Abstract
Further progress in hadron spectroscopy necessitates the phenomenological description of three particle reactions. We consider the isobar approximation, where the connected part of the mathbf {3}rightarrow mathbf {3} amplitude is first expressed as a sum over initial and final pairs, and then expanded into a truncated partial wave series. The resulting unitarity equation is automatically fulfilled by the B-matrix solution, which is an integral equation for the partial wave amplitudes, analogous to the K-matrix parameterization used to describe mathbf {2}rightarrow mathbf {2} amplitudes. We study the one particle exchange and how its analytic structure impacts rescattering solutions such as the triangle diagram. The analytic structure is compared to other parameterizations discussed in the literature. We briefly discuss the analogies with recent formalisms for extracting mathbf {3}rightarrow mathbf {3} scattering amplitudes in lattice QCD.
Highlights
Modern high-energy experiments are accumulating high quality data on three-hadron final states, that are expected to be the main decay channels of several poorly known or missing resonances
The B-matrix parameterization explicitly includes the one pion exchange as a long-range contribution required by unitarity
The singularities of the one particle exchange (OPE) directly impact the analytic structure of the B-matrix kernels, and are discussed explicitly for the triangle-like diagram
Summary
Modern high-energy experiments are accumulating high quality data on three-hadron final states, that are expected to be the main decay channels of several poorly known or missing resonances. These include, for example, the enigmatic a1, ω2, and the exotic π1 resonances that can be studied in peripheral production at COMPASS, GlueX, and CLAS12 [1,2,3,4,5,6,7]. In addition to conventional hadrons, many of the exotic X Y Z and pentaquark states observed in the heavy quarkonium sector [8,9,10], are found in three particle final states Many of these newly observed or anticipated states lie close to thresholds of their decay products. We present the 3 → 3 reaction amplitudes in a way that can be directly translated to the finite volume
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