The constrained NMSSM and Higgs near 125 GeV

We assess the extent to which various constrained versions of the NMSSM are able to describe the recent hints of a Higgs signal at the LHC corresponding to a Higgs mass in the range 123-128 GeV.

The Large Hadron Collider (LHC) data from the ATLAS [1] and CMS [2] collaborations suggests the possibility of a fairly Standard Model (SM) like Higgs boson with mass of order 123 − 128 GeV. In particular, promising hints appear of a narrow excess over background in the γγ and ZZ → 4 final states with strong supporting evidence from the W W → ν ν mode. While the ATLAS and CMS results suggest that the γγ rate may be somewhat enhanced with respect to the SM expectation, this is by at most one standard-deviation (1σ).
In this Letter, we explore the ability or lack thereof of three constrained versions of the next-to-minimal supersymmetric standard model (NMSSM) to describe these observations while remaining consistent with all relevant constraints, including those from LEP and TEVATRON searches, B-physics, the muon anomalous magnetic moment, a µ ≡ (g − 2) µ /2, and the relic density of dark matter, Ωh 2 .
The possibility of describing the LHC observations in the context of the MSSM has been explored in numerous papers, including [3][4] [5][6] [7][8] [9][10] [11]. A general conclusion seems to be that if all the constraints noted above, including a µ and Ωh 2 , are imposed rigorously, then the MSSM-especially a constrained version such as the CMSSM-is hard pressed to yield a fairly SM-like light Higgs boson at 125 GeV. This is somewhat alleviated when the a µ constraint is dropped [3] [7]. Overall, however, large mixing and large SUSY masses are needed to achieve m h ∼ 125 GeV. There has also been some exploration in the context of the NMSSM [12][13] [11], showing that for completely general parameters there is less tension between a light Higgs with mass ∼ 125 GeV and a lighter SUSY mass spectrum. The study presented here will be done in the context of several constrained versions of the NMSSM with universal or semi-universal GUT scale boundary conditions. Results for a very constrained version of the NMSSM, termed the cNMSSM, appear in [14] [15][4] -we discuss comparisons later in the paper.
The three models which we discuss here are defined in terms of grandunification (GUT) scale parameters as follows: I) a version of the constrained NMSSM (CNMSSM) in which we adopt universal m 0 , m 1/2 , A 0 = A t,b,τ values but require A λ = A κ = 0, as motivated by the U (1) R symmetry limit of the NMSSM; II) the non-universal Higgs mass (NUHM) relaxation of model I in which m Hu and m H d are chosen independently of m 0 , but still with A λ = A κ = 0; and III) universal m 0 , m 1/2 , A 0 with NUHM relaxation and general A λ and A κ .
We use NMSSMTools-3.0.2 [16] [17][18] for the numerical analysis, performing extensive scans over the parameter spaces of the models considered. The precise constraints imposed are the following. Our 'basic constraints' will be to require that an NMSSM parameter choice be such as to give a proper RGE solution, have no Landau pole, have a neutralino LSP and obey Higgs and SUSY mass limits as implemented in NMSSMTools-3.0.2 (Higgs mass limits are from LEP, TEVATRON, and early LHC data; SUSY mass limits are essentially from LEP). Regarding B physics, the constraints considered are those on BR(B s → X s γ), ∆M s , ∆M d , BR(B s → µ + µ − ), BR(B + → τ + ν τ ) and BR(B → X s µ + µ − ) at 2σ as encoded in NMSSMTools-3.0.2, except that we updated the bound on the radiative B s decay to 3.04 < BR(B s → X s γ) × 10 4 < 4.06; theoretical uncertainties in B-physics observables are taken into account as implemented in NMSSMTools-3.0.2. These combined constraints we term the 'B-physics contraints'. Regarding a µ , we require that the extra NMSSM contribution, δa µ , falls into the window defined in NMSSMTools of 8.77 × 10 −10 < δa µ < 4.61 × 10 −9 expanded to 5.77 × 10 −10 < δa µ < 4.91 × 10 −9 after allowing for a 1σ theoretical error in the NMSSM calculation of ±3×10 −10 . In fact, points that fail to fall into the above δa µ window always do so by virtue of δa µ being too small. For Ωh 2 , we declare that the relic density is consistent with WMAP data provided 0.094 < Ωh 2 < 0.136, which is the 'WMAP window' defined in NMSSMTools-3.0.2 after including theoretical and experimental systematic uncertainties. We will also consider the implications of relaxing this constraint to simply Ωh 2 < 0.136 so as to allow for scenarios in which the relic density arises at least in part from some other source. A "perfect" point will be one for which all constraints are satisfied including requring that δa µ is in the above defined window and Ωh 2 is in the WMAP window.
We find that only in models II and III is it possible for a "perfect" point to have a light scalar Higgs in the mass range 123 − 128 GeV as consistent with the hints from the recent LHC Higgs searches. The largest m h 1 achieved for perfect points is about 125 GeV. However, relaxing the a µ constraint vastly increases the number of accepted points and it is possible to have m h 1 > ∼ 126 GeV in both models II and III even if δa µ is just slightly outside (below) the allowed window. Comparing with [3], the tension between obtaining an ideal or nearly ideal δa µ while predicting a SM-like light Higgs near 125 GeV appears to be somewhat less in NUHM variants of the NMSSM than in those of the MSSM.
In the plots shown in the following, the coding for the plotted points is as follows: • grey squares pass the 'basic' constraints but fail B-physics constraints (such points are rare); • green squares pass the basic constraints and satisfy B-physics constraints; • blue plusses (+) observe B-physics constraints as above and in addition have Ωh 2 < 0.136, thereby allowing for other contributions to the dark matter density (a fraction of order 20% of these points have 0.094 < Ωh 2 < 0.136) but they do not necessarily have acceptable δa µ ; • magenta crosses (×) have satisfactory δa µ as well as satisfying B-physics constraints, but arbitrary Ωh 2 ; • golden triangle points pass all the same constraints as the magenta points and in addition have Ωh 2 < 0.136; • open black/grey 1 triangles are perfect, completely allowed points in the sense that they pass all the constraints listed earlier, including 5.77×10 −10 < δa µ < 4.91 × 10 −9 and 0.094 < Ωh 2 < 0.136; • open white diamonds are points with m h 1 ≥ 123 GeV that pass basic constraints, B-physics constraints and predict 0.094 < Ωh 2 < 0.136 but have 4.27 × 10 −10 < δa µ < 5.77 × 10 −10 , that is we allow an excursion of half the 1σ theoretical systematic uncertainty below the earlier defined window. We will call these "almost perfect" points.
The only Higgs production mechanism relevant for current LHC data is gluongluon to Higgs. For our plots it will thus be useful to employ the ratio of the gg induced Higgs cross section times the Higgs branching ratio to a given final state, X, relative to the corresponding value for the SM Higgs boson: where h i is the i th NMSSM scalar Higgs, and h SM is the SM Higgs boson. The ratio is computed in a self-consistent manner (that is, treating radiative corrections for the SM Higgs boson in the same manner as for the NMSSM Higgs bosons) using an appropriate additional routine for the SM Higgs added to the NMHDECAY component of the NMSSMTools package. To compute the SM denominator, we proceed as follows. 2 NMHDECAY computes couplings for each We then have all the information needed to compute R h i for some given final state X.
We begin by presenting the crucial plots of Fig. 1 in which we show R h 1 (γγ) as a function of m h 1 for cases I, II and III. Only in cases II and III do we find points that pass all constraints (the open black triangles) with m h 1 ∼ 124 − 125 GeV. These typically have R h 1 (γγ) of order 0.98. Somewhat surprisingly, such points were more easily found by our scanning procedure in case II than in case III. Many additional points with m h 1 ∼ 125 GeV emerge if we relax only slightly the δa µ constraint. The white diamonds show points for cases for which 4.27×10 −10 < δa µ < 5.77 × 10 −10 having m h 1 ≥ 123 GeV. As can be seen in more detail from the sample point tables presented later, the parameter choices that give the largest m h 1 values are ones for which the h 1 is really very SM-like in terms of its couplings and branching ratios. Our scans did not find parameter choices for which R h 1 (γγ) was significantly larger than 1 for m h 1 = 123 − 128 GeV, as hinted at by the ATLAS data.
As regards h 2 , if we require m h 2 ∈ [110 − 150] GeV then we find points that pass the basic constraints and the B-physics constraints, but none that pass the further constraints. So, it appears that within these models it is the h 1 that must be identified with the Higgs observed at the LHC. In contrast, if parameters are chosen at the SUSY scale without regard to GUT-scale unification, it is possible to find scenarios in which m h 2 ≈ 125 GeV and, moreover, R h 2 (γγ) > 1 [13].
In passing, we note that should the Higgs hints disappear and a low-mass SM-like Higgs be excluded then it is of interest to know if BR(h 1 → a 1 a 1 ) can be large for m h 1 in the < ∼ 130 GeV range. It turns out that, although large BR(h 1 → a 1 a 1 ) is possible while satisfying basic and B-physics constraints, once additional constraints are imposed, BR(h 1 → a 1 a 1 ) < ∼ 0.2 for all three model cases being considered. Small BR(h 1 → a 1 a 1 ) is expected [20] (see also [21]) when the a 1 is very singlet, as is the case in our scenarios once all constraints are imposed. So, in these models a light Higgs has nowhere to hide.
The points in the scatter plots were primarily obtained through random scans over the parameter spaces of the three models considered. In addition, we performed Markov Chain Monte Carlo (MCMC) scans to zero in better on points with m h 1 ∼ 125 GeV that observe all constraints. For this purpose, we defined a χ 2 (m h 1 ) = (m h 1 − 125) 2 /(1.5) 2 . The B-physics constraints were also implemented using a χ 2 approach with the 1σ errors from theory and experiment (as implemented in NMSSMTools) combined in quadrature. The global likelihood was then computed as L tot = i L i with L i = e −χ 2 i /2 for two-sided constraints i is a 95% CL upper limit. The a µ and Ωh 2 constraints were either implemented a-posteriori using the 2σ window approach of NMSSMTools, or also included in the global likelihood. Since CMSSM-like boundary conditions with A λ = A κ = 0 did not generate points anywhere near the interesting region, we have only performed this kind of scan for cases II and III. This allowed us to find additional "perfect" and "almost perfect" points for models II and III with m h 1 > ∼ 123 GeV.
We next illustrate in Fig. 2 R h 1 (V V ) (the ratio being the same for V V = W W and V V = ZZ) for boundary condition cases II and III. As for the γγ final state, for m h 1 > ∼ 123 GeV the predicted rates in the V V channels are very nearly SMlike. Overall, it is clear that, for the GUT scale boundary conditions considered here, one finds that for parameter choices yielding consistency with all constraints and yielding m h 1 close to 125 GeV, the h 1 will be very SM-like. If future data confirms a γγ rate in excess of the SM prediction, then it will be necessary to go beyond the constrained versions of the NMSSM considered here (cf. [13]). And, certainly it is very difficult within the constrained models considered here to obtain a SM-like Higgs with mass much above 126 GeV for parameter choices such that all constraints, including δa µ and Ωh 2 , are satisfied.
Should a later LHC data set prove consistent with a rather SM-like Higgs in the vicinity of m h 1 ∼ 125 GeV (rather than one with an enhanced γγ rate), it will be of interest to know the nature of the parameter choices that yield the perfect, black triangle and almost perfect white diamond points with m h 1 ∼ 125 GeV and what the other experimental signatures of these points are. We therefore present a brief summary of the most interesting features. First, one must ask if such points are consistent with current LHC limits on SUSY particles, in particular squarks and gluinos. To this end, Fig. 3 shows the distribution of squark and gluino masses for the various kinds of points for models II and III. Interestingly, all the perfect, black triangle and almost perfect, white diamond points with m h 1 > ∼ 123 GeV have squark and gluino masses above 1 TeV and thus have not yet been probed by current LHC results. (Note that since we are considering models with universal m 0 and m 1/2 for squarks and gauginos, analyses in the context of the CMSSM apply.) It is quite intriguing that the regions of parameter space that are consistent with a Higgs of mass close to 125 GeV automatically evade the current limits from LHC SUSY searches.
In order to further detail the parameters and some relevant features of perfect and almost perfect points we present in Tables 1-4 seven exemplary points with m h 1 > ∼ 124 GeV from models II and III. Some useful observations include the following: • Because of the way we initiated our model III MCMC scans, restricting |A λ,κ | ≤ 1 TeV, most of the tabulated model III points have quite modest A λ and A κ . However, a completely random scan finds almost perfect points with quite large A λ and A κ values as exemplified by tabulated point #7. The fact that the general scan over A λ and A κ did not find any perfect points with m h 1 > ∼ 124 GeV, whereas such points were fairly quickly found using the MCMC technique, suggests that such points are quite fine-tuned in the general scan sense. See Table 1 for specifics.
• In Table 2, we display various details regarding the Higgs bosons for each of our exemplary points. As already noted, for the perfect and almost perfect points the h 1 is very SM-like when m h 1 > ∼ 123 GeV. To quantify how well the LHC Higgs data is described for each of our exemplary points, we use a chisquared approach. In practice, only the ATLAS collaboration has presented the best fit values for R h (γγ, ZZ → 4 , W W → ν ν) along with 1σ upper and lower errors as a function of m h . Identifying h with the NMSSM h 1 , we have employed Fig. 8 of [1] to compute a χ 2 (ATLAS) for each point in the NMSSM parameter space (but this was not included in the global likelihood used for our MCMC scans). From Table 2 we see that the smallest χ 2 (ATLAS) values (of order 0.6 to 0.7) are obtained for m h 1 ∼ 124 GeV. This is simply because at this mass the ATLAS fits to R h (γγ) and R h (4 ) are very close to one, the natural prediction in the NMSSM context. For m h ∼ 125 GeV, the R h 's for the ATLAS data are somewhat larger than 1 leading to a discrepancy with the NMSSM SM-like prediction and a roughly doubling of χ 2 (ATLAS) to values of order 1.3 to 1.6 for our exemplary points.
In this context, we should note that at a Higgs mass of 125 GeV the CMS data is best fit if the Higgs signals are not enhanced and, indeed, are very close to SM values.
• The mass of the neutralino LSP, χ 0 1 , is rather similar, m χ 0 1 ≈ 300 − 450 GeV, for the different perfect and almost perfect points with m h 1 > ∼ 124 GeV. For all but pt. #5, the χ 0 1 is approximately an equal mixture of higgsino and bino. There is some variation in the primary annihilation mechanism, with τ 1 τ 1 and χ 0 1 χ 0 1 annihilation being the dominant channels except for pt. #2 for which ν τ ν τ and ν τ ν τ annihilations are dominant. In the case of dominant τ 1 τ 1 annihilation, the bulk of the χ 0 1 's come from those τ 's that have not annihilated against one another or co-annihilated with a χ 0 1 .
• All the tabulated points yield a spin-independent direct detection cross section of order (3.5 − 6) × 10 −8 pb. For the above m χ 0 1 values, current limits on σ SI are not that far above this mark and upcoming probes of σ SI will definitely reach this level.
• The 7 points all have mg and mq above 1.5 TeV and in some cases above 2 TeV. Detection of the superparticles may have to await the LHC upgrade to 14 TeV.
• Only thet 1 is seen to have a mass distinctly below 1 TeV for the tabulated points. Still, for all the points mt 1 is substantial, ranging from ∼ 500 GeV to above 1 TeV. For such masses, detection of thet 1 as an entity separate from the other squarks and the gluino will be quite difficult and again may require the 14 TeV LHC upgrade.
• The effective superpotential µ-term, µ eff , is small for all the exemplary points. This is interesting regarding the question of electroweak fine-tuning.
For completeness, we have run separate scans for the case of the cNMSSM of [14] [15] with completely universal m 0 = 0 and A 0 ≡ A t = A b = A τ = A λ = A κ (which is in fact a limit case of our model III). Here, one can have a singlino LSP. This requires small λ < 10 −2 . Correct relic density is achieved via co-annihilation withτ R for the rather definite choice of A 0 ∼ − 1 4 M 1/2 . For small enough m 1/2 , the h 1 is dominantly singlet, while the h 2 is SM-like. For larger m 1/2 , the h 1 is SM-like, and the h 2 is mostly singlet. The cross-over where h 1 and h 2 are highly mixed occurs roughly in the range of m 1/2 = 500 − 600 GeV, depending on λ.
Overall, we find that the h 1 can attain a mass of at most ∼ 121 GeV in this scenario in the limit of large m 1/2 . 3 The h 2 , on the other hand, can have a mass  Table 1: Input parameters for the exemplary points. We give tan β(m Z ) and GUT scale parameters, with masses in GeV and masses-squared in GeV 2 . Starred points are the perfect points satisfying all constraints, including δa µ > 5.77 × 10 −10 and 0.094 < Ωh 2 < 0.136. Unstarred points are the almost perfect points that have 4.27 × 10 −10 < δa µ < 5.77 × 10 −10 and 0.094 < Ωh 2 < 0.136.
in the 123 − 128 GeV range for not too large m 1/2 . For λ = 10 −2 , this happens in the region of the cross-over where R h 2 (γγ) is of order 0.5 − 0.6. Squark and gluino masses are around 1.2 − 1.3 TeV in this case, and hence highly pressed by LHC exclusion limits. For smaller λ, an h 2 with mass near 125 GeV is always singlet-like and its signal strength in the γγ and V V channels is very much suppressed relative to the prediction for the SM Higgs, in apparent contradiction to the ATLAS and CMS results.
In summary, we find that the fully constrained version of the NMSSM is not able to yield a Higgs boson consistent with the current hints from LHC data for a fairly SM-like Higgs with mass ∼ 125 GeV, once all experimental constraints are imposed including acceptable a µ and Ωh 2 in the WMAP window. However, by relaxing the CNMSSM to allow for non-universal Higgs soft-masses-squared (NUHM scenarios), it is possible to obtain quite perfect points in parameter space satisfying all constraints with m h 1 ∼ 125 GeV even if the attractive U (1) R symmetry limit of A λ = A κ = 0 is imposed at the GUT scale and certainly if general A λ and A κ values are allowed. We observe a mild tension between the a µ constraint and obtaining m h 1 ∼ 125 GeV; just slightly relaxing the a µ requirement makes it much easier to find viable points with m h 1 ∼ 125 GeV, thus opening up interesting regions of parameter space. We also note that our scanning suggests that relatively small A λ , A κ values are preferred for (almost) perfect points. Masses of SUSY particles for perfect/almost perfect points are such that direct detection of SUSY may have to await the 14 TeV upgrade of the LHC. However, the predicted χ 0 1 masses and associated spin-independent cross sections suggest that direct detection of the χ 0 1 will be possible with the next round of upgrades to the direct detection experiments.  total width in GeV, decay branching ratios, R h 1 (γγ), R h 1 (V V ) and χ 2 ATLAS of the lightest CP-even Higgs for the seven exemplary points.