Precise focus point scenario for a natural Higgs boson in the MSSM

A small Higgs mass parameter $m_{h_u}^2$ can be insensitive to various trial heavy stop masses, if a universal soft squared mass is assumed for the chiral superpartners and the Higgs boson at the grand unification (GUT) scale, and a focus point (FP) of $m_{h_u}^2$ appears around the stop mass scale. The challenges in the FP scenario are (1) a too heavy stop mass ($\approx 5\mathrm{TeV}$) needed for the 126 GeV Higgs mass and (2) the too high gluino mass bound ($\gtrsim 1.4\mathrm{TeV}$). For a successful FP scenario, we consider (1) a superheavy right-hand (RH) neutrino and (2) the first and second generations of hierarchically heavier chiral superpartners. The RH neutrino can move a FP in the higher energy direction in the space of $(Q,m_{h_u}^2(Q))$, where $Q$ denotes the renormalization scale. On the other hand, the hierarchically heavier chiral superpartners can lift up a FP in that space through two-loop gauge interactions. Precise focusing of $m_{h_u}^2(Q)$ is achieved with the RH neutrino mass of $\sim 10^{14}\mathrm{GeV}$ together with an order one (0.9–1.2) Yukawa coupling to the Higgs boson, and the hierarchically heavy masses of 15–20 TeV for the heavier generations of superpartners, when the $\mathrm{U}(1{)}_R$ breaking soft parameters $m_{1/2}$ and $A_0$ are set to be 1 TeV at the GUT scale. Those values can naturally explain the small neutrino mass through the seesaw mechanism, and suppress the flavor violating processes in supersymmetric models.

Figure 1

RG evolutions of $m_{h_u}^2$ for $m_0^2=(7\mathrm{TeV}{)}^2$ (red), $(5\mathrm{TeV}{)}^2$ (green), and $(3\mathrm{TeV}{)}^2$ (blue), and for (a) $\tan \beta =5$ and (b) $\tan \beta =50$, when $m_{1/2}=1\mathrm{TeV}$ and $A_0=0$. Here, we take ${\alpha }_G=1/25$. The unit of the vertical axis is $(\mathrm{GeV}{)}^2$. The dotted lines at $t\approx 0.92$ denote the assumed stop decoupling scale, $Q=5\mathrm{TeV}$. $t\approx -2.3$ [$t\approx 29.9$] corresponds to $Q=200\mathrm{GeV}$ [$Q=2\times 10^{16}\mathrm{GeV}$]. Below the stop decoupling scale, the above RG runnings must be modified. The above figures show that the extrapolated FP, where $m_{h_u}^2$ is negative, appears at a relatively higher (lower) energy scale for small (large) $\tan \beta$.

Figure 2

Desirable locations of the focus point in the $(t,m_{h_u}^2)$ space. The straight lines sketch different RG evolutions of $m_{h_u}^2$ for various $m_0^2\mathrm{s}$. $t_T$ corresponds to the assumed stop decoupling scale ($Q=5\mathrm{TeV}$). $\epsilon$ needs to be as small as possible.

Figure 3

RG evolutions of $m_{h_u}^2$ for $m_0^2=(9\mathrm{TeV}{)}^2$ (red), $(7\mathrm{TeV}{)}^2$ (green), and $(5\mathrm{TeV}{)}^2$ (blue), and for (a) $y_{NI}^2=0.8$, ${\tilde{m}}^2=(15\mathrm{TeV}{)}^2$ and (b) $y_{NI}^2=1.0$, ${\tilde{m}}^2=(20\mathrm{TeV}{)}^2$, when $\tan \beta =5$ and $m_{1/2}=A_0=1\mathrm{TeV}$ with ${\alpha }_G=1/24$. The unit of the vertical axis is $(\mathrm{GeV}{)}^2$. Below the seesaw scale, $t=t_I\approx 25.3$ [$Q\approx 2\times 10^{14}\mathrm{GeV}$], the RH neutrino is decoupled. The dotted lines at $t\approx 0.92$ denote the assumed stop decoupling scale, $Q=5\mathrm{TeV}$. Below the stop decoupling scale, the above RG runnings must be modified. The above figures show that the (extrapolated) FP appears at desirable locations.

Figure 4

RG evolutions of $m_{h_u}^2$ for $m_0^2=(9\mathrm{TeV}{)}^2$ (red), $(7\mathrm{TeV}{)}^2$ (green), and $(5\mathrm{TeV}{)}^2$ (blue), and for (a) $y_{NI}^2=1.0$, ${\tilde{m}}^2=(15\mathrm{TeV}{)}^2$ and (b) $y_{NI}^2=1.2$, ${\tilde{m}}^2=(20\mathrm{TeV}{)}^2$, when $\tan \beta =50$ and $m_{1/2}=A_0=1\mathrm{TeV}$ with ${\alpha }_G=1/24$. The unit of the vertical axis is $(\mathrm{GeV}{)}^2$. Below the seesaw scale, $t=t_I\approx 25.3$ [$Q\approx 2\times 10^{14}\mathrm{GeV}$], the RH neutrino is decoupled. The dotted lines at $t=0.92$ denote the assumed stop decoupling scale, $Q=5\mathrm{TeV}$. Below the stop decoupling scale, the above RG runnings must be modified. The above figures show that the (extrapolated) FP appears at desirable locations.