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In this paper, we investigate the Higgs Triplet Model with hypercharge$Y_{\Delta}=0$ (HTM0), an extension of the Standard model, caracterised by amore involved scalar spectrum consisting of two CP even Higgs $h^0, H^0$ andtwo charged Higgs bosons $H^\pm$. We first show that the parameter space ofHTM0, usually delimited by combined constraints originating from unitarity andBFB as well as experimental limits from LEP and LHC, is severely reduced whenthe modified Veltman conditions at one loop are also imposed. Then, we performan rigorous analysis of Higgs decays either when $h^0$ is the SM-like or whenthe heaviest neutral Higgs $H^0$ is identified to the observed $125$ GeV Higgsboson at LHC. In these scenarios, we perform an extensive parameter scan, inthe lower part of the scalar mass spectrum, with a particular focus on theHiggs to Higgs decay modes $H^0 \to h^0h^0, H^\pm\,H^\mp$ leading predominantlyto invisible Higgs decays. Finally, we also study the scenario where $h^0, H^0$are mass degenerate. We thus find that consistency with LHC signal strengthsfavours a light charged Higgs with a mass about $176\sim178$ GeV.  Our analysis shows that the diphoton Higgs decay mode and $H \to Z \gamma$are not always positively correlated as claimed in a previous study.Anti-correlation is rather seen in the scenario where $h$ is SM like, whilecorrelation is sensitive to the sign of the potential parameter $\lambda$ when$H$ is identified to $125$ GeV Higgs.

Probing the Higgs sector of $Y=0$ Higgs Triplet Model at LHC