Predictive Signatures of Supersymmetry: Measuring the Dark Matter Mass and Gluino Mass with Early LHC data

Predictive Signatures of Supersymmetry: Measuring the
Dark Matter Mass and Gluino Mass with Early LHC data

Daniel Feldman Michigan Center for Theoretical Physics, University of Michigan, Ann Arbor, MI 48109, USA    Katherine Freese Michigan Center for Theoretical Physics, University of Michigan, Ann Arbor, MI 48109, USA Texas Cosmology Center, University of Texas, Austin, TX 78712, USA    Pran Nath Department of Physics, Northeastern University, Boston, MA 02115, USA    Brent D. Nelson Department of Physics, Northeastern University, Boston, MA 02115, USA    Gregory Peim Department of Physics, Northeastern University, Boston, MA 02115, USA
Abstract

We present a focused study of a predictive unified model whose measurable consequences are immediately relevant to early discovery prospects of supersymmetry at the LHC. ATLAS and CMS have released their analysis with 35 pb of data and the model class we discuss is consistent with this data. It is shown that with an increase in luminosity the LSP dark matter mass and the gluino mass can be inferred from simple observables such as kinematic edges in leptonic channels and peak values in effective mass distributions. Specifically, we consider cases in which the neutralino is of low mass and where the relic density consistent with WMAP observations arises via the exchange of Higgs bosons in unified supergravity models. The magnitudes of the gaugino masses are sharply limited to focused regions of the parameter space, and in particular the dark matter mass lies in the range with an upper bound on the gluino mass of , with a typical mass of . We find that all model points in this paradigm are discoverable at the LHC at . We determine lower bounds on the entire sparticle spectrum in this model based on existing experimental constraints. In addition, we find the spin-independent cross section for neutralino scattering on nucleons to be generally in the range of with much higher cross sections also possible. Thus direct detection experiments such as CDMS and XENON already constrain some of the allowed parameter space of the low mass gaugino models and further data will provide important cross-checks of the model assumptions in the near future.

LHC, SUSY, Gluino, Higgs, Dark Matter, XENON

I Introduction

Unified models of supergravity with gravity mediated breaking of supersymmetry sugra () extend the standard model of particle physics and are being tested with the Large Hadron Collider experiments at CERN. As a consequence of the breaking of supersymmetry, one obtains soft masses and couplings of the form sugra (); 1992ArnowittNath ()

(1)
(2)
(3)

where at the unification scale, , there are universal mass terms for the gauginos of , denoted by , and universal mass squared terms for scalar fields denoted by (where  () stands for squarks (sleptons)), and universal cubic (trilinear) couplings which multiply the Yukawa couplings of matter fields to the Higgs fields. In addition, a (bilinear) soft Higgs mixing term proportional to of the form arises from the superpotential, where are the Higgs doublets which give mass to the up quarks (down quarks and charged leptons). The constraints of electroweak symmetry breaking allow the determination of (where is at the electroweak scale) in terms of and further one makes the replacement of by the ratio of the Higgs vacuum expectation values leaving minimally 4 parameters and one sign needed as input to define the model sugra (); 1992ArnowittNath ()

(4)

Through renormalization group evolution, one computes the predictions for all the masses of the superpartners and their couplings to each other and to the standard model fields111For recent reviews see: Nath:2010zj (); KaneFeldman (); Nath20 (); RossIbanez ().

Models of supergravity address fundamental questions in particle physics, such as the gauge hierarchy problem, the breaking of electroweak symmetry, and the unification of strong and electroweak forces. In addition, such models also provide a compelling dark matter candidate; the lightest supersymmetric particle (LSP). In particular, the neutralino is a linear combination of gauginos and Higgsinos as follows:

(5)

where is the bino, is the wino and are the Higgsinos. The neutralino can have the right cross section and mass to provide a natural candidate for the observed density of cold dark matter (CDM) in the universe. According to the analysis in WMAP (), the latter has the value

(6)

Here is the Hubble constant, , in units of 100 km/s/Mpc, and under the assumption that , one has where the neutralino density is in units of the critical density g/cm. The measurement of the relic density together with a variety of results from collider experiments provide strong constraints on models of new physics.

In this paper we study a particular region of the unified supersymmetric parameter space which satisfies all the existing experimental and astrophysical bounds and is testable in the very near future. We focus on the region where the neutralino has a mass in the range . In this mass range, which is above the -pole, when , in those models that are unconstrained by present experimental data, the relic density of neutralinos is largely governed by the presence of the light CP even Higgs pole (-pole) Nath (); Djouadi () through annihilations in the early universe, schematically:

(7)

arising from the resonance; however, other channels can contribute in general. Additionally, when the relic density can also be achieved via

(8)

through the s-channel where the heavier Higgses can play the dominant role Nath (). Such annihilations can lead to effects on the relic density when the mass of the pseudoscalar is light, of order a few hundred GeV, which corresponds to the case of large . Our analysis will find results consistent with a large range of with the possibility of both a heavy and a light pseudoscalar. We will refer to the collective region of the parameter space, with as the “Higgs-pole region”.

With universal boundary conditions at the unification scale, the mass range of the neutralino is confined by mass limits on the other particles in the spectrum. In particular the light chargino has a bound from LEP of Nakamura:2010zzi (). It is known that in models with the minimal supersymmetric field content the light CP-even Higgs mass has an upper bound of roughly  GeV Haber (). The Higgs mass is bounded from below by direct searches at LEP Barate:2003sz () and, more recently, at the Tevatron tev (). We will use a conservative lower bound of to allow for the theoretical uncertainty in computing the loop corrections to the Higgs mass. We note that a stricter imposition of would narrow the space of models but has little impact on our generic conclusions. Specifically, the low mass gaugino models we study in the Higgs-pole region will correspond to light neutralino dark matter in the range

(9)

that yields the correct relic density and obeys all other experimental constraints subject to the boundary conditions of Eq. (3).

Here we will show explicitly with a dedicated study that this class of low mass gaugino models should either be found or ruled out with early LHC data if the expected luminosity of is reached at . In addition, we will discuss current and upcoming dark matter direct detection experiments which also have the possibility of detecting the neutralino LSP in these models.

The reason the models in the Higgs-pole region can be tested soon is that several important mass scales are low enough to be within the discoverable reach of LHC-7. It is known that in minimal supergravity models the following scaling relation amongst the neutralino LSP, the chargino, next to lightest neutralino, and the gluino masses are satisfied 1992ArnowittNath () 222This relation holds for the case when all taken at the electroweak scale.

(10)

For a precise determination of the scaling relations above one must include loop corrections to the gaugino masses MV (); PBMZ (). Eq. (10) typically holds for a very pure bino LSP; whereas the scaling relations receive significant corrections when the LSP eigenstate has a non-negligible Higgsino component. The constraint of Eq. (10), which we will generalize, is an important guide regarding the types of signatures at the LHC for this class of models. In what follows we will take the scaling assumption to mean that the mass relations of Eq. (10) (or the generalization thereof, which is included in Eq. (13) in what follows) hold to a good approximation.

Remarkably, in the literature there are rather few studies of the impact on LHC physics from this Higgs-pole region with correspondingly low mass gauginos; only recently has it seen some attention. Thus, some aspects of the minimal supergravity models where the relics annihilate near the light CP-even Higgs pole have been discussed in Ref. land1 (); landb (); Feldman7Tev (); Chattopadhyay:2010vp (); land2 (); Ross (), which fall under the mass hierarchy denoted by mSP4 (supergravity mass pattern 4)  land1 (); landb (), where, in particular, a clean edge in the dilepton invariant mass in this model class was noted in Ref. landb (). In addition, the very recent work of Ref. Ross () studies electroweak symmetry breaking in an overlapping class of models with a focus on the parameter and radiative breaking.

Some of our observations and emphasis here have overlap with Refs. Chattopadhyay:2010vp () and some are rather different. In Ref. Chattopadhyay:2010vp () emphasis was given to explaining the CDMS II results and predictions for the XENON data, and in doing so, a slice of the parameter space was studied where and was fixed for a few choice values, while the analysis allowed for flavor violation, and thus constraints from and were not imposed. Our present analysis imposes these constraints and opens up new parameter space where all direct and indirect constraints are satisfied, and where the spin independent scattering cross section can lead to event rates that can be observed in the XENON detector.

When all direct search limits and indirect constraints on the parameter space are imposed a number of robust mass relations are predicted. The main points emphasized in this work are as follows:

  1. Two key observables which are directly measureable at the LHC: the peak in the effective mass distribution as well as the dilepton invariant mass edge are shown to be strongly correlated in these models. A first determination of the gluino mass can be measured from the peak value of the effective mass distribution and the dark matter mass can simultaneously be inferred from the dilepton edge due to the predicted scaling relations in the gaugino sector given in Eq. (10).

  2. The recent CMS and ATLAS data with 35 pb of integrated luminosity Khachatryan:2011tk (); AtlasSUSY () do not yet provide constraints on the models discussed in this paper. In the Higgs-pole region, even though the gluino has a low mass, the generation squark masses are larger than 1 TeV and typically of order several TeV which is the main reason these models remain unconstrained by the CMS and ATLAS data (the gluino mass bounds in the recent ATLAS analysis AtlasSUSY () do not apply to our models). However, we will show that with increased luminosity they will begin to probe such models.

  3. The gluino has a low mass which is tightly constrained to lie in the range , with most points having333This is the Gaussian peak (i.e. mean) and Gaussian width (i.e. 1 standard deviation) . The mass splitting between the gluino and the lighter gauginos is appreciable. Thus should this model class be realized in nature, the production of jets from the gluino must be seen at the LHC at with about 1 of data Feldman7Tev (),FKLN (),Lessa (),Peim (); Peim2 (),Ross ().

  4. The chargino mass is bounded from below by the LEP search limits and from above by theory, , with the second heaviest neutralino being effectively degenerate with the lightest chargino. This suggests that the associated production of is sizeable and may reveal itself in multilepton channels, in particular the trilepton channel Nath:1987sw (); BaerChenPaigeTata (). The large SUSY breaking scalar masses in the models imply that the current bounds from the Tevatron do not yet constrain the models.

  5. There is a sizable region of the parameter space in which can be large and the pseudoscalar Higgs boson is relatively light. Such model points may allow for simultaneous reconstruction of and in early LHC data collection.

  6. The constraints from the CDMS and XENON data XenonExp (); CDMS () on the spin independent scattering cross section of neutralinos on nucleons is complimentary to searches for the CP-odd Higgs at the Tevatron and at the LHC. In fact, for some models in the parameter space the XENON data already constrains models that will be tested in 2011 and 2012 at the LHC. We find many candidate models that yield large event rates in upcoming dark matter direct detection experiments.

As an aside, we note that the neutralino annihilation rate we consider is too low to produce observable cosmic signatures of positrons, antiprotons, or gamma rays; hence recent experimental bounds from a variety of cosmic ray experiments are not a concern. In principle one could boost the annihilation cross section in a number of ways in order to reach the sensitivity of the experiments, but that approach is not considered here.

Thus in this work we study a dense region of the parameter space of minimal supergravity models where the LSPs have low mass that also have low mass gluinos which will be tested at the LHC in the very near future. In addition, we find a bound on the Higgs sector from the XENON data. We explore the connection between these models and what the LHC, the Tevatron, and the dark matter scattering experiments can observe. The prominent signatures of the models under full collider simulation are discussed in detail in what follows.

Ii Analysis of the Parameter Space and Sparticle Masses

In this section we describe our targeted parameter scan over the minimal supergravity parameter space for the low mass gaugino models that lie in the Higgs-pole region. We will illustrate the various constraints we have imposed on the models, from astrophysical relic density as well as accelerator bounds. From the results of our survey of parameter space, we then obtain the viable range for sparticle masses and the relations between them.

In the analysis that follows we compute the thermal relic density as implemented in MicrOMEGAs 2.4 susypackage3 (). We demand that the resulting value of the cold dark matter relic density satisfy

(11)

The spread in (11) around the WMAP band WMAP () is chosen to allow theoretical uncertainties and sensitivity to the top pole mass, both of which enter in the sparticle spectrum under renormalization group flow and radiative electroweak symmetry breaking.

Our targeted parameter scan over the minimal supergravity parameter space is described in what follows. For models in which the gaugino masses are given by a universal parameter at the scale , the analysis of Ref. 1992ArnowittNath (); Nath:1992yr () found that Eq. (10) is consistent with ; thus in the interest of obtaining models with low mass gauginos, we restrict to the range . The universal scalar mass was allowed to vary in the range with the upper bound representing a naturalness requirement on the models. The entire allowed range of was explored and the universal trilinear parameter was allowed to vary over the range . Throughout we take and . Renormalization group evolution and calculation of the physical masses of the sparticles was performed using SuSpect susypackage1 () and SUSY-HIT susypackage2 () was used in the computation of branching ratios of the superpartners.

Our survey resulted in 12,000 parameter sets, each defining a single model. All model points were required to satisfy the requirements of radiative electroweak symmetry breaking. Accelerator constraints were applied as well. The most important bounds include the imposition of the higgs mass bound discussed in the previous section, and the bound on the chargino mass from direct searches for sparticles from LEP Nakamura:2010zzi (). In addition a number of indirect experimental constraints were imposed, which include those from the Tevatron, Belle/BaBar/Cleo and Brookhaven experiments. Specifically we impose the conservative constraints , see Djouadi:2006be (); Chen:2009cw (), (90 % C.L.) Aaltonen (), and  Barberio:2008fa (). The indirect constraints were calculated using MicrOmegas, with the Standard Model contribution in the last observable corrected according to the work of Misiak et al. Misiak:2006zs (); Chen:2009cw (). Finally, we require that the relic density satisfy Eq. (11).

The models surveyed are consistent with

(12)

with most models satisfying . Therefore, post facto, Eq. (11) and Eq. (12) together provide an effective definition of what constitutes the Higgs-pole region. From this ensemble of models we find the mass relations

(13)

and the qualitative scaling relations in Eq. (10) can be replaced by the more quantitative relations

(14)
Figure 1: Distribution of the ratio from Eq. (13). The distribution is well approximated by a Gaussian characterized by . The corresponding spread of gluino masses for the models simulated was found to be (quoted are mean values and one standard deviation about the mean).

The distribution of gluino masses for the models is well approximated by a Gaussian with a remarkably small width. In Figure 1 we plot the distribution in the dimensionless ratio from Eq. (13). We see that in general the models produce a gluino mass of

(15)

Thus consistent with Eq. (10) one finds

(16)

Predictions for the Sparticle Masses and LSP Eigencontent Mass Predictions  (GeV)   Eigencontent of the LSP () () () ()

Table 1: General predictions for the sparticle masses for the models with satisfying all phenomenological constraints discussed in the text. It is further found that , and the scalar masses are bounded as : , , , , , and  GeV.

In Table 1 we expand on the general ranges given in Eq. (13). For example, whereas in the previous paragraph and in Figure 1 the 1 error bars are quoted for the gluino mass, the full range of all gluino masses obtained in our survey is

(17)

The upper bound for the gluino mass, consistent with a low mass neutralino has very important consequences for LHC searches as discussed in the next section. Another result of our analysis is that while the LSP is dominantly bino-like it can also have a significant Higgsino component as seen from Table 1.

For the small values of that lead to a light gaugino sector it is necessary to require large and/or to satisfy the direct search limits on the light CP-even Higgs mass . We therefore found that ranges from about to and that typically is much larger than . Indeed in our survey an empirical lower bound of was obtained. A large fraction of the models thus lie on the hyperbolic branch/focus point region ArnowittNath () in which scalars are in the TeV range and is typically small. Consequently all the first and the second generation squarks and sleptons are significantly heavier than the gluino. In particular one finds the lower bounds and on squarks and sleptons of the first two generations. Third generation squarks and sleptons are also found to be generally heavy, though lower masses occasionally arise for certain combinations of and . Specifically we find the following lower bounds on third generation scalars: , and .

We further note that the parameter for most of the models lies in the range , though larger values are possible. The models with low can lead to a CP-odd Higgs mass that can be quite light – particularly when the value of is simultaneously large. We find a lower limit of over the ensemble of models studied. As we will see below, inclusion of the limits on the neutralino-proton spin independent cross section, , from the CDMS and XENON experiments further constrain the models. We discuss this in some detail in Section IV.

Finally, one might ask if charginos with masses in the range are already ruled out by direct searches at the Tevatron, given the recently quoted lower bounds of derived from the absence of trilepton events with large missing transverse energy C2 (); D2 (); SUSY09 (). Such a lower bound is due to the assumption of light slepton masses. However, as discussed above, the low mass gaugino models in the Higgs-pole region single-out scenarios in which the sleptons are generally very heavy, as in the “large ” models analyzed by DØ D2 (). Using Prospino2 Prospino () to calculate the next-to-leading order (NLO) production cross sections for the Tevatron at we find, before cuts and efficiency factors,

(18)

after simply summing over all three generations of leptonic decay products, which is the maximal case, and this result is below the reported limits from the Tevatron C2 (); D2 (); SUSY09 ().

Iii Signatures of the Low Mass Gaugino Models in the Higgs-pole region at the LHC

To study the signatures of the low mass gaugino models at LHC-7 we simulate events at for a sample of 700 model points from the larger set discussed in the previous section. The standard model (SM) backgrounds considered were those used in Peim (); Peim2 () which compare well to those given in Lessa (). The SM background was generated with MadGraph 4.4 MGraph () for parton level processes, Pythia 6.4 PYTHIA () for hadronization and PGS-4 PGS4 () for detector simulation. The total R parity-odd SUSY production cross section () for the low mass gaugino models are composed, to a first approximation, of only three contributions: production of chargino and the second lightest neutralino (i.e. ); gluino pair production (i.e. ); and chargino pair production (i.e. ). The three sparticles produced with the largest production modes, namely , , and , then decay with the dominant branching ratios shown in Table 2. The ranges shown are for the subset of 700 models. The total SUSY production cross section is relatively large for this class of models given the relatively light gluino, charginos and neutralinos () over the set of 700 models.

Branching Ratios of the Low Mass Gaugino models in the Higgs-pole region

% % %
Table 2: Typical size of dominant branching ratios of the sparticles with the largest production modes emerging from proton-proton collision at the LHC over a subset of 700 models. Here includes the first 2 generations of quarks and includes all 3 generations of leptons (hence the factors of 2 and 3 in the Table). The factor of 4 includes and the conjugate modes for the charginos. In addition to the three dominant sparticles arising from proton-proton collisions (the three cases considered in the Table), a small subset of models are found to produce light stops () at the LHC which decay via respectively, depending on the particular model point.

The rather small variances around the central values for production cross sections and branching fractions suggest that the models in the Higgs-pole region are strikingly similar in their features, at least in terms of the phenomenology associated with the gaugino sector. This is not unexpected given previous studies of sparticle mass hierarchical patterns land1 (); landb (); land2 (); land3 (). As we will demonstrate in what follows, these similarities allow predictions to be made if excesses over SM background are observed at the LHC. Furthermore, as we will see in Section IV, it is likely that these models will allow for a determination of the light gaugino masses and a partial determination of the neutralino LSP’s eigencontent should a corroborating signal be observed in dark matter direct detection experiments.

Key Spectra of Sample Models

Label
2990 148 2503 26 476 119 60 117 2959 1668 2608
1238 132 -2007 7 407 116 55 109 1250 421 1467
2463 133 -2003 50 447 118 58 117 2443 1353 423
2839 131 -2401 50 451 119 58 118 2812 1562 355
Table 3: Four benchmarks to illustrate collider and dark matter signals of the low mass gaugino models in the Higgs-pole region. All models give a suitable relic density consistent with WMAP. Masses and dimensionful input parameters are given in units of GeV. The first and second generation squarks are denoted by . The top pole mass is set to and the sign of is positive. Number in the table are rounded to the nearest integer. All values are computed with MicrOMEGAS 2.4 and SuSpect.

To illustrate the phenomenology of the low mass gaugino modelsin the Higgs-pole region we have chosen four benchmark models as presented in Table 3. For each of these models we will compute the event rates for eight supersymmetric discovery channels defined by the following sets of cuts Peim (); Peim2 ()

(19)

All eight channels involve a cut on transverse sphericity of and a missing transverse energy cut of , except for for which we impose . Leptons of the first two generations () are denoted collectively by and the number of leptons and the number of jets in an event are denoted by and respectively. Similarly, and refer to the transverse momentum of the hardest lepton or jet, respectively. The notation means that the first through the fourth hardest jets in an event each have to individually pass the cut, and does not imply a sum. If no value is specified for an object then no cut has been made for that object. In the specification of the cut , the subscripts and indicate that the two opposite sign leptons may be of different flavors; a -veto is imposed on the invariant mass of the two leptons only in the case when they are of the same flavor, so as to avoid contamination from the boson peak produced through Standard Model production modes.

We define the effective mass and by

(20)

where is a visible object (jet or lepton) and the summation, in both cases, is done over the first four hardest objects. The variable is closely related to other definitions of (see htvar () for different definitions of ). We define a model to be discoverable in a given channel (or for a given cut), , if , where is the number of SUSY events and is the number of background events. Further, we loosely refer to a excess as one which satisfies , and a lower bound of ten events is imposed in rare cases where the SM background is insignificant for a specific channel.

LHC Significance for Channel with 35 pb and 1 fb @

Jets   Leptons + Jets  
Label  CUT  CUT  CUT  CUT  CUT  CUT  CUT  CUT
(2) [12] (1) [6] (2) [9] (2) [11] (2) [11] (0) [1] (1) [3] (0) [2]
(4) [21] (3) [14] (4) [21] (4) [24] (4) [23] (0) [2] (1) [6] (0) [1]
(3) [13] (1) [10] (2) [13] (3) [15] (3) [15] (0) [2] (1) [5] (0) [2]
(2) [15] (2) [10] (2) [13] (3) [16] (3) [15] (1) [2] (1) [5] (0) [2]
Table 4: for the models of Table 3 for both (35 pb) and [1 fb] of integrated luminosity at the LHC with . The (0) in the table means a significance of less than 1. We expect the entire set of our models discussed in Table 1 to surpass the 5 significance threshold in jet-based channels early at LHC-7 with about an inverse femtobarn of data.
Figure 2: (color online) Effective mass variable for the SUSY signal plus background with cut at  TeV. The SM background alone is shown shaded for comparison. For benchmark 1 (top panel), with a gluino mass of  GeV, we see a peak at corresponding to a mass ratio of . For benchmark 2 (bottom panel), with a gluino mass of  GeV, a peak is observed at which corresponds to a mass ratio of .

In Table 4, we give an analysis of a broad range of event rates for the low mass gaugino models in the Higgs-pole region at with both 35 pb and 1 fb of luminosity under the cuts as defined in Eq. (19). None of the models reach the discovery limit for the case of 35 pb. Benchmark point 2 has the largest significance for two reasons: It has the lightest gluino mass of the benchmarks and the generation squarks are just above the TeV scale. Indeed, these models will produce discoverable signals with an increase of about a factor of 5 in luminosity, which may be expected within the next 6 to 8 months of data taking. However, any type of serious mass reconstruction will require about an inverse femtobarn of data.

We find that the models analyzed produce a significant amount of jet events. These events arise from gluino decays via off shell squarks into fermion pairs with a chargino or neutralino, that is, and with secondary 3-body decays  + 2 fermions and  + 2 fermions. Additionally, one has a significant cross section for the direct production of charginos and neutralinos which can also give leptonic final states. Our analysis finds that the distribution of the transverse momentum of the hardest lepton is peaked near and falls off quickly near 60 GeV before imposing the cuts in Eq. (19). The relatively soft leptonic decay products makes it more difficult to use leptonic signatures as discovery channels with limited data, as exhibited in Table 4. However, the lepton + jets signal can be strong (see channel ) where a large significance is achieved. Trileptonic signal is only at the level of but would become visible with an increase in luminosity by a factor of six. The above features are generic to all models in the in the sample, given the rigid properties of the gaugino sector shown in Table 1.

The strongest signal of new physics will be in the multijet channel. In Figure 2, we plot the distribution in for two of our benchmark points using the cut of Eq. (19). The heavy solid line gives the supersymmetric signal events plus the SM background while the shaded area is the SM background. The peaks in this distribution can be identified with a typical accuracy of 25 GeV, which is half the bin size. A more statistically rigorous approach gives similar results.

Figure 3: (color online) Left: Distribution of the ratio of the effective mass peak to the gluino mass. The models plotted here are the 700 model subset and the peak is found after adding the SM background and applying cut . We find the peak to be at . Right: Distribution of the ratio of effective mass peak to the mass difference between the two lightest neutralinos under the same cut. The mass difference between the two lightest neutralinos corresponds to the upper bound of the edge in the OSSF dilepton invariant mass plot. We find the peak to be at .

Several previous works msusy () have shown that there is a relationship between the effective mass peak and the minimum mass of the gluino and the first two generation squark masses. Since in the low mass gaugino models that lie in the Higgs-pole region, the first and the second generation squark masses are always heavier than the gluino mass, the peak of the effective mass gives a relationship to the gluino mass. Analyzing the effective mass peak for cut for all 700 simulated models we find in general

(21)

with the precise range being , as can be seen from the distribution in the left panel of Figure 3. We note that both of the benchmark cases in Figure 2 show this result explicitly. Thus a measurement of provides an important early clue to the size of the gluino mass. Next, defining

(22)

the mass relations found in Eq. (13) or Eq. (14) suggest that under cut the peak in the effective mass distribution will be proportional to

(23)

The distribution of is shown to be peaked in the right panel of Figure 3, a result which follows from the left panel of Figure 3 and from the distribution in shown previously in Figure 1.

Figure 4: (color online) OSSF dilepton invariant mass for the SUSY signal plus SM background using cut at  TeV. The SM background is shown separately for comparisons. For the benchmark 1 (left panel) we see an edge at and for the benchmark 2 (right panel) we see an edge at , which agree well with the mass differences between the two lightest neutralinos in both cases, which are predicted to be 60 GeV and 55 GeV from theory (see Table 3).

The mass ratio plotted in the right panel in Figure 3 is noteworthy in that the quantity is measurable from the edge of the opposite-sign, same-flavor (OSSF) dilepton invariant mass distribution, (for a recent study see Mohr ()). In Figure 4 we plot this distribution for the same two benchmark models from Figure 2 after applying the cuts  from Eq. (19). Upon reconstruction of the dilepton invariant mass for the two sample models, one observes clean edges near 55 GeV and 60 GeV for the two cases. For the complete set of the 700 simulated models one finds

(24)

In addition, from Eq. (13) we expect the upper bound of the OSSF dilepton plot to be less then 65 GeV which is the upper limit on found in the analysis which can be understood by using the appropriate predictions for the for each model point.

In addition, because , we can express the effective mass peak in terms of the edge approximately as

(25)

Thus we arrive at a very simple, but strong correlation between these two key observables at the LHC, i.e., and .

We therefore come to the conclusion that the low mass gaugino models in the Higgs-pole region are fully testable with early LHC data. If the models studied in this paper do indeed describe the supersymmetric content of our Universe, then the following three observations must follow:

  1. The dilepton invariant mass edge with an upper bound of must be found.

  2. The multi-jet effective mass must be found, which peaks in the range
    consistent with Eq.(21).

  3. The mass relation in Eq.(25) must hold.

We now emphasize

  • LHC measurements can be used to estimate the dark matter mass in this model class. The upper bound of the edge in the OSSF dilepton invariant mass allows us to estimate the neutralino mass splitting and the scaling relation of Eq. (24) allows us to infer the dark matter mass.

  • The effective mass peak and the dilepton invariant mass edge are strongly correlated via Eq. (25) and provide cross-checks of the model.

In the next section we will look for further avenues to exploit the remarkable predictivity of this model paradigm.

Iv Dark Matter Direct Detection Experiments and Connection to the LHC

The complementarity between dark matter detection experiments and collider signatures has been emphasized in many previous works (for a recent review see Nath:2010zj ()). Here we will focus on this complementarity within the context of the low mass gaugino models in the Higgs-pole region. We will show that experiments for the direct detection of dark matter such as XENON put further constraints on the parameter space of the model.

We begin by noting that the predictions of Eqs. (10,12) and the relic density constraint largely ensure that the models yield predictions in narrow corridors as exhibited in Table 1. Nevertheless, the properties of the neutralino, and in particular its scattering cross section on nucleons, will depend on parameters such as , and the resultant components which govern the wavefunction of the LSP. The features of the spin-independent neutralino-nucleon scattering are easily understood in the models as they arise for large with the s-channel squark exchange suppressed and the scattering is dominated by Higgs exchange through the -channel. Thus the spin independent scattering off target nucleus arising via the interaction , in the limit of small momentum transfer is well approximated by with with the form factors given in Ibrahim (); Gondolo:1999gu (); Ellis (); susypackage3 () and with coupling given by Ibrahim (); Gondolo:1999gu (); Ellis ()

(26)

The parameters depend on eigen components of the LSP wave function and depend on VEVs of the Higgs fields and the neutral Higgs mixing parameter . For up quarks one has and for down quarks . These simple relations reproduce numerical results of  susypackage3 () and closely match the numerical work we do in this paper.

Dark Matter and the Sample Models

Label
1.4 0.995 -0.023 0.093 -0.015 0.110
1.7 0.998 -0.029 0.058 -0.012 0.108
1.8 0.996 -0.018 0.092 -0.012 0.104
3.0 0.996 -0.016 0.085 -0.011 0.125
Table 5: Spin-independent cross section for neutralino scattering on protons for the benchmark models of Table 3. Also given is the computed thermal relic density and the components of the LSP wavefunction.

For the four benchmark models of Table 3, we present the spin-independent cross section of neutralino scattering on protons in Table 5. However, from a survey over the collection of all the models in the Higgs-pole region we find a very broad range of possible scattering cross sections

(27)

The largest of these are already ruled out experimentally from the null results of the CDMS II and XENON 100 experiments CDMS (); XenonExp (). For the purposes of this paper we will assume a hard limit of for all neutralino masses under consideration as indicated by the XENON 100 experiment; this value is extremely conservative as their reported bounds are a factor of two more stringent, but we wish to allow for some uncertainty. A large fraction of the remaining models will be probed after longer exposures with XENON, or in future at other experiments. The distribution of our 12,000 models in the plane is given in Figure 5 with both the CDMS II and XENON 100 limits indicated CDMS (); XenonExp (). Models which are being constrained by the XENON and CDMS data are those with . Note that the models in Figure 5 satisfy all the constraints discussed in Sec.(II).

Figure 5: (color online) The spin independent cross section versus neutralino mass. Points are colored according to the value of taken. Applying the XENON and CDMS limits we see that is preferred in the 120 GeV to 155 GeV region.

An important point to note is that dark matter direct detection experiments can be used to learn about soft supersymmetry breaking parameters. Figure 5 shows that once the spin independent cross section and neutralino mass are known from direct detection experiments, then can be determined directly. Let us assume that a dark matter direct detection experiment observes a signal in the near future which is compatible with a neutralino LSP in the mass range . Within the constraints of the of the Higgs-pole region even a crude measurement of the scattering cross section yields important information about the parameters of the model. The results shown in Figure 5 already demonstrate a correlation between and . For example a simultaneous estimation of and would predict due to the correlated nature of the parameters within the Higgs-pole region. This, in turn, would have testable consequences for the gaugino sector at the LHC.

Figure 6: Displayed is the small subset of the 12,000 models which are those corresponding to large