Quark loop calculation of theγ→3πform factor

The presently experimentally interesting form factor for the anomalous process $\overrightarrow{\gamma }{\pi }^{+}{\pi }^0{\pi }^{-}$ is calculated as the quark “box” amplitude where the intermediate fermion loop is the one of constituent quarks with the pseudoscalar coupling to pions. This also corresponds to the form factor, in the lowest order in pion interactions, of the $\sigma$ model and of the chiral quark model. We give the analytic expression for the form factor in terms of an expansion in the pion momenta up to the order $\mathcal{O}{(p}^8)$ relative to the soft point result, and also perform its exact numerical evaluation. We compare our predictions with those of vector meson dominance and chiral perturbation theory, as well as with the scarce data available so far.