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

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