Finite-volume formalism for physical processes with an electroweak loop integral

TL;DR

This paper develops a formalism to correct finite-volume effects in physical processes involving electroweak loops, enabling more accurate lattice QCD calculations of electroweak-related phenomena.

Contribution

It introduces a general approach for finite-volume corrections in processes with hadronic states dominated by one or two particles, including new methods for power-law and exponential effects.

Findings

Derived correction formulas for two-particle finite-volume effects.

Established the infinite-volume reconstruction (IVR) method.

Applicable to QED corrections and meson decay processes.

Abstract

This study investigates finite-volume effects in physical processes that involve the combination of long-range hadronic matrix elements with electroweak loop integrals. We adopt the approach of implementing the electroweak part as the infinite-volume version, which is denoted as the EW$_\infty$ method in this work. A general approach is established for correcting finite-volume effects in cases where the hadronic intermediate states are dominated by either a single particle or two particles. For the single-particle case, this work derives the infinite volume reconstruction (IVR) method from a new perspective. For the two-particle case, we provide the correction formulas for power-law finite-volume effects and unphysical terms with exponentially divergent time dependence. The finite-volume formalism developed in this study has broad applications, including the QED corrections in various processes and the two-photon exchange contribution in $K_L\to\mu^+\mu^-$ or $\eta\to\mu^+\mu^-$ decays.