Vacuum stability of the effective Higgs potential in the minimal supersymmetric standard model

The parameters of the Higgs potential of the minimal supersymmetric standard model (MSSM) receive large radiative corrections which lift the mass of the lightest Higgs boson to the measured value of 126 GeV. Depending on the MSSM parameters, these radiative corrections may also lead to the situation that the local minimum corresponding to the electroweak vacuum state is not the global minimum of the Higgs potential. We analyze the stability of the vacuum for the case of heavy squark masses as favored by current LHC data. To this end we first consider an effective Lagrangian obtained by integrating out the heavy squarks and then study the MSSM one-loop effective potential $V_{\text{eff}}$, which comprises all higher-dimensional Higgs couplings of the effective Lagrangian. We find that only the second method gives correct results and argue that the criterion of vacuum stability should be included in phenomenological analyses of the allowed MSSM parameter space. Discussing the cases of squark masses of 1 and 2 TeV we show that the criterion of vacuum stability excludes a portion of the MSSM parameter space in which $|\mu \tan \beta |$ and $|A_{\mathrm{t}}|$ are large.

Figure 1

The 1-loop contribution to the effective potential as the sum of all one-particle irreducible diagrams with zero external momenta.

Figure 2

Couplings of neutral Higgs fields to squarks in the MSSM. $\mu$ is the higgsino mass parameter and $Y_{\mathrm{t},\mathrm{b}}$ and $A_{\mathrm{t},\mathrm{b}}$ are the Yukawa coupling and trilinear supersymmetry-breaking term, respectively, of top or bottom (s)quarks.

Figure 3

The 1-loop effective potential $V_{\mathrm{eff}}=V_0+V_1$ for $\tan \beta =40$ and $m_{A^0}=800\mathrm{GeV}$. Soft supersymmetry-breaking masses as well as the renormalization scale $Q$ have been taken at 1 TeV. The Higgs couplings in the loops with top and bottom squarks involve $\mu =2.55\mathrm{TeV}$ and $A_{\mathrm{t}}\simeq 1.5\mathrm{TeV}$. The hatched area highlights the analytic continuation beyond the branch point at $x-y=1$. The cases with $V_{\text{eff}}$ truncated after terms of dimension $2(k+n)=4$, 8, and 12 are shown with dashed lines (from top to bottom).

Figure 4

The 1-loop effective potential as in Fig. 3, with $\mu$ lowered to 2.51 TeV.

Figure 5

Area in the $\mu \text{−}\tan \beta$ plane for which $V_{\text{eff}}$ develops an unwanted minimum as depicted in Fig. 3. The red, cross-hatched area corresponds to $M_{\tilde{Q}}=M_{\tilde{\mathrm{t}}}=M_{\tilde{\mathrm{b}}}=Q=2\mathrm{TeV}$; the light blue area is excluded if $M_{\tilde{Q},\tilde{\mathrm{t}},\tilde{\mathrm{b}}}$ is lowered to 1 TeV. $A_{\mathrm{t}}\simeq 1.5\mathrm{TeV}$ is fitted to reproduce $m_{h^0}=126\mathrm{GeV}$ in both cases; $m_{A^0}=800\mathrm{GeV}$ is chosen to comply with LHC search limits for $A^0\to \tau \tau$.