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Dark Matter versus h into gamma gamma and h into gamma Z with supersymmetric triplets

The Triplet extension of the MSSM (TMSSM) alleviates the little hierarchy problem and provides a significant enhancement of the loop-induced diphoton rate of the lightest CP-even Higgs h. In this paper we pursue the analysis of the TMSSM Higgs phenomenology by computing for the first time the h into Z + gamma decay. Interestingly we find that the rates of loop-induced decays are correlated and their signal strengths can rise up to 40% - 60% depending on the channel. We furthermore study the dark matter phenomenology of the TMSSM. The lightest neutralino is a good dark matter candidate in two regions. The first one is related to the Higgs and Z resonances and the LSP is mostly Bino. The second one is achieved for a mass larger than 90 GeV and the LSP behaves as the well-tempered neutralino. An advantage of the triplet contribution is that the well-tempered neutralino can be a Bino-Triplino mixture, relieving the problem of achieving M_2 ~ M_1 in unified scenarios. The dark matter constraints strongly affect the Higgs phenomenology, reducing the potential enhancements of the diphoton and of the Z + photon channels by 20% at most. These enhancements are however larger than the MSSM ones. In the near future, complementarity of dark matter direct searches and collider experiments will be crucial to probe most of the parameter space where the neutralino is the dark matter candidate.