τradiative decay and pion structure form factors

The decay mode of the $\tau$ lepton ${\tau }^{-}\to {\nu }_{\tau }{\pi }^{-}\gamma$ is calculated in terms of the axial-vector and vector structure-dependent form factors of the pion, which also describe the radiative $\pi$ decay ${\pi }^{-}\to l^{-}{\overline{\nu }}_l\gamma$. The isospin-rotated vector form factor is related (as usual) to the decay ${\pi }^0\to 2\gamma$, as well as to $e^{+}{e}^{-}\to {\pi }^0\gamma$. These reactions thus serve to probe the isospin structure of the vector form factor, as well as the momentum-transfer dependence of both form factors. The numbers expected in the $\tau$ radiative decay are calculated using vector-meson-dominance estimates for the form factors.