$\pi a_1$ Transitions and low-energy limit in the linear sigma model

The problem of obtaining the correct low-energy limit in the linear sigma model with participation of vector and axial-vector mesons is discussed. In a naive treatment of the tree approximation, for example, in $\pi\pi$ scattering, the diagrams with intermediate scalar and vector mesons may lead to double counting in the scattering lengths. For the case of the scattering amplitudes and the semileptonic decay $\tau\to\nu_\tau 3\pi$ it is shown that this problem can be solved by taking into account $\pi a_1$ transitions at the external ends of the corresponding meson vertices.