The Higgs mass in the MSSM as a function of the lightest top squark mass, $m_{\tilde t_1}$, with red/blue solid lines computed using Suspect/FeynHiggs. The two upper lines are for maximal top squark mixing assuming degenerate stop soft masses and yield a 126 GeV Higgs mass for $m_{\tilde t_1}$ in the range of 500--800 GeV, while the two lower lines are for zero top squark mixing and do not yield a 126 GeV Higgs mass for $m_{\tilde t_1}$ below 3 TeV. Here we have taken $\tan\beta = 20$. The shaded regions highlight the difference between the Suspect and FeynHiggs results, and may be taken as an estimate of the uncertainties in the two-loop calculation.

The Higgs mass in the NMSSM as a function of $\tan \beta$. The solid lines show the tree-level result of equation \ref{eq:hmassNMSSM} while the shaded bands bounded by dashed lines result from adding the $\lambda^2 v^2 \sin^2 2 \beta$ contribution of equation~\ref{eq:hmassNMSSM} to the two-loop Suspect/FeynHiggs MSSM result, with degenerate stop soft masses. The top contribution $\delta_t$ is sufficient to raise the Higgs mass to 126 GeV for $\lambda = 0.7$ for a top squark mass of 500 GeV; but as $\lambda$ is decreased to 0.6 a larger value of the top squark mass is needed.

The Higgs mass in $\lambda$-SUSY, as a function of the singlet soft mass $m_s$. Here, $\lambda = 2$, $\tan \beta =2$, and the other parameters are as described in Table~\ref{tab:bench}, which gives the light Higgs a mass of $m_h = 280$~GeV in the limit of heavy singlet mass. However, we see that lowering the singlet mass $m_s$ results in a lighter Higgs due to mixing of the singlet with the Higgs.

Contours of $m_h = 126$~GeV in the MSSM as a function of a common stop mass $m_{Q_3} = m_{u_3} = m_{\tilde t}$ and the stop mixing parameter $X_t$, for $\tan \beta = 20$. The red/blue lines show the result from Suspect/FeynHiggs. The left panel shows contours of the fine tuning of the Higgs mass, $\Delta_{m_h}$, and we see that $\Delta_{m_h} > 100$ in order to achieve a Higgs mass of 126 GeV. The right panel shows contours of the lightest stop mass, which is always heavier than 500 GeV when the Higgs mass is 126 GeV.

A blowup of the maximal mixing regime, $X_t \sim 2 m_{\tilde t}$, in the MSSM, with $\tan\beta = 20$ and $m_A = 1$ TeV. The purple contours show $R_{\gamma \gamma}$, the ratio of $\sigma(g g \rightarrow h) \times \mathrm{Br}(h \rightarrow \gamma \gamma)$ in the MSSM to the Standard Model, computed with FeynHiggs. The 1-loop contribution from stops depletes the rate to be $\sim80 - 85\%$ of the SM rate. Had we chosen non-degenerate squark soft masses, this effect could be larger, at the cost of increased fine-tuning. The other contours are the same as the right side of Figure~\ref{fig:MSSM}.

Contours of $m_h = 126$~GeV in the NMSSM, taking $m_{Q_3} = m_{u_3} = m_{\tilde t}$ and varying $\tan \beta = 2, 5, 10$ from left to right, and varying $\lambda$ within each plot. We add the tree-level Higgs mass (with NMSSM parameters chosen to maximize it) to the 2-loop stop contribution from Suspect. The tree-level Higgs mass is largest at lower values of $\tan \beta$ and larger values of $\lambda$, where only modestly heavy stops, $m_{\tilde t} \sim 300$~GeV, are needed to raise the Higgs to 126 GeV. Heavy stops are still required for lower values of $\lambda$ and larger values of $\tan \beta$.

Contours of Higgs mass fine tuning, $\Delta_{m_h}$, in the NMSSM with the maximal value of $\lambda = 0.7$ for $\tan \beta = 2$ and 5, moving from left to right, with $m_{Q_3} = m_{u_3} = m_{\tilde t}$. Contours of $m_h = 126$~GeV are overlaid, including loop corrections from Suspect and FeynHiggs. When $\tan \beta = 2$ the tuning can be low, $\Delta_{m_h} \lesssim 15$, while for $\tan \beta = 5$ heavier stop masses are required because the tree-level Higgs mass is lower.

The necessary stop mass (left) and fine tuning (right) in order to achieve a Higgs mass of 126 GeV in the NMSSM, as a function of $\lambda$. We see that larger values of $\lambda$ allow for lighter stops and much less fine tuning. We consider two cases for the stop mixing: (1) maximally mixed stops with negative mixing, $X_t = -2 X_t^{\rm max}$, and (2) zero mixing. For both plots, the loop corrections are computed using Suspect and FeynHiggs, and we fix $\tan \beta = 2$.

The Higgs mass in $\lambda$-SUSY varying the singlet supersymmetric mass, $M_S$, and soft mass, $\tilde m_S$. The Higgs mass contours are shown in blue, contours of Higgs fine tuning, $\Delta_{m_h}$, are shown in red, and the region where the Higgs is tachyonic, due to Higgs-singlet mixing, is shown in purple. The fine tuning is increased when the Higgs mass drops, however, a Higgs mass of 126 GeV is achieved in a region of low fine tuning, $\Delta_{m_h} \sim 5$. The orange region is where the lightest neutralino is lighter than half the Higgs mass, and in this region the Higgs would dominantly decay invisibly.

The Higgs mass and fine tuning contours, $\Delta_{m_h}$ in $\lambda$-SUSY. On the left, we vary $\lambda$ and $\tan \beta$ and on the right we vary $\lambda$ and the singlet soft mass, $\tilde m_s$. The rest of the parameters are fixed as in table~\ref{tab:bench}. We find that there is a preference for large $\lambda$, small $\tan \beta$, and moderate values of the singlet soft mass, $m_s \sim 500$~GeV. Overall, there is a large region of parameter space where a Higgs mass of 126 is consistent with very mild tuning, $\Delta_{m_h} \sim 5$. Within the purple region, the Higgs is driven tachyonic due to Higgs-singlet mixing, and in the orange region on the right plot, there is a light neutralino and the Higgs dominantly decays invisibly.

The maximum values of the stop, charged Higgs, and Higgsino masses, before the fine tuning of the electroweak vev becomes worse than 10\% (5\%) is shown with solid (dashed) lines. We vary $\lambda$ along the horizontal axis and for each choice of $\lambda$ (and $\mu$, $m_{H^+}$), we choose $M_s$ in such a way as to fix the Higgs mass to 126 GeV. We see that larger values of $\lambda$ allow for heavier sparticles while maintaining naturalness. For $\lambda=2$, the stops can be as heavy as 1500 GeV, the charged Higgs can be 1 TeV, and the Higgsinos can be around 350 GeV.

The ratio of Higgs couplings squared relative to the Standard Model for $b \bar b$, $t \bar t$, $W^- W^+$ and $\gamma \gamma$ as a function of the charged Higgs mass, $m_{H^\pm}$. $\lambda$-SUSY is shown to the left and the MSSM is shown to the right, with the couplings computed in both theories at tree-level. We see that in $\lambda$-SUSY, unlike the MSSM, the Higgs coupling to the bottom quark drops dramatically away from the decoupling limit, leading to a depleted Higgs width and an enhanced $\gamma \gamma$ signal. The $\lambda$-SUSY parameters, other than the charged Higgs mass, are as in table~\ref{tab:bench}; and for the MSSM we choose $\tan \beta = 20$.

The ratio $R_{\gamma \gamma}$ of $\sigma \times Br$ in $\lambda$-SUSY relative to the SM for the process $g g \rightarrow h \rightarrow \gamma \gamma$. The red contours show $R_{\gamma \gamma}$ and the blue contours show the Higgs mass in the $\lambda, \tan \beta$ plane. We see that this process generically has a larger rate in $\lambda$-SUSY than in the SM, this is due to the depletion of the Higgs coupling to bottom quarks in the non-decoupling limit. The left and right panels correspond to charged Higgs masses of 350 and 495 GeV, respectively.

An example of a natural SUSY spectrum in $\lambda$SUSY with $\lambda \sim 2$. The fine tuning of the Higgs mass, and electroweak symmetry breaking, can remain milder than 10\% with the Higgsinos at 350 GeV, the stops at 1.5 TeV, and the gluino at 3 TeV. Mixing between the Higgs and the singlet lowers the Higgs mass to 126 GeV.