h Ka-Tp-01-2012
h Sfb/cpp-12-02
h Uh511-1188-2012
NMSSM Higgs Benchmarks Near 125 GeV

The ATLAS and CMS Collaborations have recently presented the first indication for a Higgs boson with a mass in the region $\sim 124-126$ GeV [1, 2]. An excess of events is observed by the ATLAS experiment for a Higgs boson mass hypothesis close to 126 GeV with a maximum local statistical significance of 3.6$\sigma$ above the expected SM background and by the CMS experiment at 124 GeV with 2.6$\sigma$ maximum local significance. If the ATLAS and CMS signals are combined the statistical significance increases, but is still less than the 5$\sigma$ required to claim a discovery. Interestingly, the ATLAS signal in the $\gamma \gamma$ decay channel by itself has a local significance of 2.8$\sigma$ whereas a SM-like Higgs boson would only have a significance of half this value, leading to speculation that the observed Higgs boson is arising from beyond SM physics. In general, these results have generated much excitement in the community, and already there are a number of papers discussing the implications of such a Higgs boson [3, 4, 5, 6, 7].

In the Minimal Supersymmetric Standard Model (MSSM) the lightest Higgs boson is lighter than about 130-135 GeV, depending on top squark parameters (see e.g. [8] and references therein). A 125 GeV SM-like Higgs boson is consistent with the MSSM in the decoupling limit. In the limit of decoupling the light Higgs mass is given by

where $\Delta m_h^2$ is dominated by loops of heavy top quarks and top squarks and $\tan \beta$ is the ratio of the vacuum expectation values (VEVs) of the two Higgs doublets introduced in the MSSM Higgs sector. At large $\tan \beta$, we require $\Delta m_h\approx 85$ GeV which means that a very substantial loop contribution, nearly as large as the tree-level mass, is needed to raise the Higgs boson mass to 125 GeV. The rather complicated parameter dependence has been studied in [4] where it was shown that, with “maximal stop mixing”, the lightest stop mass must be $m_{{\tilde{t}}_1}\text{raisebox}-3.698858pt\text{shortstack}\$>\$[-0.07cm]\$\sim \$500$ GeV (with the second stop mass considerably larger) in the MSSM in order to achieve a 125 GeV Higgs boson. However one of the motivations for SUSY is to solve the hierarchy or fine-tuning problem of the SM [9]. It is well known that such large stop masses typically require a tuning at least of order 1% in the MSSM, depending on the parameter choice and the definition of fine-tuning [10].

In the light of such fine-tuning considerations, it has been known for some time, even after the LEP limit on the Higgs boson mass of 114 GeV, that the fine-tuning of the MSSM could be ameliorated in the Next-to-Minimal Supersymmetric Standard Model (NMSSM) [11]. With a 125 GeV Higgs boson, this conclusion is greatly strengthened and the NMSSM appears to be a much more natural alternative. In the NMSSM, the spectrum of the MSSM is extended by one singlet superfield [12, 13, 14] (for reviews see [15, 16]). In the NMSSM the supersymmetric Higgs mass parameter $\mu$ is promoted to a gauge-singlet superfield, $S$, with a coupling to the Higgs doublets, $\lambda S{H}_u{H}_d$, that is perturbative up to unified scales. In the pure NMSSM values of $\lambda \sim 0.7$ do not spoil the validity of perturbation theory up to the GUT scale only providing $\tan \beta \gtrsim 4$, however the presence of additional extra matter [17] allows smaller values of $\tan \beta$ to be achieved. The maximum mass of the lightest Higgs boson is

where here we use $v=174$ GeV. For $\lambda v>M_Z$, the tree-level contributions to $m_h$ are maximized for moderate values of $\tan \beta$ rather than by large values of $\tan \beta$ as in the MSSM. For example, taking $\lambda =0.7$ and $\tan \beta =2$, these tree-level contributions raise the Higgs boson mass to about 112 GeV, and $\Delta m_h\gtrsim 55\text{GeV}$ is required. This is to be compared to the MSSM requirement $\Delta m_h\gtrsim 85\text{GeV}$. The difference between these two values (numerically about 30 GeV) is significant since $\Delta m_h$ depends logarithmically on the stop masses as well as receiving an important contribution from stop mixing. This means for example, that, unlike the MSSM, in the case of the NMSSM maximal stop mixing is not required to get the Higgs heavy enough.

The NMSSM has in fact several attractive features as compared to the widely studied MSSM. Firstly, the NMSSM naturally solves in an elegant way the so–called $\mu$ problem [18] of the MSSM, namely that the phenomenologically required value for the supersymmetric Higgsino mass $\mu$ in the vicinity of the electroweak or SUSY breaking scale is not explained. This is automatically achieved in the NMSSM, since an effective $\mu$-parameter is dynamically generated when the singlet Higgs field acquires a vacuum expectation value of the order of the SUSY breaking scale, leading to a fundamental Lagrangian that contains no dimensionful parameters apart from the soft SUSY breaking terms. Secondly, as compared to the MSSM, the NMSSM can induce a richer phenomenology in the Higgs and neutralino sectors, both in collider and dark matter (DM) experiments. For example, heavier Higgs states can decay into lighter ones with sizable rates [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. In addition, a new possibility appears for achieving the correct cosmological relic density [31] through the so-called “singlino”, i.e. the fifth neutralino of the model, which can have weaker-than-usual couplings to SM particles. The NMSSM Higgs boson near 125 GeV may also have significantly different properties than the SM Higgs boson [5, 6], as we shall discuss in some detail here. Thirdly, as already discussed at length above, the NMSSM requires less fine-tuning than the MSSM [11].

Since the NMSSM (as in the case for the MSSM) has a rich parameter space, it is convenient to consider the standard approach of so-called “benchmark points” in the SUSY parameter space. These consist of a few “discrete” parameter configurations of a given SUSY model, which in our case are supposed to lead to typical phenomenological features. Using discrete points avoids scanning the entire parameter space, focusing instead on representative choices that reflect the new interesting features of the model, such as new signals, peculiar mass spectra, and so on. A reduced number of points can then be subject to full experimental investigation, without loss of substantial theoretical information. While several such benchmark scenarios have been devised for the MSSM [32, 33] and thoroughly studied in both the collider and the DM contexts, and even for the NMSSM [34], these need to be revised in the light of the 125 GeV Higgs signal. The motivation for finding such NMSSM benchmarks is to enable the characteristics of the NMSSM Higgs boson near 125 GeV to be identified, so that it may eventually be resolved from that of the SM Higgs boson.

In this paper we present a set of NMSSM benchmark points with a SM-like Higgs boson mass near 125 GeV, focussing on the cases where both stop masses are as light as possible in order to reduce the fine-tuning, in accordance with the above discussion. The tools to calculate the Higgs and SUSY particle spectra in the NMSSM, in particular NMSSMTools, have been available for some time [21, 35, 36], although these will be supplemented by other codes as discussed later. The goal of the paper is to firstly find NMSSM benchmark points which contain a 125 GeV SM-like Higgs boson and in addition involve relatively light stops. Since the stops receive radiative corrections to their masses from gluinos, we shall also require relatively light gluinos as well as having as small an effective ${\mu }_{eff}$ parameter (${\mu }_{eff}=\lambda ⟨S⟩$) as possible [37]. Having relatively light stops (and sbottoms), care has to be taken not to be in conflict with direct searches at the Tevatron and recent searches at the LHC. In addition, light stops appearing in the triangle loop diagrams can significantly affect Higgs production via gluon fusion as well as Higgs decay into two photons. Once a parameter set leading to a 125 GeV NMSSM Higgs boson has been found it has to be checked if the production cross-sections and branching ratios are such that they lead to a total Higgs production cross-section times branching ratios which is compatible with the LHC searches. The obtained branching ratios can be used to ultimately distinguish the NMSSM Higgs boson from the SM Higgs state. However, we do not attempt to simulate the LHC search strategies, our goal being the much more modest one of providing benchmarks with different characteristic types of NMSSM Higgs bosons, which can eventually be studied in more detail. The types of benchmark points considered here all involve relatively large values of $\lambda >0.5$ and small values of $\tan \beta \le 3$, in order to allow for the least fine-tuning possible involving light stops as discussed above. However we are careful to respect the perturbativity bounds on $\lambda$ up to the GUT scale, either for the pure NMSSM, or the NMSSM supplemented by three copies of extra $SU\left(5\right)$ $5+\overline{5}$ states, where such bounds are calculated using two-loop renormalisation group equations (RGEs). We find that the NMSSM Higgs boson near 125 GeV can come in many guises. It may be very SM-like, practically indistinguishable from the SM Higgs boson. Or it may have different Higgs production cross-sections and widely varying decay branching ratios which enable it to be resolved from the SM Higgs boson. The key distinguishing feature of the NMSSM, compared to the MSSM, is the presence of the singlet $S$ which may mix into the 125 GeV Higgs mass state to a greater or lesser extent. As the singlet component of the 125 GeV Higgs boson is increased, either the $H_d$ or $H_u$ components (or both) must be reduced. If the $H_d$ component is reduced then this reduces the decay rate into bottom quarks and can allow other rare decays like $\gamma \gamma$ to have larger branching ratios [6]. In addition there is the effect of SUSY particle and charged Higgs boson loops in the Higgs coupling to photons which can increase or decrease the rate of decays into $\gamma \gamma$.

In order to put the present work in context, it is worth to emphasise that this is the first paper to propose “natural” (i.e. involving light stops) NMSSM benchmark points with a SM-like Higgs boson near 125 GeV. For example, the previous NMSSM benchmark paper [34] was concerned with the constrained NMSSM which does not allow light stops consistent with current LHC SUSY limits. Also, while this paper was being prepared there have appeared a number of other related papers, none of which however are focussed on the task of providing benchmark points. For example, the results in [4] were mainly concerned with fine-tuning issues in SUSY models with a $\lambda S{H}_u{H}_d$ coupling for a SM-like Higgs boson near 126 GeV and the actual NMSSM was strictly not considered since an explicit $\mu$ term was included whereas the trilinear singlet coupling $\frac{\kappa }{3}{S}^3$ was neglected. By contrast another recent paper [5] did consider the actual NMSSM although no benchmarks were proposed. As this paper was being finalised, further dedicated NMSSM papers with a Higgs boson near 125 GeV have started to appear, in particular [6] in which the two photon enhancement was emphasised but without benchmark points, and [7] where (versions of) the constrained NMSSM were considered. It should be clear that the present paper is both complementary and contemporary to all these papers.

The layout of the remainder of the paper is as follows. In Section 2 we discuss the MSSM and fine-tuning, showing that it leads to constraints on the mass of the heavier stop quark. In Section 3 we briefly review the NMSSM, and provide perturbativity limits of the coupling $\lambda$ depending on the coupling $\kappa$ as well as $\tan \beta$. We show how the limit on $\lambda$ may be increased if there is extra matter in the SUSY desert. Section 4 is concerned with constraints from Higgs searches, SUSY particle searches and Dark Matter. Section 5 contains the four sets of NMSSM Higgs benchmarks near 125 GeV that we are proposing, including the Higgs production cross-sections and branching ratios which ultimately will enable it to be distinguished from the SM Higgs boson. Section 6 summarises and concludes the paper. Appendix A contains the two-loop RGEs from which the perturbativity bounds on $\lambda$ were obtained.

In the MSSM, at the 1-loop level, stops contribute to the Higgs boson mass and three more parameters become important, the stop soft masses, $m_{{\tilde{Q}}_3}$ and $m_{{\tilde{t}}_R}$, and the stop mixing parameter

The dominant one-loop contribution to the Higgs boson mass depends on the geometric mean of the stop masses, $m_{\tilde{t}}^2=m_{{\tilde{Q}}_3}{m}_{{\tilde{t}}_R}$, and is given by,

The Higgs mass is sensitive to the degree of stop mixing through the second term in the brackets, and is maximized for $|X_t|=X_t^{max}=\sqrt{6}{m}_{\tilde{t}}$, which was referred to as “maximal mixing” above.

It has been noted that large or maximal stop mixing is associated with large fine-tuning. This also follows from Fig.1 and Eq. (2.12). Indeed, Fig.1 demonstrates that the contribution of one–loop corrections to Eq. (2.11) increases when the mixing angle in the stop sector becomes larger. In fact when ${\theta }_t$ is close to $\pi /4$ the last term in Eq. (2.12) gives the dominant contribution to $\Delta$ enhancing the overall contribution of loop corrections in the minimization condition (2.11) which determines the mass of the $Z$–boson.

Eq. (2.11) also indicates that in order to avoid tuning one has to ensure that the parameter $\mu$ has a reasonably small value. To avoid tuning entirely one should expect $\mu$ to be less than $M_Z$. However, so small values of the parameter $\mu$ are ruled out by chargino searches at LEP. Therefore in our analysis we allow the effective ${\mu }_{eff}$ parameter to be as large as $200\text{GeV}$ that does not result in enormous fine-tuning.

In this paper, we only consider the NMSSM with a scale invariant superpotential. Alternative models known as the minimal non-minimal supersymmetric SM (MNSSM), new minimally-extended supersymmetric SM or nearly-minimal supersymmetric SM (nMSSM) or with additional $U(1{)}^{\prime }$ gauge symmetries have been considered elsewhere [38], as has the case of explicit CP violation [39].

The growth of the Yukawa couplings $h_t$, $\lambda$ and $\kappa$ at the electroweak (EW) scale entails the increase of their values at the Grand Unification scale $M_X$ resulting in the appearance of the Landau pole. Large values of $h_t\left(M_X\right)$, $\lambda \left(M_X\right)$ and $\kappa \left(M_X\right)$ spoil the applicability of perturbation theory at high energies so that the RGEs cannot be used for an adequate description of the evolution of gauge and Yukawa couplings at high scales $Q\sim M_X$. The requirement of validity of perturbation theory up to the Grand Unification scale restricts the interval of variations of Yukawa couplings at the EW scale. In particular, the assumption that perturbative physics continues up to the scale $M_X$ sets an upper limit on the low energy value of $\lambda \left(M_Z\right)$ for each fixed set of $\kappa \left(M_Z\right)$ and $h_t\left(M_t\right)$ (or $\tan \beta$). With decreasing (increasing) $\kappa \left(M_Z\right)$ the maximal possible value of $\lambda \left(M_Z\right)$, which is consistent with perturbative gauge coupling unification, increases (decreases) for each particular value of $\tan \beta$. In Table 1 we display two-loop upper bounds on $\lambda \left(M_Z\right)$ for different values of $\kappa \left(M_Z\right)$ and $\tan \beta$ in the NMSSM. As one can see the allowed range for the Yukawa couplings varies when $\tan \beta$ changes. Indeed, for $\tan \beta =2$ the value of $\lambda \left(M_Z\right)$ should be smaller than $0.62$ to ensure the validity of perturbation theory up to the scale $M_X$. At large $\tan \beta$ the allowed range for the Yukawa couplings enlarges. The upper bound on $\lambda \left(M_Z\right)$ grows with increasing $\tan \beta$ because the top–quark Yukawa coupling decreases. At large $\tan \beta$ (i.e. $\tan \beta >4-5$) the upper bound on $\lambda \left(M_Z\right)$ approaches the saturation limit where ${\lambda }_{max}\simeq 0.72$.

The renormalisation group (RG) flow of the Yukawa couplings depends rather strongly on the values of the gauge couplings at the intermediate scales. To demonstrate this we examine the RG flow of gauge and Yukawa couplings within the SUSY model that contains three extra $SU\left(5\right)$ $(5+\overline{5})$–plets that survive to low energies and can form three $10$–plets in the SUSY-GUT model based on the $SO\left(10\right)$ gauge group. In this SUSY model the strong gauge coupling has a zero one–loop beta function whereas at two–loop level the coupling has a mild growth as the renormalisation scale increases. Since extra states form complete $SU\left(5\right)$ multiplets the high-energy scale where the unification of the gauge couplings takes place remains almost the same as in the MSSM. At the same time extra multiplets of matter change the running of the gauge couplings so that their values at the intermediate scale rise substantially. In fact the two–loop beta functions of the SM gauge couplings are quite close to their saturation limits when these couplings blow up at the GUT scale. Further enlargement of the particle content can lead to the appearance of the Landau pole during the evolution of the gauge couplings from $M_Z$ to $M_X$. Because the beta functions are so close to the saturation limits the RG flow of the gauge couplings (i.e. their values at the intermediate scale) also depends on the masses of the extra exotic states. Since $g_i\left(Q\right)$ occurs in the right–hand side of the RGEs for the Yukawa couplings with negative sign the growth of the gauge couplings prevents the appearance of the Landau pole in the evolution of these couplings. It means that for each value of $h_t\left(M_t\right)$ (or $\tan \beta$) and $\kappa \left(M_Z\right)$ the upper limit on $\lambda \left(M_Z\right)$ increases as compared with the NMSSM. The two-loop upper bounds on $\lambda \left(M_Z\right)$ for different values of $\kappa \left(M_Z\right)$ and $\tan \beta$ in the NMSSM supplemented by three $SO\left(10\right)$ $10$–plets, or equivalently three $SU\left(5\right)$ $(5+\overline{5})$–plets are displayed in Table 2 for the case that the masses of all extra exotic states are set to be equal to $300\text{GeV}$ and in Table 3 in case they are set to be equal to $1\text{TeV}$. The two-loop RGEs used to obtain these results are given in Appendix A. Because the RG flow of the gauge couplings depends on the masses of extra exotic states the upper bounds on $\lambda \left(M_Z\right)$ presented in Tables 2 and 3 are slightly different. The restrictions on $\lambda \left(M_Z\right)$ obtained in this section are useful for the phenomenological analysis which we are going to consider next.

Our scenarios are subject to constraints on the Higgs boson masses from the direct searches at LEP, Tevatron and the LHC. Also the SUSY particle masses have to be compatible with the limits given by the experiments. Finally, the currently measured value of the relic density shall be reproduced. Further constraints arise from the low-energy observables.