Three-Loop Corrections to the Higgs Boson Mass and Implications for Supersymmetry at the LHC

In supersymmetric models with minimal particle content and without left-right squark mixing, the conventional wisdom is that the 125.6 GeV Higgs boson mass implies top squark masses of $\mathcal{O}(10)\mathrm{TeV}$, far beyond the reach of colliders. This conclusion is subject to significant theoretical uncertainties, however, and we provide evidence that it may be far too pessimistic. We evaluate the Higgs boson mass, including the dominant three-loop terms at $\mathcal{O}({\alpha }_t{\alpha }_s^2)$, in currently viable models. For multi-TeV top squarks, the three-loop corrections can increase the Higgs boson mass by as much as 3 GeV and lower the required top-squark masses to 3–4 TeV, greatly improving prospects for supersymmetry discovery at the upcoming run of the LHC and its high-luminosity upgrade.

Figure 1

The Higgs boson mass $m_h$ from H3m at one, two, and three loops for nearly degenerate ($m_{{\tilde{t}}_L}=m_{{\tilde{t}}_R}$), unmixed ($X_t=0$) top squarks, as a function of the physical mass $m_{{\tilde{t}}_1}$. The renormalization scale is fixed to $M_S=\sqrt{m_{{\tilde{t}}_1}{m}_{{\tilde{t}}_2}}$, we set $\tan \beta =20$, $\mu =200\mathrm{GeV}$, all other sfermion soft parameters equal to $m_{{\tilde{t}}_{L,R}}+1\mathrm{TeV}$, and assume gaugino mass unification with $m_{\tilde{g}}=1.5\mathrm{TeV}$. The bands indicate the parametric uncertainty from $m_t^{\mathrm{pole}}=173.3\pm 1.8\mathrm{GeV}$ and ${\alpha }_s(m_Z)=0.1184\pm 0.0007$. The horizontal bar is the experimentally allowed range $m_h=125.6\pm 0.4\mathrm{GeV}$.

Figure 2

Comparison of H3m results with the two-loop results of FeynHiggs [10, 11, 12, 13], SOFTSUSY [14], SuSpect [15], and SPheno [16, 17]. The H3m bands indicate the uncertainty from varying the renormalization scale between $M_S/2$ and $2M_S$. The supersymmetry parameters are as in Fig. 1.

Figure 3

Three-loop H3m $m_h$ contours in two ($m_0$, $M_{1/2}$) planes of mSUGRA, with $\tan \beta$, $A_0$, and $\mathrm{sign}(\mu )$ as indicated. In the dark blue (light green) shaded regions, the theoretical prediction is within ${\Delta }_{\mathrm{th}}$ ($2{\Delta }_{\mathrm{th}}$) of the experimental central value. On the ${\Omega }_{\chi }={\Omega }_{\mathrm{DM}}$ contour, thermal relic neutralinos are all the dark matter. Top: Negligible top squark mixing, with current exclusion contour from CMS [31], and projected sensitivities of the 14 TeV LHC and its high-luminosity upgrade [32]. Bottom: Significant top squark mixing, with current exclusion contour from ATLAS [33], and projected sensitivities of the 14 TeV LHC and its high-luminosity upgrade [34].