Connection of "Parabolic" Mass and Width Trajectories with High-Energy Scattering

Abstract

An infinite sequence of direct-channel resonances is explicitly summed by the Sommerfeld-Watson method. This is quite different from the problem heretofore considered of summing $t$-channel resonances and requires different physical assumptions. In fact, the "parabolic" mass trajectory ${\ensuremath{\mu}}_{J}\ensuremath{\propto}J$ and a slow growth of width (on resonance) ${\ensuremath{\Gamma}}_{J}\ensuremath{\propto}J$ as $J\ensuremath{\rightarrow}\ensuremath{\infty}$ are necessary to reproduce in the Regge limit the form $\mathrm{const}\ifmmode\times\else\texttimes\fi{}{J}_{\ensuremath{\Delta}\ensuremath{\lambda}}(R\sqrt{\ensuremath{-}t})$, $R\ensuremath{\equiv}\mathrm{interaction}\mathrm{range}$, $\ensuremath{\Delta}\ensuremath{\lambda}\ensuremath{\equiv}\mathrm{total}\mathrm{helicity}\mathrm{change}$, for the imaginary part of this (nondiffractive) part of the c.m. two-body reaction amplitude, in agreement with experiment, and Harari's qualitative theory, for $\ensuremath{\pi}N$ and $\mathrm{KN}$ entrance channels. The above width ansatz is of central importance in the derivation and is also experimentally supported. Linear and parabolic mass trajectories in the direct channel are compared. Small $s$-dependent deviations from fixed crossover $\ensuremath{-}t\ensuremath{\approx}0.2$ ${(\mathrm{G}\mathrm{e}\mathrm{V}/\mathit{c})}^{2}$ and first dip $\ensuremath{-}t\ensuremath{\approx}0.6$ ${(\mathrm{G}\mathrm{e}\mathrm{V}/\mathit{c})}^{2}$ points are predicted.