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

R. B. Paris, M. A. Shpot

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    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
    Issue number5
    Early online date7 Feb 2018
    Publication statusPublished - 30 Mar 2018


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