the interval $m_{f_1}+m_\pi\leq \sqrt{s}\leq m_\tau$, so it is too low to apply the QCD perturbation theory, but it is too large that the original chiral perturbation theory (ChPT) of QCD would be applicable [@Weinberg79; @Gasser84; @Gasser85a; @Gasser85b]. In addition, an order of magnitude of energies involved seems to indicate that not only the ground axial-vector $a_1(1260)$ state contributes to the pertinent hadronic axial-vector current. It is not excluded that the first radial excitation, $a_1(1640)$, of the ground resonance state may be also important. We have found that both resonances affect the form of the spectral function, giving an interesting interference picture.

Presently there is no clear understanding of the nature of the $a_1(1260)$ and $f_1(1285)$ mesons. In our work we hold the view that these resonances are the standard quark-antiquark bound states. Our reasoning is based on the large-$N_c$ expansion of QCD [@Hooft74; @Witten79], indicating that at $N_c=\infty$ mesons are pure $q\bar q$ states. Phenomenologically these states can be described by the local effective meson Lagrangians [@Schwinger67; @Wess67; @Weinberg68; @Gasiorowicz69]. These Lagrangians are not known from first principles, however, they basically can be constructed on the chiral symmetry grounds. Comprehensive reviews of such attempts can be found in [@Meissner88; @Bando88]. 