  Estimating the contribution from axial-vector intermediate states to hadronic
light-by-light scattering requires input on their transition form factors
(TFFs). Due to the Landau-Yang theorem, any experiment sensitive to these TFFs
needs to involve at least one virtual photon, which complicates their
measurement. Phenomenologically, the situation is best for the $f_1(1285)$
resonance, for which information is available from $e^+e^-\to e^+e^- f_1$,
$f_1\to 4\pi$, $f_1\to \rho \gamma$, $f_1\to \phi \gamma$, and $f_1\to e^+e^-$.
We provide a comprehensive analysis of the $f_1$ TFFs in the framework of
vector meson dominance, including short-distance constraints, to determine to
which extent the three independent TFFs can be constrained from the available
experimental input, a prerequisite for improved calculations of the
axial-vector contribution to hadronic light-by-light scattering. In particular,
we focus on the process $f_1\to e^+e^-$, evidence for which has been reported
recently by SND for the first time, and discuss the impact that future improved
measurements will have on the determination of the $f_1$ TFFs.
