Soft-Pion Production in Electron-Positron Collisions

The processes $\gamma \to {\pi }^{+}{\pi }^{-}$, $\gamma \to {\pi }^{+}{\pi }^{+}{\pi }^{-}{\pi }^{-}$, $\gamma \gamma \to {\pi }^{+}{\pi }^{-}$, and $\gamma \gamma \to {\pi }^{+}{\pi }^{+}{\pi }^{-}{\pi }^{-}$ can now be studied in $e^{+}-e^{-}$ colliding-beam experiments. Amplitudes for these processes are considered in the low-energy region where simple analyticity may be used to extrapolate from the zero-energy values determined by gauge invariance and current algebra. It is argued that in the $S$-wave two-pion state a broad $\sigma$ resonance will appear as an enhancement in $\gamma \gamma \to {\pi }^{+}{\pi }^{-}$ at an energy much lower than the $\sigma$ mass. For the process $\gamma \to {\pi }^{+}{\pi }^{+}{\pi }^{-}{\pi }^{-}$ an approximation is made (which amounts to an extreme interpretation of $\rho$ dominance) to predict a cross section of about 1.6 × ${10}^{-34\mathrm{}}$ ${\mathrm{cm}}^2$ at energies just above the $\rho$ mass. A similar approximation is used to estimate the nonresonant cross section for $\gamma \gamma \to {\pi }^{+}{\pi }^{+}{\pi }^{-}{\pi }^{-}$, in substantial accord with a recent result by Terazawa. We find this cross section to be too small [$O$(${10}^{-36\mathrm{}}$ ${\mathrm{cm}}^2$)] to be of present experimental interest.