kept (in the case of anomalies, which lead to the finite result we take the first term of such expansion). This approximation is qualitatively justified by the $1/N_c$ expansion which states that meson physics in the large $N_c$ limit is described by the local vertices [@Hooft74; @Witten79]. It is also well known from the sigma model that divergent parts of radiative corrections have a strictly chiral-symmetric structure [@Eguchi78].

The amplitude corresponding to the graphs shown in Fig.\[Contact\] and Fig.\[Intermediate\] is presented in an Appendix. Here we show the result obtained after derivative expansion of quark vertices, as it was discussed above. Note also that from (\[angles\]) it follows that the angle $\theta_\pi^-$ is small, and $\theta_\pi\simeq \theta^0_\pi$. Neglecting $\theta_\pi^-$, one concludes from (\[coefficients\]) that $A_{\pi} \simeq g_{\pi}$. As a result, an amplitude takes the form $$\begin{aligned}
\label{M-without}
\mathcal{M} & = & 4mG_{F}V_{ud}\,g_\pi l^\mu e_{\mu\nu\alpha\beta}\epsilon^{\nu}(p_{f_1})p_{f_1}^\alpha p_{\pi}^\beta\left\{ I_{3}^{A_{f_{1}}} \right. \nonumber \\
&+& \frac{C_{a_1}}{g_{a_{1}}} I_{3}^{A_{f_{1}}A_{a_{1}}} \frac{s - 6m^2}{M_{a_{1}}^2 - s - i \sqrt{s}\Gamma_{a_{1}}} \nonumber\\
& + & \left. \frac{C_{a'_1}}{g_{a_{1}}} I_{3}^{A_{f_{1}}B_{a_{1}}} \frac{s - 6m^2}{M_{a'_{1}}^2 - s - i \sqrt{s}\Gamma_{a'_{1}}} \right\}\!.\end{aligned}$$ Here $G_F = 1.1663787(6) \times 10^{-11}$ MeV$^{-2}$ is the Fermi coupling constant; $V_{ud} = 0.97417\pm 0.00021$ is the Cabbibo-Kobayashi-Maskawa quark-mixing matrix element; $l_{\mu}=\bar\nu_\tau(Q')\gamma_\mu(1-\gamma_5)\tau (Q)$ 