$(P_c)^{i i'}_{k' k} ~(c = 1,..,4)$ can be taken from Ref. [@egt]: $$\begin{aligned}
\label{pr}
(P_1)_{kk'}^{ii'} = \frac12\delta_k^{i'} \delta^i_{k'},
 &&(P_2)_{kk'}^{ii'} = 
2 (t_c)_k^{i}(t_c)^{i'}_{k'}, \\ \nonumber
(P_3)_{kk'}^{ii'} = \frac{1}{2} \left[\delta^i_k \delta^{i'}_{k'}-
\delta^{i'}_{k} \delta^i_{k'}\right]~,
 &&(P_4)_{kk'}^{ii'} = \frac{1}{2} \left[\delta^i_k \delta_{k'}^{i'} +
\delta^{i'}_{k} \delta^i_{k'}\right] ~.\end{aligned}$$

According to the results of Ref. [@egt], the forward $(A_F)$ and backward $(A_B)$ amplitudes of the $e^+e^-$ annihilation into quarks are expressed through invariant amplitudes $A_j$ as follows: $$\begin{aligned}
\label{amplquark}
A_F(e^+e^- \to u\bar{u}) = (A_3 + A_4)/2,
~A_B(e^+e^- \to u\bar{u}) = A_4 , \\ \nonumber
~A_F(e^+e^- \to d\bar{d}) = (A_1 + A_2)/2, 
~A_B(e^+e^- \to d\bar{d}) = A_2~.\end{aligned}$$ and the annihilation into leptons is expressed through the leptonic invariant amplitudes very similarly: $$\begin{aligned}
\label{ampllept}
A_F(e^+e^- \to \mu^+\mu^-) = (A_1 + A_2)/2, 
~A_B(e^+e^- \to \mu^+\mu^-) = A_2 ~, \\ \nonumber
A_F(e^+e^- \to \nu_{\mu} \bar{\nu_{\mu}}) = (A_3 + A_4)/2,
~~A_B(e^+e^- \to \nu_{\mu} \bar{\nu_{\mu}}) = A_4 ~.\end{aligned}$$ We have used the general notation $A_j$ for the invariant amplitudes in Eqs. (\[amplquark\], \[ampllept\]) and we will keep using this notation until Sect. 4.

In order to calculate the amplitudes $A_j$ to all orders in the electroweak couplings in the DLA, we construct and solve some infrared evolution equations (IREE). These equations describe the evolution of 