Supersymmetric custodial Higgs triplets and the breaking of universality

Higgs triplet models are known to have difficulties obtaining agreement with electroweak precision data and in particular constraints on the $\rho$ parameter. Either a global $SU(2{)}_L\otimes SU(2{)}_R$ symmetry has to be imposed on the scalar potential at the electroweak scale, as done in the well-known Georgi-Machacek (GM) model, or the triplet vacuum expectation values must be very small. We construct a supersymmetric model that can satisfy constraints on the $\rho$ parameter, even if these two conditions are not fulfilled. We supersymmetrize the GM model by imposing the $SU(2{)}_L\otimes SU(2{)}_R$ symmetry at a scale $\mathcal{M}$, which we argue should be at or above the messenger scale, where supersymmetry breaking is transmitted to the observable sector. We show that scales $\mathcal{M}$ well above 100 TeV and sizable contributions from the triplets to electroweak symmetry breaking can be comfortably accommodated. We discuss the main phenomenological properties of the model and demonstrate that the departure from custodial symmetry at the electroweak scale, due to radiative breaking, can show up at the LHC as a deviation in the “universal” relation for the Higgs couplings to $WW$ and $ZZ$. As a by-product of supersymmetry, we also show that one can easily obtain both large tree-level and one-loop corrections to the Higgs mass. This allows for top squarks that can be significantly lighter and with smaller mixing than those needed in the MSSM.

Figure 1

Running of $(m_{H_1}^2,m_{H_2}^2)$ (red lines from top to bottom) and $(m_{{\mathrm{\Sigma }}_0}^2,m_{{\mathrm{\Sigma }}_{+}}^2,m_{{\mathrm{\Sigma }}_{-}}^2)$ (black lines from bottom to top), normalized to their values at the scale $\mathcal{M}={10}^5\mathrm{GeV}$ for $m_{1/2}=1.2\mathrm{TeV}$ and $v_{\mathrm{\Delta }}=20\mathrm{GeV}$.

Figure 2

Regions allowed by the $T$ parameter as a function of $\mathcal{M}$ and $v_{\mathrm{\Delta }}$. The region between the solid lines corresponds to the allowed 95% C.L. interval, having fixed the parameters as in Eq. (12) and for $m_{1/2}=1$ (solid black lines), 1.1 (solid blue lines), 1.2 (solid red lines) and 1.3 (solid orange lines) TeV at the scale $\mathcal{M}$. The corresponding dashed lines are for $T=0$.

Figure 3

Contours of $\tan \beta$ (blue dashed), $\tan {\theta }_0$ (black solid), and $\tan {\theta }_1$ (dark green dotted) are shown, having fixed the parameters as in Eq. (12) and for $m_{1/2}=1.2\mathrm{TeV}$. The shaded pink region is allowed at 95% C.L. by the $T$ parameter.

Figure 4

Contours of $\lambda$, defined at the high scale $\mathcal{M}$, reproducing the observed value of the Higgs mass $\sim 125\mathrm{GeV}$ for the $SU(2{)}_L\otimes SU(2{)}_R$ symmetric parameters in Eq. (12) and $m_{1/2}=1.2\mathrm{TeV}$.

Figure 5

The solid black lines represent the region producing a Higgs mass of $125.5\pm 1.0\mathrm{GeV}$ in the $X_t/m_{\tilde{t}}-m_{\tilde{t}}$ plane, where $m_{\tilde{t}}$ is the physical mass of the lightest top squark and $X_t\equiv A_t-\mu /\tan \beta$. The shaded pink band is the region allowed by constraints on the $\rho$ parameter. We have fixed the parameters $\lambda =0.45$, $\mathcal{M}=65\mathrm{TeV}$, $m_{1/2}=1.2\mathrm{TeV}$, $v_{\mathrm{\Delta }}=10\mathrm{GeV}$ while the rest are given in Eq. (12), except we now allow $m_0\in [500,1000]\mathrm{GeV}$ and $A_0\in [-250,500]$. We do not explicitly show the region favored by the MSSM since it arises only at much heavier top squark masses ($m_{\tilde{t}}\gtrsim 6\mathrm{TeV}$ [44, 45, 46, 47]).

Figure 6

Contours of $r_{\mathcal{H}WW}$ (dark green dotted), $r_{\mathcal{H}ZZ}$ (blue dashed), and $r_{\mathcal{H}bb}$ (black solid) in the $(v_{\mathrm{\Delta }},\mathcal{M})$ plane for the values of the parameters given in Eq. (12) and $m_{1/2}=1.2\mathrm{TeV}$.

Figure 7

Deviation from the universal condition ${\lambda }_{WZ}=1$ along the $2v_{\varphi }^2=v_{\chi }^2+v_{\psi }^2$ direction (or $\tan {\theta }_0=1$, which provides $\mathrm{\Delta }\rho =0$) as a function of $m_{1/2}$ (black solid line) and $v_{\mathrm{\Delta }}$ (red dashed line) for parameter values given in Eq. (12) and $\mathcal{M}=850\mathrm{TeV}$.