Prospects for a lattice calculation of the rare decay Σ+ → pℓ+ℓ−

We present a strategy for calculating the rare decay of a Σ+(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 → 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 proton and proton-pion intermediate states. These lead to growing Euclidean time dependence and, in the case of the two-particle intermediate states, to power-like volume effects. We discuss how to treat these issues in the context of the Σ+ → pℓ+ℓ− 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 already described in the literature.