$\pi -a_{1}$ transitions). To avoid this mixing the unphysical axial-vector fields $\vec a_{1\mu}(x)$ should be redefined $$\label{pa-trans}
\vec a_{1\mu}(x)=\vec a_{1\mu}^{phys}(x)+\sqrt{\frac{2Z_\pi}{3}}\kappa m\partial_\mu\vec\pi (x),$$ where $$\begin{aligned}
Z_{\pi} = \left(1 - \frac{6m^{2}}{M^{2}_{a_{1}}}\right)^{-1} \approx 1.45. \end{aligned}$$ A dimensional parameter $\kappa$ is fixed by requiring that the free meson Lagrangian does not contain the unphysical $\vec\pi-\vec a_{1\mu}$ transitions; it gives $\kappa =3/M_{a_1}^2$, where $M_{a_1}$ is a mass of the $a_1(1260)$ meson, $M_{a_1}=1230\pm 40\,\mbox{MeV}$. The replacement (\[pa-trans\]) does not affect the Green function of the axial-vector field, but it affects the kinetic term of a pion free Lagrangian. Consequently, the pion field wave-function is additionally renormalized by the factor $\sqrt{Z_\pi}$ in (\[pa-trans\]).

There are other mixings between $J^P=1^+$ and $J^P=0^-$ states. The $\pi' -a_1'$ mixing does not contribute to the $\tau\to f_1\pi\nu_\tau$ decay. In the following we neglect the $a_1'\to\pi$ mixing due to a heavy mass of the $a_1'$ state which is associated with $a_1(1640)$ meson.

Eq. (\[coefficients\]) includes the angles, $\theta_{M}^{0}$ and $\theta_{M}$ [@Volkov97; @Volkov17]. These parameters appear due to $M\to M'$ mixing ($\theta_{M}^{0}$), and as a result of diagonalization, which aimed to avoid such a mixing ($\theta_{M}$). We arranged them in the following combinations: $A_M^0=1/\sin(2\theta_{M}^{0})$, $\theta_M^\pm = \theta_{M} \pm \theta_{M}^{0}$. The numerical values of 