$ \rho\to \pi\pi $ Hadronic Decay in the Nambu-Jona-Lasinio Model: Mass-Width Interplay and Beyond-RPA Corrections

TL;DR

This paper introduces a new method using Green's functions and dispersion relations to analyze the decay of the $ ho$ meson into pions within the NJL model, effectively linking resonance mass and decay width.

Contribution

It develops a framework that incorporates decay processes into the NJL model, addressing limitations of traditional four-quark interaction descriptions for unstable particles.

Findings

Calculated $ ho$ meson mass and width agree with experimental data

Demonstrated the importance of decay width corrections on resonance mass

Provided insights into the mass-width relationship in hadron physics

Abstract

We present a novel framework for analyzing unstable composite particles using Green's functions and dispersion relations. As an illustrative example, we explore the $\rho$ vector meson decay process $\rho\to\pi\pi$ within the Nambu -- Jona - Lasinio (NJL) model. Our approach addresses a key limitation of the four-quark interaction description, which adequately describes two-quark bound states but fails to describe decay processes. The Bethe-Salpeter(BS) wave function of the $\rho$ meson exhibits time evolution that leads to the physical mass $M$ incorporating a correction $\Delta M $. This correction depends on the decay width $\Gamma(M)$. This work provides crucial insights into the dynamical relationship between resonance masses and their decay properties, addressing a long-standing challenge in hadron physics. The calculated mass and width are in good agreement with the experimental values, demonstrating the effectiveness of this approach for studying unstable hadronic systems beyond conventional bound-state approximations.