Partial-wave projection of the one-particle exchange in three-body scattering amplitudes

As the study of three-hadron physics from lattice QCD matures, it is necessary to develop proper analysis tools in order to reliably study a variety of phenomena, including resonance spectroscopy and nuclear structure. Reconstructing the three-particle scattering amplitude requires solving integral equations, which can be written in terms of data-constrained dynamical functions and physical on-shell quantities. The driving term in these equations is the so-called one-particle exchange, which leads to a kinematic divergence for particles on-mass-shell. A vital component in defining three-particle amplitudes with definite parity and total angular momentum, which are used in spectroscopic studies, is to project the one-particle exchange into definite partial waves. We present a general procedure to construct exact analytic partial wave projections of the one-particle exchange contribution for any system composed of three spinless hadrons. Our result allows one full control over the analytic structure of the projection, which we explore for some low-lying partial waves with applications to three pions.