The impact of a 126 GeV Higgs on the neutralino mass

We highlight the differences of the dark matter sector between the constrained minimal supersymmetric SM (CMSSM) and the next-to-minimal supersymmetric SM (NMSSM) including the 126 GeV Higgs boson using GUT scale parameters. In the dark matter sector the two models are quite orthogonal: in the CMSSM the WIMP is largely a bino and requires large masses from the LHC constraints. In the NMSSM the WIMP has a large singlino component and is therefore independent of the LHC SUSY mass limits. The light NMSSM neutralino mass range is of interest for the hints concerning light WIMPs in the Fermi data. Such low mass WIMPs cannot be explained in the CMSSM. Furthermore, prospects for discovery of XENON1T and LHC at 14 TeV are given.


Introduction
The discovery of the Higgs boson completes the standard model (SM) of particle physics but there are still some inconsistencies which cannot be explained within this theory. The lack of a dark matter candidate is still an unsolved issue within the SM. Recent measurements by the Planck satellite yielded the most precise measurement of the relic density Ωh 2 = 0.1199 ± 0.002 [1]. Since physics beyond the SM is necessary, one possible extension is supersymmetry (SUSY), a symmetry between fermions and bosons [2][3][4]. The lightest supersymmetric particle (LSP) is stable assuming Email addresses: conny.beskidt@kit.edu (C. Beskidt), wim.de.boer@kit.edu (W. de Boer) R-Parity conservation. If the LSP is represented by the neutralino, the supersymmetric partner of the neutral gauge bosons, supersymmetry provides a weakly interacting massive particle (WIMP) candidate with all needed properties to get the right amount of dark matter [5]. Besides the WIMP candidate, SUSY provides at the same time nice features like the unification of the couplings and electroweak symmetry breaking because of radiative corrections. A light Higgs boson below 130 GeV is predicted, so the discovery of a Higgs-like boson with a mass of 126 GeV [6,7] is up to now the strongest support for supersymmetry despite of the fact that no SUSY particle has been found so far. However, the predicted value for each Higgs mass depends on the model. Within the CMSSM a 126 GeV Higgs boson is only possible for heavy stop masses, which excludes stops below 1 TeV, see e.g. [8][9][10][11] and references therein. However, a 126 GeV Higgs boson is easily fulfilled in the NMSSM for small and moderate stop masses because the mixing with the additional Higgs singlet increases the Higgs mass at tree level [12][13][14][15][16][17][18][19]. Compared to the CMSSM, neutralinos now have an additional singlino component, which changes the relic abundance and the neutralino-nucleon cross section. In this letter we study the differences in the neutralino and dark matter sector considering the allowed parameter space of both models. For this analysis we use GUT scale parameter, which implies including complete radiative corrections between the GUT and the SUSY scale via RGEs including fixed point solutions. We start with a summary of the Higgs and neutralino sector for each model including the relic density constraint. We conclude by discussing the consequences of the global fits to all available data and give future prospects for the discovery reach in direct dark matter searches and LHC at 14 TeV (LHC14).

The CMSSM neutralino sector
The minimal supersymmetric extension of the SM would lead to over 100 free SUSY parameters. The number is reduced to a few within the simplest supersymmetric model by assuming the unification of the gauge couplings and masses of the spin 0 and spin 1/2 particles at the GUT scale. With this assumption, only four free parameters and one free sign are left within the constrained minimal supersymmetric SM (CMSSM): m 0 the common spin 0 mass, m 1/2 the common spin 1/2 mass, A 0 the trilinear coupling, tan β the ratio of the vacuum expectation values of the two Higgs fields and the sign of the Higgs mixing parameter µ [2][3][4]. The Higgs sector of the MSSM requires two Higgs doublets. After the expansion around the vacuum expectation value and the absorption of the longitudinal polarization of the heavy gauge bosons, five Higgs bosons remain: two neutral CP-even Higgs bosons H and h, one CP-odd neutral Higgs A and two CP-even charged Higgs bosons H ± . In the decoupling limit, corresponding to a large ratio of M A /M Z , the lightest Higgs boson h represents the SM Higgs. The tree level mass of h has to be below the Z boson mass, but can be shifted by radiative corrections [20], which need to be large to lift the mass up to the measured value of about 126 GeV. Such large corrections can be fulfilled by either a minimal/maximal mixing scenario in the stop sector or heavy stop masses [21,22]. Such maximal/minimal mixing scenarios are not possible within the CMSSM because of the fixed point solutions for the RGEs running from the GUT to the low scale [23]. A 126 GeV Higgs therefore requires stops in the multi-TeV range [18].

The LSP in the CMSSM
The lightest neutralino is a perfect WIMP candidate, since it is heavy, neutral and stable and the annihilation cross section leads to the correct relic density for typical SUSY parameters [5]. Neutralinos are superpositions of the neutral gauginos and higgsinos. In the gauge-eigenstate basis χ 0 = To obtain the mass eigenstates the mass matrix has to be diagonalized. The relic density is inversely proportional to the neutralino annihilation cross section. The dominant annihilation channel depends on the corresponding parameter space within the m 0 -m 1/2 plane which refers to a different neutralino mixing, mass and next to LSP (NLSP) mass [24]. Most of the parameter space is dominated by the annihilation through the s-channel exchange of a pseudo-scalar Higgs boson. The annihilation cross section which provides the correct relic density leads to a neutralino mass which is close to the A-resonance m χ ≈ m A /2, see e.g. [25]. In addition m A is sensitive to tan β, so the right amount of dark matter can be obtained by adjusting tan β. Large values of tan β of about 50 turned out to be favoured in almost the whole m 0 -m 1/2 plane. The lightest neutralino is bino-like as can be seen from Fig.  1, which shows the neutralino content for typical SUSY parameters. This means that its mass is approximately M 1 ≈ 0.4m 1/2 [26]. The distribution of the neutralino mass which is proportional to m 1/2 is shown in Fig. 2 in the m 0 -m 1/2 plane.

The NMSSM neutralino sector
Within the NMSSM an additional Higgs singlet is added to the two Higgs doublets, which solves the µ problem [27,28]. Because of the resulting Higgs self-coupling terms the number of free parameters increases even in the constrained model. We use the semi-constrained model, where the mixing parameter µ itself is a free parameter and is related to the vacuum expectation value of the singlet s µ ef f = λ s , where λ is a coupling of the Higgses with the singlet. In addition to the mixing term of the Higgs doublets and singlet, the singlet self interaction described by κS 3 /3 appears in the NMSSM superpotential with its coupling κ.
The corresponding soft breaking terms include the trilinear couplings A λ and A κ . This model, which allows a splitting of the Higgs boson masses at the GUT scale, is one of the options of the publicly available software package NMSSMTools [29], which was used to calculate the NMSSM particle spectrum. Combined with the four free CMSSM parameter, the semi-constrained NMSSM contains in total nine free parameters: m 0 , m 1/2 , tan β, A 0 , A κ , A λ , κ, λ and µ ef f . For small values of the couplings λ and κ, the NMSSM approaches the MSSM limit. The additional degrees of freedom in the Higgs sector lead to seven Higgs bosons, so a further Higgs boson is added both to the scalar and the pseudoscalar sector. The mixing of the Higgs doublet and singlet leads to additional terms in the 3×3 mass matrix, which lift the light Higgs boson masses [12,13,[15][16][17]. Within the NMSSM either the lightest or the second lightest Higgs boson represents the SM Higgs boson, whereas the other light Higgs boson is singlet-like with reduced couplings to SM particles. It is possible to obtain m h = 126 GeV for small values of tan β in the whole m 0 -m 1/2 plane independent of the SUSY masses. Especially light and moderate stop masses are possible for different NMSSM scenarios, as discussed previously in [18] and references therein.
The first 4 × 4 elements of the neutralino mixing matrix are similar to Eq. 1, but the main difference among others concerns the Higgs mixing parameter µ ef f , since it is not calculated anymore as in the CMSSM, but is an input parameter as mentioned in section 3. In addition, other NMSSM input parameters such as λ and κ enter into Eq. 3. This influences directly the neutralino content. Typically, the neutralino is a mixture of a higgsino and singlino as shown on the right-hand side in Fig. 1, which means that the (5,5) term in Eq. 3 is small. This is the case if the vev of the singlet s is taken to be of the order of the electroweak scale. The LSP is then independent of m 0 and m 1/2 as shown on the right-hand side of Fig. 2. The small higgsino component allows to fulfill the relic density. If one chooses s to be of the order of m 1/2 in the range allowed by the LHC SUSY searches (see Fig. 2), i.e. well above the electroweak scale, the bino component of the LSP becomes significant and the LSP mass starts to depend on m 1/2 . A bino-like LSP is only possible in the semi-constrained NMSSM if the lightest Higgs represents the SM Higgs, otherwise the relic density is too small. However, the singlet has usually a mass below the SM-like Higgs boson in the semi-constrained NMSSM. A singlet Higgs boson above the SM Higgs boson requires a strong fine tuning of the rather large trilinear couplings as discussed in [18].

LSP-nucleon cross section in the NMSSM
To detect dark matter one has to measure the recoil of a WIMP scatter on a nuclei. Several experiments try to measure these rare events, but no dark matter particle has been detected so far. The best limit for the spin independent (SI) WIMP-nucleon cross section is given so far by the LUX experiment [54], but other experiments give low limits as well [55][56][57][58]. They exclude discovery claims by DAMA/LIBRA [59] and CoGeNT [60]. The main contribution to the scalar elastic-scattering amplitude of a neutralino from quarks comes from scalar Higgs boson t-channel exchange. The pseudo-scalar Higgs boson exchange is suppressed because of the γ 5 factor, whereas heavy squark exchange as well as the heavy Higgs boson exchange is suppressed by their mass. So the scattering via the lightest H 1 and the second lightest Higgs boson H 2 is dominant. These diagrams have a negative interference, which can lead to very small cross sections. An example is shown in Fig. 3.
Here the spin independent cross section is plotted versus the low scale value of the trilinear coupling A λ , which influences the Higgs masses. The cross section is calculated within the software package NMSSMTools [29], which uses micrOmegas [61] for the relic density and cross section calculation. The steep decrease at the edges of the distribution is caused by the increase of the lightest Higgs mass m H 1 , which varies one order of magnitude in this range of A λ . If m H 1 and m H 2 are getting close, the two amplitude cancel which leads to the dip in the cross section distribution for scattering on protons and neutrons. For a fine-tuning of the H 1 and H 2 masses the cross section for a single nucleon can actually become zero, but this does not happen simultaneously for proton and neutron. Therefore, the average cross section stays finite in Fig. 3. The horizontal red, dashed line corresponds to the LUX limit, which excludes a wide range of A λ at low scale, but the allowed  values indicated by the green, vertical band are typically within the range of the quasi fixed-point solution of the RGE for A λ , as shown in Fig. 3 on the right-hand side.

Results of the χ 2 optimization
We perform global fits to all available data, like data from cosmology and accelerators to determine the allowed parameter space within the CMSSM and NMSSM, see Ref. [8] and [18], respectively. We apply our multi-step fitting method to cope with the strong correlations of the free CMSSM and NMSSM parameters. By scanning over all m 0 and m 1/2 values in the range between 100 GeV and 3/1.5 TeV for the CMSSM/NMSSM, we obtain from the constrained fit the preferred values of the other parameters (2 in the CMSSM, 7 in the NMSSM). For these points we calculate the spin independent WIMP-nucleon cross section. The results are shown in Fig 4. A large fraction of the points is already excluded by the direct dark matter searches from LUX (above solid line) and the LHC constraint (light (red) points). In the CMSSM the WIMP mass can reach large values since m W IM P ∝ m 1/2 , as shown before in Fig. 2. Masses of the lightest neutralino below 450 GeV are excluded. For the NMSSM the WIMP is typically a singlino with a mass of the order of the electroweak scale (∝ to the singlet vev s ) which is independent of the other SUSY mass parameters. The mass of the lightest neutralino ranges from 30 to 200 GeV, but the masses below 60 GeV are excluded by the LHC SUSY searches, as indicated by the red (light) points. These points have typically m 1/2 values in the white region of Fig. 2. Future direct dark matter and LHC sensitivities are shown in the last row of Fig. 4. The XENON1T limit is expected to reach a sensitivity two orders of magnitude better than XENON100. The current results of the SUSY searches at the LHC have been extrapolated to 3000f b −1 and 14 TeV. One observes that a large range of parameter space will be covered for both models, both by the LHC and the direct dark matter searches.

Summary
In this letter we compared the dark matter sector of the CMSSM and the NMSSM using GUT scale input parameters. Within the CMSSM the lightest neutralino is bino-like and has a mass proportional to m 1/2 . The relic density can be obtained for large values of tan β. The Higgs mass requires low values of tan β within the NMSSM. The lightest neutralino is typically a mixture of a higgsino and singlino to obtain the correct relic density. Small cross sections for the WIMP nucleon cross section are possible within the NMSSM because of the negative interference of the amplitudes of the two lightest Higgs bosons. The allowed mass range for the neutralino within the NMSSM covers the region around 60-200 GeV. Within the CMSSM the lower limit on the neutralino mass is around 450 GeV. The complementary mass ranges of the two models can be used to distinguish the models, since a light neutralino is not possible within the CMSSM, but is favoured within the NMSSM and vice versa. We show the sensitivities of future dark matter searches like XENON1T and searches for SUSY particles at the LHC at 14 TeV. They will be sensitive to a large part of the parameter space, as shown in Fig. 4.