Electroweak and supersymmetry breaking from the Higgs boson discovery

We will explore the consequences on the electroweak breaking condition, the mass of supersymmetric partners and the scale at which supersymmetry breaking is transmitted, for arbitrary values of the supersymmetric parameters $\tan \beta$ and the stop mixing $X_t$, which follow from the Higgs discovery with a mass $m_H\simeq 126\mathrm{GeV}$ at the LHC. Within the present uncertainty on the top quark mass we deduce that radiative breaking requires $\tan \beta \gtrsim 8$ for maximal mixing $X_t\simeq \sqrt{6}$, and $\tan \beta \gtrsim 20$ for small mixing $X_t\lesssim 1.8$. The scale at which supersymmetry breaking is transmitted $\mathcal{M}$ can be of order the unification or Planck scale only for large values of $\tan \beta$ and negligible mixing $X_t\simeq 0$. On the other hand for maximal mixing and large values of $\tan \beta$ supersymmetry should break at scales as low as $\mathcal{M}\simeq {10}^5\mathrm{GeV}$. The uncertainty in those predictions stemming from the uncertainty in the top quark mass, i.e. the top Yukawa coupling, is small (large) for large (small) values of $\tan \beta$. In fact for $\tan \beta =1$ the uncertainty on the value of $\mathcal{M}$ is several orders of magnitude.

Figure 1

Left panel: Contour lines of ${\log }_{10}[{\mathcal{Q}}_0/\mathrm{GeV}]$ (for the values specified in the plot) in the plane $(\tan \beta ,X_t)$. Right panel: Contour line of $m_2^2({\mathcal{Q}}_0)=0$, as given by Eq. (2.13), in the plane $(\tan \beta ,X_t)$. The inner region corresponds to radiative electroweak breaking.

Figure 2

Left panel: Plot of ${\mathcal{Q}}_0$ as a function of $\tan \beta$ for $X_t=0$ (upper band) and $X_t=\sqrt{6}$ (lower band). The width of bands corresponds to the experimental error $\mathrm{\Delta }{\overline{m}}_t=\pm 2\mathrm{GeV}$. Right panel: Plot of ${\mathcal{Q}}_0$ as a function of $X_t$ for $\tan \beta =2$ (upper band) and 15 (lower band).

Figure 3

Left panel: Plot of $|m_2({\mathcal{Q}}_0)|$ as a function of $\tan \beta$ for $X_t=0$ (upper band) and $X_t=\sqrt{6}$ (lower band). The width of bands corresponds to the experimental error $\mathrm{\Delta }{\overline{m}}_t=\pm 2\mathrm{GeV}$. Right panel: Plot of $|m_2({\mathcal{Q}}_0)|$ as a function of $X_t$ for $\tan \beta =2$ (upper band) and 15 (lower band).

Figure 4

Contour lines of constant ${\log }_{10}[\mathcal{M}/\mathrm{GeV}]$ in the $(\tan \beta ,X_t)$ plane for $X_t\ge 0$ (left panel) and $X_t<0$ (right panel).

Figure 5

Left panel: Plot of ${\log }_{10}[\mathcal{M}/\mathrm{GeV}]$ as a function of $\tan \beta$ for $X_t=0$ (upper band), $X_t=0.5$ (central band) and $X_t=\sqrt{6}$ (lower band). The width of bands corresponds to the experimental error $\mathrm{\Delta }{\overline{m}}_t=\pm 2\mathrm{GeV}$. Right panel: Plot of $\mathcal{M}$ as a function of $X_t$ for $\tan \beta =2$ (wider band) and 15 (narrower band).

Figure 6

RGE running between $\mathcal{M}$ and ${\mathcal{Q}}_0$ of the supersymmetric spectrum for the case $\tan \beta =10$, $X_t=0$ (left panel) and $X_t=2$ (right panel) with universal boundary conditions.

Figure 7

RGE running between $\mathcal{M}$ and ${\mathcal{Q}}_0$ of the supersymmetric spectrum for the case $A_t(\mathcal{M})=0$ and $\tan \beta =15$ (left panel) and $\tan \beta =8$ (right panel) with gauge mediated boundary conditions.