Finite System Size Correction to the Effective Coupling in $\phi^4$ Scattering

We compute and explore numerically the finite system size correction to NLO $2\to2$ scattering in massive scalar $\phi^4$ theory. The derivation uses "denominator regularization" (instead of the usual dimensional regularization) on a spacetime with spatial directions compactified to a torus, with characteristic lengths not necessarily of equal size. We determine a useful analytic continuation of the generalized Epstein zeta function to isolate the usual UV divergence. Self-consistently, the renormalized finite system size correction reduces to zero as the system size goes to infinity and, further, satisfies the optical theorem. One of our checks of unitarity leads to a generalization of a number theoretic result from Hardy and Ramanujan. Precise numerical exploration of the finite system size correction to the amplitude and coupling when two spatial dimensions are finite requires the exploitation of the analytic structure of the finite system size result via a dispersion relation. We find that the finite system size scattering amplitude exhibits "geometric" bound states. Even away from these bound states, the finite system size correction to the effective coupling can be large.