A Feynman integral in Lifshitz-point and Lorentz-violating theories in $R^D\oplus R^m$

TL;DR

This paper computes a one-loop Feynman integral in anisotropic Lifshitz and Lorentz-violating theories, providing series and asymptotic expansions valid for generic dimensions and confirming results through numerical checks.

Contribution

It introduces a comprehensive evaluation of the integral $I_{D,m}(p,q)$ in anisotropic spaces, including series expansions, hypergeometric representations, and asymptotic analysis, extending previous special case results.

Findings

Series expansions in powers of $X$ for $I_{D,m}(p,q)$

Analytic continuation of the integral for $X \,\geq\, 1$

Asymptotic expansion in inverse powers of $X^{1/2}$

Abstract

We evaluate a one-loop, two-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 $\mathbb R^D\oplus\mathbb R^m$. Our results are valid in the whole convergence region of the integral for generic (non-integer) co-dimensions $D$ and $m$. We obtain series expansions of $I_{D,m}(p,q)$ in terms of powers of the variable $X:=4p^2/q^4$, where $p=|\bm p|$, $q=|\bm q|$, $\bm p\in\mathbb R^D$, $\bm q\in\mathbb R^m$, and in terms of generalised hypergeometric functions $_3F_2(-X)$, when $X<1$. These are subsequently analytically continued to the complementary region $X\ge 1$. 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.