Dark photon manifestation in the triplet-like QED processes $\gamma +{\mathrm{\ell }}_i\to {\mathrm{\ell }}_j^{+}{\mathrm{\ell }}_j^{-}+{\mathrm{\ell }}_i$, $i\mathrm{\ne }j,$ $i=e,\mu ,$ $j=e,\mu ,\tau$

The triplet-like QED processes $\gamma + \ell_i \to \ell^+_j \ell^-_j + \ell_i$with$i\neq j,$and$i=e, \mu,$ $j=e,\mu ,\tau$ has been investigated as the reactions where a dark photon, $A^{\mathrm{\prime }}$, is produced as a virtual state with subsequent decay into a ${\mathrm{\ell }}_j^{+}{\mathrm{\ell }}_j^{-}-$-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 $A^{\mathrm{\prime }}$ in these processes is that the background to the $A^{\mathrm{\prime }}$ signal is pure QED. Concerning $A^{\mathrm{\prime }}$, 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, $A^{\mathrm{\prime }}$ can manifest itself. The contribution of $A^{\mathrm{\prime }}$ in the Borsellino diagrams is negligible since, in this case, the virtual dark photon is space-like, the $A^{\mathrm{\prime }}$ 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 ${\mathrm{\ell }}_j^{+}{\mathrm{\ell }}_j^{-}-$ 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.