Finite-volume formalism for Nππ at maximal isospin

Abstract

We extend the relativistic field theoretic finite-volume formalism to Nππ scattering states at maximal isospin, I = 5/2. As in previous work using the relativistic field theory approach, we work to all orders in a generic low-energy effective theory, and determine the quantization condition that relates finite-volume energies to intermediate K matrices, and the integral equations connecting the latter to the physical scattering amplitudes. We discuss the parametrization of the K matrices, and explain in detail the new features that arise in implementing the quantization condition due to the spin of the nucleon in combination with the use of non-degenerate particles. As a concrete example, we provide a sample numerical application including the ∆ resonance in the Nπ subchannel. The extension to the I = 3/2 and 1/2 channels is more involved, due to mixing with Nπ states, and we do not provide a complete formalism for these cases. We explain why Nπ states cannot be included by treating the nucleon as a pole in p-wave Nπ scattering, an approach that has been successful in studying DD* scattering using the three-particle DDπ formalism. We additionally provide results for all isospins under the assumption of no two-to-three mixing, thereby laying the groundwork for a follow-up paper in which all Nππ → Nπ systems are fully treated. Finally, we study the singularities in Nππ amplitudes arising from Nπππ intermediate states, and find that our subthreshold cutoff functions must be modified to avoid such singularities.