Lepton Flavour Violating Heavy Higgs Decays Within the nuMSSM and Their Detection at the LHC

Within the $\nu$MSSM, a Minimal Supersymmetric neutrino See-saw Model, Lepton Flavour Violating Higgs couplings are strongly enhanced at large $\tan\beta$ ($\gsim30$), which can lead to BR$(H^0/A^0 \to \tau\mu) \simeq O(10^{-4})$, for $M_{H^0/A^0}\gsim 160$ GeV. Enhancements on the production of Higgs bosons, through the gluon fusion mechanism, $gg\to H^0/A^0$, and the associated production channel $gg,q\bar q\to b\bar bH^0/A^0$, whose rates grow with $\tan\beta$, as well as the mass degeneracy that occurs between the $H^0$ and $A^0$ states in this regime, also contribute to further the possibilities to detect a heavy Higgs signal into $\tau\mu$ pairs. We show that the separation of $\tau\mu$ Higgs events from the background at the upcoming CERN Large Hadron Collider could be done for Higgs masses up to about 600 GeV for 300 fb$^{-1}$ of luminosity, for large $\tan\beta$ values. However, even with as little as 10 fb$^{-1}$ one can probe $H^0/A^0$ masses up to 400 GeV or so, if $\tan\beta=60$. Altogether, these processes then provide a new Higgs discovery mode as well as an independent test of flavour physics.