Dark Matter Phenomenology of GUT-less SUSY Breaking
We study models in which supersymmetry breaking appears at an intermediate scale, , below the GUT scale. That is, that the soft supersymmetry-breaking parameters of the MSSM are universal at . We demand that the lightest neutralino be the LSP, and that the relic neutralino density not conflict with measurements by WMAP and others, and study the morphology of this constraint as the universality scale is reduced from the GUT scale. At moderate values of , we find that the allowed regions of the plane are squeezed by the requirements of electroweak symmetry breaking and that the lightest neutralino be the LSP, whereas the constraint on the relic density is less severe. At very low , the electroweak vacuum conditions become the dominant constraint, and a secondary source of astrophysical cold dark matter would be necessary to explain the measured relic density for nearly all values of the soft SUSY-breaking parameters and .
pacs:12.60.JvSupersymmetric models and 95.35.+dDark matter
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It is well known that TeV scale SUSY offers a compelling solution to the related hierarchy and naturalness problems of the Standard Model. It also facilitates unification of the gauge couplings, as expected in Grand Unified Theories (GUTs) and predicts a light Higgs boson, as favored by precision electroweak data. If R-parity is assumed to be conserved, the lightest supersymetric particle (LSP) is stable, and, if uncharged, is therefore a natural particle candidate for astrophysical cold dark matter.
SUSY is, of course, broken, however the mechanism of this breaking and how it is communicated to the observable sector are unknown. One option is to parametrize the breaking at a high scale with soft SUSY-breaking mass parameters and use the renormalization group equations (RGEs) of the low-energy effective theory to evolve them down to lower scales. In the Constrained Minimal Supersymmetric Standard Model (CMSSM), the soft SUSY-breaking parameters are taken to be universal at the SUSY GUT scale, GeV. The RGEs of the MSSM determine the weak scale observables, given five inputs at the GUT scale, the scalar mass, , the gaugino mass, , the trilinear coupling, , the ratio of the Higgs vevs, , and the sign of the Higgs mass parameter, . In some models it may be more appropriate, however, to assume universality of the soft SUSY-breaking parameters at some scale intermediate between the GUT scale and the electroweak scale.
Here, we present the results of recent studies on the effect of lowering the universality scale on dark matter phenomenology eos1 (); eos2 (). We begin with a discussion of renormalization of the soft SUSY-breaking parameters and expectations based on simple one-loop approximations. Section 3 contains the core of our results, though we limit ourselves here to the scenario, with only a brief comment on . In Section 4 we discuss the prospects for direct detection. Finally, conclusions are given in Section 5.
2 GUT-less Renormalization
The consequences of lowering the universality scale can be easily understood by examining the changes to the running of the soft mass parameters at the one-loop level111Note that all calculations carried out in making the following plots incorporate the full two-loop RGEs.. In the GUT-scale CMSSM, the one-loop renormalizations of the gaugino masses are identical to those of the corresponding gauge couplings. The gaugino masses, , where , 2, 3, and is some low scale, are then proportional to , with the proportionality defined by the ratio of the corresponding gauge coupling at the low scale to the coupling at the GUT scale. The running of the gauge couplings is unaffected by the unversality scale of the soft SUSY-breaking parameters, so in the GUT-less CMSSM, the gaugino masses become
Since will be closer to for , the low-scale guagino masses will be closer to as is lowered.
In Figure 1, we show several low energy sparticle masses, calculated using the full two-loop RGEs, as functions of the universality scale for GeV and GeV. As is lowered, , the bino mass, approaches its input scale value of 800 GeV.
Renormalization of the soft SUSY-breaking scalar masses comes from both gauge and Yukawa interactions, so the running is more complicated. To one loop, the effects can be summarized as
for , where and are the universal scalar and gaugino masses at the input scale. The renormalization coefficients, vanish as . As is decreased from , these coefficients diminish, and the low-scale scalar masses become closer to their input value and therefore less separated.
This modification in the running of the soft scalar masses has a significant impact on the calculated value of the Higgs mass parameter. At tree-level, the electroweak vacuum conditions imply
where and are the soft Higgs masses. As is lowered, these scalar masses become closer to each other, and consequently becomes smaller for fixed , , and , as seen in Figure 1.
Another important consequence of this modification to the running of the sparticle masses concerns the composition of the neutralino LSP. In the CMSSM, the neutralino is commonly bino-like, and . When , however, the neutralino is Higgsino-like, and annihilations to vector bosons are enhanced. As is lowered, we expect that the LSP becomes more Higgsino-like over all of parameter space. Again turning to Figure 1, we see that for this point in parameter space, the LSP mass tracks that of the bino for GeV. Below this value, however, becomes much less than the bino mass, and the LSP eventually becomes Higgsino-like with .
3 Evolution of the Relic Density
We assume that the neutralino LSP constitutes the cold dark matter in the universe, and demand that the relic density of neutralinos falls within the range
in accordance with the WMAP observations wmap (). In Figure 2 we show how the shape of this constraint changes in the plane as the universality scale is lowered from the GUT scale. Here we take , , and GeV.
Panel (a) of Figure 2 shows the constraints from cosmology and collider experiments in the plane in the normal GUT-scale CMSSM. Collider constraints are described in the figure caption. The only allowed regions where the relic density of neutralinos falls within the cosmologically preferred range are the focus point, which borders the region excluded by the electroweak symmetry breaking condition at large , and the coannihilation strip, which lies along the border of the forbidden -LSP region. The relic density of neutralinos is much too large to explain the WMAP measurement over most of the plane.
For GeV, as shown in Panel (b), the unphysical region where has encroached further into the plane, as has the forbidden -LSP region. has become smaller over the plane, leading to a more Higgsino-like (and lighter) LSP. The cosmologically preferred strips have pulled away from these excluded regions, and the two strips on the lower side of the “C” shape indicate the two walls of a rapid annihilation funnel. Inside these two walls, , and neutralino annihilation proceeds extremely efficiently through the s-channel -pole, resulting in a very low relic density.
As the universality scale is lowered to GeV in Panel (c), the “C” has closed into a small island and the rapid annihilation funnel has broadened and moved to lower . The lower funnel wall has taken on a peculiar shape, the sharp plunge indicating the threshold. For this value of , the LSP is Higgsino-like and the relic density is below the WMAP range over most of the plane.
In Panel (d), where GeV, the island has evaporated and the lower funnel wall has dipped into the -LSP region such that the relic density of neutralinos is too low to fully account for the WMAP measurement over nearly all of the plane. Very low values of are therefore disfavored due to the necessity of a secondary source of astrophysical cold dark matter in order to explain the measured abundance.
At large , a similar evolution presents itself, the key difference being that the rapid annihilation funnel is present already at . As in the case, the upper funnel wall and what had been the focus point region approach, merge, and eventually evaporate, at which point the lower funnel wall falls into the -LSP region so that the relic density of neutralinos is too low over the entire plane when GeV.
4 Direct Detection
Many direct searches for dark matter particles such as CDMS and XENON10 look for signatures of weakly-interacting massive particles (WIMPs) scattering on nuclei. The neutralino-nucleon scattering cross section can be broken into spin-dependent and spin-independent (scalar) parts. In Figure 3, we plot the neutralino-nucleon elastic scattering cross sections as functions of neutralino mass for regions in our parameter space that pass all constraints (blue) and that fail only the relaxed LEP Higgs constraint222See eos2 () for details. (green) with in Panel (a) and GeV in Panel (b). If the relic density is below the central value, , we plot the cross section scaled by the ratio of the relic density of neutralinos to the measured value such that our cross sections can be compared with the exclusion limits from direct detection experiments. In particular, we show the limits from CDMS II and XENON10 for the scalar part of the neutralino-nucleon cross section cdms (); xenon10 ().
In Panel (a), where , the regions that pass all constraints are separated into a longer strip of lower cross sections and a smaller island of larger cross sections. The longer strip corresponds to points in the - coannihilation region, while the smaller island corresponds to the focus point region. We note that this island would extend to larger had we considered values of GeV.
When GeV, as shown in Panel (b), the most striking difference is that for a given value of , there are a wide range of potential cross sections. This is due to the fact that the relic density of neutralinos is below the WMAP range over most of the plane, so cross sections are scaled accordingly. The upper boundaries of the regions pictured in Panel (b) come from points in the plane where the relic density is the largest and is the lowest, so one can map the upper edges in Panel (b) to cosmologically preferred regions in Panel (c) of Figure 2.
The next generation of direct detection experiments will probe much of the range of scalar cross sections pictured here. SuperCDMS Phase A with seven towers deployed is expected to be sensitive to scalar cross sections as low as pb at GeV supercdms (). Detectors that will use liquid nobles such as Xenon or Argon expect sensitivities as low as pb for one year of operation of a one ton detector ardm (). We look forward to increased sensitivities to neutralino-nucleon cross sections as a useful complement to collider searches.
Lowering the unification scale of the soft SUSY-breaking mass parameters significantly changes the appearance of the constraint on the relic density of neutralinos. Intermediate universality results in the merging and evaporation of the focus point and rapid annihilation funnel regions. For some critical value of , dependent on and other factors, the relic density of neutralinos becomes too low over all or nearly all of the plane to explain the measured relic density of cold dark matter. The dark matter phenomenology of GUT-less SUSY represents a substantial departure from that of the standard GUT-scale CMSSM.
The work of P.S. was supported in part by DOE grant DE–FG02–94ER–40823.
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