Next-to-minimal supersymmetric model solution to the fine-tuning problem, precision electroweak constraints, and the largest CERN LEP Higgs event excess

Radovan Dermíšek and John F. Gunion

Phys. Rev. D 76, 095006 – Published 14 November 2007

Abstract

We present an extended study of how the next-to-minimal supersymmetric model easily avoids fine-tuning in electroweak symmetry breaking for a SM-like light Higgs with mass in the vicinity of 100 GeV, as beautifully consistent with precision electroweak data, while escaping LEP constraints due to the dominance of $h \to a a$ decays with $m_{a} < 2 m_{b}$ so that $a \to τ^{+} τ^{-}$ or jets. The residual $\sim 10 %$ branching ratio for $h \to b \bar{b}$ explains perfectly the well-known LEP excess at $m_{h} \sim 100 GeV$ . Details of model parameter correlations and requirements are discussed as a function $\tan β$ . Comparisons of fine-tuning in the NMSSM to that in the MSSM are presented. We also discuss fine-tuning associated with scenarios in which the $a$ is essentially pure singlet, has mass $m_{a} > 30 GeV$ , and decays primarily to $γ γ$ leading to an $h \to a a \to 4 γ$ Higgs signal.

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Received 14 June 2007

DOI:https://doi.org/10.1103/PhysRevD.76.095006

Authors & Affiliations

Radovan Dermíšek¹ and John F. Gunion²

¹School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey 08540, USA
²Department of Physics, University of California at Davis, Davis, California 95616, USA

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Vol. 76, Iss. 9 — 1 November 2007

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Figure 1
Expected and observed 95% CL limits on $C_{eff}^{2 b}$ from Ref. 1 are shown vs. $m_{h}$ . Also plotted are the predictions for the NMSSM parameter cases discussed in 3 having fixed $\tan β = 10$ , $M_{1, 2, 3} (m_{Z}) = 100$ , 200, 300 GeV that give fine-tuning measure $F < 25$ and $m_{a_{1}} < 2 m_{b}$ and that are consistent with Higgs constraints obtained using the preliminary LHWG analysis code [4].Reuse & Permissions
Figure 2
Fine-tuning vs ${\bar{m}}_{\tilde{t}}$ (top), $m_{h}$ (middle) and $A_{t}$ (bottom) for randomly generated MSSM parameter choices with $\tan β = 10$ and $M_{1, 2, 3} (m_{Z}) = 100$ , 200, 300 GeV. Blue pluses correspond to parameter choices such that $m_{h} < 114 GeV$ . Red crosses are points with $m_{h} > 114 GeV$ .Reuse & Permissions
Figure 3
For the NMSSM, we plot the fine-tuning measure $F$ vs ${\bar{m}}_{\tilde{t}}$ , $m_{h_{1}}$ and $A_{t}$ for NMHDECAY-accepted scenarios with $\tan β = 10$ and $M_{1, 2, 3} (m_{Z}) = 100$ , 200, 300 GeV. Points marked by blue $+$ ’s are consistent with LEP limits on the $Z + 2 b$ channel and the $Z + 4 b$ channel [11], considered separately, but not necessarily with LEP limits on the combined $Z + 2 b$ and $Z + 4 b$ channels). Points marked by red $\times$ ’s escape LEP limits due to $m_{h_{1}} > 114 GeV$ .Reuse & Permissions
Figure 4
For the NMSSM, we plot $C_{eff}^{2 b}$ and $C_{eff}^{4 b}$ as a functions of $m_{h_{1}}$ for NMHDECAY-accepted scenarios with $\tan β = 10$ and $M_{1, 2, 3} (m_{Z}) = 100$ , 200, 300 GeV. Point notation as in Fig. 3.Reuse & Permissions
Figure 5
For the NMSSM, we plot the fine-tuning measure $F$ vs $B R (h_{1} \to a_{1} a_{1})$ and vs $m_{a_{1}}$ for NMHDECAY-accepted scenarios with $\tan β = 10$ and $M_{1, 2, 3} (m_{Z}) = 100$ , 200, 300 GeV. Point notation as in Fig. 3.Reuse & Permissions
Figure 6
Fine-tuning vs $m_{h_{1}}$ for $\tan β = 10$ , $\tan β = 3$ , and $\tan β = 50$ for points with $F < 50$ , taking $M_{1, 2, 3} (m_{Z}) = 100$ , 200, 300 GeV. Small black $\times$ points are those obtained after requiring a global and local minimum, no Landau pole before $M_{U}$ and a neutralino LSP. The green O’s are those that in addition satisfy experimental limits on stops and charginos, but not necessarily Higgs limits. The blue $□$ ’s are the points that remain after imposing all LEP single channel Higgs limits, in particular, limits [11] on the $Z + 2 b$ and $Z + 4 b$ channels considered separately. The yellow fancy crosses are the blue square points that remain after requiring $m_{a_{1}} < 2 m_{b}$ , so that LEP limits on $Z + b^{'} s$ , where $b^{'} s = 2 b + 4 b$ , are not violated.Reuse & Permissions
Figure 7
For the $F < 15$ scenarios that are fully consistent with all LEP constraints, we plot $G$ vs $\cos θ_{A}$ taking $M_{1, 2, 3} = 100$ , 200, 300 GeV and $\tan β = 10$ (top), 3 (middle), and 50 (bottom). The point coding is: $black = 8.8 GeV < m_{a_{1}} < m_{h_{1}} / 2$ ; $dark grey (red) = 2 m_{τ} < m_{a_{1}} < 7.5 GeV$ ; $light grey (green) = 7.5 GeV < m_{a_{1}} < 8.8 GeV$ ; and $darkest grey (blue) = m_{a_{1}} < 2 m_{τ}$ .Reuse & Permissions
Figure 8
Fine-tuning vs the Higgs mass for randomly generated NMSSM parameter choices except for fixed $M_{1, 2, 3} (m_{Z}) = 100$ , 200, 600 GeV and a fixed value of $\tan β = 10$ . Notation as in Fig. 6.Reuse & Permissions
Figure 9
Fine-tuning vs $m_{a_{1}}$ (top) and vs $m_{h^{+}}$ (bottom) for points with $F < 50$ taking $M_{1, 2, 3} (m_{Z}) = 100$ , 200, 300 GeV and $\tan β = 10$ . Point notation as in Fig. 6.Reuse & Permissions
Figure 10
Fine-tuning vs $m_{a_{2}}$ (top), $m_{h_{2}}$ (middle), and $m_{h_{3}}$ (bottom) for points with $F < 50$ taking $M_{1, 2, 3} (m_{Z}) = 100$ , 200, 300 GeV and $\tan β = 10$ . Point notation as in Fig. 6.Reuse & Permissions
Figure 11
Fine-tuning vs $m_{a_{1}}$ (top) and vs $m_{h^{+}}$ (bottom) for points with $F < 50$ taking $M_{1, 2, 3} (m_{Z}) = 100$ , 200, 300 GeV and $\tan β = 50$ . Point notation as in Fig. 6.Reuse & Permissions
Figure 12
Fine-tuning vs $m_{a_{2}}$ (top), $m_{h_{2}}$ (middle), and $m_{h_{3}}$ (bottom) for points with $F < 50$ taking $M_{1, 2, 3} (m_{Z}) = 100$ , 200, 300 GeV and $\tan β = 50$ . Point notation as in Fig. 6.Reuse & Permissions
Figure 13
Fine-tuning vs $A_{t} (m_{Z})$ (top) and vs $A_{t} (M_{U})$ (bottom) for points with $F < 50$ taking $M_{1, 2, 3} (m_{Z}) = 100$ , 200, 300 GeV and $\tan β = 10$ . Point notation as in Fig. 6.Reuse & Permissions
Figure 14
Fine-tuning vs $A_{t} (m_{Z})$ (top) and vs $A_{t} (M_{U})$ (bottom) for points with $F < 50$ taking $M_{1, 2, 3} (m_{Z}) = 100$ , 200, 300 GeV and $\tan β = 50$ . Point notation as in Fig. 6.Reuse & Permissions
Figure 15
Fine-tuning vs $μ_{eff}$ for points with $F < 50$ taking $M_{1, 2, 3} (m_{Z}) = 100$ , 200, 300 GeV and $\tan β = 10$ . Point notation as in Fig. 6.Reuse & Permissions
Figure 16
Fine-tuning vs $μ_{eff}$ for points with $F < 50$ taking $M_{1, 2, 3} (m_{Z}) = 100$ , 200, 300 GeV and $\tan β = 50$ . Point notation as in Fig. 6.Reuse & Permissions
Figure 17
Fine-tuning vs $M_{Q} (m_{Z})$ , $M_{U} (m_{Z})$ , and $M_{D} (m_{Z})$ for points with $F < 50$ taking $M_{1, 2, 3} (m_{Z}) = 100$ , 200, 300 GeV and $\tan β = 10$ . Point notation as in Fig. 6.Reuse & Permissions
Figure 18
Fine-tuning vs $M_{Q} (m_{Z})$ , $M_{U} (m_{Z})$ , and $M_{D} (m_{Z})$ for points with $F < 50$ taking $M_{1, 2, 3} (m_{Z}) = 100$ , 200, 300 GeV and $\tan β = 50$ . Point notation as in Fig. 6.Reuse & Permissions
Figure 19
Fine-tuning vs $M_{Q} (M_{U})$ , $M_{U} (M_{U})$ , and $M_{D} (M_{U})$ for points with $F < 50$ taking $M_{1, 2, 3} (m_{Z}) = 100$ , 200, 300 GeV and $\tan β = 10$ . Point notation as in Fig. 6. Negative values indicate cases for which $M_{Q, U, D}^{2} < 0$ in which case the plot gives $- \sqrt{- M_{Q, U, D}^{2}}$ .Reuse & Permissions
Figure 20
Fine-tuning vs $M_{Q} (M_{U})$ , $M_{U} (M_{U})$ , and $M_{D} (M_{U})$ for points with $F < 50$ taking $M_{1, 2, 3} (m_{Z}) = 100$ , 200, 300 GeV and $\tan β = 50$ . Point notation as in Fig. 6. Negative values indicate cases for which $M_{Q, U, D}^{2} < 0$ in which case the plot gives $- \sqrt{- M_{Q, U, D}^{2}}$ .Reuse & Permissions
Figure 21
$F$ vs $A_{λ} (m_{Z})$ and $A_{κ} (m_{Z})$ (upper two plots) and $A_{κ} (m_{Z})$ vs $A_{λ} (m_{Z})$ (lower plot) for points with $F < 50$ taking $M_{1, 2, 3} (m_{Z}) = 100$ , 200, 300 GeV and $\tan β = 10$ . Point notation as in Fig. 6.Reuse & Permissions
Figure 22
$m_{a_{1}}$ vs $A_{λ} (m_{Z})$ and $A_{κ} (m_{Z})$ for points with $F < 50$ taking $M_{1, 2, 3} (m_{Z}) = 100$ , 200, 300 GeV and $\tan β = 10$ . Point notation as in Fig. 6.Reuse & Permissions
Figure 23
$F$ vs $A_{λ} (M_{U})$ and $A_{κ} (M_{U})$ (upper two plots) and $A_{κ} (M_{U})$ vs $A_{λ} (M_{U})$ (lower plot) for points with $F < 50$ taking $M_{1, 2, 3} (m_{Z}) = 100$ , 200, 300 GeV and $\tan β = 10$ . Point notation as in Fig. 6.Reuse & Permissions
Figure 24
Fine-tuning vs $m_{H_{u}}^{2} (M_{U})$ , $m_{H_{d}}^{2} (M_{U})$ , and $m_{S}^{2} (M_{U})$ for points with $F < 50$ taking $M_{1, 2, 3} (m_{Z}) = 100$ , 200, 300 GeV and $\tan β = 10$ . Point notation as in Fig. 6.Reuse & Permissions
Figure 25
Fine-tuning vs $m_{H_{u}}^{2} (M_{U})$ , $m_{H_{d}}^{2} (M_{U})$ , and $m_{S}^{2} (M_{U})$ for points with $F < 50$ taking $M_{1, 2, 3} (m_{Z}) = 100$ , 200, 300 GeV and $\tan β = 50$ . Point notation as in Fig. 6.Reuse & Permissions
Figure 26
Fine-tuning vs the GUT-scale soft Higgs masses squared for points with $F < 50$ taking $M_{1, 2, 3} (m_{Z}) = 100$ , 200, 300 GeV and $\tan β = 10$ . Point notation: dark (red) $m_{H_{u}}^{2} (M_{U})$ ; light gray (cyan) $m_{H_{d}}^{2} (M_{U})$ ; darker gray (green) $m_{S}^{2} (M_{U})$ . Points plotted are the yellow fancy cross points from Fig. 24.Reuse & Permissions
Figure 27
Fine-tuning vs the relative coupling strength, $C_{V} \equiv g_{h_{1} W W} / g_{h_{SM} W W}$ for $\tan β = 10$ . Point notation as in Fig. 6.Reuse & Permissions
Figure 28
The upper plot shows fine-tuning vs $m_{h_{1}}$ for the large (yellow) cross points (for clarity, we use blue $+$ ’s in their place in this and succeeding plots) with $F < 10$ taking $M_{1, 2, 3} (m_{Z}) = 100$ , 200, 300 GeV and $\tan β = 10$ . The lower plot shows $F$ vs $m_{a_{1}}$ .Reuse & Permissions
Figure 29
The plot shows $ξ^{2} (Z + b^{'} s)$ (which equals $C_{eff}^{2 b}$ since $C_{eff}^{4 b} = 0$ when $m_{a_{1}} < 2 m_{b}$ ) vs $F$ for the large (yellow) cross points with $F < 10$ taking $M_{1, 2, 3} (m_{Z}) = 100$ , 200, 300 GeV and $\tan β = 10$ .Reuse & Permissions
Figure 30
The plots show $F$ vs $\cos θ_{A}$ (upper) and $F$ vs $\sin θ_{S}$ (lower) for the large (yellow) cross points with $F < 10$ taking $M_{1, 2, 3} (m_{Z}) = 100$ , 200, 300 GeV and $\tan β = 10$ .Reuse & Permissions
Figure 31
The upper plot shows fine-tuning vs $λ (M_{U})$ for large (yellow) cross points with $F < 10$ taking $M_{1, 2, 3} (m_{Z}) = 100$ , 200, 300 GeV and $\tan β = 10$ . The lower plot shows $κ (M_{U})$ as a function of $λ (M_{U})$ .Reuse & Permissions
Figure 32
The upper plot shows fine-tuning vs $A_{λ} (M_{U})$ for fully okay points with $F < 10$ taking $M_{1, 2, 3} (m_{Z}) = 100$ , 200, 300 GeV and $\tan β = 10$ . The lower plot shows $A_{κ} (M_{U})$ as a function of $A_{λ} (M_{U})$ .Reuse & Permissions
Figure 33
The upper plot shows fine-tuning vs $m_{H_{u}} (M_{U})$ for fully okay points with $F < 10$ taking $M_{1, 2, 3} (m_{Z}) = 100$ , 200, 300 GeV and $\tan β = 10$ . The middle plot shows $m_{H_{d}} (M_{U})$ as a function of $m_{H_{u}} (M_{U})$ . The bottom plot shows $m_{S} (M_{U})$ as a function of $m_{H_{u}} (M_{U})$ . Our convention is that if an $M^{2}$ is negative then we plot $- \sqrt{- M^{2}}$ .Reuse & Permissions
Figure 34
For the points with dominant $a_{1} \to γ γ$ decays and low $F$ , we plot: $F$ vs $m_{h_{1}}$ (top); and $m_{a_{1}}$ vs $m_{h_{1}}$ (bottom). Note that there are a number of degeneracies where exactly the same $F$ and $m_{h_{1}}$ are predicted for somewhat different parameter choices in the scan.Reuse & Permissions
Figure 35
For the points with dominant $a_{1} \to γ γ$ decays and low $F$ , we plot: $F$ vs $\cos θ_{A}$ (top); $B (a_{1} \to γ γ)$ vs $\cos θ_{A}$ (middle); and $ξ^{2} (Z + b^{'} s)$ vs $m_{h_{1}}$ (bottom). Since $F$ depends primarily on $\cos θ_{A}$ , there are a number of points in the $F$ vs $\cos θ_{A}$ plot that are actually multiple repetitions of exactly the same $F$ at a given $\cos θ_{A}$ value but with $B (a_{1} \to γ γ)$ and $ξ^{2} (Z + b^{'} s)$ varying slightly because of sensitivity to other parameters of the scan.Reuse & Permissions
Figure 36
For the points with dominant $a_{1} \to γ γ$ decays and low $F$ , we plot: $M_{12}^{2} / m_{a_{1}}^{2}$ vs $\cos θ_{A}$ (top); and $F_{A_{λ}}^{\cos θ_{A}}$ as a function of $\cos θ_{A}$ (bottom). (Many of the points have the same $F_{A_{λ}}^{\cos θ_{A}}$ value.)Reuse & Permissions
Figure 37
For the points with dominant $a_{1} \to γ γ$ decays and low $F$ , we plot: $κ$ vs $λ$ (top); and $A_{κ}$ vs $A_{λ}$ (bottom).Reuse & Permissions

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