when $\omega$, $\eta$ are replaced by new variables $$\label{xy}
x = \omega/\lambda_j,~~~y = \lambda_j \eta~,$$ with $\lambda_j = \sqrt{ b_j^{(\gamma)}/(8\pi^2)}$. Changing to the new variables, we arrive again at the Riccati equation: $$\label{riccatij}
\frac{\partial F_j}{\partial \tau} = 
\big( \sigma - \tau \big) F_j -2q_j f^{(0)}_j F_j - q_j F^2_j ~,$$ where $\sigma = (x + y)/2,~~ \tau = (x - y)/2$ and $q_j = c_j /\lambda_j$.

The general solution to Eqs. (\[riccatij\]) is $$\label{generalj}
F_j = \frac{P_j(\sigma, \tau)}{ C(\sigma) + q_j Q_j(\sigma, \tau)}~,$$ where $C(\sigma)$ should be specified, $$\label{P}
P_j(\sigma, \tau) = \exp \Big( \sigma\tau - \tau^2/2 
 - 2q_j \int_{\sigma}^{\sigma + \tau} d \zeta f_j^{(0)}(\zeta) \Big)$$ and $$\label{Q}
Q_j (\sigma, \tau) = 
\int_{\sigma}^{\sigma + \tau} d \zeta P_j(\sigma, \zeta) ~.$$

The QED amplitudes $f^{(0)}_j$ can be obtained from the known expressions for the backward, $f^{(0)}_B$ and forward $f^{(0)}_j$ QED scattering amplitudes: $$\begin{aligned}
\label{fzerob}
 f^{(0)}_B(x) = (4\pi\alpha e_q/ p_B^{(0)})
d \ln (e^{x^2/4} D_{p_B^{(0)}}
(x)) /d x ~, \end{aligned}$$ where $D_p$ are the Parabolic cylinder functions with $ p_B^{(0)} = -2e_q/(1 + e_q)^2$ and $e_q = 1$ for the annihilation into muons, $e_q = 1/3 ~(2/3)$ for the annihilation into $d ~(u)$- quarks. Similarly, the QED forward scattering amplitudes for the annihilation 