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

R. B. Paris, M. A. Shpot

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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.
Original languageEnglish
Pages (from-to)2220-2246
Number of pages26
JournalMathematical Methods in the Applied Sciences
Volume41
Issue number5
Early online date7 Feb 2018
DOIs
Publication statusPublished - 30 Mar 2018

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Feynman Integrals
D-space
Series Expansion
Direct Sum
Numerical Calculation
Asymptotic Expansion
Correctness
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Paris, R. B. ; Shpot, M. A. / A Feynman integral in Lifshitz-point and Lorentz-violating theories in RD ⨁ Rm. In: Mathematical Methods in the Applied Sciences. 2018 ; Vol. 41, No. 5. pp. 2220-2246.
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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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Paris, RB & Shpot, MA 2018, 'A Feynman integral in Lifshitz-point and Lorentz-violating theories in RD ⨁ Rm', Mathematical Methods in the Applied Sciences, vol. 41, no. 5, pp. 2220-2246. https://doi.org/10.1002/mma.4763

A Feynman integral in Lifshitz-point and Lorentz-violating theories in RD ⨁ Rm. / Paris, R. B.; Shpot, M. A.

In: Mathematical Methods in the Applied Sciences, Vol. 41, No. 5, 30.03.2018, p. 2220-2246.

Research output: Contribution to journal › Article

TY - JOUR

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

AU - Paris, R. B.

AU - Shpot, M. A.

PY - 2018/3/30

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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

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VL - 41

SP - 2220

EP - 2246

JO - Mathematical Methods in the Applied Sciences

JF - Mathematical Methods in the Applied Sciences

SN - 0170-4214

IS - 5

ER -

Paris RB, Shpot MA. A Feynman integral in Lifshitz-point and Lorentz-violating theories in RD ⨁ Rm. Mathematical Methods in the Applied Sciences. 2018 Mar 30;41(5):2220-2246. https://doi.org/10.1002/mma.4763

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