Isotopic spin conservation and charge distribution in multipion production

A general reaction of the type $A+B\to C+N\pi$ 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 $\Gamma \mathrm{}(N,q,T)\mathrm{}$ 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 $T=0\mathrm{} \mathrm{and} 1$ and new bounds are obtained for $T=2\mathrm{} \mathrm{and} 3$. 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) ⊃ O(3) missing-label problem. An example of the type of bounds we obtain is for the reaction ${\pi }^{+}+p\to n+N\pi$, where $T=q=2\mathrm{}$. For $N\ge 4$ and even, we obtain $\frac{\mathrm{}(N-2\mathrm{})\mathrm{}}{\mathrm{}(6N+2)\mathrm{}}<~\Gamma <~\frac{\{7N+2\mathrm{}{\mathrm{}(2N-1)\mathrm{}}^2-8\mathrm{}{\mathrm{}(N-4\mathrm{})\mathrm{}}^{\frac{1}{2}}\}}{\{14N-2\mathrm{}{[{\mathrm{}(2N-1)\mathrm{}}^2-8\mathrm{}(N-4\mathrm{}\left)\right]}^{\frac{1}{2}}}\}$. Similar results are given for $N$ odd and also for $T=3\mathrm{}$ reactions like ${\pi }^{+}+p\to {\Delta }^{-}+N\pi$.