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 ${\cal F}_{3\pi}$. We compare a full calculation at ${\cal O}(p^6)$ 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 ${\cal O}(p^6)$ 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, which have to be compared with the low-energy theorem ${\cal F}_{3\pi}=e/(4\pi^2 F_\pi^3)=9.72 {GeV}^{-3}$. We emphasize the need for new data to allow for a comparison of experimental and theoretical distributions and to obtain ${\cal F}_{3\pi}$ with smaller uncertainty.