Fluctuations of the charged pion number ine+e−annihilation

It is shown how to find bounds on $〈n_{\mathrm{ch}}^{}{}_{}{}^k\mathrm{}\left(N\right)\mathrm{}〉$, the average of the k th power of the number of charged pions produced in the one-quantum annihilation $e^{+}{e}^{-}\to N\pi$. The second moment, $k=2\mathrm{}$, is studied in detail, also in its dependence on information on the first moment, $k=1\mathrm{}$. The lower bound for $k=2\mathrm{}$ appears to be of an experimentally significant magnitude. The $I^G=0^{-}$ bounds are much tighter than those for $1^{+}$. It is further shown that the optimal bounds can be violated if $I^G=1^{-}$ and especially if $0^{+}$-$2^{+}$ interference plays a sizable role (as may happen in the two-photon process). A brief comment on $\overline{p}p$ annihilation is also made, and an asymptotic (large $N$) formula is given for the upper bound of the neutral-to-charged pion ratio of a neutral system, for any pure isospin.