Coulomb interaction within the modified Lüscher framework

Determine whether the Coulomb interaction in finite-volume QCD+QED can be treated consistently using the modified Lüscher equation derived from the modified effective-range expansion that separates long-range and short-range interactions, and ascertain whether this approach yields substantive results beyond existing QED-corrected finite-volume formulations by Beane and Savage (2014), Cai and Davoudi (2018), and Christ et al. (2022).

Background

The paper derives a modified Lüscher equation that isolates known long-range forces (e.g., one-pion exchange) from short-range interactions, improving applicability near left-hand cuts and mitigating large partial-wave mixing. Throughout, the authors deliberately exclude massless photons and defer QED to future work.

In the conclusions, they highlight an explicit uncertainty about extending this framework to Coulomb interactions. Prior studies have incorporated QED effects perturbatively or via specialized finite-volume treatments, and periodic boundary conditions lead to removal of the zero mode, rendering the effective Lagrangian non-local and potentially complicating EFT matching. The open question is whether the present long-range/short-range separation can be applied consistently to Coulomb forces and whether it provides advantages beyond existing approaches.