Unitarity of the infinite-volume three-particle scattering amplitude arising from a finite-volume formalism

Hansen and Sharpe [Phys. Rev. D 92, 114509 (2015)PRVDAQ1550-799810.1103/PhysRevD.92.114509] derived a relation between the scattering amplitude of three identical bosons, M3, and a real function referred to as the divergence-free K matrix and denoted Kdf,3. The result arose in the context of a relation between finite-volume energies and Kdf,3, derived to all orders in the perturbative expansion of a generic low-energy effective field theory. In this work we set aside the role of the finite volume and focus on the infinite-volume relation between Kdf,3 and M3. We show that, for any real choice of Kdf,3, M3 satisfies the three-particle unitarity constraint to all orders. Given that Kdf,3 is also free of a class of kinematic divergences, the function may provide a useful tool for parametrizing three-body scattering data. Applications include the phenomenological analysis of experimental data (where the connection to the finite volume is irrelevant) as well as calculations in lattice quantum chromodynamics (where the volume plays a key role).