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Here be Dragons: The Unexplored Continents of the CMSSM

The Higgs boson mass and the abundance of dark matter constrain the CMSSM/mSUGRA supersymmetry breaking inputs. A complete map of the CMSSM that is consistent with these two measured quantities is provided. Various"continents,"consisting of non-excluded models, can be organized by their dark matter dynamics. The following mechanisms manifest: well-tempering, resonant pseudo-scalar Higgs annihilation, neutralino/stau coannihilations and neutralino/stop coannihilations. Benchmark models are chosen in order to characterize the viable regions. The expected visible signals of each are described, demonstrating a wide range of predictions for the 13 TeV LHC and a high degree of complementarity between dark matter and collider experiments. The parameter space spans a finite volume, which can be probed in its entirety with experiments currently under consideration.


INTRODUCTION
The discovery of the Higgs boson [1,2] confirmed the Standard Model. But open questions remain: Why is the W -boson mass so far below the Planck scale? What non-baryonic substance makes up roughly 80% of the matter content for our Universe [3]? Is the Standard Model prediction that the gauge couplings nearly unify at high scales a hint of new physics?
The leading framework that can address all of these outstanding issues with the Standard Model is supersymmetry (SUSY). The Minimal Supersymmetric Standard Model (MSSM) [4] predicts precision gauge coupling unification [5], has a stable particle which can freeze-out to the observed abundance of dark matter (under the assumption of a simple (thermal) cosmological history) [6,7], and provides a TeV scale cutoff for quadratic divergences in the Higgs mass.
Despite all of its theoretical successes, the observed value of the Higgs mass is a challenge to accommodate inside the MSSM. At the same time the non-observation of superpartners at the LHC calls into question the existence of low-scale supersymmetry. Arguably, these two lessons from the LHC may be related to each other. The tree level prediction for the Higgs boson mass m h in the MSSM is m h ≤ m Z , where m Z is the Z 0 -boson mass. However, as the superpartners become heavy there are sizable one-loop radiative corrections [8][9][10][11]: where m W is the W ± -boson mass, g is the SU (2) standard model gauge coupling, tan β is the ratio of the Higgs vevs, m t is the top quark mass, mt i are the physical stop masses, and  This demonstrates that going from the LEP2 limit on the Higgs mass of m h ≥ 114.4 GeV [12] to the observed value of m h 125 GeV [1,2] requires quadrupling the top squark masses.
Taken at face value, the Higgs discovery has profound implications on the expectation of the mass scale for the supersymmetric particles. 1 One goal of the SUSY phenomenology community is to understand the consequences of m h 125 GeV on the ∼ 120 dimensional MSSM parameter space. In practice this is intractable due to the immense size of the MSSM, not to mention all the possible extensions.
The resulting space of phenomenological signatures is enormous. It is an unrealistic task to "exclude the MSSM." The desire to chart all possible experimental implications of the MSSM has motivated many different approaches. Evoking a top-down perspective, many models of the SUSY breaking parameters have been constructed. These proposals derive the low energy 1 An arbitrarily light t 1 is possible with a 125 GeV Higgs with careful choices of A t , m 2 q3 and m 2 u c 3 . [13,14].
parameters from far fewer inputs. Specific frameworks tend to be highly predictive; large classes of SUSY signatures can be forbidden. In some cases, it is conceivable to test the full parameter space. In contrast, a bottom-up motivated reduction of the full MSSM parameters to a set of 19 phenomenologically motivated inputs was proposed [15,16]. Even with this dramatic decrease in complexity, it is not possible to map all possible signals.
The CMSSM parameter space is compact. Stated precisely, every direction in parameter space is bounded using four minimal assumptions [38]: • The Higgs mass is less than 128 GeV.
• The lifetime of the Universe is longer than the observed value.
• The bottom quark Yukawa coupling remains perturbative to the unification scale.
• The LSP is a thermal relic that does not overclose the Universe.
One purpose of this article is to quantify the extent of the CMSSM. 2 For a previous attempt to map the full CMSSM, see [37].
With the exception of the last, these are unarguable constraints on the CMSSM. There are several possibilities beyond a neutralino WIMP χ -R-parity violation would cause the Lightest SuperPartner (LSP) to decay [39]; a non-trivial cosmological history such as a late-stage entropy production from moduli decay or a low reheat temperature could alter the freeze-out prediction [40][41][42]. Nevertheless, requiring a thermal history and stable χ require no additional assumptions and together stand as one of the main motivators for the MSSM in the first place. This scenario is incorporated in all that follows.
The full extent of the CMSSM will be demonstrated. It will be argued that it is in principle possible to experimentally access all of it. However, the range of allowed masses for the supersymmetric particles extends significantly further than is usually discussed. While tremendous progress toward excluding this slice of the MSSM (or discovering something like it) will be made, there exist regions which will remain beyond the combined reach of the 13 TeV LHC, 1 ton scale direct detection experiments, and telescopes which target gamma rays from dark matter annihilations. One goal of this work will be to enumerate what will remain of the CMSSM once near-term experiments have competed their searches. The entire CMSSM parameter space can be probed using experiments currently under consideration.
Most of the parameter space can be reach with the 33 TeV HE-LHC and the remaining regions can be completely covered with the 100 TeV VHE-LHC.
As is often done, the different islands of the CMSSM will be classified by the (dominant) process which sets the relic density. Several mechanisms manifest: • Light χ -Z 0 and h pole annihilation determines the relic density. This channel is active for dark matter masses m χ < ∼ 70 GeV. Since the LSP is dominantly bino and gaugino masses unify, there is a corresponding bound on the gluino mass of mg 450 GeV. This region has been excluded by LHC7 searches for gluinos.
• Well-tempered -the dark matter has a non-trivial Higgsino component [43]. This is the home of the "focus point" supersymmetry region [44][45][46][47]. Most of this region can be probed using 1 ton scale direct detection [48][49][50]. Once the lightest neutralino masses reach O(1 TeV), the relic density requirement forces a dominantly Higgsino admixture; in this limit, the tree-level direct detection cross section becomes suppressed, although it remains within reach of multi-ton scale experiments.
• A 0 -pole annihilation 3 -the strongest dark matter annihilation channel is through an s-channel pseudo-scalar Higgs resonance. This region tends to have heavy colored superpartners which limits the ability of the 13 TeV LHC to explore this entire region.
Direct detection can also be highly suppressed. There is some hope for indirect detection since the annihilation cross section today is also dominated by s-channel A 0 exchange yielding a b b final state.
• Stau coannihilation -this is being tested by a combination of searches for colored particles and direct detection. In some of this parameter space the staus decay length becomes macroscopic; it can be useful to search for charged tracks or displaced vertices to test these models [51].
• Stop coannihilation [52] -this region is largely untested. Furthermore, much of this parameter space will remain even after the full run of the 13 TeV LHC.
All of these annihilation mechanisms have been previously discovered within the CMSSM parameter space; however, the number of disconnected regions existing in the post-Higgs discovery era has not been discussed. Furthermore, many studies are based on specific slices (one common strategy is to fix A 0 and tan β and explore the M 1 2 −M 0 plane). While this can be an instructive exercise, it can lead to incorrect inferences about the general predictions of the CMSSM ansatz. This work will serve to clarify many of these issues.
There exists a large literature on mapping the CMSSM where a likelihood value is applied to each point in parameter space, see for example [37,[53][54][55][56][57][58][59]. These approaches are extremely powerful and can allow for exhaustive studies of high dimensional parameter spaces. In order to compute the likelihood, a variety constraints with associated error bars must be compiled. This can lead to conclusions about the "allowed" parameter space which are driven by measurements on quantities such as (g − 2) of the muon, b → s γ, and so on.
For example, stop coannihilation was shown to be strongly disfavored using these methods [53]. The approach taken here does not attempt to assign any statistical significance to any point in parameter space. This draws a distinction between the results presented here and these other studies, e.g. we determine that large regions of stop coannihilation are allowed.
Therefore, this work provides a compliment to the existing literature.
The organization of the paper is as follows. Sec. 2 provides our map of the viable CMSSM parameter space. Sec. 3 discusses the specifics of how the spectra and processes are calculated from the CMSSM inputs. Sec. 4 provides an in depth look at the separate regions and discusses their properties including detailed descriptions of of the expected first signals.
Sec. 5 summarizes our findings and gives a rough idea of what regions will remain unexplored after the full run of the 13 TeV LHC and ton scale direct detection experiments. An appendix is given which provides the details of how to reproduce our maps of the CMSSM using files which are available on the arXiv. Also available on the arXiv are the relevant cross sections and decay tables for all of the presented benchmark models.

CMSSM CARTOGRAPHY
This section presents four two-dimensional slices of the CMSSM parameter space which allow one to infer the location of each continent in the coordinates M 0 , M1 2 , A 0 , tan β, and the sign of µ.
A 3 GeV error bar for the Higgs mass is used; this is an estimate of the uncertainty in the theoretical calculation [60]. The spread in allowed dark matter relic density is taken to account for the O(10%) uncertainty in the calculation of Ω h 2 , e.g. from the fact that only two-to-two tree processes are included when computing this quantity. A naive bound on charginos, m χ ± > 100 GeV [61]  In practice, to determine if a point should be classified into one of these categories, the following scheme is employed: where Z B is the bino-LSP mixing angle and All CMSSM points which were generated and satisfied the Higgs mass and relic density constraints fall into one of these categories. While we have tested that this scheme matches closely with the actual processes that contribute to the neutralino annihilation cross section in the early Universe, there are some overlapping regions where the actual classification of a point is ambiguous, e.g. the cut on σ ann v in step 3 above is to separate the overlapping A 0 -pole annihilation and stau coannihilation regions in the second quadrant. This will not have a qualitative impact on any of our conclusions below.
The next section discusses the specifics of the assumptions and the tools used to make these plots. Sec. 4 discusses the observable consequences of each CMSSM continent including benchmarks which exemplify how one would search for these classes of models.

APPROACH
This section discusses the assumptions made in this study. The task of mapping the full parameter space required developing a novel scan strategy which will be discussed below. This section also explains why unfolding the parameter space into (sign(A 0 ), sign(µ)) quadrants leads to a clean presentation of the results.  SoftSUSY, an on-shell mass is computed using the two-loop massless DR scheme, including one-loop DR finite corrections [64] which are added at the scale M S = √ mt 1 mt 2 .
Given the low energy spectrum, the lightest CP even Higgs boson mass is constrained to lie between which is based on a 3 GeV uncertainty estimate [60]. One non-trival contribution to this range is the error bar on the measured value of the top quark Yukawa coupling. In addition, there are 3-loop investigations [65][66][67]

Scan Strategy
Building the maps presented in Figs. 2.1-2.4 required a more targeted computational strategy than simply randomly scanning. The results in this article began from an extensive random scan of the input parameter space. This "seed" scan was performed until a few points on all continents were discovered. Many of the regions had partner disconnected components in the other quadrants and this helped discover several of the continents. Ultimately, the discovery of any isolated region is limited by this original seed scan and there is no way to guarantee that every island was discovered. Nevertheless, O(10 7 ) random points were attempted with the following bounding box 0 ≤ M 0 ≤ 10 TeV; 0 ≤ M 1 2 ≤ 10 TeV; which limits the size of the undiscovered regions which limits the size of any islands not discovered at the 95% confidence level to be smaller than The smaller islands of parameter space are more susceptible to the implicit uncertainty in the numerical calculations used to identify the regions, which limits the potential relevance of searching for arbitrarily small regions. This article focuses on finding the full extent of the large regions of parameter space and the methods below use the continuity of the parameter space to find the range in the parameter space.
A variety of methods were used to extend the 4-dimensional parameter space after the random seed scans. For example, a useful method of filling in sparsely populated parameter space is to take two valid points; draw lines connecting the values of each set of parameters for each of these points; and perform a scan which was restricted to these lines, both between the original points and extrapolated beyond them in either direction.
Once this seed data was established, the remaining parameter space was filled in with a more efficient algorithm. The key was to target a specific slice which would ultimately be scatter plotted. Given a two-dimensional plane, a uniformly spaced grid can be applied to the plot and all viable CMSSM models then associated to a grid point. All squares on the grid which contain at least a single viable point would be filled in. Any empty point in the grid with two or more nearest neighbors would be attempted. Keeping two of its coordinates fixed by their position in the grid, the other two directions were randomly scanned using the parameters of the filled neighbors to determine the range. In order to ensure that the boundary was being appropriately sampled, this range was extended by O(10%) beyond the minimum and maximum value of its bounding neighbors. The number of attempts to fill an empty square was proportional to the number of neighbors -a point with more filled neighbors would be more likely to itself be valid. Once a point was found, the grid was updated and the algorithm continued. Besides filling in the bulk of the continents, this strategy allowed for a systematic test of all discovered boundaries.
Recently, it was pointed out that public MSSM spectrum generators including SoftSUSY do not provide unique solutions to the RGEs [62]. This is a pathology of the algorithm used by these programs, namely that boundary conditions are imposed at three different scales, m Z , √ mt 1 mt 2 , and M GUT . In fact, this behavior was found in the scans in the light χ region; by imposing the 100 GeV bound on the charginos mass these points with spurious behavior were removed. 4 Given the extent of the scans, all physically relevant regions of the low energy parameter space have been explored. However, the caveat remains that there could be additional regions which are not found using the default implementation of SoftSUSY.

Quadrants
When visualizing the allowed regions, it instructive to unfold the parameter space by weighting the x and y axes by the signs of A 0 and µ respectively. The canonical convention is that the gaugino masses and Higgs bilinear soft mass B µ are positive. This yields four distinct phase combinations that are specified in terms of sign(µ) and sign(A 0 ). It would be more economical to take µ > 0 and allow M1 2 to take either sign. However, as this is just a U (1) R rotation of the standard choice, there would result a sign flip redefinition for A 0 . In order to avoid confusion with the existing extensive literature, the standard sign convention is maintained and the explicit signs of µ and A 0 are kept explicit throughout.
The 1-loop the renormalization group equations for the A-terms and the B-term take the following schematic form 16 where M is a gaugino mass, y is a supersymmetric Yukawa coupling associated with the A-term, g is a gauge coupling, and t is the log of the renormalization scale [73]. Clearly each choice of phase leads to an independent renormalization group trajectory. The physics between quadrants are not simply related to each other.
As an example, consider the case when A 0 and |y| 2 > g 2 . Then Eq. (6) implies that the presence of a non-zero gaugino masses will suppress the A-term as it is evolved to lower energies -the magnitude of the low energy A-term will be smaller than the unification scale input. This should be contrasted with A 0 where the magnitude grows as it is evolved to the weak scale.
If M 0 is small and µ A > 0 (µ A < 0), the B-term is suppressed (enhanced) at low energy.
Given a set of inputs which yield broken electroweak symmetry breaking, changing the sign of µ can result in low energy parameters which violate the requirement of a stable, non-zero electroweak vacuum expectation value.
These two considerations motivate designating quadrants. In practice, one can see in  Working in the D-flat direction H u = q = u c , it is straightforward to find a constraint on the A-term such that the color and charge breaking minima are not the absolute minima of the potential. For the top squark this bound on A t is [74,75]: Since Notice that this allows for A t /mt √ 6 which is the condition for "maximal mixing" in the top squark sector. There is an analogous condition for the stau direction in the scalar potential.
The constraint in Eq. (8) is too restrictive. Absolute stability is not a sufficient requirement. As long as the tunneling rate from the standard model vacua to the color and charge breaking minima is longer than the age of the Universe, the theory is phenomenologically viable. This more complicated condition does not yield a simple analytic constraint [76][77][78]. However by performing a scan over a limited subset of the CMSSM parameter space, it has been argued that the metastability requirement relaxes the bound to [78] |A t | 2 < 7.5 m 2 This article will use this less restrictive requirement, though most regions satisfy the absolute stability bound.
The charge/color breaking minima are typically close to the origin in field space.
Therefore it is appropriate to evaluate this condition at low energies. In practice the DR values from SoftSUSY are evaluated at the scale M S = √ mt 1 mt 2 for checking the condition in Eq. (9).

THE MULTIPLE CONTINENTS
The CMSSM contains numerous disconnected regions of parameter space. This article classifies each continent by dark matter annihilation mechanism and quadrant. The purpose of this section is to present several benchmark models. These are chosen to exhibit some of the distinctive signatures which are possible within each CMSSM region. Additional, data files provided with the arXiv submission give a set of CMSSM inputs which can be used to reproduce all of the points in the plots.
The goal is to give a rough picture for how to discover any point within the entire CMSSM. Given the scope of this task, only the roughest description of the phenomenology is presented. Many constraints, which are traditionally used to explore the CMSSM, are Cascade decays have relevant mass scales beyond the parent particle and LSP masses.
For a one-step cascade decay, i.e., one which proceeds by emitting an additional particle, a useful dimensionless variable to describe the amount of phase space available is [83,84] when the parent particle P decays into an intermediate particle I and a Standard Model state, followed by the decay of I into another Standard Model state and the daughter particle where r = 0 corresponds to the intermediate particle and daughter particle being degenerate and r = 1 corresponds to the intermediate particle and parent particle being degenerate. A canonical example is In what follows, the values for r are specified for all cascade decays. This allows easy comparison of the benchmarks with Simplified Model results from the LHC collaborations.
One ton scale spin-independent direct detection are projected to reach [85][86][87][88][89] σ 1 ton SI ∼ 10 −11 pb at m χ = 300 GeV. (13) In the following discussion we will use the projected limit obtainable for a one ton Xenon experiment from [89] to estimate the future reach of direct detection.
One caveat to consider when comparing direct detection limits to predictions is the range for the plausible size of the Higgs-nucleon effective Yukawa coupling. This can imply up to an order of magnitude variation in the predictions for direct detection [90,91]. The main point of contention is the determination of the strange quark content of the nucleon.
There is a discrepancy when comparing lattice determinations with the value derived from a combination of chiral perturbation theory and measurements of the pion-nucleon scattering sigma term. It is worth noting that there seems to be a consensus among the lattice community [92]. For concreteness, we take the default values in DarkSUSY -if we had used the lattice values instead our predictions for the spin-independent cross section would be a factor of a few lower. This should not have a qualitative impact on any of the statements we make below.
This article will not emphasize the fine-tuning associated with any benchmark because the tolerance of fine tuning is a subjective preference. Since the entire CMSSM augmented with plausible theoretical constraints is bounded in all directions, it is not necessary to impose a bound on fine tuning. However, since it is of general interest to the community values of the canonical Barbieri-Giudice tuning measure [93] ∆ where X ∈ {M 0 , M 1 2 , A 0 , B µ } will be provided. We use the built in SoftSUSY routines to compute this value. We find that 250 < ∼ ∆ v < ∼ 60000 for viable points (before applying any LHC or direct detection bounds).
We also present the fine-tuning associated with the relic density ∆ Ω . This must be considered when discussing naturalness given that this is an "orthogonal" tuning to ∆ v .
Furthermore, some of the regions require a conspiracy in the spectrum to reproduce the observed value of Ω h 2 . Analogous to Eq. (14) we define where again X ∈ {M 0 , M 1 2 , A 0 , B µ }. We perform this derivative numerically by interfacing SoftSUSY and MicrOmegas. Given that determining ∆ Ω is computationally expensive, we have only explored this tuning for the benchmarks. We find that 22 < ∼ ∆ v < ∼ 1100 for the models presented below.
Before discussing the detailed regions and benchmarks we need to quickly clarify our notation. We will interchangeably use the terms LSP, lightest neutralino, χ, and χ 0 1 . The other neutralinos will be denoted with χ 0 i and we will call the charginos χ ± i . All other superpartners will be demarcated with a tilde. When they are nearly pure we will also refer to the electroweakino states as bino B, wino W , and Higgsino H. We will often refer to the light flavor squarks q which includes the superpartners of the u, d, c, and s quarks.  However, as is demonstrated by the following benchmark, this is non-trivial to show as the gluino tends to have many competing decay modes. Direct electroweakino production can also be constraining. As the scalar superpartners decouple, the electroweak tuning can also be large. This region spans the range 260 The second impact of having a bino LSP is that spin-independent direct detection tends to be small. This is shown explicitly in Fig. 4.1.3. Since many of these models rely on coupling to the Z 0 , it is possible that spin-dependent direct detection could be important -we find that these cross sections are outside the reach of near term experiments. None of this is relevant for phenomenology as these models have already been excluded by the LHC.    This benchmark provides a concrete example of the issues that arise when attempting to exclude a model with a variety of competing decay modes. Both M1 2 and µ are a factor of 15 to 30 smaller than M 0 ; this is a CMSSM realization of "Mini-Split Supersymmetry" [94].
While this benchmark does not have a small A term, there exist points that do. Given the low energy spectrum shown in Table 4.1.1, it is clear that the squarks and sleptons are well out of range of colliders and play no role in determining the signatures of this model. The most promising avenue for discovery is via gluino pair production. The gluino mass is 409 GeV which implies that the 7 TeV gluino pair production cross section is σ(p p → g g) 9.0 pb. Hence, it is likely that this model has already been excluded by the LHC. In order to determine if this is true, we need the full neutralino spectrum This allows the determination of the gluino branching ratios; the rates involving Higgsinos are suppressed. However, there are a variety of gluino cascades involving the winos which make limit setting non-trivial for this benchmark. The signatures of this benchmark are well-approximated by three of the standard Simplified Models: So far the only applicable LHC results which provide the limits on Simplified Models with BR < 1 have been released using 7 TeV data. There are many searches with sensitivity to this model [96][97][98]. The most relevant of these is an ATLAS search for jets, / E T , and no high p T electrons or muons using 4.7 fb −1 of data [96]. The collaboration has recast this search for the final state providing a limit of σ × BR < ∼ 1 pb. The corresponding prediction for this benchmark is 1.8 pb when all combinations for the sign of the W ± bosons are included. Furthermore, the decay involving Z 0 bosons will have a very similar efficiency for this search [83,84,99].
Finally, we note that a direct search for wino-like charginos and neutralinos decaying to electroweak gauge bosons and bino-like neutralinos using the full 8 TeV data set is also sensitive to this model [100]. These considerations exclude this benchmark.

Well-tempered
The well-tempered region of the CMSSM is characterized by a non-trivially mixed LSP.
Specifically, the lightest neutralino has a non-trivial wino and/or Higgsino component. This implies that the dominant process which determines the relic density is the annihilation channel χ χ → W + W − as shown in Fig. 4.2.1. The well-tempered region encompasses the socalled focus point region [44][45][46][47].  squark masses tend to be larger than the gluino mass because larger values of M 0 are required in order to achieve a Higgs boson mass of 125 GeV. The squarks in these models will lie outside the range of the 13 TeV LHC. Only a small range of these models will be testable at colliders through the direct production of gluinos.
Well-tempered The most effective way to discover or exclude this region is through direct detection. As shown in Fig. 4.2.3, the ton scale limits on spin independent scattering will cover most welltempered models. Multi-ton scale direct detection should have sensitivity to this remaining sliver of parameter space. Even accounting for the uncertainty in the nucleon form factor, the well-tempered region is probable utilizing near-term direct detection proposed experiments.
The LSP is dominantly bino dark matter with a non-trivial wino ad-mixture -this is clearly a well-tempered neutralino. This model a sizable annihilation cross section to W + W − in the early Universe so that the computed relic abundance can match the observation.
The gluino has a mass of m g = 1333 GeV and currently is too heavy to have been directly produced. Its cross section at the 13 TeV LHC is σ(p p → g g) = 30 fb.
The most important gluino decays for phenomenology are cascades involving the electroweakinos that have a large Higgsino fraction, χ ± 1 χ 0 2 , and χ 0 3 .
The majority of the gluinos decay into heavy flavor. The cumulative light flavor branching ratio is g → q q X 11.5% (23) where X is a neutralino or chargino. This pattern of branching ratios can be understood from the pattern of the squark masses, Since the gluino decays are mediated through off-shell squarks, the branching ratios are proportional to 1/m 4 q i . The process with an off-shell right handed stop is enhanced by a factor of 7.7 over light flavored squarks. All of the properties of the spectra will have to be inferred from these decay widths since the direct production of these squarks is beyond the reach of the 13 TeV LHC. They should be accessible at the 33 TeV LHC.
The LHC phenomenology of this model is dominated by the following simplified models involving the lightest Higgsinos The Higgsino decays to the bino via an off-shell W ± or Z 0 . The Higgsino cross sections are sufficiently large such that they will be discoverable at the 13 TeV LHC; for example at √ s = 13 TeV.
Direct detection is a good way to discover this model with a spin independent cross section of This is within a factor of two of current sensitivity. Therefore, this benchmark will likely be discovered using direct detection instead of at the LHC. Given a direct detection signal the LHC signatures will provide a crucial compliment for understanding the properties of this model. This benchmark provides an example of a well-tempered neutralino with a very heavy squark and gluino spectrum. Clearly from Table 4 The decays of the lighter electroweakinos occur via off-shell decays mediated by the W ± and Z 0 . The cross sections for producing these states directly at the 13 TeV LHC are below 1 fb. The wino decays yield boosted W ± , Z 0 and h with p T ∼ 600 GeV. However, given the large diboson background, these states may not be observable at the 13 TeV LHC.
The direct detection cross section is at the edge of the current XENON100 exclusion: This benchmark will be probed by direct detection using existing technology. This benchmark is provided as an example of a model in the "pure Higgsino" limit of the CMSSM. From Table 4 The LSP is very nearly Higgsino. Given this model will require a multi-ton scale direct detection experiment to be convinced it has been probed. This model demonstrates that even the most difficult limit of the well-tempered region can eventually be explored.    The story for direct detection is more favorable. Fig. 4.3.3 shows that some of these models are already in tension with the XENON100 limit. The 1 st and 2 nd quadrants can be almost entirely covered using ton scale direct detection. However, the 4 th quadrant will remain outside the capabilities of these experiments. We note that the shape of the region In order to probe the remaining models, another class of experiment is necessary. One promising avenue is indirect direct detection. In particular, searches for continuum γ-rays gives the current bound from XENON100 [95] and the dashed line is a projected limit for a ton scale Xenon experiment [89].
from dark matter annihilations could be sensitive to these models in the future [101,102].
The relic density in this region is determined by annihilation to bottom quarks. Annihilations from a survey of 10 dwarf galaxies results in the solid line plotted in Fig. 4.3.4 [103]. One can also derive complimentary limits using the Fermi LAT galactic center data [104]. This appears to be the most promising way to test these 4 th quadrant points. Future experiments such as the proposed Cherenkov Telescope Array (CTA) will be relevant [105,106].
As discussed above, DarkSUSY and MicrOmegas tend to disagree by roughly 30% in this region. All of our benchmarks match the observed relic density according to both calculations; we provide both values below. Low energy spectrum mg mq mt It is clear that the gluino decays will involve heavy flavor. In fact The The lightest stop decays will dominantly involve the bino and winos: The neutral wino decays to h χ 0 1 over 90% of the time while the charged wino decays to W + χ 0 1 with a 100% branching ratio. Therefore, the dominant simplified models to describe the first LHC signals of this benchmark are The decay involving Higgs bosons is an interesting feature of this model.
Since they are pure winos, there is a large 13 TeV cross section for The χ 0 2 decays will involve boosted Higgs bosons. There may be a possibility of distinguishing this signal from the electroweak backgrounds [107][108][109].
It should also be possible to test this model using ton scale direct detection experiments: σ SI = 6.1 × 10 −10 pb. (41) Finally, there is a complimentary signal for indirect detection. The annihilation cross section to bottom quarks is It is possible that CTA would have sensitivity to this model [105,106]. However, given the uncertainty associated with the dark matter profile it is unlikely that CTA can conclusively exclude a cross section of this size.
All together, this benchmark involves many interesting signatures with a high degree of complimentarily between many experiments.  σ SI = 6.7 × 10 −10 pb. (43) Note that it also has an annihilation cross section of which, given optimistic assumptions about the dark matter profile, would be also possible to explore with CTA [105, 106].

Benchmark 3.3
Input parameters   3 presents an A 0 -pole annihilation benchmark which will be unobservable at the LHC and outside the reach of ton scale direct detection. The squarks are far beyond the reach of the 13 TeV LHC. The gluino is 3 TeV which will also be difficult for the LHC to discover since the large g q production channel will not be active. The next to lightest electroweakino is a pure wino at 1.1 TeV. The spin-independent direct detection cross section is also beyond the reach of a 1 ton scale experiment. The annihilation cross section is and is dominated by the b b channel. Therefore the only possibility for probing this model will be indirect detection with the CTA experiment [105,106].
This is an example of a model within the CMSSM which will be very difficult to test.

Stau Coannihilation
Stau co-annihilation is a commonly studied mechanism for setting the dark matter abundance within the CMSSM [110,111]. If the stau mass is where T f.o is the LSP freeze-out temperature, the staus may annihilate with the otherwise inert LSP. For a range of input parameters, the appropriate rate to achieve the measured relic abundance can be found.  The LSP mass is less than 800 GeV within the stau coannihilation regions. Therefore, the stau must be light and the scalar masses must be low. This in turn forces a fairly light supersymmetric spectrum. It is clear from the plot of the squark mass versus gluino mass plane shown in Fig. 4.4.2 that nearly all of this region will be observable at the LHC.
As shown in Fig. 4.4.3, the entire 2 nd quadrant region should be visible to ton scale direct detection experiments. This implies that the few points in the 2 nd quadrant with squark mass above 3.5 TeV which may remain unprobed by the 13 TeV LHC will be tested other ways (see Table 4.4.3 for a benchmark).
One final characteristic of this region is that the stau is the NLSP. Depending on the mass splitting between the stau and the LSP, the stau will decay promptly, inside the detector, or long after it has passed through [51]. An example of the first and last possibilities are given in the following discussion and the variety of LHC phenomenology that can result is presented.  The first stau coannihilation benchmark features a promptly decaying stau; the mass splitting between the lightest stau and the LSP is mτ 1 − m χ = 3.36 GeV so that the decay τ → τ χ can proceed on-shell. However, given that this splitting is small, the resulting τ will be very soft and essentially invisible at the LHC. In practice, any τ 1 which is produced

Stau Coannihilation
In principle one would like a direct confirmation that the stau and LSP were degenerate in order to determine that the stau coannihilation was the dominant process for determining the relic density. The cleanest channel would be through direct stau production. At the 13 TeV this process has a 0.55 fb cross section. However, since the stau decays only produce / E T , this would be essentially impossible to distinguish from the background.
Another avenue would be to produce staus in the decays of neutralinos and charginos.
The electroweakinos are nearly pure with the following spectrum The degeneracy between τ 1 and the LSP is driven by the presence of a large A-term for the stau at low energies. Therefore, the stau is the only slepton with a mass below the lightest chargino; the other sleptons have masses between 800-900 GeV. This impacts the decay pattern for the lightest chargino and second lightest neutralino. Specifically The χ + 1 decay effectively results in / E T while the χ 0 2 decay yields a τ and / E T . The best channel for observing these states at the 13 TeV LHC would be though χ 0 2 χ ± 1 production Given that the final state is a τ and / E T this will be very challenging to observe. However, it should be possible to discover these states at a TeV e + e − collider.
These two contributions will both provide similar signatures so for the purpose of arguing the first signals of this model, these can be combined. The right handed squarks decay to q + χ while left handed squarks decay to q + χ + 1 . As described above, χ + 1 yields / E T so the signature will jets + / E T with the caveat that the phase space distribution for the left handed squark decay will be distorted by the cascade.
The stop is the lightest squark due to the large A-term.
Therefore, the gluino decay patters are dominated by heavy flavor: The rest of the gluinos decay to light flavor quarks and squarks.
The large A-term impacts the decays of the stops and sbottoms since it gives these states a large coupling to the Higgs boson. Given a gluino produces heavy flavor and a boosted Higgs in its decays 18% of the time. Relying on the gluino-squark associated production channel results in a production cross section of which is O(100) events at 100 fb −1 . This is a distinctive signal to search for at the 13 TeV LHC that currently has not been targeted. This stau coannihilation benchmark features a long lived stau; the mass splitting between the lightest stau and the LSP is mτ 1 − m χ = 0.28 GeV. The stau lifetime will be O(10 −2 s) [51]. It is stable on detector time scales and will manifest as a CHArged Massive Particle (CHAMP).
Using 7 TeV data, ATLAS has already placed bound of mτ 1 < ∼ 280 GeV on the direct production of long lived staus [112]. At the 13 TeV LHC, Since these particles are CHAMPs it should only require a handful of event to discover them.
Decays involving the staus and the tau-sneutrinos are particularly relevant for LHC searches. Their masses are The electroweakinos are very pure: Given that the LSP is very nearly pure bino, direct detection is too small to be observed.
The winos will play an interesting role in the potential collider signatures.
The gluino and squarks are observable at the 13 TeV LHC. The patterns of squark masses determines the gluino branching ratios into different flavors: The gluino has a similar decay pattern to the previous benchmark with decays involving stops and sbottoms 73% of the time.
The most interesting signature of this model is the presence of a CHAMP in the gluino and squark decays. In fact, the discovery mode seems likely to be squark pair production: σ(p p → q q) 10 fb; (66) σ(p p → q q * ) 1.9 fb.
To understand the light flavor squark decay patterns, take the up squark as an example: The decay pattern of u L is more interesting since this can result in CHAMPs. To determine the fraction of u L decay modes that have CHAMPs requires knowledge of the following Noting that squark pair production leads to at least one left handed squark roughly 75% of the time, all of this information can be combined together to give σ(p p → CHAMP + j + X) 1.4 fb.
This cross section is larger than that for direct stau production by more than a factor of two. This benchmark would provide an an early discovery for the LHC. 5 The A-term is not as large in this model as it was in the previous benchmark. This This benchmark provides an example which would likely be observed first in direct detection.
Given mτ 1 − m χ = 0.13 GeV, the τ 1 is a CHAMP. The stau pair production cross section is at the 13 TeV LHC. Given 100 fb −1 of data, there would be roughly 2 events. Assuming a decent efficiency for CHAMP searches, it is likely that this model could also be probed at the LHC.

Stop Coannihilation
The remaining regions are characterized by stop coannihilation. 6 If the stop mass is within Since the relic abundance is dominated by coannihilation, the LSP can be very binolike. Hence, the tree-level direct detection cross sections can be very small, as shown in  are far too low to ever be discovered. This behavior results because the LSP is approaching the limit of pure bino. In this limit, the µ term is becoming heavy which implies that the scalar masses are also becoming large. Therefore, the low energy A-term for the stop must also become large in this region to result in a t 1 eigenvalue which is nearly degenerate with the LSP.
These large A-terms have another important physical consequence -they can contribute to direct detection at 1 loop via the diagram in Fig. 4.5.4. The appropriate 1-loop calculation has been performed [114][115][116]. However, the region of parameter space resulting in stop coannihilation has not been fully explored. While a full reevaluation of the 1-loop diagrams are beyond the scope of this work, it is possible to estimate the size of these contributions.
Consider the effective operator for Higgs mediated spin-independent direct detection after the Higgs boson has been integrated out: where y χ is the effective coupling between the dark matter and the Higgs and y q is the quark Yukawa coupling. The estimate for the size of the correction from Fig. 4.5.4 is where g is the hypercharge gauge coupling, N c = 3 is the number of colors, Q i Y is the hypercharge of the particle i, and C is a numerical constant that has not been computed.
Note that for this region mt 1 m t so that it is safe to ignore the contribution to the estimate that is proportional to m t .  Following the conventions of [48], this yields The value of A t /mt 1 can range from In the absence of a accidental suppression, there is a class of models for which this process would be observable using ton scale technology. This will be discussed for both benchmarks below. The lightest stop mass in this region is driven by the non-zero contribution from A t × H .
If the Higgs vev is changing during freeze-out, the stop mass can be drastically different and the naive freeze-out calculation breaks down. There are a variety of effects to consider. For example, the neutralino-stop system can undergo a second period of annihilation after the Higgs field settles into its final value -this is reminiscent of "changing dark matter" models [118].
The largest freeze-out temperature in the stop coannihilation region of the CMSSM is This is benchmark provides an example of a point in the stop coannihilation region that is observable at the 13 TeV LHC. The electroweakinos are nearly pure with masses and orderings given by The lightest stop decays are The gluinos decay to one final state: The production largest cross section is for t 1 pair production. At 13 TeV However, given that the stop and LSP are incredibly degenerate, the decay products would be extremely soft. This process will be unobservable. Furthermore, direct squark production is O(10 −2 fb), and would therefore be difficult to observe.
Given the squark spectrum in Eq. (79), it is clear that associated production will dominantly include the right handed up-type squarks.
The squark decays are Due to the stop-neutralino degeneracy, any t 1 in the final state will manifest as missing energy. The relevant electroweakino decays are Putting all of this together gives σ(p p → t t / E T X) = 0.22 fb (89) σ(p p → t t / E T X) = 0.22 fb (90) This benchmark motivates the study of a "like-sign tops plus / E T " simplified model [119] which is not currently being searched for at the LHC.
The direct production cross sections for χ + 1 χ − 1 and χ + 1 χ 0 2 are 1.6 fb and 2.6 fb respectively. The charged winos decay to b+ / E T and the neutral wino gives t+ / E T . The only electroweakino production signature which would be potentially observable is single top + / E T with cross section of a few fb. This is a very challenging signal to observe. This benchmark serves as an example of something that is impossible to see at the 13 TeV LHC. The gluino mass is 5 TeV and the squark masses are Given that m χ 1 TeV, this will likely require multi-ton scale experiments to be observed.
However, further study is warranted to determine the precise value of this cross section.
Taken together, this point provides an example of a CMSSM benchmark that will be extremely difficult to probe without an energy upgrade for the LHC.

DISCUSSION
This article has mapped out the entire parameter space of the CMSSM ansatz in the post-Higgs discovery era. The constructed maps of the regions that are consistent with the measured values of the Higgs mass and dark matter relic density demonstrate that the CMSSM is compact. The inputs can range from O(100 GeV) to O(10 TeV). While the Giudice-Barbieri definition of fine-tuning indicates that the CMSSM is at least tuned to a part in 200, this quantity is bounded to be less than a part in 60,000.
The near-term discovery or exclusion of the CMSSM shows an interesting interplay between the three common searches for physics beyond the Standard Model: direct production of superpartners, direct detection of the LSP, and indirect detection of the LSP annihilation products. While it is not possible to fully exclude this ansatz by the end of the decade, a large portion of the CMSSM will be discovered or excluded. Going through each of the five regions • Light χ: LHC 7 and LHC 8 -completely excluded; • Well-tempered: multi-ton scale direct detection -most likely discover or exclude; • A 0 -pole annihilation: LHC 13, ton scale direct detection, and indirect detectionsome region will remain; • Stau coannihilation: LHC 13 and multi-ton direct detection -most likely discover or exclude; • Stop coannihilation: LHC 13 and direct detection -some region will remain.
After the full run of the 13 TeV LHC with 300 fb −1 and ton-scale direct detection only portions of the A 0 -pole annihilation and stop coannihilation regions will go untouched.
This article provided benchmarks and discussed a wide variety of the Simplified Models which can result from the CMSSM including some with the following features: • gluino cascade decays involving heavy flavor and electroweak gauge bosons; • gluino cascade decays to heavy flavor and Higgs bosons; • electroweakino production resulting in boosted Higgs bosons; • colored production with stable charged particles in the final state; • same sign top production with missing energy.
This paper demonstrates a general philosophy that can be taken when attempting to exclude the entire parameter space of any restrictive slice of a model such as the MSSM.
The CMSSM serves as a nice example for demonstrating the importance of complimentary experiments -it is our job to be sure we are looking under every possible rock as we search for the signs of beyond the Standard Model physics.
Going into the future, due to the compactness of the CMSSM parameter space, the masses of the heaviest particles are all beneath 30 TeV and the heaviest particles are colored. Since all R-odd superparticles can be made through the decays of colored particles, it is possible to discover all of these states at a "human-buildable," e.g. √ s = 100 TeV, hadron collider in the foreseeable future. Fortunately, for most of the parameter space, discoveries of physics beyond the Standard Model should have occurred beforehand.

SoftSUSY Input Card.
This is the input card we use to compute the spectra using SoftSUSY v3.3.7: The CMSSM inputs are specified by replacing the variables with the desired input values in the "MINPAR" block.

Format of Data Files
Data files are available in the tarball on arXiv in connection to the preprint of this paper.
We have generated a data file corresponding to each region for all figures. The files are named by the dark matter classification and the corresponding figure number. The data is organized so that each row is one set of CMSSM inputs, separated by a comma and a space.
The order is