Phenomenology of relativistic $$\mathbf {3}\rightarrow \mathbf {3}$$ reaction amplitudes within the isobar approximation

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 3 → 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 2 → 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 3 → 3 scattering amplitudes in lattice QCD.

Three-body scattering: ladders and resonances

“…(16). Using the terminology introduced in this section, we now compare several models that are currently discussed in the literature [12,13,48]. The present model, as well as the one from Ref.…”

“…[12] and Ref. [13] explore a similar construction: the kernel function B includes the one-pion-exchange and the contact term. A difference between these models can be found in the lower integration limit in Eq.…”

“…The value 4m 2 π is employed in Ref. [13], the lower limit is set to −∞ in [12] (see the detailed discussion in [13]). Treating the long-range exchanges as a separate ladder term allowed us to focus on the short-range physics (i.e.…”

Equivalence of three-particle scattering formalisms

“…The first approach is referred to as the B-matrix representation, and was studied in Refs. [25,26]. Motivated by unitarity of the S matrix, the B matrix refers to a kernel in a linear integral equation for an elastic 3 → 3 connected amplitude.…”

“…Aspects of its analytic properties were studied in Ref. [26], showing how, besides the unitarity branch point, there are other singularities near the physical region generated by the one-particle exchange, e.g., triangle singularities. Applying the B-matrix formalism in a finite volume leads to the quantization condition of Ref.…”