A new determination of the $γπ\to ππ$ anomalous amplitude via $π^- e^- \to π^- e^- π^{0}$ data
We discuss the reaction $π^- e^- \to π^- e^- π^{0}$ with the purpose of obtaining information on the $γπ\to ππ$ anomalous amplitude ${\cal F}_{3π}$. 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 $σ= 2.05 $ nb at ${\cal O}(p^6)$ and $σ= 2.17 $ nb after including the dominant electromagnetic correction. Both results are in good agreement with the experimental cross section of $σ= (2.11 \pm 0.47)$ nb. On the basis of the ChPT results one would extract from the the experimental cross section as amplitudes ${\cal F}_{3π}^{(0)\rm extr} = (9.9 \pm 1.1)$ GeV$^{-3}$ and ${\cal F}_{3π}^{(0)\rm extr} = (9.6 \pm 1.1)$ GeV$^{-3}$, respectively, which have to be compared with the low-energy theorem ${\cal F}_{3π}=e/(4π^2 F_π^3)=9.72 {GeV}^{-3}$. We emphasize the need for new data to allow for a comparison of experimental and theoretical distributions and to obtain ${\cal F}_{3π}$ with smaller uncertainty.