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

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.