Finite volume formalism of 2 to 3 scattering

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In this work we consider formal, quantum field-theoretic relations that are used to extract first principles scattering predictions from quantum chromodynamics (QCD), the theory of the strong nuclear force. Specifically, this work concerns formalism that can be combined with the numerical approach of lattice QCD to calculate 2→3 scattering amplitudes of QCD bound-states called hadrons. Because of limitations of the numerical calculations, the scattering observables can only be calculated indirectly, by using the finite system size to probe the interactions. This is known as the Luscher finite-volume-formalism in the context of 2→2 scattering and has been generalized by others to more complicated systems. The formalism is considered here for 2→2 scattering in both one and three spatial dimensions, 3→3 scattering in three spatial dimensions. Working in the center of mass frame and considering identical scalar identical particles of mass m, the relation between the finite-volume energies and the 2→3 scattering K-matrices is studied numerically using Python 3.7. The relation is studied using a quantization condition derived in [Phys.Rev.D 95 (2017) 7, 074510] for a coupled two- and three-particle system in a finite volume of length L. The final section is dedicated to a numerical investigation that is new to this work, where the energy spectrum is extracted for 2→3 scattering that exhibits an avoided level crossings to illustrate the coupling of the two- and three-particle system