Possible search for Majorana neutrinos at future lepton colliders

We discuss the process ${{\ell }^{ + }}{{\ell }^{ - }} \to N{{W}^{ \pm }}{{\ell }^{ \mp }}$, where $N$ is a heavy Majorana neutrino and $\ell = e,\mu $. Large cross sections are expected for these processes at high center-of-mass energies, which can be reached at future lepton–lepton colliders. The Monte Carlo simulation of the studied processes is produced within the framework of the seesaw type-I model, where the Majorana neutrinos (or heavy neutral leptons, HNL), are introduced in the standard leptonic sector. Recently the possibility to search for the direct HNL production was studied in the ${{\ell }^{ + }}{{\ell }^{ - }} \to N{{\nu }_{\ell }}$ process with the subsequent decay $N \to {{W}^{ \pm }}{{\ell }^{ \mp }}$. In this paper we investigate an alternative process ${{\ell }^{ + }}{{\ell }^{ - }} \to N{{W}^{ \pm }}{{\ell }^{ \mp }} \to {{W}^{ \pm }}{{W}^{ \pm }}{\kern 1pt} {{\ell }^{ \mp }}{\kern 1pt} {{\ell }^{ \mp }}$ with the lepton number violation by two units. The similar processes appear in collisions with the same-sign beams, ${{e}^{ - }}{{e}^{ - }} \to N{{W}^{ - }}{\kern 1pt} {{e}^{ - }} \to {{W}^{ - }}{\kern 1pt} {{W}^{ - }} {{e}^{ + }}{\kern 1pt} {{e}^{ - }}$ or ${{\mu }^{ + }}{{\mu }^{ + }}$ $ \to $ $N{{W}^{ + }}{\kern 1pt} {{\mu }^{ + }} \to {{W}^{ + }}{\kern 1pt} {{W}^{ + }}{\kern 1pt} {{\mu }^{ + }}{\kern 1pt} {{\mu }^{ - }}$. The cross sections of the processes under consideration are enhanced by the soft photon exchange in the t-channel. We calculate the cross sections for the signals and potential Standard Model backgrounds for the ${{e}^{ + }}{{e}^{ - }}$ beam collisions at the $1$ TeV center-of-mass energy and the ${{\mu }^{ + }}{{\mu }^{ - }}$ collisions at $3$ and $10$ TeV. Due to the diagrams with soft t-channel photons and respective interference the promptly emitted leptons are produced in the direction close to the corresponding beam. These leptons will be lost in the beam pipe or badly measured by forward detectors. However, the signal events can be well separated from backgrounds using the rest of the event containing the $WW\ell $ particles. Finally, the expected upper limits on the mixing parameters ${{\left| {{{V}_{{\ell N}}}} \right|}^{2}}$ as a function of $M(N)$ are calculated.