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We consider flavor-changing decays of neutral Higgs bosons in the context of CP-conserving BGL model — a variant of 2HDM Type 3 model suggested by Branco, Grimus and Lavoura — in which tree-level FCNC couplings are suppressed by elements of known fermion mixing matrices. The relevant regions of parameter space compatible with experimental restrictions on the SM Higgs properties are studied. We also include current bounds on $h\to \mu \tau$ into consideration. In addition, different FCNC decay modes are analyzed for heavier Higgs states $(H/A)$ and conservative estimates for $\mathrm{\text{Br}}(A/H\to \mu \tau )$ are provided. We updated previous studies and found that it cannot be more than 30% for heavy Higgses with masses around 350 GeV.

In this paper, we present the novel analytical expressions for the bounded-from-below or the vacuum stability conditions of scalar potential for a general two-Higgs-doublet model by using the concepts of co-positivity and the gauge orbit spaces. More precisely, several sufficient conditions and necessary conditions are established for the vacuum stability of the general 2HDM potential, respectively. We also give an equivalent condition of the vacuum stability of the general 2HDM potential in theory, and then, apply it to derive the analytical necessary conditions of the general 2HDM potential. Meanwhile, the semi-positive definiteness is proved for a class of 4th-order two-dimensional complex tensor.

The Higgs condensates are proven along with the minimal versions of the hyper-symmetric 2HDM extensions when containing Supersymmetry that can play the role of oscillators for the CPC and CPV interferences, up to via ${\tau }_2$. With the uncharged Higgs inclusion (considering the uncharged Goldstone Bosons compared to the spontaneously broken SM-like Higgs), the symmetric $H_T$ and anti-symmetric $H_S$ linear combinations of an ‘$H_u$’ and a rescaled $H_d^{\prime }={(\frac{3}{2})}^{\frac{2}{3}}{H}_d$, where masses of $m_{H_d^{\prime }}\cong m_{H_{\mathrm{\text{SM}}}}\approx 125$GeV and $m_{H_d}={(\frac{2}{3})}^{\frac{2}{3}}{m}_{H_u}\approx 95$GeV, such as under the condensate’s boundary impacts. The range of the $H_d^{\prime }$s would be up to where the composed singlet and triplet can interchange, then, they approach the Custodial SUSY at a Triple point transition with a Goldstone-type order mass $m_{S_0}\approx 0.34$GeV, as well into their Dark Matter sectors’ cut-off $\mathrm{\Lambda }$, (including $m_T$) up to order 1.7TeV. That will lead to opening the way to measure the existence of Supersymmetry, indirectly at the least, through any dominance of $Z_2$-violating Higgs decays.