Naturality vs perturbativity, $B_s$ physics, and LHC data in triplet extension of MSSM

In this study we investigate the phenomenological viability of the $Y=0$ Triplet Extended Supersymmetric Standard Model (TESSM) by comparing its predictions with the current Higgs data from ATLAS, CMS, and Tevatron, as well as the measured value of the $B_s\to X_s \gamma$ branching ratio. We scan numerically the parameter space for data points generating the measured particle mass spectrum and also satisfying current direct search constraints on new particles. We require all the couplings to be perturbative up to the scale $\Lambda_{\rm UV}=10^4$ TeV, by running them with newly calculated two loop beta functions, and find that TESSM retains perturbativity as long as $\lambda$, the triplet coupling to the two Higgs doublets, is smaller than 1.34 in absolute value. For $|\lambda|\gtrsim 0.8$ we show that the fine-tuning associated to each viable data point can be greatly reduced as compared to values attainable in MSSM. We also find that for perturbatively viable data points it is possible to obtain either enhancement or suppression in $h\rightarrow \gamma \gamma$ decay rate depending mostly on the relative sign between $M_2$ and $\mu_D$. Finally, we perform a fit by taking into account 58 Higgs physics observables along with $\mathcal{B}r(B_s\to X_s \gamma)$, for which we calculate the NLO prediction within TESSM. We find that, although naturality prefers a large $|\lambda|$, the experimental data disfavors it compared to the small $|\lambda|$ region, because of the low energy observable $\mathcal{B}r(B_s\to X_s \gamma)$. We notice, though, that this situation might change with the second run of LHC at 14 TeV, in case the ATLAS or CMS results confirm, with smaller uncertainty, a large enhancement in the Higgs decay channel to diphoton, given that this scenario strongly favours a large value of $|\lambda|$.