Prospects for a lattice calculation of the rare decay $\Sigma^+\to p\ell^+\ell^-$

TL;DR

This paper proposes a lattice QCD strategy to compute the rare decay of a Sigma+ baryon into a proton and lepton pair, addressing complex intermediate state effects and finite-volume issues.

Contribution

It introduces a detailed lattice QCD approach for calculating the Sigma+ decay, including handling of intermediate states and finite-volume corrections, which are crucial for accurate results.

Findings

Framework for relating finite-volume lattice results to physical amplitudes

Analysis of nucleon and nucleon-pion intermediate state effects

Discussion of volume effects and complex amplitude extraction

Abstract

We present a strategy for calculating the rare decay of a $\Sigma^+ (uus)$ baryon to a proton $(uud)$ and di-lepton pair using lattice QCD. To determine this observable one needs to numerically evaluate baryonic two-, three-, and four-point correlation functions related to the target process. In particular, the four-point function arises from the insertion of incoming and outgoing baryons, together with a weak Hamiltonian mediating the $s \to d$ transition and an electromagnetic current creating the outgoing leptons. As is described in previous work in other contexts, this four-point function has a highly non-trivial relation to the physical observable, due to nucleon and nucleon-pion intermediate states. These lead to growing Euclidean time dependence and, in the case of the nucleon-pion states, to power-like volume effects. We discuss how to treat these issues in the context of the $\Sigma^+\rightarrow p\ell^+\ell^-$ decay and, in particular, detail the relation between the finite-volume estimator and the physical, complex-valued amplitude. In doing so, we also make connections between various approaches in the literature.