CP violation in rare lepton-number-violating $W$ decays at the LHC

Some models of leptogenesis involve a quasi-degenerate pair of heavy neutrinos N$_{1,2}$ whose masses can be small, O(GeV). Such neutrinos can contribute to the rare lepton-number-violating (LNV) decay $ {W}^{\pm}\to {\mathrm{\ell}}_1^{\pm }{\mathrm{\ell}}_2^{\pm }{\left({q}^{\prime}\overline{q}\right)}^{\mp } $. If both N$_{1}$ and N$_{2}$ contribute, there can be a CP-violating rate difference between the LNV decay of a W$^{−}$ and its CP-conjugate decay. In this paper, we examine the prospects for measuring such a CP asymmetry ACP at the LHC. We assume a value for the heavy-light neutrino mixing parameter |B$_{ℓN}$|$^{2}$ = 10$^{−5}$, which is allowed by the present experimental constraints, and consider 5 GeV ≤ M$_{N}$≤ 80 GeV. We consider three versions of the LHC — HL-LHC, HE-LHC, FCC-hh — and show that small values of the CP asymmetry can be measured at 3σ, in the range 1% ≲ A$_{CP}$ ≲ 15%.