Naturalness of scale-invariant NMSSMs with and without extra matter

We present a comparative and systematic study of the fine-tuning in Higgs sectors in three scale-invariant next-to-minimal supersymmetric standard model (NMSSM) models: the first is the standard $Z_3$-invariant NMSSM; the second is the NMSSM plus additional matter filling $3(5+\overline{5})$ representations of $SU(5)$ and is called the $\mathrm{NMSSM}+$; while the third model comprises $4(5+\overline{5})$ and is called the $\mathrm{NMSSM}++$. Naively, one would expect the fine-tuning in the plus-type models to be smaller than that in the NMSSM since the presence of extra matter relaxes the perturbativity bound on $\lambda$ at the low scale. This, in turn, allows larger tree-level Higgs mass and smaller loop contribution from the top squarks. However we find that LHC limits on the masses of sparticles, especially the gluino mass, can play an indirect, but vital, role in controlling the fine-tuning. In particular, working in a semiconstrained framework at the grand unified theory scale, we find that the masses of third generation top squarks are always larger in the plus-type models than in the NMSSM without extra matter. This is a renormalization group equation effect which cannot be avoided, and as a consequence the fine-tuning in the $\mathrm{NMSSM}+$ ($\mathrm{\Delta }\sim 200$) is significantly larger than in the NMSSM ($\mathrm{\Delta }\sim 100$), with fine-tuning in the $\mathrm{NMSSM}++$ ($\mathrm{\Delta }\sim 600$) being significantly larger than in the $\mathrm{NMSSM}+$.

Figure 1

Gauge coupling unification to two loop in the $\mathrm{NMSSM}+$ (left) and the $\mathrm{NMSSM}++$ (right). The mass scale of the extra matter in the $\mathrm{NMSSM}+$ ($\mathrm{NMSSM}++$) is taken here to be 3 (6) TeV. Below that scale, the NMSSM without extra matter is assumed. At $M_t=173.6\mathrm{GeV}$, we set $g_{1,\mathrm{SM}}=0.35940$, $g_2=0.64754$, $g_3=1.1666$, $h_t=1.01685$, $\tan \beta =5$, $\lambda =0.7$, and $\kappa =0.1$.

Figure 2

Left panel: $\lambda (1\mathrm{TeV})$ as a function of $\tan \beta$ in the three different models [for $\kappa (1\mathrm{TeV})\sim 0.002$]. Right panel: The one-loop running of the strong coupling ${\alpha }_3(\equiv {\alpha }_s)$, where the running from $M_Z$ (leftmost vertical line), passing through $m_t=173.6\mathrm{GeV}$ (second vertical line), to a fixed SUSY scale at 1 TeV (third vertical line) is performed using SM RGE. Next, the running from 1 TeV to the GUT scale is performed using the NMSSM RGE, and at 3 TeV (rightmost vertical line) the $\mathrm{NMSSM}+$ and the $\mathrm{NMSSM}++$ RGEs are used to run up to the GUT scale.

Figure 3

A schematic two-loop diagram illustrating how extra matter loops can modify $\mathcal{O}(g_a^4)$ terms of the running scalars, gauginos, trilinears, and Yukawas (see the Appendix).

Figure 4

The left panel shows the fine-tuning while the right panel shows the gluino mass, both in the $m_0\text{−}{m}_{1/2}$ plane in the NMSSM when $m_{h_1}$ is SM-like.

Figure 5

The left panel shows the rms top squark mass, while the right panel shows the lightest top squark mass, both in the $m_0\text{−}{m}_{1/2}$ plane in the NMSSM when $m_{h_1}$ is SM-like.

Figure 6

Fine-tuning as a function of $m_{\tilde{g}},M_S$, and $m_{{\text{stop}}_1}$ in the NMSSM when $m_{h_1}$ is SM-like.

Figure 7

The left panel shows the fine-tuning while the right panel shows the gluino mass, both in the $m_0\text{−}{m}_{1/2}$ plane in the NMSSM when $m_{h_2}$ is SM-like.

Figure 8

The left panel shows the rms top squark mass, while the right panel shows the lightest top squark mass, both in the $m_0\text{−}{m}_{1/2}$ plane in the NMSSM when $m_{h_2}$ is SM-like.

Figure 9

Fine-tuning as a function of $m_{\tilde{g}},M_S$, and $m_{{\text{stop}}_1}$ in the NMSSM when $m_{h_2}$ is SM-like.

Figure 10

The left panel shows the fine-tuning while the right panel shows the gluino mass, both in the $m_0\text{−}{m}_{1/2}$ plane in the $\mathrm{NMSSM}+$ when $m_{h_1}$ is SM-like.

Figure 11

The left panel shows the rms top squark mass, while the right panel shows the lightest top squark mass, both in the $m_0\text{−}{m}_{1/2}$ plane in the $\mathrm{NMSSM}+$ when $m_{h_1}$ is SM-like.

Figure 12

Fine-tuning as a function of $m_{\tilde{g}},M_S$, and $m_{{\text{stop}}_1}$ in the $\mathrm{NMSSM}+$ when $m_{h_1}$ is SM-like.

Figure 13

The left panel shows the fine-tuning while the right panel shows the gluino mass, both in the $m_0\text{−}{m}_{1/2}$ plane in the $\mathrm{NMSSM}+$ when $m_{h_2}$ is SM-like.

Figure 14

The left panel shows the rms top squark mass, while the right panel shows the lightest top squark mass, both in the $m_0\text{−}{m}_{1/2}$ plane in the $\mathrm{NMSSM}+$ when $m_{h_2}$ is SM-like.

Figure 15

Fine-tuning as a function of $m_{\tilde{g}},M_S$, and $m_{{\text{stop}}_1}$ in the $\mathrm{NMSSM}+$ when $m_{h_2}$ is SM-like.

Figure 16

The left panel shows the fine-tuning while the right panel shows the gluino mass, both in the $m_0\text{−}{m}_{1/2}$ plane in the $\mathrm{NMSSM}++$ when $m_{h_1}$ is SM-like.

Figure 17

The left panel shows the rms top squark mass, while the right panel shows the lightest top squark mass, both in the $m_0\text{−}{m}_{1/2}$ plane in the $\mathrm{NMSSM}++$ when $m_{h_1}$ is SM-like.

Figure 18

Fine-tuning as a function of $m_{\tilde{g}},M_S$, and $m_{{\text{stop}}_1}$ in the $\mathrm{NMSSM}++$ when $m_{h_1}$ is SM-like.

Figure 19

The left panel shows the fine-tuning while the right panel shows the gluino mass, both in the $m_0\text{−}{m}_{1/2}$ plane in the $\mathrm{NMSSM}++$ when $m_{h_2}$ is SM-like.

Figure 20

The left panel shows the rms top squark mass, while the right panel shows the lightest top squark mass, both in the $m_0\text{−}{m}_{1/2}$ plane in the $\mathrm{NMSSM}++$ when $m_{h_2}$ is SM-like.

Figure 21

Fine-tuning as a function of $m_{\tilde{g}},M_S$, and $m_{{\text{stop}}_1}$ in the $\mathrm{NMSSM}++$ when $m_{h_2}$ is SM-like.

Figure 22

The left panel shows the correlation between the fine-tuning and the gluino mass for the three models. The right panel shows the correlation between the rms top squark mass and the gluino mass for the three models.