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

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 $p=|\mathbf{p}|$, $q=|\mathbf{q}|$, $\mathbf{p}\in {\mathbb{R}}^D$, $\mathbf{q}\in {\mathbb{R}}^m$, and in terms of generalised hypergeometric functions ${}_3{F}_2(-X)$, 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.

33 pages, 4 figures

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Cited by in corpus (3)

• Indications of isotropic Lifshitz points in four dimensions
• Isotropic Lifshitz Scaling in four dimensions
• Ultraviolet properties of Lifshitz-type scalar field theories