantum mechanics for point particles might follow from the quantum field theory of general relativity in a similar manner as the classical equations of motion for point particles could be derived from the classical field theory. Even though Bergmann and Wheeler were pursuing quite different approaches, their common interest in EIH can be explained rather simply: EIH held the promise that general relativity might have something to contribute to the microphysics of particles. And for Wheeler, who had now been trying unsuccessfully to reconstruct general relativity from microscopic particle trajectories for quite some time, this prospect, which at the same time let him keep the central notion of the world line, was naturally very interesting.\n\nFor Einstein, the representation of matter particles as singular world lines in EIH had not been intended as final. It was a place holder for an ultimate (field theoretic) description of matter, no better (but also no worse) than the energy-momentum tensor on the right-hand side of the Einstein equations.\\footnote{This assessment is based on \\citep{lehmkuhl_2017_general}.} For Wheeler, on the other hand, coming from the pure world line approach, singular world lines appeared as a perfectly adequate description of material particles. The different status accorded to the world lines determined their assumed properties beyond mere approval or disapproval:  For Einstein the properties of the singularities could only be determined by the field equations. These did not determine the mass or the charge of the singularities, which were consequently free parameters, independently choosable for each individual singularity; much to Einstein's dismay, it should be added, as he hoped that the final theory would be able to explain why only two different masses (electron and proton) occur for the elementary particles \\citep{einstein_1935_the-particle}. For Wheeler, in contrast, the world lines were still entities in and of themselves, and the default assumption (at least in Wheeler's `everything as electrons' tradition) was that they would be identical:\n\n\\begin{quote}\n[T]here is no place for the $e\/m$ of a particle to enter, and all particles should have the same $e\/m$. (WR1, p. 150)\n\\end{quote}\n\nThis presented challenges of its own, since there was of course more than one type of particle in the world. Wheeler reported that his physicist colleague Hartland Snyder ``was inclined to pooh-pooh it all [on] acc't of existence of mesons, etc., in the world.'' (WR1, p. 150). Still, Wheeler was optimistic and had some ideas on how to produce a larger variety of particles with just one type of world line: Anti-particles were to be explained as world lines with the opposite orientation in time (an idea he had proposed to Feynman already a decade earlier); and he hoped to include spin in the picture by somehow taking into account the duality introduced by the two-sheeted Einstein-Rosen metric:\n\n\\begin{quote}\nTheir [Einstein and Rosen's] bridge idea is most intriguing -- two sheets of $g$ meeting at each singularity, get neutrino? (WR1, p. 151)\n\\end{quote}\n\nOn 13 May 1953, Wheeler then took his new idea of combining a (ideally unified, i.e., gravitational and electromagnetic) field theory with particles explicitly described as singular world lines to Einstein himself, when he visited him in his house on Mercer Street together with his entire relativity class. Ten years after his first discussion on AAD gravity with Wheeler, Einstein's reaction appears to have been mixed. As opposed to most of the others that Wheeler had spoken to, ``Einstein agreed [the] universe had to be closed to make [Mach's] principle valid'' (p. 11 of Wheeler's Notebook Relativity 2, henceforth referred to as WR2), but believed this to be merely a necessary but not a sufficient condition.\\footnote{According to the recollections of Wheeler's student Marcel Wellner, Einstein had apparently not thought about Mach's principle in a long time \\citep{wheeler_1979_mercer} when it came up during the visit of Wheeler's class. But less than a year after that visit, Einstein was asked about the matter again, by Felix Pirani. Einstein expressed his surprise at the renewed interest, opening his letter of 2 February 1954 (Einstein Papers, Jerusalem) with the words: ``There is a lot of talk about Mach's principle.\" By that time, apparently having rethought the matter following the meeting with Wheeler and his students, Einstein had convinced himself that the principle was obsolete, telling Pirani: ``In my opinion, one should not speak of Mach's principle at all any more.''}\n\nEinstein's reactions to the specifics of Wheeler's research plan were even more lukewarm: He declared that he ``was not interested in singularities'' (WR2, p. 11) and that the idea expressed of ``connecting [an Einstein-Rosen bridge] up with spin of electron, neutrino is no good.''\\footnote{Arthur Komar offered a more specific account of Einstein's dismissal of Einstein-Rosen bridges, recalling: ``John Wheeler asked him about the Einstein-Rosen bridge. Why had he first introduced it and then dropped it again? Einstein answered that he had initially believed that the bridge connects two almost plane surfaces in a unique manner. When he, however, discovered that they did not have a unique structure, the bridge seemed to him to be too cumbersome, unattractive, and ambiguous.'' (\\emph{John Wheeler fragte ihn \\\"uber die Einstein-Rosen-Br\\\"ucke. Warum habe er sie zun\\\"achst eingef\\\"uhrt und dann wieder fallengelassen? Einstein antwortete, dass er zun\\\"achst glaubte, die Br\\\"ucke verbinde zwei fast ebene Fl\\\"achen in eindeutiger Weise. Als er jedoch entdeckte, dass sie keine eindeutige Struktur war, schien ihm die Br\\\"ucke zu schwerf\\\"allig, unattraktiv und vieldeutig.}) These still rather vague recollections might be of Einstein referring to the fact that he had hoped that multi-bridge solutions to the Einstein equations might be so constrained as to enforce equal masses for the individual bridges, thereby addressing the problem discussed earlier of explaining why only a few different mass values for elementary particles were observed. He ultimately appears to have concluded that no such constraints would arise, as stated in a letter to Richard Tolman of 23 May 1935 (Einstein Papers): ``One does not see why the ponderable and electric masses cannot be arbitrarily large or different, when several are present.'' Many thanks to Dennis Lehmkuhl for discussions on the Einstein-Rosen paper and for making this letter available to us.} (WR2, p. 11) Wheeler's general relativity class ended two weeks later with a final exam on 28 May. His interest in general relativity was unbroken and his notebook contains notes on cosmology, gravitational radiation, and long passages in French copied from Lichnerowicz's 1948 lecture notes ``G\\'eom\\`etrie diff\\'erentielle et topologie''  before having to return them to the library (WR2, pp. 29-34). But for two months after the visit to Einstein, the notebook contains nothing new on Wheeler's foundational ideas and the question of how to turn singular world lines in general relativity into full-fledged particles. Wheeler did take Einstein's negative remarks with a grain of salt, in particular attributing Einstein's negative attitude toward singularities to the fact that recent work by Infeld had shown that applying the EIH method to Einstein's unified field theory did not return the correct equations of motion, i.e., the Lorentz force law in curved space-time \\citep{infeld_1950_the-new}. But it was his preparations to give a talk at the International Conference of Theoretical Physics in Japan, to be held in September 1953, that gave Wheeler a new impulse.\n\n\n\\section{Daring Conservatism and the Field Program}\n\\label{sec:fields}\n\nWheeler ended up giving three talks in Japan: two rather technical ones on the origin of cosmic rays \\citep{wheeler_1954_the-origin} and on collective models for nuclei \\citep{wheeler_1954_collective}, published in the conference proceedings; and one more programmatic talk, which he held on 10 September 1953, before the conference, at the Physical Society of Japan and which was only published in Japanese translation\\footnote{According to the notebook (Wheeler Papers) that Wheeler kept during his stay in Japan, the translation was done by Takahiko Yamanouchi; Japan Notebook p. 51.}  in the Proceedings of the Society. We provide a retranslation into English of this talk (the original manuscript and recording are lost) in the appendix. It is this talk which is of central importance to our story, and it is this talk that one finds Wheeler preparing in his notebook on 18 July 1953 under the heading: ``Philosophy of approach to elementary particle problem''. From the start, Wheeler was very eager to establish a clear connection to Japan in his talk, noting in the margins: ``Each one of us finds himself reflected in the countries he visits.'' But he also took the opportunity to reflect on his overall methodology. We have seen several times Wheeler's predilection for taking existing theories and using and extrapolating them outside their established domain of applicability, the paradigmatic example being his attempts to explain the nuclear forces electromagnetically. In the notes for the Tokyo talk, this methodology, which he would later characterize as ``daring conservatism'' is now, for the first time, made explicit as the ``Tokyo Program'':\n\n\\begin{quote}\nProposed Tokyo program: Be as conservative as possible about introducing new elements into description. Make basics as clear \\& simple as possible. Is only the consequences that are complicated: ice; elem. particles; meteorology; geology. [...] Strengths of this approach. Its weaknesses. Einstein's May '53 remark to JAW: `The Lord may have made the universe with five fields. I don't think so. But if he did, I am not interested in the universe.' Quote as a Princeton physicist, nameless. An extreme attitude, not fully open minded. Surely much good.\n\\end{quote}\n\nAs he outlined his guiding methodology explicitly for the first time, Wheeler was clearly becoming excited, referring to himself in the margins as ``Tokyo Wheeler'', drawing an admittedly somewhat bizarre analogy between his new ideas on elementary particle physics and the demoralizing propaganda spread to the American troops by ``Tokyo Rose'' (Iva Toguri), host of the WWII Japanese English-language radio show ``The Zero Hour''.\n\nAt this point Wheeler's notes shift away from a lecture sketch to an inner monologue about the foundations of his research program:\n\n\\begin{quote}\nEvidently have in mind something more fundamental. Out with it! Desert island philosophy: imagine selves cast up on Wake Island with library of all theory \\& exp[erimen]t up to now, to solve elem. particle problem -- What to use as starting points? -- Others not ambitious enough? Go whole hog now!\n\\end{quote}\n\nWhat follows is a long list of elements (of existing theory) that might be of importance in his attempts at crafting a theory of elementary particles. We explicitly see Wheeler assessing the potential of existing theory, in particular general relativity. The list contains familiar tropes (action at a distance -- point 2; Mach's principle -- point 7), but also some novel elements, indicating how Wheeler was reordering his vision of how to think of elementary particles. The central new element is an emphasis on fundamental masslessness, the vision of a theory without intrinsic mass parameters that would ideally include``\\emph{no natural constants}. Nothing but $e$, $\\pi$, etc.'' (point 1).\\footnote{In a manuscript entitled ``The Zero Rest Mass Fundamental Field Hypothesis'' (WR, p. 101), which we shall discuss later in more detail, Wheeler ascribes this vision of a theory with no free parameters to Einstein. We have not been able to find relevant statements in Einstein's work.} Where then was mass to come from? In point 8 of his list, Wheeler remarked on the ``Electromagnetic origin of mass and the self energy story'', jotting down the first order radiative corrections to the electron mass, as first derived by Weisskopf in 1939. These terms were, in modern theory, simply absorbed in a renormalization of the electron mass, ultimately implying total agnosticism about the origin of mass. But in view of the proposed masslessness of the fundamental pointlike particles, Wheeler was highlighting the electromagnetic origin of mass, advocating (point 11) that one ``should apply electrodynamics to very small distances''.\\footnote{Wheeler here also invoked, for the first time, Bohr as the godfather of daring conservatism, because Bohr had applied ``electrostatics to very small distances'' in his atomic model.} With mass externalized from the point-like singular particles to the surrounding field, Wheeler could consider all particles as composite (point 13), as ``structures held together by radiative, electrodynamic and gravitational forces.'' (point 14) Wheeler's new vision thus really amalgamated all existing theory by proposing particles with a singular point-like core and field-generated structure. \n\nThis new focus on masslessness temporarily moved the neutrino to the center of Wheeler's theorizing as he emphasized the ``importance of the \\emph{neutrino} in the scheme of things'' (point 3). We again encounter the ambiguity in the conceptualization of the neutrino: At one point it appears as the fundamental point-like entity, with the electron to be thought of as a ``neutrino with a charge loaded on its back''. At other times, it clearly appears as a field-like entity, possibly arising through ``spinorization'' of the metric of general relativity, that is taking the ``square root'' of the (vacuum) Einstein equations in a manner analogous to that which generates the Dirac from the Klein-Gordon equation. While Wheeler saw this as a major challenge, he was rather optimistic, remarking: ``Spinorize, fit all together, and listen for the harmony.''\n\nWheeler's novel emphasis on neutrinos was apparently also fueled by first results of the efforts by Frederick Reines and Clyde Cowan to directly detect these elusive particles. At this time, in the summer of 1953, Reines and Cowan were performing first background checks with their liquid scintillator detector at the nuclear reactor in Hanford, WA. They had found a source-independent background, which they thought might be due to ``natural neutrinos'' (what one might call cosmic relic neutrinos in big bang cosmology). Wheeler was aware of these results, referring in his notebook to ``Reines-Cowan radiation'' when emphasizing the importance of the neutrino. Wheeler was briefly envisioning the neutrino not only as the fundamental constituent of all particles, but also as the prime component of the energy density of the universe, and his notes of 6 August 1953 show him studying the Friedmann equations in a neutrino-dominated universe.\\footnote{No correspondence between Wheeler and Reines or Cowan from 1953 is extant, but Reines in turn was clearly aware of Wheeler's contemporaneous elevation of the neutrino to central stage. In his Nobel lecture, Reines makes an inside joke, remarking without mentioning Wheeler: ``While we were engaged in this background test, some theorists were rumored to be constructing a world made predominantly of neutrinos!''} These calculations were interrupted by a phone call from Reines, informing Wheeler that they had identified their source-independent background as due to nuclear capture of cosmic ray muons.\n\nStill, the neutrino kept an important role, also in the talk that Wheeler eventually held in September 1953 in Tokyo. The talk is set up as a dialogue between Wheeler and two figures from Japanese history, Saigo Takamori and Sugawara no Michizane.  Saigo, an important 19th Century Samurai, is given the role of the daring modernizer and presents the current state of the art in particle physics, the discovery of new particles at accelerators, and the meson theory of nuclear interaction. Sugawara no Michizane, a Ninth Century scholar and poet, is given the role of reflective traditionalist, who presents Wheelers Tokyo program, though the program is not actually named in the talk. It is merely characterized as ``the principle, which is the basis of the scientific method, of not introducing a new hypothesis until it is clearly and undoubtedly necessary.''\n\nSugawara begins by lauding general relativity as a model field theory: On the one hand, there\nis the point we have already discussed extensively, that it allows for the integration and the\nderivation of the equations of motion of point particles. But more importantly, general relativity, viewed as Einstein's formalization of Mach's principle, was supposed to provide an\naccount how a field theory (or more generally an interaction, which could also be a theory\nof action at a distance) could generate mass in a massless theory, or rather inertial mass in a\ntheory without inertial mass. The argument as presented in the talk (or at least as presented\nin the Japanese translation) is somewhat elliptic. It is formulated not in terms of general relativity, but in terms of an AAD theory. Clearly, such an AAD theory could not be equivalent\nto GR; we know that Wheeler had been searching for such an AAD formulation of GR for several years, but had not been able to construct one. Instead, the AAD theory he used in the\nTokyo talk was a slight modification of Newtonian theory, where the usual Coulomb field is supplemented by a second field that falls off only as $1\/r$ and thus dominates at long\ndistances. Wheeler gives this field explicitly as $G m_g a\/c^2 r$, where $m_g$ is the particle's \\emph{gravitational} mass and $a$ is its acceleration. This expression is analogous to the long-distance Li\\'enard-Wiechert field of an accelerating charge in electrodynamics, and since it was not to be expected that the analogy between electrodynamics and gravity would be that perfect, Wheeler\/Sugawara put the expression in scare quotes. With the long-distance interaction established, Wheeler then introduced what he called the \"whole idea of gravity theory\", namely that the total gravitational force on a particle is zero: a particle subject only to gravity is not moved by forces, but by the curvature of spacetime. In the modified Newtonian AAD theory the same idea is appropriate to express the expectation that inertia is provided by interaction and an \"intrinsic inertia\" term ($m_{\\mathrm{inert}}a$) is absent from the equation of motion:\n\n\\begin{equation}\n\\frac{G m_1 m_2}{r^2} - \\sum_k \\frac{G m_1 a m_k}{c^2 r_k} = 0\n\\end{equation}\n\nThe equation's second term can instead be understood as the reaction on mass $m_1$ to the force that $m_1$'s acceleration exerts on the masses $m_k$ through the new, long-distance, Li\\'enard-Wiechert-type interaction. One gets the usual (unmodified) Newtonian equation of motion for $m_1$ in gravitational interaction with $m_2$ (with $m_{\\mathrm{inert}} = m_g$), under the condition that\n\n\\begin{equation}\n\\label{eq:mach}\n\\frac{G}{c^2} \\sum_k \\frac{m_k}{r_k} = 1\n\\end{equation}\n\nwhere the sum extends over all of the distant masses $m_k$ which are at distances $r_k$ from the mass $m_1$.\n\nIt is appropriate at this point to point out the intimate relation between Wheeler's argument and a sketch of the origin of inertial mass published by Dennis \\citet{sciama_1953_on-the-origin} just a few months before Wheeler's talk. Sciama's argument was field-theoretical, but also built on the electromagnetic analogy, explicitly employed vector fields obeying the Maxwell equations as gravitational fields and obtaining long-distance Li\\'enard-Wiechert potentials that correspond to Wheeler's long-distance force. Sciama also introduced an analogous principle to Wheelers ``whole point'', which in his field-theory language reads that ``the total gravitational field at the body arising from all other matter in the universe is zero'', but Sciama explicitly labels this as a postulate and specifies that it holds in that body's rest frame. In this rest frame, the whole exterior universe is moving with acceleration $-a$, and the total field from the distant matter (the $1\/r$ term) should exactly cancel the short-distance gravitational field (the $1\/r^2$ term) of the particle with mass $m_2$. Rewriting this equation of cancellation, Sciama gets the usual Newtonian force law for the gravitational interaction between the masses $m_1$ and $m_2$  under the same condition as Wheeler (Equation \\ref{eq:mach}) obtained in field-theoretical terms (Equation 6\/7 of Sciama). It is unclear whether Wheeler knew of Sciama's argument and merely rephrased it\nin AAD terms, or whether he had found it independently in his attempts at constructing an AAD version of gravity, building on an AAD formulation of electrodynamics. Both stories seem plausible, and if Wheeler really did not mention Sciama in his talk (and this is not just an omission of the transcription that was then translated into Japanese) the second one\nseems the more likely. Wheeler did eventually learn of Sciama's paper, as he jotted the reference down on the last page of his second relativity notebook (which covers the period up to April\n1954), but since this last page appears to have served as a general place to note miscellaneous references, it is impossible to date. In any case, the Machian argument in the Japan talk was merely to serve as a proof of principle how mass might arise in a theory in which it is not a primary attribute of matter.\n\nA similar proof of principle was given for the electrodynamic generation of mass through the radiative corrections calculated by Weisskopf, which we have already mentioned above. Wheeler's treatment in the Tokyo talk is somewhat problematic. Following Weisskopf, he (or rather Sugawara) gave the radiative correction $\\delta_m$ to the electron mass as\n\n\\begin{equation}\n\\label{eq:weisskopf}\n\\frac{\\delta m}{m} = \\frac{3}{2 \\pi} \\frac{e^2}{\\hbar c} \\ln{\\frac{\\lambda_{max}}{\\lambda_{min}}}\n\\end{equation}\n\nLeaving the question of the infrared and ultraviolet cutoffs in the logarithm aside for the moment, the parameter $m$ is here the electron's bare mass, which should be zero according to Wheeler's assumptions. Wheeler, however, takes it to be the electron's physical mass, assumes this to arise entirely from radiative corrections (i.e., from the field), and thus sets the lefthand side of the equation to 1. Today it is well established that perturbatively a massless fermion cannot gain mass from its electromagnetic interaction, precisely because the radiative corrections are always proportional to the bare mass (due to chiral symmetry). However, chiral symmetry may well be broken through non-perturbative effects, so that the general idea of a purely electromagnetic mass is not implausible. And again, Wheeler appears to merely have been floating some rough ideas for how mass might arise in a fundamentally massless theory and how one might obtain a unique value for the fine structure constant.\\footnote{Here Wheeler was following in the footsteps of a number of famous physicists who had attempted to derive the value of the fine structure constant (which for a long time looked like it might be precisely 1\/137) in the preceding decades. See \\citep{kragh_2003_magic}.}\n\nAll of this was thus an elaboration of the program he had outlined in his preparatory notes. The conservative Tokyo Program was now personified by the measured statesman and poet who was filled with a ``love of Japanese beauty and harmony'', who wished to work only with well-established entities and theories and to introduce no free parameters, such as masses, into his considerations; though the end of the talk saw Sugawara reconciled with the audacious Samurai Saigo, already heralding the reformulation of Wheeler's program as not merely conservatism but ``daring conservatism'' several months later. The part of Wheeler's program that was most in flux, however, as witnessed not only by the Tokyo talk but also by the notebook entries of the time, was the exact role of the neutrino. While the talk clearly focussed on the field-theoretical aspect of the neutrino, it explicitly raised the question whether it was to be thought of as a massless field that joined the electromagnetic and gravitational fields in giving structure to the elementary particles, or whether it was only a derivative of the gravitational field, arising upon spinorization. \n\nThrough his study of the literature on spin in general relativity (specifically \\citep{pauli_1933_uber-die-formulierung}), and through discussions with the Princeton mathematician Oswald Veblen (30 October 1953), Wheeler reached the conclusion that the last point was true, but that this spinorization could only occur upon quantization:\n\n\\begin{quote}\nMy conclusion? I know that the neutrino obeys Pauli statistics, therefore cannot come into a classical theory, therefore ought to show up only after quantization, therefore I should look for the classical theory \\& then quantize it a la Feynman, but with a square root, antisym, spinor character all put in at that time.\n\\end{quote}\n\nThe neutrino and the issue of spin, which had temporarily been at the focus of Wheeler's interest and of the Japan talk, was thus temporarily set aside and relegated to the quantum realm. This further strengthened the focus on the classical fields of electrodynamics and gravitation, which, despite the persistence of singular point particles, were doing the work. It was the fields that had the potential to clarify the question of elementary particles, that would generate masses and define equations of motion, classically and in quantum theory. Wheeler's main focus was thus now on Einstein-Maxwell theory, a classical field theory that would, at least after quantization, give a full account of the physics of elementary particles:\n\n\\begin{quote}\nIf $\\nu$ is somehow contained in em+grav., and if we are right saying that only fields of zero mass count (no meson fields, etc.), and if we have the \\emph{right} theory of em+grav., and if Feynman procedure [path integral quantization] is legitimate for such fields, then \\emph{here's where we start}.\n\\end{quote}\n\nIn Einstein-Maxwell theory, the electromagnetic and gravitational fields appear as separate entities and are simply minimally coupled. Wheeler referred to it as the ``un-unified field theory''. The contrast with the unified field theory program of Einstein and others was clear and indeed these were to be viewed at the time as legitimate competitors of Einstein-Maxwell theory as classical descriptions of electrodynamics and gravitation. Wheeler thus felt the need to consider their merits, before further pursuing his program. \n\nHow now to judge these merits? Einstein's unified field theory \\citep{einstein_1950_the-meaning} was out, because, as we have already mentioned, one could not EIH-derive the Lorentz force from it. But Wheeler's student Arthur Komar (23 October) had pointed him to an alternative unified theory that gave, through the EIH method, the correct equations of motion, i.e., including the Lorentz force. This was the unified field theory of Behram \\citet{kursunoglu_1952_einsteins}. There was, however, a different problem with Kursunoglu's approach for Wheeler: It relied on the introduction of a fundamental length, i.e., a dimensionful parameter into the theory, which was of course in strict opposition to Wheeler's program of having no natural constants. Wheeler asserted that ``conservative me'' (30 October) had to try out what would happen in Kursunoglu's theory when one let the fundamental length go to zero: Would one still have a unified field theory or would one merely obtain general relativity without electrodynamics? The above quote thus continues:\n\n\\begin{quote}\nOnly one question \\emph{before we start} --- what about so-called unified field theory? Einstein's variety no good. Therefore try Kursunoglu's variety --- in case where his $p$ [inverse of fundamental length] is set equal to $\\infty$ --- just to test whether we have any \\emph{conservative} alternative to what we are doing.\n\\end{quote}\n\nWheeler was thus now explicitly using conservatism (in the sense of no natural constants) as a criterion for theory selection. On 1 November, he came to the conclusion that Kursunoglu's theory, in the limit where the fundamental length goes to zero, merely reproduced Einstein-Maxwell theory. His assessment of unified field theory thus ended with a ``bronze plaque'' in his notebook, reading: ``Unified Field Theories died here'' and a letter to Kursunoglu, on 3 November, in which Wheeler wrote:\n\n\\begin{quote}\nI am writing to ask if a conservative physicist who wants to deal with gravitation and electromagnetism within the framework of general relativity has nowadays any acceptable choice but to use as action the expression [action of Einstein-Maxwell theory]. By ``conservative'' I mean unwilling to  introduce new ideas, new concepts, and particularly unwilling to introduce any quantity with the character of a fundamental length except as called for by inescapable evidence.\\\\\nWill not one who adopts the conservative point of view, as just defined, have to abandon unified field theory as it stands at present?\n\\end{quote}\n\nWheeler had thus firmly convinced himself that the theory he needed to quantize was the conservative, minimal Einstein-Maxwell theory; he had found the new focus of his research in an attempt to quantize gravity, minimally coupled to electrodynamics. While quantum gravity nowadays, with all of the technical and conceptual difficulties it entails all too clear, hardly seems a conservative endeavor, to Wheeler it certainly seemed as such; it was based merely on a combination of the well-established principles of general relativity, Maxwell electrodynamics, and quantum theory. After 20 years of private speculations, he felt he was now ready to publicly elaborate on his vision for the foundation of physics, a vision that was built on general relativity, a theory that was not only coherent and well-established, but also, through its unique features, such as Mach's principle and the EIH determination of equations of motion, had the potential to resolve the great open questions of microscopic physics. On 4 November 1953, we thus find in Wheeler's notebook ``Points for proposed article `Elementary particles from Massless Fields --- An Assessment.''\n\nAround this time, Wheeler suddenly appears to have remembered a central point, which indeed was absent at least from his notebook entries for a while: the point particles. For November 8, we find the following short entry:\n\n\\begin{quote}\nThe big question\\\\\nLet's forget about electromagnetism for present. In quantum transcription of the pure gravitation theory with the variation principle based on $\\psi = \\sum e^{\\frac{i c^3}{16 \\pi G \\hbar} \\int \\int \\int \\int R \\sqrt{-g} dx^1 dx^2 dx^3 dx^4}$ [i.e., the path integral for the Hilbert action] how do we take into account the existence of singularities?\n\\end{quote}\n\nThe singular world lines had, over the course of the year 1953, been transformed from the central element of the theory into a problematic embarrassment in the promising program of quantizing general relativity. Like Bergmann several years earlier, Wheeler realized that quantization and point singularities in the field did not really mesh. Bergmann had resigned himself to studying pure general relativity, but this was hardly an option for Wheeler who was after all trying to solve the problem of elementary particles. And indeed, the fields in Wheeler's approach were still mainly meant to provide services to the point particles: give them mass, define their equations of motion. When he met with Einstein once more, in the morning of 13 November 1953, Einstein asked (WR2, p. 83): ``What about matter term in Lagrangian''  to which Wheeler replied that ``matter was to originate from singularities.'' However, when Wheeler then went on to explain Feynman quantization to Einstein, he remarked that in this setup ``the singularities in field get eliminated, never have to be talked about.'' This seems to be in reference to the assumption that singular field configurations would have measure zero in the path integral, which seems like a problem for describing matter by singularities, but is of course a good thing when talking about pathological singularities.\\footnote{Indeed, Einstein appears to have been impressed. While first remarking that he ``abhorred'' the idea of first constructing the classical field theory and then quantizing it, he then conceded (according to Wheeler's notes) that ``it was the first time he had ever heard describe a way that [quantum theory] might get through, found it very attractive.''}\n\nBut an even more severe difficulty with the singular point particle notion lay in its relation to the field concept. EIH determination of the equations of motion, of course, offered the prospect of reconciling  the notions of field and point particle; this fact had originally led Wheeler to reintroduce fields into his worldview and endorse a dualistic ontology. As soon, however, as mass generation through the field entered into the picture, the fundamental incompatibility of point particles and local fields, which had haunted fundamental physics ever since Hendrik Lorentz had first tried to think the two together in his electron theory, again became visible. Indeed, already in his Tokyo lecture, Wheeler had been forced to introduce an ultraviolet cutoff ($\\lambda_{\\mathrm{min}}$ of Equation \\ref{eq:weisskopf}) to make the field-generated mass finite. This essentially meant abandoning the idea of a point particle and introducing a finite size for the electron. It is important here that Wheeler (or Sugawara) had hypothesized that this finite size would be given by the gravitational (Schwarzschild) radius of the electron, and not the Planck length. So the necessary mass scale that one needed to make a length using the gravitational constant $G$ and the speed of light $c$ was provided by the mass $m$ of the electron and not by Planck's constant $h$. This clearly indicated that the cutoff was to arise not as a quantum effect, but due to the presence of the particle. By introducing the notion of field-generated masses, Wheeler had thus effectively abandoned the notion of a point particle that had been a mainstay of his research program for a long time. This was not a problem for the EIH determination of the equations of motion, as the use of point particles in that derivation could well be viewed as a mere approximation.\\footnote{While \\citet{wheeler_1961_geometrodynamics} would later conclude that point singularities were not a valid approximation for any reasonable model of matter (which by that time for him meant geons and wormholes), there is no indication that he (or anybody else) harbored such doubts in 1953\/54, given that the concepts and in particular the conception of matter that these conclusions were based on had not been developed yet.} But it ultimately undermined Wheeler's briefly-kept hopes for a dual theory of point particles and fields and forced him to consider novel conceptions of matter.\n\nWhile he spent the next weeks thinking about how to spinorize Einstein-Maxwell theory by taking the square root of the Lagrangian in the action (WR2, p. 88), the pressing question of the constitution of matter moved to the center in a working paper entitled ``The Zero Rest Mass Fundamental Field Hypothesis'' and dated 19 January 1954.\\footnote{The paper is included in WR2, p. 101, as an insert. This copy is noteworthy also for some remarks in the margins in which Wheeler explicitly connects his conservative heuristic in physics with conservatism in politics, noting: ``To defend well established physical ideas as unpopular as defending well established political parties. People like to criticize. Religion the great defender.'' In this connection it appears pertinent to mention that Wheeler's conservative stance (in physics), as outlined in the Tokyo talk, was explicitly criticized by the Japanese physicist Shoichi Sakata, an outspoken Marxist \\citep{staley_2004_lost}. In discussions on September 18 at the conference in Kyoto, a week after Wheeler's lecture, Sakata remarked: ``I am convinced the future theory should not be the progressive improvement of the present theory. At the Tokyo meeting Professor Wheeler pointed out that there are two methods of approaching the truth; that is Saigo Takamori's method and Sugawara Michizane's method. But in Japan Professor Tomonaga had pointed out that there are two ways, namely a non-reactionary conservative way and also a revolutionary way. This is our common sense.'' \\citep[p. 34-35]{japanproceedings}} Here, Wheeler addressed the central question that any theory of extended (i.e., not pointlike) particles would have to answer. While the spatial extension of the particles avoided the issue of divergent field strengths, it brought with it a different issue, which had a long tradition going back to first attempts at a solution by Poincar\\'e: the issue of stability. Given that there would be no more singular point-like cores, all that was left for constructing a particle were the electromagnetic, gravitational, and possibly neutrino fields (the ``zero rest mass fundamental fields'' of the manuscript's title), and ``an elementary particle is held together by the balance of gravitational, neutrino, and electromagnetic forces'' (p. 7 of the manuscript). But how to envision such an object? In the manuscript, Wheeler explored the possibility of comparing elementary particles with a (collapsing) star---the analogy being based on both objects (star and particle) being held together by gravitational forces.\n\nBut the big breakthrough for how to model elementary particles only occurred about a week later, when Wheeler attended the Fourth Rochester Conference on High Energy Nuclear Physics from 25-27 January 1954 \\citep{rochester_1954}. It is the last one of Wheeler's breakthroughs that we shall discuss in this paper, as it finally brings us to  Wheeler's geon paper \\citep{wheeler_1955_geons} and his embrace of a pure field theory, from which also the singularities representing matter had been removed. Up until now, Wheeler had mainly attempted to use the untapped potential of general relativity as it related to mass points: The ability to derive their equations of motion from the field equation, the possibility of generating mass for them from fields or interactions. In late January 1954, Wheeler seized upon a feature of general relativity, which he had hardly engaged with so far: the non-linearity of the field equations, which in principle allowed for solutions describing a localized and (meta)stable concentration of energy, an idea which had been in the back of Einstein's head for a long while. \n\nOn his manuscript of 19 January (which was never published), Wheeler had noted that he was distributing it to a small number of physicists, including Einstein, Bohr, and Wightman. Wightman was also attending the Rochester conference, and Wheeler appears to have discussed his ideas with him there, for on 25 January 1954, we find the notebook entry (WR2, p. 96):\n\n\\begin{quote}\nBall of light held together by gravitational forces as classical model for an elementary particle = fireball = (Wightman name) Kugelblitz\n\\end{quote}\n\nimmediately followed by calculations for a spherically symmetric graviational potential fulfilling the vacuum Einstein-Maxwell equations (i.e., the Einstein equations with only an electromagnetic energy-momentum tensor as a source), with all of the electromagnetic energy constrained to a sphere of finite radius. Such a field configuration, which could only exist in a non-linear field theory such as Einstein-Maxwell theory and which Wheeler would soon label a ``geon'' (first found in WR2, p. 104, in an entry dated 19 February 1954), was thus the new model for elementary particles that Wheeler would pursue for the next few years. Everything point-like had been expelled from the model, in favor of a spatially extended pure zero-mass-field configuration.\n\nThere were of course many open questions to tackle, some of which Wheeler listed in the entries of the next two days (WR2, p. 100ff), such as whether such entities really existed, how to incorporate charge,\\footnote{Here Wheeler already pondered the possibility of having ``outgoing lines of force [...] understood in terms of lines coming in from an `internal universe''', an idea that would later mature into his notion of a wormhole.} the still unsettled role of the neutrino and the square root of the Einstein-Maxwell Lagrangian, and the role of quantum theory and self energies,\\footnote{Here Wheeler encountered some conservative resistance from Wightman, who objected to Wheeler's predilection for path integrals, arguing instead that one should ``improve \\& understand present formalism'', i.e., pursue axiomatic quantum field theory. Even Feynman appears to have been doubtful about the ``general utility'' of the path integral, as he had not yet been able to properly accommodate fermions.} in particular concerning the quantization of general relativity, in which context Wheeler noted (WR2, p. 103):\n\n\\begin{quote}\nTry to understand whether Gupta or anyone else really know what he's talking about on the quantization of gravitation theory, esp. the comm'n. rel'ns at small distances.\n\\end{quote}\n\nBut while the new geon model of elementary particles brought with it a host of unanswered questions, an entire research program as it were, just days after the Rochester conference (where he had talked on charged meson decay) Wheeler certainly felt confident enough to publicly present his new idea in New York City, where he held the annual Richtmyer Memorial Lecture of the American Association of Physics Teachers (AAPT).\\footnote{\\label{fn:long} The AAPT was conducting its winter meeting in parallel with the American Physical Society, which conducted its annual meeting at Columbia University from 28-30 January 1954 (Physical Review, Volume 94, pp. 742ff), so that there were also many research physicists in the audience.} This Lecture, entitled ``Fields and Particles\", is the last text we shall be discussing and is, as we shall see, in many ways the sum of the development in Wheeler's thinking that we have reconstructed in this paper.\\footnote{The lecture was never published, but there is an extant transcript in the Wheeler Papers, in a folder entitled ``Fields and Particles.'' The Richtmyer Lecture Memorial Award had been established in 1941 to honor Floyd Richtmyer, one of the founders of the AAPT (\\url{https:\/\/www.aapt.org\/Programs\/awards\/richtmyer.cfm}). Many of the previous lectures had ben published in the AAPT's journal, the American Journal of Physics (e.g., \\citep{slater_1951_the-electron, vleck_1950_landmarks, dubridge_1949_the-effects}). Wheeler had plans to publish his lecture there as well, and the folder contains two revised versions of the original lecture transcripts, which were clearly supposed to lead up to a publication. The folder also contains some correspondence between Wheeler and Thomas Osgood, editor of the American Journal of Physics, such as a letter from Osgood of 28 January 1957, which begins: ``Here is my annual letter of inquiry about the manuscript of the paper ``Fields and Particles'' that you gave as Richtmyer Memorial Lecture during the meeting of the American Association of Physics Teachers in New York, January 28-30, 1954 It ought to be published without delay.'' Wheeler in fact cited the paper in the first footnote of the Geon paper as ``to be published''. That long footnote (a specialty of Wheeler, to which this footnote here is a sort of tribute) also contained a reference to Wheeler's Tokyo talk and ``the point of view ascribed by the author to Sugawara-no-Michizane,'' making the entire footnote rather enigmatic for the average American reader of the Physical Review.} \n\nThe Richtmyer Lecture began with Wheeler's most explicit elaboration of his conservative methodology, which he now labelled ``daring conservatism'' and couched in religious terms, citing the apostle Paul:\n\n\\begin{quote}\n``Whatsoever things are true, whatsoever things are honored, whatsoever things are judged, whatsoever things are pure, whatsoever things are lovely, whatsoever things are of good repute. If there be any virtue and if there be any praise, think on these things.''\\footnote{This passage is from Philippians 4:8, where it reads ``honest'' instead of ``honored'', ``just'' instead of ``judged'', and ``report'' instead of ``repute.'' We have given the quote as it appears in the lecture transcript, and it is to be assumed that the transcriber simply misheard these three words. Wheeler corrected all three in the later manuscripts of the Richtmyer Lecture mentioned in Footnote \\ref{fn:long}.} Following these words of Paul, I would like to dedicate this occasion [...] to an appreciation of the great truth of physics in the saying that from them we will receive guidance in this elementary particle problem beyond anything that we now imagine.\n\\end{quote}\n\n Wheeler then went on to highlight the role of general relativity among the ``already well established ideas'' of physics on which the conservative physicist should build by daringly ``following out [its] consequences'' to the ``utter most extreme.'' He then went on to outline the great potential (``exciting new possibilities'') of general relativity both ``in the realm of what might be called astrophysics'' and for the ``elementary particle problem'', introducing his geon\\footnote{Then still referred to as a ``Kugelblitz'' or, in the words of the person who transcribed the lecture, ``cugoflix''.} idea to the world and presenting it as a new research program:\n \n \\begin{quote}\n In my view following out the philosophy of the conservative daring [sic], it's an inescapable obligation of our present-day physics to continue the investigation of these objects and to see what boundary line if any separates them from the elementary particle problem. The full investigation of both electromagnetism and gravitation of course has to take place within the frame work of quantum theory.\n \\end{quote}\n \nWheeler had thus publicly outlined his new research program in general relativity, which consisted of studying stable, localized solutions of the Einstein-Maxwell equations, their modification through quantum theory and their relation to elementary particles, as well as the inclusion of further elements into this picture, such as charge and the neutrino\/spin. Wheeler's transition to a full-blown ``relativist'' was completed, and the research program outlined in the Richtmyer lecture would occupy him and his graduate students for years to come. So fruitful was this approach that Princeton and the Wheeler School, despite being the youngest of the relativity centers soon to be connected in the Renaissance, became one of the central hubs of that process.\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nIn this paper we have reconstructed John Wheeler's turn to general relativity in the years ca. 1941-1954 and how it was driven by what we have called the untapped potential of general relativity, thereby corroborating and filling with meaning the claim of \\citep{blum_2015_the-reinvention} that this untapped potential was one of the motors of the renaissance of general relativity. Our reconstruction has shown that Wheeler's general methodology, ultimately branded ``daring conservatism'', precisely consisted in seeking out the potential of existing theories, rather than constructing new ones. It should, however, be added that the general notion of daring conservatism can be read in two ways, both of which Wheeler endorsed. One is to extrapolate existing theory in order to make predictions for new, unexpected phenomena and then trust those predictions, even though they are made outside the domain for which the theory has been experimentally corroborated. This view of daring conservatism is to be found in an example that Wheeler gave in the Richtmyer lecture, where he claimed that he could have predicted nuclear fission two years before its experimental discovery, had he only trusted the extreme predictions of 1930s nuclear modelling. This view also applies to the use of general relativity in making novel predictions for astrophysics.\n\nBut as we have seen, it was another reading of daring conservatism that was initially more central to Wheeler's thinking: Using existing theory not to predict novel phenomena, but to solve existing (theoretical) problems and paradoxes that one might otherwise have been tempted to solve by introducing new theories. The central issue that Wheeler came to believe general relativity had the potential to solve was what he called the ``elementary particle problem''. A precise definition of this ``problem'' is hard to come by, but it meant something along the lines of obtaining a consistent description of the internal structure of elementary particles (which originally of course implied finding a consistent theory of point-like particles without structure). The solutions that Wheeler considered to this problem were shaped by several convictions, in particular that (i) the general idea of the solution should be expressible in classical language, (ii) the solution should be monistic, or at least not gratuitously introduce various types of particles, and (iii) the solution should ideally not involve any free parameters. All of these three conditions favored Wheeler's turn to GR, which was (i) a classical theory, (ii) dealing in universal substance (space-time), (iii) involving no free parameters beside the gravitational constant (which could be set to 1 in what Wheeler would later call Planck units).\n\nWe thus see that also the further development of Wheeler's career in relativity closely paralleled the overall development, as questions relativistic astrophysics (and thus the first reading of daring conservatism) gradually supplanted (or merged with) his original foundationalist aspirations, in what Roberto Lalli, J\\\"{u}rgen Renn and one of the authors (AB) have called the astrophysical turn of the late renaissance \\citep[p. 540f]{blum_2018_gravitational}. It turns out then that an important factor in assessing the relevance of the epistemic potential of GR in the renaissance is the question of ``potential for what?''. This is true not only with regards to what problems to solve, but also to what kind of work to generate. For we have clearly seen the strong pedagogical bent in the way in which Wheeler tackled general relativity, and the focus on problems to be solved; the general relativity that Wheeler was exploring was swarming with future PhD theses, theses in physics, that is, connecting the heretofore isolated field of general relativity to particle physics, quantum theory, and astrophysics. It was this aspect which turned Princeton from a research center among several to the home of the ``Wheeler School'' \\citep{christensen_2009_john, misner_2010_john}\n\nThis brings us to a final paradox: How to explain the great impact of Wheeler's approach to general relativity, given that the various solutions to the elementary particle problem that we have discussed in this paper were all eventually viewed as misguided. Neither worldlines and liaisons nor geons are nowadays regarded as fruitful ways for thinking about the structure of particles, and also the quantization of gravity did not yield to Wheeler's simple path integral vision. Our study at least suggests an answer to this paradox: The important thing was not so much the specific manner(s) in which Wheeler tried to resolve the elementary particle problem, but rather his keen sense for which elements of general relativity would turn out to be the most fruitful. \n\nLooking at Wheeler's trajectory thus also provides insight into where exactly the epistemic potential of general relativity lay, namely in its unique features as a theory: the determination of the equations of motion through the field equations, the non-linearity of the field equations, and that its quantization will lead to non-trivial new physics. Conceptual studies on the role of point particles in GR could thus segue into studies on the so-called problem of motion, studies on geons into studies of exact solutions of the full Einstein equations, studies on path integral quantization would come to be regarded as important puzzle pieces in the ongoing search for a quantum theory of gravity. Here too, we observe Wheeler's trajectory closely mirroring general trends, where isolated research centers originally focusing on GR-based speculative theorizing move, in the course of the Renaissance, to the study of important conceptual questions within general relativity, relevant to the emerging community at large. The question remains to what extent Wheeler's original interests actually shaped the problems considered important in the GR community of the renaissance and beyond. But this question is beyond the scope of our study, which focused on an individual intellectual trajectory and on a conversion from particle to field theory that turned out to be far more gradual than expected. If the reader thus takes home just one fact from our story, it might be this: For a few months there, in late 1953, John Wheeler believed in both particles and fields.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nIt is generally believed that QCD undergoes chiral restoration\nat sufficiently high temperatures. This is supported by lattice\nsimulations \\cite{K95}, as well as by a variety of model \ncalculations. As the temperature grows, the value\nof the quark condensate increases from its negative $T=0$ value \nand approaches zero. As shown in Ref. \\cite{GL}, in the exact chiral \nlimit (zero current quark masses) chiral symmetry dictates the form of the\nfirst two terms ($\\sim T^2$ and $T^4$) in the low-temperature expansion \nof the quark condensate. At higher temperatures we do not have\nfundamental knowledge of the behavior of $\\langle \\overline{q}q \\rangle$, \nhowever most of model calculations show a phase transition at\ntemperatures $T \\sim {\\rm 150 - 200}$~MeV. Lattice calculations\nalso show a dramatic change of $\\langle \\overline{q}q \\rangle$ \nat similar temperatures. \n\nIn this letter we study the temperature dependence of the quark condensate \nin the two-flavor Nambu--Jona-Lasinio (NJL) model \\cite{NJL}. There have \nalready been several studies \\cite{HK85,BMZ87,Lutz92} of chiral restoration \nin this model. Our investigation brings a new important element: it includes \n{\\em meson loops} in a self-consistent way. \nPrevious studies have been performed at the quark-loop level only. \nAttempts have been made to include \nmeson loops in the NJL model, but self-consistency was not \ncompletely \nfulfilled \\cite{Heid}. \nOur approximation is symmetry-conserving \\cite{DSTL95,NBCRG96}, \nhence it is consistent \nwith all requirements of chiral symmetry.\nThe key ingredient is the self-consistency \nin solving the equation for the scalar density with meson loops present.\nThis makes the approach consistent with the requirements of \nchiral symmetry, such as the Goldstone theorem, Gell-Mann--Oaks--Renner\nand Goldberger-Treiman relations, or one-loop chiral expansions.\n\nWe find important qualitative and quantitative differences in\nthe temperature dependence of the quark condensate \nin our calculation with meson loops \ncompared to the case with quark loops only.\nWith quark loops only, at low temperatures the condensate\nremains flat, whereas in our case it changes considerably. We show that  \nin the exact chiral limit the change agrees with the prediction\nof the chiral perturbation theory \\cite{GL}. We also find that \nmeson loops decrease the temperature of chiral restoration by about 10\\%. \n\n\n\n\\section{Definition of the model}\n\\label{sec:model}\n\nThe Lagrangian of the two-flavor NJL model with scalar-isoscalar and\npseu\\-do\\-sca\\-lar-isovector interactions is\n\\begin{equation}\n{\\cal L}=\\bar q({\\rm i}\\partial^\\mu \\gamma_\\mu - m)q + \n {\\frac{1}{2 a^2}}\\left( (\\bar q q)^2 + \n (\\bar q{\\rm i}\\gamma_5 \\mbox{\\boldmath $\\tau$} q)^2\\right) \\; ,  \n \\label{eq:lagr}\n\\end{equation}\nwhere $q$ is the quark field, $m$ is the current quark mass, and \n$1\/a^2$ is the coupling constant. It is convenient to apply the \nformalism of effective action \\cite{ItzZub} to Lagrangian (\\ref{eq:lagr}). \nDetails of this procedure are given in Ref. \\cite{NBCRG96}. Meson fields \nare introduced in the usual way (partial bosonization), with $\\Phi = \n(\\Phi_0, \\mbox{\\boldmath $\\Phi$})$ related to the sigma and \nthe pion mean field. At the quark-loop level the \neffective action is\n\\begin{equation}\n{I}(\\Phi) = \\int d^4x \\left ( \\half{a^2} \\Phi^2 \n - a^2 m \\Phi_0 + \\half{a^2} m^2 \\right )\n - \\half{\\rm Tr}\\,\\ln (D^{\\dagger }D)  \\;,  \n\\label{eq:Seffq}\n\\end{equation}\nwhere  $D$ is the Dirac operator , \n $D = \\partial_\\tau - {\\rm i}{\\mbox{\\boldmath $\\alpha$}\n  \\cdot \\mbox{\\boldmath $\\nabla$} } \n    + \\beta \\Phi_0 + {\\rm i} \\beta \\gamma_5 \n   {\\mbox{\\boldmath $\\tau$}} \\cdot {\\mbox{\\boldmath $\\Phi$}}$.\nWe work in Euclidean space-time ($\\tau$, ${\\mbox{\\boldmath $x$}}$). \nIn Eq.~(\\ref{eq:Seffq}) we have replaced the usual ${\\rm Tr}\\,\\ln D$ term \nwith $\\half{\\rm Tr}\\,\\ln (D^{\\dagger }D)$, which is allowed in the \nabsence of anomalies. In fact, this replacement is necessary for the\nintroduction of the proper-time regulator \\cite{pt} used in many NJL \ncalculations, and also in this paper.\n\nMeson loops bring an additional term to the effective action \n\\cite{NBCRG96,ItzZub}\n\\begin{equation}\n{\\Gamma}(\\Phi) =  {I}(\\Phi) + \\half {\\rm Tr}\\,\\ln ({K}^{-1})  \\; . \n \\label{eq:Seffm}\n\\end{equation}\nThe inverse {\\em meson propagator} matrix $K$ is defined as \n $K^{-1}_{ab}(x,y) = \\frac{\\delta^2 I \\left( \\Phi \\right)}\n  {\\delta \\Phi_a(x) \\delta \\Phi_b(y)}$.\nIn Eqs.~(\\ref{eq:Seffq},\\ref{eq:Seffm}) \n${\\rm Tr}$ denotes the full trace, including functional space, \nisospin, and in addition color and spinor trace for quarks.\nIn the $N_c$-counting scheme, the quark loop term ${I}(\\Phi)$\nis the leading contribution of order ${\\cal O}(N_c)$, and the \nmeson loop term $\\frac{1}{2}{\\rm Tr}\\,\\ln {K}$ is of order ${\\cal O}(1)$.\nThus the one-meson-loop contributions give the first correction to\nthe leading-$N_c$ results. \n\n\nUsing standard methods, Green's functions can be obtained from \nEq.~(\\ref{eq:Seffm}) via differentiation with respect to mean \nmeson fields.\nOf particular importance is the one-point function, which gives the\nexpectation value of the sigma field.  The condition\n\\begin{eqnarray}\n\\label{GAP} \n\\frac{\\delta \\Gamma(\\Phi)} {\\delta \\Phi_0(x)}_{\\mid \\Phi_0(x)=S} \n= && a^2 (S-m) - \\half {\\rm Tr}\n\\left( (D^{\\dagger} D)^{-1} \\frac{\\delta (D^{\\dagger }D)}{\\delta \\Phi_0(x)}\n\\right )_{\\Phi_0(x)=S}  \\nonumber \\\\ \n&& + \\half {\\rm Tr} \\left ( K \\frac{\\delta K^{-1}}\n{\\delta \\Phi_0(x)} \\right)_{\\Phi_0(x)=S} = 0\n\\end{eqnarray}\nyields the equation for the vacuum expectation value of $\\Phi_0$, \nwhich we denote by $S$. \nIntroducing \n\\begin{eqnarray}\n\\label{prop}\nK_\\sigma(S,Q^2) & = & \\left ( 4 N_c f(S,Q^2)(Q^2+4S^2) + \na^2 m\/S \\right )^{-1} \\; , \\nonumber \\\\\nK_\\pi(S,Q^2) & = & \\left ( 4 N_c f(S,Q^2) Q^2 + a^2 m\/S  \\right )^{-1} \\;,\n\\end{eqnarray}\nand retaining terms up to order ${\\cal O}(N_c^0)$, \nEq.~(\\ref{GAP}) can be written in the form \\cite{NBCRG96}\n\\begin{eqnarray}\n\\label{gap0}\n& & a^2 \\left(S - m \\right) - 8 N_c \\, S g(S) \\nonumber \\\\\n& & + S \\frac{N_c}{4 \\pi^4} \\int d^4 Q \n   \\left\\{ \\left [2 f(S,0) + \\frac{d}{dS^2} \n \\left (f(S,Q^2)(Q^2 + 4 S^2) \\right ) \\right]\n   K_\\sigma(S,Q^2) \\right. \\nonumber \\\\\n& & + \\left. 3 \\left [2 f(S,0) + \\frac{d}{dS^2} f(S,Q^2) Q^2 \\right]  \n K_\\pi(S,Q^2) \\right\\} = 0.\n\\end{eqnarray}\nFunctions $g$ and $f$ in the above expressions are the {\\em quark \nbubble functions}. Their form is very simple if no cut-offs were present.\nIn this case we would have \n\\mbox{$g(S) = \\int {d^4k \\over (2\\pi)^4} {1 \\over k^2 + S^2}$} and\n\\mbox{$f(S,Q^2) = \\int {d^4k \\over (2\\pi)^4} {1 \\over k^2 + S^2}\n{1 \\over (k+Q)^2 + S^2 }$}, and Eq.~(\\ref{gap0}) could be interpreted\nvia standard Feynman diagrams (see Fig.~\\ref{fig:0}).\nIn the presence of a cut-off these functions are complicated. \nIn the case of the proper-time cut-off \\cite{pt} used here\nwe have \\cite{NBCRG96}\n\\begin{equation}\n\\label{g0r}\ng(S) = \\int {d^4k \\over (2\\pi)^4} \\int\\limits_{\\Lambda_f^{-2}}^{\\infty}\nds \\, \\exp\\left\\{-s [k^2+S^2]\\right\\} = {\\Lambda_f^2 \\over 16 \\pi^2} \n\\, E_2\\left[{S^2 \\over \\Lambda_f^2} \\right]\n\\end{equation}\nand\n\\begin{eqnarray}\n\\label{f0r}\nf(S,Q^2) &=& \\int {d^4k \\over (2\\pi)^4} \\int\\limits_{\\Lambda_f^{-2}}^{\\infty}\nds \\, s \\int\\limits_0^1 du \\exp\\left\\{-s [k^2 + S^2 + u(1-u) Q^2] \\right\\} \n \\nonumber \\\\\n&=& {1 \\over 16 \\pi^2} \\int\\limits_0^1 du \\, E_1\\left[{S^2 \\over \\Lambda_f^2}\n+ u (1-u) {Q^2 \\over \\Lambda_f^2} \\right],\n\\end{eqnarray}\nwhere $\\Lambda_f$ is the quark cut-off, and \nthe exponential integral is defined as \n\\mbox{$E_n(x) \\equiv \\int\\limits_1^{\\infty} dt \\, {e^{-xt} \\over t^n}$}.\n\nThe one meson-loop gap equation (\\ref{gap0}) requires also the\nintroduction of a regulator for meson momenta. In other words,  \nwe have to regularize the divergent integral over $d^4Q$.\nIn Ref. \\cite{NBCRG96} this was achieved by the substitution\n\\mbox{$\\int d^4Q  \\longrightarrow \\pi^2 \\int\\limits_0^{\\Lambda_b^2} \n dQ^2 \\, Q^2$}, where $\\Lambda_b$ was the four-dimensional Euclidean meson\nmomentum cutoff. \nIn the present study at finite temperatures, \nwe employ the\nthree-dimensional cutoff procedure, i.e., we make the replacement\n\\begin{equation}\n\\label{3dc}\n\\int d^4Q  \\longrightarrow 4 \\pi \\int d\\omega \n\\int\\limits_0^{\\Lambda_b} dq \\, q^2 \\,\\,\\,\\,\\,\\, ,\n\\end{equation}\nwhere $Q=(\\omega,{\\bf q})$ and $q = |{\\bf q}|$. The form\n(\\ref{3dc}) is convenient for the implementation of the boundary\nconditions satisfied by temperature Green's functions.\n\n\\vfill\n\n\\begin{figure}[b]\n\\xslide{fig0.ps}{3cm}{30}{370}{560}{490}\n\\caption{Diagramatic representation of Eq.~(\\ref{gap0}). The cross\nrepresents the first term, the one-quark-loop contribution \ncorresponds to the second term, and meson-loop terms \nrepresent subsequent terms. The solid lines represent the \nquark propagator $1\/(D^{\\dagger} D)$, the dased lines correspond to \nthe meson propagators $K$ of Eqs.~(\\ref{prop}), the external \ndased line represents scalar-isoscalar coupling, and the vertices follow\nfrom the form of $(D^{\\dagger} D)$.} \n\\label{fig:0}\n\\end{figure}\n\n\n\\section{Finite temperature}\n\\label{sec:gap}\n\nFor calculations at finite temperature $T$ we shall adopt the imaginary time\nformalism \\cite{Kapusta}. \nThis can be done by making the following replacement in the\nquark momentum integrals\n\\begin{equation}\n\\label{itf}\n\\int {d^4k \\over (2\\pi)^4} F(k) = \\int {dE \\over 2\\pi} \\int\n{d^3k \\over (2\\pi)^3} F(E,{\\bf k}) \\rightarrow T \\sum_{j=-\\infty}^{\\infty}\n\\int {d^3k \\over (2\\pi)^3} F(E_j,{\\bf k}).\n\\end{equation}\nHere $F(k)=F(E,{\\bf k})$ is an arbitrary integrand, \nand the sum runs over the fermionic \nMatsubara frequencies $E_j = (2j+1)\\pi T$. The integral over the meson\nfour-momenta should be also replaced by the sum of the form (\\ref{itf}).\nIn this case, however, the sum runs over the bosonic Matsubara\nfrequencies $\\omega_n = 2\\pi n T$.\nWith this prescription we can turn to\nthe calculation of the functions which are the finite temperature\nanalogs of $g(S)$ and  $f(S,Q^2)$.\nWe find\n\\begin{eqnarray}\ng(S,T) &=& T \\sum_j \\int {d^3k \\over (2\\pi)^3} \n\\int\\limits_{\\Lambda_f^{-2}}^{\\infty} ds \\, \n\\exp\\left\\{ -s \\left[ E_j^2\n+ {\\bf k}^2 + S^2 \\right] \\right\\} \\nonumber \\\\\n&=& {T \\Lambda_f \\over 8 \\pi^{{3 \\over 2}} } \\sum_j\nE_{3 \\over 2} \\left[{{S^2+E_j^2} \\over \\Lambda_f^2} \\right]\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!& &f(S,n,q,T) = \nT \\sum_j \\int {d^3k \\over (2\\pi)^3} \n\\int_{\\Lambda_f^{-2}}^{\\infty} ds \\, s \\int\\limits_0^1 du \\times \n\\nonumber \\\\ \n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!& & \\exp\\left\\{ -s \\left[ \nS^2 + u(1-u)(\\omega_n^2 + {\\bf q}^2) + \\left[ {\\bf k}\n- {\\bf q} (1-u) \\right]^2 + \\left[E_j - \\omega_n (1-u) \\right]^2\n\\right] \\right\\}  \\nonumber \\\\ \n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!& &=  {T \\over 8 \\pi^{{3 \\over 2}} \\Lambda_f} \\sum_j\n\\int\\limits_0^1 du \\, E_{1\\over 2}\\left[{S^2 \\over \\Lambda_f^2} +\nu(1-u) {\\omega_n^2 + {\\bf q}^2 \\over \\Lambda_f^2} \\right. \n  +  \\left. {[E_j-\\omega_n(1-u)]^2 \\over \\Lambda_f^2} \\right] \\;.\n\\end{eqnarray}\nAnalogously, the inverse meson propagators become\n\\begin{eqnarray}\n\\label{propT}\n K_\\sigma(S,n,q,T) & = & \\left ( \n4 N_c f(S,n,q,T)(\\omega_n^2 + {\\bf q}^2+4S^2) + \n  a^2 m\/S \\right )^{-1} \\; , \\nonumber \\\\ \n K_\\pi(S,n,q,T) & = & \\left \n ( 4 N_c f(S,n,q,T)(\\omega_n^2 + {\\bf q}^2) + a^2 m\/S  \\right )^{-1} \\;.\n\\end{eqnarray}\nFinally, we can write the finite-temperature analog\nof Eq.~(\\ref{gap0}):\n\\begin{eqnarray}\n\\label{gapT}\n& & a^2 \\left(S - {m} \\right) - 8 N_c S \\, g(S,T) + \n {2 S N_c T \\over \\pi^2} \\sum_n \\int\\limits_0^{\\Lambda_b} dq \\, q^2 \\times \n\\nonumber \\\\\n& & \\left\\{ \\left [2 f(S,0,0,T) + \\frac{d}{dS^2} \\left (f(S,n,q,T)\n(\\omega_n^2 + {\\bf q}^2 + 4S^2) \\right ) \\right]\n K_\\sigma(S,n,q,T) \\right. \\nonumber \\\\\n& & + \\left. 3 \\left [2 f(S,0,0,T) + \\frac{d}{dS^2} f(S,n,q,T) \n(\\omega_n^2 + {\\bf q}^2) \\right]  \n K_\\pi(S,n,q,T) \\right\\} = 0 \\;.\\nonumber \\\\\n\\end{eqnarray}\nIf chiral symmetry is broken, then the above equation has a \nnontrivial solution for $S$. \nThe quark condensate and $S$ are related by the formula\n\\begin{equation}\n\\label{qq}\n\\langle \\overline{q}q \\rangle = - a^2 (S - m) \\;,\n\\end{equation}\nwhich follows immediately from the fact that \n $\\langle \\overline{q}q \\rangle = \\delta \\Gamma(\\Phi)\/\\delta m$\nand Eq.~(\\ref{eq:Seffm}).\n\n\n\\section{Low-temperature expansion in the chiral limit}\n\\label{sec:lowT}\n\nBefore presenting our numerical results for $\\langle {\\overline q} q\n\\rangle_T$\nlet us consider the low-temperature expansion. As shown by\nGasser and Leutwyler \\cite{GL}, {\\em in the chiral limit}\nthe low-temperature expansion of the quark condensate has the form\n\\begin{equation}\n\\label{eq:gl}\n\\langle \\overline{q} q \\rangle_T = \\langle \\overline{q} q \\rangle_0 \n\\left ( 1 - \\frac{T^2}{8 F_\\pi^2} - \\frac{T^4}{384 F_\\pi^4} + ... \\right ) .\n\\end{equation}\nFirst, let us do the $N_c$ counting in this formula. Since\n $F_\\pi \\sim {\\cal O}(\\sqrt{N_c})$, subsequent terms in the expansion are \nsuppressed by $1\/N_c$. Since our one-meson-loop \ncalculation accounts for first\nsubleading effects in the  $1\/N_c$ expansion, we can hope for reproducing \nonly the $T^2$ term in Eq.~(\\ref{eq:gl}). Further terms would require\nmore loops.\n\nUsing standard techniques \\cite{Kapusta}, \nthe sum over the bosonic Matsubara\nfrequencies in Eq.~(\\ref{gapT}) can be converted to a contour integral\nin the complex energy plane. By deforming this contour\nwe collect all contributions from the singularities of the\nintegrand, weighted with the thermal Bose distribution. At low\ntemperatures, the dominant contribution comes from the lowest\nlying pion pole, and other singularities are negligible. \nThus, the third term in\n(\\ref{gapT}) becomes\n\\begin{equation}\n\\label{ae1}\n{3T \\over \\pi^2} \\sum_n \\int\\limits_0^{\\Lambda_b} dq \\, q^2\n{1 \\over \\omega_n^2 + q^2} = {3 \\over 2\\pi^2 }\n\\int\\limits_0^{\\Lambda_b} dq \\, q \\left[ 1 + {2 \\over e^{q\/T} - 1} \\right].\n\\end{equation}\nWriting Eq. (\\ref{ae1}) we have approximated the function \n$f(M,n,q,T)$, appearing in the pion propagator, by its value at $n=q=0$. \nFor sufficiently large cutoff $\\Lambda_b$, the integral over the thermal \ndistribution function in (\\ref{ae1}) can be expressed by the Riemann\nzeta function $\\zeta(2) = \\pi^2\/6$. Thus, the final result for\n(\\ref{ae1}) is $ 3\\Lambda^2_b\/4 \\pi^2 + T^2\/2$.\nInserting the above result into the gap equation (\\ref{gapT}) we find,\nwith $m=0$, the following equality:\n\\begin{equation}\n\\label{ae2}\nh(S,T) \\equiv a^2 - 8N_c g(S,T) + {3\\Lambda_b^2 \\over\n4 \\pi^2} + \\half T^2 = 0.\n\\end{equation}\nEq. (\\ref{ae2}) defines implicitly the function $S(T)$,\nwhich satisfies the equation\n\\begin{equation}\n\\label{ae3}\n{dS \\over dT^2} = - \n{ \\partial h(S,T) \/ \\partial T^2 \\over \\partial h(S,T)   \/  \\partial S}\n= \\left[ 16 N_c\n{\\partial g(S,T) \\over \\partial S} \\right]^{-1}.\n\\end{equation}\nHere we have neglected the term $\\partial g(S,T) \/ \n\\partial T^2$, since it is exponentially suppressed by the factor\n $\\exp(-S\/T)$. Furthermore, the leading-$N_c$ term on the \nright hand side of (\\ref{ae3}) can be rewritten using the \nrelations \\cite{NBCRG96}\n $\\partial g(S,T)\/ \\partial S = -2S f(S,0)$ and \n \\mbox{$4 N_c f(S,0) = \\overline{F}_{\\pi}^2\/S^2$}, \n where $\\overline{F}_{\\pi}$ is the leading-$N_c$ \npiece of the pion decay constant. \nCollecting these equalities we \narrive at $dS\/dT^2 = -S\/(8\\overline{F}_{\\pi}^2)$, which finally gives\n\\begin{equation}\n\\label{ae4}\nS(T) = S(0) \\left[ 1 - {T^2 \\over 8\\overline{F}_{\\pi}^2} \\right].\n\\end{equation}\nProportionality (\\ref{qq}) implies that the above expression\ncoincides (in the large $N_c$ limit) with Eq.~(\\ref{eq:gl}).\nHence our method is consistent with a basic requirement of chiral\nsymmetry at the one-meson-loop level.\n\n\\section{Results}\n\\label{sec:res}\n\nIn the exact chiral limit the \nmodel has 3 parameters: $a$, $\\Lambda_f$, and $\\Lambda_b$. \nIn this paper we fix arbitrarily $\\Lambda_b\/\\Lambda_f = \\half$.\nThe remaining\n2 parameters are fixed by reproducing the physical value of\n $F_\\pi=93{\\rm MeV}$ and \na chosen value for $\\langle \\overline{q} q \\rangle_0$. \nFor the case of $m \\neq 0$ we have an extra parameter, $m$, which is fitted\nby requiring that the pion has its physical mass.\nWe compare results with meson loops to results with the quark loop\nonly ($\\Lambda_b = 0$).  Parameters for the two calculations\nare adjusted in such a way, that the values of $F_\\pi$, \n $\\langle \\overline{q} q \\rangle_0$, and $m_\\pi$ are the same.\n\nThe calculation of $F_\\pi$ with meson loops, although \nstraightforward, is rather tedious, so we \ndo not present it here. The method has been presented in detail in \nRef.~\\cite{NBCRG96,thesis}. The only difference in our calculation \nis that the three-momentum cut-off (\\ref{3dc}) rather than\nthe four-momentum cut-off of Ref.~\\cite{NBCRG96} is used.\n\n\\begin{figure}[b]\n\\xslide{fig1.ps}{13.5cm}{45}{160}{550}{680}\n\\caption{Dependence of of the quark condensate on $T^2$ \nin the chiral limit $m_\\pi=0$. The curves correspond to \nthe calculation\nwith meson loops (solid line), with quark loops only (dashed line),\nand the lowest-order chiral expansion (dotted line). The parameters\nfor the solid line and dashed line are adjusted in such a way that\n$F_\\pi = 93{\\rm MeV}$ and $\\langle \\overline{q} q \\rangle_0 \n =-(184{\\rm MeV})^3$. For the solid line $a=175{\\rm MeV}$, \n $\\Lambda_f = 723{\\rm MeV}$, and $\\Lambda_b = \\half \\Lambda_f$, \nwhereas for the\ndashed line $a=201{\\rm MeV}$, $\\Lambda_f = 682{\\rm MeV}$, and\n $\\Lambda_b = 0$. }\n\\label{fig:1}\n\\end{figure}\nFigure \\ref{fig:1} shows the dependence of $\\langle \\overline{q} q \\rangle$\non $T^2$. The solid line represents the case with meson loops. We note that\nat low temperatures the curve has a finite slope, as requested by\nEq.~(\\ref{ae4}). The slope is close to the leading-order \nGasser-Leutwyler result (dotted curve). As explained earlier, \nthe slopes would overlap in the large-$N_c$ limit. \nThis behavior is radically different from the case with quark loops \nonly (dashed curve). In this case at low temperatures \n \\mbox{$\\langle \\overline{q} q \\rangle_T - \n  \\langle \\overline{q} q \\rangle_0 \\sim e^{-M\/T}$}, \nwhere $M$ is the \nmass of the constituent quark. All derivatives of this function vanish\nat $T=0$, and $\\langle \\overline{q} q \\rangle$ is flat at the origin.\nWe can also see from the figure that the fall-off of the condensate\nis faster when the meson loops are included. \nIn fact, for the parameters of Fig.~\\ref{fig:1} we have an interesting\nphenomenon. At $T = 162{\\rm MeV}$ the condensate abruptly jumps to 0. There is \na first-order phase transition, with a latent heat necessary to melt\nthe quark condensate. Such a behavior is not present \nin the case of calculations without meson loops \\cite{HK85,BMZ87,Lutz92}.\nWe note that with meson loops present the chiral restoration \ntemperature is $162{\\rm MeV}$, {\\em i.e.} about 10\\% \nless than $176{\\rm MeV}$ of the quark-loop-only case.\n\\begin{figure}[b]\n\\xslide{fig2.ps}{13.5cm}{45}{160}{550}{680}\n\\caption{Same as Fig.~\\ref{fig:1} for $m_\\pi=139~{\\rm MeV}$, \n $F_\\pi = 93{\\rm MeV}$, and $\\langle \\overline{q} q \\rangle_0 \n =-(174{\\rm MeV})^3$.\nFor the solid line $a=164{\\rm MeV}$, \n $\\Lambda_f = 678{\\rm MeV}$, $\\Lambda_b = \\half \\Lambda_f$, and \n $m = 15{\\rm MeV}$, whereas for the\ndashed line $a=175{\\rm MeV}$, \n $\\Lambda_f = 645{\\rm MeV}$, $\\Lambda_b = 0$, and $m = 15{\\rm MeV}$. }\n\\label{fig:2}\n\\end{figure}\n\nFigure \\ref{fig:2} shows the same study, but for the physical\nvalue of $m_\\pi$. We note that now $\\langle \\overline{q} q \\rangle$ \n(solid line) is also flat at the origin, since the pion is\nno more massless, and at low $T$ we have \n \\mbox{$\\langle \\overline{q} q \\rangle_T - \\langle \\overline{q} q \\rangle_0\n \\sim e^{-m_\\pi\/T}$}.\nNevertheless, the region of this flatness is small, and at \nintermediate temperatures the curve remains close to the\nGasser-Leutwyler expansion. We note again that meson loops \nconsiderably speed up\nthe melting of the condensate compared to the case of quark loops only.\nHowever, there is no first-order phase transition such as in \nFig.~\\ref{fig:1}. Instead, we observe a smooth cross-over typical for the case\nof $m \\neq 0$. \n\nThe faster change of the quark condensate in our study\nis not surprising.\nIt is caused by the presence of light pions which \nare known to play a dominant role\nat low-temperatures \\cite{Heid}. \nThe behavior of  $\\langle \\overline{q} q \\rangle$ \nreflects this general feature.\nConcluding, we stress that the inclusion of meson loops \nin the NJL model \nqualitatively and quantitatively changes the results \nin comparison to the \ncalculations at the quark-loop level. In particular, we find finite slope of\n $\\langle \\overline{q} q \\rangle$ vs. $T^2$ at the origin in the chiral\nlimit, faster melting of the condensate, and lower chiral restoration \ntemperature. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}