Finite System Size Correction to NLO Scattering in ${\varphi }^4$ Theory

Abstract

We compute $2\to 2$ scattering in massive ${\varphi }^4$ theory on ${\mathbb{R}}^{1,m}\times T^n$ to NLO. We perform the calculations using "denominator regularization" instead of the usual dimensional regularization, which allows for asymmetric configurations of the $T^n$. We give a transparent derivation of and equation for the analytic continuation of the generalized Epstein zeta function. We show that the Optical Theorem is satisfied and generalize a conjecture by Hardy on square counting functions. We comment on the implications.

Comments: 6 pages. Minor revisions to align the arXiv version with the published Phys.Rev.D Letter version