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, $O({\rm GeV})$. Such neutrinos can contribute to the rare lepton-number-violating (LNV) decay $W^\pm \to \ell_1^\pm \ell_2^\pm (q'{\bar q})^\mp$. 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 $A_{\rm CP}$ at the LHC. We assume a value for the heavy-light neutrino mixing parameter $|B_{\ell N}|^2 = 10^{-5}$, which is allowed by the present experimental constraints, and consider $5~{\rm GeV} \le M_N \le 80~{\rm GeV}$. 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 $3\sigma$, in the range $1\% \lesssim A_{\rm CP} \lesssim 15\%$.