$Z^{\prime }$ mass limits and the naturalness of supersymmetry

The discovery of a 125 GeV Higgs boson and rising lower bounds on the masses of superpartners have led to concerns that supersymmetric models are now fine-tuned. Large stop masses, required for a 125 GeV Higgs, feed into the electroweak symmetry breaking conditions through renormalization group equations forcing one to fine-tune these parameters to obtain the correct electroweak vacuum expectation value. Nonetheless, this fine-tuning depends crucially on our assumptions about the supersymmetry breaking scale. At the same time, $U(1)$ extensions provide the most compelling solution to the $\mu$ problem, which is also a naturalness issue, and allow the tree-level Higgs mass to be raised substantially above $M_Z$. These very well-motivated supersymmetric models predict a new $Z^{\prime }$ boson which could be discovered at the LHC, and the naturalness of the model requires that the $Z^{\prime }$ boson mass should not be too far above the TeV scale. Moreover, this fine-tuning appears at the tree level, making it less dependent on assumptions about the supersymmetry breaking mechanism. Here we study this fine-tuning for several $U(1)$ supersymmetric extensions of the Standard Model and compare it to the situation in the MSSM where the most direct tree-level fine-tuning can be probed through chargino mass limits. We show that future LHC $Z^{\prime }$ searches are extremely important for challenging the most natural scenarios in these models.

Figure 1

Left panel: Scatter plot of fine-tuning in the MSSM as a function of the lightest stop mass, $m_{{\tilde{t}}_1}$, for the cutoff scales (from bottom to top) $M_X=20\mathrm{TeV}$, $M_X=50\mathrm{TeV}$, $M_X=100\mathrm{TeV}$ and $M_X={10}^{16}\mathrm{GeV}$. Right panel: Scatter plot of fine-tuning in the MSSM as a function of the lightest Higgs mass, $m_{h_1}$, for the cutoff scales (from bottom to top) $M_X=20\mathrm{TeV}$, $M_X=50\mathrm{TeV}$, $M_X=100\mathrm{TeV}$ and $M_X={10}^{16}\mathrm{GeV}$.

Figure 2

Scatter plot of fine-tuning vs lightest Higgs mass for the MSSM (light blue, bottom band), ${\mathrm{E}}_6\mathrm{SSM}$ with $M_{Z^{\prime }}=2.5\mathrm{TeV}$ (dark blue, middle band) and ${\mathrm{E}}_6\mathrm{SSM}$ with $M_{Z^{\prime }}=4.5\mathrm{TeV}$ (dark yellow, top band). Note that there are points for which the fine-tuning in the MSSM and ${\mathrm{E}}_6\mathrm{SSM}$ with $M_{Z^{\prime }}=2.5\mathrm{TeV}$ is larger than is visible on this plot and those below; however, these points are obscured by the overlaid data for the ${\mathrm{E}}_6\mathrm{SSM}$ with $M_{Z^{\prime }}=4.5\mathrm{TeV}$, and it is the lower bound on the achievable tuning that is of interest here.

Figure 3

Scatter plot of fine-tuning vs lightest Higgs mass in the MSSM with $200\mathrm{GeV}\le m_{{\tilde{\chi }}_1^{\pm }}\le 400\mathrm{GeV}$ shown in light blue (bottom band), $500\mathrm{GeV}\le m_{{\tilde{\chi }}_1^{\pm }}\le 600\mathrm{GeV}$ in dark blue (middle band), and $700\mathrm{GeV}\le m_{{\tilde{\chi }}_1^{\pm }}\le 800\mathrm{GeV}$ in dark yellow (top band).

Figure 4

Top left panel: Scatter plot of the fine-tuning vs lightest Higgs mass in the $U(1{)}_I$ model. Top right panel: Scatter plot of the fine-tuning vs lightest Higgs mass in the $U(1{)}_{\psi }$ model. Bottom panel: Scatter plot of the fine-tuning vs lightest Higgs mass in the $U(1{)}_{\eta }$ model. In each plot points with $M_{Z^{\prime }}=2.5\mathrm{TeV}$ are shown in dark blue (bottom band), and points with $M_{Z^{\prime }}=4.5\mathrm{TeV}$ are shown in dark yellow (top band).

Figure 5

Top panel: Scatter plot of the fine-tuning as a function of the cutoff scale $M_X$ for the four benchmark points given in Table 3. Middle left panel: Individual sensitivities for MSSM BM1 plotted against the high scale $M_X$ which give the overall tuning shown by the dotted line in the top panel. Middle right panel: Individual sensitivities for ${\mathrm{E}}_6\mathrm{SSM}$ BM1 plotted against the high scale $M_X$ which give the overall tuning shown by the solid line in the top panel. Bottom left panel: Individual sensitivities for MSSM BM2 plotted against the high scale $M_X$ which give the overall tuning shown by the dash-dotted line in the top panel. Bottom right panel: Individual sensitivities for ${\mathrm{E}}_6\mathrm{SSM}$ BM2 plotted against the high scale $M_X$ which give the overall tuning shown by the dashed line in the top panel.