Pomeranchuk Theorem for Partial Waves and Narrowing of Resonance Widths: TheϱMeson

Abstract

A model is proposed, within the framework of single-channel Frye-Warnock formalism, in which it is assumed that at high energies low partial waves become purely imagninary. A Pomeranchuk-type theorem then implies that the Frye-Warnock Born term must vanish asymptotically. Thus no cutoffs are needed in the integral equations. For the $P$-wave $\ensuremath{\pi}\ensuremath{\pi}$ amplitude we assume that only the nearby part of the left cut (i.e., the part up to a finite negative value ---${s}_{A}$) is given by the discontinuity due to elementary $\ensuremath{\rho}$ exchange. The rest of the left cut is assumed to be given by the asymptotic right-hand discontinuity which, because of unitarity, is positive and \ensuremath{\le}\textonehalf{}, and is smaller than the abnormally large discontinuity ($\ensuremath{\simeq}3$) given, in the earlier models, by the elementary $\ensuremath{\rho}$ exchange. For the right cut of the Frye-Warnock Born term, two cases have been considered: (a) The absorption parameter $\ensuremath{\eta}(s)$ is assumed constant beyond $s=100{{m}_{\ensuremath{\pi}}}^{2}$. (b) The $\ensuremath{\eta}(s)$ is assumed constant ($={\ensuremath{\eta}}_{d}$) between $100{{m}_{\ensuremath{\pi}}}^{2}$ and $300{{m}_{\ensuremath{\pi}}}^{2}$, and beyond $300{{m}_{\ensuremath{\pi}}}^{2}$ it is assumed to be $\ensuremath{\simeq}0.5$ as given by Regge fits with a flat Pomeranchuk trajectory. Unlike earlier models, the width of the $\ensuremath{\rho}$ can be controlled. Its value is very sensitive to $\ensuremath{\eta}$. For case (a) with $\ensuremath{\eta}=0.2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}1}$ and ${s}_{A}=225{{m}_{\ensuremath{\pi}}}^{2}$, and for case (b) with ${\ensuremath{\eta}}_{d}=0.2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ and ${s}_{A}=380{{m}_{\ensuremath{\pi}}}^{2}$, we obtain ${m}_{\ensuremath{\rho}}=760$ MeV and the $\ensuremath{\rho}$ width ${\ensuremath{\Gamma}}_{\ensuremath{\rho}}$ as small as 100 MeV.