Wave functions for dynamically generated resonances; the two $\Lambda(1405)$ and $\Lambda(1670)$

TL;DR

This paper develops a formalism to evaluate wave functions of dynamically generated resonances, providing insights into their structure, couplings, and decay mechanisms, with applications to the two $ ext{Lambda}(1405)$ states.

Contribution

It introduces a new formalism for calculating wave functions of resonances in coupled channel approaches, linking couplings to wave functions and decay properties.

Findings

Wave functions of the two $ ext{Lambda}(1405)$ states are evaluated.

The formalism offers practical rules for coupling and decay analysis.

Provides a method to study three-body systems with resonant clusters.

Abstract

In this work we develop a formalism to evaluate wave functions in momentum and coordinate space for the resonant states dynamically generated in a unitary coupled channel approach. The on shell approach for the scattering matrix, commonly used, is also obtained in Quantum Mechanics with a separable potential, which allows one to write wave functions in a trivial way. We develop useful relationships among the couplings of the dynamically generated resonances to the different channels and the wave functions at the origin. The formalism provides an intuitive picture of the resonances in the coupled channel approach, as bound states of one bound channel, which decays into open ones. It also provides an insight and practical rules for evaluating couplings of the resonances to external sources and how to deal with final state interaction in production processes. As an application of the formalism we evaluate the wave functions of the two $\Lambda(1405)$ states in the $\pi \Sigma$, $\bar{K} N$ and other coupled channels. It also offers a practical way to study three body systems when two of them cluster into a resonance.