A new determination of the $\gamma \pi \to \pi \pi$ anomalous amplitude via ${\pi }^{-}{e}^{-}\to {\pi }^{-}{e}^{-}{\pi }^0$ data

We discuss the reaction ${\pi }^{-}{e}^{-}\to {\pi }^{-}{e}^{-}{\pi }^0$ with the purpose of obtaining information on the $\gamma \pi \to \pi \pi$ anomalous amplitude ${\cal F}{3\pi}$\mathrm{.}Wecompareafullcalculationat${\cal O}(p^6)$ in chiral perturbation theory and various phenomenological predictions with the existing data of Amendolia {\em et al}. By integrating our theory results using Monte Carlo techniques we obtain $\sigma =2.05$ nb at $\mathcal{O}(p^6)$ and $\sigma= 2.17 $ nb after including the dominant electromagnetic correction. Both results are in good agreement with the experimental cross section of $\sigma= (2.11 \pm 0.47)$ nb. On the basis of the ChPT results one would extract from the the experimental cross section as amplitudes ${\cal F}{3\pi}^{(0)\rm extr} = (9.9 \pm 1.1)$GeV$^{-3}$and${\cal F}{3\pi}^{(0)\rm extr} = (9.6 \pm 1.1)$GeV$^{-3}$,respectively,whichhavetobecomparedwiththelow-energytheorem${\cal F}{3\pi}=e/(4\pi^2 F_\pi^3)=9.72 {GeV}^{-3}$. We emphasize the need for new data to allow for a comparison of experimental and theoretical distributions and to obtain ${\mathcal{F}}_{3\pi }$ with smaller uncertainty.