Light dark matter in the singlet-extended MSSM

We discuss the possibility of light dark matter in a general singlet extension of the MSSM. Singlino LSPs with masses of a few GeV can explain the signals reported by the CRESST, CoGeNT and possibly also DAMA experiments. The interactions between singlinos and nuclei are mediated by a scalar whose properties coincide with those of the SM Higgs up to two crucial differences: the scalar has a mass of a few GeV and its interaction strengths are suppressed by a universal factor. We show that such a scalar can be consistent with current experimental constraints, and that annihilation of singlinos into such scalars in the early universe can naturally lead to a relic abundance consistent with the observed density of cold dark matter.


Introduction
Supersymmetry offers a very attractive solution to the dark matter puzzle since the lightest superpartner (LSP) is a quite compelling candidate for the observed cold dark matter (CDM). The LSP is usually assumed to be stable or, at least, long lived. Most studies on scenarios of supersymmetric dark matter focus on the LSPs with electroweak scale masses and cross sections, whose relic density can match the measured CDM density. However, recent results from the direct detection experiments CoGeNT [1] and CRESST [2] seem to hint at somewhat lighter dark matter particles with masses of a few GeV. This interpretation is also consistent with the DAMA signal [3,4]. Do we expect to have such particles in supersymmetric extensions of the standard model (SM)? Certainly, in the minimal supersymmetric SM (MSSM), such masses appear hardly justifiable for particles interacting strong enough to explain the above signals [5][6][7]. This is because, in the MSSM, such particles will typically contribute to the Z boson decay width. On the other hand, in singlet extensions of the MSSM this problem may be circumvented. While in the usual NMSSM it still appears difficult [8,9], but not impossible [10], to obtain particles with the desired properties, generalized singlet extensions of the MSSM [11] can indeed give rise to settings with light dark matter candidates whose interactions with nuclei are mediated by weakly coupled light scalars.
In this paper we focus on a particularly simple scenario based on such a singlet extension of the MSSM in which the singlet sector is only weakly coupled to the MSSM. As we shall see, this will lead to a Higgs-like scalar h 1 which behaves similar to the SM Higgs with two crucial differences. First, all couplings to SM matter are suppressed by an overall factor, and second its mass is in the GeV range. This scalar is accompanied by a singlino superpartner, which, as we shall see, has naturally the correct abundance to explain the observed CDM. Further, h 1 mediated scatterings of nuclei with the singlino can give rise to the signals reported by CRESST, CoGeNT and possibly also DAMA.

Light Singlets in the S-MSSM
We consider the MSSM Higgs sector extended by a gauge singlet superfield S. The most general renormalizable superpotential reads [12] In the so-called NMSSM the µ and µ s terms are set to zero. On the other hand, it is well-known that there are mechanisms that explain a suppressed µ term [13,14]; such mechanisms may also give rise to a µ s parameter of the order of the electroweak scale.
Recently the resulting scheme has been investigated in a different context and was dubbed 'S-MSSM' [15]; we will adopt this terminology. A simple setting where the smallness of µ and µ s finds an explanation will be discussed elsewhere [16]. We include both dimensionful parameters in our analysis. In addition the scalar potential includes the following soft terms The singlet superfield S contains a complex scalar s and a Majorana fermion, the singlino s.
The important feature of the resulting model is that all interactions between the MSSM and the singlet sectors are controlled by a single parameter λ. As we shall see, in the region where λ is of the order 10 −2...3 and the singlet fields are light a simple explanation of the direct detection signals mentioned in the introduction emerges. To obtain light singlets we shall assume that all singlet mass terms are set by a scale m singlet ∼ 10 GeV. Relatively suppressed soft terms for the singlet can be motivated in settings in which the MSSM soft masses are dominated by the gaugino contribution in the renormalization group. In what follows, we start by discussing the limit λ → 0, and then explore what happens if we switch on a finite but small λ.
Limit λ → 0. All terms which mix the singlets with the MSSM contain the parameter λ, i.e. in the case λ = 0 both sectors are completely decoupled. The singlino mass is simply given by m s = µ s , the complex scalar s receives additional mass contributions from the soft terms which also split its real and imaginary components. If we introduce the real scalar h s and pseudoscalar a s through the relation s The following discussion is based on this assumption.
Small λ. Switching on a small λ leads to couplings to and mixings with the MSSM fields. Through the F -and soft terms of the MSSM Higgs fields there arises a linear term in s of the form The linear term induces a vacuum expectation value x = s which can be estimated as There are two competing effects, the smallness of λ and the m 2 singlet in the denominator, such that x can be of the order of the electroweak scale. Note, however, that the impact of x on the SM Higgs masses is almost negligible. In the presence of the singlet VEV there will be new singlet mass terms such as κ 2 s 2 and κ A κ s. Therefore, in order to keep the singlet sector light, we assume that the self-coupling κ is not too large, κ 0.1, and that the trilinear coupling A κ m singlet .
Let us look at the masses and mixings of the singlets. As λ is small we can treat MSSM and singlet sector separately and consider mixing as a perturbation. To simplify our analysis, we impose the decoupling limit on the MSSM Higgs fields. This allows us to ignore mixing of the singlets with the pseudoscalar and the heavy scalar MSSM Higgs. We, however, keep track of the mixing between the light MSSM Higgs h and h s . We further use the minimization conditions for the Higgs potential in order to eliminate the soft masses.
The mass of the singlet pseudoscalar is then given by The scalar mass matrix in the basis (h, h s ) reads and m 2 h as in the usual MSSM. Note that with the assumptions made all contribution to m as and m hs are of the order m singlet or smaller, i.e. we obtain m as , m hs ∼ m singlet . Given our assumptions, m hhs ∼ m singlet .
As m 2 h ≫ m 2 hhs , m 2 hs there is little mixing between h and h s . The light physical mass eigenstate is mainly singlet with a small admixture from h, The heavier state h 2 essentially coincides with the MSSM Higgs h. The mass of h 1 is given by In the fermion sector there is mixing between the singlino and the MSSM neutralinos. This mixing can maximally reach the size of sin θ if the higgsinos are relatively light. 2 As such a small mixing in the fermion sector does not play a role in the following discussion, we ignore it and take the LSP to be a pure singlino with mass The couplings in the singlet sector are all controlled by κ. Most relevant for the following discussion are the trilinear interaction terms which comprise 2 The possible exception of a bino with a mass close to m s is not considered here. with The coupling of h 1 to quarks and leptons is the SM Higgs coupling suppressed by a factor of sin θ.
In summary we have obtained a light scalar which shares the properties of the SM Higgs with two crucial differences: its mass can be in the GeV range and its couplings to SM matter are suppressed by a universal factor, essentially sin θ. The second feature is robust to the extent that the MSSM decoupling limit can be applied. As we shall see in section 4.2, sin θ can be so large that the interactions of the singlino with the light scalar h 1 lead to a coherent picture of singlino CDM in which the recent direct detection signals find an explanation. Before explaining these statements in detail, we will discuss in section 3 that the required values of sin θ can be consistent with experimental constraints.

Experimental Constraints on Light Singlets
We start with a comment on the heavier scalar h 2 : as mixing with the light singlets is suppressed, h 2 decays like the SM Higgs boson. Therefore the usual LEP limit m h 2 > 114.4 GeV applies.
Let us now study the light h 1 . In experiments the light scalar behaves as a light SM Higgs with its coupling reduced by the mixing angle sin θ. Higgs searches by LEP -especially the data set from the L3 collaboration [17] -set strong constraints on the cross section for e + e − → Z + h 1 which can be translated directly in limits on sin θ. 3 Processes in which the resulting Z decays further into neutrinos are treated separately.
As we consider values of m h 1 in the GeV range we also have to consider the production and subsequent decay of h 1 in meson decays. The CLEO [18], and BABAR [19,20] collaborations have measured the branching fractions of the radiative decays Υ → γ + ℓ + ℓ − with ℓ = τ, e. Currently, the limits on sin θ from Υ decay are rather weak (see e.g. [21]), but they may improve in the future.
Below the B meson threshold h 1 can further contribute to the inclusive and exclusive decay modes of B. Strong limits are set by the inclusive process B → h 1 + X s followed by the decay h 1 → µ + µ − . The branching ratio for this process can be taken from [22], the branching ratio for Br(h 1 → µ + +µ − ) can be extracted from [21]. Measurements of the inclusive B decay by Belle [23] together with the calculation of the SM background [24,25] suggest that Br(B → h 1 + X s ) × Br(h 1 → µ + + µ − ) < 2.5 × 10 −6 . This sets limits on sin θ which we show together with the LEP constraints in figure 1.

Singlinos as Dark Matter
Let us now discuss whether the singlino discussed above is a viable dark matter candidate. We start by showing that the singlino has the right relic abundance to constitute the observed dark matter, continue by discussing how the interactions with the singlino may explain the current anomalies in direct detection experiments and finally present a benchmark scenario which is consistent with present data.

Relic Abundance
As singlinos are only very weakly coupled to the MSSM sector, annihilation into SM particles is suppressed. However, singlinos can efficiently annihilate into the light singlet scalars / pseudoscalars provided that m s > m h 1 , m as (see figure 2). It is convenient to expand the cross section in powers of the relative singlino velocity v rel , As an approximation we will only consider the leading contribution to σv rel which is the term σ 0 for a final state with one scalar and one pseudoscalar. For final states with either two scalars or two pseudoscalars σ 0 vanishes and the term σ 1 dominates. Evaluating the Feynman diagrams shown in figure 2 we find the following leading coefficients in the v rel expansion (16): These formulas strictly apply if m h 1 , m as ≪ m s , but they remain a good approximation as long as h 1 , a s and s are not degenerate in mass.
The relic singlino density can be obtained from the annihilation cross section by numerically solving the corresponding Boltzmann equation. An analytic formula which reproduces our numerical results with good accuracy is [26] where g * denotes the effective number of relativistic degrees of freedom and T F the freezeout temperature. For reasonable parameter choices we find T F ≃ m s /20. Note that in our setup the singlinos typically freeze out at a temperature close to the QCD phase transition temperature where the quantity g * changes rapidly [27]. This induces an O(1) uncertainty in our estimate of the relic density.
Given that the singlet coupling is sizable (κ = O(0.1)) and at least one of the discussed annihilation channels is available, it is possible to obtain a relic singlino density which matches the observed dark matter density.

Direct Detection
At the same time, singlinos can explain the recently observed anomalies in direct dark matter detection experiments. The CoGeNT collaboration has reported an excess of low energy scattering events in their germanium detector [1]. This signal is consistent with light weakly interacting massive particles (WIMPs) (m ∼ 5 − 15 GeV) which exhibit a rather large cross section with nucleons, σ n ∼ 10 −40 cm 2 . Preliminary data from the CRESST collaboration seem to support this interpretation [2], although one should await precise information on their backgrounds. Particularly strong limits on light dark matter particles are set by the Xenon10/100 experiment [28,29]. At the moment, however, due to experimental uncertainties these cannot rule out WIMPs with mass m 10 GeV as an explanation for CoGeNT and CRESST (see discussion in [30]). A recent analysis [31] (a) Input parameters.

A Benchmark Scenario
To illustrate our results, let us look at some benchmark values (table 1). From (18) it follows that the relic abundance of the singlino LSP has the appropriate value to match the observed CDM density. Due to the mass relation m as > m s > m h 1 annihilation can only proceed into h 1 h 1 final states and the cross section is determined by (17a). Equation (22) shows that the singlino nucleon cross section is in a range where the CoGeNT, CRESST and DAMA signals can potentially be explained.

Conclusion
We have discussed a simple singlet extension of the MSSM in which the singlino LSP can constitute the observed cold dark matter of the universe. There is a scalar particle h 1 with mass in the few GeV region which behaves like the SM Higgs with universally reduced couplings. We have checked that the light h 1 is consistent with collider and flavor physics constraints. An important ingredient of our scenario is the h 1 singlino coupling κ, which is of the order 0.1. This facilitates efficient annihilation of singlinos into h 1 pairs, which decay further into quark and lepton pairs, such that the correct relic abundance can be obtained. The same coupling κ enters h 1 mediated interactions with nuclei, which can potentially explain the CoGeNT, CRESST and DAMA anomalies.
Our scenario will soon be tested in various experiments. Future direct detection experiments will confirm or rule out the dark matter interpretation of CoGeNT, CRESST and DAMA. Neutrino telescopes will soon reach the sensitivity where they can probe singlino annihilation in the sun, especially if a significant fraction of the annihilation products are taus. The hypothesis of a singlino LSP can be tested at the LHC. Promising signatures include the measurement of missing energy which is reduced against what one expects in the usual neutralino case. Further, the next-to-lightest superpartner may be charged, which can result in charged tracks and other interesting signatures. Finally, B factories offer the possibility to look for the light scalar h 1 in decays of Υ and B mesons.