Light dark matter in the NMSSM: upper bounds on direct detection cross sections

In the Next-to-Minimal Supersymmetric Standard Model, a bino-like LSP can be as light as a few GeV and satisfy WMAP constraints on the dark matter relic density in the presence of a light CP-odd Higgs scalar. We study upper bounds on the direct detection cross sections for such a light LSP in the mass range 2-20 GeV in the NMSSM, respecting all constraints from B-physics and LEP. The OPAL constraints on e^+ e^- ->\chi^0_1 \chi^0_i (i>1) play an important role and are discussed in some detail. The resulting upper bounds on the spin-independent and spin-dependent nucleon cross sections are ~ 10^{-42} cm^{-2} and ~ 4\times 10^{-40} cm^{-2}, respectively. Hence the upper bound on the spin-independent cross section is below the DAMA and CoGeNT regions, but could be compatible with the two events observed by CDMS-II.


Introduction
The DAMA [1] and CoGeNT [2] dark matter detection experiments have reported events in excess of the expected background, which would be compatible with a WIMP mass of a few GeV. Also the CDMS-II experiment [3] has reported two events, which could be explained by a WIMP mass of > ∼ 10 GeV (or background). On the other hand, exclusion limits from the Xenon10 [4], Xenon100 [5] and CDMS-Si [6] experiments set upper bounds on the spin independent detection cross sections for this mass range of a WIMP, which seem incompatible with the reported hints for a signal.
In any case, it is important to know whether specific models for dark matter with a WIMP mass of a few GeV can produce direct detection cross sections compatible with the reported excesses, and/or whether regions in the parameter space of such models can be tested by present and future exclusion limits.
Supersymmetric (Susy) extensions of the Standard Model are popular, amongst others, since they predict naturally (for unbroken R-parity, and for a neutral Lightest Supersymmetric Particle (LSP)) a candidate for dark matter, with a relic density compatible with WMAP constraints [7]. Within the Minimal Supersymmetric extension of the Standard Model (MSSM) four neutral fermions (neutralinos χ 0 i , i = 1 . . . 4) exist, which are composed of the bino (superpartner of the U(1) Y gauge boson), the wino (superpartner of the W 3 µ gauge boson) and two higgsinos (superpartners of neutral Higgs bosons). These states mix, and the lightest neutralino χ 0 1 , which is the lightest eigenvalue of the 4×4 mass matrix, will be the LSP (leaving aside the possibility of a sneutrino LSP).
Often the LSP is dominantly bino-like, whose mass m χ 0 1 is approximately given by the soft Susy breaking gaugino mass ∼ M 1 . Assuming unification of the three gaugino masses for the bino (M 1 ), the winos (M 2 ) and the gluino (M 3 ) at the scale of Grand Unification, M 1 is naturally the smallest among these mass terms at the electroweak scale. However, due to the lower bound of ∼ 100 GeV on M 2 from the lower bound on chargino masses, the assumption of unification of the three gaugino masses implies M 1 > ∼ 50 GeV and a similar lower bound on the mass of the LSP.
The assumption of unification of the three gaugino masses can be dropped, however, in that case M 1 and hence the LSP mass m χ 0 1 can be arbitrarily small. Then, on the other hand it can become difficult to satisfy the WMAP constraint on the dark matter relic density i.e., to reduce the dark matter relic density after the Big Bang to an acceptable value compatible with this constraint. To this end, dark matter annihilation processes have to be sufficiently effective. For a LSP mass > ∼ 50 GeV the following pair annihilation processes can be relevant: exchange of Susy partners of fermions (sfermions, in particular sleptons) in the t-channel, and Z-exchange or Higgs-exchange in the s-channel (if the LSP has a sufficiently large higgsino component). In addition, neutralinos can co-annihilate with other sparticles if they have similar masses, but co-annihilation processes will not be relevant for a light LSP as considered here. In the MSSM, sufficiently effective dark matter annihilation processes impose constraints on a light LSP: Considering LSP annihilation via slepton exchange in the t-channel, a lower bound m χ 0 1 > ∼ 18 GeV was derived in [8,9] from the lower bound of ∼ 100 GeV on the slepton masses. (Relaxing this bound to > ∼ 80 GeV for stau masses, one obtains m χ 0 1 > ∼ 13 GeV [8][9][10], unless the LSP mass is very small corresponding to hot dark matter [10].) Allow-ing for LSP annihilation via CP-odd Higgs (A) exchange in the s-channel, a lower limit m χ 0 1 > ∼ 6 GeV was given in [11][12][13][14] from m A > ∼ 90 GeV for large values of tan β > ∼ 25. However, as noted in [15], this region of the parameter space of the MSSM is now strongly constrained by the bounds on B s → µ + µ − . A LSP with a mass in the 5 − 15 GeV range in the MSSM has been considered in [16] without, however, asking for a correct relic density.
In the Next-to-Minimal Supersymmetric Standard Model (NMSSM, see [17,18] for recent reviews), which can solve the µ-problem of the MSSM, the Higgs and neutralino sectors are extended by gauge singlet states. As noticed in [19][20][21][22], the mass of the LSP can be considerably smaller in the NMSSM than in the MSSM and can still be compatible with the WMAP constraint on the relic density. This is a consequence of a light CP-odd Higgs boson in the spectrum (on top of the CP-odd Higgs boson of the MSSM), which can be mostly singlet-like and which is not ruled out by LEP-constraints. Then, sufficiently large LSP annihilation cross sections via the exchange of this additional CP-odd Higgs boson in the s-channel may be possible even for a light LSP with mass of a few GeV.
A light LSP in the NMSSM could be a (dominantly) singlet-like state; in this case, however, its direct detection cross sections would be tiny. On the other hand, as in the MSSM, a light LSP in the NMSSM can originate from a small value of M 1 in which case it will be dominantly bino-like and can have larger direct detection cross sections. These have been estimated in [21], where also constraints on the corresponding parameter space from B-physics, LEP and Υ-physics were discussed. However, the corresponding points in the parameter space given as examples in [21] suffer from a negative effective µ-parameter (which is in conflict with the measured anomalous magnetic moment of the muon), and not all experimental constraints considered below are taken into account.
In view of the interest in a light LSP with a mass in the 2 − 20 GeV range, we find it appropriate to study upper bounds on its direct detection cross sections in the NMSSM. Direct detection cross sections in the NMSSM including WMAP constraints have been studied before in [23][24][25][26][27][28][29], but not for the LSP mass range considered here. Apart from WMAP constraints, we take care of a lengthy list of experimental constraints from Bphysics (important for large tan β and/or relatively light charged and CP-odd Higgs bosons as relevant here), Υ-physics and LEP-constraints on neutralino production. Among the latter, OPAL limits on e + e − → χ 0 1 χ 0 i (i > 1) turn out to be very important. Since these are also relevant for the MSSM, but have hardly been discussed before (a notable exception is [10]), we study their consequences in some detail. For the numerical analysis we use the code NMSSMTools [30,31] coupled to micrOMEGAs [32][33][34]. As a result we obtain upper bounds on the spin-independent and spin-dependent LSP-nucleon cross sections of σ SI < ∼ 10 −42 cm −2 and σ SD < ∼ 4 × 10 −40 cm −2 , varying somewhat with the LSP mass in the 2 − 20 GeV range. The maximal value for σ SI is indeed near the estimate given in [21].
In the next section (2) we present the relevant parameters of the NMSSM and their impact on the LSP cross sections. In section (3), we discuss the relevant experimental constraints. The consequences of the OPAL constraints on e + e − → χ 0 1 χ 0 i (i > 1) on the parameter space (implying a lower bound on µ ef f ) are estimated in an analytic approximation, which reproduces well the full numerical results. Section (4) is devoted to our results and conclusions.

The NMSSM and the impact of its parameters on the LSP cross sections
In the NMSSM the µ parameter of the MSSM is replaced by a Yukawa coupling λ to a gauge singlet (super-) field S. Then, the vacuum expectation value (vev) s of the real scalar component of S generates an effective µ-term (Occasionally one considers the so-called nMSSM [22,25,28] without the trilinear coupling ∼ κ 3 S 3 , which is replaced by a tadpole-term ∼ ξ F S.) Compared to the MSSM, the gauge singlet superfield S adds additional degrees of freedom to the CP-even and CP-odd Higgs sectors as well as to the neutralino sector. Hence the spectrum contains • 3 CP-even neutral Higgs bosons H i , i = 1, 2, 3, which mix in general; • 2 CP-odd neutral Higgs bosons A 1 and A 2 ; • one charged Higgs boson H ± ; • five neutralinos χ 0 i , i = 1 . . . 5, which are mixtures of the bino, the neutral wino, the neutral higgsinos and the singlino; • two charginos which are mixtures of the charged winos and the charged higgsinos.
Apart from the Susy generalisations of the Standard-Model-like gauge and Yukawa couplings and the superpotential in Eq. (2), the Lagrangian of the NMSSM contains soft Susy breaking terms in the form of gaugino masses M 1 , M 2 and M 3 for the bino, the winos and the gluino, respectively, mass terms for all scalars (squarks, sleptons, Higgs bosons including the singlet S) as well as trilinear scalar self-couplings as λA λ SH u H d + κ 3 A κ S 3 , which reflect the trilinear couplings among the superfields in the superpotential.
Expressions for the mass matrices for all Higgs-and neutralino states can be found in [17,18]; below we confine ourselves to those which are of relevance subsequently. Dropping the Goldstone mode, the 2 × 2 mass matrix for the CP-odd Higgs bosons M 2 P in the basis (A M SSM , S I ) has the elements where GeV and, as usual, tan β = v u /v d . The matrix element M 2 P,11 would be the mass squared of the MSSMlike CP-odd scalar A M SSM , if the singlet sector were absent; subsequently we will denote it simply by M 2 A . (This parameter can replace the parameter A λ .) For any (possibly large) value of M 2 A , M 2 P can have another small eigenvalue corresponding to an additional light CP-odd Higgs boson A 1 which is mostly singlet-like. This state will be relevant for the LSP annihilation cross section below.
The mass of the charged Higgs scalar is given by note that it decreases with increasing λ. As is well known, too small values of M H ± can cause disagreements between measurements and corresponding contributions to B-physicsobservables as b → sγ; this will be of importance below.
Notably for large M A , one of the 3 CP-even Higgs bosons will have a mass close to M A . In the MSSM, the corresponding CP-even state is denoted by H, and we will maintain this denomination. The spin-independent LSP-nucleon cross section will be dominated by the exchange of this CP-even scalar H, since its couplings to down-type quarks (particularly the strange quark) are enhanced for large values of tan β.
Also, the mass of the charged Higgs scalar is close to M A for large M A ; then the states H, A M SSM and H ± form a nearly degenerate SU(2) doublet. In fact this approximate degeneracy holds down to fairly low values of M A ∼ 300 GeV.
In the neutralino sector, the bino λ 1 and the neutral wino λ 3 2 mix with the neutral higgsinos ψ 0 d , ψ 0 u and the singlino ψ S , and generate a symmetric 5 × 5 mass matrix M 0 . In the basis It can be diagonalized by an orthogonal real matrix N ij such that the physical masses m χ 0 i ordered in |m χ 0 i | are real (but not necessarily positive). Denoting the 5 eigenstates by χ 0 i , we have Finally, the chargino masses are described by a 2 × 2 mass matrix containing M 2 and µ ef f as diagonal entries. The lower bound of ∼ 103 GeV on the lightest chargino implies at least the constraint (one can choose M 2 > 0 by convention). Next, we discuss the dominant contribution to the spin-independent LSP-nucleon cross section σ SI . Leaving aside scenarios with light squark masses of ∼ 100 GeV (which are difficult to reconcile with Tevatron constraints), σ SI is dominated by the exchange of CPeven Higgs bosons, which couple mostly to the strange quark sea. Among the CP-even Higgs bosons, the coupling of the state H to down-type quarks (as the strange quark) increases with tan β. Hence, although its mass is generally larger than the mass of the Standard-Model-like Higgs boson h, H-exchange provides the leading contribution to σ SI for large values of tan β. Then, the dominant component of H is given by H d .
The dominant coupling of H ∼ H d to the LSP is induced by the bino-higgsino-Higgs vertex ∼ g 1 and hence proportional to g 1 N 11 N 13 , where N 11 denotes the bino-and N 13 the ψ 0 d -higgsino-component of the LSP. All in all one finds which shows that the largest values of σ SI are obtained for a large product N 11 N 13 , large tan β and low values of m H .
The dominant contribution to the spin-dependent LSP-nucleon cross section σ SD originates, as in the MSSM, from Z-exchange. At first sight one could imagine that, for a light CP-odd Higgs boson A 1 , its exchange could also give important contributions to σ SD . However, a light A 1 is dominantly singlet-like and, moreover, the coupling of its doublet component to strange quarks is always tiny compared to the Z-boson coupling.
The coupling of the Z-boson to the LSP originates from the gauge couplings of the higgsino components ψ 0 u and ψ 0 d . Since no additional free parameters intervene, the spindependent cross section σ SD is proportional to Finally the LSP annihilation cross section σ ann is dominated, for the LSP mass range 2 − 20 GeV under consideration, by the exchange of a light A 1 in the s-channel. The dominant contribution to the A 1 χ 0 1 χ 0 1 coupling is induced by the doublet component of A 1 and the bino-higgsino components of χ 0 1 as in the case of the Hχ 0 1 χ 0 1 coupling above; the singlet components of A 1 and χ 0 1 play a minor rôle here. In any case one has (neglecting the finite width of A 1 and the velocity of χ 0 1 near the freeze-out temperature) and hence σ ann can be sufficiently large for suitable values of m 2 A 1 , the lightest eigenvalue of M 2 P in Eq. (3).

Experimental constraints on the parameter space
In this section we discuss various constraints on the parameters of the NMSSM, notably (but not exclusively) from LEP and B-physics, separately in various subsections.

Constraints from sparticle and Higgs searches
As we have seen in Eq. (8), a large spin-independent detection cross section σ SI requires bino-components N 11 and higgsino-components N 13 of the LSP. For small M 1 such that m χ 0 1 is in the 2 − 20 GeV range, the bino component of χ 0 1 is automatically large. However, a large higgsino component of χ 0 1 require relatively small values for µ ef f (below ∼ 160 GeV) in the mass matrix M 0 in Eq. (5). Consequently the neutralino states χ 0 2 and χ 0 3 (for M 2 , 2κs > µ ef f ) are higgsino-like with masses of the order of µ ef f . Then, the production process e + e − → χ 0 1 χ 0 i (i = 2, 3) was kinematically possible at LEP2, and corresponding limits from DELPHI [35] and OPAL [36] have to be taken into account.
The strongest limits come from OPAL at 208 GeV, where we can assume 100% Z * branching ratios for the χ 0 i decays (see Fig. 10 in [36]). Upper bounds on the cross section are given in 5 GeV-wide bins of m χ 0 i . Since we will find m χ 0 3 − m χ 0 2 ∼ 40 GeV, the bounds apply for χ 0 2 and χ 0 3 separately. For m χ 0 1 < 20 GeV, at least one of the χ 0 2 or χ 0 3 production cross sections (in association with χ 0 1 ) is bounded from above by 0.05 pb. In principle, both Z * -exchange in the s-channel and selectron exchange in the t-channel contribute to this cross section. However, the interference between these channels is positive, hence the most conservative bounds on the parameters are obtained by assuming heavy selectrons and that e + e − → χ 0 1 χ 0 i originates from Z * -exchange only. The expression for σ Z (e + e − → χ 0 1 χ 0 i ) is given, e.g., in [37] and can be written as (note the different basis for the neutralinos in [37]) with λ(s) = s 2 + m 4 In order to obtain an approximate expression for the resulting constraints on the parameters, we first neglect m χ 0 1 everywhere in (11). Using numerical values for the gauge couplings, (11) simplifies to with s and the masses in GeV. Next we look for approximations for the relevant neutralino mixing parameters N ij . For simplification we assume M 2 , 2κs ≫ |µ ef f | such that the wino-and singlino-sectors in the mass matrix M 0 in Eq. (5) decouple. (The wino-and singlino-components of the LSP hardly contribute to the spin-independent cross section.) Assuming, in addition, large tan β such that v d ≪ v u , M 0 can be diagonalised analytically with the results (we define u = g 1 v u / √ 2 ∼ 43 GeV and write µ ≡ µ ef f ) Replacing these expressions into (13), using the numerical values for s and M Z in the denominator and, notably, approximating m χ 0 i ∼ µ, one ends up with where s, µ and u are in GeV. Then the upper OPAL bound on σ Z (e + e − → χ 0 1 χ 0 i ) of 0.05 pb becomes a lower bound on |µ| (≡ µ ef f ), A somewhat stronger version of the OPAL bound (σ Z < 0.01 pb) is implemented in the default version of NMSSMTools [30,31]. We replace it by the published value of 0.05 pb [36] for our numerical analysis. From this, without any approximations, we obtain |µ ef f | > ∼ 114 GeV (varying somewhat with M 2 and tan β) for small values of m χ 0 1 in good agreement with the previous estimation. We remark that, within the approximations used in Eqs. (14), this implies an upper bound on N 13 of N 2 13 < ∼ 0.12. Next, we consider constraints from the upper bound on the invisible Z decay width, to which the decay Z → χ 0 1 χ 0 1 would contribute. From [38] we obtain ∆Γ inv Z < ∼ 2.0 MeV (a value slightly above the one used in [10], but below the value used in [21]). The expression for the contribution to ∆Γ inv Z from χ 0 1 reads where the last expression holds for small m χ 0 1 . Then the upper bound on ∆Γ inv Z implies N 2 13 − N 2 14 < 0.11 .
For large tan β, where N 2 14 ≪ N 2 13 , this bound on N 2 13 is very similar to the bound obtained above from the OPAL limits. According to the numerical analysis without approximations we find that the constraints on the parameter space from e + e − → χ 0 1 χ 0 i are mostly somewhat stronger than those from ∆Γ inv Z ; from (8) and (9) it should be clear, that these constraints are relevant for upper bounds on the spin-independent and spin-dependent LSP-nucleon cross sections.
For the chargino masses we require a lower bound of 103 GeV [39], which implies lower limits on combinations of the parameters M 2 and µ ef f . In the neutral Higgs sector we apply the various constraints from [40]. Since the lightest CP-even Higgs boson h is mostly Standard-Model-like in our case, these constraints reduce to the well-known bound m h > 114 GeV. On the other hand the constraints from B-physics, as described below, will imply charged Higgs masses above ∼ 200 GeV, hence additional bounds from direct charged Higgs production are not required.
It should be noted that charged Higgs boson exchange contributes to BR(b → sγ) and BR(B + → τ + ν τ ), hence the corresponding limits impose lower bounds on m H ± . On the other hand, Susy diagrams also contribute to these observables which depend on parameters like M 2 , µ ef f , M squark and A top [46]. For specific choices of these parameters (notably not too large positive values of A top ), the charged Higgs boson contributions can be partially cancelled. This will be relevant below, since the spin-independent LSP-nucleon cross section (8) is maximal for small m H and, as noted above, m H ∼ m H ± .
At large tan β, the observables ∆M s , ∆M d and BR(B s → µ + µ − ) can receive large contributions from a light CP-odd Higgs boson A 1 [45] which, in turn, plays an important rôle for the LSP annihilation cross section (10) for a small LSP mass m χ 0 1 . Again, additional Susy contributions (box diagrams) exist, leading to a complicated combination of constraints in the parameter space. We find that, for a small LSP mass (light A 1 ), practically all these observables impose bounds on various corners in the parameter space.

Additional constraints
On the dark matter relic density we impose the 3 σ WMAP bound [7] 0.091 < Ωh 2 < 0.129 , which requires a sufficiently large LSP annihilation rate (10). A light CP-odd Higgs boson A 1 with a mass below ∼ 9.3 GeV can appear in radiative Υ → A 1 γ decays, on which CLEO [47] and BaBar [48,49] have obtained upper bounds. These can be translated into the parameter space (couplings of A 1 ) of the NMSSM [50][51][52] and are implemented, together with constraints from possible A 1 − η b mixing effects [51], in NMSSMTools. We find that these constraints are so strong (imposing, essentially, strong upper bounds on the A 1 bb coupling for m A 1 < ∼ 10 GeV) that it becomes very difficult to obtain a LSP annihilation rate compatible with (19) for m χ 0 1 < ∼ 2 GeV. Finally we require that the Susy contributions to the anomalous magnetic moment of the muon (see [53,54] for such contributions in the NMSSM) improve the disagreement between the result of the E821 experiment [55] and the Standard Model; as in the MSSM, this implies a positive value for µ ef f .
we choose a small value λ = 0.05 such that its negative effect on M 2 H ± as in Eq. (4) remains negligible while a non-zero doublet component of A 1 induced by the off-diagonal term in Eq. (3). A large value for κ = 0.55 makes the singlino (and the singlet-like CP-even Higgs state) heavy such that perturbing mixing effects in these sectors are avoided.
For the Susy breaking squark and slepton masses we use 1 TeV such that sleptons hardly contribute to e + e − → χ 0 1 χ 0 i . A top varies from 300 to 650 GeV where H ± -induced and Susy-induced contributions to BR(b → sγ) tend to cancel [45]. The Susy breaking gluino and the wino masses are chosen as M 3 = 350 GeV and M 2 = 180 GeV, respectively. (These parameters appear in the loop-induced flavour changing A 1 -quark vertices [46], which should be small in order to allow for a light A 1 consistent with the constraints from BR(B s → µ + µ − ).) Although Eq. (8) suggests that σ SI is maximised for very large values of tan β, the best compromise with B-physics is obtained for reasonable values of tan β ∼ 35 − 44. Likewise, Eq. (14) suggests that µ ef f should be as small as possible in order to maximise N 11 N 13 , but we find that the best compromise in parameter space is obtained for µ ef f ≃ 128 GeV. Eq. (8) also suggests that σ SI is maximised for m H as small as possible. However, we recall that m H ∼ M A ∼ m H ± and that m H ± is bounded from below by several B-physics processes. We choose M A as an input parameter of the NMSSM (instead of A λ ) and  (19). Due to the relatively large couplings involved in the χ 0 1 pair annihilation process, A 1 must actually be off-shell and hence m A 1 substantially larger than 2m χ 0 1 ; otherwise the relic density is too small. In Fig. 1 we show the result for m A 1 as function of m χ 0 1 . For m A 1 < ∼ 40 GeV (m χ 0 1 < ∼ 5 GeV) the constraints from BR(B s → µ + µ − ) (where A 1 appears in the s-channel) become particularly strong and require a somewhat smaller doublet component of A 1 . Denoting its doublet component by sin θ A , we have sin θ A ∼ 0.8 for m A 1 > ∼ 50 GeV, but sin θ A ∼ 0.45 for m A 1 ∼ 10 GeV. We note that for m A 1 < ∼ 40 GeV the value of A κ has to be chosen within a precision less than 1% such that the relic density of χ 0 1 is below the WMAP bound (possibly smaller), but m 2 A 1 > 0; hence this region in the parameter space requires considerable fine tuning. For m A 1 < 10 GeV (m χ 0 1 < ∼ 2 GeV) the constraints from CLEO and BaBar become so strong that sin θ A must be much smaller requiring an even stronger fine tuning of parameters, therefore we will not consider this range of parameters subsequently.
The components of χ 0 1 (the coefficients N 1i , see Eq. (6)) hardly change in the range m χ 0 1 = 2 − 20 GeV considered here, once we maximise the product N 11 N 13 in order to maximise σ SI . We have The masses of the mostly higgsino-like states χ 0 2 and χ 0 3 are ∼ 105 and ∼ 145 GeV, respec- tively, and hence as stated before, the limits on σ Z (e + e − → χ 0 1 χ 0 i ) are relevant. The scattering rates of χ 0 1 depend somewhat on astrophysical parameters as the escape velocity v max and the dark matter density ρ 0 near the sun and, more importantly, on nuclear form factors (quark matrix elements) as the pion-nucleon sigma term σ πN and the size of SU(3) symmetry breaking parametrized by σ 0 . (The difference σ 0 − σ πN is proportional to the strange quark matrix element.) For the astrophysical parameters we use the default values of micrOMEGAs v max = 544 km/s and ρ 0 = 0.3 GeV/cm 3 [34]. The default values in micrOMEGAs for σ πN and σ 0 are σ πN = 55 MeV and σ 0 = 35 MeV.
The corresponding results for the upper bound on the spin-independent cross section of χ 0 1 off protons σ SI p in the NMSSM are shown in Fig. 2 as a function of m χ 0 1 as a full red line. (The spin-independent cross section off neutrons is nearly the same.) In order to indicate the variation of this upper bound with σ πN and σ 0 , we show a red dashed line as the upper bound on σ SI p for σ πN = 73 MeV and σ 0 = 30 MeV, which would correspond to a larger strange quark matrix element and hence an increase of σ SI by a factor ∼ 3.3.
Also shown in Fig. 2 are the regions compatible with the excesses of events reported by DAMA [1] (without channeling (dark blue) and with channeling (light blue)), CoGeNT [2] (light green) and a fit to the two events observed by CDMS-II [56] (denoted as CDMS-09 fit surrounded in dashed green; these events are also compatible with background). Exclusion limits are shown from Xenon10 [4] (violet), Xenon100 [5] (black) and CDMS-II [3,6] (magenta, assuming that the two observed events originate from background). Fig. 2 is our main result, which leads to the following conslusions: • It seems difficult to explain the excesses of events reported by DAMA and CoGeNT within the general NMSSM (without unification constraints on M 1 ). Hence, as stated in [21], significant modifications of parameters like a larger local dark matter density ρ 0 would be required to this end. On the other hand, the two events observed by CDMS-II (within the contour denoted as CDMS-09 fit) could be explained in the NMSSM.
• Actual limits of Xenon10, Xenon100 and CDMS-II on spin-independent cross sections of WIMPS in the 2 − 20 GeV mass range test regions of the parameter space of the NMSSM.
For completeness we have also considered the spin-dependent cross section σ SD in the NMSSM, which is maximal for tan β > ∼ 20 (such that N 2 14 ≪ N 2 13 in Eq. (9)), large values of M A (since m H is irrelevant here), and µ ef f ∼ 121 − 129 GeV. In Fig. 3 we show the maximum of the spin-dependent cross section off protons σ SD p for the same range of m χ 0 1 = 2 − 20 GeV. Note that σ SD originates from Z-exchange, hence the spin-dependent cross section off neutrons σ SD n is given by σ SD n ≃ 0.78 × σ SD p . The actual experimental upper limits on σ SD are one to two orders of magnitude larger [57] than the upper bounds in the NMSSM and not shown in Fig. 3. To conclude, we have performed a detailed analysis of the parameter space of the NMSSM for general values of M 1 , which allows for WIMP masses in the 2 -20 GeV range. In contrast to the MSSM, light bino-like WIMPs can have a relic density compatible with WMAP constraints due to a light NMSSM-specific CP-odd Higgs state which can be exchanged in the s-channel. Due to reported excesses of events compatible with WIMP masses below 20 GeV, this region is of particular interest.
We have studied in detail the constraints on this region of the parameter space of the NMSSM from LEP and B-physics, and the regions of parameter space which give rise to maximal direct detection cross sections while not contradicting experimental limits. The resulting upper bounds on σ SI < ∼ 10 −42 cm 2 = 10 −6 pb make it difficult to explain the excesses of events reported by DAMA and CoGeNT within the NMSSM for small values of M 1 . On the other hand, the two events observed by CDMS-II could be explained in the NMSSM.
Notably the Xenon10 limits [4] on σ SI for WIMP masses below 20 GeV start to test corresponding regions of the NMSSM parameter space. Future results from Xenon100 could confirm the presence of a light WIMP compatible with the NMSSM, or impose further constraints on its parameter space.