A Feynman integral in Lifshitz-point and Lorentz-violating theories in RD ⨁ Rm

R. B. Paris; M. A. Shpot

doi:10.1002/mma.4763

A Feynman integral in Lifshitz-point and Lorentz-violating theories in R^D ⨁ R^m

R. B. Paris, M. A. Shpot

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Abstract

We evaluate a 1-loop, 2-point, massless Feynman integral I_D,m(p,q) relevant for perturbative field theoretic calculations in strongly anisotropic d=D+m dimensional spaces given by the direct sum R^D ⨁ R^m . Our results are valid in the whole convergence region of the integral for generic (noninteger) codimensions D and m. We obtain series expansions of I_D,m(p,q) in terms of powers of the variable X:=4p²/q⁴, where p=|p|, q=|q|, p Є R^D, q Є R^m, and in terms of generalised hypergeometric functions ₃F₂(−X), when X<1. These are subsequently analytically continued to the complementary region X≥1. The asymptotic expansion in inverse powers of X^1/2 is derived. The correctness of the results is supported by agreement with previously known special cases and extensive numerical calculations.

Original language	English
Pages (from-to)	2220-2246
Number of pages	26
Journal	Mathematical Methods in the Applied Sciences
Volume	41
Issue number	5
Early online date	7 Feb 2018
DOIs	https://doi.org/10.1002/mma.4763
Publication status	Published - 30 Mar 2018

Keywords

Analytic continuation
Asymptotic expansions
Feynman integrals and graphs
Hypergeometric functions
Lifshitz point
Lorentz-violating theory

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10.1002/mma.4763

Paris_AFeynmaIntegralInLifshitz-PointAndLorentz-ViolatingTheories_Author_2018Accepted author manuscript, 552 KB

Cite this

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title = "A Feynman integral in Lifshitz-point and Lorentz-violating theories in RD ⨁ Rm",

abstract = "We evaluate a 1-loop, 2-point, massless Feynman integral ID,m(p,q) relevant for perturbative field theoretic calculations in strongly anisotropic d=D+m dimensional spaces given by the direct sum RD ⨁ Rm . Our results are valid in the whole convergence region of the integral for generic (noninteger) codimensions D and m. We obtain series expansions of ID,m(p,q) in terms of powers of the variable X:=4p2/q4, where p=|p|, q=|q|, p Є RD, q Є Rm, and in terms of generalised hypergeometric functions 3F2(−X), when X<1. These are subsequently analytically continued to the complementary region X≥1. The asymptotic expansion in inverse powers of X1/2 is derived. The correctness of the results is supported by agreement with previously known special cases and extensive numerical calculations.",

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T1 - A Feynman integral in Lifshitz-point and Lorentz-violating theories in RD ⨁ Rm

AU - Paris, R. B.

AU - Shpot, M. A.

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N2 - We evaluate a 1-loop, 2-point, massless Feynman integral ID,m(p,q) relevant for perturbative field theoretic calculations in strongly anisotropic d=D+m dimensional spaces given by the direct sum RD ⨁ Rm . Our results are valid in the whole convergence region of the integral for generic (noninteger) codimensions D and m. We obtain series expansions of ID,m(p,q) in terms of powers of the variable X:=4p2/q4, where p=|p|, q=|q|, p Є RD, q Є Rm, and in terms of generalised hypergeometric functions 3F2(−X), when X<1. These are subsequently analytically continued to the complementary region X≥1. The asymptotic expansion in inverse powers of X1/2 is derived. The correctness of the results is supported by agreement with previously known special cases and extensive numerical calculations.

AB - We evaluate a 1-loop, 2-point, massless Feynman integral ID,m(p,q) relevant for perturbative field theoretic calculations in strongly anisotropic d=D+m dimensional spaces given by the direct sum RD ⨁ Rm . Our results are valid in the whole convergence region of the integral for generic (noninteger) codimensions D and m. We obtain series expansions of ID,m(p,q) in terms of powers of the variable X:=4p2/q4, where p=|p|, q=|q|, p Є RD, q Є Rm, and in terms of generalised hypergeometric functions 3F2(−X), when X<1. These are subsequently analytically continued to the complementary region X≥1. The asymptotic expansion in inverse powers of X1/2 is derived. The correctness of the results is supported by agreement with previously known special cases and extensive numerical calculations.

U2 - 10.1002/mma.4763

DO - 10.1002/mma.4763

M3 - Article

SN - 0170-4214

VL - 41

SP - 2220

EP - 2246

JO - Mathematical Methods in the Applied Sciences

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