Abstract
In previous work, we have developed a relativistic, model-independent three-particle quantization condition, but only under the assumption that no poles are present in the two-particle K matrices that appear as scattering subprocesses. Here we lift this restriction, by deriving the quantization condition for identical scalar particles with a G-parity symmetry, in the case that the two-particle K matrix has a pole in the kinematic regime of interest. As in earlier work, our result involves intermediate infinite-volume quantities with no direct physical interpretation, and we show how these are related to the physical three-to-three scattering amplitude by integral equations. This work opens the door to study processes such as $a_2 \to \rho \pi \to \pi \pi \pi$, in which the $\rho$ is rigorously treated as a resonance state.
Highlights
Studies of hadronic resonances using lattice QCD (LQCD) have progressed rapidly in recent years.1 The present frontier of this effort involves resonances that have significant branching ratios into channels with three particles
In order to keep track of these singularities, we find it convenient to express the problem in terms of two effective channels: one containing the physical three-particle state, and a second built from a particle and a pseudoparticle arising from the K2 pole, which we refer to as the “ρπ channel.”
III we present the derivation of the quantization condition, with technical details given in Appendix B
Summary
Studies of hadronic resonances using lattice QCD (LQCD) have progressed rapidly in recent years. The present frontier of this effort involves resonances that have significant branching ratios into channels with three (or more) particles. In the case of a single channel of identical scalar particles, the relation between finite-volume energies and the scattering amplitude was first derived by Lüscher [18,19]. For resonances with threeparticle decay channels, a further step is required, in which intermediate infinite-volume quantities are related to the scattering amplitudes. [7], and it is the purpose of the present paper to lift the second restriction, i.e., to allow arbitrary interactions in the twoparticle subsystems This removes the last major theoretical obstacle to general implementation of the formalism. One would have to develop a framework to address finite-volume effects associated with all possible scenarios With these considerations in mind, we find it preferable to work with K2 and properly treat its poles in the kinematic window of interest. In Appendix A we demonstrate these two results using constraints from unitarity and all-orders perturbation theory
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