Strong evidence of $\rho(1250)$ from a unitary multichannel reanalysis of elastic scattering data with crossing-symmetry constraints

Abstract

An analysis is presented of elastic $P$-wave $\pi\pi$ phase shifts and inelasticities up to 2 GeV, aimed at identifying the corresponding $J^{PC}=1^{--}$ excited $\rho$ resonances and focusing on the $\rho(1250)$ vs. $\rho(1450)$ controversy. The approach employs an improved parametrization in terms of a manifestly unitary and analytic three-channel $S$-matrix with its complex-energy pole positions. The included channels are $\pi\pi$, $\rho2\pi$, and $\rho\rho$, the latter two being effective in the sense that they mimic several experimentally observed decay modes with nearby thresholds. In an alternative fit, the $\rho2\pi$ mode is replaced by $\omega\pi$, which is also an experimentally relevant channel. The improvement with respect to prior work amounts to the enforcement of maximum crossing symmetry through once-subtracted dispersion relations called GKPY equations. A separate analysis concerns the pion electromagnetic form factor, which again demonstrates the enormous importance of guaranteeing unitarity and analyticity when dealing with very broad and highly inelastic resonances. In the case of $\rho(1250)$ vs. $\rho(1450)$, the failure to do so is shown to give rise to an error in the predicted mass of about 170 MeV. A clear picture emerges from these analyses, identifying five vector $\rho$ states below 2 GeV, viz. $\rho(770)$, $\rho(1250)$, $\rho(1450)$, $\rho(1600)$, and $\rho(1800)$, with $\rho(1250)$ being indisputably the most important excited $\rho$ resonance. The stability of the fits as well as the imposition of unitarity, analyticity, and approximate crossing symmetry in the analyses lend very strong support to these assignments. The possibly far-reaching consequences for meson spectroscopy are discussed.