V0→π+π−γdecay rates

In the radiative decays of the neutral vector mesons into charged pions there are contributions from both bremsstrahlung and diagrams with internal vector mesons. We have calculated both these contributions for the processes $\rho \to {\pi }^{+}{\pi }^{-}\gamma$, $\omega \to {\pi }^{+}{\pi }^{-}\gamma$, and $\phi \to {\pi }^{+}{\pi }^{-}\gamma$. As expected, bremsstrahlung dominates $\rho \to {\pi }^{+}{\pi }^{-}\gamma$, in agreement with the data. However, for $\omega \to {\pi }^{+}{\pi }^{-}\gamma$ the vector-meson pole diagrams arising from gauging the anomalous Wess-Zumino terms are comparable to the bremsstrahlung contributions and for $\phi \to {\pi }^{+}{\pi }^{-}\gamma$ they even dominate the bremsstrahlung contributions. In these decays, both contributions are $O\left({\alpha }^3\right)$ since the nonradiative process is isospin nonconserving. However, $\omega -\rho$ mixing is quite important while $\phi -\rho$ mixing is a smaller effect. Numerical results are given for the decay width of these processes for various values of the cutoff on the minimum photon energy measured.