Raising the Higgs mass in supersymmetry with $t-t^{\prime }$ mixing

In this article we propose a new strategy to address the little hierarchy problem. We show that the addition of a fourth generation with vectorlike quarks to the minimal supersymmetric standard model can raise the predicted value of the physical Higgs mass by mixing with the top sector. The mixing requires a larger top quark Yukawa coupling (by up to $\sim 6\%$) to produce the same top mass. Since loop corrections to $m_h$ go as $y_{\text{top}}^4$, this will in turn increase the predicted value of the physical Higgs mass, a point not previously emphasized in the literature. In the presence of mixing, for A terms and soft masses around 900 GeV, a Higgs mass of 125 GeV can be generated while retaining perturbativity of the gauge couplings, evading constraints from electroweak precision measurements and recent LHC searches, and pushing the Landau pole for the top Yukawa above the GUT scale. Soft masses can be as low as 800 GeV in parts of parameter space with a Landau pole at $\sim {10}^{10}\mathrm{GeV}$. However, the Landau pole can still be pushed above the GUT scale if one sacrifices perturbative unification by adding fields in a $\mathbf{5}+\overline{\mathbf{5}}$ representation. With a ratio of weak-scale vector masses $\ne 1$, soft masses may be slightly below 800 GeV. The model predicts new quarks and squarks with masses $⪆750\mathrm{GeV}$. We briefly discuss potential paths for discovery or exclusion at the LHC.

Figure 1

We plot the values of the Yukawa couplings at the weak scale necessary to obtain $m_h=125.5\pm 0.5\mathrm{GeV}$, as a function of $\mathrm{\Delta }m$. We take $A=\mathrm{\Delta }m$, ${\mu }_4=900\mathrm{GeV}$. When either $y_{34}$ or $y_{43}$ dominates, the same value of the dominant Yukawa is required to get $m_h=125.5\mathrm{GeV}$ so both scenarios are represented by one black line. The dotted lines show the maximum values allowed by EWPM for each mixing scenario (see Sec. 4). Since $y_{34}$ and $y_{43}$ contribute to the oblique parameters differently they have different constraints on their maximum values, represented by the green and orange dotted lines, respectively. Above the dotted line requires Yukawas larger than allowed by EWPM and is thus ruled out.

Figure 2

We plot the scale $\mathrm{\Lambda }$ where the $y_{33}$ required to get $m_h=125.5\mathrm{GeV}$ hits a Landau pole, as a function of the soft mass $\mathrm{\Delta }m$. We set $A=\mathrm{\Delta }m$, ${\mu }_4=900\mathrm{GeV}$, and $n_5=0$. Soft masses to the left of the dotted lines can only yield $m_h=125.5\mathrm{GeV}$ with Yukawa couplings larger than allowed by EWPM and are thus ruled out (see Sec. 4). Physically uninteresting values of $\mathrm{\Lambda }<1\mathrm{TeV}$ are not plotted. The presence of mixing decreases significantly the value of the soft masses needed. As can be seen from the plot, the scale of the Landau pole in the cases with sizable mixing are all comparable. The cases where either $y_{34}$ or $y_{43}$ dominate (shown in black) yield identical values since each contributes to the top Yukawa beta function in the same way. However, their differing effects on the oblique parameters lead to different minimum values for the soft masses.

Figure 3

We plot the scale $\mathrm{\Lambda }$ where the $y_{33}$ required to get $m_h=125.5\mathrm{GeV}$ hits a Landau pole, as a function of the soft mass $\mathrm{\Delta }m$. We set $y_{34}\gg y_{44},y_{43}$, ${\mu }_4=900\mathrm{GeV}$, and $n_5=0$. Soft masses to the left of the dotted lines can only yield $m_h=125.5\mathrm{GeV}$ with Yukawa couplings larger than allowed by EWPM and are thus ruled out (see Sec. 4). There is only one line here since these limits are independent of the $A$ terms). For a given soft mass the implied Landau pole gets significantly pushed up by the presence of $A$ terms.

Figure 4

We plot the scale $\mathrm{\Lambda }$ where the $y_{33}$ required to get $m_h=125.5\mathrm{GeV}$ hits a Landau pole, as a function of the soft mass $\mathrm{\Delta }m$. We set $y_{34}\gg y_{44},y_{43}$, $A=\mathrm{\Delta }m$, and $n_5=0$. Soft masses to the left of the dotted lines can only yield $m_h=125.5\mathrm{GeV}$ with Yukawa couplings larger than allowed by EWPM and are thus ruled out (see Sec. 4). For a given soft mass, the implied Landau pole increases as the vector mass decreases.

Figure 5

We plot the scale $\mathrm{\Lambda }$ where the $y_{33}$ required to get $m_h=125.5\mathrm{GeV}$ hits a Landau pole, as a function of the soft mass $\mathrm{\Delta }m$. We set $y_{34}\gg y_{44},y_{43}$, $A=\mathrm{\Delta }m$, ${\mu }_4=900\mathrm{GeV}$. Here the dotted lines indicate where the gauge couplings become nonperturbative for $n_5=2$ and $n_5=1$. They remain perturbative all the way to the GUT scale for $n_5=0$.

Figure 6

Present status of heavy vectorlike top searches with $19.5{\mathrm{fb}}^{-1}$ of 8 TeV data with the CMS detector (figure taken from [48]). A branching-fraction triangle is shown with expected (left) and observed 95% C.L. (right) on the mass. Every point in the triangle corresponds to a specific set of branching-fraction values subject to the constraint that all three add up to 1.

Figure 7

$S_{\text{new}}$ versus ${\mu }_4$ for $y_{34}=0.6$ and $y_{44}=y_{43}=0$. $S_{\text{new}}$ remains small as ${\mu }_4\to \infty$.

Figure 8

$T_{\text{new}}$ versus ${\mu }_4$ for $y_{34}=0.6$ and $y_{44}=y_{43}=0$. $T_{\text{new}}$ remains small as ${\mu }_4\to \infty$.

Figure 9

$S_{\text{new}}$ versus $y_{34}$ for the benchmark scenario $y_{34}\gg y_{44},y_{43}$, ${\mu }_4=900\mathrm{GeV}$, $A=\mathrm{\Delta }m=800\mathrm{GeV}$. $S_{\text{new}}$ remains small in this region. As $y_{34}$ increases from 0.5 to 1, $S_{\text{new}}$ increases by a negligible amount of the order of ${10}^{-4}$. The region $y_{34}\gtrsim 0.8$ to the right of the dashed line is disfavored by EWPM due to the $T$ parameter (see Fig. 10).

Figure 10

$T_{\text{new}}$ versus $y_{34}$ for the benchmark scenario $y_{34}\gg y_{44},y_{43}$, ${\mu }_4=900\mathrm{GeV}$, $A=\mathrm{\Delta }m=800\mathrm{GeV}$. As $y_{34}$ increases from 0.5 to 1, $T_{\text{new}}$ increases from $\sim 0.05$ to $\sim 0.25$. The region $y_{34}\gtrsim 0.8$ to the right of the dashed line is disfavored by EWPM as can be seen in Fig. 11.

Figure 11

We calculate $S_{\text{new}}$ and $T_{\text{new}}$ for each of the benchmark scenarios: $y_{34}\gg y_{43},y_{44}$; $y_{43}\gg y_{34},y_{44}$; and $y_{34}=-y_{43}\gg y_{44}$. Within each scenario ${\mu }_4=900\mathrm{GeV}$, $A=600\mathrm{GeV}$, and we vary $\mathrm{\Delta }m$ from 300 to 1500 GeV. Each of these points satisfies current mass bounds (see Sec. 4c) and gives a Higgs mass $m_h=125.5\pm .5\mathrm{GeV}$ while yielding new quarks discoverable at the LHC. The points corresponding to very low $\mathrm{\Delta }m$ and larger Yukawas lie farthest from the best fit, with the agreement improving as $\mathrm{\Delta }m$ grows and the Yukawas decrease. For many of these points the net effect from the new sector falls within the 95% or 68% confidence limits on the electroweak observables. The experimental best fit corresponds to the center of the ellipses, at (0.00,0.02) [52]. The light (dark) grey ellipse denotes the 95% (65%) C.L. on the EW observables. The origin is defined to be the standard model prediction with a 125 GeV Higgs. In concert with the results of Sec. 3b, precision electroweak observables permit sufficiently large Yukawa mixing to obtain a Higgs mass $\sim 125\mathrm{GeV}$ with soft terms below a TeV.

Figure 12

We plot the $S_{\text{new}},T_{\text{new}}$ for ratios ${\mu }_U/{\mu }_Q=0.9,1.0,1.1$, and Yukawa values $y_{34}=-y_{43}$ ranging from 0.01 to 0.56 in steps of 0.05. Each of these points satisfies current mass bounds (see Sec. 4c) and gives a Higgs mass $m_h=125.5\pm .5\mathrm{GeV}$ while yielding new quarks discoverable at the LHC. The points corresponding to very low $\mathrm{\Delta }m$ and larger Yukawas lie farthest from the best fit, with the agreement improving as $\mathrm{\Delta }m$ grows and the Yukawas decrease. For many of these points the net effect from the new sector falls within the 95% or 68% confidence limits on the electroweak observables. The experimental best fit corresponds to the center of the ellipses, at (0.00,0.02) [52]. The light (dark) grey ellipse denotes the 95% (65%) C.L. on the EW observables. The origin is defined to be the Standard Model prediction with a 125 GeV Higgs. In concert with the results of Sec. 3b, precision electroweak observables permit sufficiently large Yukawa mixing to obtain a Higgs mass $\sim 125\mathrm{GeV}$ with soft terms below a TeV.