Summing Soft Pions. II. Statistical Approach to Unitarity Damping

Abstract

The effects of multiple-pion exchange in hard-hadron reactions are explored in a general chiral-invariant model, without neglect of multipion interactions or loops. It is found that the term in the $S$ matrix belonging to the ($\frac{N}{2},\frac{N}{2}$) representation of algebraic SU(2) \ifmmode\times\else\texttimes\fi{}SU(2) is damped by a factor ${D}_{N}<1$. For $N$ going to infinity through even or odd values, ${D}_{N}$ becomes proportional to ${(N+1)}^{\ensuremath{-}1}$, with a remainder which vanishes rapidly as $N\ensuremath{\rightarrow}\ensuremath{\infty}$. It is suggested that when the ultraviolet cutoff on the virtual-pion momenta is allowed to go to infinity, the damping factors ${D}_{N}$ approach zero for $N$ odd and $\frac{1}{(N+1)}$ for $N$ even. As a consequence, hard-hadron reaction amplitudes may conserve a new multiplicative quantum number.