On kinematical constraints in the hadrogenesis conjecture for the baryon resonance spectrum

TL;DR

This paper develops a covariant partial-wave amplitude framework for baryon resonance spectra, addressing kinematical constraints in reaction dynamics involving bosons and fermions with specific parities and spins.

Contribution

It introduces a novel projection algebra and covariant amplitudes that are free from kinematical constraints, simplifying the analysis of scattering amplitudes in the hadrogenesis conjecture.

Findings

Derived covariant partial-wave amplitudes free from Kibble constraints

Presented a decomposition into invariant functions satisfying dispersion relations

Provided analytical solutions to Bethe-Salpeter equations with short-range interactions

Abstract

We consider the reaction dynamics of bosons with negative parity and spin $0$ or $1$ and fermions with positive parity and spin $\frac{1}{2}$ or $\frac{3}{2}$. Such systems are of central importance for the computation of the baryon resonance spectrum in the hadrogenesis conjecture. Based on a chiral Lagrangian the coupled-channel partial-wave scattering amplitudes have to be computed. We study the generic properties of such amplitudes. A decomposition of the various scattering amplitudes into suitable sets of invariant functions expected to satisfy Mandelstam's dispersion-integral representation is presented. Sets are identified that are free from kinematical constraints and that can be computed efficiently in terms of a novel projection algebra. From such a representation one can deduce the analytic structure of the partial-wave amplitudes. The helicity and the conventional angular-momentum partial-wave amplitudes are kinematically constrained at the Kibble conditions. Therefore an application of a dispersion-integral representation is prohibitively cumbersome. We derive covariant partial-wave amplitudes that are free from kinematical constraints at the Kibble conditions. They correspond to specific polynomials in the 4-momenta and Dirac matrices that solve the various Bethe-Salpeter equations in the presence of short-range interactions analytically.