data_all_eng_slimpj/shuffled/split2/finalzlxd

{"text":"\\section{Introduction}\n  \\hspace{\\parindent}\nModified theories of gravity in the form of braneworld models can in\nprinciple account for the phenomenon of dark energy as well as for\nnontrivial compactifications of multi-dimensional string models. It\nbecomes increasingly more obvious that one should include in such\nmodels the analysis of quantum effects beyond the tree-level\napproximation \\cite{quantumDGP}. This is the only way to reach an\nultimate conclusion on the resolution of such problems as the\npresence of ghosts \\cite{ghosts} and low strong-coupling scale\n\\cite{scale}. Quantum effects in brane models are also important for\nthe stabilization of extra dimensions \\cite{GarPujTan}, fixing the\ncross-over scale in the Brans-Dicke modification of the DGP model\n\\cite{Pujolas} and in the recently suggested mechanism of the\ncosmological acceleration generated by the four-dimensional\nconformal anomaly \\cite{slih}.\n\nA general framework for treating quantum effective actions in brane\nmodels (or, more generally, models with timelike and spacelike\nboundaries) was recently suggested in\n\\cite{BKRK,gospel,qeastb,toyDGP}. The main peculiarity of these\nmodels is that due to quantum field fluctuations on the branes the\nfield propagator is subject to generalized Neumann boundary\nconditions involving normal and tangential derivatives on the\nbrane\/boundary surfaces. This presents both technical and conceptual\ndifficulties, because such boundary conditions are much harder to\nhandle than the simple Dirichlet ones. The method of \\cite{qeastb}\nprovides a systematic reduction of the generalized Neumann boundary\nconditions to Dirichlet conditions. As a byproduct it disentangles\nfrom the quantum effective action the contribution of the surface\nmodes mediating the brane-to-brane propagation, which play a very\nimportant role in the zero-mode localization mechanism of the\nRandall-Sundrum type \\cite{RS}. The purpose of this work is to make\nthe next step --- to extend a well-known Schwinger-DeWitt technique\n\\cite{DeWitt,PhysRep,McKean-Singer,Vassilevich} to the calculation\nof this contribution in the DGP model in a weakly curved spacetime\nin the form of the {\\em covariant} curvature expansion.\n\nBriefly the method of \\cite{qeastb} looks as follows. The action of\na (free field) brane model generally contains the bulk and the brane\nparts,\n    \\begin{eqnarray}\n    S[\\,\\phi\\,]=\\frac12\\int_{\\rm\\bf B} d^{d+1}X\\,G^{1\/2}\\phi(X)\n    F(\\nabla_X)\\,\\phi(X)\n    +\\frac12\\int_{\\rm \\bf b}\n    d^dx \\,g^{1\/2}\\varphi(x)\\,\n    \\kappa(\\nabla_x)\\,\\varphi(x) \\ ,                         \\label{1}\n    \\end{eqnarray}\nwhere the $(d\\!+\\!1)$-dimensional bulk and the $d$-dimensional brane\ncoordinates are labeled respectively by $X=X^A$ and $x=x^\\mu$, and\nthe boundary values of bulk fields $\\phi(X)$ on the brane\/boundary\n${\\rm\\bf b}=\\partial\\rm\\bf B$ are denoted by $\\varphi(x)$,\n    \\begin{eqnarray}\n    \\phi(X)\\,\\Big|_{\\,\\rm\\bf b}=\\varphi(x),        \\label{2}\n    \\end{eqnarray}\n$G$ and $g$ are the determinants of the bulk $G_{AB}$ and\n$g_{\\mu\\nu}$ metrics respectively.\n\nThe kernel of the bulk Lagrangian is given by the second order\ndifferential operator $F(\\nabla_X)$, whose covariant derivatives\n$\\nabla_X$ are integrated by parts in such a way that they form\nbilinear combinations of first order derivatives acting on two\ndifferent fields. Integration by parts in the bulk gives nontrivial\nsurface terms on the brane\/boundary. In particular, this operation\nresults in the Wronskian relation for generic test functions\n$\\phi_{1,\\,2}(X)$,\n    \\begin{eqnarray}\n    \\int_{\\rm\\bf B} d^{\\,d+1}X\\,G^{1\/2}\n    \\left(\\,\\phi_1\\stackrel{\\rightarrow}{F}\\!(\\nabla_X)\\phi_2-\n    \\phi_1\\!\\stackrel{\\leftarrow}{F}\\!(\\nabla_X)\\,\\phi_2\\right)=\n    -\\int_{\\partial{\\rm\\bf B}} d^{\\,d}x\\,g^{1\/2}\n    \\left(\\,\\phi_1\\stackrel{\\rightarrow}{W}\\!\n    \\phi_2-\n    \\phi_1\\stackrel{\\leftarrow}{W}\\!\n    \\phi_2\\right).                          \\label{3a}\n    \\end{eqnarray}        \nArrows everywhere here indicate the direction of action of\nderivatives either on $\\phi_1$ or $\\phi_2$.\n\nThe brane part of the action contains as a kernel some local\noperator $\\kappa(\\nabla)$, $\\nabla\\equiv\\nabla_x$. Its order in\nderivatives depends on the model in question. In the Randall-Sundrum\nmodel \\cite{RS}, for example, it is for certain gauges just an\nultralocal multiplication operator generated by the tension term on\nthe brane. In the Dvali-Gabadadze-Porrati (DGP) model \\cite{DGP}\nthis is a second order operator induced by the brane Einstein term\non the brane, $\\kappa(\\nabla)\\sim\\nabla\\nabla\/m$, where $m$ is the\nDGP scale which is of the order of magnitude of the horizon\nscale, being responsible for the cosmological acceleration\n\\cite{Deffayet}. In the context of the Born-Infeld action in D-brane\nstring theory with vector gauge fields, $\\kappa(\\nabla)$ is a\nfirst-order operator \\cite{open}.\n\nIn all these cases the variational procedure for the action\n(\\ref{1}) with dynamical (not fixed) fields on the boundary\n$\\varphi(x)$ naturally leads to generalized Neumann boundary\nconditions of the form\n    \\begin{eqnarray}\n    \\left.\\Big(\\stackrel{\\rightarrow}{W}\\!(\\nabla_X)\n    +\\kappa(\\nabla)\\Big)\\,\\phi\\,\\right|_{\\,\\rm\\bf b}\n    =0,                                                     \\label{3}\n    \\end{eqnarray}\nwhich uniquely specify the propagator of quantum fields and,\ntherefore, a complete Feynman diagrammatic technique for the system\nin question. The method of \\cite{qeastb} allows one to\nsystematically reduce this diagrammatic technique to the one subject\nto the Dirichlet boundary conditions $\\phi|_{\\,\\rm\\bf b}=0$. The\nmain additional ingredient of this reduction procedure is the brane\noperator $\\mbox{\\boldmath$F$}^{\\,\\rm brane}(x,x')$ which is\nconstructed from the Dirichlet Green's function $G_D(X,X')$ of the\noperator $F(\\nabla)$ in the bulk,\n    \\begin{eqnarray}\n    \\mbox{\\boldmath$F$}^{\\,\\rm brane}(x,x')=-\n    \\stackrel{\\rightarrow}{W}\\!(\\nabla_X\\!)\\,G_{D}(X,X')\\!\n    \\stackrel{\\leftarrow}{W}\\!(\\nabla_{X'}\\!)\n    \\,\\Big|_{\\,X=e(x),\\,X'=e(x')}\n    +\\kappa(\\nabla)\\,\\delta(x,x')\\ .           \\label{5}\n    \\end{eqnarray}\nThis expression expresses the fact that the kernel of the Dirichlet\nGreen's function is being acted upon both arguments by the Wronskian\noperators with a subsequent restriction to the brane, with $X=e(x)$\ndenoting the brane embedding function.\n\nAs shown in \\cite{qeastb}, this operator determines the\nbrane-to-brane propagation of the physical modes in the system with\nthe classical action (\\ref{1}) (its inverse is the brane-to-brane\npropagator) and additively contributes to its full one-loop\neffective action according to\n    \\begin{eqnarray}\n    \\mbox{\\boldmath$\\varGamma$}_{\\rm 1-loop}\\equiv\\frac12\\;\n    {\\rm Tr}_N^{(d+1)}\\ln F=\\frac12\\;{\\rm Tr}_D^{(d+1)}\\ln F\n    +\\frac12\\;{\\rm Tr}^{(d)}\n    \\ln \\mbox{\\boldmath$F$}^{\\,\\rm brane},    \\label{11}\n    \\end{eqnarray}\nwhere ${\\rm Tr}_{D,\\,N}^{(d+1)}$ denotes functional traces of the\nbulk theory subject to Dirichlet and Neumann boundary conditions,\nrespectively, while ${\\rm Tr}^{(d)}$ is a functional trace in the\nboundary $d$-dimensional theory. The full quantum effective action\nof this model is obviously given by the functional determinant of\nthe operator $F(\\nabla_X)$ subject to the generalized Neumann\nboundary conditions (\\ref{5}), and the above equation reduces its\ncalculation to that of the Dirichlet boundary conditions plus the\ncontribution of the brane-to-brane propagation.\n\nHere we apply (\\ref{11}) to a simple model of a scalar field which\nmimics in particular the properties of the brane-induced gravity\nmodels and the DGP model \\cite{DGP}. This is the $(d\\!+\\!1)$-dimensional\nmassive scalar field $\\phi(X)=\\phi(x,y)$ with mass $M$ living in the\n{\\em curved} half-space $y\\geq 0$ with the additional\n$d$-dimensional kinetic term for $\\varphi(x)\\equiv\\phi(x,0)$\nlocalized at the brane (boundary) at $y=0$,\n    \\begin{eqnarray}\n    S[\\,\\phi\\,]=\\frac12\\int\\limits_{y\\geq 0}\n    d^{d+1}X\\,G^{1\/2}\\Big((\\nabla_X\\phi(X))^2\n    +M^2\\phi^2(X)\\Big)\n    +\\frac1{4m}\\int\n    d^dx \\,g^{1\/2}\\,(\\nabla_x\\varphi(x))^2.           \\label{1.1}\n    \\end{eqnarray}\nHere and in what follows we work in a Euclidean\n(positive-signature) spacetime. Therefore, this action corresponds\nto the following choice of $F(\\nabla_X)$ in terms of\n$(d\\!+\\!1)$-dimensional and $d$-dimensional covariant D'Alembertians\n(Laplacians)\n    \\begin{eqnarray}\n    &&F(\\nabla_X)=M^2-\\Box^{(\\,d+1)}=\n    M^2-G^{AB}\\nabla_A\\nabla_B,.              \\label{6}\n    \\end{eqnarray}\nIn the normal Gaussian coordinates its Wronskian operator is given\nby $W=-\\partial_y$ --- the normal derivative with respect to\noutward-pointing normal to the brane,  and the boundary operator\n$\\kappa(\\nabla)$ equals\n    \\begin{eqnarray}\n    \\kappa(\\nabla)=-\\frac1{2m}\\,\\Box,\\, \\,\\,\\,\\,\n    \\Box=\\Box^{(d)}\\equiv g^{\\mu\\nu}\\nabla_\\mu\\nabla_\\nu, \\label{8}\n    \\end{eqnarray}\nwhere the dimensional parameter $m$ mimics the role of the DGP\nscale \\cite{DGP}. Thus, the generalized Neumann boundary conditions\nin this model involve second-order derivatives tangential to the\nbrane,\n    \\begin{eqnarray}\n    \\Big(\\partial_y\n    +\\frac1{2m}\\,\\Box\\Big)\\,\\phi\\,\n    \\Big|_{\\,\\rm\\bf b}=0,               \\label{1.3}\n    \\end{eqnarray}\ncf. (\\ref{3}) with $W=-\\partial_y$ and $\\kappa=-\\Box\/2m$.\n\nAs was shown \\cite{toyDGP}, the flat space brane-to-brane operator for\nsuch a model has the form of the pseudodifferential operator with\nthe flat-space $\\Box$,\n    \\begin{eqnarray}\n    \\mbox{\\boldmath$F$}^{\\,\\rm brane}(\\nabla)\n    =\\frac1{2m}\\,(-\\Box+2m\\sqrt{M^2-\\Box}).          \\label{9}\n    \\end{eqnarray}\nIn the massless case of the DGP model \\cite{DGP}, $M=0$, this\noperator is known to mediate the gravitational interaction on the\nbrane, interpolating between the four-dimensional Newtonian law at\nintermediate distances and the five-dimensional law at the horizon\nscale $\\sim 1\/m$ \\cite{scale}.\n\nHere we generalize this construction to a curved spacetime and\nexpand the brane-to-brane operator and its effective action in\ncovariant curvature series. This is the expansion in powers of the\nbulk curvature symbolically denoted below as $R$, extrinsic\ncurvature of the brane denoted by $K$ and their covariant\nderivatives --- all taken at the location of the brane. The\nexpansion starts with the approximation (\\ref{9}) based on the {\\em\nfull covariant} d'Alembertian on the brane. We present a systematic\ntechnique of calculating curvature corrections to (\\ref{9}) and\nrewrite their nonlocal operator coefficients --- functions of the\ncovariant $\\Box$\n--- in the form of the generalized (weighted) proper time\nrepresentation.\n\nThe success of the conventional Schwinger-DeWitt method is based on\nthe fact that the one-loop effective action of the operator, say\n$M^2-\\Box$, has a proper time representation\n    \\begin{eqnarray}\n    &&\\frac12\\,{\\rm Tr}\\,\\ln\\,\\Big(M^2-\\Box\\Big)\n    =-\\frac12 \\int_0^\\infty\n    \\frac{ds}s\\;e^{-s\\,M^2}\\,\n    {\\rm Tr}\\:e^{s\\,\\Box}.               \\label{proptime}\n    \\end{eqnarray}\nIn view of the well-known small time expansion for the heat kernel\n\\cite{DeWitt,PhysRep},\n    \\begin{eqnarray}\n    e^{\\,s\\,\\Box}\\delta(x,x')=\\frac1{(4\\pi s)^{d\/2}}\n    D^{1\/2}(x,x')\\,e^{-\\sigma(x,x')\/2s}\\sum_{n=0}^\\infty\\,s^n\\,\n    a_n(x,x'),\n    \\end{eqnarray}\n($\\sigma(x,x')$ is the geodetic world function, $D(x,x')$ is the\nassociated Van Vleck determinant and $a_n(x,x')$ are the\nSchwinger-DeWitt or Gilkey-Seely coefficients) the curvature\nexpansion eventually reduces to the calculation of the coincidence\nlimits of $a_n(x,x')$ and a trivial proper time integration\nresulting in the inverse mass expansion\n    \\begin{eqnarray}\n    \\frac12\\,{\\rm Tr}\\,\\ln\\,\\Big(M^2-\\Box\\Big)\n    =-\\frac12 \\frac1{(4\\pi)^{d\/2}}\n    \\sum_{n=0}^\\infty\\frac{\\Gamma(n\\!-\\!d\/2)}{M^{2n-d}}\\,\\int\n    dx\\,g^{1\/2}\\,a_n(x,x).\n    \\end{eqnarray}\n\nAs we will show below, the calculation of the brane effective action\ndiffers from the conventional Schwinger-DeWitt case in that the\nproper time integral (\\ref{proptime}) contains in the integrand a\ncertain extra weight function $w(s)$ and instead of just ${\\rm\nTr}\\,e^{s\\Box}$ one has to calculate the trace of the heat kernel\nacted upon by a certain local differential operator ${\\rm\nTr}\\,\\big(W(\\nabla)e^{s\\Box}\\big)$. This again reduces to the\ncalculation of the coincidence limits --- this time of the multiple\ncovariant derivatives of $a_n(x,x')$,\n$\\nabla_{\\mu_1}...\\nabla_{\\mu_n}a_n(x,x')|_{x'=x}$ --- the task\neasily doable within a conventional DeWitt recurrence procedure for\n$a_n(x,x')$.\n\n\n\\section{Perturbation theory for the bulk Green's function and brane\neffective action}\n  \\hspace{\\parindent}\nIn normal Gaussian coordinates the covariant bulk d'Alembertian\ndecomposes as $\\Box^{\\,(d+1)}_X=\\partial_y^2+\\Box(y)+...$, where\nellipses denote depending on spin terms at most linear in\nderivatives\\footnote{This term for a general spin structurally has the form\n$K\\nabla_X+K^2+(\\nabla K)+R$ where $K$ is the extrinsic curvature of\n$y={\\rm const}$ slices and $R$ is the bulk curvature.} and $\\Box(y)$\nis a covariant d'Alembertian on the slice of constant coordinate\n$y$. Therefore the full bulk operator takes the form\n    \\begin{eqnarray}\n    F(\\nabla)=M^2-\\Box^{(\\,d+1)}_X+P(X)=\n    M^2-\\Box-\\partial_y^2-V(X\\,|\\,\\partial_y,\\nabla)\\equiv F^0-V,\n    \\end{eqnarray}\nin which all nontrivial $y$-dependence is isolated as a perturbation\nterm $V(X\\,|\\,\\partial_y,\\nabla)\\equiv V(y,\\partial_y)$ --- a\nfirst-order differential operator in $y$, proportional to the\nextrinsic and bulk curvatures, and of second order in brane\nderivatives $\\nabla$ which we do not explicitly indicate here by\nassuming that they are encoded in the operator structure of\n$V(y,\\partial_y)$. In particular, it includes the difference\n$\\Box(0)-\\Box(y)\\equiv\\Box-\\Box(y)$ expandable in Taylor series in\n$y$.\n\nThe kernel of the bulk Green's function can formally be written as a\n$y$-dependent nonlocal operator acting on the $d$-dimensional brane\n--- some non-polynomial function of the brane covariant derivative\n    \\begin{eqnarray}\n    G_D(X,X')=G_D(y,y'|\\,\\nabla)\\,\\delta(x,x').\n    \\end{eqnarray}\nThe perturbation expansion for $G_D(y,y'|\\,\\nabla)$ is usual\n    \\begin{eqnarray}\n    G_D=G_D^0+G^0_D V G^0_D+...=\n    G^0_D\\sum_{n=0}^\\infty \\big( G^0_D\\,\n    V \\big)^n,                           \\label{Gpert}\n    \\end{eqnarray}\nwhere $G_D^0$ is the propagator for operator $F^0$ obeying Dirichlet boundary conditions\nand  the composition law includes the integration over the bulk\ncoordinates, like for example in the first subleading term\n    \\begin{eqnarray}\n    G^0_D V G^0_D(y,y')=\\int_0^\\infty dy''\\,\n    G^0_D(y,y'') V(y'',\\partial_{y''}) G^0_D(y'',y').\n    \\end{eqnarray}\n\nThe lowest order Green's function in the half-space of the DGP model\nsetting --- the Green's function of $F^0=M^2-\\Box-\\partial_y^2$\nsubject to Dirichlet conditions on the brane $y=0$ and at infinity\n--- reads as follows\n    \\begin{eqnarray}\n    &&G_{D}^0(y,y')=\\frac{e^{-|\\,y-y'|\\,\\sqrt{M^2-\\Box}}\n    -e^{-(y+y')\\,\\sqrt{M^2-\\Box}}}{2\\sqrt{M^2-\\Box}}.\n    \\end{eqnarray}\nWe want to stress that here we assume the exact (curved)\n$d$-dimensional d'Alembertian $\\Box$ depending on the induced metric\nof the brane $g_{\\mu\\nu}(x)$. This means that in the lowest order\napproximation the underlying spacetime is not flat, but rather has a\nnontrivial but constant in $y$ metric of constant $y$ slices.\nCorrespondingly in the zeroth order we have\n    \\begin{eqnarray}\n    \\big[\\stackrel{\\rightarrow}{W} G_{D}(y,y')\\!\n    \\stackrel{\\leftarrow}{W}\n    \\big]_{\\,y=y'=0}^{\\,0}=\\,\\,\n    \\stackrel{\\rightarrow}{\\partial_y}G_{D}^0(y,y')\n    \\stackrel{\\leftarrow}{\\partial_y}\\!\n    \\,\\Big|_{\\,y=y'=0}=-\\sqrt{M^2-\\Box}. \\label{G0}\n    \\end{eqnarray}\n\nThe perturbation of the bulk operator can be expanded in Taylor\nseries in $y$, so that it reads\n    \\begin{eqnarray}\n    V(y,\\partial_y)=\n    \\sum_{k=0}^\\infty y^k\\,V_k(\\partial_y),       \\label{V}\n    \\end{eqnarray}\nwhere $V_k(\\partial_y)=V_k(\\partial_y|\\nabla)$ is a set of\n$y$-independent {\\em local $d$-dimensional covariant} operators of\nsecond order in $\\nabla_x$ and first order in $\\partial_y$.\n\nOn substitution of (\\ref{G0}) and (\\ref{V}) into (\\ref{Gpert})\nexactly calculable integrals over $y$ result in a nonlocal series in\ninverse powers of $\\sqrt{M^2-\\Box}$, and the perturbation expansion\ntakes the form\n    \\begin{eqnarray}\n    \\big[\\stackrel{\\rightarrow}{W} G_{D}(y,y')\\!\n    \\stackrel{\\leftarrow}{W}\n    \\big]_{\\,y=y'=0}=-\\sqrt{M^2-\\Box}\n    +\\sum_{k=0}^\\infty U_k(\\nabla)\\frac1{(M^2-\\Box)^{k\/2}},\n    \\end{eqnarray}\nwhere $U_k(\\nabla)$ is a set of local covariant differential\noperators acting on the brane\\footnote{Strictly speaking each $k$-th\norder of this series arises in the form of the following nonlocal\nchain of square root ``propagators\",\n$\\frac1{(M^2-\\Box)^{l_1\/2}}\\,U_1\\frac1{(M^2-\\Box)^{l_2\/2}}\\,U_2...\nU_{p-1}\\frac1{(M^2-\\Box)^{l_p\/2}}$, $l_1+l_2+...+l_p=k$, with\ndifferential operators $U_i$ as its vertices, but all these\npropagators can be systematically commuted either to the uppermost\nright or left by the price of extra commutator terms of the same\nstructure.}. The dimensionality of each $U_k(\\nabla)$ is the inverse\nlength to the power $k\\!+\\!1$, which is composed of the dimensionalities\nof bulk and extrinsic curvatures and covariant derivatives all taken\non the brane at $y=0$.\n\nWith $\\kappa(\\nabla)=-\\Box\/2m$ the brane-to-brane operator reads\n    \\begin{eqnarray}\n    2m\\mbox{\\boldmath$F$}^{\\,\\rm brane}(\\nabla)\n    =-\\Box+2m\\sqrt{M^2-\\Box}\n    -2m\\sum_{k=0}^\\infty U_k(\\nabla)\\frac1{(M^2-\\Box)^{k\/2}}.\n    \\end{eqnarray}\nThen we consider the perturbation series for the functional trace of\nits logarithm in powers of the full $U_k$-series. After reexpansion\nin powers of two sets of nonlocal propagators $1\/\\sqrt{M^2-\\Box}$\nand $1\/(-\\Box+2m\\sqrt{M^2-\\Box})$ the brane effective action finally\ntakes the form\n    \\begin{eqnarray}\n    &&\\frac12\\;{\\rm Tr}\n    \\ln \\mbox{\\boldmath$F$}^{\\,\\rm brane}=\n    \\frac12 {\\rm Tr}\n    \\ln\\Big(-\\Box+2m\\sqrt{M^2-\\Box}\\,\\Big)\\nonumber\\\\\n    &&\\qquad\\qquad\\qquad\n    +\\sum_{k\\geq 0,\\,l\\geq 1} {\\rm Tr}\\; W_{kl}(\\nabla)\n    \\frac1{(M^2-\\Box)^{k\/2}}\\,\n    \\frac1{(-\\Box+2m\\sqrt{M^2-\\Box}\\,)^l}   \\label{efacpert}\n    \\end{eqnarray}\nwith a new set of local covariant differential operators\n$W_{kl}(\\nabla)$ acting on the brane. The dimensionality of\n$W_{kl}(\\nabla)$ is $k+2l$ in units of inverse length. One should\nalso remember that each power of $1\/(-\\Box+2m\\sqrt{M^2-\\Box})$ is\naccompanied by one power of $m$ in the numerator, so that\nstructurally\n    \\begin{eqnarray}\n    W_{kl}(\\nabla)\\sim m^l\\,\\nabla^a\\,R^b\\,K^c,\n    \\end{eqnarray}\nwhere the integer overall powers of the covariant derivatives\n$\\nabla$, bulk curvatures $R$ and extrinsic curvatures $K$ are\nconstrained by the relation $a+2b+c=k+l$.\n\n \\section{Generalized proper time method}\n  \\hspace{\\parindent}\nOur goal now is to find the proper time representation of nonlocal\noperators in Eq.(\\ref{efacpert}) in the form of the exponentiated\n$\\Box$. A systematic way to do this consists in the following\nfactorization of the brane-to-brane operator as\n    \\begin{eqnarray}\n    2m\\mbox{\\boldmath$F$}^{\\,\\rm brane}_0(\\nabla)\n    =-\\Box+2m\\sqrt{M^2-\\Box}=\n    (\\sqrt{M^2-\\Box}-m_+)(\\sqrt{M^2-\\Box}-m_-).   \\label{factorization}\n    \\end{eqnarray}\nHere the masses $m_\\pm$ are the roots of the relevant quadratic\nequation, $x^2+2mx-M^2=0$, $x=\\sqrt{M^2-\\Box}$,\n    \\begin{eqnarray}\n    m_\\pm=-m\\pm\\sqrt{M^2+m^2},\\qquad m_-\\!<\\!-M<0<m_+\\!<M,\n    \\end{eqnarray}\nwhich determine the poles of the propagator of\n$\\mbox{\\boldmath$F$}^{\\,\\rm brane}_0(\\nabla)$ in spacetime with the\nLorentzian signature\\footnote{The pole at $\\Box_-$ is formally\ntachyonic, but it is located on the unphysical sheet of the Riemann\nsurface for the propagator in the complex plane of $\\Box$\n\\cite{GabShif} (which is indicated in (\\ref{phase}) by the\nnontrivial phase). Moreover, its residue is identically vanishing in\nview of $m_-<0$. Therefore this pole does not correspond to a real\nparticle. For $M\\neq 0$ only $\\Box_+$ gives rise to a particle with\nthe decreasing mass as $M\\to 0$, $M^2-m_+^2\\to 0$, which also\ndisappears in the DGP limit because the pole residue also vanishes\nat $M=0$. In this limit only the continuum spectrum of massive\nintermediate states survives forming the spectral representation for\nthe DGP propagator \\cite{DHK}\n    \\begin{eqnarray}\n    \\frac1{-\\Box+2m\\sqrt{-\\Box}}=\\frac{4m}\\pi\\int_0^\\infty\n    \\frac{d\\mu}{\\mu^2+4m^2}\\,\\frac1{\\mu^2-\\Box}.\n    \\end{eqnarray}}\n    \\begin{eqnarray}\n    &&\\Box_+=M^2-m_+^2>0,        \\label{realparticle}\\\\\n    &&\\Box_-=|M^2-m_-^2|\\,e^{3i\\pi}<0.   \\label{phase}\n    \\end{eqnarray}\n\nThe factorization (\\ref{factorization}) allows one to rewrite the\n$l$-th power of the brane-to-brane propagator in (\\ref{efacpert}) as\n    \\begin{eqnarray}\n    \\frac1{(-\\Box+2m\\sqrt{M^2-\\Box}\\,)^l}=\n    \\frac1{(\\sqrt{M^2-\\Box}-m_+)^l}\\,\\frac1{(\\sqrt{M^2-\\Box}-m_-)^l}\n    \\end{eqnarray}\nand then decompose the resulting fraction into the sum of simple\nfractions for which one has explicit proper time representations.\nThese representations begin with the following relations \\cite{Abramowitz_Stegun}\n    \\begin{eqnarray}\n    &&\\frac1{(M^2-\\Box)^{k\/2}}=\n    \\frac1{\\varGamma({k\/2})}\n    \\int\\limits_0^\\infty ds\\;s^{k\/2-1}\\,\n    e^{\\,s(\\Box-M^2)},                    \\label{proptimerep1}\\\\\n    &&\\frac1{\\sqrt{M^2-\\Box}-m}=\n    \\int\\limits_0^\\infty ds\\;e^{\\,s(\\Box-M^2)}\\,\n    \\left(\\frac1{\\sqrt{\\pi s}}\n    +m\\, w(-m\\sqrt{s})\\right),\\,\\,\\,m<M,  \\label{proptimerep2}\\\\\n    &&\\frac1{\\sqrt{M^2-\\Box}\\,\\big(\\sqrt{M^2-\\Box}-m\\big)}=\n    \\int\\limits_0^\\infty\n    ds\\;e^{\\,s(\\Box-M^2)}\\,\n    w(-m\\sqrt{s}),\\,\\,\\,m<M,                  \\label{proptimerep3}\n    \\end{eqnarray}\nwhich generate (by differentiating with respect to $m$, $\\Box$, $M$\nand linear recombining the results) the list of fractions with all\npossible powers of the factors $\\sqrt{M^2-\\Box}$ and\n$\\sqrt{M^2-\\Box}-m_\\pm$ in their denominators. Here the weight\nfunction $w(s)$ is given in terms of the error function \\cite{Abramowitz_Stegun}\n$\\mathrm{erf}(x)=\\frac2{\\sqrt\\pi}\\int_0^x dy\\,\\exp(-y^2)$ and has the\nfollowing ultraviolet and infrared asymptotics\n    \\begin{eqnarray}\n    w(x)\\equiv e^{x^2}\n    \\Big(1-\\mathrm{erf}(x)\\Big)\n    \\to\\left\\{\\begin{array}{ll}\\,\\,\\;\\,1\\,,\\,\\,&\\,x\\to 0,\\\\\n    \\,\\frac1{x\\sqrt\\pi}\\,,\\,\\,\\,\\,\\,&\\,x\n    \\to +\\infty,\\\\\n    \\,2\\,e^{x^2},\\,\\,\\,\\,\\,&\\,x\n    \\to -\\infty\\end{array}\\right.     \\label{wasymp}\n    \\end{eqnarray}\n\nThe last two proper time integrals above are defined only for $m<M$\n(for any negative $m$ and for $m<M$ if $m$ is positive), because in\nview of these asymptotics they are convergent at infinity only in\nthis range. Interestingly, the forbidden domain corresponds to the\nreal tachyon, because for $m>M>0$ the pole $\\Box=M^2-m^2$ belongs to\nthe physical sheet of the propagator, and its residue is\nnonvanishing.\n\nFrom (\\ref{proptimerep2})-(\\ref{proptimerep3}) it immediately\nfollows that the zeroth order brane-to-brane propagator and its\none-loop functional determinant read\n    \\begin{eqnarray}\n    &&\\frac1{-\\Box+2m\\sqrt{M^2-\\Box}}=\n    \\int\\limits_0^\\infty ds\\;e^{s\\,(\\Box-M^2)}\\;\n    \\frac{m_+w(-m_+\\sqrt{s})-m_-w(-m_-\\sqrt{s})}{m_+-m_-},  \\label{wpm}\\\\\n    &&{\\rm Tr}\\,\\ln\\Big(\\!-\\Box+2m\\sqrt{M^2-\\Box}\\,\\Big)\\nonumber\\\\\n    &&\\qquad\\qquad\\qquad\\qquad=\n    -\\frac12\\,{\\rm Tr}\\,\n    \\int\\limits_0^\\infty\\frac{ds}s\\,\n    e^{s\\,(\\Box-M^2)}\\,\\Big(w(-m_+\\sqrt{s})\n    +w(-m_-\\sqrt{s})\\Big),                        \\label{zeroefac}\n    \\end{eqnarray}\n\nFor the DGP model case with $M^2=0$ and $m_+=0$, $m_-=-2\\,m$ these\nrepresentations simplify to the equations derived in \\cite{toyDGP}\n    \\begin{eqnarray}\n    &&\\frac1{-\\Box+2m\\sqrt{-\\Box}}=\n    \\int\\limits_0^\\infty ds\\,e^{s\\,\\Box}\\,w(\\,2m\\sqrt{s}),  \\label{7.1}\\\\\n    &&{\\rm Tr}\\,\\ln\\Big(\\!-\\Box+2m\\sqrt{-\\Box}\\,\\Big)=\n    -{\\rm Tr}\\,\n    \\int\\limits_0^\\infty\\frac{ds}s\\,\n    e^{s\\,\\Box}\\,\\frac{1+w(\\,2m\\sqrt{s})}2.       \\label{7.2}\n    \\end{eqnarray}\nThe interpretation of the weight contribution here is very\ntransparent. It interpolates between the ultraviolet and infrared\ndomains where the brane operator and its logarithm have\nqualitatively different behaviors. In the domain of a small proper\ntime $m\\sqrt{s}\\ll 1$ it approximates the brane operator by a large\n$-\\Box\\gg m^2$ (hence the overall weight $(1+w)\/2\\simeq 1$), whereas\nin the infrared domain $m\\sqrt{s}\\gg 1$ it approximates the operator\nby $2m\\sqrt{-\\Box}$ (hence the weight is $(1+w)\/2\\simeq 1\/2$\ncorresponding to $\\,\\ln\\sqrt{-\\Box}=(1\/2)\\ln(-\\Box)\\,$).\n\nBy decomposing the nonlocal fractions of (\\ref{efacpert}) into the\nsum of simple fractions and using the weighted proper time\nrepresentations (\\ref{proptimerep1})-(\\ref{proptimerep3}) and their\nderivatives with respect to mass parameters $m_\\pm$ we obtain the following expression\n    \\begin{eqnarray}\n    \\frac1{(M^2-\\Box)^{k\/2}}\\,\n    \\frac1{(-\\Box+2m\\sqrt{M^2-\\Box}\\,)^l}=\n    \\int\\limits_0^\\infty\\frac{ds}s\\,\n    e^{-s(M^2-\\Box)}\\,w_{kl}(s,m,M),     \\label{w1}\n    \\end{eqnarray}\nwith some weight function $w_{kl}(s,m,M)$.\\footnote{Alternatively\nthis weight function can of course be obtained as a Mellin transform\nof the function of $\\Box$ in the left hand side, but this simple\nfraction decomposition method gives a more regular and systematic\nway to achieve the needed goal.}\n\nUsing (\\ref{zeroefac}) and (\\ref{w1}) we finally obtain for the\nperturbative expansion (\\ref{efacpert})\n    \\begin{eqnarray}\n    &&\\frac12\\;{\\rm Tr}\n    \\ln \\mbox{\\boldmath$F$}^{\\,\\rm brane}=\n    -\\frac12\\,\n    \\int\\limits_0^\\infty\\frac{ds}s\\,\n    e^{-sM^2}\\,\\frac{w(-m_+\\sqrt{s})+w(-m_-\\sqrt{s})}2\\;\n    {\\rm Tr}\\,e^{\\,s\\,\\Box}\\nonumber\\\\\n    &&\\qquad\\qquad\\qquad\\qquad+\\sum_{k\\geq 0,\\,l\\geq 1}\\;\n    \\int\\limits_0^\\infty\\frac{ds}s\\,\n    e^{-s\\,M^2}\\,w_{kl}(s,m,M)\\;{\\rm Tr}\\Big[\n    W_{kl}(\\nabla)\\,e^{\\,s\\,\\Box}\\Big].         \\label{finalresult}\n    \\end{eqnarray}\nThis formally solves the problem of constructing the\nSchwinger-DeWitt expansion for the brane effective action, because\nas it was expected all the remaining calculations reduce to the\nconventional calculation of the coincidence limits of the\nSchwinger-DeWitt coefficients and their covariant derivatives in\n    \\begin{eqnarray}\n    {\\rm Tr}\\Big[\n    W_{kl}(\\nabla)\\,e^{\\,s\\,\\Box}\\Big]=\\frac1{(4\\pi s)^{d\/2}}\\int\n    d^dx\\,g^{1\/2}\\sum_{n=0}^\\infty\\,s^n\\,\n    {\\rm tr}\\,W_{kl}\\Big(\\nabla^x_\\mu-\\frac{\\sigma_\\mu(x,x')}{2s}\\Big)\\,\n    \\hat a_n(x,x')\\,\\Big|_{\\,x'=x}.\n    \\end{eqnarray}\nRemember that every $W_{kl}(\\nabla)$ is a finite order covariant\ndifferential operator with the coefficients built of the powers of\nthe bulk curvature, extrinsic curvature of the brane and their\ncovariant derivatives. Here lengthening of the derivatives in\n$W_{kl}(\\nabla)$ originates from commuting them with the the\nexponential factor $\\exp(-\\sigma(x,x')\/2s)$ contained in the kernel\nof $\\exp(s\\Box)$, $\\sigma_\\mu(x,x')\\equiv \\nabla_\\mu^x\\sigma(x,x')$.\nThis of course brings to life world function coincidence limits\n$\\nabla_{\\mu_1}...\\nabla_{\\mu_p}\\sigma(x,x')|_{x'=x}$ also easily\ncalculable by the DeWitt recurrence procedure\n\\cite{DeWitt,PhysRep}.\n\nIt is important that the expansion (\\ref{finalresult}) is efficient\nfor the purpose of obtaining the asymptotic $1\/M$-expansion. Indeed,\nin view of the weight function asymptotics (\\ref{wasymp}) the\n$w(-m_-\\sqrt{s})$-parts of the overall $w_{kl}(s,m,M)$ (cf.\nEq.(\\ref{wpm})) with $m_-<0$ are suppressed at $s\\to\\infty$ by\n$e^{-sM^2}$ and, therefore, generate after the integration over $s$\nthe needed $1\/M^2$-series. In the $w(-m_+\\sqrt{s})$-parts the\nintegrand behaves as $e^{-s(M^2-m_+^2)}$, and generates the\n$1\/(M^2-m_+^2)\\sim 1\/2mM$-series also appropriate for the\n$M\\to\\infty$ limit, though converging slower than the $1\/M^2$ one.\nThis is the expansion in inverse squared masses of the real particle\nassociated with the pole (\\ref{realparticle}). Unfortunately, the\npowers of $1\/M$ are accompanied by those of $1\/m$, which comprises\nin the DGP model the problem of low strong-coupling scale\n\\cite{scale} for small DGP crossover scale $m$.\n\n\n\n\n\\section{Conclusions}\n  \\hspace{\\parindent}\nThis is obvious that the Schwinger-DeWitt technique in brane models\nis much more complicated than in models without spacetime\nboundaries. It does not reduce to a simple bookkeeping of local\nsurface terms like the one for simple boundary conditions reviewed\nin \\cite{Vassilevich}. Nevertheless it looks complete and\nself-contained, because it provides in a systematic way a manifestly\ncovariant calculational procedure for a wide class of boundary\nconditions including tangential derivatives (in fact of any order).\nOn the other hand, the calculational strategy of the above type is\nthus far nothing but a set of blueprints for the Schwinger-DeWitt\ntechnique in brane models, because there is still a large set of\nissues and possible generalizations to be resolved in concrete\nproblems.\n\nOne important generalization is a physically most interesting limit\nof the vanishing bulk mass $M^2$. Local curvature expansion is\nperfect and nonsingular for nonvanishing $M^2$ and applicable in the\nrange of curvatures and magnitudes of spacetime derivatives\n$(R,\\,K^2,\\nabla K)\\ll M^2$, $\\nabla\\nabla R\\ll M^4$, etc. However,\nfor $M^2\\to 0$ it obviously breaks down, because the proper time\nintegrals start diverging at the upper limit. These infrared\ndivergences can be avoided by a nonlocal curvature expansion of the\nheat kernel of \\cite{CPT}. Up to the cubic order in curvatures this\nexpansion explicitly exists for ${\\rm Tr}\\,e^{s\\Box}$ \\cite{CPT3},\nbut for the structure involving a local differential operator ${\\rm\nTr}\\,W(\\nabla) e^{s\\Box}$ it still has to be developed.\n\nAnother important generalization is the extension of these\ncalculations to the cases when already the lowest order\napproximation involves a curved spacetime background (i.e. dS or\nAdS bulk geometry, deSitter rather than flat brane, etc.). The\nsuccess of the above technique is obviously based on the exact\nknowledge of the $y$-dependence in the lowest order Green's function\nin the bulk and the possibility to perform exactly (or\nasymptotically for large $M^2$) the integration over $y$. All these\ngeneralizations and open issues are currently under study.\n\nTo summarize, we developed a new scheme of calculating quantum\neffective action for the braneworld DGP-type system in curved\nspacetime. This scheme gives a systematic curvature expansion by\nmeans of a manifestly covariant technique. Combined with the method\nof fixing the background covariant gauge for diffeomorphism\ninvariance developed in \\cite{gospel,qeastbg} this gives the\nuniversal background field method of the Schwinger-DeWitt type in\ngravitational brane systems.\n\n\n\n\n\n\\section*{Acknowledgements}\n  \\hspace{\\parindent}\nThe work of A.B. was supported by the Russian Foundation for Basic\nResearch under the grant No 08-01-00737. The work of D.N. was partly\nsupported by the RFBR grant No 08-02-00725. D.N. is also grateful to\nthe Center of Science and Education of Lebedev Institute and Russian\nScience Support Foundation. This work was also supported by the LSS\ngrant No 1615.2008.2.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\nWith the isolation of graphene,\\cite{Novo1,Zhang} a single layer of carbon\natoms on a honeycomb lattice, the study of the properties of relativistic\nDirac fermions has continued to intensify and has been extended to many\nother materials. An important example of a new class of materials which\nsupport massless helical Dirac fermions are topological insulators(TI).\\cit\n{Hasan,Moore1,Qi,Bernevig,Fu,Moore2,Hsieh,Chen1,Hsieh2,Hsieh3} These\nmaterials are insulators in their bulk but have metallic topological surface\nstates consisting of an odd number of Dirac cones protected by time reversal\ninvariance (TRI). Dirac fermions with finite mass are also found in some\nsystems. A prominent example which has only relatively recently come to the\nforefront are single layer group VI dichalcogenides such as $MoS_{2}$. \\cit\n{Novo2} It consists of a layer of molybdenum atoms between two layers of\nsulfur in a trigonal arrangement without inversion symmetry. At the K and -K\npoints of the honeycomb lattice Brillouin zone the Dirac valence and\nconduction bands are separated by a direct band gap which fall in the\ninfrared and there is spin polarization of the bands due to a large spin\norbit coupling. This material is considered ideal for valleytronics\\cit\n{Xiao,Zeng,Mak,Cao} in which the valley index is manipulated in direct\nanalogy to the spin degree of freedom for spintronics.\\cite{Wolf,Fabian} A\nclosely related single layered material is silicene\\cit\n{Drum,Aufray,Stille,Ezawa1,Ezawa2} which is made of silicon atoms on\na honeycomb lattice with one sublattice slightly shifted out of the\nplane of the other sublattice i.e. there is a buckling. Its model\nHamiltonian can be considered as a subcase of that for $MoS_{2}$\ninvolving however very different energy scales. Massive Dirac\nfermions are also found in bilayer graphene.\\cite{Nicol1, Nicol2}\nThe Dirac fermions seen in topological insulators can become massive\nwhen time reversal symmetry is violated through the introduction of\nmagnetic dopants. This was done by Chen et.al. \\cite{Chen2} in\n$Bi_{2}Se_{3}$ with $Mn$ as magnetic dopants. A composition\n$(Bi_{0.99}Mn_{0.01})_{2}Se_{3}$ can put the chemical potential\ninside the surface Dirac gap. An alternative is to use sufficiently\nthin topological insulator that top and bottom surfaces communicate\nthrough tunneling and this gaps the Dirac fermions on each of the\ntwo surfaces as discussed by Lu et.al.\\cite{Lu} A very different\nproposal to produce massive Dirac fermion was made by Ojanen and\nKitagawa\\cite{Ojanen} using irradiation of a two dimensional spin\norbit coupled electron gas with circular polarized THz\nelectromagnetic waves. Finally we mention that in their\nangle-resolved photoemission spectroscopy (ARPES) study of the TI,\n$Tl Bi(S_{1-x}Se_{x})_{2} $, Sato et.al \\cite{Sato} go from a\ntopological massless Dirac state in $Tl Bi Se_{2}$ to a ordinary\nnon-topological massive state in $Tl Bi S_{2}$. As the system goes\nthrough a quantum phase transition (QPT) from topological to\nnon-topological the Dirac fermions acquire a mass before reaching\nthe non-topological state. While the mechanism by which mass is\nacquired is not yet known the authors suggest the possibility of\nspontaneous symmetry breaking as in a Higgs mechanism.\n\nCoupling of massive Dirac fermions to an Einstein phonon modeled\nwith a Holstein hamiltonian leads not only to the usual modification\nof the bare dispersion relations through an electron-phonon self\nenergy but also modifies\\cite{Li1,asgari}the gap itself which\nbecomes complex and acquires energy dependence. This finding is\nanalogous to Eliashberg theory\\cite{Carbotte1,Carbotte2,Carbotte3}\nof superconductivity which is a generalization of BCS theory. In\nEliashberg theory, the details of the electron-phonon interaction\nresponsible for the condensation into Cooper pairs are explicitly\nincluded rather than modeled by a constant pair potential of BCS\ntheory. This leads to renormalizations in the single particle\nchannel which gives a shift in the quasiparticle energies as well as\nprovides a lifetime. In addition the pairing or gap channel is\ndirectly modified by the electron-phonon term and the gap which\nwould be constant and real in BCS theory is now complex and energy\ndependent. In this paper we calculate the effect of electron-phonon\ncoupling on the gapped Dirac dispersion curves measured in angular\nresolved photo emission spectroscopy (ARPES) and in the density of\nstate $N(\\omega)$ which can be measured in scanning tunneling\nmicroscopy (STM).\n\nIn section II we provide the necessary formalism. Our numerical results for\nthe case of Topological Insulators are found in Section III. Results\nspecific to the single layer $MoS_{2}$ membrane and silicene are presented\nin section IV with a summary and conclusion in section V.\n\n\\section{Formalism}\nWe begin with a two by two matrix Hamiltonian for the bare bands\nwhich is sufficiently general to describe topological insulators as\nwell as single layer $MoS_{2}$ and silicene. This Hamiltonian is\ngiven by Eq.~(\\ref{Ham})\n\\begin{equation}\nH_{0}=at[\\tau k_{x}\\sigma _{x}+k_{y}\\sigma _{y}]+\\frac{\\Delta\n}{2}\\sigma _{z}-\\lambda \\tau \\frac{\\sigma _{z}-1}{2}s_{z}\n\\label{Ham}\n\\end{equation\nwhere $\\tau =\\pm 1$ is a valley index, $k$ is momentum and $\\sigma _{x}$ \n\\sigma _{y}$ $\\sigma _{z}$ are Pauli matrices with $\\Delta $ the gap\nparameter. To describe single layer $MoS_{2}$ the spin orbit coupling \n2\\lambda =0.15eV$, $t$ is the nearest neighbor hopping of $1.1eV$\nwith $a$ the lattice parameter $3.193\\mathring{A}$ and $s_{z}$ is\nthe z-component of spin. The same form of this Hamiltonian applies\nto silicene dropping the $1$ in the last term which leads to equal\nspin splitting in valence and conduction band in contrast to\n$MoS_{2}$ where it is large in the valence band and small in the\nconduction band. The energy scales are\nhowever very different with $\\lambda $ of order $meV$ as is also the gap \n\\Delta $ which is now associated here with an electric field $E_{z}$ applied\nperpendicular to the two sublattice planes. To describe a topological\ninsulator surface the last term is to be dropped ($\\lambda =0$) and the \n\\sigma $'s are to be understood as real spins rather than the pseudospin of\ngraphene, $MoS_{2}$ and silicene. Also the valley index $\\tau $ is to take\non a single value and spin degeneracy no longer arises. A Holstein\nhamiltonian has been widely used to describe the coupling of electrons to an\nEinstein phonon of energy $\\omega _{E}$ and this will be sufficient here. It\ntakes the for\n\\begin{equation}\nH_{e-ph}=-g\\omega _{E}\\sum_{\\mathbf{k},\\mathbf{k}^{\\prime },s}c_{\\mathbf{k\n,s}^{\\dag }c_{\\mathbf{k}^{\\prime },s}(b_{\\mathbf{k}^{\\prime }-\\mathbf{k\n}^{\\dag }+b_{\\mathbf{k}-\\mathbf{k}^{\\prime }})  \\label{phonon}\n\\end{equation\nwhere $b_{\\mathbf{q}}^{\\dag }$ create a phonon of energy $\\omega\n_{E}$ and momentum $\\mathbf{q}$ and $c_{\\mathbf{k},s}^{\\dag }$ an\nelectron of momentum $\\mathbf{k}$ and spin $s$. The coupling\nstrength is $g$. For this very simple model the interacting two by\ntwo matrix Matsubara Green's function for the charge carriers\n\\begin{equation}\nG(\\mathbf{k},i\\omega _{n})=\\frac{1}{2}\\sum_{s=\\pm }(1+s\\mathbf{H}_{\\mathbf{k\n}\\cdot \\mathbf{\\sigma })G(k,s,i\\omega _{n}) \\label{Greenf}\n\\end{equation\nwit\n\\begin{equation}\n\\mathbf{H}_{\\mathbf{k}}=\\frac{(at\\tau k_{x},atk_{y},\\frac{\\Delta ^{\\prime }}\n2}+\\Sigma ^{Z\\ast }(i\\omega _{n}))}{\\sqrt{a^{2}t^{2}k^{2}+|\\frac{\\Delta\n^{\\prime }}{2}+\\Sigma ^{Z}(i\\omega _{n})|^{2}}}  \\label{Hk}\n\\end{equation\nand\n\\begin{widetext}\n\\begin{equation}\nG(k,s,i\\omega _{n})=\\frac{1}{i\\omega _{n}+\\mu -\\lambda \\tau\ns_{z}\/2-\\Sigma ^{I}(i\\omega _{n})-s\\sqrt{|\\frac{\\Delta ^{\\prime\n}}{2}+\\Sigma ^{z}(i\\omega _{n})|^{2}+a^{2}t^{2}k^{2}}} \\label{Green}\n\\end{equation}\n\\end{widetext}In these formulas $\\Delta ^{\\prime }=\\Delta -\\lambda \\tau s_{z}\n$ and $\\Sigma ^{I}(i\\omega _{n})$ and $\\Sigma ^{Z}(i\\omega _{n})$\nare the quasiparticle and gap self energy corrections at Matsubara\nfrequency $i\\omega _{n}$. These quantities at\ntemperature $T$ can be written a\n\\begin{eqnarray}\n&&\\Sigma ^{I}(i\\omega _{n})=\\frac{g^{2}\\omega _{E}^{2}}{2}\\sum_{\\mathbf{q,}s}\n\\notag \\\\\n&&\\left[ \\frac{f_{F}(\\varepsilon _{q,s})+f_{B}(\\omega _{E})}{i\\omega\n_{n}+\\mu +\\omega _{E}-\\varepsilon _{q,s}}+\\frac{f_{B}(\\omega\n_{E})+1-f_{F}(\\varepsilon _{q,s})}{i\\omega _{n}+\\mu -\\omega _{E}-\\varepsilon\n_{q,s}}\\right]   \\label{sigmaI}\n\\end{eqnarray\nand\n\\begin{eqnarray}\n&&\\Sigma ^{Z}(i\\omega _{n})=\\frac{g^{2}\\omega _{E}^{2}}{2}\\sum_{\\mathbf{q,}s\n\\frac{s\\frac{\\Delta ^{\\prime }}{2}}{\\sqrt{a^{2}t^{2}q^{2}+(\\frac{\\Delta\n^{\\prime }}{2})^{2}}}\\times   \\notag \\\\\n&&\\left[ \\frac{f_{F}(\\varepsilon _{q,s})+f_{B}(\\omega _{E})}{i\\omega\n_{n}+\\mu +\\omega _{E}-\\varepsilon _{q,s}}+\\frac{f_{B}(\\omega\n_{E})+1-f_{F}(\\varepsilon _{q,s})}{i\\omega _{n}+\\mu -\\omega _{E}-\\varepsilon\n_{q,s}}\\right]   \\label{sigmaZ}\n\\end{eqnarray\nHere $f_{F}$ and $f_{B}$ are fermion and boson distribution functions \n1\/[e^{(\\omega -\\mu )\/k_{B}T}\\pm 1]$ respectively and $\\varepsilon _{k,s}$ is\nthe bare band quasiparticle energy $\\varepsilon _{k,s}=\\lambda \\tau s_{z}\/2+\n\\sqrt{a^{2}t^{2}k^{2}+(\\frac{\\Delta ^{\\prime }}{2})^{2}}$. Note that\n$\\Sigma ^{Z}(i\\omega _{n})$ in Eq.~(\\ref{sigmaZ}) is directly\nproportional to $\\Delta^{\\prime}$ which appears linearly on the\nright hand side of the equation.\n\n\\section{Numerical results}\n\\begin{figure}[tp]\n\\begin{center}\n\\includegraphics[height=4in,width=3.0in]{Fig1a.eps}\n\\includegraphics[height=2.5in,width=1.5in,angle=-90]{Fig1b.eps}\n\\end{center}\n\\caption{The real(solid) and imaginary(dashed) part of the self\nenergy of a massive Dirac Fermion as a function of energy $\\omega$.\nThe top frame of the top panel gives the mass renormalization\n$\\Sigma^{Z}(\\omega)$ while the lower frame is for the quasiparticle renormalization $\\Sigma^{I}\n\\omega)$. The middle panel is the charge carrier density of states\nDOS as a function of $\\omega$. It compares the bare band case (solid\nline) with the dressed case (dashed line). The bare band chemical\npotential was set at $\\mu=30meV$ and the bare band gap extends from\n-55 meV to -15 meV (solid line). Coupling to a phonon at\n$\\omega_{E}=7.5$ meV\nintroduces structure at $-\\mu-\\Delta\/2-\\omega_{E}$, $\n\\mu+\\Delta\/2-\\omega_{E}$, $-\\omega_{E}$ and \n\\omega_{E}$. The gap for the interacting case is shifted with\nrespect to its bare value and its magnitude is effectively reduced\nto about 25 meV. In the bottom panel we show a schematic of the band\nstructure of ungapped (left) and gapped (right) Dirac fermions. The\nblue dot in the middle of the Brillouin zone is the $\\Gamma$ point.\n} \\label{fig1}\n\\end{figure}\n\n\\begin{figure}[tp]\n\\begin{center}\n\\includegraphics[height=4in,width=3.0in]{Fig2.eps}\n\\end{center}\n\\caption{(Color online) The Dirac fermion spectral density $A(k\n\\omega)$ vs. $\\omega$ in eV for various values of momentum as\nlabeled. Each curve is restricted to the region below 600 for ease\nof viewing. The chemical potential is 25 meV and both gap and\nquasiparticle self energies $\\Sigma^{Z}(\\omega)$ and\n$\\Sigma^{I}(\\omega)$ are included. The vertical dotted lines mark\nimportant energies 7.5, -7.5(blue), -12.5 and -52.5 meV(red).}\n\\label{fig2}\n\\end{figure}\n\n\\begin{figure}[tp]\n\\begin{center}\n\\includegraphics[height=3.0in,width=3.0in]{Fig3a.eps}\n\\includegraphics[height=3.0in,width=3.0in]{Fig3b.eps}\n\\end{center}\n\\caption{(Color online) Color plots of the dressed Dirac fermion\ndispersion curves $\\omega=E(k)$ (left frame) as a function of\nmomentum k compared with the bare case (right frame). In this last\ncase a small constant residual scattering rate of $\\Gamma =0.1meV$\nwas included. The chemical potential is set at 25 meV and the bare\ngap $\\Delta $ is 40 meV. For ease of comparison between left and\nright frames we have used for the bare case a value of chemical\npotential shifted by the value of $Re\\Sigma ^{I}(\\omega )$ at \n\\omega =0$. Fig. 3b is same as Fig. 3a but the electron-phonon\ncoupling has been halved to show the progression from bare to\ndressed case.} \\label{fig3}\n\\end{figure}\n\n\\begin{figure}[tp]\n\\begin{center}\n\\includegraphics[height=4in,width=3in]{Fig4a.eps}\n\\includegraphics[height=1.5in,width=1.5in]{Fig4b.eps}\n\\end{center}\n\\caption{(Color online) The dressed Dirac fermion density of state\nas a function of $\\omega$ for the case of a single layer $MoS_{2}$\nwith spin polarized bands. In the top frame of the top panel the\nchemical potential has been chosen to be equal to -0.845 eV which\nfalls below the top of the spin up and above the top of the spin\ndown valence band. The chemical potential is shown by the vertical\nblack dashed line. The phonon structures are at\n$-\\Delta\/2-\\lambda-\\omega_{E}$, $\\mu-\\omega_{E}$, $\\mu+\\omega_{E}$\nand $-\\Delta\/2+\\lambda+\\omega _{E}$. In the lower frame of the top\npanel $\\mu$ = -0.995 eV and so falls below the top of both spin up\nand down bands as shown. The phonon structures are at\n$\\mu-\\omega_{E}$, $\\mu+\\omega_{E}$,$-\\Delta\/2-\\lambda+\\omega_{E}$\nand $-\\Delta\/2+\\lambda+\\omega _{E}$. The dotted red curves are for\nthe spin up band and the solid black for sum of up and down. The\nbottom panel is a schematic of the band structure of $MoS_{2}$. The\nspin splitting of the conduction band is small and not visible in\nthe schematic. The electron-phonon mass renormalization was set at\n$\\lambda_{ep}$=0.1. } \\label{fig4}\n\\end{figure}\nAs we saw in the previous section the electron-phonon interaction\ngiven by Eq.~(\\ref{phonon}) leads to two self energy\nrenormalizations in the Green's function of Eq.~(\\ref{Greenf}). The\nquasiparticle self energy $\\Sigma ^{I}(i\\omega _{n})$ given by\nEq.~(\\ref{sigmaI}) renormalizes the bare energies in the denominator\nof Eq.~(\\ref{Green}) and remains even when the gap is set to zero.\nHowever there is also a second self energy correction $\\Sigma\n^{Z}(i\\omega _{n})$ of Eq.~(\\ref{sigmaZ}), not present for the case\nof massless Dirac fermions, which directly modifies the gap. It has\nboth real and imaginary part and is frequency dependent. It is this\nfrequency dependence in both $\\Sigma ^{I}(i\\omega _{n}\\rightarrow\n\\omega +i\\delta )$ and $\\Sigma ^{Z}(i\\omega _{n}\\rightarrow \\omega\n+i\\delta )$ which carries the information about phonon structure and\non how this structure manifests itself in the dynamics of the Dirac\nfermions. For simplicity we start with the case of massive Dirac\nfermions but with $\\lambda $=0 in the bare band Hamiltonian of\nEq.~(\\ref{Ham}). In the top two frames of Fig.~\\ref{fig1} we show\nrespectively the renormalized self energy $\\Sigma ^{Z}$ and $\\Sigma\n^{I}$ as a function of $\\omega $. For illustrative purpose the\nchemical potential was set at $\\mu $=0.03 eV and the Einstein phonon\nenergy $\\omega _{E}$=7.5 meV with a gap of 40 meV. Our choice of\nphonon frequency is motivated by the experimental finding of La\nForge et.al\\cite{Forge} who found phonon absorption feature in their\ninfrared optical work at $61cm^{-1}$ and $133cm^{-1}$. The\ntheoretical work of Zhu et.al\\cite{Zhu1} identifies a dispersive\nsurface phonon branch ending at 1.8THz which they associate with a\nstrong Kohn anomaly indicative of a large electron-phonon\ninteraction. This observation is further supported by the\nangular-resolved photoemission study of $Bi_{2}Se_{3}$ where the\nelectron-phonon mass enhancement parameter $\\lambda_{ep}$ is\nfound\\cite{Hofm} to be 0.25 so that we can expect significant\neffects of the electron-phonon interaction in the properties of\ntopological insulators. This provides a motivation for the present\nwork. The electron-phonon coupling constant is set to be about 0.3,\nwell within the range of the reported value of 0.25 in the reference\n[\\onlinecite{Hofm}] and 0.43 in the reference [\\onlinecite{Zhu2}].\nThe fermi surface falls, by arrangement, at $\\omega $=0. We see\nprominent logarithmic structure in the real part (solid curve) at\n$\\omega =\\pm \\omega _{E}$ with corresponding small jumps in the\nimaginary part (dashed curve). There are additional phonon induced\nsignature at $-\\mu +\\Delta \/2-\\omega _{E}$ and $-\\mu -\\Delta\n\/2-\\omega _{E}$. We note from the mathematical structure of\nEq.~(\\ref{sigmaI}) and Eq.~(\\ref{sigmaZ}) that these boson\nstructures fall at precisely the same energies in both $\\Sigma ^{Z}$\nand $\\Sigma ^{I}$. For the imaginary part there is a Dirac Delta\nfunction of the form $\\delta (\\omega +\\mu \\pm \\omega\n_{E}-\\varepsilon _{q,s})$ which leads to jumps, as we have noted. By\nKramers-Kronig relations these imply logarithmic type singularities\nin the real part. These phonon structure get directly mirrored in\nthe Dirac spectral function $A_{s}(k,\\omega )$ which is given by\n\\begin{equation}\nA_{s}(k,\\omega )=-\\frac{1}{\\pi }ImG(k,s,i\\omega _{n}\\rightarrow \\omega\n+i\\delta )\n\\end{equation\nand works out to b\n\\begin{equation}\nA_{s}(k,\\omega )=\\frac{1}{\\pi }\\frac{Im\\Sigma ^{I}(\\omega )}{[\\tilde{\\omega\n-s\\sqrt{M}]^{2}+[Im\\Sigma ^{I}(\\omega )]^{2}} \\label{AI}\n\\end{equation\nwhere $\\tilde{\\omega}=\\omega +\\mu -\\frac{\\lambda \\tau s_{z}}{2}-Re\\Sigma\n^{I}(\\omega )$ and $M=[\\frac{\\Delta ^{\\prime }}{2}+Re\\Sigma ^{Z}(\\omega\n)]^{2}+[Im\\Sigma ^{Z}(\\omega )]^{2}+a^{2}t^{2}k^{2}$. The density of states \nN(\\omega )$ follows as\n\\begin{equation}\nN(\\omega )=\\sum_{\\mathbf{k},s}A_{s}(k,\\omega ). \\label{DOS}\n\\end{equation\nWe note that it is only the imaginary part of the quasiparticle self\nenergy $Im\\Sigma ^{I}(\\omega )$ which broadens the Lorentzian form\nof Eq.~(\\ref{AI}). However both real and imaginary part of the gap\nself energy $\\Sigma ^{Z}(\\omega )$ modify the gap which becomes an\neffective gap\n\\begin{equation}\n\\frac{\\Delta _{eff}(\\omega )}{2}=\\sqrt{[\\frac{\\Delta ^{\\prime }}{2}+Re\\Sigma\n^{Z}(\\omega )]^{2}+[Im\\Sigma ^{Z}(\\omega )]^{2}}\n\\end{equation\nand is now a frequency dependent quantity in sharp contrast to the\nbare band case for which it is a constant and equal to $|\\frac{\\Delta ^{\\prime }}\n2}|$ in magnitude. Also, in general the bare band density of state\nis independent of filling factor i.e. of the chemical potential $\\mu\n$. Introducing the electron-phonon interaction lifts this simplicity\nand the DOS is, in principle, different for each value of $\\mu $.\n\nIn the lower frame of Fig.~\\ref{fig1} we present results for the\ndensity of states of gapped Dirac fermions $N(\\omega )$ vs. $\\omega\n$ given in Eq.~(\\ref{DOS}). The parameters are, chemical potential\n$\\mu$=30 meV above the center of the gap with $\\Delta \/2=$ 20 meV.\nThe dashed curve includes renormalizations due to the coupling to\nphonons while the solid curve, given for comparison, is for the bare\nband. Some care is required in making such a comparison. The\nchemical potential of the interacting system ($\\mu$) is not the same\nas for the bare band ($\\mu_{0}$). The two are related by the\nequation $\\mu =\\mu_{0}+ Re\\Sigma ^{I}(\\omega =0)$. From the lower\nframe of the top panel of Fig.~\\ref{fig1} we find $Re\\Sigma\n^{I}(\\omega =0)\\approx-5$ meV. This gives $\\mu_{0}$=35 meV so the\ngap in the bare band case falls between $-15$ meV (bottom of\nconduction band) to $-55$ meV (top of valence band). Including\ninteractions further shifts the bottom of the conduction band to\nlower energies and the top of the valence band to higher energies.\nThese shifts effectively reduce the band gap in the interacting case\nto about 25 meV as compared with a bare band value of 40 meV. A\nfeature to be noted is that the value of the dressed density of\nstates at $\\omega$=0 (vertical dashed line) remains unchanged from\nits bare band value as is known for conventional\nsystems.\\cite{Carbotte1,Carbotte2,Carbotte3} In addition we note\nsharp phonon structures originating from both $\\Sigma ^{I}(\\omega )$\nand $\\Sigma ^{Z}(\\omega )$. These structures in the self energies\n$\\Sigma ^{Z}(\\omega )$ and $\\Sigma ^{I}(\\omega )$ of top and middle\nframes are at $ -\\mu -\\Delta \/2-\\omega _{E}$, $-\\mu +\\Delta\n\/2-\\omega _{E}$, $-\\omega _{E}$ and $\\omega _{E}$ as identified in\nthe figure. We emphasized that the last two structures, placed\nsymmetrically around the Fermi surface at $\\omega =\\pm \\omega _{E}$,\nare very familiar in metal physics. The other two at $-\\mu -\\Delta\n\/2-\\omega _{E}$ and $-\\mu +\\Delta \/2-\\omega _{E}$ are not. As noted,\nthe top of the valence band has been shifted to higher energy by\ncorrelation effects but the shape of its profile is not very\ndifferent. By contrast, the onset of the conduction band has a very\nmuch altered shape. In particular it shows sharp phonon structure at\n$-\\mu +\\Delta \/2-\\omega _{E}$. Further, there is a prominent dip\naround $\\omega =-\\omega _{E}$. In the energy region between these\ntwo structures, electrons and phonon are strongly mixed by the\ninteractions. This can be seen clearly in Fig.~\\ref{fig2} where we\nshow a plot of the Dirac fermion spectral function $A(k,\\omega )$\nvs. $\\omega $ of Eq.~(\\ref{AI}) for various values of momentum $k$.\nThis function is measured directly in angular-resolved photoemission\nspectroscopy. Fourteen values of $k$ are considered as indicated and\neach curve is restricted to the region below 600 for clarity. The\nsolid vertical pink line identifies the fermi energy placed at\n$\\omega =0$. The vertical dotted blue lines are at $\\omega =\\pm\n\\omega _{E}$ and the dotted red lines identify $\\omega =-\\mu \\pm\n\\Delta \/2-\\omega _{E}$. The two curves close to $k=k_{F}$ are shown\nas red. In both curves we see a well defined quasiparticle peak near\n$\\omega =0$. For the bare band we would have a Dirac delta function\nat $\\omega =s\\sqrt{a^{2}t^{2}k^{2}+(\\frac{\\Delta }{2})^{2}}$. If\nsome residual scattering rate is included, the delta function\nbroadens into a Lorentzian form. When the electron-phonon\ninteraction of Eq.~(\\ref{phonon}) is included there are further\nshifts associated with the real part of $\\Sigma ^{I}(\\omega )$ and\nthe gap is modified by both real and imaginary part of $\\Sigma\n^{Z}(\\omega )$(see top and middle frame of Fig.1). In addition to\nthe quasiparticle peak seen in the red curves of Fig.~\\ref{fig2}\nthere are also small incoherent phonon assisted side bands with\nonset at $\\omega =\\pm \\omega _{E}$. As $k$ is increased beyond\n$k=k_{F}$ the phonon side band on the right hand side of the main\nquasiparticle peak becomes more prominent and as $k$ is decreased\nbelow $k_{F}$ it is the left side phonon assisted band which becomes\nstronger. As the quasiparticle peak falls at smaller energy with\ndecreasing $k$, the onset of the boson structure remains fixed in\nenergy and eventually they meet. When this happens the phonon and\nthe electron lose individual identity and merge into a composite\nwhole. As $k$ is reduced towards zero the quasiparticle peak crosses\nthe boson structure and we see that it re-emerges on the opposite\nside. It is clear from this discussion that the bottom of the\nconduction band falls precisely in the region where electron\nquasiparticle and phonon are not separately well defined. Instead\nthe electron spectral density consists of a composite incoherent\nentity with no identifiable sharp quasiparticle peak. This situation\nis very different when the top of the valence band is considered. In\nthis case, there appears a well defined quasiparticle peak near\n$k=0$ and consequently this region looks very much quasiparticle\nlike. Of course as $k$ is increased the spectral density of the\nvalence band starts to show a sideband with onset at $-\\mu -\\Delta\n\/2-\\omega _{E}$, as the quasiparticle peak approaches more closely\nthis energy.\n\nIn frame (a) of Fig.~\\ref{fig3} we show a color plot of the spectral\ndensity $A(k,\\omega)$ ($\\omega=E(k)$, the dressed dispersion curves)\nfor the bottom of the conduction band (top left) and for the top of\nthe valence band (bottom left). In the right frames we show results\nfor the corresponding bare band case with residual smearing $\\Gamma\n=0.1$ meV. For the bare band the chemical potential has been shifted\nbecause renormalized ($\\mu$) and bare ($\\mu_{0}$) chemical potential\nare related by $\\mu_{0}=\\mu - Re\\Sigma ^{I}(\\omega =0)$. In frame\n(b) we provide similar results but have halved the value of the\nelectron-phonon coupling $g$ in Eq.~(\\ref{phonon}), consequently the\nrenormalization effects are smaller but still quite significant.\nReturning to the top left frame of panel Fig.~\\ref{fig3}(a) we noted\nthe region of the conduction band between $\\omega =-7.5$ meV (dotted\nblue in Fig.~\\ref{fig2}) and $\\omega =-12.5$ meV (dotted red in\nFig.~\\ref{fig2}) which showed complex changes due to the phonon.\nBelow this region however we see a much more conventional type of\ngapped Dirac fermion dispersion curve which is recognizable as a\ndistortion of a bare dispersion curves (shown on the right). This is\nalso true for the valence band dispersion in the bottom left frame.\nHere it is only below -52.5 meV that significant phonon distortions\ncan be seen. Analogous modifications of the Dirac fermion dispersion\ncurves due to correlations have been observed in\ngraphene.\\cite{Bost1,Bost2,Walter} When electron-electron\ninteraction are accounted for in a random phase approximation to\nlowest order perturbation theory, one finds that the Dirac point at\nthe intersection of valence and conductance band splits into two\nDirac points with a plasmaron ring inserted in between. This\nrepresents resonant scattering between Dirac quasiparticles and\nplasmons (collective modes of the charge fluid) which are called\nplasmarons. Here it is the electron-phonon interaction which is\ninvolved instead. We hope the region where quasiparticles cease to\nbe well defined excitations can be observed in future ARPES\nexperiments.\n\n\\section{Results specific to single layer $MoS_{2}$ and silicene}\n\\begin{figure}[tp]\n\\begin{center}\n\\includegraphics[height=3in,width=3in]{Fig5.eps}\n\\end{center}\n\\caption{(Color online) The dressed density of state for the massive\nDirac fermions of single layer $MoS_{2}$. The chemical potential is\n$\\mu$ =-0.995 eV and falls below the top of both spin up (left top\nframe) and down (left bottom frame) valence bands as illustrated in\nthe inset of Fig. 4. In each frame the solid curves are the bare\nband results shown for comparison with red dashed curve for which we\nhave included only the quasiparticle self energy\n$\\Sigma^{I}(\\omega)$, and the blue dotted curves which involves both\n$\\Sigma^{I}(\\omega)$ and $\\Sigma^{Z}(\\omega)$. The bare case has\nbeen shifted by the constant $Re\\Sigma^{I}(\\omega)$ at $\\omega$=0.\nThe right frames show the corresponding spectral density\n$A(k=0,\\omega)$ vs. $\\omega$.} \\label{fig5}\n\\end{figure}\n\n\\begin{figure}[tp]\n\\begin{center}\n\\includegraphics[height=4in,width=3in]{Fig6a.eps}\n\\includegraphics[height=1.6in,width=1.6in]{Fig6b.eps}\n\\end{center}\n\\caption{(Color online) The density of state vs. $\\omega$ for\nmassive Dirac fermions described by the Hamiltonian Eq.~(\\ref{Ham})\n(with the -1 in the last term left out) and coupling to phonons\ndescribed by Eq.~(\\ref{phonon}). The parameters used are much\nsmaller than those for $MoS_{2}$. They are illustrative of silicene\nwith $\\Delta\/2$= 20 meV, $\\omega_{E}$= 7.5 meV and $\\lambda\/2$= 10\nmeV. Both conduction and valence bands are shown. Solid curves are\nfor spin down and dotted for spin up (green is for the conduction\nband and blue for the valence band). The bare band edges are shown\nas heavy vertical red lines and an arrow indicates how they are\nshifted by interactions. The phonon structures are identified in\nterms of $\\Delta$, $\\lambda$ and $\\omega_{E}$. In the lower frame of\nthe top panel the phonon energy has been shifted to 16.5 meV. The\nbottom panel is a schematic of the bands in silicene. The spin\nsplitting is the same size in both valence and conduction bands. The\nelectron-phonon mass renormalization is $\\lambda_{ep}$=0.3 for\n$\\omega_{E}$=7.5 meV and $\\lambda_{ep}$=0.45 for $\\omega_{E}$=16.5\nmeV.} \\label{fig6}\n\\end{figure}\nWe turn next to the specific case of single layer\n$MoS_{2}$.\\cite{Li2,Chei,Zhu,Stille} In this material both valence\nand conduction bands are spin polarized. This is shown schematically\nin the inset of the top panel of Fig.~\\ref{fig4} for the valence\nband as well as in the bottom panel where valence and conduction\nbands are both shown schematically in color, spin up valence band in\nblue and down in red. We note that the spin splitting in the\nconduction band (in gold) is small and does not appear in the\nschematic. In a related material silicene\\cite{Ezawa1,Ezawa2} the\nsplitting is the same size in both bands (see lower panel of\nFig.~\\ref{fig6}) as we will discuss later. It is the -1 in the last\nterm of Eq.~(\\ref{Ham}) which controls the amount of spin\npolarization seen in the conduction band and this term is missing in\nsilicene. Our numerical results for the energy dependence of the\nvalence band density of states (DOS) are shown in the top and bottom\nframe (top panel) for two values of chemical potential (see heavy\nblack dashed vertical lines) respectively -0.845 and -0.995 mev. The\nelectron-phonon coupling constant is set so that the mass\nenhancement parameter $\\lambda_{ep}$=0.1. In the first instance the\nchemical potential falls below the top of the spin up band but above\nthe top of the spin down band while in the second instance the\nchemical potential falls below the top of both up and down valence\nbands. The density of states for the spin up band alone is shown as\nthe heavy dotted red curve and for the combined up and down band by\nthe solid black curve. The phonon structures are\nat $-\\Delta\/2+\\lambda+\\omega_{E}$, $\\mu+\\omega_{E}$, $\\mu-\\omega_{E}$ and \n-\\Delta\/2-\\lambda-\\omega_{E}$ in the top frame and at $-\\Delta\/2+\\lambda\n\\omega_{E}$, $-\\Delta\/2-\\lambda+\\omega_{E}$, $\\mu+\\omega_{E}$ and \n\\mu-\\omega_{E}$ in the bottom frame. They are ordered according to\nthe larger energy first. In addition to phonon kinks, the\nelectron-phonon interaction has also shifted and modified the top of\neach of the two bands. As we have already seen there are two\ndistinctly different behaviors which characterize the shape of the\ntop (bottom) of these renormalized bands. We refer to the first as\nquasiparticle like. This designation applies to the spin up band in\nthe top frame. The band edge rises smoothly although rather sharply.\nThe second behavior seen is referred to as correlation dominated. It\napplies to the other three cases. Here a quasiparticle description\nceases to be possible and a sharp phonon induced structure is\nassociated with the ending of the band.\n\nThese issues are elaborated upon and better illustrated in\nFig.~\\ref{fig5} where we show a blow up of the band edges. For both\ntop and bottom frame the chemical potential has a value of -0.995 eV\nand thus falls below the top of both spin up and spin down valence\nbands. The solid black curve is the bare band case and is shown for\ncomparison with the dressed case. The long dashed red curves include\nonly the quasiparticle self energy $\\Sigma^{I}$ while the dotted\nblue curve includes in addition the gap renormalization\n$\\Sigma^{Z}$. In both top and bottom frames we note that when we\ninclude only the quasiparticle self energy, the edge shifts slightly\nto higher energy but does not change its shape which remains\ncharacteristic of the existence of good Dirac quasiparticles as is\nthe case for the bare bands. On the other hand, the dotted blue\ncurve which includes gap renormalizations as well as quasiparticle\nself energy corrections has change radically as compared to the bare\nband. The shift of the edge to higher energies is greater and\nits shape is also very different. There is a sharp rise at \n-\\Delta\/2+\\lambda+\\omega_{E}$ and at $-\\Delta\/2-\\lambda+\\omega_{E}$\nfor up and down bands respectively which is followed by a second\nrise of magnitude and shape much more comparable to that for the\nbare band. In the lower frame there is also a second phonon kink at\n$\\mu+\\omega_{E}$.\n\nIn the right panel of Fig.~\\ref{fig5} we show results for the\nspectral density $A(k,\\omega)$ at $k=0$ for the valence bands of the\nleft hand frame. The solid black curve is for the bare band but\nincludes a small constant scattering rate $\\Gamma$=1 meV so as to\nbroaden the Dirac delta function of the pure case. The long dashed\nred curve is for the electron-phonon dressed band where we include\nonly the quasiparticle self energy correction which renormalizes\ndirectly the bare quasiparticle energies. When gap self energy\nrenormalization is additionally included we get the dotted blue\ncurve which has entirely lost its sharp quasiparticle peak. The\nspectral density also shows a phonon peak at\n$-\\Delta\/2+\\lambda+\\omega_{E}-\\mu$ and\nat$-\\Delta\/2-\\lambda+\\omega_{E}-\\mu$ in top and bottom frame\nrespectively. These kinks are directly mirrored in the DOS of the\nleft hand frames. Of course we expect that it is not just the k=0\nvalue of the spectral density that contributes to the density of\nstate around the band edge but the results given are enough to\nunderstand how the band edge becomes modified from its bare band\nbehavior.\n\nIn Fig.~\\ref{fig6} (top panel) we present additional results for the\ndressed density of state when the -1 in the last term of our\nHamiltonian Eq.~(\\ref{Ham}) is left out and we use much reduced\nenergy scales with $\\Delta\/2$= 20 meV and $\\lambda\/2$=10 meV which\nis more representative of silicene (see illustrative figure in right\nhand frame). In both upper and lower frames the black solid line\ngives the contribution to the total DOS of the spin down band and\nthe dotted of the spin up band. In this case the green and blue\ncolor apply to the conduction and valence band respectively. In the\nupper frame we employ a phonon energy $\\omega_{E}$ of 7.5 meV and in\nthe lower frame the Einstein energy is increased to 16.5 meV. We\nkeep the electron-phonon coupling $g$ in Eq.~(\\ref{phonon}) fixed.\nComparing upper and lower frame shows that this increase in\n$\\omega_{E}$ has lead to much more filling of the gap between\nvalence and conduction band. In fact the band gap in the lower frame\nhas almost closed. For comparison with the dressed case the band\nedges in the bare bands is shown as the heavy red vertical lines. It\nshould be emphasized that increasing the value of the Einstein\nfrequency ($\\omega_{E}$) effectively increases the electron-phonon\ncoupling because of the $\\omega^{2}_{E}$ factor which appears in the\nnumerator of Eq.~(\\ref{sigmaI}) and Eq.~(\\ref{sigmaZ}) although\nthere are certain amount of cancelations from the denominator\ncontaining $\\omega_{E}$. The electron-phonon mass renormalization is\n$\\lambda_{ep}$=0.3 for $\\omega_{E}$=7.5 meV and $\\lambda_{ep}$=0.45\nfor $\\omega_{E}$=16.5 meV. This change in the electron-phonon\ncoupling is largely responsible for the near closing of the gap\nnoted above.\n\\section{Summary and conclusions}\nCoupling of Dirac fermions to a phonon bath changes their dynamical\nproperties. For momentum $k$ near the Fermi momentum $k_{F}$ well\ndefined quasiparticles exist in the conduction band with energies\nshifted from bare to dressed value controlled by the real part of\nthis self energy. The imaginary part of the quasiparticle self\nenergy gives them a finite lifetime. In addition there are phonon\nsidebands to which part of the spectral weight under the spectral\ndensity $A(k,\\omega )$ has been transferred. The onsets of these\nsidebands are determined by the singularities in the self energy.\nFor an Einstein optical phonon with energy $\\omega _{E}$, these\nonsets are found to be at $\\omega _{E}$, -$\\omega _{E}$ on either\nside of the chemical potential. As momentum $k$ is moved away from\n$k_{F}$, the energy of the renormalized quasiparticle peak\napproaches more closely these onset energies and the spectral weight\ntransfer to the sidebands increases. Eventually the quasiparticle\npicture itself breaks down entirely and the spectral density takes\non the appearance of a broad incoherent background with no sharp\nquasiparticle peaks. Electron and phonon are no longer individually\ndefined and a Green's function formalism as we have used here is\nneeded for a proper description. Near $k=0$, which corresponds to\nstates near the top of the valence band and bottom of the conduction\nband with a gap in between for massive Fermions, we find that the\nspectral function can remain largely coherent and\nquasiparticle-like, while in other instances it can be quite\nincoherent. In the first case the density of states (DOS) near the\ngap edge retains the general characteristic of bare bands, while in\nthe second case, the shape of the band edge becomes much more\ncomplex reflecting the incoherence due to correlations and the\nabsence of dominant quasiparticle peaks.\n\nIn addition to the familiar quasiparticle self energy, in the case\nof gapped Dirac fermions, the electron-phonon interaction also\nrenormalizes directly the gap through a new self energy which is\ncomplex and energy dependent. This self energy shifts up the top of\nthe valence band, and shifts down the bottom of the conduction band,\nthus reduces the effective gap. It does not however introduce\nadditional damping of the spectral density. We found that it is this\nself energy that plays the major role in reshaping the gap edges in\nthe DOS making it go from coherent to incoherent behavior. The\neffects described here can be measured in angular-resolved\nphotoemission spectroscopy (ARPES) and in scanning tunneling\nmicroscopy (STM).\n\n\\begin{acknowledgments}\nThis work was supported by the Natural Sciences and Engineering\nResearch Council of Canada (NSERC) and the Canadian Institute for\nAdvanced Research (CIFAR).\n\\end{acknowledgments}\n\n\\section*{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Background}\n\n The experimentally documented physical relevance of vacuum energy \ninvolves the electromagnetic field (the Casimir effect).\n Other theoretical work also usually considers massless fields\n (scalar fields for calculational and conceptual simplicity;\n neutrinos or gluons for more exotic or speculative applications).\n Nevertheless, the vacuum energy of a quantum field with mass is of \ntheoretical importance.\n Adding the mass term is the simplest generalization of the basic \nmodel of a massless scalar field in an empty flat cavity,\n a step toward the more complicated scenarios with space-dependent \n external potentials or curved background geometries.\nAlthough massive Casimir-type effects in the simplest geometries \n tend to be exponentially \ndamped relative to their massless counterparts, they can be less\nnegligible in other geometries \\cite[Secs.\\ 7.4--7.5]{K}.\n \n The massive field in one-dimensional space was studied by \n Hays \\cite{Hays}.\n (See also [\\onlinecite{BH,Kay,AW}].)\n Perhaps the most notable feature of the results is the presence of \na logarithmic divergence, absent from the massless case.\n Although physically harmless in the context of the one-dimensional \nbag model \\cite{BH,Hays}, a logarithmic divergence is regarded as \nparticularly problematic by some theorists because it is not \nautomatically eliminated by dimensional or zeta-function \nregularization.\n\n As I have stressed elsewhere \\cite{FG,systemat,norman},\n vacuum energy is one of a series of moments of the spectral \ndistribution of the differential operator appearing in the field \nequation; \n  these quantities arise as the \ncoefficients in the short-time asymptotics of a certain Green \nfunction, the ``cylinder kernel''\\negthinspace.\n The diagonal value  of the latter is,\n up to a differentiation and a numerical factor, the regularized \nvacuum energy defined by an ultraviolet cutoff (with the time \nvariable as the cutoff parameter) \\cite{BH,Hays,CVZ}.\n Thus the vacuum energy (regardless of experimental relevance) is \nof mathematical interest as a tool of spectral analysis.\n The cylinder-kernel coefficients incorporate nonlocal geometrical \ninformation that is not extractible from the much-studied \n short-time asymptotic expansion of the heat kernel.\n(The cylinder kernel is minus twice the $t$-derivative of the \nGreen function \nemployed in [\\onlinecite{Hays,BH}].)\n\n \\section{Notation}\n\n\n Consider a field equation of the type \n $\\frac{\\partial^2\\phi}{\\partial t^2} = -H\\phi$, where\n \\begin{equation}\n H=H_0 + \\mu, \\quad \\mu\\equiv m^2,\n\\label{operator} \\end{equation}\n and $H_0$ is a self-adjoint second-order differential operator in \nthe spatial variables, such as $-\\nabla^2$.\n For simplicity (although the treatment of local energy density is \nactually more general) assume that the spatial domain is compact \nand $H$ has a discrete, positive spectrum $\\{\\omega_n^2\\}$ with \northonormal eigenfunctions $\\{\\phi_n(x)\\}$.\n\n  For auxiliary mathematical purposes one studies the \n \\emph{heat kernel}\n \\begin{equation}\n K(t,x,y) = \\langle x|e^{-tH}|y\\rangle\n = \\sum_{n=1}^\\infty e^{-t\\omega_n^2} \\phi_n(x) \\phi_n(y)^*\n \\label{heat}\\end{equation}\n and the \\emph{cylinder (Poisson) kernel}\n \\begin{equation}\n T(t,x,y) = \\langle x|e^{-t\\sqrt{H}}|y\\rangle\n = \\sum_{n=1}^\\infty e^{-t\\omega_n} \\phi_n(x) \\phi_n(y)^* .\n \\label{cyl}\\end{equation}\nEach of these can be ``traced'' over space; for example,\n \\begin{equation}\n \\mathop{\\rm Tr} T \n = \\int \\langle x|e^{-t\\sqrt{H}}|x\\rangle \\, dx\n = \\sum_{n=1}^\\infty e^{-t\\omega_n}.\n\\label{cyltrace}\\end{equation}\n Formally,\n the vacuum energy of the quantized field configuration is\n \\begin{equation}\n E = \\frac12 \\sum_{n=1}^\\infty \\omega_n\n = -\\, \\frac12 \\lim_{t\\to 0} \\frac{\\partial}{\\partial t}\n  \\mathop{\\rm Tr} T,\n \\label{energy}\\end{equation}\n and one possible definition (see [\\onlinecite{systemat,norman}])\n of the vacuum energy density is\n\\begin{equation}\n T_{00}(x) \n = \\frac12 \\sum_{n=1}^\\infty \\omega_n \\phi_n(x) \\phi_n(y)^*\n =-\\,\\frac12 \\lim_{t\\to 0}\\frac{\\partial}{\\partial t} T(t,x,x).\n \\label{density}\\end{equation}\n\n  In reality, the limits in (\\ref{energy}) and (\\ref{density}) do \nnot exist,  but    \n  $\\,\\mathop{\\rm Tr} T$ and $T(t,x,x)$ possess asymptotic \nexpansions as $t\\downarrow 0$ of the form \\cite{CVZ,GG,FG,BM}\n\\begin{equation}  T \\sim\n\\sum_{s=0}^\\infty e_s t^{-d+s}\n+\\sum^\\infty_{\\scriptstyle s=d+1\\atop\n\\scriptstyle s-d \\mathrm{\\ odd}} f_s t^{-d+s} \\ln t,\n \\label{cylseries}\\end{equation}\nwhere the coefficients of the divergent terms are simple, local \nobjects that can be absorbed by renormalization.\n (Here $d$ is the spatial dimension.)\n Therefore, one regards (\\ref{cyl}) and  (\\ref{cyltrace}),\n after operation by $-\\frac12 \\frac{\\partial}{\\partial t}\\,$,\n  as the \\emph{regularized} energy and energy density, and\n one regards\n $-\\frac12$ times the coefficient of the term of order $t$ in \n(\\ref{cyl}) and  (\\ref{cyltrace}) as the \n \\emph{renormalized} energy and energy density:\n \\begin{equation}\n E   \\mbox{ or } T_{00} = - {\\textstyle \\frac12} e_{d+1}.\n \\label{renorm}\\end{equation}\n\n Similarly, if $K$ stands for either  \n $\\,\\mathop{\\rm Tr} K$ or $K(t,x,x)$,\n it has an expansion of the form\n\\begin{equation}\n K \\sim \\sum_{s=0}^\\infty b_s t^{(-d+s)\/2}.\n \\label{heatseries}\\end{equation}\n\n\n  \\section{The main equation} \n\nThe coefficients in (\\ref{cylseries}) and (\\ref{heatseries}) are\n functions of $\\mu$.\n Let us write $T(\\mu,t)$ and $K(\\mu,t)$ for the quantities being \nexpanded and write $e_s(\\mu)$, etc., for the coefficients.\n In the case of the heat kernel, it is elementary that\n \\begin{equation}\n K(\\mu,t)= K(0,t) e^{-\\mu t},\n \\label{heatfactor}\\end{equation}\nand\n from (\\ref{heatfactor}) it is routine to find formulas for the \n$b_s(\\mu)$ in terms of $b_{s'}(0)$ ($s'\\le s$).\n For the cylinder kernel it is clear that no elementary \nfactorization like (\\ref{heatfactor}) occurs, \n and hence the mass dependence is much more interesting and nontrivial.\n\n\nOn the other hand, (\\ref{heatfactor}) is equivalent to the \ndifferential equation\n \\begin{equation}\n \\frac{\\partial K}{\\partial \\mu} = -tK.\n \\label{heatpde}\\end{equation}\n The goal of the present paper is to find, as nearly as possible, \nan analogue of (\\ref{heatpde}) for the \n quantities $T(\\mu,t)$ related to vacuum energy. \n Since $\\omega_n(\\mu) = \\sqrt{\\omega_n(0)^2+\\mu}$,\n it is easy to show from (\\ref{cyltrace}) or (\\ref{cyl}) that\n \\begin{equation}\n\\frac{\\partial^2}{\\partial\\mu\\,\\partial t} \\left(\\frac Tt \\right)\n =\\frac T2 \\,,\n \\label{cylpde}\\end{equation}\nwhich is the central equation of this paper.\n  The variables $t$ and $\\mu$ naturally range from $0$ to \n  $+\\infty$.\n\n If its right side  were zero, (\\ref{cylpde}) would \nbe mathematically equivalent to the massless wave equation in \n two-dimensional space-time written in null (light cone) \ncoordinates; as is well known, its general solution would then be\n $T(\\mu,t)\/t=A(t) + B(\\mu)$, where $A$ and $B$ are arbitrary \nfunctions. \nThe full equation (\\ref{cylpde}) is of the same hyperbolic type, \nand one can again expect the general solution to involve two \narbitrary one-variable functions.\n One of these should be the ``initial value'' $T(0,t)$,  in analogy \nwith  (\\ref{heatfactor}).\n The remaining boundary condition is \n (cf.\\ (\\ref{cyl})--(\\ref{cyltrace}))\n \\begin{equation}\n \\lim_{t\\to+\\infty} T(\\mu,t) =0.\n \\label{zeroinfty}\\end{equation}\n\n \\section{Solution by Laplace transform}\n\n Let $F(s,t)$ be the Laplace transform of $T(\\mu,t)\/t$ with respect \nto~$\\mu$.\n Then (\\ref{cylpde}) is equivalent to\n \\[\n s\\,\\frac {dF}{dt} - \\frac{\\partial}{\\partial t}\\,\n  {T(0,t)\\over t}= \\frac t2\\, F\\,;\n \\]\n i.e.,\n \\begin{equation}\n\\frac {dF}{dt} - \\frac t{2s}\\, F = \\frac{\\partial}{\\partial t}\\, \n {T(0,t)\\over st}\\,.\n \\label{ltpde}\\end{equation}\n The solution of (\\ref{ltpde}) consistent with\n  (\\ref{zeroinfty}) is \n \\begin{equation}\n  F(s,t) = -e^{t^2\/4s}\\int_t^\\infty e^{-v^2\/4s}\n\\frac{\\partial}{\\partial v}\\, {T(0,v)\\over sv}   \\,dv. \n \\label{lapcyl}\\end{equation}\n Equivalently,\n  \\begin{equation}\n  F(s,t) = {T(0,t)\\over st}\n -\\frac1{2s^2} e^{t^2\/4s}\\int_t^\\infty e^{-v^2\/4s}\nT(0,v)\\,dv. \n \\label{lapcylbp}\\end{equation}\nThus, in principle, $T(\\mu,t)$ can be calculated from $T(0,v)$.\n \n\n Indeed, the inverse Laplace transform can be performed at the \nkernel level (under the integral sign in (\\ref{lapcyl}) or\n(\\ref{lapcylbp})) \\cite[p.\\ 1026]{AS}:\n \\begin{equation} \n {T(\\mu,t)\\over t} =\n -\\int_t^\\infty J_0\\bigl(m\\sqrt{v^2-t^2}\\bigr) \n\\frac{\\partial}{\\partial v}\\left({T(0,v)\\over v}\\right) dv\n \\label{bescyl} \\end{equation}\n or\n \\begin{equation}\n T(\\mu,t)= T(0,t)\n  -t\\int_t^\\infty \\frac {m\\,dv}{\\sqrt{v^2-t^2}} \\, \n J_1\\bigl(m\\sqrt{v^2-t^2}\\bigr) T(0,v).\n \\label{bescylbp}\\end{equation}\n A change of variable\n ($w^2=v^2-t^2$)\n  converts (\\ref{bescylbp}) to\n \\begin{equation}\n T(\\mu,t)= T(0,t)\n -t\\int_0^\\infty \\frac {m\\,dw}{\\sqrt{w^2+t^2}} \\, J_1(mw) \nT\\bigl(0,\\sqrt{w^2+t^2}\\bigr),\n \\label{bescylvar}\\end{equation}\n and there is a similar variant of (\\ref{bescyl}).\n \n\n \\subsection{Example 1}  The cylinder kernel of the free massless \nscalar field in spatial dimension~$d$ is \n \\begin{equation}\nT(0,t,\\mathbf x,\\mathbf y)= {\\Gamma(c)\\pi^{-c} t\\over (t^2+z^2)^c}\n \\label{freemassless}\\end{equation}\n where $z\\equiv |\\mathbf x-\\mathbf y|$ is the spatial separation\n and $c = \\frac12(d+1)$.\n According to (\\ref{bescyl}), therefore,\n \\begin{equation}\nT(\\mu,t,\\mathbf x,\\mathbf y)=2c\\Gamma(c)\\pi^{-c} t \\int_0^\\infty\n {w J_0(mw) \\,dw\\over (w^2 + t^2 + z^2)^{c+1}}\\,.\n\\label{freebes}\\end{equation}\n From \\cite[p.~425]{W} follows\n \\begin{equation}\nT(m^2,t,\\mathbf x,\\mathbf y)=   2^{1-c} \\pi^{-c} m^c t (t^2+z^2)^{-c\/2}\nK_c(m\\sqrt{t^2+z^2}),\n \\label{free}\\end{equation}\n which is the correct formula\\footnote{The Bender--Hays Green \nfunction \\cite{Hays,BH}\nis the Green function of the Helmholtz equation in one higher \ndimension,\n $[-\\nabla^2 -\\frac{\\partial^2}{\\partial t^2} +m^2]G = \n\\delta(t)\\delta^d(\\mathbf{z})$.\n In the free case this kernel is known \\cite[(4.25)]{OCF} and is \nproportional to $K_{c-1}(m\\sqrt{t^2+z^2})$.\n Differentiation and a recursion relation for the modified Bessel \nfunction then lead to (\\ref{free}).}\n for the cylinder kernel of the free field of mass $m$. \n All the solution formulas for problems with infinite flat boundaries\n now follow by the method of images.\n\n \\subsection{Example 2} Let $d=1$ and consider an interval of \nlength~$L$ with Dirichlet boundary conditions.\n For the massless case the solution by images can be summed in \nclosed form \\cite[(23) and (27)]{FG} with the result\n \\begin{equation}\n  \\mathop{\\rm Tr} T(0,t) = \\frac12 \\,{\\sinh (\\pi t\/L) \\over\n \\cosh(\\pi t\/L) -1} -\\frac12\\,.\n \\label{cyltr1}\\end{equation}\n We apply (\\ref{bescylvar}) and (\\ref{energy})\n  and compare with the conclusions \nof Hays \\cite{Hays} about the renormalized total energy in the \nmassive case\n (finding complete agreement).\n It is convenient to separate out the contribution of the free Green \nfunction (\\ref{freemassless})  (with $c=1$, $z=0$, and\n integrated over $0<x<L$),\n   because that is where all the divergences lie.\n That is, in both places in (\\ref{bescylvar}) write\n \\[\n T(0,v) = {L\\over\\pi v} +\\left[T(0,v) - {L\\over\\pi v}\\right].\n \\]\n When we apply the operator\n $-\\frac12 \\lim_{t\\to0}\\frac{\\partial}{\\partial t}$ to \n(\\ref{bescylvar})\nwe thus encounter four terms:\n\\begin{itemize}\n\n\\item The divergent term \n \\begin{equation}\n\\frac L{2\\pi t^2}\n \\label{masslessren}\\end{equation}\n (present already in empty \nspace) is the mass-independent \npart of the renormalization of the bag constant in \n[\\onlinecite[(3.10)]{Hays}].\n\n\\item The remaining (bracket) contribution of the first term in \n(\\ref{bescylvar}) is the familiar massless Casimir energy,\n \\begin{equation}\n-\\,\\frac{\\pi}{24L}\\,. \n \\label{massless}\\end{equation}\n  (It comes from the $O(t^2)$ term in the Taylor \nexpansion of (\\ref{cyltr1}).  Regrettably, that crucial term is \nwritten in [\\onlinecite[(27)]{FG}] with the wrong sign.) \n\n\\item The contribution of the free Green function to the integral \nin (\\ref{bescylvar}) works out to\n\\[\n{m^2Lt\\over 2\\pi}\\,\\ln\\left({mt\\over2}\\right) \n+(2C-1){m^2Lt\\over 4\\pi}\n\\]\n($C= 0.577\\ldots\\,$, Euler's constant).\nThe corresponding term in the regularized energy,\n\\begin{equation} \n-\\, {m^2L\\over 4\\pi}\\,\\ln\\left({mt\\over2}\\right) \n-(2C+1){m^2L\\over 8\\pi} \\,,\n\\label{massren}\\end{equation} \n  is the \nmass-dependent part of the renormalization  \n[\\onlinecite[(3.10)]{Hays}].\nIt, also, is present in empty space.\n(It includes a finite term, proportional to $m^2L$, which is \nactually ambiguous in the sense that the scale factor in the \nargument of the logarithm function is arbitrary.)\n\n\\item The remaining (bracket)  part of the integral splits into \ntwo disparate pieces.\n\n\\begin{itemize}\n\\item The term $-\\frac12$ in (\\ref{cyltr1}) \ncontributes\\footnote{This follows from [\\onlinecite[(6.552.1)]{GR}]\n and the power series of the Bessel functions.}\n\\begin{equation}\n -\\,\\frac14 \\lim_{t\\to0} \\frac{\\partial}{\\partial t}(1-e^{-mt})\n =-\\,\\frac m4\n\\label{massconst}\\end{equation}\nto the energy, in agreement with [\\onlinecite[(3.15)]{Hays}].\nThis constant term, associated with paths that reflect from the \nboundaries an odd number of times \\cite[Sec.~4]{Hays}, represents \nthe energy of interaction of the massive field with the two \nboundaries separately (i.e, it survives when $L$ approaches \ninfinity, and it does not contribute to the Casimir force).\n\n\\item What remains is the contribution of the paths that reflect \nan even number of times; it is the mass-dependent part of the \ntrue Casimir energy.\nIn our present approach it equals\\footnote{The \nerror made by setting $t=0$ inside the \nintegrand of this term \nof (\\ref{bescylvar}) before differentiating is of order $t^2\\ln t$, \nso it vanishes in the limit.}\n\\[\n\\frac m4 \\int_0^\\infty {J_1(mw)\\over w}\\,\n\\left[ {\\sinh(\\pi w\/L)\\over \\cosh(\\pi w\/L)-1} -\n{2L\\over \\pi w}\\right] dw \n \\]\n \\begin{equation}\n{} =\\frac m4 \\int_0^\\infty J_1\\left({mLu\\over \\pi}\\right)\n \\left[\\coth \\left(\\frac{u}2\\right) -\\frac2u \\right] {du\\over u}\n\\,.\n\\label{energyint}\\end{equation}\nIn Hays's approach [\\onlinecite[(3.11) and (3.13)]{Hays}] it \nappears as\n\\begin{equation}\n-\\,\\frac m{2\\pi} \\sum_{s=1}^\\infty {K_1(2mLs)\\over s} + {\\pi\\over \n24L}\n\\label{energysum}\\end{equation}\n(since Hays's sum includes the massless Casimir energy, \n(\\ref{massless})).\n The equivalence of (\\ref{energyint}) and (\\ref{energysum}) is not \nobvious, but it can be verified by the method of\n \\cite[p.~427]{W}.\n(It has also been tested numerically.)\n   We claim no practical advantage for the integral, since \nthe sum converges faster.\n\\end{itemize}\n\\end{itemize}\n\n\n\n\n\n \\section{Partial solution by  recursion}\n\n  Rarely will one of the integrals \n (\\ref{lapcyl})--(\\ref{bescylvar})  be evaluatable analytically \nin any particular case.\n Moreover, $T(0,t)$ often will not be available for \narbitrarily large~$t$. \nIt is worthwhile, therefore, to see how much information can be\n obtained from the known asymptotic structure (\\ref{cylseries})\n if the coefficients for $m=0$ are known.\n\n By substituting (\\ref{cylseries}) into (\\ref{cylpde}) one obtains\n \\begin{widetext}\n \\[\n \\sum_{s=0}^\\infty (-d+s-1)\\frac{\\partial e_{s}}{\\partial\\mu} t^{-\nd+s-2}\n+\\sum^\\infty_{\\scriptstyle s=d+1\\atop\n\\scriptstyle s-d \\mathrm{\\ odd}} (-d+s-1) \\frac{\\partial f_{s}}{\\partial\\mu} \n t^{-d+s-2} \\ln t\n+\\sum^\\infty_{\\scriptstyle s=d+1\\atop\n\\scriptstyle s-d \\mathrm{\\ odd}}  \\frac{\\partial f_{s}}{\\partial\\mu} \n t^{-d+s-2} \n \\]\n\\begin{equation} {} =\n\\sum_{s=2}^\\infty \\frac{e_{s-2}}2 t^{-d+s-2}\n+\\sum^\\infty_{\\scriptstyle s=d+3\\atop\n\\scriptstyle s-d \\mathrm{\\ odd}} \\frac{f_{s-2}}2 t^{-d+s-2} \\ln t.\n \\label{cylrr}\\end{equation}\n \\end{widetext}\n Therefore, we have the recursion relations\n \\begin{equation}\n(-d+s-1)\\frac{\\partial f_{s}}{\\partial\\mu} = \\frac{f_{s-2}}2\n \\label{rrf}\\end{equation}\n for $s  -d$ odd and positive, and\n \\begin{equation}\n(-d+s-1)\\frac{\\partial e_{s}}{\\partial\\mu}\n  =\\frac{e_{s-2}}2 -\\frac{\\partial f_{s}}{\\partial\\mu}\n \\label{rre}\\end{equation}\n for all nonnegative integers $s  $, \n the  terms being set to $0$ when not defined.\n Generically these equations can be solved recursively for \n$f_{s  }(\\mu)$ and $e_{s  }(\\mu)\\,$, respectively\n (the initial data $f_{s  }(0)$ and $e_{s  }(0)$ being presumed \nknown).\nExceptions occur when $-d+s-1=0$ (i.e., $s=d+1$);\n   then  (\\ref{rrf}) becomes a tautology and\n (\\ref{rre}),\n\\begin{equation}\n\\frac{\\partial f_{d+1}}{\\partial\\mu} = \\frac{e_{d-1}}2 \\,,\n\\label{excep}\\end{equation}\ntakes its place as the equation determining $f_{d+1}$.\n\n\n Thus there is no equation to determine $e_{d+1}(\\mu)$.\n The reason is that there is no way in this approach to \nimpose the second boundary condition (\\ref{zeroinfty}), so \nthe solution must involve an arbitrary function.\n Ironically, that function turns out to be naturally \nidentified with the renormalized vacuum energy \n(\\ref{renorm}), precisely the quantity of greatest \nphysical interest.\n In fact, the only coefficients that have been completely \ndetermined by this exercise are the ones that are \nequivalent to heat-kernel coefficients \n\\cite{FG,systemat,norman}.\n Nevertheless, the calculation clarifies the structure of \nthe problem and shows that once $e_{d+1}(\\mu)$ is known\n (along with the mass-zero coefficients), all the higher \ncylinder-kernel coefficients are computable.\n\n\\bigskip\n \\begin{acknowledgments}\n I thank Stuart Dowker and Klaus Kirsten for comments on the first \ndraft, and Todd Zapata for help with the numerical verification \nof ``(\\ref{energyint}) = (\\ref{energysum})''\\negthinspace.\n \\end{acknowledgments}\n\n\\goodbreak\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section*{\\LARGE\\bf #1}%\n  \\stepcounter{section}%\n  \\addcontentsline{toc}{section}{#1}%\n  \\large\\itshape%\n  #2\\\\\\vspace{0.1pt}\\\\%\n  #3%\n  \\normalsize\\upshape%\n  \\bigskip}\n\n\\begin{document}\n\\head{Cherenkov radiation by neutrinos}\n     {Ara N.~Ioannisian$^{1,2}$, Georg G.~Raffelt$^2$}\n     {$1$ Yerevan Physics Institute, Yerevan 375036, Armenia \\\\\n      $2$ Max-Planck-Institut f\\\"ur Physik (Werner-Heisenberg-Institut), \nF\\\"ohringer Ring 6, 80805 M\\\"unchen, Germany}\n\\subsection*{Abstract}\nWe discuss the Cherenkov process $\\nu\\to\\nu\\gamma$ in the presence\nof a homogeneous magnetic field. The neutrinos are taken to be\nmassless with only standard-model couplings.  The magnetic field\nfulfills the dual purpose of inducing an effective neutrino-photon\nvertex and of modifying the photon dispersion relation such that the\nCherenkov condition $\\omega<|{\\bf k}|$ is fulfilled.  \nFor a field strength $B_{\\rm\ncrit}=m_e^2\/e=4.41\\times10^{13}~{\\rm Gauss}$ and for $E=2m_e$ the\nCherenkov rate is about $6\\times10^{-11}~{\\rm s}^{-1}$.\n\\vspace{0.5cm}\n\nIn many astrophysical environments the absorption, emission, or\nscattering of neutrinos occurs in dense media or in the presence of\nstrong magnetic fields \\cite{ara1}. Of particular conceptual\ninterest are those reactions which have no counterpart in vacuum,\nnotably the decay $\\gamma\\to\\bar\\nu\\nu$ and the Cherenkov\nprocess $\\nu\\to\\nu\\gamma$. These reactions do not occur in vacuum\nbecause they are kinematically forbidden and because neutrinos do not\ncouple to photons. In the presence of a medium or $B$-field, neutrinos\nacquire an effective coupling to photons by virtue of intermediate\ncharged particles. \nIn addition, media or external fields modify the dispersion relations of\nall particles so that phase space is opened for neutrino-photon\nreactions of the type $1\\to 2+3$.\n\nIf neutrinos are exactly massless as we will always assume, and if\nmedium-induced modifications of their dispersion relation can be\nneglected, the Cherenkov decay $\\nu\\to\\nu\\gamma$ is kinematically\npossible whenever the photon four momentum $k=(\\omega,{\\bf k})$ is\nspace-like, i.e.\\ ${\\bf k}^2-\\omega^2>0$.\nOften the dispersion relation is expressed by $|{\\bf k}|=n\\omega$ in\nterms of the refractive index~$n$. In this language the Cherenkov decay\nis kinematically possible whenever $n>1$. \n\nAround pulsars field strengths around the critical value $B_{\\rm\ncrit}=m_e^2\/e=4.41\\times10^{13}~{\\rm Gauss}$.\nThe Cherenkov condition is satisfied for significant ranges of photon\nfrequencies. In addition, the magnetic field itself causes an\neffective $\\nu$-$\\gamma$-vertex by standard-model neutrino couplings\nto virtual electrons and positrons. Therefore, we study the Cherenkov\neffect entirely within the particle-physics standard model.\n\nThis process has been calculated earlier in\n~\\cite{ara2}. However, we do not agree with their\nresults. \n\nOur work is closely related to a recent series of papers ~\\cite{ara3}\nwho studied the neutrino radiative decay $\\nu\\to\\nu'\\gamma$ in the\npresence of magnetic fields.\n\nOur work is also related to the process of photon splitting that may\noccur in magnetic fields as discussed, for example, in\nRefs.~\\cite{ara4,ara5}.  \n\nPhotons couple to neutrinos by the amplitudes\nshown in Figs.~1(a) and (b). \nWe limit our discussion to field\nstrengths not very much larger than $B_{\\rm crit}=m_e^2\/e$. \nTherefore, we keep only electron in the loop.  \nMoreover, we are\ninterested in neutrino energies very much smaller than the $W$- and\n$Z$-boson masses, allowing us to use the limit of infinitely heavy\ngauge bosons and thus an effective four-fermion interaction (Fig.~1(c)).\nThe matrix element has the form\n\\begin{figure}\n\\centering\\leavevmode\n\\vbox{\n\\unitlength=0.8mm\n\\begin{picture}(60,25)\n\\put(8,15){\\line(-1,1){8}}\n\\put(8,15){\\line(-1,-1){8}}\n\\put(0,7){\\vector(1,1){4}}\n\\put(8,15){\\vector(-1,1){6}}\n\\multiput(9.5,15)(6,0){3}{\\oval(3,3)[t]}\n\\multiput(12.5,15)(6,0){3}{\\oval(3,3)[b]}\n\\put(31,15){\\circle{10}}\n\\put(31,15){\\circle{9}}\n\\multiput(37.5,15)(6,0){3}{\\oval(3,3)[t]}\n\\multiput(40.5,15)(6,0){3}{\\oval(3,3)[b]}\n\\put(0,10){\\shortstack{{}$\\nu$}}\n\\put(18,18){\\shortstack{{$Z$}}}\n\\put(43,18){\\shortstack{{$\\gamma$}}}\n\\put(30,11){\\shortstack{{e}}}\n\\put(40,5){\\shortstack{{(a)}}}\n\\end{picture}\n\\hspace{0.5cm}\n\\unitlength=0.8mm\n\\begin{picture}(60,32)\n\\put(16,15){\\line(-1,1){7.5}}\n\\put(16,15){\\line(-1,-1){7.5}}\n\\put(15,15){\\line(-1,1){7}}\n\\put(15,15){\\line(-1,-1){7}}\n\\put(16,15){\\line(-1,1){16}}\n\\put(16,15){\\line(-1,-1){16}}\n\\put(1,0){\\vector(1,1){6}}\n\\put(16,15){\\vector(-1,1){10}}\n\\multiput(17.5,15)(6,0){3}{\\oval(3,3)[t]}\n\\multiput(20.5,15)(6,0){3}{\\oval(3,3)[b]}\n\\multiput(8.2,8.7)(0,6){3}{\\oval(3,3)[l]}\n\\multiput(8.2,11.7)(0,6){2}{\\oval(3,3)[r]}\n\\put(2,15){\\shortstack{{}$W$}}\n\\put(0,3){\\shortstack{{}$\\nu$}}\n\\put(23,18){\\shortstack{{$\\gamma$}}}\n\\put(25,5){\\shortstack{{(b)}}}\n\\end{picture}\n\\unitlength=0.8mm\n\\begin{picture}(60,25)\n\\put(8,15){\\line(-1,1){8}}\n\\put(8,15){\\line(-1,-1){8}}\n\\put(0,7){\\vector(1,1){4}}\n\\put(8,15){\\vector(-1,1){6}}\n\\put(13,15){\\circle{10}}\n\\put(13,15){\\circle{9}}\n\\multiput(19.5,15)(6,0){3}{\\oval(3,3)[t]}\n\\multiput(22.5,15)(6,0){3}{\\oval(3,3)[b]}\n\\put(0,10){\\shortstack{{}$\\nu$}}\n\\put(26,18){\\shortstack{{$\\gamma$}}}\n\\put(12,11){\\shortstack{{e}}}\n\\put(25,5){\\shortstack{{(c)}}}\n\\end{picture}\n}\n\\smallskip\n\\caption[...]{Neutrino-photon coupling in an external magnetic field.\nThe double line represents the electron propagator in the presence of\na $B$-field. \n(a)~$Z$-$A$-mixing. (b)~Penguin diagram (only for $\\nu_e$).\n(c)~Effective coupling in the limit of infinite gauge-boson masses.\n\\label{Fig1}}\n\\end{figure}\n\\begin{equation}\n\\label{m}\n{\\cal M}=-\\frac{G_F}{\\sqrt{2}\\,e}Z\\varepsilon_{\\mu}\n\\bar{\\nu}\\gamma_{\\nu}(1-\\gamma_5)\\nu\\,\n(g_V\\Pi^{\\mu \\nu}-g_A\\Pi_5^{\\mu \\nu}) ,\n\\end{equation} \nwhere $\\varepsilon$ is the photon\npolarization vector and $Z$ its wave-function renormalization\nfactor. For the physical circumstances of interest to us, the photon\nrefractive index will be very close to unity so that we will be able\nto use the vacuum approximation $Z=1$.  \n$g_V=2\\sin^2\\theta_W+\\frac{1}{2}$ and \n$g_A=\\frac{1}{2}$ for $\\nu_e$, and\n$g_V=2\\sin^2\\theta_W-\\frac{1}{2}$ and\n$g_A=-\\frac{1}{2}$ for $\\nu_{\\mu,\\tau}$.\n\nFollowing Refs.~\\cite{ara4,ara6,ara7,ara8} $\\Pi^{\\mu\\nu}$ and\n$\\Pi_5^{\\mu\\nu}$ are\n\\begin{eqnarray}\n\\Pi^{\\mu\\nu}(k) &=& \\frac{e^3B}{(4\\pi)^2}\n\\Bigl[(g^{\\mu \\nu}k^2-k^{\\mu}k^{\\nu})N_0\n-\\,(g^{\\mu \\nu}_{\\|}k^2_{\\|}-k_{\\|}^{\\mu}k^{\\nu}_{\\|})N_{\\|}+\n(g^{\\mu\\nu}_{\\bot}k^2_{\\bot}-k^{\\mu}_{\\bot}k^{\\nu}_{\\bot})N_{\\bot}\n\\Bigr], \\nonumber\\\\\n\\Pi_5^{\\mu \\nu}(k) &=& \\frac{e^3}{(4\\pi)^2m_e^2}\n\\Bigl\\{-C_\\|\\,k_{\\|}^{\\nu}(\\widetilde{F} k)^{\\mu}\\\n+ \\ C_\\bot\\,\\Bigl[k_{\\bot}^{\\nu}(k\\widetilde{F})^{\\mu}\n+k_{\\bot}^{\\mu}(k\\widetilde{F})^{\\nu}-\nk_{\\bot}^2\\widetilde{F}^{\\mu \\nu}\\Bigr]\\Bigr\\},\n\\end{eqnarray} \nhere  $\\widetilde{F}^{\\mu \\nu}=\n\\frac{1}{2}\\epsilon^{\\mu \\nu \\rho \\sigma}F_{\\rho\n\\sigma}$, where $F_{12}=-F_{21}=B$. The $\\|$ and $\\bot$ decomposition of\nthe metric is\n$g_\\|={\\rm diag}(-,0,0,+)$ and\n$g_\\bot=g-g_\\|={\\rm diag}(0,+,+,0)$. $k$ is the four\nmomentum of the photon.\n$N_0$, $N_{\\bot}$,$N_{\\|}$, $C_\\bot$ and $C_\\|$ are\nfunctions on $B$,$k^2_{\\|}$ and $k^2_{\\bot}$.\nThey  are real for $\\omega<2m_e$, i.e.\\ below the pair-production\nthreshold.  \n\nThe four-momenta conservation constrains the photon emission angle to have\nthe\nvalue\n\\begin{equation}\\label{emissionangle}\n\\cos \\theta = \\frac{1}{n} \\\n\\left[1+(n^2-1)\\frac{\\omega}{2E}\\right],\n\\end{equation}\nwhere $\\theta$ is the angle between the emitted photon and incoming\nneutrino.\nIt turns out that for all situations of practical interest we have\n$|n-1|\\ll 1$ ~\\cite{ara4,ara9}.\nThis reveals that the outgoing photon\npropagates parallel to the original neutrino direction.\n\nIt is easy to see that the parity-conserving part of the effective vertex\n($\\Pi^{\\mu \\nu}$) is proportional to the small parameter\n$(n-1)^2 \\ll 1$ and the parity-violating part ($\\Pi_5^{\\mu \\nu}$) is {\\it\nnot\\\/}.\nIt is interesting to compare this finding with the standard \nplasma decay process $\\gamma\\to\\bar\\nu\\nu$ which is dominated by the\n$\\Pi^{\\mu \\nu}$. Therefore, in the approximation\n$\\sin^2\\theta_W=\\frac{1}{4}$ only the electron\nflavor contributes to plasmon decay. Here the Cherenkov rate is equal for \n(anti)neutrinos of all flavors.\n\nWe consider at first neutrino energies below the pair-production\nthreshold $E<2m_e$. For $\\omega<2m_e$ the photon refractive\nindex always obeys the Cherenkov condition $n>1$ ~\\cite{ara4,ara9}.\nFurther, it turns out that in the range\n$0<\\omega< 2m_e$ $C_\\|$,$C_\\perp$ depend only\nweakly on $\\omega$ so that it is well approximated by its value at\n$\\omega=0$.  \nFor neutrinos which propagate perpendicular to the magnetic\nfield, a Cherenkov emission rate can be written in the form\n\\begin{eqnarray}\\label{finalresult}\n\\Gamma\\ \\approx \\ \\frac{4\\alpha G_F^2E^5}{135(4\\pi)^4}\\,\n\\left(\\frac{B}{B_{\\rm crit}}\\right)^2 h(B)\\\n= \\ \n2.0\\times10^{-9}~{\\rm s}^{-1}~\\left(\\frac{E}{2m_e}\\right)^5\n\\left(\\frac{B}{B_{\\rm crit}}\\right)^2 h(B),\n\\end{eqnarray}\nwhere \n\\begin{equation}\nh(B)= \n\\cases{(4\/25)\\,(B\/B_{\\rm crit})^4&for $B\\ll B_{\\rm crit}$,\\cr\n1&for $B\\gg B_{\\rm crit}$.\\cr}\n\\end{equation}\nTurning next to the case $E>2m_e$ we note that in the presence of a\nmagnetic field the electron and positron wavefunctions are Landau\nstates so that the process $\\nu\\to\\nu e^+e^-$ becomes kinematically\nallowed. Therefore, neutrinos with such large energies will\nlose energy primarily by pair production rather than by Cherenkov\nradiation (for recent calculations see ~\\cite{ara10}).\n\n\nThe\nstrongest magnetic fields known in nature are near pulsars. However,\nthey have a spatial extent of only tens of kilometers. Therefore, even\nif the field strength is as large as the critical one, most neutrinos\nescaping from the pulsar or passing through its magnetosphere will not\nemit Cherenkov photons. Thus, the magnetosphere of a pulsar is quite\ntransparent to neutrinos as one might have expected.\n\\subsection*{Acknowledgments}\nIt is pleasure to thanks the organizers of the Neutrino Workshop at the\nRingberg Castle for organizing a very interesting and enjoyable\nworkshop. \n\\bbib\n\\bibitem{ara1} G.~G.~Raffelt, {\\it Stars as Laboratories for\n  Fundamental Physics\\\/} (University of Chicago Press, Chicago, 1996).\n\\bibitem{ara2} D.~V.~Galtsov and N.~S.~Nikitina,\n  Sov. Phys. JETP 35, 1047 (1972); \n  V.~V.~Skobelev, Sov. Phys. JETP 44, 660 (1976). \n\\bibitem{ara3} \n  A.~A.~Gvozdev et al.,\n  Phys. Rev. D {\\bf 54}, 5674 (1996);\n  V.~V.~Skobelev, JETP 81, 1 (1995);  \n  M.~Kachelriess and G.~Wunner, \n  Phys. Lett. B {\\bf 390}, 263 (1997). \n\\bibitem{ara4} S.~L.~Adler, Ann. Phys. (N.Y.) {\\bf 67}, 599 (1971).\n\\bibitem{ara5} S.~L.~Adler and C.~Schubert, Phys. Rev. Lett. {\\bf 77},\n  1695 (1996). \n\\bibitem{ara6} W.-Y.~Tsai, Phys. Rev. D {\\bf 10}, 2699 (1974).\n\\bibitem{ara7} L.~L.~DeRaad Jr., K.~A.~Milton, and N.~D.~Hari Dass, \n  Phys. Rev. D {\\bf 14}, 3326 (1976).\n\\bibitem{ara8} A.~Ioannisian, and G.~Raffelt, Phys. Rev. D {\\bf 55}, 7038\n  (1997).\n\\bibitem{ara9} W.-Y.~Tsai and T.~Erber, Phys. Rev. D {\\bf 10}, 492\n  (1974); {\\bf 12}, 1132 (1975); Act. Phys. Austr. {\\bf 45}, 245\n  (1976). \n\\bibitem{ara10} A.~V.~Borisov, A.~I.~Ternov, and V.~Ch.~Zhukovsky,\n  Phys. Lett. B {\\bf 318}, 489 (1993).\n  A.~V.~Kuznetsov and N.~V.~Mikheev, \n  Phys. Lett. B {\\bf 394}, 123 (1997). \n\\ebib\n\\end{document}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\n\\section{Introduction}\\label{sec:intro}\n\nCausal effects of spatially-varying exposures on spatially-varying outcomes may be subject to \\emph{non-local confounding} (NLC), which occurs when the treatments and outcomes for a given unit are affected by \\emph{covariates} of other nearby units \\citep{cohen2008obesity, florax1992specification, chaix2010neighborhood, elhorst2010applied}. In simple cases, NLC can be resolved using simple summaries of non-local data, such as the averages of the covariates over pre-specified neighborhoods. But in many realistic settings, NLC is caused by the complex interaction of spatial factors, and thus it cannot be resolved using simple \\emph{ad hoc} summaries of neighboring covariates. For such scenarios, we propose {\\em weather2vec}\\xspace, a framework that uses a U-net \\citep{ronneberger2015u} to learn representations that encode NLC information and can be used in conjunction with standard causal inference tools. The method is broadly applicable to settings where the covariates are available over a grid of spatial units, and where the outcome and treatment are observed in some subset of the grid. \n\nThe name {\\em weather2vec}\\xspace stems from its motivation to address limitations in current methods for estimating causal effects in environmental studies where meteorological processes are known confounders, aiming to contribute to the development of new flexible machine learning tools to assess the effect of policies and climate-related events on health-relevant outcomes: a task which has been recently identified by \\citet{rolnick2022tackling} as a pressing outstanding challenge for tackling the effects of climate change. \n\nTwo applications will be discussed in detail. The first application follows an earlier analysis by \\citet{papadogeorgou2019adjusting}, who estimated the air quality impact of power plant emissions controls. This case study evaluates the method's ability to reduce NLC under sparsely observed treatments (in combination with with propensity matching methods \\citep{rubin2005causal}).\nThe second example is an application to the problem of meteorological detrending \\cite{wells2021improved}, and uses {\\em weather2vec}\\xspace to deconvolve climate variability from policy changes when characterizing long-term air quality trends.\nThese two examples are accompanied by a simulation study comparing alternative adjustments to account for NLC. \n\nIn summary, this article has three aims:\n\\begin{enumerate}[itemsep=0pt, parsep=0pt]\n  \\item Provide a rigorous characterization of NLC using the potential outcomes framework \\citep{rubin2005causal}, clarifying some connections with interference and related methods \\citep{tchetgen2012causal, forastiere2021identification, sobel2006randomized}.\n  \\item Expand the library of NN methods in causal inference by proposing a U-net \\citep{ronneberger2015u} as a viable model to account for NLC in conjunction with standard causal inference tools.\n   \\item Establish a promising research direction for addressing NLC in scientific studies of air pollution exposure -- in which NLC is a common problem (driven by meteorology) for which widely applicable tools are lacking. \n\\end{enumerate}\n\nWe investigate two mechanisms to obtain the representations: one supervised, and one self-supervised.  The supervised one formally links the representation of NLC to the balancing property of propensity (and prognostic) scores in the causal inference literature \\citep{rubin_for_2008,hansen2008prognostic}. This approach requires that the outcome and treatment are densely available throughout the covariates' grid. By contrast, the self-supervised approach first learns representations encoding neighboring covariate information into a low-dimensional vector, which can subsequently be included as confounders in downstream causal analyses when the outcomes and treatments are sparsely observed on the grid.\n\n\n\n\\paragraph{Related work} \nPrevious research has investigated NNs for the (non-spatial) estimation of balancing scores \\citep{keller2015neural, westreich2010propensity, setoguchi2008evaluating} and counterfactual estimation  \\citep{shalit2017estimating, johansson2016learning, shi2019adapting}.\nNone of these works, however, specifically consider NLC. \n\nRelevant applications of U-nets in environmental studies include forecasting \\citep{larraondo2019data, sadeghi2020improving}, estimating spatial data distributions from satellite images \\citep{hanna2021multitask,fan2021resolving}, indicating that U-nets are powerful tools to manipulate rasterized weather data. Also relevant, \\citet{lu2005meteorologically} give a specific application of NNs for meteorological detrending, although without considering adjusting for neighboring covariates.\n\nApproaches to learn summaries of neighboring covariates for regression-based causal inference have been investigated in the econometrics literature. For example, WX-regression models \\citep{elhorst2010applied} formulate the outcome as a linear function of the treatment and the covariates of some pre-specified neighborhood.  \nSimilarly, CRAE \\citep{blier2020encoding} uses an autoencoder to encode pre-extracted patches of regional census data into a lower-dimensional vector that is fed into an econometric regression, somewhat analogous to the dimensionality reduction step in the self-supervised formulation of {\\em weather2vec}\\xspace, but requiring to pre-process the data into patches. \nIn contrast to these regression-based approaches, {\\em weather2vec}\\xspace aims at learning balancing scores, which have known benefits that include the ability to empirically assess the threat of residual confounding and the offer of protection against model misspecification that arises when modeling outcomes directly \\citep{rubin_for_2008}. \n\n\nThere is also a maturing literature on adjusting for unobserved spatially-varying confounding \\citep{reich2021review, veitch2019using, papadogeorgou2019adjusting}.\nSpatial random effect methods are popular in practice, although \\citet{khan2020restricted} have highlighted their sensitivity to misspecification for the purposes of confounding adjustment. As an alternative, the distance adjusted propensity score matching (DAPSm) \\citep{papadogeorgou2019adjusting} matches units based jointly on estimated propensity scores and spatial proximity under the rationale that spatial proximity can serve as a proxy for similarity in spatially-varying covariates. We use this model as a baseline in one of our applications.  \\citet{veitch2019using} take a related approach in a context where network proximity is viewed analogously to spatial proximity, and show that, under certain regularity conditions, network proximity can be used as a proxy for a network-level unobserved confounder. They propose a mechanism to learn embeddings that capture confounding information and used them together with augmented inverse probability weighting to obtain unbiased causal estimates. Importantly, they only consider the ``pure homophily'' case \\citep{shalizi2011homophily}, where the entirety of the confounding is assumed to be encoded by relative position in the network. While some of these methods could be useful for NLC, they all primarily target settings where confounding is local.\n\n\n\n\nFinally, NLC is distinct from, but notionally similar to, \\emph{causal interference} \\citep{tchetgen2012causal, forastiere2021identification, sobel2006randomized, zigler2021bipartite, ogburn2014causal,bhattacharya2020causal}. Both interference and NLC arise from spatial (or network) interaction, and they both impose limitations on standard causal inference methods. Various works in this literature have discussed the role of conditional ignorability given neighboring covariates; for instance, in \\citet{vansteelandt2007confounding} and \\citet{ forastiere2021identification}. However, to the best of our knowledge, flexible statistical methods specifically addressing NLC by learning the dependencies with respect to neighboring covariates have been ignored.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Potential outcomes and NLC}\\label{sec:rubin}\n\n\nThe potential outcomes framework, also known as the Rubin Causal Model (RCM) \\citep{rubin_for_2008}, distinguishes between the observed outcome $Y_s$ at unit $s$ and those that would be observed under counterfactual (potential) treatments $Y_s(a)$ (formally defined below). We start with some notaton.  The assigned treatment is denoted $A_s$. It is assumed to be binary for ease of presentation, although the ideas generalize to more general treatments. Next, ${\\mathbb{S}}$ is the set where the outcome and treatment are measured; ${\\mathbb{G}} \\supset {\\mathbb{S}}$ is a grid that contains the rasterized covariates $\\{{\\bm{X}}_s \\in \\mathbb{R}^d \\colon s\\in{\\mathbb{G}}\\}$; denoting ${\\tens{X}}_{B}$ where $B\\subset{\\mathbb{G}}$ is a set means ${\\tens{X}}_{B}= \\{{\\bm{X}}_s \\mid s\\in B\\}$; and $X \\perp \\!\\!\\!\\! \\perp Y \\mid Z$ means that $X$ and $Y$ are conditionally independent given $Z$.  Throughout ${\\bm{X}}_s$ is assumed to consist of pre-treatment covariates only, meaning they are not affected by the treatment or outcome. Finally, we use the generic notation $p(U)$ to denote the density or probability function of a random variable $U$.\n\n\n\n\\begin{mydef}[Potential outcomes]\\label{def:potential-outcomes} The potential outcome $Y_s({\\bm{a}})$ is the outcome value that \\textit{would} be observed at location $s$ under the global treatment assignment ${\\bm{a}}=(a_1,\\hdots, a_{|{\\mathbb{S}}|})$.\n\\end{mydef}\n\nFor $Y_s({\\bm{a}})$ to depend only on $a_s$, the RCM needs an additional condition called the \\textit{stable unit treatment value assumption}, widely known as SUTVA, and encompassing notions of {\\it consistency} and ruling out {\\it interference}.\n\n\\begin{assumption}[SUTVA] \\label{def:sutva} (1) Consistency: there is only one version of the treatment. (2) No interference: the potential outcomes for one location do not depend on treatments of other locations. Together, these conditions imply that $Y_s({\\bm{a}})=Y_s(a_s)$ for any assignment vector ${\\bm{a}}\\in\\{0,1\\}^{|{\\mathbb{S}}|}$, and that the observed outcome is the potential outcome for the observed treatment, i.e., $Y_s=Y_s(A_s)$.\n\\end{assumption}\n\n\n\n\n\n\n\nThe potential outcomes and SUTVA allow to define an important estimand of interest: the average treatment effect.\n\n\n\\begin{mydef}[ATE]\\label{def:ate} The average treatment effect (ATE) is the quantity\n$\\tau_\\text{ATE} = |{\\mathbb{S}}|^{-1}\\textstyle{\\sum_{s\\in{\\mathbb{S}}}} \\left\\{Y_s(1) - Y_s(0)\\right\\}$.\n\\end{mydef}\n\n\n\n\n\\begin{figure}[htb]\n    \\centering\n    \\begin{subfigure}[t]{.3\\linewidth}\\centering\n    \\includegraphics[width=.85\\linewidth]{img\/causal_diagram_local.png}\n    \\subcaption{ Local Confounding.}\n    \\label{fig:confounding-types:local}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{.3\\linewidth}\\centering\n    \\includegraphics[width=.85\\linewidth]{img\/causal_diagram_nonlocal.png}\n    \\subcaption{ NLC, no interference.}\n    \\label{fig:confounding-types:nlc}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{.3\\linewidth}\\centering\n    \\includegraphics[width=.85\\linewidth]{img\/causal_diagram_interf.png}\n    \\subcaption{ Interference, no NLC.}\n    \\label{fig:confounding-types:interf}\n    \\end{subfigure}\n    \\caption{Confounding types.}\n    \\label{fig:confounding-types}\n\\end{figure}\n\nOne cannot estimate the ATE directly since one never simultaneously observes $Y_s(0)$ and $Y_s(1)$. The next assumption in the RCM formalizes conditions for estimating the ATE, (or other causal estimands) with observed data by stating that any  observed association between $A_s$ and $Y_s$ is not due to an unobserved factor.\n\n\\begin{assumption}[Treatment Ignorability]\\label{def:ignorability} The treatment $A_s$ is ignorable with respect to some vector of controls ${\\bm{L}}_s$ if and only if $Y_s(1), Y_s(0) \\perp \\!\\!\\!\\! \\perp A_s \\mid {\\bm{L}}_s$.\n\\end{assumption}\n\nFor the sake of brevity, we will say that ${\\bm{L}}_s$ is \\emph{sufficient} to mean that the treatment is ignorable conditional on ${\\bm{L}}_s$. NLC occurs when local covariates are not sufficient. It is formally stated as follows:\n\n\\begin{mydef}[Non-local confounding] We say there is non-local confounding (NLC) when there exist neighborhoods $\\{{\\mathcal{N}}_s \\subset {\\mathbb{G}} \\mid s\\in{\\mathbb{S}} \\}$ such that ${\\bm{L}}_s={\\tens{X}}_{{\\mathcal{N}}_s}$ is sufficient and the neighborhoods are necessarily non-trivial (${\\mathcal{N}}_s\\neq\\{s\\}$).\n\\end{mydef}\n\nFigures \\ref{fig:confounding-types:local} and \\ref{fig:confounding-types:nlc} show a graphical representation of local confounding and NLC respectively.  Horizontal dotted lines emphasize that there may be spatial correlations in the covariate, treatment and outcome processes that do not result in confounding. For contrast, figure \\ref{fig:confounding-types:interf} shows the distinct phenomenon of (direct) interference, in which $A_{s'}$ affects $A_{s}$ \\citep{ogburn2014causal}. We remark that this depiction of is only on of the forms that interference can take. For instance, it may also happen through contagion \\citep{ogburn2014causal}. But they key point is that \\ref{fig:confounding-types:nlc} shows a scenario where the neighboring covariates suffice for ignorability and SUTVA still holds, whereas this is generally not the case with interference, which usually requires other causal estimands and implies the violation of SUTVA \\citep{ogburn2014causal, forastiere2021identification}. Additional discussion of interference is in section \\ref{sec:interference}.\n\n\n\nSubsequent discussion of the size of the NLC neighborhood, ${\\mathcal{N}}_s$, will make use of the following proposition stating that a neighborhood containing sufficient confounders can be enlarged without sacrificing the sufficiency.\n\n\\begin{proposition}\\label{prop:pre-treatment}\nLet ${\\bm{L}}_s$ be a sufficient set of controls including only pre-treatment covariates. and let ${\\bm{L}}_s'$ be another set of controls satisfying ${\\bm{L}}_s' \\supset {\\bm{L}}_s$. Then, ${\\bm{L}}_s'$ is also sufficient.\n\\end{proposition}\n\nAll the proofs are in Appendix \\ref{appendix:proofs}. We conclude this section with a classic result stating that any sufficient ${\\bm{L}}_s$ can be used to estimate the ATE from quantities and relations in the observed data.\n\n\\begin{proposition}\\label{prop:unbiasedness} Assume SUTVA holds and that ${\\bm{L}}_s$ is sufficient. Then\n\\begin{equation}\\label{eq:ate}\n  \\begin{aligned}\n      \\mathbb{E}\\left[\\mathbb{E}[Y_s \\mid {\\bm{L}}_s, A_s=1] - \\mathbb{E}[Y_s \\mid {\\bm{L}}_s, A_s=0]\\right],\n  \\end{aligned}\n\\end{equation}\nis an unbiased estimator of $ \\tau_\\text{ATE}$ (where $s$ is taken uniformly at random from ${\\mathbb{S}}$).\n\\end{proposition}\n\n\n\\section{Adjustment for NLC with {\\em weather2vec}\\xspace}\n\nAccounting for NLC would be fairly straightforward provided infinite data and the right set of confounders.  By virtue of Proposition \\ref{prop:pre-treatment}, one could, in principle, specify a non-linear regression $Y_s \\approx f(A_s, {\\tens{X}}_{\\mathbb{G}}, s)$ that includes every non-local covariate $\n{\\bm{X}}_{s'}\\in {\\tens{X}}_{{\\mathbb{G}}}$ as part of the regressors. With large model capacity and infinite repeated samples per location, this regression would perfectly estimate $\\mathbb{E}[\\mathbb{E}[Y_s \\mid {\\bm{L}}_s, A_s=a]$ and thus be able to estimate the ATE using Proposition \\ref{prop:unbiasedness}. But this scenario is far from realistic. Most commonly, there will be only be one observation for each $s$, and ${\\mathbb{S}}$ can also be small, requiring additional structure to enable statistical estimation.\nThus, we consider the question: what kind of statistical and functional model (e.g., to predict the probability of treatment) reflects the causal structure of NLC and allows for flexible statistical models under such restrictions?\n\n\nOne desirable statistical property to consider is \\emph{spatial stationarity}. Intuitively, it entails that the distributions of $Y_s$ and $A_s$ with respect to a neighboring covariate ${\\bm{X}}_{s'}$ should only depend on $s-s'$ (their relative position). Formally, it requires that for any set $B\\subset {\\mathbb{G}}$, displacement vector $\\delta$, and $s\\in {\\mathbb{G}}$, the following identity holds $p(A_s, Y_s \\mid {\\tens{X}}_B=x)\\overset{}{=} p(A_{s + \\delta}, Y_{s + \\delta} \\mid {\\tens{X}}_{B + \\delta}=x)$. \n\nThis paper propose to use U-nets \\citep{ronneberger2015u} to learn representations of NLC, since they are spatially stationary and adhere to the causal structure of NLC shown in figure \\ref{fig:confounding-types:nlc}. An overview of U-nets is provided in the next section for completeness. A key property is that a U-net $f_\\theta$ can transform the input covariates ${\\tens{X}}_{\\mathbb{G}}$ onto an output grid ${\\tens{Z}}_{\\theta,{\\mathbb{G}}}:=f_\\theta({\\tens{X}}_{\\mathbb{G}})$ of same spatial dimensions that each scalar or vector ${\\bm{Z}}_{\\theta, s}\\in {\\tens{Z}}_{\\theta,{\\mathbb{G}}}$ localizes contextual spatial information from the input grid.  Notice that U-nets are not the only NN architecture with these properties. For instance, residual architectures \\citep{he2016deep} could also be considered.\nNonetheless, our focus here is not on choosing an optimal architecture, but to highlight some fundamental statistical properties that an architecture must satisfy to be useful in accounting for NLC. The essence of {\\em weather2vec}\\xspace is to define appropriate learning tasks to obtain the NN weights $\\theta$. Two such tasks are considered, summarized below and described in detail in subsequent sections.\n\n\\begin{enumerate}[itemsep=0pt, itemindent=0pt, parsep=0pt]\n  \\item (\\textbf{Supervised)}\\quad Assuming the treatment and outcome are densely available over ${\\mathbb{G}}$, regress $A_s$ on ${\\bm{Z}}_{\\theta, s}$ (propensity score) or, for a subset of untreated units, regress $Y_s$ on ${\\bm{Z}}_{\\theta, s}$ (prognostic score).\n  \\item (\\textbf{Self-supervised)}\\quad If the treatment and outcome are not densely available over ${\\mathbb{G}}$, then learn $\\theta$ so that ${\\bm{Z}}_{\\theta, s}$ is highly predictive of ${\\bm{X}}_{s'}$ for any $s'$ within a specified radius of $s$.\n\\end{enumerate}\n\n\n\\subsection{An overview of the U-net for summarizing NLC information}\n\n\nIn this section, we provide a brief description of the U-net's functional form, referring the reader to \\citet{ronneberger2015u} for additional details. The U-net transformation involves two parts: a \\emph{contractive} stage and an symmetric \\emph{expansive stage}. Both of these steps use convolutions with learnable parameters and non-linear functions to aggregate information from the input grid spatially and create rich high-level features. The convolutions in the contractive path duplicate the number of latent features at each layer. Then, these intermediate outputs go through \\textit{pooling} layers which halve the spatial dimensions. Together, these operations augment the dimensionality of each point of the grid, combining information at many spatial points to richer information contained at fewer points. Convolutions propagate information spatially, and the deeper they are in the contractive path, the larger their propagation reach (in the original scale of the input grid).\nThe expansive path, on the other hand, uses \\textit{up-sampling} to progressively interpolate the deep higher-level features back to a finer spatial lattice, and then uses convolutions to reduce back the latent dimensionality at each grid point; with the characteristic that, in contrast to the input grid, every point now localizes spatial information. The output vector can have any arbitrary dimension after possibly applying an additional linear or convolutional layer followings the expansive path (or before the contractive path, or both). The unknown weights $\\theta$ dictate the size -- the ``radius of influence\" -- and what non-local information is summarized by ${\\bm{Z}}_{\\theta, s}$. \nThe functional model is spatially stationary because all of its components are (convolutions, pointwise activations, pooling, and upsampling). Figure \\ref{fig:architecture} provides a visual example of the U-net architecture.\n\n\n\n\\subsection{Learning NLC representations via supervision}\\label{sec:supervised}\n\nThe supervised approach links the proposed representation learning to the procedure of learning a balancing score \\citep{rubin2005causal, hansen2008prognostic} in causal inference. We recall the definition here for completeness.\n  \n  \\begin{mydef}[Balancing score]\\label{def:balancing} $b({\\bm{L}}_s)$ is a {\\em balancing score} iff $A_s \\perp \\!\\!\\!\\! \\perp {\\bm{L}}_s \\mid b({\\bm{L}}_s)$. The coarsest balancing score is $b({\\bm{L}}_s) := p(A_s=1 \\mid {\\bm{L}}_s)$, widely known as the \\emph{propensity score}.\n  \\end{mydef}\n  \n  \\begin{mydef}[Prognostic score]\\label{def:prognostic}  $b({\\bm{L}}_s)$ is a {\\em prognostic score} iff $Y_s(0) \\perp \\!\\!\\!\\! \\perp {\\bm{L}}_s \\mid b({\\bm{L}}_s)$. The coarsest prognostic score is $b({\\bm{L}}_s) := \\mathbb{E}[Y_s(0) \\mid {\\bm{L}}_s]$. \n  \\end{mydef}\n  \n  \n  The propensity score blocks confounding through the treatment \\citep{rubin2005causal}; prognostic scores do so through the outcome \\citep{hansen2008prognostic}. The importance of these definitions is summarized by the next well-known result.\n  \n  \\begin{proposition}\\label{prop:bs-reduction}\n  If $b({\\bm{L}}_s)$ is a balancing score, then ${\\bm{L}}_s$ suffices to control for confounding iff $b({\\bm{L}}_s)$ does. The same result holds for the prognostic score under the additional assumption of no effect modification.\n  \\end{proposition}\n\n\n\n\n\n  \n  \n  Thus, using one-dimensional latent dimension output for ${\\bm{Z}}_{\\theta,s}$, the straightforward regression loss functions for the propensity score and prognostic score regression are given respectively by\n\n\\begin{equation}\\label{eq:sup-prop}\n\\begin{aligned}\n{\\mathcal{L}}_\\text{sup}^\\text{prop}(\\theta) = \n\\sum_{s\\in {\\mathbb{S}}} \\text{CrossEnt}(A_s, {\\bm{Z}}_{\\theta,s})\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\\label{eq:sup-prog}\n\\begin{aligned}\n{\\mathcal{L}}_\\text{sup}^\\text{prog}(\\theta) =\n\\sum_{s\\in {\\mathbb{S}}\\colon A_s = 0} (Y_s - {\\bm{Z}}_{\\theta,s})^2\n\\end{aligned}\n\\end{equation}\nwhere CrossEnt is the binary cross-entropy loss. Algorithm \\ref{alg:sup} summarizes the supervised {\\em weather2vec}\\xspace approach.\n\n\\begin{algorithm}[tb]\n\\caption{Supervised representation learning and causal estimation}\n\\label{alg:sup}\n\\textbf{Input}: Covariates ${\\bm{X}}_{{\\mathbb{G}}}$; outcome and treatment $Y_s$ and $A_s$ at a dense subset ${\\mathbb{S}}\\subset {\\mathbb{G}}$; and U-net model $f_\\theta$ with 1-dimensional outputs. \n\\begin{algorithmic}[1] %\n\\STATE Obtain neural network weights $\\hat{\\theta}$ minimizing the supervised loss (equations \\ref{eq:sup-prop} or \\ref{eq:sup-prog}) and compute output balancing score grid from optimal weights $\\hat{{\\tens{Z}}}_{{\\mathbb{G}}}=f_{\\hat{\\theta}}({\\bm{X}}_{{\\mathbb{G}}})$.\n\\STATE Use standard causal inference methods (e.g., IPTW in Appendix \\ref{app:iptw}) to obtain $\\hat{\\tau}_\\text{ATE}$ adjusting for $\\hat{{\\bm{Z}}}_{s}$ and other relevant local confounders  at each unit $s$.\n\\end{algorithmic}\n\\end{algorithm}\n\n\nLearning $\\theta$ through supervision results in an efficient scalar ${\\bm{Z}}_{\\theta, s}$ compressing NLC information, allowing for $\\theta$ to just attend to relevant neighboring covariate information that pertains to confounding. However, supervision may not be possible with small-data studies where $Y_s$ and $A_s$ are only measured sparsely. In such cases, the supervised model will likely overfit to the data. For example, in application 1 in section \\ref{sec:applications}, ${\\mathbb{S}}$ consists only of measurements at 473 power plants, while the size of ${\\mathbb{G}}$ is $128 \\times 256$. An over-fitted propensity score would result in insufficient ``overlap'' \\citep{stuart2010matching} by assigning zero probability to the unobserved treatment, resulting in causal inferences that would rely on model extrapolation to areas where covariate information is not represented in  both treatment groups. \nTo avoid this, the self-supervised approach\ntargets scenarios with sparse ${\\mathbb{S}}$.\n\n\n\n\n\\subsection{Representations via self-supervised dimensionality reduction}\\label{sec:self-supervised}\n\nSelf-supervision frames the representation learning problem as dimension reduction without reference to the treatment or outcome. The representations are then used to learn a balancing score for causal effect estimation in a second analysis stage.  This approach requires specification of a fixed neighborhood ${\\mathcal{N}}_s$ (parameterized by a radius $R$) and latent dimension $k$, resulting on different representations for different hyper-parameter choices, which can be selected using standard model selection techniques (such as AIC) in the second stage. The dimension reduction's objective is that ${\\bm{Z}}_{\\theta, s}$ encodes predictive information of any ${\\bm{X}}_{s + \\delta}$ for $(s + \\delta) \\in {\\mathcal{N}}_s$.\nA simple predictive model ${\\bm{X}}_{s + \\delta} \\approx g_\\phi({\\bm{Z}}_{\\theta, s}, \\delta)$ is proposed. First, let ${\\bm{\\Gamma}}_{\\phi}(\\cdot)$ be a function taking an offset $\\delta$ as an input and yielding a $k\\times k$ matrix, and let $h_\\psi(\\cdot)\\colon \\mathbb{R}^k \\to \\mathbb{R}^d$ be a decoder with output values in the covariate space. The idea is to consider ${\\bm{\\Gamma}}_{\\phi}(\\delta)$ as a selection operator acting on ${\\bm{Z}}_{\\theta, s}$. The task loss function can be written succinctly as\n\n\\begin{equation}\\label{eq:self}\n\\begin{aligned}\n{\\mathcal{L}}_\\text{self}(\\theta, \\phi,\\psi\\mid R) &= \\\\\n\\sum_{s\\in {\\mathbb{G}}}\\sum_{ \\{\\delta\\colon \\lVert \\delta \\rVert \\leq R\\}} & \\left({\\bm{X}}_{s + \\delta} - h_\\psi(  {\\bm{\\Gamma}}_\\phi(\\delta){\\bm{Z}}_{\\theta, s})\\right)^2.\n\\end{aligned}\n\\end{equation}\n Algorithm \\ref{alg:self} summarizes the causal estimation procedure. See Appendix \\ref{appendix:motivation-selfsup} for additional intuition. A connection with PCA is described in Appendix \\ref{appendix:pca}. \n\n\\begin{algorithm}[tb]\n\\caption{Self-supervised representation learning and causal estimation}\n\\label{alg:self}\n\\textbf{Input}: Covariates ${\\bm{X}}_{{\\mathbb{G}}}$; outcome and treatment $Y_s$ and $A_s$ at any subset ${\\mathbb{S}}\\subset {\\mathbb{G}}$; U-net model $f_\\theta$ with $k$-dimensional outputs; and grid of candidate radii ${\\mathcal{R}}=\\{R_1,\\hdots,R_\\text{max}\\}$. \n\\begin{algorithmic}[1] %\n\\STATE For each $R \\in {\\mathcal{R}}$, obtain neural network weights $\\hat{\\theta}^R$ minimizing the self-supervised loss (equations \\ref{eq:self}) and compute $\\hat{{\\tens{Z}}}_{R, {\\mathbb{G}}}=f_{\\hat{\\theta}^R}({\\bm{X}}_{{\\mathbb{G}}})$.\n\\STATE Use a propensity\/prognostic score model to choose the optimal $r$. For example, use a logistic regression taking inputs $\\hat{{\\bm{Z}}}_{R,s}$ (and other relevant local confounders) and compute the Akaike information criterion (AIC). Then choose $\\hat{r}$ that minimizes the AIC.\n\\STATE Use standard causal inference methods (e.g., IPTW in Appendix \\ref{app:iptw}) to obtain $\\hat{\\tau}_\\text{ATE}$ adjusting for $\\hat{{\\bm{Z}}}_{\\hat{R}, s}$ and other local confounders at each unit $s$.\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{NLC and Interference}\\label{sec:interference}\n\nSection \\ref{sec:intro} briefly contrasted NLC with the related problem of interference, a topic that we expand here. We first formalize the concept of interference. To ground the discussion, we follow closely the form of interference considered in \\citet{forastiere2021identification}, which replaces SUTVA with the following neighborhood-level assumption, termed the \\textit{stable unit neighborhood treatment value assignment} (SUTNVA).\n\n\\begin{assumption}[SUTNVA] \\label{def:sutnva} (1) Consistency: there is only one version the treatment. (2) Neighborhood-level interference: for each location $s$, there is a neighborhood ${\\mathcal{N}}_s$ such that the potential outcomes depend only on the treatments at ${\\mathcal{N}}_s$. Together, these conditions imply that $Y_s({\\bm{a}})=Y_s({\\bm{a}}_{{\\mathcal{N}}_s})$ for any assignment vector ${\\bm{a}}\\in\\{0,1\\}^{|{\\mathbb{S}}|}$, and that the observed outcome is the potential outcome for the observed treatment, i.e., $Y_s=Y_s({\\tens{A}}_{{\\mathcal{N}}_s})$.\n\\end{assumption}\nThis definition of interference only considers \\emph{direct} interference \\citep{ogburn2014causal}, leaving aside indirect mechanisms such as contagion \\citep{ogburn2018challenges, shalizi2011homophily}. Investigating the role of NLC in such scenarios is left for future work. We now describe one generalization of the ATE for this type of direct interference.  The statement uses potential outcomes of the form  $Y_s(a_s=a, {\\tens{A}}_{{\\mathcal{N}}_s\\setminus \\{s\\}})$ -- a short-hand notation for the potential outcome that assigns the treatments of all the neighbors of $s$ to their observed treatments in the data.\n\n\\begin{mydef}[DATE] The direct average treatment effect (DATE) is the quantity $\n\\tau_\\text{DATE} =  |{\\mathbb{S}}|^{-1}\\textstyle{\\sum_{s\\in{\\mathbb{S}}}} \\{Y_s(a_s=1, {\\tens{A}}_{{\\mathcal{N}}_s\\setminus \\{s\\}})  -Y_s(a_s=0, {\\tens{A}}_{{\\mathcal{N}}_s\\setminus \\{s\\}}) \\}.$\n\\end{mydef}\n\nNotice that $\\tau_\\text{DATE}$ is not the only estimand of interest. For instance, the literature often considers \\emph{spill-over} effects \\citep{ogburn2018challenges}. But for this work, we only focus on $\\tau_\\text{DATE}$, leaving other estimands for future work. Now consider the question: how does NLC affect the estimation of $\\tau_\\text{DATE}$ based on \\eqref{eq:ate}? We provide a partial answer based on the following result by \\citet{forastiere2021identification}, which states two conditions under which \\eqref{eq:ate} is an unbiased estimator of $\\tau_\\text{DATE}$. \n\n\\begin{proposition}\\label{prop:interference} Assume SUTNVA.\nIf (1) ${\\tens{A}}_{{\\mathcal{N}}_s} \\perp \\!\\!\\!\\! \\perp (Y_s({\\bm{a}}))_{{\\bm{a}}\\in\\{0,1\\}^{|{\\mathcal{N}}_s|}} \\mid {\\bm{L}}_s$  and (2) $A_s \\perp \\!\\!\\!\\! \\perp A_{s'} \\mid {\\bm{L}}_s$ for all $s\\in{\\mathbb{S}}, s'\\in {\\mathcal{N}}_s$. Then\n\\eqref{eq:ate} is an unbiased estimator of $\\tau_\\text{DATE}.$\n\\end{proposition}\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=.3\\linewidth]{img\/causal_diagram_interf_nlc.png}\n    \\caption{ Interference + NLC.}\n    \\vspace*{-10pt}\n    \\label{fig:interf2}\n\\end{figure}\n\nConditions (1) and (2) correspond, respectively, to the notions of neighborhood-level ignorability and conditional independence of the neighboring treatments. It turns out that\nwhen NLC is present (the arrows from $X_{s'}$ in figure \\ref{fig:confounding-types:nlc})), conditions (1) and (2) in the proposition can be violated. To see this, consider Figure \\ref{fig:interf2} representing the co-occurrence of interference and NLC. Adjusting only for local covariates would violate neighborhood ignorability condition (1) with a spurious correlation between $Y_s$ and $A_{s'}$ (through the backdoor path $Y_s \\leftarrow {\\bm{X}}_{s'} \\to A_{s'}$). Similarly, a spurious correlation between $A_s$ and $A_{s'}$ would persist (via the the path $A_s \\leftarrow {\\bm{X}}_{s'} \\to A_{s'}$). For such cases, {\\em weather2vec}\\xspace can play an important role in satisfying (1) and (2) since, after controlling for NLC (consisting in Figure \\ref{fig:interf2} of adjusting for both ${\\bm{X}}_s$ and ${\\bm{X}}_{s'}$ and blocking the incoming arrows from neighboring covariates into one's treatments and outcomes), the residual dependencies would more closely resemble those of Figure \\ref{fig:confounding-types:interf}. In summary, adjusting for NLC with {\\em weather2vec}\\xspace can aid satisfaction of the conditional independencies required to estimate causal effects with the same estimator used to estimate the ATE absent interference. \n\n\n\n\\begin{figure*}[htb]\n\\centering\n\\includegraphics[width=0.9\\textwidth]{img\/simstudy_results.png}\n\\caption{Simulation study results. (a) the basic linear task. (b) the non-linear task. Both tasks use outcomes and treatments densely available on the grid (${\\mathbb{S}}={\\mathbb{G}}$). (c) and (d) are the same tasks respectively, but only 500 random locations are in ${\\mathbb{S}}$.}\n\\label{fig:simulation}\n\\end{figure*}\n\n\\begin{figure*}[htb]\n  \\centering\n  \\begin{subfigure}[t]{.31\\textwidth}\n    \\centering\n    \\includegraphics[width=0.99\\textwidth]{img\/explained_variance.pdf}\n   \\caption{Self-supervision, NARR data}\n   \\label{fig:app1-self-supervision} \n  \\end{subfigure}%\n  \\begin{subfigure}[t]{.36\\textwidth}\n    \\centering\n    \\includegraphics[width=0.99\\textwidth]{img\/ex1-metrics.png}\n    \\caption{Fit metrics, propensity score model}\n    \\label{fig:app1-fit-metrics}\n  \\end{subfigure}%\n  \\begin{subfigure}[t]{.27\\textwidth} \\centering\n    \\includegraphics[width=0.99\\textwidth]{img\/dapsm_estimates.png}\n    \\caption{Estimated causal effects}\n    \\label{fig:app1-effects}\n  \\end{subfigure} %\n  \\caption{Application 1: The effectiveness of catalytic devices to reduce power plant ozone emissions.\n  }\n  \\label{fig:app1}\n\\end{figure*}\n\n\\section{Simulation study}\\label{sec:simulation}\n\nWe conduct a simulation study that roughly mimics a dataset where pollution is dispersed in accordance with non-local meteorological covariates as in our applications. We briefly describe the setup here and the results. Appendix \\ref{appendix:simulation} contains a detailed explanation and additional visualizations.\n\nThe covariates (simulating wind vectors) are generated from the gradient field of a random spatial process. The treatment probability and the outcome (simulating air pollution) are non-local functions of the covariates such that areas with lower outcomes have a higher probability of treatment, with a fixed treatment effect of 0.1. Two varying factors are considered: whether ${\\mathbb{S}}$ is dense or sparse; and whether the simulated data is linear or non-linear on the covariates. We then use the self-supervised (W2V-self) and supervised (W2V-sup) variants in a propensity score model for causal estimation using IPTW (Appendix \\ref{app:iptw}). Five additional baselines are considered: no controls (Unadjusted); controlling for local covariates (Local only); controlling for local and averages of neighboring covariates (Local$+$Averages); spatial random effects (spatial RE only), and an ablation of the supervised variant adding spatial random effects (W2V-sup$+$spatial RE). A total of 10 experiments are conducted for each configuration. The results are shown in Figure \\ref{fig:simulation}. When ${\\mathbb{S}}$ is dense, W2V-sup outperforms all other methods (panels a and b), exhibiting near zero bias in the linear case and a small amount of finite-sample bias for the non-linear task. When ${\\mathbb{S}}$ is sparse (panels c and d), W2V-self gives the best results, noting further the poor performance of W2V-sup due to overfitting. Alone, spatial RE outperforms Local and Local$+$Averages, but hurts in W2V-sup$+$spatial RE.\n\n\n\n\n\\section{Applications in Air Pollution and Climate}\\label{sec:applications}\n\n\\paragraph{Application 1: Quantifying the impact of power plant emission reduction technologies}\n\n\n\nThe study aims to quantify the impact of SCR\/SNCR catalytic devices \\citep{muzio2002overview} to reduce emissions among coal-fired power plants in the U.S \\citep{DVN\/M3D2NR_2016}. Appendix \\ref{appendix:app1} provides a description of the dataset. Since air quality regulations are inherently regional and power plants are concentrated in regions with similar weather and economic demand factors, regional weather correlates with the assignment of the intervention. Further, weather patterns (such as wind vectors, precipitation and humidity) dictate regional differences in the formation and dispersion of ambient air pollution. Thus, the weather is a potential confounding factor which cannot be entirely characterized by local measurements. \ns\n\n\n\n\\begin{figure*}[htb]\n  \\centering\n  \\begin{subfigure}[t]{.3\\textwidth}\n    \\centering\n    \\includegraphics[width=0.99\\textwidth]{img\/prognostic_score.pdf}\n   \\caption{Prognostic score fit averaged over the entire grid ${\\mathbb{G}}$.}\n   \\label{fig:app2-prognostic} \n  \\end{subfigure}\\hfill%\n  \\begin{subfigure}[t]{.68\\textwidth}\n    \\centering\n    \\includegraphics[width=0.5\\textwidth]{img\/app2_yearly.png}\\hfill\n    \\includegraphics[width=0.5\\textwidth]{img\/app2_monthly.png}\n    \\caption{Detrended series at ${\\mathbb{S}}^*$ resembles power plant emissions. (\\emph{Left}) Yearly trend $\\delta_{\\textrm{year}(t)}$. (\\emph{Right}) Monthly trend $\\delta_{\\text{year}(t)} + \\gamma_{\\text{month}(t)}$}\n    \\label{fig:app2-trends}\n  \\end{subfigure} %\n  \\caption{Application 2: Meteorological detrending of $\\textrm{SO}_4$.\n  }\n  \\label{fig:app2}\n\\end{figure*}\n\n\n\n\\emph{Self-supervised features from NARR.} We downloaded monthly NARR data \\citep{mesinger2006north} containing averages of gridded atmospheric covariates across mainland U.S. for the period 2000-2014. We considered 5 covariates: temperature at 2m, relative humidity, total precipitation, and north-sound and east-west wind vector components. For each variable we also include its year-to-year average. Each grid cell covers roughly a $32\\times 32$ km area and the lattice size is $128 \\times 256$. We implemented the self-supervised {\\em weather2vec}\\xspace with a lightweight U-net of depth 2, 32 hidden units, and only one convolution per level. See Appendix \\ref{appendix:app1} for more details and schematic of the U-net architecture. To measure the quality of the encoding, Figure \\ref{fig:app1-self-supervision} shows the percentage of variance explained ($R^2$), comparing with neighbor averaging and local values. The results shows that the 32-dimensional self-supervised features provide a better reconstruction than averaging and using the local values. For instance, the 300km averages only capture 82\\% of the variance, while the self-supervised {\\em weather2vec}\\xspace features capture 95\\%. See Appendix \\ref{appendix:app2} for details on the calculation of the $R^2$ and neural network architecture.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n \n\n\n\\emph{Estimated pollution reduction.}  We evaluate different propensity score models for different neighborhood sizes of the June 2004 NARR {\\em weather2vec}\\xspace-learned features with the same logistic model and other covariates as in DAPSm, augmented with the self-supervised features. We selected the representation using features within a 300km radius on the basis of its accuracy, recall, and AIC in the propensity score model relative to other considered neighborhood sizes (Figure \\ref{fig:app1-fit-metrics}).  \nThe causal effects are then obtained by performing 1:1 nearest neighbor matching on the estimated propensity score as in DAPSm. \nFigure \\ref{fig:app1-effects} compares treatment effect estimates for different estimation procedures. Overall, standard (naive) matching using the self-supervised features is comparable to DAPSm, but without requiring the additional spatial adjustments introduced by DAPSm. The same conclusion does not hold when using local weather only\n, which (as in the most naive adjustment) provides the scientifically un-credible result that emissions reduction systems significantly {\\it increase} ozone pollution. Do notice the wide confidence intervals which are constructed using conditional linear models fit to the matched data sets \\citep{ho2007matching}. Thus, while the mean estimate shows a clear improvement, the intervals shows substantial overlap, warranting  caution.\n\n\n\n\n\\paragraph{Application 2: Meteorological detrending of sulfate}\\label{sec:detrending}\n\n\nWe investigate meteorological detrending of the U.S. sulfate ($\\textrm{SO}_4$) time series with the goal (common to the regulatory policy and atmospheric science literature) of adjusting long-term pollution trends by factoring out meteorologically-induced changes and isolating impacts of emission reduction policies \\citep{wells2021improved}. We focus on $\\textrm{SO}_4$ because it is known that its predominant source in the U.S. is $\\textrm{SO}_2$ emissions from coal-fired power plants, on which observed data are available for comparison. Thus, we hypothesize that an effectively detrended $\\textrm{SO}_4$ time series will closely resemble that of the power plant emissions. \n\n\\emph{Prognostic score.} We obtained gridded $\\textrm{SO}_4$ concentration data publicly available from the Atmospheric Composition Analysis Group \\citep{so4data, van2021monthly}, consisting of average monthly value for each raster cell in the mainland U.S. for the period of study 2000--2014. The data is aggregated into 32km-by-32km grids to match the resolution of atmospheric covariates. The model uses a U-net with quadratic loss for the (log) concentrations of $\\text{SO}_4$.\nSince the prognostic score is defined based on outcome data in the absence of treatment, we leverage the fact that the power plant emissions were relatively constant for the period 2000-2005 and using 2006 as test data -- regarding this period as absent of treatment. The model predictions, aggregated by all points in the grid is shown in Figure \\ref{fig:app2-prognostic}.\nThe difference between the red line (the prognostic score fit) and the black dotted line (the $\\textrm{SO}_4$) observations during 2000 - 2006 is a proxy for the meteorology-induced changes in the absence of treatment.\n\n\\emph{Trend estimation.} For comparability we adhere to the recommended detrending model by \\citep{wells2021improved}.  Accordingly, we specify a regression with a year and seasonal fixed-effect term. Rather than pursue an entirely new methodology for detrending, we intentionally adhere to standard best practices and merely aim to evaluate whether augmenting this approach with the {\\em weather2vec}\\xspace representation of the prognostic score offers improvement. The outcome $\\log(Y_{s,t})$ for untreated units is regressed using the predictive model\n\\begin{equation}\\label{eq:detrending}\n\\begin{aligned}\n\\mu_{s,t} &=\\alpha + \\delta_{\\text{year}(t)} + \\gamma_{\\text{month}(t)} + \\textstyle{\\sum_{j=1}^p} \\beta_{p} X_{st}^p\\\\\n \\end{aligned}\n\\end{equation}\nfor all $s\\in{\\mathbb{S}}^*$ and $t=1,\\hdots, T$; and where $\\delta_\\ell$ is the year effect for $\\ell=2000,\\hdots,2014$; $\\gamma_\\kappa$ is the seasonal (monthly) effect for $\\kappa=1,\\hdots,12$; ${\\mathbb{S}}^* \\subset {\\mathbb{S}}$ are the locations of the power plants; and $X_{st}^p$ are the controls with linear coefficients $\\beta_{s,p}$. These controls are obtained from a B-spline basis of degree 3 using: 1) local weather only, and 2) local weather plus the {\\em weather2vec}\\xspace prognostic score. The model is fitted using Bayesian inference with a Gibbs sampler. Figure \\ref{fig:app2-trends} shows the fitted (posterior median) yearly and monthly trends, which resemble the power plant emissions trends much more closely than the predicted trends from models that include local or neighborhood average weather. Notice the ``double peak'' per year in the monthly power plant emissions (owing to seasonal power demand), which is only captured by the detrended {\\em weather2vec}\\xspace series.\n\n\\section{Discussion and Future Work}\n\n\nWhile notions of NLC have been acknowledged in causal inference (most explicitly in spatial econometrics but also alluded to in literature on spatial confounding and interference), potential-outcomes formalization of NLC and flexible tools to address it are lacking. We offer such a formalization, along with a flexible representation learning approach to account for NLC with gridded covariates and treatments and outcomes measured (possibly sparsely) on the same grid.  Our proposal is most closely tailored to problems in air pollution and climate science, where key relationships may be confounded by meteorological features, and promising results from two case studies evidence the potential of {\\em weather2vec}\\xspace to improve causal analyses over those with more typical accounts of local weather. A limitation of the approach is that the learned {\\em weather2vec}\\xspace representations are not as interpretable as direct weather covariates and using them could impede transparency when incorporated in policy decisions. Future work could explore new methods for interpretability. Other extensions could include additional data domains, such as graphs and longitudinal data with high temporal resolution. The links to causal interference explored in Section \\ref{sec:interference} also offer clear directions for future work to formally account for NLC in the context of estimating causal effects with interference and spill-over.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}