Isotopic spin conservation and charge distribution in multipion production

Abstract

A general reaction of the type $A+B\ensuremath{\rightarrow}C+N\ensuremath{\pi}$ is considered in which the final $N$ pions are produced in a state with definite isospin $T$ and charge $q$. Rigorous bounds on the ratio $\ensuremath{\Gamma}(N,q,T)$ of the number of neutral to the number of charged pions are derived, following from isospin invariance and Bose statistics alone. Previously known results are recovered for $T=0 \mathrm{and} 1$ and new bounds are obtained for $T=2 \mathrm{and} 3$. The $N$-pion state with fixed $T$ is appropriately symmetrized using the representation theory of SU(3) in an O(3) basis. We can deal with arbitrary values of $T$, since we have resolved the SU(3) \ensuremath{\supset} O(3) missing-label problem. An example of the type of bounds we obtain is for the reaction ${\ensuremath{\pi}}^{+}+p\ensuremath{\rightarrow}n+N\ensuremath{\pi}$, where $T=q=2$. For $N\ensuremath{\ge}4$ and even, we obtain $\frac{(N\ensuremath{-}2)}{(6N+2)}<~\ensuremath{\Gamma}<~\frac{{7N+2{(2N\ensuremath{-}1)}^{2}\ensuremath{-}8{(N\ensuremath{-}4)}^{\frac{1}{2}}}}{{14N\ensuremath{-}2{[{(2N\ensuremath{-}1)}^{2}\ensuremath{-}8(N\ensuremath{-}4)]}^{\frac{1}{2}}}}$. Similar results are given for $N$ odd and also for $T=3$ reactions like ${\ensuremath{\pi}}^{+}+p\ensuremath{\rightarrow}{\ensuremath{\Delta}}^{\ensuremath{-}}+N\ensuremath{\pi}$.