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Dark photon manifestation in the triplet-like QED processes $γ+ \ell_i \to \ell^+_j \ell^-_j + \ell_i$, $i\neq j,$ $i=e, μ,$ $j=e, μ,τ$

The triplet-like QED processes $γ+ \ell_i \to \ell^+_j \ell^-_j + \ell_i$ with $i\neq j,$ and $i=e, μ,$ $j=e, μ,τ$ 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 $ε$ 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 $ε^2$. 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 $ε$ is estimated as a function of the dark photon mass for a given number of events.