Relativistic Parametrization of Resonances: theρMeson

Abstract

We construct a simple formula for a resonant scattering amplitude which employs relativistic phase space and possesses reasonable analytic properties in the total-energy variable. This formula is a direct analog of the nonrelativistic Breit-Wigner formula and contains the same number of parameters. With it, one can immediately calculate electromagnetic form factors and determine the relative contribution of each channel to the formation of the resonance. Application to the $\rho$ meson indicates that the $\rho$ resonance alone is not sufficient to explain the isovector form factor even when many channels are taken into account. It is also noted that the dominant channels for making the $\rho$ are the $N\overline{N}$ and $\pi \omega$, but not the $\pi \pi$. The coupling of the $\pi \omega$ channel to the $\rho$ produces a $\pi \pi \to \pi \omega$ cross section which has an enhancement near the $B$-meson mass due solely to the decrease of the $\rho$ pole with increasing energy and the phase space of the $\pi \omega$ channel. The value of the isovector form factor in the timelike region is also calculated. Values for the $p$-wave $\pi \pi$ phase shift are computed and compared with the nonrelativistic prediction.