f^{(0)}_j (x + y)\big) P_j(\sigma, \tau)}
{P_j(\sigma, \sigma) - (f_j(x + y) - f^{(0)}_j (x + y)) 
\big( Q_j(\sigma, \sigma) - Q_j(\sigma, \tau) \big)} ~. \end{aligned}$$ It is easy to check that when $\mu = M$, $A_j(s/M^2, \eta)$ coincides with the amplitude $\tilde{A}_j(s, M^2)$ obtained in Ref. [@egt].

Eqs. (\[aleptf\], \[solutionaj\]) describe all invariant amplitudes for $e^+e^-$ -annihilation into a quark or a lepton pair in the collinear kinematics (\[tmu\], \[umu\]).

Scattering amplitudes at large values of $t$ and $u$  {#FB}
=====================================================

In this section we calculate the scattering amplitudes $A$ when the restriction of Eqs. (\[tmu\], \[umu\]) for the kinematical configurations (\[tkin\], \[ukin\]) are replaced by $$\label{tm}
s \gg M^2 \geq -t \gg \mu^2$$ and $$\label{um}
s \gg M^2 \geq -u \gg \mu^2 ~.$$

In this kinematical regions it is more convenient to study the scattering amplitudes $A$ directly, rather than using the invariant amplitudes $A_j$. In order to unify the discussion for both kinematics (\[tm\], \[um\]), let us introduce $$\label{kappat}
\kappa = -t ~,$$ when (\[tm\]) is considered and $$\label{kappau}
\kappa = -u$$ for the other case (\[um\]). Using this notation, the same parameterization $A = A(s, \mu^2, M^2, \kappa)$ holds for both kinematics (\[tm\], \[um\]). Let us 