The $\rho$ parameter and $H^0\to \ell_i \ell_j$ in models with TeV sterile neutrinos

The presence of massive sterile neutrinos $N$ mixed with the active ones induces flavor violating processes in the charged lepton sector at the loop level. In particular, the amplitude of $H^0\to \bar{\ell}_i\ell_j$ is expected to be proportional to the product of heavy-light Yukawa couplings $y_iy_j=2 \,s_{\nu_i} s_{\nu_j}\, m_N^2/v^2$, where $s_{\nu_{i,j}}$ express the heavy-light neutrino mixings. Here we revisit these Higgs decays in the most generic extension of the neutrino sector, focusing on large values of $y_{i}$. We show that decoupling effects and a cancellation between the two dominant contributions to these processes makes the amplitude about one hundred times smaller than anticipated. We find that perturbative values of $y_{i}$ giving an acceptable contribution to the $\rho$ parameter imply ${\cal B}(H^0\to \bar{\ell}_i\ell_j)<10^{-8}$ for any lepton flavors, a rate that is not accessible at current colliders.