Finite System Size Correction to NLO Scattering in $\phi^4$ Theory

TL;DR

This paper calculates next-to-leading order scattering amplitudes in massive phi^4 theory on a space with compact dimensions, using a novel regularization method, and explores implications for analytic continuation and mathematical conjectures.

Contribution

It introduces denominator regularization for NLO scattering calculations on compact spaces and derives a new equation for the Epstein zeta function's analytic continuation.

Findings

Optical Theorem is satisfied in the calculations

Derived a new equation for Epstein zeta function continuation

Generalized Hardy's conjecture on square counting functions

Abstract

We compute $2\rightarrow2$ scattering in massive $\phi^4$ 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.