Prospects for constrained supersymmetry at $\sqrt{s}=33$ TeV and $\sqrt{s}=100$ TeV proton-proton super-colliders

Discussions are underway for a high-energy proton-proton collider. Two preliminary ideas are the $\sqrt{s}=33$ TeV HE-LHC and the $\sqrt{s}=100$ TeV VLHC. With Bayesian statistics, we calculate the probabilities that the LHC, HE-LHC and VLHC discover SUSY in the future, assuming that nature is described by the CMSSM and given the experimental data from the LHC, LUX and Planck. We find that the LHC with $300$/fb at $\sqrt{s}=14$ TeV has a $15$-$75%$ probability of discovering SUSY. Should that run fail to discover SUSY, the probability of discovering SUSY with $3000$/fb is merely $1$-$10%$. Were SUSY to remain undetected at the LHC, the HE-LHC would have a $35$-$85%$ probability of discovering SUSY with $3000$/fb. The VLHC, on the other hand, ought to be definitive; the probability of it discovering SUSY, assuming that the CMSSM is the correct model, is $100%$.


I. INTRODUCTION
Supersymmetry [1][2][3] (SUSY) is the most popular scenario of new physics beyond the Standard Model (SM). It stabilizes the electroweak scale against radiative corrections from any high-scale physics such as a Grand Unified Theory [4,5] (GUT), predicts gauge coupling unification, and provides a natural framework for explaining the observed amount of dark matter (DM) via a thermal relic density of the lightest supersymmetric particle [6]. Despite LHC, HE-LHC and VLHC, given that previous experiments found no discrepancies with the predictions of the SM. We choose to work with the CMSSM because it is the best-known SUSY model with non-degenerate mass spectra. We believe that our results help to motivate building the HE-LHC and the VLHC, and contribute to forming research programs for those colliders.
Given our assumptions, we find that there is an appreciable probability that the LHC or HE-LHC could discover the CMSSM. However, the VLHC would be definitive -either the VLHC discovers the CMSSM or one has to relax some of the assumptions we have made.
This conclusion results from the fact that the CMSSM mass spectrum cannot be arbitrarily heavy, else the Higgs mass would be greater than ∼ 125 GeV and the DM abundance could not be explained with thermal freeze-out processes. Since those are physical requirements, a similar conclusion must be obtained for other SUSY models with similar numbers of free parameters. Our results do not apply to SUSY models with compressed mass spectra [9], which could evade searches for missing transverse momentum, or to models with extended particle content, such as the NMSSM [10,11], that have additional tree-level contributions to the Higgs boson mass as well as new candidates for DM. Those models deserve separate dedicated studies.
The paper is organized as follows. In the next section we present the basics of Bayesian statistics on which our methodology is based. In Sec. III, we calculate the posterior pdf and in Sec. IV, we discuss which parts of the CMSSM parameter space might be discoverable at the LHC √ s = 14 TeV, a √ s = 100 TeV VLHC or a √ s = 33 TeV HE-LHC and complete our calculation, finding the probability that a collider could find constrained supersymmetry.

II. BAYESIAN APPROACH
To quantify statements such as, "the probability that the HE-LHC could discover constrained supersymmetry," we invoke Bayesian statistics. We remind the reader that in Bayesian statistics, probability is a numerical measure of belief in an hypothesis. See e.g., Ref. [12] for a pedagogical introduction.
From an experiment, one can construct a "likelihood function," describing the probability of obtaining the data, given a particular point, x, in a model's parameter space, The likelihood function for a measurement is typically a Gaussian function (by the central limit theorem). Note that the likelihood function, however, is not a probability distribution function (pdf). Bayes' theorem, permits us to "invert" the likelihood function to find the probability density of the model's parameter space, given the data; This quantity, which we shall call the "posterior," is central to our work; it is a numerical measure of our belief in the parameter space after seeing the experimental data. On the other hand, π( x) ≡ p( x | model), the "prior," is a numerical measure of our belief in the parameter space before seeing the experimental data. For our purposes, the denominator is merely a normalization factor. One can see that the likelihood function "updates" our prior beliefs with experimental data, resulting in our posterior beliefs.
The posterior is a pdf of the continuous model parameters x. As such, if we wish to find the probability within a region of parameter space, we integrate (in the lexicon of Bayesian statistics, "marginalize"), e.g., In our analysis, the model is the Constrained Minimal Supersymmetric Standard Model (CMSSM) [13], the data is that from, inter alia, the LHC, LUX and Planck, and x is a parameter point in the CMSSM. The CMSSM has four continuous parameters, that is three soft-breaking parameters (m 1/2 , m 0 and A 0 ) and the ratio of the Higgs vevs (tan β), and one discrete parameter, the sign of the Higgs parameter in the superpotential (sign µ).
We assume that particular regions of our model's parameter space would be discoverable at the e.g., VLHC, whereas the complement of those regions would be entirely inaccessible; We find the probability that a point is discoverable by marginalizing; for which we rely on the fact that i.e., whether a point in a model's parameter space is discoverable is dependent on only the point itself, and not on any previous measurements. Combining our Eq. (5) with Eq. (6), we find our desired result, It can be shown that but it is an approximation that is reasonable only if no particular models are favored by the experimental data, i.e., one should not, in general, multiply probabilities from Eq. (8) by his prior for the model in question; the normalization of the result will be incorrect.

III. POSTERIOR MAPS OF THE CMSSM
We wish to calculate the posterior density of the CMSSM, given the experimental data in Table I, which includes the Higgs mass, dark matter constraints, which we assume to be the lightest neutralino, b-physics observables, electroweak precision observables and the anomalous magnetic moment of the muon (see e.g., Ref. [14][15][16][17][18][19][20][21][22][23][24][25] for similar analyses). The two ingredients that we require are the likelihood function and the priors in Eq. (3). We will supply these ingredients to the nested sampling algorithm implemented in MultiNest [26], which will return the posterior. See e.g., Ref. [27] for a detailed introduction to the methodology.
We construct Gaussian likelihoods for these experiments, including theory errors in quadrature. Our likelihood function is thus the product of these Gaussian functions, where p( x) is our model's prediction at parameter point x, µ, σ and τ are the mean, experimental error and theory error, respectively, and the product is taken over the data in Table I. We calculate the CMSSM's predictions with SOFTSUSY [28], micrOMEGAS [29,30] and FeynHiggs [31][32][33][34]. Furthermore, we veto CMSSM points excluded at 95% by LHC direct searches. We apply the 95% exclusion contour from the ATLAS search in 20.1/fb at √ s = 8 TeV [35] as a hard-cut on the (m 0 , m 1/2 ) plane. We implement the LUX direct search for dark matter [36] with an exclusion contour on the (m χ 0 1 , σ SI p ) plane, however; the CMSSM's prediction for σ SI p contains a factor of 10 uncertainty [37], which we incorporate with a method identical to that in e.g., Ref. [27].
Because we are, a priori, ignorant of the supersymmetry breaking scale, our priors for the soft parameters are invariant under rescalings, i.e., we pick logarithmic priors for the soft masses m 0 and m 1/2 ,   We will, however, investigate linear priors by re-weighting our posterior. We pick linear priors for the remaining parameters A 0 and tan β. The principle motivation for supersymmetry is that it solves the "fine-tuning" problem that afflicts the SM [51], if the soft-breaking masses are sufficiently light [52,53]. For this reason, we restrict soft-breaking masses in our priors to less than 20 TeV, in accordance with what one might have believed prior to seeing experimental data from e.g., the LHC. Furthermore, Ref. [19] indicates that this choice omits no credible parts of the parameter space. We include the SM nuisance parameters (m t , m b , α s and 1/α em ) with informative Gaussian priors, with their experimental means and variances from the Particle Data Group (PDG) [41]. Our priors are listed in Table II.
Furthermore, because our prior ranges and constraints are similar to those in Ref. [54], in which a simple "hard-cut" scanning algorithm was applied to the CMSSM, we can make a fair comparison between a Bayesian analysis and a simple method.
Parameter Distribution m 0 Log, 0.3-20 TeV We plot the credible regions of the marginalized posterior on the (m 0 , m 1/2 ) plane in MSSM models, in which m A is a free parameter [55]. Note that with lighter m A , supersymmetric contributions to BR(B s → µµ) are appreciable.
This region is similar in this regard to the focus-point. With heavier µ 1 TeV, the relic density is increased. Because µ M W , the mass splitting between the higgsinolike chargino and higgsino-like neutralinos is negligible. The higgsino-like chargino is "parasitic" in the language of Ref. [56] and increases the relic density. With lighter µ 1 TeV, the Higgs mass is lighter than its measured value.
Because the neutralino is higgsino-like, the spin-independent scattering cross-section with a proton, σ SI p , is enhanced via an s-channel Higgs diagram, with higgsino-bino-Higgs vertices. Although with µ < 0, light and heavy Higgs bosons could destructively interfere, the heavy Higgs is so heavy that significant cancellations rarely occur. This region is disfavored by LUX, which at m χ 0 1 1 TeV restricts σ SI p 10 −44 cm 2 .
These mechanisms are not disjoint; A-funnel annihilation is present, though somewhat offresonance, at heavier m 1/2 above the "1 TeV higgsino region." A line of fine-tuned solutions exist from the A-funnel mode along the top of the "1 TeV higgsino region." Although m χ 0 1 − m A /2 is increasing along this line, so too is the pseudoscalar's width. Regardless of the dark matter annihilation mechanism and regardless of our priors, we find that the neutralino mass must be less than 1160 GeV at 95% (one tail). In Fig. 2a, we see that the sizes of the A-funnel and"1 TeV higgsino region" on the (m 0 , m 1/2 ) plane are considerably increased if one uses linear priors, rather than logarithmic priors, and both are present at 1σ rather than 2σ.
Other mechanisms, e.g., Z/h resonances and bulk annihilation, which could annihilate dark matter at a sufficient rate, are excluded by LHC direct searches for sparticles. Our posterior is, by eye, in good agreement with that in Ref. [19], which omits direct detection experiments, with ∆χ 2 plots in Ref. [17] and with crude results in Ref. [54]. Ref. [19] identifies a "focus-point" region at m 0 ∼ 4 TeV and m 1/2 ∼ 1 TeV. Whilst we see a few such points in Fig. 1b at the beginning of the "1 TeV higgsino region," their combined  posterior weight is negligible, and they are absent in Fig. 1a. The "focus-point" struggles to produce m h ∼ 125 GeV and is disfavored by the LUX experiment. It might have an appreciable posterior weight in Ref. [19] because that analysis included a 3 GeV theory error (rather than our 2 GeV theory error) in the m h calculation and omitted direct detection experiments.
Ref. [54] identified stop-coannihilation as a possible dark matter annihilation mechanism.
Because the MultiNest algorithm found few such solutions (although we confirm that such points with reasonable χ 2 exist), the stop-coannihilation regions are fine-tuned so much so that their posterior weight is negligible. We plot stop-coannihilation solutions in Fig. 1b.
On the (A 0 , tan β) plane in Fig. 3, we identify two modes that correspond to the staucoannihilation and A-funnel regions on the (m 0 , m 1/2 ) plane. The stau-coannihilation region prefers tan β 30 and negative A 0 to achieve the required mass degeneracy and stop mixing.  Within the stau-coannihilation region, µ > 0 is preferred to enhance δa µ with light sparticle loops. The A-funnel requires tan β 50 and prefers A 0 1.5 TeV so that the pseudo-scalar Higgs is sufficiently light.
Our posterior on this plane is, by eye, in agreement with ∆χ 2 plots in Ref. [17], but in poor agreement with that in Ref. [19]. In each case, one can identify a stau-coannihilation region at small tan β. We, however, fail to see modes at A 0 ∼ ±8 TeV present in Ref. [19], corresponding to the "1 TeV higgsino region." The higgsino-like neutralino is disfavored by the LUX direct search for dark matter. No direct detection constraint was applied in Ref. [19], though its potential impact was investigated, and found to be substantial in the  region," the posterior weight is enhanced by the many (A 0 , tan β) solutions, whereas, on the (A 0 , tan β) plane in the "1 TeV higgsino region," the posterior weight is the sum of few (m 0 , m 1/2 ) solutions.
The "1 TeV higgsino region" is, however, visible on the (A 0 , tan β) plane with linear priors in Fig. 2b with tan β ∼ 45 and 10 TeV A 0 20 TeV. Large A 0 and tan β are preferred to achieve µ ∼ 1 TeV and to insure that radiative electroweak symmetry breaking with m 2 Hu < 0 is achieved, in spite of such heavy m 0 . Because the stop-coannihilation mechanism requires large a t at the electroweak scale to split the stops so that the lightest is degenerate with the neutralino, it prefers large negative A 0 .
On the (m χ 0 1 , σ SI p ) plane in Fig. 4, which is relevant to direct searches for dark matter, we again identify three modes characterized by their dark matter annihilation mechanism: the stau-coannihilation region at m χ 0 1 ∼ 500 GeV, the A-funnel at m χ 0 1 ∼ 700 GeV and the "1 TeV higgsino region." As anticipated, the scattering cross-section is largest in the "1 TeV  higgsino region," because the higgsino-bino-Higgs vertices are enhanced by the neutralino's composition. Linear priors (Fig. 4b) favor the "1 TeV higgsino region." The LUX limit was, of course, included in our likelihood function; prima facie it is surprising that the "1 TeV higgsino region" lies above that limit. This is because there is significant theoretical uncertainty in the scattering cross-sections. There is an appreciable chance that regions with calculated scattering cross-sections that are above the LUX limit in fact have true scattering cross-sections that are below the LUX limit. Our Bayesian methodology and results reflect this fact.
To conclude, let us summarize the impact of the experimental measurements in our likelihood function in Table I: • If the relic density, Ωh 2 , is to be sufficiently small, the neutralino's mass or composition must be fine-tuned to enhance a particular annihilation mechanism. This selects narrow regions of parameter space. • The null result of the LUX direct search for dark matter disfavors higgsino-like neutralinos.
• The effect of LHC direct searches is somewhat trivial; a low-mass portion of the (m 0 , m 1/2 ) plane is excluded.

IV. RESULTS AND DISCUSSION
For concreteness, we say that a CMSSM point is discoverable if, were nature described by that CMSSM point, it is expected that the SM background hypothesis could be rejected at 5σ. We assume that two simplified channels [58] studied in Ref. [59] describe the hadron collider phenomenology of the CMSSM's favored regions: • If gluinos and squarks are light, we assume a simplified gluino-squark-neutralino channel in which gluinos and squarks are pair produced and decay to a neutralino and a quark or quark pair. Because the neutralinos must be less than approximately 1 TeV at 95% from our posterior, one can expect a discovery potential of mg mq 2.7 TeV (mg mq 3.0 TeV) at the LHC with 300/fb (3000/fb), mg mq 6.6 TeV at the HE-LHC and mg mq 15 TeV at the VLHC with 3000/fb [59].
• If squarks are heavy, we assume a simplified gluino-neutralino channel in which gluinos are pair produced and decay to a neutralino and a quark pair via an off-shell squark.
Because the neutralinos must be less than approximately 1 TeV at 95% from our posterior, one can expect a discovery potential of mg 1.9 TeV (mg 2.2 TeV) at the LHC with 300/fb (3000/fb), mg 5.0 TeV at the HE-LHC and mg 11 TeV at the VLHC with 3000/fb [59].
We neglect electroweak production, which is unlikely to significantly extend the discovery potential at a hadron collider.
The discovery potential of the various experiments is shown on the (m 0 , m 1/2 ) planes in Fig. 1 and Fig. 2 for logarithmic and linear priors, respectively. For logarithmic priors, the LHC with 300/fb could discover all of the CMSSM's stau-coannihilation region and a light fraction of the A-funnel region while the HE-LHC could discover all of the A-funnel. The "1 TeV higgsino region," however, is entirely beyond the reach of the LHC, and partially beyond that of the HE-LHC. For linear priors, large part of the A-funnel region remains uncovered also by the HE-LHC. However, for the VLHC our conclusion is the same for both choices of the prior: the CMSSM can be discovered for all favored dark matter annihilation mechanisms. We stress once more the complementarity between collider searches and direct detection, as seen in Fig. 4. Improving direct detection constraints and/or reducing the associated nuclear uncertainties could entirely exclude the "1 TeV higgsino region." It is evident that our results, which we calculate from Eq. (8)    Our probabilities were, of course, dependent on our assumption that the dark matter is entirely neutralino. We recalculated the probabilities with the minimal assumption that the CMSSM accounted for only the Higgs boson mass, by including only the measurement of the Higgs mass in our likelihood function. We permitted, a priori, tan β 1, which is disfavored by e.g., the dark matter relic density measurement, but reduces the tree-level Higgs mass. In all our results, however, we vetoed points for which Yukawa couplings were non-perturbative below the GUT scale or points that were otherwise unphysical.
The resulting posterior is plotted in Fig. 5 for the (m 0 , m 1/2 ) and for stop-gluino mass planes. The effect of giving up the requirement of obtaining the observed amount of DM is evident: part of the high mass parameter region remains uncovered by all the planned colliders. Whilst the credible regions suggest that the VLHC has an appreciable (85%) probability of discovering the CMSSM, this conclusion is somewhat dependent on our priors.
Soft-breaking masses of greater than ∼ 10 TeV typically result in a Higgs mass that is greater than its measured value, but are also disfavored by our logarithmic priors. With linear priors for m 0 and m 1/2 , that probability is moderate (50%). The results plotted in Fig. 5 demonstrate that DM annihilation processes do constrain SUSY parameters significantly.

V. CONCLUSIONS
Whilst in the near future experiments at the LHC are expected to continue at √ s = 14 TeV, discussions are underway for a high-energy proton-proton collider. Two preliminary ideas are the √ s = 100 TeV VLHC and the √ s = 33 TeV HE-LHC. We calculated the Bayesian posterior density of the CMSSM's parameter space, given experimental data from the LHC, LUX and Planck. Our result was in agreement with similar previous analyses and could be understood with reference to possible dark matter annihilation mechanisms.
With Bayesian statistics, we calculated the probabilities that the LHC, HE-LHC and VLHC discover SUSY in the future, assuming that nature is described by the CMSSM and given the experimental data from the LHC, LUX and Planck (e.g., assuming that dark matter is entirely neutralino).
We found that the LHC with 300/fb at √ s = 14 TeV has a 15-75% probability of discovering SUSY. If that experiment does not discover SUSY, the probability of discovering SUSY with 3000/fb is merely 1-10%. Were SUSY to remain undetected at the LHC, the HE-LHC would have a 35-85% probability of discovering SUSY with 3000/fb. The VLHC, on the other hand, ought to be definitive; it has a 100% probability of discovering SUSY with 3000/fb. All probabilities, summarized in Table III, are conditional on the experimental data obtained thus far and on the assumption that the CMSSM is the true model. We remind the reader that in Bayesian statistics, probability is a numerical measure of belief, and is sensitive to one's prior beliefs, e.g., our priors for the CMSSM in Table II, though we believe our priors were fair. We checked that our conclusions were robust with respect to systematic errors in the expected performance of the collider experiments.
Our results were primarily determined by the CMSSM's predictions for the Higgs mass and the DM abundance, which we assumed to be entirely neutralinos. If, however, DM has a different origin, the CMSSM could be unreachable even at the VLHC. Our conclusions are applicable to only the CMSSM; in relaxed models, one can tune the pseudo-scalar resonance so that heavy neutralinos are annihilated at a sufficient rate.