How to understand $\rho$ Resonance from the Quark Model and $\pi\pi$ $P$-wave phase shift

As the lightest isovector vector meson, the $\rho$ meson is an important object for investigating the structure of resonant states in strong interactions. Owing to its strong coupling to the $\pi\pi$ channel and its large decay width, the conventional constituent quark model treatment, in which it is simply regarded as a pure $q\bar q$ bound state while the hadronic-channel coupling effects are neglected, is insufficient to fully characterize its physical properties. To this end, in the present work we establish a unified framework for studying the structure and resonant properties of the $\rho$ meson by combining the quark-gluon and hadronic degrees of freedom. At the quark-gluon level, we first determine the parameters of the chiral quark model by refitting a set of narrow mesons for which open OZI-allowed strong-decay channels are absent or strongly suppressed. With these parameters fixed, the bare mass of the $\rho$ meson is obtained and used as the input for the subsequent hadronic-level analysis. At the hadronic level, based on inverse scattering theory, we construct a model including the coupling between the bare state and the $\pi\pi$ continuum, extract the $\rho_0-\pi\pi$ interaction using the $P$-wave $\pi\pi$ scattering phase-shift data, and further calculate the width of the $\rho$ meson as well as the bare-state component in the physical state. The present work also provides a generalizable analytical framework for further studies of other hadronic resonances with significant channel-coupling effects.