  In this work, the $\gamma\to3\pi$ form factor is calculated within the
Dyson-Schwinger equations framework using a contact interaction model within
the so-called modified rainbow ladder truncation. The present calculation takes
into account the pseudovector component in the pion Bethe-Salpeter amplitude
(BSA) and $\pi-\pi$ scattering effects, producing a $\gamma\to3\pi$ anomaly
which is $1+6\mathcal{R}_\pi^2$ larger than the low energy prediction. Here
$\mathcal{R_\pi}$ is the relative ratio of the pseudovector and pseudoscalar
components in the pion BSA; with our parameters input, this correction raises
the $\gamma\to3\pi$ anomaly by around $10\%$. The main outcome of this work is
the unveiling of the origin of such correction, which could be a possible
explanation of the discrepancy between the existing experimental data and the
low energy prediction. Moreover, it is highlighted how the magnitude of the
anomaly is affected in effective theories that require an irremovable
ultraviolet cutoff. We find that for both the anomalous processes
$\pi\to2\gamma$ and $\gamma\to 3\pi$, the missing contribution to the anomaly
can be compensated by the additional structures related with the quark
anomalous magnetic moment.
