Lepton flavour violating semileptonic tau decays in constrained MSSM-seesaw scenarios

In this work we study the Lepton Flavour Violating semileptonic $tau$ decays: 1) $tau to mu PP$ with $PP= pi^+pi^-, pi^0pi^0, K^+K^-, K^0 {bar K}^0$; 2) $tau to mu P$ with $P=pi^0, eta, eta$ and 3) $tau to mu V$ with $V = rho^0, phi$. We work within the context of two constrained MSSM scenarios: the CMSSM and NUHM. A full one-(SUSY)loop computation is presented and the importance of the various contributions, the $gamma$-, $Z$-, and Higgs bosons mediated ones are analysed. The hadronization of quark bilinears is performed within the chiral framework. Some discrepancies in the predicted rates for BR($tau to mu eta$), BR($tau to mu eta$) and BR($tau to mu K^+K^-$) are found with respect to previous estimates, which will be commented here. These three channels will be shown to be the most competitive ones to test simultaneously the Lepton Flavour Violation and the Higgs sector. We further present here a set of approximate formulas for all the semileptonic channels which we believe can be useful for further comparison with present and future data.