Total Cross Section for Electron-Positron Annihilation and Hadronic Contribution to the Muon Magnetic Moment

A class of rigorous inequalities are derived for the integrated cross section for $e^{-}{e}^{+}$ annihilation into hadrons through one photon. They involve the $p$-wave $\pi \pi$ cross section and a slope of the electromagnetic form factor of the pion. A lower bound on the hadronic contribution to the muon magnetic moment is estimated in terms of the electromagnetic charge radius $r_{\pi }$. The bound is twice as large as the $\rho$-meson dominance value if $r_{\pi }=0.86\mathrm{}$ F as indicated by old data, but no meaningful result is obtained if one substitutes the values for $r_{\pi }$ obtained in a new experiment. The inequalities can also be used to test various predictions concerning the asymptotic behavior of the $e^{-}{e}^{+}$ annihilation cross section. A slowly decreasing behavior is favored over a rapidly decreasing one like ${\sigma }^{e^{+}{e}^{-}}\mathrm{}\left(t\right)\mathrm{}\propto t^{-3\mathrm{}}$, but more accurate data are needed to draw a definite conclusion.