LFV Higgs and $Z$-boson decays: leptonic CPV phases and CP asymmetries

Heavy neutral leptons are motivated by several extensions of the Standard Model and their presence induces modifications in the lepton mixing matrix, including new Dirac and Majorana CP violating phases. It has been recently shown that these phases play an important role in lepton number and lepton flavour violating decays and transitions, with a striking impact for the predicted rates of certain observables. In this work, we now consider the potential role of the heavy neutral fermions and of the new CP violating phases in Higgs and $Z$-boson lepton flavour violating decays $H,Z\to \ell_\alpha^\pm\ell_\beta^\mp$, as well as in the corresponding CP asymmetries. In order to allow for a thorough analysis, we derive the full analytical expressions of the latter observables, taking into account the effect of the (final state) charged lepton masses. A comprehensive exploration of lepton flavour violating $Z$ and Higgs decays confirms that these are very sensitive to the presence of the additional heavy neutral leptons. Moreover, the new CPV phases have a clear impact on the decay rates, leading to both destructive and constructive interferences. Interestingly, in the $\mu\tau$ sector, the $Z\to \ell_\alpha^\pm\ell_\beta^\mp$ rates are within reach of FCC-ee, with associated CP asymmetries that can potentially reach up to $20\% - 30\%$.