Neutralino Dark Matter in MSSM Models with Non-Universal Higgs Masses
We consider the Minimal Supersymmetric Standard Model (MSSM) with varying amounts of non-universality in the soft supersymmetry-breaking contributions to the Higgs scalar masses. In addition to the constrained MSSM (CMSSM) in which these are universal with the soft supersymmetry-breaking contributions to the squark and slepton masses at the input GUT scale, we consider scenarios in which both the Higgs masses are non-universal by the same amount (NUHM1), and scenarios in which they are independently non-universal (NUHM2). As the lightest neutralino is a dark matter candidate, we demand that the relic density of neutralinos not be in conflict with measurements by WMAP and others, and examine the viable regions of parameter space. Prospects for direct detection of neutralino dark matter via elastic scattering in these scenarios are discussed.
address=Theory Group and Texas Cosmology Center,
The University of Texas at Austin, TX 78712, USA
TeV-scale Supersymmetry (SUSY) is one of the most compelling theories of physics beyond the Standard Model for several reasons: it facilitates unification of the gauge couplings, as expected in Grand Unified Theories (GUTs); it stabilizes the Higgs vacuum expectation value, offering a solution to the related heirarchy and naturalness problems of the Standard Model; and it predicts a light Higgs boson, as expected from electroweak precision measurements. In addition, if R-parity is conserved, the lightest supersymmetric particle (LSP) is stable, and, if uncharged, is therefore a natural particle candidate for astrophysical cold dark matter.
Phenomenologically, one of the most-studied versions of the Minimal Supersymmetric Standard Model (MSSM) is the constrained MSSM (CMSSM), in which the soft supersymmetry-breaking contributions to the masses of the SUSY partners of the quarks and leptons and the Higgs scalars are universal at some GUT input scale, as motivated by minimal supergravity. However, while the CMSSM may be the simplest scenario, it is by no means the most plausible version of the MSSM. Here, we present the results of recent studies of models with non-universal supersymmetry-breaking contributions to the Higgs masses [1, 2].
The CMSSM is parametrized by four continuous parameters specified at the SUSY GUT scale and a sign choice: the universal gaugino mass, , the universal scalar mass, , the universal value for the trilinear couplings, , the ratio of the Higgs vacuum expectation values, , and the sign of the Higgs mixing parameter, . In this scenario, the GUT-scale values of the effective Higgs masses are , and the electroweak vacuum conditions fix and the pseudoscalar Higgs mass, . In the NUHM1, the effective Higgs masses are assumed to be universal at the GUT scale, though the universality with is relaxed. One more input parameter, in addition to those of the CMSSM, is required, and may be taken as , , or the GUT-scale value of the Higgs masses . Similarly, in NUHM2 models, no relation between , , and is assumed. This scenario may be parametrized by additional inputs and , or by both GUT-scale effective Higgs masses.
In each scenario, the renormalization group equations of the MSSM are used to determine the low-scale observables, and constraints from colliders and cosmology are applied. We assume that the lightest neutralino is the dark matter candidate, calculate its abundance, and determine the prospects for direct detection for models which are phenomenologically and cosmologically viable, i.e. models not excluded by constraints from colliders, and where the upper limit on the relic density of neutralinos is respected. Direct detection experiments derive limits on the dark matter-nucleon scattering cross section under the assumption that the local density of cold dark matter comes from a single source, that is, one particle species is responsible for the entire dark matter abundance, . However, this may not be the case. If the calculated neutralino abundance is less than the WMAP central value , , we assume a secondary source of cold dark matter, and rescale the neutralino-nucleon cross sections by a factor in order to compare our calculated elastic scattering cross sections with limits from direct searches.
2 The CMSSM
In Fig. 1, we present the neutralino-nucleon elastic scattering cross section as a function of neutralino mass for one slice of the CMSSM parameter space. In the left panel, we show the plane for , , and , with the corresponding neutralino-nucleon elastic scattering cross sections for phenomenologically-viable regions in the panel on the right. The region excluded because the LSP is a charged is shaded brown, and, at large , that where electroweak symmetry breaking cannot be obtained (resulting in unphysical ), in pink. The red dot-dashed contour corresponds to a Higgs mass of 114 GeV. At lower the Higgs boson would be lighter, which is excluded by its non-observation at LEP . We also plot a black dashed contour for GeV, with the region at lower also excluded by LEP. The green shaded region at very low and is disfavored by the measured branching ratio for , while the light pink shaded region is favored by the measurement of the muon anomalous magnetic moment at the 2- level . Finally, in the turquoise shaded regions, the relic density of neutralinos falls within the WMAP range . For the value presented here, the only cosmologically-preferred regions are the coannihilation strip, bordering the -LSP region, and the focus-point region at large , where is small and the LSP is a mixed bino-Higgsino state. Over the bulk of the plane, the relic density of neutralinos exceeds the value measured by WMAP by more than 2-. There are, however, slim strips where is below the WMAP range, which lie between the WMAP-preferred strips and the excluded regions they border.
Turning to the right panel of Fig. 1, we have separated out the cross section for spin-dependent (SD) scattering (red/orange) and the scalar or spin-independent (SI) cross section (blue/green). Here and throughout, we assume the current limits on the branching ratios of and are respected, as well as the lower limit on the chargino mass, and differentiate regions in which the scalar Higgs mass is below the LEP limit (green/orange) and above the LEP limit (blue/red). Here, the Higgs mass is below the LEP limit in the coannihilation strip at low , corresponding to low . Both the SI and SD cross sections in the upper right panel contain two distinct regions, that arising from the focus-point region at GeV and relatively large , and a more extended region due to the coannihilation strip. In the coannihilation strip, 50 GeV 400 GeV, where the lower limit on is a result of the LEP constraint on the chargino mass, and the upper limit on corresponds to the end-point of the coannihilation strip for . At larger values of , we also find a rapid annihilation funnel in the CMSSM parameter space.
We also plot in the right panel of Fig. 1 the limits on the SI cross section from CDMS II  (solid black line) and XENON10  (solid pink line), as well as the sensitivities projected for XENON100  (or a similar 100-kg liquid noble-gas detector such as LUX, dashed pink line) and SuperCDMS at the Soudan Mine  (dashed black line). One can see that the SD cross sections are generally larger than SI cross sections, though current upper limits are larger than pb.
At very low , CDMS II and XENON10 have definitively excluded some of the region where is below the LEP limit. We show this explicitly in the left panel by plotting the reach of current and future direct detection experiments in the parameter space. Here and in subsequent parameter space scans, we display contours of scalar neutralino-nucleon scattering cross section, scaled by if necessary, of pb (solid green contours) and pb (dashed green contours). A cross section of pb is currently excluded by XENON10 for GeV and by CDMS II for GeV, and will be probed by SuperCDMS for up to GeV. Tonne-scale liquid noble-gas detectors such as the proposed XENON1T or a similar detector mass for LUX/ZEP will be sensitive to scalar cross sections below pb for all neutralino masses in the range 10 GeV a few TeV . These detectors will be sensitive to cross sections below pb over much of the preferred mass range GeV.
Of course, we would like to know the range of potential neutralino-nucleon elastic scattering cross sections that are possible in the CMSSM. In Fig. 2, we scan over all CMSSM parameters, with SD neutralino-nucleon elastic scattering cross sections in the left panel, and SI cross sections in the right panel. One can see that, in both the SI and SD cases, there is an upper limit on the cross section as a function of neutralino mass. Focusing on the SI elastic scattering cross sections, at low GeV, cross sections generally exceed pb. The largest cross sections, already excluded by CDMS and XENON10, come primarily from the focus point region at large .
3 NUHM1 Models
In the NUHM1, the parameter space is expanded by one dimension, resulting in additional regions where all constraints are satisfied: selectron coahhihilation strips and crossover regions. In the left panel of Fig. 3, we show an example slice of NUHM1 parameter space, a plane. The cosmologically-preferred (turquoise) shaded strips, from large to small , are: rapid annihilation funnels, where , rising up at GeV and arcing outwards (note that for the entire funnel is excluded by the branching ratio of ); selectron coannihilation strips bordering the black excluded selectron-LSP regions at low ; stau coannihilation strips between GeV and GeV at low ; and vertical crossover strips at GeV, where the LSP is in a mixed bino-Higgsino state. The resulting neutralino-nucleon cross sections are shown in the right panel of Fig. 3. Most cross sections are clustered at GeV due to the fact that the lightest neutralino is bino-like over most of the plane. The smallest cross sections shown come from the funnel or other regions where the relic density is suppressed (and therefore the cross section is scaled). The largest cross sections come from the crossover strip, near its intersection with the coannihilation strip, and for . Between the crossover strips, moving to smaller , decreases, as does the relic density of neutralinos, leading to strips in the right panel of scaled cross sections, with decreasing to GeV.
In Fig. 4, we show a different slice of parameter space, this time the plane, for GeV, , , and . The triangular allowed region is bounded by the constraint on the branching ratio of at small , the appearance of a slepton LSP at large , and the absence of a consistent electroweak vacuum at larger and smaller . The diagonal blue line indicates where . On each side of this contour there is a narrow rapid-annihilation funnel strip where the relic density falls within the WMAP range. The funnel extends only to GeV, and therefore GeV for this region. There is another WMAP strip in the focus-point region close to the electroweak vacuum boundary, where the LSP is more Higgsino-like. The displayed part of the focus-point strip is cut off at GeV, corresponding to GeV: larger values of would be allowed if one considered larger values of . Apart from the region between this strip and the boundary, and between the funnel strips, the relic LSP density would exceed the WMAP range. The limit on the branching ratio of is important at very low and , but it is the constraint on the Higgs mass (red dot-dashed curve, roughly horizontal at GeV) that places a lower limit on the expected LSP mass for the funnel region of GeV.
The green dashed contour in the left panel of Fig. 4, indicating a future sensitivity to spin-independent dark matter scattering at the level of picobarns, runs mainly through the region where the relic density exceeds the WMAP upper limit. SuperCDMS at Soudan would be sensitive to much of the focus-point region shown in the panel on the left, but a more advanced detector would be required if the relic density of neutralinos is below the WMAP range. Unfortunately, even if neutralinos make up all the dark matter in the universe, much of the funnel region will remain out of reach, even to next-generation direct detection experiments. Points associated with the funnel with cross sections larger than pb fail the Higgs mass constraint.
The right panel of Fig. 4 displays the scattering cross sections for phenomenologically-viable points in the left plane. The fact that scalar cross sections in the funnel region are generally smaller than those in the focus-point region is reflected in the lower cutoff on the LSP mass, GeV, seen in the right panel for the points with few pb. In the focus-point region, the LSP is in a mixed state with GeV, and the cross section may be much larger: by two orders of magnitude for the SI cross section, and four orders of magnitude for the SD cross section. These large cross sections at large have no counterparts in the CMSSM, where mixed states in the focus-point region are much lighter. Furthermore, in the CMSSM, the focus point is reached only at large , whereas here it appears even at fixed GeV. CDMS II and XENON10 are already beginning to probe he large scalar cross sections in the focus-point region, with a few points at very low already being excluded, as one can see directly in the right panel111We note these points also have GeV and are within the solid green contour in the left panel of Fig. 4 at very low values of and .. We also note that there are points in the right panel with very low cross sections even though they have GeV, which also have no counterparts in the CMSSM. These points are associated with the funnel and their cross sections are scaled down due to the low relic density in that region.
A new feature seen in this plot is the near-vanishing of the spin-dependent cross section when GeV. This feature is associated with the funnel region where elastic scattering cross sections are substantially lower than those of the focus-point region. However, near GeV there is the possibility for a complete cancellation in the SD neutralino-nucleon elastic scattering matrix elements when the spin contribution from up quarks cancels that due to down and strange quarks. The exact position of the cancellation is sensitive to the values of the spin matrix elements adopted, but the existence of the cancellation is robust. Of course, as we present here the cross sections for scattering on protons, the cross sections for scattering on neutrons will not exhibit a cancellation in the same place.
4 NUHM2 Models
As already discussed, the NUHM2 has two parameters in addition to those already present in the CMSSM, which may be taken as free choices of both the quantities and . As the number of parameters is relatively large, a systematic survey of the NUHM2 parameter space is quite complex. Here we exhibit one plane of parameter space whose features we compare with the CMSSM and NUHM1. For further discussions of direct detection cross sections in the NUHM2, see [2, 11].
We display in the left panel of Fig. 5 a sample NUHM2 plane with and fixed GeV and GeV, which reveals a few interesting new features. As in the CMSSM, there is a region in the plane at large and small which is forbidden because the lighter stau is the LSP. Just above this forbidden region, as in the CMSSM, there is a stau-coannihilation strip. However, jutting up from this strip at GeV and GeV, there are vertical strips where the relic density of neutralinos falls within the WMAP range. The double strips at GeV form a rapid-annihilation funnel on either side of the (solid blue) contour where . Again, such funnels appear only at large in the CMSSM, but the freedom to choose different values of in the NUHM2 permits the appearance of a rapid-annihilation funnel also at the low value shown here. The other vertical WMAP strip appears because, as increases relative to (which is fixed here), the Higgsino fraction of the lightest neutralino increases, which in turn increases the annihilation rate, decreasing the relic density. In this case, this results in a crossover region when GeV. At slightly larger GeV, the lightest and next-to-lightest neutralinos are nearly degenerate in mass, and rapid coannihilations (through -exchange) result in a narrow region with suppressed relic density, and therefore a suppressed scalar cross section.
These novel regions are clearly visible in the right panel of Fig. 5. The elastic scattering cross sections do not vary rapidly as increases, until the funnel region at GeV is reached. The ”V”-shaped suppression in the cross section arises from the increasing value of as one rises up the funnel. The two sides of the funnel approach each other as increases, eventually joining together and resulting in a minimum value where the two sides of the funnel meet (at a value of GeV). After this excitement, the cross section continues to rise gradually as one follows the coannihilation strip, until the crossover strip is reached at GeV. Here the cross section decreases again as increases, to values even smaller than in the rapid-annihilation funnel, before rising again and finally declining towards the end of the coannihilation strip. Note that the entire region to the right of the transition strip is viable, albeit with a relic density below the WMAP range. The cross section is therefore reduced due to scaling. Because the neutralino is predominantly Higgsino-like here, its mass is given by rather than , and so we see in the right panel that the largest values of correspond to our choice of fixed .
Finally, in Fig. 6 we show the entire potential ranges for the SI neutralino-nucleon elastic scattering cross section as a function of neutralino mass in the NUHM1 (left) and NUHM2 (right). Because of the additional freedom in the Higgs sector, and since the SI elastic scattering cross sections receive important contributions from Higgs exchange, there is more variability in the cross sections in the NUHM1 and NUHM2 than in the CMSSM. This is clear from comparison with the corresponding CMSSM panel in Fig. 2. Differences are due primarily to the appearance of additional regions of parameter space in NUHM models where the lightest neutralino is in a mixed bino-Higgsino state, such as the crossover region.
In both the NUHM1 and NUHM2, we find significantly larger cross sections at larger than would be expected in the CMSSM. These cross sections may be probed by the next generation of direct detection experiments for as large as GeV, while in the CMSSM there is no hope for detection with these instruments for GeV. In addition, we find viable points at very low with rather low cross sections. If nature has given us a neutralino of this character, then we may not directly detect dark matter for some time, but the LHC may find supersymmetric particles fairly quickly. In either scenario, we will know that the CMSSM is not an adequate description of nature. We look forward to results from both collider experiments and the next generation of dark matter direct detection experiments, which may give us some interesting hints about the mechanism of supersymmetry breaking.
Acknowledgements.This material is based upon work supported by the National Science Foundation under Grant No. PHY-0455649.
-  J. R. Ellis, K. A. Olive and P. Sandick, Phys. Rev. D 78, 075012 (2008) [arXiv:0805.2343 [hep-ph]].
-  J. Ellis, K. A. Olive and P. Sandick, arXiv:0905.0107 [hep-ph].
-  J. Dunkley et al. [WMAP Collaboration], Astrophys. J. Suppl. 180, 306 (2009) [arXiv:0803.0586 [astro-ph]]; E. Komatsu et al. [WMAP Collaboration], Astrophys. J. Suppl. 180, 330 (2009).
Joint LEP 2 Supersymmetry Working Group,
Combined LEP Chargino Results, up to 208 GeV,
http://lepsusy.web.cern.ch/lepsusy/www/inos_ moriond01/charginos_pub.html; LEP Higgs Working Group for Higgs boson searches, OPAL Collaboration, ALEPH Collaboration, DELPHI Collaboration and L3 Collaboration, Phys. Lett. B 565 (2003) 61 [arXiv:hep-ex/0306033]. Search for neutral Higgs bosons at LEP, paper submitted to ICHEP04, Beijing, LHWG-NOTE-2004-01, ALEPH-2004-008, DELPHI-2004-042, L3-NOTE-2820, OPAL-TN-744,
-  S. Chen et al. [CLEO Collaboration], Phys. Rev. Lett. 87 (2001) 251807 [arXiv:hep-ex/0108032]; P. Koppenburg et al. [Belle Collaboration], Phys. Rev. Lett. 93 (2004) 061803 [arXiv:hep-ex/0403004]. B. Aubert et al. [BaBar Collaboration], arXiv:hep-ex/0207076; E. Barberio et al. [Heavy Flavor Averaging Group (HFAG)], arXiv:hep-ex/0603003.
-  G. Bennett et al. [The Muon g-2 Collaboration], Phys. Rev. Lett. 92 (2004) 161802, hep-ex/0401008; G. Bennett et al. [The Muon g-2 Collaboration], Phys. Rev. D 73 (2006) 072003 [arXiv:hep-ex/0602035].
-  Z. Ahmed et al. [CDMS Collaboration], Phys. Rev. Lett. 102, 011301 (2009).
-  J. Angle et al. [XENON Collaboration], Phys. Rev. Lett. 100, 021303 (2008).
-  E. Aprile, L. Baudis and f. t. X. Collaboration, arXiv:0902.4253 [astro-ph.IM].
-  SuperCDMS Development Project, Fermilab Proposal 0947, October 2004.
-  J. R. Ellis, A. Ferstl, K. A. Olive and Y. Santoso, Phys. Rev. D 67 (2003) 123502 [arXiv:hep-ph/0302032].