Dark photon manifestation in the triplet-like QED processes $gamma + ell_i to ell^+_j ell^-_j + ell_i$, $i eq j,$ $i=e, mu,$ $j=e, mu,tau$

The triplet-like QED processes $gamma + ell_i to ell^+_j ell^-_j + ell_i$ with $i eq j,$ and $i=e, mu,$ $j=e, mu,tau$ has been investigated as the reactions where a dark photon, $A$, is produced as a virtual state with subsequent decay into a $ell^+_j ell^-_j-$-pair. This effect arises due to the so-called kinetic mixing and is characterized by the small parameter $epsilon$ describing the coupling strength relative to the electric charge $e$. The main advantage of searching $A$ in these processes is that the background to the $A$ signal is pure QED. Concerning $A$, we consider its contribution in the Compton-type diagrams only since, in this case, the virtual dark photon has time-like nature and its propagator has the Breit-Wigner form. So, near the resonance, $A$ can manifest itself. The contribution of $A$ in the Borsellino diagrams is negligible since, in this case, the virtual dark photon is space-like, the $A$ propagator does not peak and the effect is proportional, at least, to $epsilon^2$. We calculate the distributions over the invariant mass of the produced $ell^+_j ell^-_j-$ pair and search for the kinematical region where the Compton-type diagrams contribution is not suppressed with respect to the Borsellino ones. The value of the parameter $epsilon$ is estimated as a function of the dark photon mass for a given number of events.