# A Comparative Study of Dark Matter in the MSSM and Its Singlet Extensions: A Mini Review

Authors: Wenyu Wang
Journal: Advances in High Energy Physics
 (2012)

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## Abstract

In this note we briefly review the recent studies of dark matter in the MSSM and its singlet
extensions: the NMSSM, the nMSSM, and the general singlet extension. Under the new detection
results of CDMS II, XENON, CoGeNT, and PAMELA, we find that (i) the latest detection results
can exclude a large part of the parameter space which is allowed by current collider constraints in
these models. The future SuperCDMS and XENON can cover most of the allowed parameter
space; (ii) the singlet sector will decouple from the MSSM-like sector in the NMSSM; however,
singlet sector makes the nMSSM quite different from the MSSM; (iii) the NMSSM can allow light
dark matter at several GeV to exist. Light CP-even or CP-odd Higgs boson must be present so as
to satisfy the measured dark matter relic density. In case of the presence of a light CP-even Higgs
boson, the light neutralino dark matter can explain the CoGeNT and DAMA/LIBRA results; (iv)
the general singlet extension of the MSSM gives a perfect explanation for both the relic density
and the PAMELA result through the Sommerfeld-enhanced annihilation. Higgs decays in different
scenario are also studied.


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## Body

## 1. Introduction

Although there are many theoretical or aesthetical arguments for the necessity of TeV-scale new physics, the most convincing evidence is from the (Wilkinson Microwave Anisotropy Probe) WMAP  observation of the cosmic cold dark matter, which naturally indicates the existence of (weakly interacting massive particles) WIMPs beyond the prediction of the standard model (SM). By contrast, the neutrino oscillations may rather imply trivial new physics (plainly adding right-handed neutrinos to the SM) or new physics at some very high see-saw scale unaccessible to any foreseeable colliders. Therefore, the TeV-scale new physics to be unraveled at the large hadron collider (LHC) is the most likely related to the WIMP dark matter.If WIMP dark matter is chosen by nature, it will give a strong argument for low-energy supersymmetry (SUSY) with R parity which can give a good candidate. Nevertheless, SUSY is motivated for solving the hierarchy problem elegantly. It can also solve other puzzles of the SM, such as the3σ deviation of the muon anomalous magnetic moment from the SM prediction. In the framework of SUSY, the most intensively studied model is the minimal supersymmetric standard model (MSSM) [1], which is the most economical realization of SUSY. However, this model suffers from the μ-problem. The μ-parameter is the only dimensional parameter in the SUSY conserving sector. From a top-down view, one would expect the μ to be either zero or at the Planck scale. But in the MSSM, the relation of the electroweak (EW) scale soft parameters (m~d2,m~u2) [2]
(1.1)12MZ2=m~d2-m~u2tan2βtan2β-1-μ2
makes that μ must be at the EW scale, while LEP constraints on the chargino mass require μ to be nonzero [3]. A simple solution is to promote μ to a dynamical field in extensions of the MSSM that contain an additional singlet superfield S^ which does not interact with the MSSM fields other than the two Higgs doublets. An effective μ can be reasonably getten at EW scale when S^ denotes the vacuum expectation value (VEV) of the singlet field. Among these extension models the next-to-minimal supersymmetric model (NMSSM) [4–27] and the nearly minimal supersymmetric model (nMSSM) [28–37] caused much attention recently. Note that the little hierarchy problem which is also a trouble of the MSSM is relaxed greatly in the NMSSM.If we introduce a singlet superfield to the MSSM, the Higgs sector will have one more CP even component and one more CP odd component, and the neutralino sector will have one more singlino component. These singlet multiplets compose a “singlet sector” of the MSSM. It can make the phenomenologies of SUSY dark matter and Higgs different from the MSSM. More and more precision results of dark matter detection give us an opportunity to test if this singlet sector really exists. For example, experiments for the underground direct detection of cold dark matterχ~ have recently made significant progress while the null observation of χ~ in the CDMS and XENON100 experiments has set rather tight upper limits on the spin-independent (SI) cross-section of χ~-nucleon scattering [38–40]. The CoGeNT experiment [41] reported an excess which cannot be explained by any known background sources but seems to be consistent with the signal of a light χ~ with mass around 10 GeV and scattering rate (1-2)×10-40cm2. Intriguingly, this range of mass and scattering rate is compatible with dark matter explanation for both the DAMA/LIBRA data and the preliminary CRESST data [42]. Though CoGeNT result is not consistent with the CDMS or XENON results, it implies that the mass of dark matter can range a very long scope at EW scale, that is, from a few GeV to several TeV. The indirect detection PAMELA also observed an excess of the cosmic ray positron in the energy range 10–100 GeV [43, 44] which may be explained by dark matter.In this paper, we will give a short review on the difference between the MSSM and the MSSM with a singlet sector under the constraints of new dark matter detection results. As the Higgs hunting on colliders has delicate relation with dark matter detections, the implication on Higgs searching is also reviewed. The content is based on our previous work [45–47]. The paper is organized as follows: in Section 2, we will give a short review on the structures of the MSSM, the NMSSM, and the nMSSM. In Section 3 we will give a comparison on the models under the constraints of CDMS, XENON, and CoGeNT. In Section 4, a general singlet extension of the MSSM is discussed, and a summary is given in Section 5.


## 2. The MSSM and Its Singlet Extensions

As an economical realization of supersymmetry, the MSSM has the minimal content of particles, while the NMSSM and the nMSSM extend the MSSM by only adding one singlet Higgs superfieldS^. The difference between these models is reflected in their superpotential:
(2.1)WMSSM=WF+μH^u·H^d,WNMSSM=WF+λH^u·H^dS^+13κS^3,WnMSSM=WF+λH^u·H^dS^+ξFMn2S^,
where WF=YuQ^·H^uU^-YdQ^·H^dD^-YeL^·H^dE^ with Q^, U^, and D^ being the squark superfields, and L^ and E^ being the slepton superfields. H^u and H^d are the Higgs doublet superfields, λ, κ, and ξF are dimensionless coefficients, and μ and Mn are parameters with mass dimension. Note that there is no explicit μ-term in the NMSSM or the nMSSM, and an effective μ-parameter (denoted as μeff) can be generated when the scalar component (S) of S^ develops a VEV. Also note that the nMSSM differs from the NMSSM in the last term with the trilinear singlet term κS^3 of the NMSSM replaced by the tadpole term ξFMn2S^. As pointed out in [28–36], such a tadpole term can be generated at a high loop level and naturally be of the SUSY breaking scale. The advantage of such replacement is the nMSSM that has no discrete symmetry thus free of the domain wall problem which the NMSSM suffers from.Corresponding to the superpotential, the Higgs soft terms in the scalar potentials are also different between the three models (the soft terms for gauginos and sfermions are the same thus not listed here)(2.2)VsoftMSSM=m~d2|Hd|2+m~u2|Hu|2+(BμHu·Hd+h.c.),(2.3)VsoftNMSSM=m~d2|Hd|2+m~u2|Hu|2+m~s2|S|2+(AλλSHd·Hu+κ3AκS3+h.c.),(2.4)VsoftnMSSM=m~d2|Hd|2+m~u2|Hu|2+m~s2|S|2+(AλλSHd·Hu+ξSMn3S+h.c.).
After the scalar fields Hu, Hd, and S develop their VEVs vu, vd, and s, respectively, they can be expanded as
(2.5)Hd=(12(vd+ϕd+iφd)Hd-),Hu=(Hu+12(vu+ϕu+iφu)),S=12(s+σ+iξ).
The mass eigenstates can be obtained by unitary rotations
(2.6)(h1h2h3)=UH(ϕdϕuσ),(a1a2G0)=UA(φdφuξ),(G+H+)=UH+(Hd+Hu+),
where h1,2,3 and a1,2 are, respectively, the CP-even and CP-odd neutral Higgs bosons, G0 and G+ are Goldstone bosons, and H+ is the charged Higgs boson. Including the scalar part of the singlet sector in the NMSSM and the nMSSM leads to a pair of charged Higgs bosons, three CP-even and two CP-odd neutral Higgs bosons. In the MSSM, we only have two CP-even and one CP-odd neutral Higgs bosons in addition to a pair of charged Higgs bosons.The MSSM predicts four neutralinosχi0 (i=1,2,3,4), that is, the mixture of neutral gauginos (bino λ' and neutral wino λ3) and neutral higgsinos (ψHu0,ψHd0), while the NMSSM and the nMSSM predict one more neutralino corresponding to the singlino ψS from the fermion part of singlet sector. In the basis (-iλ',-iλ32,ψHu0,ψHd0,ψS) (for the MSSM ψS is absent) the neutralino mass matrix is given by
(2.7)(M10mZsWsb-mZsWcb00M2-mZcWsbmZcWcb0mZsWsb-mZsWsb0-μ-λvcb-mZsWcb-mZcWcb-μ0-λvsb00-λvcb-λvsb{2κλμfortheNMSSM0forthenMSSM),
where M1 and M2 are, respectively, U(1) and SU(2) soft gaugino mass parameters, sW=sinθW, cW=cosθW, sb=sinβ, and cb=cosβ with tanβ≡vu/vd. The lightest neutralino χ~10 is assumed to be the lightest supersymmetric particle (LSP), serving as the SUSY dark matter particle. It is composed by
(2.8)χ~10=N11(-iλ′)+N12(-iλ3)+N13ψHu0+N14ψHd0+N15ψS,
where N is the unitary matrix (N15 is zero for the MSSM) to diagonalize the mass matrix in (2.7). For the mass matrices above we should note the following two points.  (1)
For a moderate value ofκ, the neutralino sector of the NMSSM can go back to the MSSM when λ approaches to zero. This is because in such case the singlino component will become super heavy and decouple from EW scale. The singlet scalar will not mix with the two Higgs doublet, and then the NMSSM will be almost the same as the MSSM at EW scale.(2)
Since theψSψS element of (2.7) is zero in the nMSSM, the singlino will not decouple when λ approaches to zero. In fact, in the nMSSM the mass of the LSP can be written as
(2.9)mχ10≃2μeffλ2(vu2+vd2)2μeff2+λ2(vu2+vd2)tanβtan2β+1.
This formula shows that, to get a heavy χ~10, we need a large λ, a small tanβ as well as a moderate μeff.The chargino sector of these three models is the same except that in the NMSSM/nMSSM the parameterμ is replaced by μeff. The charginos χ~1,2± (mχ1±≤mχ2±) are the mixture of charged Higgsinos ψHu,d± and winos λ±=(λ1±λ2)/2, whose mass matrix in the basis of (-iλ±,ψHu,d±) is given by
(2.10)(M22mWsb2mWcbμeff).
So the chargino χ~1± can be wino dominant (when M2 is much smaller than μ) or higgsino dominant (when μ is much smaller than M2). Since the composing property (wino-like, bino-like, higgsino-like, or singlinolike) of the LSP and the chargino χ~1± is very important in SUSY phenomenologies, we will show such a property in our following study.


## 3. Comparison with the MSSM and the MSSM with a Singlet Sector

### 3.1. In Light of CDMS II and XENON

First let us see the MSSM, the NMSSM, and the nMSSM under the constraints of results of CDMS II and XENON100. As both current and future limits ofχ~-nucleon of CDMS and XENON are similar to each other, we will show only one of them. Nevertheless, as a good substitute of the SM, SUSY model must satisfy all the results of current collider and detector measurements. In our study we consider the following experimental constraints [48]: (1) we require χ~10 to account for dark matter relic density 0.105<Ωh2<0.119; (2) we require the SUSY contribution to explain the deviation of the muon aμ, that is, aμexp-aμSM=(25.5±8.0)×10-10, at 2σ level; (3) the LEP-I bound on the invisible Z-decay, Γ(Z→χ~10χ~10)<1.76 MeV, and the LEP-II upper bound on σ(e+e-→χ~10χ~i0), which is 5×10-2pb for i>1, as well as the lower mass bounds on sparticles from direct searches at LEP and the Tevatron; (4) the constraints from the direct search for Higgs bosons at LEP-II, including the decay modes h→h1h1,a1a1→4f, which limit all possible channels for the production of the Higgs bosons; (5) the constraints from B physics observable such as B→Xsγ, Bs→μ+μ-, B+→τ+ν, ϒ→γa1, the a1-ηb mixing, and the mass difference ΔMd and ΔMs; (6) the constraints from the precision EW observable such as ρlept, sin2θefflept, mW, and Rb; (7) the constraints from the decay ϒ→γh1, and the Tevatron search for a light Higgs boson via 4μ and 2μ2τ signals [49]. The constraints (1.1)–(2.2) have been encoded in the package NMSSMTools [50]. We use this package in our calculation and extend it by adding the constraints (2.3), (2.4). As pointed out in [49], the constraints (2.4) are important for a light Higgs boson. In addition to the above experimental limits, we also consider the constraint from the stability of the Higgs potential, which requires that the physical vacuum of the Higgs potential with nonvanishing VEVs of Higgs scalars should be lower than any local minima.For the calculation of cross-section ofχ~-nucleon scattering, we use the formulas in [51, 52] for the MSSM and extend them to the NMSSM/nMSSM. It is sufficient to consider only the SI interactions between χ~10 and nucleon (denoted by fp for proton and fn for neutron [52]) in the calculation. The leading order of these interactions is induced by exchanging the SM-like Higgs boson at tree level. For moderately light Higgs bosons, fp is approximated by [52] (similarly forfn)
(3.1)fp≃∑q=u,d,s‍fqHmqmpfTq(p)+227fTG∑q=c,b,t‍fqHmqmp,
where fTq(p) denotes the fraction of mp (proton mass) from the light quark q while fTG=1-∑u,d,s‍fTq(p) is the heavy quark contribution through gluon exchange. fqH is the coefficient of the effective scalar operator. The χ~0-nucleus scattering rate is then given by [52]
(3.2)σSI=4π(mχ~0mTmχ~0+mT)2×(npfp+nnfn)2,
where mT is the mass of target nucleus, and np(nn) is the number of proton (neutron) in the target nucleus. In our numerical calculations we take fTu(n)=0.023, fTd(n)=0.034, fTu(p)=0.019, fTd(p)=0.041, and fTs(p)=fTs(n)=0.38. Note that the scattering rate is very sensitive to the value of fTs [53, 54]. Recent lattice simulation [55–57] gave a much smaller value of fTs (0.020), and it reduces the scattering rate significantly which can be seen in [58, 59].Considering all the constraints listed above, we scan over the parameters in the following ranges:(3.3)100GeV≤(Mq~,Mℓ~,mA,μ)≤1TeV,50GeV≤M1≤1TeV,1≤tanβ≤40,(|λ|,|κ|)≤0.7,|Aκ|≤1TeV,
where Mq~ and Mℓ~ are the universal soft mass parameters of the first two generations of squarks and the three generations of sleptons, respectively. To reduce the number of the relevant soft parameters, we worked in the so-called mhmax scenario with the following choice of the soft masses for the third generation squarks: MQ~3=MU~3=MD~3=800GeV, and Xt=At-μcotβ=-1600GeV. The advantage of such a choice is that other SUSY parameters more easily survive the constraints (so that the bounds we obtain are conservative). Moreover, we assume the grand unification relation for the gaugino masses:
(3.4)M1:M2:M3≃1:1.83:5.26.
This relation is often assumed in studies of SUSY at the TeV scale for it can be easily generated in the mSUGRA model [60]. Note that relaxing this relation will give a large effect on the light neutralino scenario [61–64].The surviving points for the three model are displayed in Figure1 for the spin-independent elastic cross-section of χ~-nucleon scattering. We see that for each model the CDMS II limits can exclude a large part of the parameter space allowed by current collider constraints, and the future SuperCDMS (25 kg) limits can cover most of the allowed parameter space. For the MSSM and the NMSSM dark matter mass is roughly in range of 50–400 GeV, while for the nMSSM dark matter mass is constrained below 40 GeV by current experiments and further constrained below 20 GeV by SuperCDMS in case of nonobservation.The scatter plots (taken for [46]) for the spin-independent elastic cross-section of χ~-nucleon scattering. The “+” points (red) are excluded by CDMS limits (solid line), and the “×” (blue) would be further excluded by SuperCDMS 25 kg [65] in case of nonobservation (dash-dotted line), and the “∘” (green) are beyond the SuperCDMS sensitivity.
(a)
(b)
(c)From Figure1, we can see that the χ~-nucleon scattering plot of the MSSM and the NMSSM is very similar to each other, but very different from nMSSM. This implies that, under the experiment constraints, the singlet sector will decouple from the MSSM-like sector in the NMSSM, and then the NMSSM will perform almost the same as the MSSM. However, the singlet components change EW scale phenomenology greatly in the nMSSM. This can also be seen in Figures 2 and 3. We can see that for both the MSSM and the NMSSM χ~10 is bino dominant, while for the nMSSM χ~10 is singlino-dominant, and the region allowed by CDMS limits (and SuperCDMS limits in case of nonobservation) favors a more bino-like χ~10 for the MSSM/NMSSM and a more singlinolike χ~10 for the nMSSM. For the MSSM/NMSSM the LSP lower bound around 50 GeV is from the chargino lower bound of 103.5 GeV plus the assumed GUT relation M1≃0.5M2, while the upper bound around 400 GeV is from the binonature of the LSP (M1 cannot be too large and must be much smaller than other relevant parameters) plus the experimental constraints like the muon g-2 and B-physics. If we do not assume the GUT relation M1≃0.5M2, then M1 can be as small as 40 GeV, and the LSP lower bound in the MSSM/NMSSM will not be sharply at 50 GeV. (We will talk about it in the following section.) For both the MSSM and the NMSSM, the CDMS limits tend to favor a heavier chargino and ultimately the SuperCDMS limits tend to favor a wino-dominant chargino with mass about 2mχ10. Note that there still can be a singlino-dominant LSP in some parameter space of the NMSSM [66], but, in the scan range (3.3) listed above, getting such singlino-dominant LSP needs some fine-tuning, thus we do not focus on it.Same as Figure1, but projected on the plane of |N11|2 and |N15|2 versus dark matter mass (taken for [46]).
(a)
(b)
(c)Same as Figure1, but showing the chargino mass mχ1+ versus the LSP mass. The dashed lines indicate mχ1+=mχ10 (taken for [46]).
(a)
(b)
(c)In Figure4 we show the value of |λ| versus the charged Higgs mass in the NMSSM and the nMSSM. This figure indicates that λ larger than 0.4 is disfavored by the NMSSM. The underlying reason is that h1χ~10χ~10 depends on λ explicitly, and large λ can enhance χ~-nucleon scattering rate. By contrast, although CDMS has excluded some points with large λ in the nMSSM, there are still many surviving points with λ as large as 0.7. We have talked about the reason above to get a heavy χ10, one needs a large λ, a small tanβ as well as a moderate μeff.Same as Figure1, but projected on the plane of |λ| versus the charged Higgs mass in the NMSSM and the nMSSM (taken for [46]).
(a)
(b)From the survived parameter space for all the model above, we should know that the Higgs decay will be similar for the MSSM and the NMSSM, but quite different from the nMSSM. This can be seen in Figure5 which shows decay branching ratio of h1→χ~10χ~10 versus the mass of the SM-like Higgs boson hSM (which is h1 here, and it is Higgs doublet H^u and H^d dominant). Such a decay is strongly correlated to the χ~-nucleon scattering because the coupling h1χ~10χ~10 is involved in both processes. We see that in the MSSM and the NMSSM this decay mode can open only in a very narrow parameter space since χ~10 cannot be so light, and in the allowed region this decay has a very small branching ratio (below 10%). However, in the nMSSM this decay can open in a large part of the parameter space since the LSP can be very light, and its branching ratio can be quite large (over 80% or 90%).Same as Figure1, but projected for the decay branching ratio of hSM→χ10χ10 versus the mass of the Higgs boson hSM (taken for [46]).
(a)
(b)
(c)


### 3.2. Light Dark Matter in the NMSSM

As we talked in the introduction, the data of CoGeNT experiment favors a light dark matter around 10 GeV. However, we scan the parameter space in the MSSM and find that it is very difficult to find a neutralinoχ~10 lighter than about 28 GeV, unless when it is associated with a light stau as the next to the lightest supersymmetric particle (NLSP), but such scenario always needs a fine tuning in the parameter space [67–70]. The main reason for the absence of a lighter χ~10 is that the dominant annihilation channel for χ~10 in the early universe is χ~10χ~10→bb- through s-channel exchange of the pseudoscalar Higgs boson (A), and the measured dark matter relic density requires mA~(90-100) GeV and tanβ~50 and this is in conflict with the constraints from the LEP experiment and B physics [71–74]. The LHC data gives an even more stronger constraint on the light pseudoscalar scenario [75, 76] such that light dark matter seems impossible in the MSSM. Though in the nMSSM the neutralino χ~10 can be as light as 10 GeV (shown in Figure 1), the scattering rate is much lower under the CoGeNT-favored region. In the NMSSM, however, with the participation of singlet sector one can get very light [4–27] Higgs. This feature is particularly useful for light χ~10 scenario since it opens up new important annihilation channels for χ~10, that is, either into a pair of h1 (or a1) or into a pair of fermions via s-channel exchange of h1 (or a1) [74, 77–79]. For the former case, χ~10 must be heavier than h1 (a1), while, for the latter case, due to the very weak couplings of h1 (a1) with χ~10 and with the SM fermions, a resonance enhancement (i.e., mh1 or ma1 must be close to 2mχ~10) is needed to accelerate the annihilation. So a light χ~10 may be necessarily accompanied by a light h1 or a1 to provide the required dark matter relic density. From the discussion in the upper section, light χ~10 can be obtained by releasing the GUT relation (3.4); thus LSP in the NMSSM may explain the detection of CoGeNT. Note that as the LSP in the nMSSM is singlino dominant, relaxing the GUT relation will not the change the phenomenology of dark matter and Higgs too much.Now we discuss how to get a lighth1 or a1 in the NMSSM. A light a1 can be easily obtained when the theory is close to the U(1)R or U(1)PQ symmetry limit, which can be realized by setting the product κAκ to be negatively small [4–27]. In contrast, a light h1 cannot be obtained easily. However, as shown below, it can still be achieved by somewhat subtle cancelation via tuning the value of Aκ. We note that, for any theory with multiple Higgs fields, the existence of a massless Higgs boson implies the vanishing of the determinant of its squared mass matrix and vice versa. For the NMSSM, at tree level the parameter Aκ only enters the mass term of the singlet Higgs bosons, so the determinant (Detℳ2) of the mass matrix of the CP-even Higgs bosons depends on Aκ linearly [4–27]. When other relevant parameters are fixed, one can then obtain a light h1 by varying Aκ around the value A~κ which is the solution to the equation Detℳ2=0. In practice, one must include the important radiative corrections to the Higgs mass matrix, which will complicate the dependence of ℳ2 on Aκ. However, we checked that the linear dependence is approximately maintained by choosing the other relevant parameters at the SUSY scale, and one can solve the equation iteratively to get the solution A~κ.In Figure6 we display the surviving parameter samples, showing the χ~-nucleon scattering cross-section versus the neutralino dark matter mass (left frame) and versus the mass of h1 or a1 (right frame). It shows that the scattering rate of the light dark matter can reach the sensitivity of CDMS, and, consequently, a sizable parameter space is excluded by the CDMS data [65]. The future CDMS experiment can further explore (but cannot completely cover) the remained parameter space. Note that in the light-h1 case the scattering rate can be large enough to reach the sensitivity of CoGeNT and to cover the CoGeNT-favored region. The underlying reason is that the χ~-nucleon scattering can proceed through the t-channel exchange of the CP-even Higgs bosons, which can be enhanced by a factor 1/mh14 for a light h1 [77, 78], while a light a1 can not give such an enhancement because the CP-odd Higgs bosons do not contribute to the scattering in this way. We noticed that the studies in [73, 80] claimed that the NMSSM is unable to explain the CoGeNT data because they did not consider the light-h1 case.The scatter plots (taken for [47]) of the parameter samples which survive all constraints, with “×” (red) and “▲” (green) corresponding to a light h1 and a light a1, respectively. The left frame is projected on the σSI-mχ plane, while the right frame is projected on the σSI-mh1 plane (denoted by “×”) and the σSI-ma1 plane (denoted by “▲”). The curves are the limits from CoGeNT [41] and CDMS [38], while the contour is the CoGeNT-favored region [41].
(a)
(b)In the lightχ~10 scenario, hSM may decay exotically into χ~i0χ~j0, h1h1 or a1a1, and consequently the conventional decays are reduced. This feature is illustrated in Figure 7, which shows that the sum of the exotic decay branching ratios may exceed 50% and the traditional decays hSM→bb-,ττ-,WW*,γγ can be severely suppressed. Numerically, we find that the branching ratio of hSM→bb- is suppressed to be below 30% for all the surviving samples in the light-h1 (h2 is hSM) case and for about 96% of the surviving samples in the light-a1 (h1 is hSM) case (for the remaining 4% of the surviving samples in the light-a1 case, the decay hSM→a1a1 is usually kinematically forbidden so that the ratio of hSM→bb- may exceed 60%). Another interesting feature shown in Figure 7 is that, due to the open-up of the exotic decays, hSM may be significantly lighter than the LEP bound. This situation is favored by the fit of the precision electroweak data and is of great theoretical interest [81].Same as Figure6, but showing the decay branching ratios of the SM-like Higgs boson hSM. Here Br(hSM→χ~i0χ~j0) denotes the total rates for all possible hSM→χ~i0χ~j0 decays (taken for [47]).
(a)
(b)Since the conventional decay modes ofhSM may be greatly suppressed, especially in the light-h1 case which can give a rather large χ~-nucleon scattering rate, the LHC search for hSM via the traditional channels may become difficult. Now the LHC observed a new particle in the mass region around 125-126 GeV which is the most probable long sought Higgs boson [82]. In this mass range, the most important discovering channel of hSM at the LHC is the di-photon signal. In Figure 8 we give the ratio of the di-photon production rate to the SM at the LHC with s=7 TeV. In calculating the rate, we used the narrow width approximation and only considered the leading contributions to pp→hSM from top quark, bottom quark, and the squark loops.Figure 8
Same as Figure6, but showing the diphoton production rate of the SM-like Higgs boson at the LHC.Figure8 indicates that, compared with the SM prediction, the ratio in the NMSSM in the light χ~10 scenario is suppressed to be less than 0.4 for the light-h1 case. For the light-a1 case, most samples (about 96%) predict the same conclusion. Since in the light-h1 case the χ~-nucleon scattering rate can reach the CoGeNT sensitivity, this means that in the framework of the NMSSM the CoGeNT search for the light dark matter will be correlated with the LHC search for the Higgs boson via the di-photon channel. We checked that once the future XENON experiment fails in observing dark matter, less than 1% of the surviving samples in light a1 case predict the ratio of di-photon signal larger than 0.4.


## 3.1. In Light of CDMS II and XENON

First let us see the MSSM, the NMSSM, and the nMSSM under the constraints of results of CDMS II and XENON100. As both current and future limits ofχ~-nucleon of CDMS and XENON are similar to each other, we will show only one of them. Nevertheless, as a good substitute of the SM, SUSY model must satisfy all the results of current collider and detector measurements. In our study we consider the following experimental constraints [48]: (1) we require χ~10 to account for dark matter relic density 0.105<Ωh2<0.119; (2) we require the SUSY contribution to explain the deviation of the muon aμ, that is, aμexp-aμSM=(25.5±8.0)×10-10, at 2σ level; (3) the LEP-I bound on the invisible Z-decay, Γ(Z→χ~10χ~10)<1.76 MeV, and the LEP-II upper bound on σ(e+e-→χ~10χ~i0), which is 5×10-2pb for i>1, as well as the lower mass bounds on sparticles from direct searches at LEP and the Tevatron; (4) the constraints from the direct search for Higgs bosons at LEP-II, including the decay modes h→h1h1,a1a1→4f, which limit all possible channels for the production of the Higgs bosons; (5) the constraints from B physics observable such as B→Xsγ, Bs→μ+μ-, B+→τ+ν, ϒ→γa1, the a1-ηb mixing, and the mass difference ΔMd and ΔMs; (6) the constraints from the precision EW observable such as ρlept, sin2θefflept, mW, and Rb; (7) the constraints from the decay ϒ→γh1, and the Tevatron search for a light Higgs boson via 4μ and 2μ2τ signals [49]. The constraints (1.1)–(2.2) have been encoded in the package NMSSMTools [50]. We use this package in our calculation and extend it by adding the constraints (2.3), (2.4). As pointed out in [49], the constraints (2.4) are important for a light Higgs boson. In addition to the above experimental limits, we also consider the constraint from the stability of the Higgs potential, which requires that the physical vacuum of the Higgs potential with nonvanishing VEVs of Higgs scalars should be lower than any local minima.For the calculation of cross-section ofχ~-nucleon scattering, we use the formulas in [51, 52] for the MSSM and extend them to the NMSSM/nMSSM. It is sufficient to consider only the SI interactions between χ~10 and nucleon (denoted by fp for proton and fn for neutron [52]) in the calculation. The leading order of these interactions is induced by exchanging the SM-like Higgs boson at tree level. For moderately light Higgs bosons, fp is approximated by [52] (similarly forfn)
(3.1)fp≃∑q=u,d,s‍fqHmqmpfTq(p)+227fTG∑q=c,b,t‍fqHmqmp,
where fTq(p) denotes the fraction of mp (proton mass) from the light quark q while fTG=1-∑u,d,s‍fTq(p) is the heavy quark contribution through gluon exchange. fqH is the coefficient of the effective scalar operator. The χ~0-nucleus scattering rate is then given by [52]
(3.2)σSI=4π(mχ~0mTmχ~0+mT)2×(npfp+nnfn)2,
where mT is the mass of target nucleus, and np(nn) is the number of proton (neutron) in the target nucleus. In our numerical calculations we take fTu(n)=0.023, fTd(n)=0.034, fTu(p)=0.019, fTd(p)=0.041, and fTs(p)=fTs(n)=0.38. Note that the scattering rate is very sensitive to the value of fTs [53, 54]. Recent lattice simulation [55–57] gave a much smaller value of fTs (0.020), and it reduces the scattering rate significantly which can be seen in [58, 59].Considering all the constraints listed above, we scan over the parameters in the following ranges:(3.3)100GeV≤(Mq~,Mℓ~,mA,μ)≤1TeV,50GeV≤M1≤1TeV,1≤tanβ≤40,(|λ|,|κ|)≤0.7,|Aκ|≤1TeV,
where Mq~ and Mℓ~ are the universal soft mass parameters of the first two generations of squarks and the three generations of sleptons, respectively. To reduce the number of the relevant soft parameters, we worked in the so-called mhmax scenario with the following choice of the soft masses for the third generation squarks: MQ~3=MU~3=MD~3=800GeV, and Xt=At-μcotβ=-1600GeV. The advantage of such a choice is that other SUSY parameters more easily survive the constraints (so that the bounds we obtain are conservative). Moreover, we assume the grand unification relation for the gaugino masses:
(3.4)M1:M2:M3≃1:1.83:5.26.
This relation is often assumed in studies of SUSY at the TeV scale for it can be easily generated in the mSUGRA model [60]. Note that relaxing this relation will give a large effect on the light neutralino scenario [61–64].The surviving points for the three model are displayed in Figure1 for the spin-independent elastic cross-section of χ~-nucleon scattering. We see that for each model the CDMS II limits can exclude a large part of the parameter space allowed by current collider constraints, and the future SuperCDMS (25 kg) limits can cover most of the allowed parameter space. For the MSSM and the NMSSM dark matter mass is roughly in range of 50–400 GeV, while for the nMSSM dark matter mass is constrained below 40 GeV by current experiments and further constrained below 20 GeV by SuperCDMS in case of nonobservation.The scatter plots (taken for [46]) for the spin-independent elastic cross-section of χ~-nucleon scattering. The “+” points (red) are excluded by CDMS limits (solid line), and the “×” (blue) would be further excluded by SuperCDMS 25 kg [65] in case of nonobservation (dash-dotted line), and the “∘” (green) are beyond the SuperCDMS sensitivity.
(a)
(b)
(c)From Figure1, we can see that the χ~-nucleon scattering plot of the MSSM and the NMSSM is very similar to each other, but very different from nMSSM. This implies that, under the experiment constraints, the singlet sector will decouple from the MSSM-like sector in the NMSSM, and then the NMSSM will perform almost the same as the MSSM. However, the singlet components change EW scale phenomenology greatly in the nMSSM. This can also be seen in Figures 2 and 3. We can see that for both the MSSM and the NMSSM χ~10 is bino dominant, while for the nMSSM χ~10 is singlino-dominant, and the region allowed by CDMS limits (and SuperCDMS limits in case of nonobservation) favors a more bino-like χ~10 for the MSSM/NMSSM and a more singlinolike χ~10 for the nMSSM. For the MSSM/NMSSM the LSP lower bound around 50 GeV is from the chargino lower bound of 103.5 GeV plus the assumed GUT relation M1≃0.5M2, while the upper bound around 400 GeV is from the binonature of the LSP (M1 cannot be too large and must be much smaller than other relevant parameters) plus the experimental constraints like the muon g-2 and B-physics. If we do not assume the GUT relation M1≃0.5M2, then M1 can be as small as 40 GeV, and the LSP lower bound in the MSSM/NMSSM will not be sharply at 50 GeV. (We will talk about it in the following section.) For both the MSSM and the NMSSM, the CDMS limits tend to favor a heavier chargino and ultimately the SuperCDMS limits tend to favor a wino-dominant chargino with mass about 2mχ10. Note that there still can be a singlino-dominant LSP in some parameter space of the NMSSM [66], but, in the scan range (3.3) listed above, getting such singlino-dominant LSP needs some fine-tuning, thus we do not focus on it.Same as Figure1, but projected on the plane of |N11|2 and |N15|2 versus dark matter mass (taken for [46]).
(a)
(b)
(c)Same as Figure1, but showing the chargino mass mχ1+ versus the LSP mass. The dashed lines indicate mχ1+=mχ10 (taken for [46]).
(a)
(b)
(c)In Figure4 we show the value of |λ| versus the charged Higgs mass in the NMSSM and the nMSSM. This figure indicates that λ larger than 0.4 is disfavored by the NMSSM. The underlying reason is that h1χ~10χ~10 depends on λ explicitly, and large λ can enhance χ~-nucleon scattering rate. By contrast, although CDMS has excluded some points with large λ in the nMSSM, there are still many surviving points with λ as large as 0.7. We have talked about the reason above to get a heavy χ10, one needs a large λ, a small tanβ as well as a moderate μeff.Same as Figure1, but projected on the plane of |λ| versus the charged Higgs mass in the NMSSM and the nMSSM (taken for [46]).
(a)
(b)From the survived parameter space for all the model above, we should know that the Higgs decay will be similar for the MSSM and the NMSSM, but quite different from the nMSSM. This can be seen in Figure5 which shows decay branching ratio of h1→χ~10χ~10 versus the mass of the SM-like Higgs boson hSM (which is h1 here, and it is Higgs doublet H^u and H^d dominant). Such a decay is strongly correlated to the χ~-nucleon scattering because the coupling h1χ~10χ~10 is involved in both processes. We see that in the MSSM and the NMSSM this decay mode can open only in a very narrow parameter space since χ~10 cannot be so light, and in the allowed region this decay has a very small branching ratio (below 10%). However, in the nMSSM this decay can open in a large part of the parameter space since the LSP can be very light, and its branching ratio can be quite large (over 80% or 90%).Same as Figure1, but projected for the decay branching ratio of hSM→χ10χ10 versus the mass of the Higgs boson hSM (taken for [46]).
(a)
(b)
(c)


## 3.2. Light Dark Matter in the NMSSM

As we talked in the introduction, the data of CoGeNT experiment favors a light dark matter around 10 GeV. However, we scan the parameter space in the MSSM and find that it is very difficult to find a neutralinoχ~10 lighter than about 28 GeV, unless when it is associated with a light stau as the next to the lightest supersymmetric particle (NLSP), but such scenario always needs a fine tuning in the parameter space [67–70]. The main reason for the absence of a lighter χ~10 is that the dominant annihilation channel for χ~10 in the early universe is χ~10χ~10→bb- through s-channel exchange of the pseudoscalar Higgs boson (A), and the measured dark matter relic density requires mA~(90-100) GeV and tanβ~50 and this is in conflict with the constraints from the LEP experiment and B physics [71–74]. The LHC data gives an even more stronger constraint on the light pseudoscalar scenario [75, 76] such that light dark matter seems impossible in the MSSM. Though in the nMSSM the neutralino χ~10 can be as light as 10 GeV (shown in Figure 1), the scattering rate is much lower under the CoGeNT-favored region. In the NMSSM, however, with the participation of singlet sector one can get very light [4–27] Higgs. This feature is particularly useful for light χ~10 scenario since it opens up new important annihilation channels for χ~10, that is, either into a pair of h1 (or a1) or into a pair of fermions via s-channel exchange of h1 (or a1) [74, 77–79]. For the former case, χ~10 must be heavier than h1 (a1), while, for the latter case, due to the very weak couplings of h1 (a1) with χ~10 and with the SM fermions, a resonance enhancement (i.e., mh1 or ma1 must be close to 2mχ~10) is needed to accelerate the annihilation. So a light χ~10 may be necessarily accompanied by a light h1 or a1 to provide the required dark matter relic density. From the discussion in the upper section, light χ~10 can be obtained by releasing the GUT relation (3.4); thus LSP in the NMSSM may explain the detection of CoGeNT. Note that as the LSP in the nMSSM is singlino dominant, relaxing the GUT relation will not the change the phenomenology of dark matter and Higgs too much.Now we discuss how to get a lighth1 or a1 in the NMSSM. A light a1 can be easily obtained when the theory is close to the U(1)R or U(1)PQ symmetry limit, which can be realized by setting the product κAκ to be negatively small [4–27]. In contrast, a light h1 cannot be obtained easily. However, as shown below, it can still be achieved by somewhat subtle cancelation via tuning the value of Aκ. We note that, for any theory with multiple Higgs fields, the existence of a massless Higgs boson implies the vanishing of the determinant of its squared mass matrix and vice versa. For the NMSSM, at tree level the parameter Aκ only enters the mass term of the singlet Higgs bosons, so the determinant (Detℳ2) of the mass matrix of the CP-even Higgs bosons depends on Aκ linearly [4–27]. When other relevant parameters are fixed, one can then obtain a light h1 by varying Aκ around the value A~κ which is the solution to the equation Detℳ2=0. In practice, one must include the important radiative corrections to the Higgs mass matrix, which will complicate the dependence of ℳ2 on Aκ. However, we checked that the linear dependence is approximately maintained by choosing the other relevant parameters at the SUSY scale, and one can solve the equation iteratively to get the solution A~κ.In Figure6 we display the surviving parameter samples, showing the χ~-nucleon scattering cross-section versus the neutralino dark matter mass (left frame) and versus the mass of h1 or a1 (right frame). It shows that the scattering rate of the light dark matter can reach the sensitivity of CDMS, and, consequently, a sizable parameter space is excluded by the CDMS data [65]. The future CDMS experiment can further explore (but cannot completely cover) the remained parameter space. Note that in the light-h1 case the scattering rate can be large enough to reach the sensitivity of CoGeNT and to cover the CoGeNT-favored region. The underlying reason is that the χ~-nucleon scattering can proceed through the t-channel exchange of the CP-even Higgs bosons, which can be enhanced by a factor 1/mh14 for a light h1 [77, 78], while a light a1 can not give such an enhancement because the CP-odd Higgs bosons do not contribute to the scattering in this way. We noticed that the studies in [73, 80] claimed that the NMSSM is unable to explain the CoGeNT data because they did not consider the light-h1 case.The scatter plots (taken for [47]) of the parameter samples which survive all constraints, with “×” (red) and “▲” (green) corresponding to a light h1 and a light a1, respectively. The left frame is projected on the σSI-mχ plane, while the right frame is projected on the σSI-mh1 plane (denoted by “×”) and the σSI-ma1 plane (denoted by “▲”). The curves are the limits from CoGeNT [41] and CDMS [38], while the contour is the CoGeNT-favored region [41].
(a)
(b)In the lightχ~10 scenario, hSM may decay exotically into χ~i0χ~j0, h1h1 or a1a1, and consequently the conventional decays are reduced. This feature is illustrated in Figure 7, which shows that the sum of the exotic decay branching ratios may exceed 50% and the traditional decays hSM→bb-,ττ-,WW*,γγ can be severely suppressed. Numerically, we find that the branching ratio of hSM→bb- is suppressed to be below 30% for all the surviving samples in the light-h1 (h2 is hSM) case and for about 96% of the surviving samples in the light-a1 (h1 is hSM) case (for the remaining 4% of the surviving samples in the light-a1 case, the decay hSM→a1a1 is usually kinematically forbidden so that the ratio of hSM→bb- may exceed 60%). Another interesting feature shown in Figure 7 is that, due to the open-up of the exotic decays, hSM may be significantly lighter than the LEP bound. This situation is favored by the fit of the precision electroweak data and is of great theoretical interest [81].Same as Figure6, but showing the decay branching ratios of the SM-like Higgs boson hSM. Here Br(hSM→χ~i0χ~j0) denotes the total rates for all possible hSM→χ~i0χ~j0 decays (taken for [47]).
(a)
(b)Since the conventional decay modes ofhSM may be greatly suppressed, especially in the light-h1 case which can give a rather large χ~-nucleon scattering rate, the LHC search for hSM via the traditional channels may become difficult. Now the LHC observed a new particle in the mass region around 125-126 GeV which is the most probable long sought Higgs boson [82]. In this mass range, the most important discovering channel of hSM at the LHC is the di-photon signal. In Figure 8 we give the ratio of the di-photon production rate to the SM at the LHC with s=7 TeV. In calculating the rate, we used the narrow width approximation and only considered the leading contributions to pp→hSM from top quark, bottom quark, and the squark loops.Figure 8
Same as Figure6, but showing the diphoton production rate of the SM-like Higgs boson at the LHC.Figure8 indicates that, compared with the SM prediction, the ratio in the NMSSM in the light χ~10 scenario is suppressed to be less than 0.4 for the light-h1 case. For the light-a1 case, most samples (about 96%) predict the same conclusion. Since in the light-h1 case the χ~-nucleon scattering rate can reach the CoGeNT sensitivity, this means that in the framework of the NMSSM the CoGeNT search for the light dark matter will be correlated with the LHC search for the Higgs boson via the di-photon channel. We checked that once the future XENON experiment fails in observing dark matter, less than 1% of the surviving samples in light a1 case predict the ratio of di-photon signal larger than 0.4.


## 4. General Extension for the Explanation to PAMELA

To explain the PAMELA excess by dark matter annihilation, there are some challenges. First, dark matter must annihilate dominantly into leptons since PAMELA has observed no excess of antiprotons [43, 44] (however, as pointed in [83], this statement may be not so solid due to the significant astrophysical uncertainties associated with their propagation). Second, the explanation of PAMELA excess requires an annihilation rate which is too large to explain the relic abundance if dark matter is produced thermally in the early universe. To tackle these difficulties, a new theory of dark matter was proposed in [84]. In this new theory the Sommerfeld effect of a new force in the dark sector can greatly enhance the annihilation rate when the velocity of dark matter is much smaller than the velocity at freeze-out in the early universe, and dark matter annihilates into light particles which are kinematically allowed to decay to muons or electrons.The above fancy idea is hard to realize in the MSSM, because there is not a new force in the neutralino dark matter sector to induce the Sommerfeld enhancement, and neutralino dark matter annihilates largely to final states consisting of heavy quarks or gauge and/or Higgs bosons [52, 85–88]. However, as discussed in [89], in a general extension of the MSSM by introducing a singlet Higgs superfield, the idea in [84] can be realized by the singlinolike neutralino dark matter.  (i)
The singlino dark matter annihilates to the light singlet Higgs bosons, and the relic density can be naturally obtained from the interaction between singlino and singlet Higgs bosons.(ii)
The singlet Higgs bosons, not related to electroweak symmetry breaking, can be light enough to be kinematically allowed to decay dominantly into muons or electrons through the tiny mixing with the Higgs doublets.(iii)
The Sommerfeld enhancement needed in dark matter annihilation for the explanation of PAMELA result can be induced by the light singlet Higgs boson. In the following section, we will show how does this happen: the Higgs decay is also investigated.


### 4.1. Higgs and Neutralinos Spectrum

If we introduce a singlet Higgs to the MSSM in general, the renormalizable holomorphic superpotential of Higgs is given by [89]
(4.1)W=μH^u·H^d+λS^H^u·H^d+ηS^+12μsS^2+13κS^3,
which include linear term, quadratic term, cubic term of singlet superfield (like Wess-Zumino model [90]). Note that, in such case, we do not require the singlet to solve the μ problem. The soft SUSY-breaking terms are given by
(4.2)Vsoft=m~u2|Hu|2+m~d2|Hd|2+m~s2|S|2+(BμHu·Hd+λAλHu·HdS+CηS+12BsμsS2+13κAκS3+h.c.).
After the Higgs fields develop the VEVs vu, vd, and s, that is, we get the similar Higgs spectrum as the NMSSM and the nMSSM which is the following.  (1)
The CP-even Higgs mass matrix in the basis(ϕu,ϕd,σ) is given by
(4.3)ℳh,11=g2vu2+cotβ[λs(Aλ+κs+μs)+Bμ],(4.4)ℳh,22=g2vd2+tanβ[λs(Aλ+κs+μs)+Bμ],(4.5)ℳh,33=λ(Aλ+μs)vuvds-λμs(vu2+vd2)+κs(Aκ+4κs+3μs)-Cηs,(4.6)ℳh,12=(2λ2-g2)vuvd-λs(Aλ+κs+μs)-Bμ,(4.7)ℳh,13=2λ(μ+λs)vu-λvd(Aλ+2κs+μs),(4.8)ℳh,23=2λ(μ+λs)vd-λvu(Aλ+2κs+μs),where g2=(g12+g22)/2 with g1 and g2 being, respectively, the coupling constant of SU(2) and U(1) in the SM. (2)
The CP-odd Higgs mass matrixℳa is given by
(4.9)ℳa,11=(tanβ+cotβ)[λs(Aλ+κs+μs)+Bμ],(4.10)ℳa,22=4λκvuvd+λ(Aλ+μs)vuvds-λμs(vu2+vd2)-κs(3Aκ+μs)-Cηs-2Bsμs,(4.11)ℳa,12=λvu2+vd2(Aλ-2κs-μs).Note that here we have dropped the Goldstone mode; thus, there left a 2×2 mass matrix in the basis (A~,ξ), and it can be diagonalized by an orthogonal 2×2 matrix P' and the physical CP-odd states ai are given by (ordered as ma1<ma2)
(4.12)a1=P11'A~+P12'SI=P11′(cosβφu+sinβφd)+P12'ξ,a2=P21'A~+P22'SI=P21′(cosβφu+sinβφd)+P22'ξ.(3)
The charged Higgs mass matrixℳ± in the basis (Hu+,Hd+) is given by
(4.13)ℳ±=(λs(Aλ+κs+μs)+Bμ+huhd(g222-λ2))(cotβ11tanβ).(4)
The neutralino mass matrix is(4.14)ℳ0=(M10mZsWsb-mZsWcb00M2-mZcWsbmZcWcb0mZsWsb-mZsWsb0-μ-λvcb-mZsWcb-mZcWcb-μ0-λvsb00-λvcb-λvsb2κs+μs).


### 4.2. Explanation of PAMELA and Implication on Higgs Decays

To explain the observation of PAMELA,a1 is singlet dominant, while h1 is singlet-dominant, and the next-to-lightest h2 is doublet-dominant (hSM). We use the notation:
(4.15)a≡a1,h≡h1,hSM≡h2.
As discussed in [89], when the lightest neutralino χ~10 in (2.8) is singlino-dominant, it can be a perfect candidate for dark matter. As shown in Figure 9, such singlino dark matter annihilates to a pair of light singlet Higgs bosons followed by the decay h→aa (h has very small mixing with the Higgs doublets and thus has very small couplings to the SM fermions). In order to decay dominantly into muons, a must be light enough. Further, in order to induce the Sommerfeld enhancement, h must also be light enough. From the superpotential term κS^3 we know that the couplings hχ~10χ~10 and aχ~10χ~10 are proportional to κ. To obtain the relic density of dark matter, κ should be 𝒪(1). h,a are singlet-dominant, and χ~10 is singlino-dominant, this implies small mixing between singlet and doublet Higgs fields. From the superpotential in (4.1) we see that this means that the mixing parameter λ must be small enough. On the other hand, from (4.5) and (4.10) lightness of h1 and a1 also requires λ and other term approaching to zero. Therefore, in our scan we require that parameters Aκ and Bs have the relation:
(4.16)Aκ~(-4κs-3μs+Cηκs2),2Bsμs~(-3Aκκs-μsκs-Cηs),
to realize light h1 and a1.Feynman diagrams for singlino dark matter annihilation where Sommerfeld enhancement is induced by exchangingh (taken from [45]).
(a)
(b)The numerical results of this model are displayed in different planes in Figures10–12. We see from Figure 10 that, in the range 2mμ<ma<2mπ, a decays dominantly into muons. It is clear that h can be as light as a few GeV, which is light enough to induce the necessary Sommerfeld enhancement as shown in Figure 11.Figure 10
The scatter plots showing the decay branching ratiosa→μ+μ- (muon), a→gg (gluon), and a→ss- (s-quark) versus ma for λ=10-3 (taken from [45]).Figure 11
Same as Figure10, but showing the Sommerfeld enhancement factor induced by h (taken from [45]).Same as Figure10, but showing branching ratio of hSM→aa,hh versus mhSM and |Aκ| versus the branching ratio of hSM→aa,hh (taken from [45]).
(a)
(b)In left plot of Figure12, we show the branching ratios of hSM decays. We see that in the allowed parameter space hSM tends to decay into aa or hh instead of bb-. This can be understood as follows: the MSSM parameter space is stringently constrained by the LEP experiments if hSM is relatively light and decays dominantly to bb-, and to escape such stringent constraints hSM tends to have exotic decays into aa or hh. As a result, the allowed parameter space tends to favor a large Aκ, as shown in right plot of Figure 12, which greatly enhances the couplings hSMaa and hSMhh through the soft term κAκS3 although S has a small mixing with the doublet Higgs bosons. Such an enhancement can be easily seen. Take the coupling hSMhh as an example, the soft term κAκS3 gives a term κAκσ3 which then gives the interaction κAκ(U13H)2U23HhSMhh because σ=U13Hh1+U23Hh2+U33Hh3 with h1≡h and h2≡hSM (see (2.6) and (4.15)). Although the mixing (U13H)2U23H is small for a small λ, a large Aκ can enhance the coupling hSMhh. Note that as the mass of the observed Higgs boson at the LHC is around 125 GeV, thus in the MSSM, the dominant decay mode of hSM is bb-. In this general singlet extension of the MSSM, its dominant decay mode may be changed to aa or hh, as shown in our above results.Finally, we note that, for the specified singlet extensions like the nMSSM and the NMSSM, the explanation of PAMELA and relic density through Sommerfeld enhancement is not possible. The reason is that the parameter space of such models is stringently constrained by various experiments and dark matter relic density as shown in the above section, and, as a result, the neutralino dark matter may explain either the relic density or PAMELA, but impossible to explain both via Sommerfeld enhancement. For example, in the nMSSM various experiments and dark matter relic density constrain the neutralino dark matter particle in a narrow mass range [91], which is too light to explain PAMELA.


## 4.1. Higgs and Neutralinos Spectrum

If we introduce a singlet Higgs to the MSSM in general, the renormalizable holomorphic superpotential of Higgs is given by [89]
(4.1)W=μH^u·H^d+λS^H^u·H^d+ηS^+12μsS^2+13κS^3,
which include linear term, quadratic term, cubic term of singlet superfield (like Wess-Zumino model [90]). Note that, in such case, we do not require the singlet to solve the μ problem. The soft SUSY-breaking terms are given by
(4.2)Vsoft=m~u2|Hu|2+m~d2|Hd|2+m~s2|S|2+(BμHu·Hd+λAλHu·HdS+CηS+12BsμsS2+13κAκS3+h.c.).
After the Higgs fields develop the VEVs vu, vd, and s, that is, we get the similar Higgs spectrum as the NMSSM and the nMSSM which is the following.  (1)
The CP-even Higgs mass matrix in the basis(ϕu,ϕd,σ) is given by
(4.3)ℳh,11=g2vu2+cotβ[λs(Aλ+κs+μs)+Bμ],(4.4)ℳh,22=g2vd2+tanβ[λs(Aλ+κs+μs)+Bμ],(4.5)ℳh,33=λ(Aλ+μs)vuvds-λμs(vu2+vd2)+κs(Aκ+4κs+3μs)-Cηs,(4.6)ℳh,12=(2λ2-g2)vuvd-λs(Aλ+κs+μs)-Bμ,(4.7)ℳh,13=2λ(μ+λs)vu-λvd(Aλ+2κs+μs),(4.8)ℳh,23=2λ(μ+λs)vd-λvu(Aλ+2κs+μs),where g2=(g12+g22)/2 with g1 and g2 being, respectively, the coupling constant of SU(2) and U(1) in the SM. (2)
The CP-odd Higgs mass matrixℳa is given by
(4.9)ℳa,11=(tanβ+cotβ)[λs(Aλ+κs+μs)+Bμ],(4.10)ℳa,22=4λκvuvd+λ(Aλ+μs)vuvds-λμs(vu2+vd2)-κs(3Aκ+μs)-Cηs-2Bsμs,(4.11)ℳa,12=λvu2+vd2(Aλ-2κs-μs).Note that here we have dropped the Goldstone mode; thus, there left a 2×2 mass matrix in the basis (A~,ξ), and it can be diagonalized by an orthogonal 2×2 matrix P' and the physical CP-odd states ai are given by (ordered as ma1<ma2)
(4.12)a1=P11'A~+P12'SI=P11′(cosβφu+sinβφd)+P12'ξ,a2=P21'A~+P22'SI=P21′(cosβφu+sinβφd)+P22'ξ.(3)
The charged Higgs mass matrixℳ± in the basis (Hu+,Hd+) is given by
(4.13)ℳ±=(λs(Aλ+κs+μs)+Bμ+huhd(g222-λ2))(cotβ11tanβ).(4)
The neutralino mass matrix is(4.14)ℳ0=(M10mZsWsb-mZsWcb00M2-mZcWsbmZcWcb0mZsWsb-mZsWsb0-μ-λvcb-mZsWcb-mZcWcb-μ0-λvsb00-λvcb-λvsb2κs+μs).


## 4.2. Explanation of PAMELA and Implication on Higgs Decays

To explain the observation of PAMELA,a1 is singlet dominant, while h1 is singlet-dominant, and the next-to-lightest h2 is doublet-dominant (hSM). We use the notation:
(4.15)a≡a1,h≡h1,hSM≡h2.
As discussed in [89], when the lightest neutralino χ~10 in (2.8) is singlino-dominant, it can be a perfect candidate for dark matter. As shown in Figure 9, such singlino dark matter annihilates to a pair of light singlet Higgs bosons followed by the decay h→aa (h has very small mixing with the Higgs doublets and thus has very small couplings to the SM fermions). In order to decay dominantly into muons, a must be light enough. Further, in order to induce the Sommerfeld enhancement, h must also be light enough. From the superpotential term κS^3 we know that the couplings hχ~10χ~10 and aχ~10χ~10 are proportional to κ. To obtain the relic density of dark matter, κ should be 𝒪(1). h,a are singlet-dominant, and χ~10 is singlino-dominant, this implies small mixing between singlet and doublet Higgs fields. From the superpotential in (4.1) we see that this means that the mixing parameter λ must be small enough. On the other hand, from (4.5) and (4.10) lightness of h1 and a1 also requires λ and other term approaching to zero. Therefore, in our scan we require that parameters Aκ and Bs have the relation:
(4.16)Aκ~(-4κs-3μs+Cηκs2),2Bsμs~(-3Aκκs-μsκs-Cηs),
to realize light h1 and a1.Feynman diagrams for singlino dark matter annihilation where Sommerfeld enhancement is induced by exchangingh (taken from [45]).
(a)
(b)The numerical results of this model are displayed in different planes in Figures10–12. We see from Figure 10 that, in the range 2mμ<ma<2mπ, a decays dominantly into muons. It is clear that h can be as light as a few GeV, which is light enough to induce the necessary Sommerfeld enhancement as shown in Figure 11.Figure 10
The scatter plots showing the decay branching ratiosa→μ+μ- (muon), a→gg (gluon), and a→ss- (s-quark) versus ma for λ=10-3 (taken from [45]).Figure 11
Same as Figure10, but showing the Sommerfeld enhancement factor induced by h (taken from [45]).Same as Figure10, but showing branching ratio of hSM→aa,hh versus mhSM and |Aκ| versus the branching ratio of hSM→aa,hh (taken from [45]).
(a)
(b)In left plot of Figure12, we show the branching ratios of hSM decays. We see that in the allowed parameter space hSM tends to decay into aa or hh instead of bb-. This can be understood as follows: the MSSM parameter space is stringently constrained by the LEP experiments if hSM is relatively light and decays dominantly to bb-, and to escape such stringent constraints hSM tends to have exotic decays into aa or hh. As a result, the allowed parameter space tends to favor a large Aκ, as shown in right plot of Figure 12, which greatly enhances the couplings hSMaa and hSMhh through the soft term κAκS3 although S has a small mixing with the doublet Higgs bosons. Such an enhancement can be easily seen. Take the coupling hSMhh as an example, the soft term κAκS3 gives a term κAκσ3 which then gives the interaction κAκ(U13H)2U23HhSMhh because σ=U13Hh1+U23Hh2+U33Hh3 with h1≡h and h2≡hSM (see (2.6) and (4.15)). Although the mixing (U13H)2U23H is small for a small λ, a large Aκ can enhance the coupling hSMhh. Note that as the mass of the observed Higgs boson at the LHC is around 125 GeV, thus in the MSSM, the dominant decay mode of hSM is bb-. In this general singlet extension of the MSSM, its dominant decay mode may be changed to aa or hh, as shown in our above results.Finally, we note that, for the specified singlet extensions like the nMSSM and the NMSSM, the explanation of PAMELA and relic density through Sommerfeld enhancement is not possible. The reason is that the parameter space of such models is stringently constrained by various experiments and dark matter relic density as shown in the above section, and, as a result, the neutralino dark matter may explain either the relic density or PAMELA, but impossible to explain both via Sommerfeld enhancement. For example, in the nMSSM various experiments and dark matter relic density constrain the neutralino dark matter particle in a narrow mass range [91], which is too light to explain PAMELA.


## 5. Summary

At last we summarize here the SUSY dark matter, and Higgs physics will be changed if we introduce a singlet to the MSSM. Under the latest results of dark matter detection, we have the following.(1)
In the MSSM, the NMSSM, and the nMSSM, the latest detection result can exclude a large part of the parameter space allowed by current collider constraints, and the future SuperCDMS and XENON can cover most of the allowed parameter space.(2)
Under the new dark matter constraints, the singlet sector will decouple from the MSSM-like sector in the NMSSM; thus, the phenomenologies of dark matter and Higgs are similar to the MSSM. The singlet sector makes the nMSSM quite different from the MSSM, the LSP in the nMSSM is singlet dominant, and the SM-like Higgs will mainly decay into the singlet sector. Future precision measurements will give us an opportunity to determine whether the new scalar is from standard model or from SUSY. Perhaps the nMSSM will be the first model excluded for its much larger branching ratio of invisible Higgs decay.(3)
The NMSSM can allow light dark matter at several GeV to exist. Light CP-even or CP-odd Higgs boson must be present so as to satisfy the measured dark matter relic density. In case of the presence of a light CP-even Higgs boson, the light neutralino dark matter can explain the CoGeNT and DAMA/LIBRA results. Further, we find that in such a scenario the SM-like Higgs boson will decay predominantly into a pair of light Higgs bosons or a pair of neutralinos, and the conventional decay modes will be greatly suppressed.(4)
The general singlet extension of the MSSM gives a perfect explanation for both the relic density, and the PAMELA result through the Sommerfeld enhanced annihilation into singlet Higgs bosons (a or h followed by h→aa) with a being light enough to decay dominantly to muons or electrons. Although the light singlet Higgs bosons have small mixing with the Higgs doublets in the allowed parameter space, their couplings with the SM-like Higgs boson hSM can be enhanced by the soft parameter Aκ. In order to meet the stringent LEP constraints, the hSM tends to decay into the singlet Higgs pairs aa or hh instead of bb-.



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*Source: 216941-2012-10-24.xml*