Systematics of high temperature perturbation theory: The two-loop electron self-energy in QED

Emil Mottola, Z. Szép

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Abstract

In order to investigate the systematics of the loop expansion in high temperature gauge theories beyond the leading order hard thermal loop (HTL) approximation, we calculate the two-loop electron proper self-energy "Sigma; in high temperature QED. The two-loop bubble diagram of "Sigma; contains a linear infrared divergence. Even if regulated with a nonzero photon mass M of order of the Debye mass, this infrared sensitivity implies that the two-loop self-energy contributes terms to the fermion dispersion relation that are comparable to or even larger than the next-to-leading order (NLO) contributions of the one-loop "Sigma;. Additional evidence for the necessity of a systematic restructuring of the loop expansion comes from the explicit gauge-parameter dependence of the fermion damping rate at both one and two loops. The leading terms in the high temperature expansion of the two-loop self-energy for all topologies arise from an explicit hard-soft factorization pattern, in which one of the loop integrals is hard (p"lsim;T), nested inside a second loop integral which is soft (0"le;"lsim;T for real parts; p"lsim;eT for imaginary parts). There are no hard-hard contributions to the two-loop "Sigma; at leading order at high T. Provided the same factorization pattern holds for arbitrary " loops, the NLO high temperature contributions to the electron self-energy come from "-1 hard loops factorized with one soft loop integral. This hard-soft pattern is a necessary condition for the resummation over " to coincide with the one-loop self-energy calculated with HTL dressed propagators and vertices, and to yield the complete NLO correction to "Sigma; at scales "sim;eT, which is both infrared finite and gauge invariant. We employ spectral representations and the Gaudin method for evaluating finite temperature Matsubara sums, which facilitates the analysis of multiloop diagrams at high T. "copy; 2010 The American Physical Society.

Original languageEnglish
Article number025014
JournalPhysical Review D - Particles, Fields, Gravitation and Cosmology
Volume81
Issue number2
DOIs
Publication statusPublished - Jan 21 2010

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perturbation theory
electrons
energy
factorization
expansion
fermions
diagrams
gauge theory
divergence
apexes
bubbles
topology
damping

ASJC Scopus subject areas

  • Nuclear and High Energy Physics

Cite this

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title = "Systematics of high temperature perturbation theory: The two-loop electron self-energy in QED",
abstract = "In order to investigate the systematics of the loop expansion in high temperature gauge theories beyond the leading order hard thermal loop (HTL) approximation, we calculate the two-loop electron proper self-energy {"}Sigma; in high temperature QED. The two-loop bubble diagram of {"}Sigma; contains a linear infrared divergence. Even if regulated with a nonzero photon mass M of order of the Debye mass, this infrared sensitivity implies that the two-loop self-energy contributes terms to the fermion dispersion relation that are comparable to or even larger than the next-to-leading order (NLO) contributions of the one-loop {"}Sigma;. Additional evidence for the necessity of a systematic restructuring of the loop expansion comes from the explicit gauge-parameter dependence of the fermion damping rate at both one and two loops. The leading terms in the high temperature expansion of the two-loop self-energy for all topologies arise from an explicit hard-soft factorization pattern, in which one of the loop integrals is hard (p{"}lsim;T), nested inside a second loop integral which is soft (0{"}le;{"}lsim;T for real parts; p{"}lsim;eT for imaginary parts). There are no hard-hard contributions to the two-loop {"}Sigma; at leading order at high T. Provided the same factorization pattern holds for arbitrary {"} loops, the NLO high temperature contributions to the electron self-energy come from {"}-1 hard loops factorized with one soft loop integral. This hard-soft pattern is a necessary condition for the resummation over {"} to coincide with the one-loop self-energy calculated with HTL dressed propagators and vertices, and to yield the complete NLO correction to {"}Sigma; at scales {"}sim;eT, which is both infrared finite and gauge invariant. We employ spectral representations and the Gaudin method for evaluating finite temperature Matsubara sums, which facilitates the analysis of multiloop diagrams at high T. {"}copy; 2010 The American Physical Society.",
author = "Emil Mottola and Z. Sz{\'e}p",
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T2 - The two-loop electron self-energy in QED

AU - Mottola, Emil

AU - Szép, Z.

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N2 - In order to investigate the systematics of the loop expansion in high temperature gauge theories beyond the leading order hard thermal loop (HTL) approximation, we calculate the two-loop electron proper self-energy "Sigma; in high temperature QED. The two-loop bubble diagram of "Sigma; contains a linear infrared divergence. Even if regulated with a nonzero photon mass M of order of the Debye mass, this infrared sensitivity implies that the two-loop self-energy contributes terms to the fermion dispersion relation that are comparable to or even larger than the next-to-leading order (NLO) contributions of the one-loop "Sigma;. Additional evidence for the necessity of a systematic restructuring of the loop expansion comes from the explicit gauge-parameter dependence of the fermion damping rate at both one and two loops. The leading terms in the high temperature expansion of the two-loop self-energy for all topologies arise from an explicit hard-soft factorization pattern, in which one of the loop integrals is hard (p"lsim;T), nested inside a second loop integral which is soft (0"le;"lsim;T for real parts; p"lsim;eT for imaginary parts). There are no hard-hard contributions to the two-loop "Sigma; at leading order at high T. Provided the same factorization pattern holds for arbitrary " loops, the NLO high temperature contributions to the electron self-energy come from "-1 hard loops factorized with one soft loop integral. This hard-soft pattern is a necessary condition for the resummation over " to coincide with the one-loop self-energy calculated with HTL dressed propagators and vertices, and to yield the complete NLO correction to "Sigma; at scales "sim;eT, which is both infrared finite and gauge invariant. We employ spectral representations and the Gaudin method for evaluating finite temperature Matsubara sums, which facilitates the analysis of multiloop diagrams at high T. "copy; 2010 The American Physical Society.

AB - In order to investigate the systematics of the loop expansion in high temperature gauge theories beyond the leading order hard thermal loop (HTL) approximation, we calculate the two-loop electron proper self-energy "Sigma; in high temperature QED. The two-loop bubble diagram of "Sigma; contains a linear infrared divergence. Even if regulated with a nonzero photon mass M of order of the Debye mass, this infrared sensitivity implies that the two-loop self-energy contributes terms to the fermion dispersion relation that are comparable to or even larger than the next-to-leading order (NLO) contributions of the one-loop "Sigma;. Additional evidence for the necessity of a systematic restructuring of the loop expansion comes from the explicit gauge-parameter dependence of the fermion damping rate at both one and two loops. The leading terms in the high temperature expansion of the two-loop self-energy for all topologies arise from an explicit hard-soft factorization pattern, in which one of the loop integrals is hard (p"lsim;T), nested inside a second loop integral which is soft (0"le;"lsim;T for real parts; p"lsim;eT for imaginary parts). There are no hard-hard contributions to the two-loop "Sigma; at leading order at high T. Provided the same factorization pattern holds for arbitrary " loops, the NLO high temperature contributions to the electron self-energy come from "-1 hard loops factorized with one soft loop integral. This hard-soft pattern is a necessary condition for the resummation over " to coincide with the one-loop self-energy calculated with HTL dressed propagators and vertices, and to yield the complete NLO correction to "Sigma; at scales "sim;eT, which is both infrared finite and gauge invariant. We employ spectral representations and the Gaudin method for evaluating finite temperature Matsubara sums, which facilitates the analysis of multiloop diagrams at high T. "copy; 2010 The American Physical Society.

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