\(\tau\) vector and axial-vector spectral functions in the extended linear sigma model

The extended linear sigma model describes the vacuum phenomenology of scalar, pseudoscalar, vector and axial-vector mesons at energies $\simeq 1\text{ GeV}$. We combine the chiral $U(2)_L\times U(2)_R$ symmetry of this model with a local $SU(2)_L\times U(1)_Y$ symmetry and obtain a gauge invariant effective description for electroweak interaction of hadrons in the vacuum. Vector and axial-vector spectral functions can be described well by two intermediate resonances $\rho$ and $a_1$. They are implemented into this model as chiral partners and yield the predominant contributions to both spectral functions. However, the contributions that arise from the non-resonant decay channels of the weak charged $W$ bosons are essential for reproducing the lineshapes of the spectral functions.