$\tau$ Vector Spectral Function from a Chirally Invariant Hadronic Model with Weak Interaction

The linear sigma model we present here describes the vacuum phenomenology of the $N_F=2$ scalar, pseudoscalar, vector and axial-vector mesons at energies $\simeq 1\text{ GeV}$. Together with a local $SU(2)_L\times U(1)_Y$ symmetry obtained from a local transformation of the fields we obtain a gauge invariant effective description of electroweak interaction with hadrons in the vacuum. We show that the vector channel is described well and that the contributions from the direct decay of $W^- \rightarrow \pi^- \pi^0$ are, although quantitatively small, necessary to reproduce the line shape of the vector spectral function.