Masses and couplings of vector mesons from the pion electromagnetic, weak, and πγ transition form factors

We analyse the pion electromagnetic, charged-current, and $\pi\gamma$ transition form factors at timelike momentum transfers $q$, $q^2=s\le 1.4$ GeV$^2$, using a dispersion approach. We discuss in detail the propagator matrix of the photon-vector-meson system and define certain reduced amplitudes, or vertex functions, describing the coupling of this system to final states. We then apply the derived analytic expressions to the analysis of the recent $e^+e^-\to \pi^+\pi^-$, $\tau^-\to \pi^-\pi^0\nu_\tau$, and $e^+e^-\to \pi^0\gamma$ data. We find the reduced amplitudes for the coupling of the photon and vector mesons to two pseudoscalars to be constant, independent of $s$, in the range considered, indicating a "freezing" of the amplitudes for $s\le 1$ GeV. The fit to the form factor data leads to the following values of the Breit-Wigner resonance masses m_{\rho^-}=775.3\pm 0.8 MeV, m_{\rho^0}=773.7\pm 0.6 MeV and m_\omega=782.43\pm 0.05 MeV, where the errors are only statistical.