Model Independent Approach to Focus Point Supersymmetry: from Dark Matter to Collider Searches
Abstract:
The focus point region of supersymmetric models is compelling in that it simultaneously features low finetuning, provides a decoupling solution to the SUSY flavor and CP problems, suppresses proton decay rates and can accommodate the WMAP measured cold dark matter (DM) relic density through a mixed binohiggsino dark matter particle. We present the focus point region in terms of a weak scale parameterization, which allows for a relatively model independent compilation of phenomenological constraints and prospects. We present direct and indirect neutralino dark matter detection rates for two different halo density profiles, and show that prospects for direct DM detection and indirect detection via neutrino telescopes such as IceCube and antideuteron searches by GAPS are especially promising. We also present LHC reach prospects via gluino and squark cascade decay searches, and also via clean trilepton signatures arising from charginoneutralino production. Both methods provide a reach out to TeV. At a TeVscale linear collider (LC), the maximal reach is attained in the or channels. In the DM allowed region of parameter space, a TeV LC has a reach which is comparable to that of the LHC. However, the reach of a 1 TeV LC extends out to TeV.
1 Introduction
Supersymmetric models of particle physics provide a compelling case for physics beyond the Standard Model (SM). However, in spite of their many successes, they also provide a long list of potential problems. For instance, supersymmetric models are supposed to provide a solution to the finetuning problem which arises when the SM is embedded in theories with high mass scales beyond a few TeV. However, the generally excellent agreement of precision EW observables with SM predictions, along with null search results for various rare decays and other loop induced processes, points to a rather heavy sparticle mass spectrum, with sparticles typically in the TeV regime. These observations are supported by recent search results from LEP2 that the chargino mass GeV and the SM Higgs mass GeV. The latter limit, when applied to the Higgs bosons of the MSSM, also implies relatively heavy top squarks. A rather heavy sparticle mass spectrum, on the other hand, naively seems to reintroduce the finetuning problem into supersymmetric models.
In addition, in the 124 parameter Minimal Supersymmetric Standard Model (MSSM), unsuppressed FCNC effects arise from general Lagrangian parameters, as do large contributions to the electric dipole moments of the electron and neutron from possibly large violating phases. It has been noted by many authors that scalar masses in the multiTeV regime can act to suppress most or all of these undesired effects to within levels tolerated by data[1]. MultiTeV scalar masses can be reconciled with a nofinetuning requirement in two cases. In one case, inverted scalar mass hierarchy models[2] (IMH) require scalars of the first two generations to be in the multiTeV regime, while scalars of the third generation, which enter finetuning calculations, remain at subTeV levels. In practice, the radiatively driven IMH models require Yukawa coupling unification and nonuniversal soft SUSY breaking (SSB) Higgs masses to be viable[3, 4]. The second case, the topic of this paper, is that of hyperbolic branch[5] or focus point models[6] (HB/FP), wherein all three generations of scalars can be in the multiTeV regime, while finetuning is respected for low values of the GUT scale universal gaugino mass .
The HB/FP region appears already in Ref. [7] as a region in the plane of the minimal supergravity (mSUGRA) model where is in the multiTeV regime, but where the absolute value of the superpotential parameter becomes small, adjacent to regions where radiative electroweak symmetry breaking (REWSB) fails to occur (where ). The small value of leads to a mixed higgsinobino LSP (a mixed higgsino dark matter (DM) candidate) and a rather light spectrum of charginos and neutralinos. This mixed higgsino region with large scalar masses was investigated more thoroughly by Chan, Chattopadhyay and Nath in Ref. [5], where it was noted that the parameter can be regarded as a measure of finetuning, and where the trajectory of constant was found to form a hyperbolic branch trajectory in the mSUGRA parameter space. In Ref. [6], Feng, Matchev and Moroi noted the focusing behavior of the renormalization group (RG) trajectory, wherein a variety of GUT scale values would be “focused” to a common weak scale value of . Since REWSB leads to at the weak scale, the focused solutions gave rise to small values. These authors moreover performed a sophisticated finetuning analysis, and showed that finetuning was small in the HB/FP region as long as was not too large. It could be seen in Ref. [8], and more fully in [9, 10, 11], that the relic density is indeed low in the HB/FP region, as is typical of mixed higgsino dark matter. Further, in Ref. [9], it was shown that a variety of direct[12, 13] and indirect[14, 15] DM detection rates were large in this region, due to the large neutralinonucleon scattering cross sections, and also due to the large neutralinoneutralino annihilation rates. In addition, collider reaches in the HB/FP region were found for the Fermilab Tevatron[16], the CERN LHC[7, 17, 18] and the International Linear Collider (ILC)[19].
All of the above HB/FP analysis were performed in the plane of the mSUGRA model. However, it was noted in Ref. [16] that the exact location of the HB/FP region in the mSUGRA plane is extremely sensitive to the assumed value of . In addition, different algorithms for predicting sparticle masses in the mSUGRA model were found to give very different portrayals of the shape and location of the HB/FP region even for the same assumed value of [20]^{1}^{1}1Even updated versions of the same computer codes would give shifted locations of the HB/FP region.. Finally, it was shown in Ref. [21, 22] that the HB/FP region occurs also in models with nonuniversal scalar masses, and that its location can shift depending on the amount and type of nonuniversality which is assumed.
In this paper, we propose a more sensible parametrization of the HB/FP region based purely on weak scale parameters and gaugino mass . Such a parametrization should be independent of the GUTtoweak scale evolution algorithm assumed, and also should be only weakly dependent on the value of or other SUGRA parameters which are assumed. In Sec. 2, we present our reparametrization of the HB/FP region, and show the regions of the weak scale plane which are allowed by the recent WMAP measurement of the relic density of cold dark matter in the universe. In Sec. 3, we show current astrophysical constraints on the HB/FP region arising from various sources, including an overproduction of Li in the early universe. We also show the prospects for exploring the HB/FP region via direct and indirect DM detection. In the HB/FP region, prospects are especially encouraging via Stage 3 direct detection experiments, detection of DM via neutrino telescopes, and via antideuteron searches at experiments such as GAPS. The latter results depend strongly on the galactic DM density profile which is assumed. In Sec. 4, we show the reach prospects of the CERN LHC and also for the ILC for SUSY in the WMAP allowed part of the HB/FP region. The CERN LHC reach is evaluated in the case of gluino and squark cascade decays, and also for clean trileptons arising from production. In both cases, the LHC reach is out to TeV assuming 100 fb of integrated luminosity. The ILC reach is maximal in the or channels, and extends out to TeV for a TeV ILC, but to TeV for a TeV collider.
2 Unraveling the Focus Point Region
The purpose of this section is to introduce the HB/FP parameter space region of the mSUGRA model which produces a sufficiently low relic neutralino abundance in the Early Universe at large values of the universal scalar mass . We motivate here why a GUTscale description of the HB/FP region is illsuited for phenomenological studies, and outline an alternative and complementary weak scale parameterization. The latter allows us to describe in detail the cosmologically allowed areas in the more physical weakscale parameter space, and will be used throughout the remainder of this report to study the corresponding phenomenology at dark matter search experiments and at colliders.
Radiative electroweak symmetry breaking is the mechanism in which EWSB is triggered by turning negative due to its RG evolution. The RGE for reads
(1) 
where
(2)  
(3) 
where at all scales in models with universality. Among the necessary conditions for the spontaneous breaking of the EWS, the one which determines the (absolute) value of the parameter reads (at tree level)
(4) 
At moderate to large values of , as implied by the LEP2 lower bound on the lightest Higgs mass, . When the universal mSUGRA GUT scale scalar mass takes values much larger than all other softbreaking masses, the RG evolution of is dominated by the term in Eq. (1). As increases, cancellations occurring in yield smaller and smaller absolute values for . Therefore, Eq. (4) leads to the possibility of achieving, in principle, arbitrarily low values for , until, eventually, , and REWSB can no longer be obtained.
Within the mSUGRA model, low values of therefore occur in the region of very large universal scalar mass TeV. This opens up a qualitatively new window in the model parameter space, as far as the cosmological abundance of thermally produced relic neutralinos is concerned. As approaches the lowscale value of the lightest softbreaking gaugino mass, in the case of mSUGRA, the higgsino component of the LSP increases, leading to an enhancement of efficient LSP pair annihilations into gauge bosons (the latter being largely suppressed in the case of a binolike LSP). Further, the lightest chargino and the two nexttolightest neutralinos get closer in mass to the LSP, contributing as well to the suppression of the LSP relic abundance through coannihilations.
As a result, the particle spectrum of the cosmologically allowed HB/FP region of the mSUGRA model is characterized by (1) a heavy scalar sector, in the multiTeV range (with the exception of the lightest CPeven Higgs boson), and (2) low values (anywhere below or around 1 TeV) of the parameter. For 1 TeV, the requirement of a sufficiently low neutralino relic density forces moreover the relation , the latter being the softbreaking hypercharge gaugino mass. The weakscale values of the three softbreaking gaugino masses, unified to a universal value at the grand unification (GUT) scale, are given by the usual GUT relations,
(5)  
(6)  
(7) 
Evidently, the decoupling of the supersymmetric sfermion and heavy Higgs sectors implies that the critical parameters entering the phenomenology of the HB/FP region at colliders and at dark matter search experiments are and , their relative size setting the LSP mass and higgsino fraction. Further, the gluino mass is determined as well by , through Eqs. (5)(7). The sign of and affect as well, though less critically, the lowenergy implications of the setup.
In the standard GUTscale parameterization, where one slices the parameter space e.g. along planes, the HB/FP region appears as a narrow line squeezed on the large region adjacent to where no REWSB is attainable. The steep and complicated behavior of as a function of makes it difficult to read out the neutralino mass and composition in that representation. Further, the HB/FP region is often plagued by numerical stability problems, which will be discussed in detail in Sec. 2.1, affecting the phenomenologically crucial lowscale value of the parameter. As a consequence, it is not easy to read out from the standard parameterization most of the phenomenologically relevant information, e.g. the neutralino mass range and composition compatible with the WMAP relic abundance, and the projected reach of collider and dark matter search experiments.
The purpose of our analysis is therefore to map the GUTscale representation of mSUGRA in terms of the universal scalar and gaugino masses onto the more physical plane on which the whole HB/FP region phenomenology sensitively depends.
From the highenergy scale point of view, depends essentially only on (see Eq. (5)), while is sensitive, in principle, to all mSUGRA GUT input parameters. For given fixed values of and of the trilinear coupling , the mSUGRA parameter space spans a limited region only, on the physical plane. The parameter features, in fact, a certain maximal value (reached at low or intermediate values of ) along slices at fixed . Values larger than the mentioned maximum cannot be obtained if REWSB is required. To illustrate this point, we show, in Fig. 1, curves at fixed on the plane, computed with Isajet 7.72 [23] at two values of and , with an input top mass GeV. The iso curves terminate where the RG code no longer converges to a stable solution, while the blue curves indicate the maximal values. We remark that the low termination point of the curves is in fact only a numerical artifact, and that the entire low region extending to is a physically viable portion of the parameter space, which is not accessible in many codes which rely on a GUTscale parameterization.
As expected from Eq. (4), a given value of is obtained, at smaller , with larger values of , and viceversa: this reflects the wellknown fact that the HB/FP region appears at smaller values of the larger is.
The sensitivity of the HB/FP region parameter space in the () plane on the value of has been widely reported, see e.g. Ref. [16, 24, 20], and can be readily understood from Eq. (1). The shape of the weak scale parameter space (the plane) will change as well, but is much less sensitive to the assumed value of . We show the analogue of Fig. 1 but at GeV in Fig. 2.
2.1 Numerical issues in the HB/FP region
As we outlined above, a first, and critical, numerical issue in the HB/FP region regards the possibility of achieving a stable and convergent solution for low values of the parameter. To investigate this problem, and to verify the consistency of numerical results in the parameter space region of interest, we compared the latest release of Isajet, v.7.72, with the latest release of another RGE evolution package, Suspect, v.2.34 (for a comparison among the results of these and other numerical codes, see also Ref. [27, 24, 20]).
Fig. 3 illustrates how the two RG evolution codes project iso lines onto the physical parameter space of the focus point region, the plane. We adopt for both numerical codes the same input top mass, which we set to GeV. The lines end, as in Fig. 1 and 2, where no stable solutions are found, in the low portion of the plots. We see that in both cases, values of smaller than 100– 200 GeV in the large region cannot be numerically resolved. Further, although both RG codes feature 2loop RG running of gauge and Yukawa couplings, the disagreement is remarkable (particularly in the low region and at low values of ).
The calculated value of the parameter as a function of along mSUGRA slices at fixed is shown in Fig. 4. While the agreement among the two codes is excellent in the low end of the plots, when approaching the HB/FP region the values of (including the maxima at fixed ) significantly differ. The largest differences are found at low and at low .
A variety of numerical issues may cloud the evaluation of the parameter in supersymmetric models connecting the GUT scale to the weak scale. For example:

One problem is that the convergence at low values of will depend in part on the initial guess for the supersymmetric masses at the very beginning of the iterative RG evolution process. The default guess for the supersymmetric masses used in the two codes respectively reads
(8) (9) Clearly, the different trial values used by the two codes can alter the final convergent solution and the value of as well, particularly in the highly finetuned region where .

A further problem occurs in that SUSPECT 2.34 uses twoloop RGEs only for gauge and Yukawa couplings, but not for soft SUSY breaking terms, while Isajet uses twoloop RGEs throughout its RG treatment.

Another difficulty occurs in that loop corrections must be included in the formulae for the minimization of the scalar potential of the theory. These loop corrections depend on the full spectrum of supersymmetric particles. However, to calculate the full spectrum, the value of must be known. It is possible in the low region that the tree level value of , while loop corrections will lift . In this case, a guess must be made as to what the loop corrected value of is, just so that a viable spectrum of SUSY particles can be calculated, and used as input, in its turn, to the loop corrections. The value of will depend on how this guess is made.

As evident from Eq. (1), at very large values of the common scalar mass, is extremely sensitive to the top Yukawa coupling . The latter is defined as
(10) where is the running top quark mass in the scheme. The computation of suffers from uncertainties related to (1) the extraction of the top mass from its pole or values (i.e. the inclusion of Standard Model threshold corrections) and (2) the implementation of the SUSY loop corrections, and the scale choice at which these are implemented. Similar ambiguities pertain to the evaluation of the bottom and tau Yukawa couplings.

The numerical results will depend on the scale choice at which different SUSY particles are integrated out of the effective theory. A related issue is the choice of scale at which various SUSY threshold corrections are implemented. For instance, SoftSUSY and Spheno assume the MSSM is valid from all the way to . When scalars have masses of several TeV, such as in the HB/FP region, this may not be a good assumption.
As a bottom line, large numerical uncertainties plague the study of the phenomenology of the HB/FP region at the GUT scale, depending on a number of assumptions in the details of the RG evolution which lead to significant discrepancies in the determination of from REWSB; on top of this computational ambiguity, a possibly larger uncertainty stems from the input value of the top quark mass: even at the level of accuracy with which will be measured at the LHC the induced variations in the lowscale parameters in the HB/FP region would give rise to completely different phenomenological scenarios at the same GUTscale input values [20]. Lastly, currently available numerical codes do not fully access the phenomenologically interesting mSUGRA region at , which can thus be only explored resorting to a lowscale parameterization.
2.2 Outline of the lowenergy parameterization
We described above how the procedure of outlining a systematic mapping between the phenomenologically relevant lowenergy parameter space and the customary mSUGRA GUTscale input setting faces a number of intrinsic issues. Moreover, those issues evidently blur the possibility of a bottomup reconstruction of mSUGRA highenergy input parameters. Our main point here is therefore that the collider and dark matter phenomenology of the focus point region is better studied through a convenient twoparameters lowenergy description, which is complementary to the usual GUTscale parameterization, and which thoroughly reproduces all phenomenological implications of the “mother theory” at the highenergy scale. This section is henceforth devoted to the construction of such a lowscale parametrization, of which we will then make use in the remainder of this report.
We argued above that the physical parameter space of the HB/FP region is given by the set . At a given value of , (and therefore of at the GUT scale) and of the input top quark mass , REWSB bounds the range of from above, through the condition given in Eq. (4). As pointed out in the previous section, the precise boundaries of the region allowed by REWSB will also depend on numerical subtleties. However, we will show below that the REWSB boundary at large lies outside the cosmologically relevant HB/FP region. As a result, these uncertainties will not affect the discussion of the HB/FP region phenomenology, since that portion of the parameter space will be disregarded after the neutralino relic density analysis. The other soft breaking gaugino masses and are given, as functions of , by the GUT relations specified in Eqs. (5)(7).
Since the heavy scalar sector is largely decoupled from the lowenergy phenomenology in the HB/FP region, the details of the sfermions and heavy Higgses spectrum play a very marginal role. In this respect, we resort in most of our plots, to setting, for simplicity, all sfermions and heavy Higgs masses to a common value TeV, much larger than all other relevant weakscale parameters, and close to the average value of the masses obtained by a full RGE treatment. Further, we set all trilinear couplings to zero.
The value of the lightest Higgs mass enters, instead, quite critically in a few quantities, in particular those related to the neutralino scattering off matter: for instance, the neutralinoquark spin independent scattering cross section , in the limit of large scalar masses, is dominated by channel light Higgs exchanges, and scales as . In mSUGRA, the value of the lightest Higgs mass depends, in principle, on all input SUSY parameters. In the HB/FP region, at large universal scalar masses, we find a critical dependence on , while the GUTscale values of and are far less important. Consistently with the assumption of a common lowscale scalar mass , we therefore expressed as a function of the parameters and , according to a phenomenological functional dependence of the form
(11) 
The formula was then fitted against mSUGRA points randomly generated at large values of . We find that and do not critically depend on , and the best fit values read GeV and GeV, respectively. The exponent depends instead more sensitively on : for instance, we find for , and for . We estimate the typical accuracy of Eq. (11) in reproducing the correct value of to be less than 1 GeV for GeV, and within 2 GeV for values of smaller than 300 GeV. The relative error induced in is therefore expected to be of .
2.3 Neutralino relic abundance
Large values of the sfermion masses in the HB/FP region help alleviate a number of wellknown phenomenological difficulties of supersymmetric extensions of the standard model, ranging from the SUSY flavor and problems, to dimensionfive proton decay operators which often appear in SUSYGUT embeddings. In the present framework, where we neglect violating phases, and assume a minimal flavor structure, the SUSY contributions to rare processes, e.g. the branching ratios , , or to the muon anomalous magnetic moment, are very suppressed, being mediated by sfermions loops, and do not constrain the HB/FP region parameter space. Further, limits from direct sfermion searches at colliders are always very distant from the sfermion mass scales of the HB/FP region.
The phenomenological constraints which apply to the HB/FP region of mSUGRA are henceforth limited to the LEP2 searches for the lightest chargino, and to gluino searches at the Tevatron. In the present setting with universal gaugino masses, the latter bound is however much less constraining than that stemming from chargino searches. We use here the mass limit GeV [28], although in the purehiggsino region () the LEP2 bound is weakened by the quasidegeneracy between the lightest chargino and the LSP (namely, GeV [28]). The () plane is further constrained, at a given value of , by the REWSB conditions which limit the range of from above. In the leftover portion of parameter space we compute the thermal LSP relic abundance with the DarkSUSY package [29]. We rule out models giving , according to the 2 upper bound on the cold dark matter abundance derived by the WMAP collaboration [30]
In Fig. 5, left, we plot the parameter space of the HB/FP region using the scheme outlined in Sec. 2.2, at and positive . In this plot only, the scalar masses are determined by the value of needed to give the appropriate value as in Fig. 4b.The region shaded in red is excluded by LEP2 chargino searches, while in the gray shaded area the value of exceeds the maximal value compatible with REWSB at and (see Fig. 1, right). The parameter space portion shaded in yellow indicates where the upper bound on is violated. The viable parameter space of the theory is thus restricted to the green band (giving within the 2 WMAP range) and to the region in white ( below the 2 WMAP range). In the lowrelic density white region we suppose that some mechanism in the Early Universe has enhanced the final relic density of neutralinos (e.g. through nonthermal neutralino production [31], or through a modified cosmic expansion at the time of neutralino freezeout, as it might be the case in scenarios with quintessence [32], with a BransDickeJordan modified theory of gravity [33, 34] or with an anisotropic primordial expansion of the Universe [33, 35]). We therefore work under the assumption that all CDM is composed of relic neutralinos.
For completeness, we also indicate in Fig. 5 the location of the stau coannihilation strip in this lowenergy representation, squeezed along the line of maximal values. Had we chosen to consider negative values for , a rapid heavy Higgs channel exchangemediated annihilation funnel would also have opened. The latter would lie at rather large values of (slightly below the coannihilation strip, since the corresponding would have been larger in the funnel than in the coannihilation region, and therefore the corresponding range would be slightly lower (see Fig. 4), possibly extending to even larger values of . In this respect, the plane is not a good representation of the coannihilation and funnel regions of mSUGRA, since it trades a physically relevant parameter for those regions () for a parameter which is instead not critical for , i.e. . We therefore stress a strong complementary role of the GUTscale and of the present lowscale parameterizations for the phenomenological study of the mSUGRA model.
The shape of the parameter space region giving exactly the central value of the CDM density determined by WMAP is indicated with a black line approximately lying in the center of the greenshaded 2 area. The shape of that line provides a nontrivial information on the mechanisms responsible for a WMAPcompatible neutralino thermal relic abundance in the HB/FP region. In the large (large ) limit, the neutralino relic abundance corresponds to that of a pure higgsinolike LSP, and is fitted, with high precision, by the formula
(12) 
This formula entails the important result that the maximal neutralino mass compatible at 95% C.L. with the WMAP upper limit on the CDM abundance in the minimal supergravity model corresponds to 1150 GeV (We recall that in the funnel and coannihilation regions is always found to be less than 900 GeV, see e.g. [36]). On the other hand, the central WMAP CDM abundance value here is attained at GeV.
The central part of the curve lies along the edge; in this region the interplay of coannihilation processes and of a mixed binohiggsino LSP cooperates to fulfill the condition . In the plane the 2 range of shrinks: this fact depends on the transition from the pure higgsino to the pure bino regime. We illustrate explicitly this phenomenon in the right panel of fig. 5, where we show as a function of along slices at fixed . The mentioned transition, which we indicate as “coannihilation regime” in the figure, starts at and ends when ; in this regime, the relic abundance is clearly a very steep function of .
As is further decreased, the neutralino mass compatible with the WMAP CDM relic abundance decreases, while the bino fraction and the mass splitting between the LSP and the lightest chargino increase. This is illustrated in Fig. 6, which also shows the isoneutralino mass contours and the isohiggsino fraction levels on the plane, which will be the object of our phenomenological analysis in the remainder of this report.
In particular, the right panel of Fig. 6 illustrates that for TeV the lightest neutralino in the HB/FP region is always a very mixed binohiggsino particle, with higgsino fractions between 0.1 and 0.9. At larger , instead, , and the “pure higgsino regime” () is reached at TeV.
Finally, let us point out the threshold effects, at GeV and GeV, respectively, corresponding to and . When one of the , , final channels closes up, the resulting suppression of the lightest neutralino pair annihilation cross section must be compensated with an increased higgsino fraction and a reduced mass splitting between the LSP and its coannihilation partners, resulting in the above mentioned bumps at the corresponding thresholds.
As a concluding remark, we stress that the picture we outlined above is very mildly dependent on the particular value of we picked, and on the sign of as well. We explicitly worked out the WMAP allowed parameter space for lower , and the emerging picture is almost indistinguishable from what we show here, providing evidence for a kind of universality in the HB/FP region phenomenology on the plane. In particular, our conclusions on the maximal LSP mass in mSUGRA are not significantly affected.
3 Dark Matter Phenomenology
In the present section we study the implications of SUSY models belonging to the HB/FP region parameter space outlined in Sec. 2.3 for dark matter searches. Following previous analysis [37, 38, 39], we make use of two extreme halo profiles, a cored halo model (the Burkert profile, see [40, 41, 42]) and an Adiabatically contracted version [43] of the cuspy halo model proposed in Ref. [44] (which we dub Adiabatically Contracted N03 profile). We claim that those two instances are indicative of the range of variations in dark matter detection rates induced by different consistent models of the dark matter distribution in the halo, respectively, giving minimal (Burkert) and maximal (Adiab.Contr.N03) rates. We refer the reader to Ref. [37] for details.
Sec. 3.1 is devoted to a parallel analysis, for the two halo models, of the current constraints coming from antimatter and gamma ray fluxes. We also impose the constraint coming from the overproduction of Li in the Early Universe [45, 46]. Sec. 3.2 is instead devoted to prospects for dark matter detection in the HB/FP region at future experiments, while in Sec. 3.3, we study the spectral features at future spaceborne antimatter experiments of models in the HB/FP region giving a thermal relic density of neutralinos consistent with the WMAP CDM abundance.
As mentioned in Sec. 2.3, we assume here that models with a low thermal relic abundance, in the region, are responsible for all the CDM in the Universe, in virtue of the above mentioned mechanisms of relic density enhancement (see e.g. Ref. [35]).
3.1 Overview of current constraints
Current dark matter detection results include the recently released direct detection exclusion limits delivered by the CDMS collaboration [47], the SuperKamiokande upper limit on the muon flux from neutralino pair annihilation in the core of the Sun [48], the antimatter fluxes as measured by balloonborne experiments [49] and the EGRET data [50] on the gamma ray flux from the Galactic Center. Besides direct and indirect dark matter searches, neutralinos are also constrained by the synthesis of Li in postfreezeout neutralino annihilations, as recently shown in Ref. [45]. This constraint can be rephrased as a constraint on the neutralino pairannihilation cross section [45], and, contrary to dark matter search results, is independent of the halo model under consideration.
In Fig. 7, we depict the above mentioned current exclusion limits in the () plane. The region shaded in gray corresponds to the parameter space excluded either by LEP2 searches for charginos, or by the lack of REWSB solutions, or by the overproduction of neutralinos in the Early Universe. The greenshaded area corresponds to the models giving at the 2 level. The left panel corresponds to the Burkert profile, while the right panel to the Adiab.Contr.N03 profile.
The Li constraint is violated on the brownshaded area, extending in the pure higgsino limit in the mass range , where the neutralino pair annihilation cross section is maximal.
Neutralino induced primary antimatter fluxes, derived according to the approach described in detail in Ref. [39], are not consistent, at 95% C.L., with the measured total (signal plus background) antimatter flux of antiprotons and positrons [49] in the regions shaded in dark and light blue, respectively. We notice that for both halo models, the antiproton flux constraint is stronger than the Li bound; the positron flux is instead in excess to the measured one only for the cuspy halo model, at GeV.
The EGRET data give an energydependent upper bound on the gamma ray flux from the galactic center, under the conservative hypothesis that the neutralino pairannihilation induced signal dominates over a negligible background. The gamma ray flux depends critically on the inner structure of the halo profile, and is therefore greatly enhanced for a cuspy profile, while it is suppressed for a cored halo model. In fact, the EGRET data do not give any constraint if one assumes the Burkert profile, while they rule out pure higgsinos as heavy as 350 GeV with the cuspy Adiab.Contr.N03 profile.
As a final remark, we point out that neither direct detection searches nor current data on the muon flux from the Sun give any constraint on the HB/FP region parameter space.
3.2 Future dark matter searches
We outline in Fig. 8 (for the Burkert profile) and 9 (for the Adiab.Contr.N03 profile) the sensitivity reach of a number of future direct and indirect dark matter detection experiments. Models in the area enclosed within each contour give a signal which is larger than the expected sensitivity of the corresponding search facility.
“Stage2” detectors refer to experiments like CDMS2 [51], Edelweiss2 [52], CRESST2 [53], ZEPLIN2 [54], which will be operative in the near future (the reference sensitivity curve we take here is that corresponding to the CDMS2 experiment [51]). We indicate as “Stage3” detectors tonsize experiments like XENON [55], Genius [56], ZEPLIN4 [57] and WARP [58] (the reference sensitivity curve is here chosen to be that of XENON [55]).
The reach of Stage2 detectors is found to be limited to a tiny region at very low masses, largely already ruled out by the chargino mass limit. A neutralino in the HB/FP region is therefore not likely to be detected at Stage 2 direct detection experiments. On the other hand, tonsized detectors look very promising, their sensitivity extending, quite independently of the halo model, and largely independently of , over the whole region compatible at 2 with the WMAP CDM neutralino thermal relic abundance, up to TeV. At larger values of the common gaugino mass, the higgsino fraction becomes exceedingly large (see Fig. 6, right), and, since , the resulting direct detection rates are suppressed.
The flux of muons from the Sun is also extremely sensitive to the degree of gauginohiggsino mixing. The equilibrium between the capture rate inside the Sun and the neutralino pair annihilation, basic in order to produce a sizable neutrino flux out of the Sun, is reached provided the neutralino features (1) a sufficiently large pair annihilation cross section and (2) a large enough spindependent neutralinoproton scattering cross section. These two conditions explain the shape of the IceCube reach contours [59], which extend along the largest area at low as far as the gaugino fraction is still nonnegligible, and in the maximal mixing region. The latter region largely overlaps the WMAP favored greenshaded region. Remarkably enough, neutralinos producing the WMAP required amount of relics to make up the CDM in the Universe tend to maximize, in the HB/FP region, the detection rate at neutrino telescopes! We find that, quite independently of the halo model, the IceCube reach along the cosmologically favored strip extends up to TeV, corresponding to a neutralino mass between 600 and 700 GeV.
The dependence of the antimatter flux on the halo model [39] is found, instead, to be indeed critical. For antiprotons and positrons, the largest fluxes correspond to the pure higgsino region, partially extending into the large binohiggsino mixing region. The shape of the reach contour for the Pamela experiment (computed following the approach outlined in Ref. [39]) reflects the role of the top threshold (below which a much larger higgsino fraction is needed) and extends into the light neutralino mass region until the second critical threshold () is reached. Assuming a cored profile, models giving the WMAP relic neutralino abundance do not yield large enough antiprotons and/or positrons fluxes, while with a cuspy profile the reach along the WMAP strip in the antiproton channel extends up to neutralino masses as large as 400 GeV.
Low energy antideuterons have been shown to provide a clean indication of new physics, since the background flux in the 0.1– 1 GeV antideuteron kinetic energy interval is extremely suppressed [60]. We assessed the antideuteron flux for the AMS02 experiment [61], which will be sensitive to a flux in the energy band GeV at the level of , and for a GAPS experiment [62] placed on a high latitude mission satellite, sensitive to antideuterons in the energy band GeV at the level of . We find that antideuteron fluxes accessible to AMS02 are in general excluded by current bounds on the antiprotons flux, independently of the assumed halo profile. The GAPS sensitivity extends instead well beyond the antiproton search reach of Pamela in the WMAP favored region, with maximal accessible neutralino masses as large as 400 GeV in the case of a cored profile and of even 700 GeV in the case of a cuspy profile. The antideuteron flux is also greatly sensitive to the gauginohiggsino mixing, thus, as in the case of the muon flux from the Sun, the maximal reach is gained exactly along the WMAP 2 region.
The parameter space reach of the GLAST experiment [63], finally, largely depends on the details of the inner structure of the halo model under consideration. As can be deduced comparing Fig. 8 and 9, the resulting GLAST reach can be either the worst or the best among all direct and indirect detection channels, depending on the amount of dark matter in the center of the Galaxy: for the two halo models under consideration, for instance, this induces a variation of more than four orders of magnitude [22]! Little can therefore be said about the sensitivity reach of future gamma rays experiments without strong astrophysical assumptions (for recent related studies see [64, 14, 15]).
3.3 Spectral features at spaceborne antimatter searches
In the preceding section we assessed the reach of future spacebased antimatter experiments by means of a statistical treatment which evaluates the possibility of disentangling a puresecondary antimatter flux from the presence of a statistically nonnegligible primary component. In case such a signal is detected at Pamela or AMS02, we will be given the opportunity of studying in some detail the spectral features of the primary component, depending on the relative signaltobackground (). In particular, the future wealth of data on antimatter fluxes will greatly reduce the uncertainties in the background determination to an unprecedented level of accuracy. For this reason, we investigate in this section the yields of models in the HB/FP region, concentrating on the parameter space slice giving equal to the central value of the WMAPinferred CDM abundance .
We show in Fig. 10 the as a function of the antiparticle’s kinetic energy for antiprotons (left) and positrons (right), at four different neutralino masses. The antiprotons features in all models a clean peak at , generated by hard decay modes, particularly from gauge boson decays, plus a low energy tail, mainly fueled by products of final state processes. Positrons show a more complex , featuring a series of peaks, corresponding to quark jets yielding and positrons from decays at low energies, and to positrons generated by gauge boson decays at higher energies. In particular, prompt and decays motivate the bump at , which is however very suppressed, and hard to recognize, particularly beyond the threshold, which tends to soften the positron spectrum.
The location of the maximal () ratios is studied in Fig. 11, where we plot on the right axis the corresponding antiparticle’s kinetic energy, and on the left axis the actual value at the maximum. In both cases, the threshold (corresponding to GeV in the plot) is clearly visible. The maxima, in that low neutralino mass end, are located around 10 GeV, for both positrons and antiprotons. For , the location of the antiproton’s maximal approximately linearly tracks , while for positrons the maxima are positioned around 10 GeV, until eventually the corresponding to the prompt positron production dominates.
In order to understand to which extent the above analyzed spectral features are specific to the HB/FP region (or to any mixed higgsinobino LSP scenario), we carried out a comparison of () in different LSP scenarios, namely that of a pure bino (mainly annihilating into and pairs), as in the coannihilation and funnel region of mSUGRA, and that of a pure wino LSP (mainly annihilating into pairs), as in the minimal anomaly mediated supersymmetric breaking model. The case of the bino is characterized by a much larger antimatter yield at low energies, which, for instance, smooths out the maximum in the antiproton pointed out above, and which shifts the positron’s maximal towards much lower energies. This LSP scenario is thus virtually distinguishable from the HB/FP region LSP scenario on the basis of a correlated analysis. The case of a winolike LSP is instead more subtle. The antiproton has a more depressed lowenergy tail (which can be however hard to disentangle due to uncertainties in the lowenergy antiproton background computation), and the location of the maximal as a function of the mass is at larger values. On the other hand, a possible handle is provided by a much stronger peak in the positron spectrum corresponding to .
The occurrence of a maximum in the antiproton’s () correlated with the neutralino mass can evidently be used as an indirect indication of the neutralino mass scale, keeping in mind the above mentioned caveat on a possible entanglement of the mixedhiggsinobino and wino LSP scenarios. Vice versa, should collider experiments or other dark matter searches point at a certain neutralino mass, the analysis of the antiproton’s can be used as a crosscheck to understand the nature of the dark matter particle. Positron fluxes appear, instead, less promising for the same task, the absolute values of the being moreover much smaller than those of antiprotons. The exciting perspective of a correlation between the antiproton’s and the positron’s spectral features, which could point to a specific LSP dark matter scenario, therefore also appear rather problematic.
4 HB/FP region at LHC and ILC
4.1 HB/FP region at the CERN LHC
The reach of the LHC in mSUGRA’s () plane was described in Ref. [18] (see [17] for related earlier work). The search strategy was based on the detection of gluino and squark cascade decay products, namely multiple jets and/or leptons and large transverse missing energy. Since sfermion masses in the HB/FP region lie in the multiTeV range, LHC will not produce squarks and sleptons at detectable levels. Gluinos can be relatively light ( TeV, left frame of Fig. 12) only in the low GeV (or, equivalently, low ) part of the HB/FP region along the line . It was found in [18] that in the HB/FP region of mSUGRA, gluino masses of up to TeV could be probed. Therefore one would not expect the method, based on the production of heavy strongly interacting superpartners, to work very well in the HB/FP region for larger values. We note that recent work by Mercadante et al.[65] employed tagging in their analysis and have achieved an up to 20% extension of the LHC reach in the HB/FP region.
However, as can be seen from the left frame of Fig. 12, charginos and neutralino (collectively dubbed as inos) can still be relatively light in the HB/FP region. Even more encouraging is the fact that the mass splitting between the lightest chargino and lightest neutralino stays GeV all the way up to GeV and the mass gap never exceeds the Z boson mass (right frame of Fig. 12). This means that 3body decays and are open, and one expects multiple leptons in the final state. We will resort to the leptonic signals when exploring ino production at the LHC, since soft jet signals have an enormous QCD background.
The () mass gap is shown in the left (right) frame of Fig. 13 for various values of and signs of . Positive is beneficial for our purposes, because the mass gap is larger than for negative , and therefore the leptons in the final state are harder. A positive value of is currently favored by the discrepancy between the measured muon value and the one calculated in the SM using data for the diagrams involving hadronic vacuum polarization [66].
For the rest of this section we restrict ourselves to the following values of input parameters: , and GeV. Unless stated otherwise, we will be working along the WMAP favored line in the HB/FP region. Along this line and are not independent, and one needs to specify only the value.
4.1.1 Reach via gluino cascade decays
We use Isajet 7.72 [23] for the simulation of signal and some of the background events at the LHC. A toy detector simulation is employed with calorimeter cell size and . The hadronic energy resolution is taken to be for and for . The electromagnetic energy resolution is assumed to be . We use a UA1like jet finding algorithm with jet cone size and GeV. Leptons are considered isolated if the visible activity within the cone is GeV. The strict isolation criterion helps reduce multilepton background from heavy quark (especially ) production. Leptons (s or s) have to satisfy the requirement GeV. We also require that leptons would have and jets would propagate within .
First, we replot the reach of the LHC in Fig. 14, using the procedure described in [18]. All events had to pass the precuts, which impose the requirement that GeV and there are at least 2 jets with GeV. The definitions of jets and leptons can be found in [18], as well as the description of the cut optimization procedure. We choose the cuts which were found to be optimal for the HB/FP region. The events are divided into several classes, characterized by the number of leptons in the final state. The upper frame of Fig. 14 shows the total supersymmetric particle production cross section and the cross section after precuts. The lower frame of Fig. 14 shows the signal cross sections after the optimal cuts in various channels. The discovery reach for 100 of integrated luminosity is shown by short horizontal lines for each channel. One can see that the cuts, which are optimized for gluino pair production and subsequent cascade decays, provide a reach by the LHC of up to GeV, corresponding to a value of TeV. At that point the total sparticle production cross section is still sizable: for GeV, the total SUSY cross section is fb. This fact motivates us to look next at ino pair production at the LHC.
4.1.2 Trilepton production at the LHC in the HB/FP region
Total production cross sections for neutralino pair production at the LHC along the line of constant are shown versus in the left frame of Fig. 15. The right frame of Fig. 15 shows total cross sections for associated production of the lightest or nexttolightest neutralino with a chargino. Similarly, we present the associated production of 3rd or 4th lightest neutralino with a chargino in the left frame of Fig. 16 and the chargino pair production cross section in the right frame of Fig. 16. Of all the ino production cross sections, , and production are generally the largest.
The gluino pair production cross section is presented in Fig. 17 versus . We have also shown the total ino production cross section for comparison. Assuming 100 integrated luminosity, LHC would produce less than 10 gluino pairs for GeV, while ino pairs would be produced.
Given the relative total production cross section rates, it might be beneficial to examine ino pair production signals in the HB/FP region, as well as gluino pair signals. Single lepton signals from where will be buried under an immense background from direct boson production. Likewise, dilepton production from reactions such as production will be buried beneath large backgrounds from and production. Four lepton signals from reactions such as production would be very distinctive and it is possible to find the cuts which reduce the SM background. However, our preliminary analysis found that the 4 lepton signal rates fall very quickly with increasing , and this channel did not provide any additional reach compared to the optimized cuts for gluino pair production. Thus we were led to consider the clean trilepton signature at the LHC in more detail[67]. This signal provides the best reach in mSUGRA at the Tevatron [16, 68, 70], due to production and subsequent decays and .
Let us first examine the dominant production processes, which could produce 3 or more leptons in the final state, at GeV, where the reach due to optimized cuts peters out:
(13)  
(14)  
(15)  
(16)  
(17) 
The relevant branching fractions are:
where stands for either or . It is now possible to estimate the maximal possible 3 lepton event rate before any cuts for 100 integrated luminosity at the LHC. We do that only for the processes (13), (14) and (15) here:
Next we proceed to the fast simulation. The dominant backgrounds for the clean trilepton signature are , and production. When evaluating production, it has been shown to be of crucial importance to evaluate the full background, which includes offshell and production, as well as other diagrams[68]. We have used Isajet 7.72 [23] to calculate the , and with backgrounds. For offshell and background calculation, we have employed an exact tree level evaluation of the processes using Madgraph[69] at the parton level. A similar calculation has been performed in Refs. [68], where soft lepton cuts were invoked[70]to try to maximize the signal when leptons originate from decays. In our case, in the LHC environment, we will require three isolated leptons each with GeV and throughout our analysis. In Ref. [68], also i. an invariant mass cut of GeV was invoked to reduce BG from the onshell boson contribution, ii. GeV was used to reduce BG from the photon pole, and iii. a transverse mass veto of GeV was used to reduce BG from onshell contributions. Finally, GeV was required. These cuts, dubbed SC2 by the Tevatron Run2 study group, will be invoked here, along with the somewhat stronger lepton cuts. The BG rates after the SC2 cuts are listed in Table 4.1.2. The signal for cuts SC2 assuming 100 fb of integrated luminosity is thus fb. In Table 4.1.2, we also list a signal point with GeV, where gluino pair production is still large and the signal is very robust.
Process  (fb)  (fb) 

4.5  
238  0.79  
758.3  0.36  
—  2.0  
Total  —  7.65 
Case study at GeV  7796  13.1 
In Fig. 18, upper frame, we show the total cross section, and in the lower frame, the clean trilepton cross section after cuts SC2 along the line of as a function of . Also shown by the horizontal mark is the limit for 100 fb of integrated luminosity. We see that the CERN LHC reach for clean trileptons is possible out to GeV, which is, in fact, comparable to the reach via conventional cascade decay signatures shown in Fig. 14, left.
While the results of Fig. 18, left, are valid along the line , it is also possible that in scenarios with mixed dark matter. In this case, values of smaller than those used in Fig. 18, left, are possible. As becomes smaller, then the becomes even more higgsinolike, and the relic density drops. The masses and drop as well, as does the mass gap . The situation is illustrated in Fig. 18, right, for the case of GeV, where various ino masses are plotted versus a variable parameter.
Thus, as drops to smaller values, many of the ino production cross sections rise. However, the trilepton energy and momentum distributions will diminish, in part due to the reduced parent particle masses, and in part due to the reduced sparticle decay mass gaps, which lead to smaller energy release in the chargino and neutralino decays. Thus, as the value of is reduced, production cross sections increase, while detection efficiency decreases. The trilepton cross section after cuts SC2 is shown versus for fixed GeV in Fig. 19, right. Here we see that at large values, the trilepton cross section after cuts is at the edge of observability. However, as decreases, the reduced detection efficiency wins out over the increasing production cross section (shown in the lower frame of Fig. 19), left, resulting in an overall diminished trilepton cross section. Thus, the trilepton cross section is actually maximal along the line , and diminishes in regions where .
In Fig. 20, we show the oppositesign/same flavor dilepton invariant mass distribution from clean trilepton events at the CERN LHC for the case where GeV. It is important to note two distinct mass edges in the plot. The first occurs from the kinematical edge from decay, and occurs at GeV. The second comes from decay, and occurs at GeV. The latter mass gap is close enough to the pole that decay matrix element effect skews the invariant mass towards the high end of the range.
4.2 Prospects for sparticle detection at the ILC
The proposed International Linear Collider is projected to operate initially at TeV with an integrated luminosity of fb. In its later stages, the CM energy should be upgraded to TeV. In Ref. [19], it has been shown that the reach of a linear collider can exceed that of the CERN LHC in the HB/FP region. This is because is always small in the HB/FP region, which forces detectable charginos and neutralinos– which should be readily accessible to colliders if the beam energy is sufficiently high– to be relatively light, even if the scalars and the gluino are relatively heavy.
In Fig. 21, we show total production cross sections for various ino pair production reactions versus along the line of constant . The lefthand frame is for a TeV collider, while the righthand frame is for a TeV collider. By examining frame a), we see that for GeV, the ino production reactions are dominated by chargino pair production, although a variety of other reactions such as , and may also be accessible. The mass spectrum shown earlier in Fig. 12 shows that and are of order , and so should be heavy enough that twobody decays are accessible. However, , and have large higgsino components and are relatively light; they should typically decay via threebody modes, where the branching fractions are dominated by the or boson propagators (since scalars are assumed quite heavy). Thus, the decays of and should be very similar in that and , aside from the size of the vs. the mass gaps. We see that the ultimate reach of the TeV machine is determined by the cross section. production is more favorable kinematically, but has a lower total cross section due to a suppressed coupling. The reach of a 0.5 TeV ILC along the line is out to GeV (corresponding to GeV), and is slightly below the reach of the CERN LHC.
In frame b), for a TeV machine, we again see that production is dominant over most of the range of , while again a variety of other ino pair production reactions should in general be present. In this case, the ultimate reach is determined by the production reaction, and extends out to GeV (corresponding to TeV), far beyond the reach of the CERN LHC.
4.3 Gluino lifetime in the HB/FP region
The gluino decay width for can be expressed as
(18) 
where represents a suitable combination of the neutralinosquarkquark couplings. The gluino lifetime, taking into account that the factor , evaluating and at , can thus be cast as
(19) 
Conservatively assuming that , we obtain
(20) 
In the case of minimal supergravity, we can draw the following general upper limit on the gluino lifetime. In mSUGRA, we always have , and, in the focus point region, where the gluino lifetime is maximal, we can safely take . We therefore have
(21) 
Since in the focus point region GeV, we find that, in minimal supergravity, sec. In order to detect a displaced vertex, the lifetime of a quasistable particle should be at least larger than sec. This therefore entails that in mSUGRA gluinos are never “stable” inside a detector, which means that if a “stable” gluino , or a displaced gluino vertex is detected, the underlying SUSY theory cannot be mSUGRA.
The largest gluino lifetimes are obtained in the focus point region at low values of and of , using a large top mass input. The spread in the gluino lifetime within mSUGRA can be significant, ranging from a minimum in the stau coannihilation region, where the squarks are the lightest possible, to a maximum in the HB/FP region.
In the stau coannihilation region, we numerically find that , which gives an estimated gluino lifetime between and sec, depending on the value of