The CMSSM survives Planck, the LHC, LUX-ZEPLIN, Fermi-LAT, H.E.S.S. and IceCube

We revisit the viability of the CMSSM, searching for regions of parameter space that yield a neutralino dark matter density compatible with Planck measurements, as well as LHC constraints including sparticle searches and the mass of the Higgs boson, recent direct limits on spin-independent and -dependent dark matter scattering from the LUX-ZEPLIN (LZ) experiment, the indirect constraints from Fermi-LAT and H.E.S.S. on dark matter annihilations to photons in dwarf spheroidal galaxies and the Galactic Centre, and the IceCube limits on muons from annihilations to neutrinos in the Sun. For representative values of tanβ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tan \beta $$\end{document} and A0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_0$$\end{document} we map in detail the Planck-compatible strips in CMSSM parameter planes, which exhibit multiple distinctive features for large tanβ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tan \beta $$\end{document}, A0=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_0 = 0$$\end{document} and μ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu > 0$$\end{document}, and identify portions of the strips that survive all the phenomenological constraints. We find that the most powerful constraint is that from mh\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_h$$\end{document}, followed by the LZ limit on spin-independent scattering, whereas sparticle searches at the LHC and indirect dark matter searches are less restrictive. Most of the surviving CMSSM parameter space features a Higgsino-like dark matter particle with a mass ∼1000-1100\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sim 1000{-}1100$$\end{document} GeV, which could best be probed with future direct searches for dark matter scattering.


Introduction
The Constrained Supersymmetric extension of the Standard Model (CMSSM) is a commonlyused template for supersymmetry phenomenology [1][2][3][4][5][6], which incorporates the (over-?)simplified assumption that the soft supersymmetry-breaking parameters are universal at the gauge coupling unification scale ∼ 10 16 GeV.The CMSSM has often been used, for example, to interpret supersymmetry searches at the LHC and both direct and indirect searches for astrophysical dark matter.
The continuing lack of success in accelerator searches for supersymmetric particles [7] as well as searches for massive astrophysical dark matter particles [8][9][10][11] has been exerting everincreasing pressure on supersymmetric models in general and the CMSSM in particular.The low dimensionality of the CMSSM parameter space enforces many links between supersymmetric observables and has relatively few unexplored corners.Under these circumstances, an ever more pressing question is whether the CMSSM survives all the phenomenological constraints.If it does, there is likely to be an ever-decreasing region of CMSSM parameter space where experimental searches should focus, which may be interesting also for more general supersymmetric models.If the CMSSM does not survive, the time has come to move on, either to more general supersymmetric models or to other scenarios for dark matter and complementary possibilities for physics beyond the Standard Model.
Our philosophy in this paper is to regard the CMSSM as the canary in the supersymmetric coalmine, whose survival is indicative of the general state of health of supersymmetric phenomenology.As we shall see, the CMSSM continues to survive the experimental onslaught in a restricted range of parameters that is quite vulnerable to upcoming direct searches for astrophysical dark matter.
The following are the phenomenological constraints that we consider in this paper.The most restrictive is the constraint on the density of dark matter provided by the Planck satellite and other measurements, which has percent-level precision [12].Our calculation of the relic density assumes standard thermal freeze-out.However, it is subject to at least two important caveats.One is that the supersymmetric relic density will in general be changed if the expansion history is modified or if the late decay of a massive particle (which comes to dominate the energy density) adds entropy to the radiation bath after freeze-out [13] and the other is that there may be other sources of dark matter, in which case the supersymmetric relic density may be less than the total dark matter density indicated by Planck and other experiments.Nevertheless, we take the Planck dark matter density constraint at face value, commenting later on the extension of our results to the possibility that the LSP contributes only a fraction of the cold dark matter density.
The second most stringent constraint is provided by the mass of the Higgs boson, which has been measured at the permille level [14].Theoretical calculations of m h within supersymmetric models have estimated accuracies at the percent level.In our analysis we use calculations with the FeynHiggs 2.18.1 code [15], taking into account their estimated uncertainties.As we shall see, the m h constraint is quite complementary to the Planck constraint.
Another set of constraints is provided by LHC searches for supersymmetric particles, which yield (so far) only lower limits on sparticle masses [7] and hence the CMSSM supersymmetry-breaking parameters m 0 (the supposedly universal scalar mass at the unification scale) and m 1/2 (the supposedly universal gaugino mass).In principle, these limits depend somewhat on the values of tan β (the ratio of supersymmetric Higgs vev's) and A 0 (the supposedly universal soft trilinear supersymmetry-breaking parameter).However, we find that the LHC constraints are not relevant in the regions of CMSSM parameter space favoured by the measurement of m h , implying that a detailed implementation is not necessary.We stress that our motivation for studying the CMSSM is the simplicity embodied in the assumptions on scalar and gaugino mass universality.Of course there are many more complicated extensions of this paradigm, such as models in which the Higgs masses differ from the scalar masses at the unification scale, which are commonly known as non-universal Higgs mass models (NUHM) [16].Additional parameters will clearly relax the impact on the model imposed by experimental constraints.To the extent that the CMSSM survives them, other models will survive them more easily.As we will see, the CMSSM survives these constraints with a supersymmetry breaking mass scale that is O (10) TeV [17,18], larger than the scale originally associated with supersymmetry when it was proposed as a solution to the hierarchy problem [19].However, supersymmetry at any scale below the GUT or Planck scale alleviates the hierarchy problem to some extent.
We implement several astrophysical constraints on dark matter interactions.The first are the direct upper limits on spin-independent and -dependent astrophysical dark matter scattering set recently by the LUX-ZEPLIN (LZ) experiment [11], which are more stringent than those set previously by the LUX [9], PANDA-X [10] and XENON [8] experiments.As we show later, the spin-independent scattering constraint excludes parts of the dark matter strips that are compatible with m h , the other LHC and indirect dark matter constraints.
We also consider several indirect constraints on astrophysical dark matter annihilations [20].The first is provided by upper limits from Fermi-LAT on γ fluxes from dwarf spheroidal satellite galaxies [21][22][23][24][25][26].As discussed in [27] and in the text, we combine the γ flux limits from all the available dwarf data sets.As in the case of the LHC constraints, we find that the combined γ limit does not impact the region of parameter space favoured by m h and the dark matter density.The same is true for the upper limits from the γ flux from the Galactic Centre [28][29][30].The final indirect limit we consider is that on dark matter annihilation in the Sun [31] set by the IceCube search for energetic neutrinos from the Sun [32][33][34].
In order to set the stage for the applications of the different constraints, we first discuss in detail in Section 2 the dark matter strips for different values of tan β, A 0 = 0 and µ > 0 and the underlying mechanisms that play dominant roles in bringing the cosmological dark matter density into the range allowed by Planck and other measurements.These include the focus-point and well-tempered neutralino mechanisms, coannihilation between the LSP and other sparticles such as heavier neutralinos, charginos and stop squarks, and annihilations via s-channel heavy Higgs bosons (H/A).As we discuss in Section 2, these different mechanisms may combine constructively in non-trivial ways.The pattern of dark matter strips is richest when tan β ≥ 40, A 0 = 0 and µ > 0, simplifying significantly to the focus-point strip for smaller tan β when A 0 = 0 and µ > 0, to the stop coannihilation strip when A 0 = 3 m 0 and µ > 0, and again to the focus-point strip when µ < 0.
Section 3 of this paper discuses the LHC constraints on the CMSSM when A 0 = 0 and µ > 0, with particular focus on the m h constraint, which is generally much more important than the direct searches for supersymmetric particles.We then discuss the astrophysical constraints on dark matter interactions when A 0 = 0 and µ > 0 in Section 4, and combine these constraints in Section 5.As we discuss in more detail below, for most of the values of tan β that we study when A 0 = 0 and µ > 0, there is a small region of the focus-point strip [35] that is allowed by all the above constraints, where the LSP is mainly Higgsinolike with a mass ∼ 1000 − 1100 GeV.However, there are some cases in which a lower neutralino mass 450 GeV may be allowed, and other cases with an allowed strip at masses 1200 GeV.In all these cases the spin-independent dark matter scattering cross section may be within reach of upcoming experiments.On the other hand, when A 0 = 3m 0 only LSP masses 6 TeV are allowed, for which the dark matter scattering cross section is well below the neutrino "floor" [36].Section 6 discusses the case when A 0 = 3 m 0 and µ > 0, when there is a single dark matter strip where stop coannihilation is dominant, the uncertainty in the m h calculation is greater, and dark matter interactions are well below the experimental limits.We find that when tan β 20 the portion of the stop coannihilation strip that is allowed by m h within the calculational uncertainty extends from high masses down to the range of stop masses excluded by LHC searches.Finally, Section 7 discusses the relatively simple case when A 0 = 0 and µ < 0, and Section 8 summarizes our conclusions.

Dark Matter Strips in the CMSSM Parameter Space
We first summarize the essential features of the CMSSM that are relevant for our study.As usual, the assumption of universality for each of the various classes of soft supersymmetrybreaking parameters at the scale ∼ 10 16 GeV where the SM couplings unify leaves us with the following 4 continuous parameters: the scalar mass, m 0 , the gaugino mass, m 1/2 , the trilinear soft supersymmetry-breaking parameter, A 0 , and the ratio of supersymmetric Higgs vev's, tan β.In addition there is the unknown sign of the Higgsino mixing parameter, µ, which we take to be positive in this section.Historically, this choice has been motivated by the sign of the difference between the SM calculation of g µ − 2 [37] and its experimental value [38][39][40].However, this choice can now be questioned on two grounds.One is that the supersymmetric contribution to g µ − 2 is negligible for the relatively large supersymmetry-breaking scales found in our CMSSM analysis [41,42], and the other is that recent lattice calculations [43] cast doubt on the magnitude of the significance between the SM calculation of g µ − 2 and its experimental value.Accordingly, we also comment on the status of the CMSSM for negative µ in Section 7.
The high precision of the determination of the cold dark matter density by Planck and other experiments [12] constrains the parameters of any supersymmetric model to thin hypersurfaces in parameter space, which appear as narrow strips in any planar slice through parameter space.Along these strips, a single mechanism usually controls the dynamics of freeze-out, and obtaining the appropriate density generally requires some specific mixing or relationship between the masses of the sparticles controlling this mechanism.When the  LSP mass is relatively large, as suggested by the non-detection of sparticles at the LHC [7] and value of m h [14], sometimes more than one mechanism plays a role.These relationships frequently take the form of a near-degeneracy between the masses of the LSP and the next-to-lightest supersymmetric particle (NLSP), so as to suppress the LSP density via coannihilation [5,[44][45][46][47][48], a direct-channel resonance with mass ∼ 2m LSP [1,49], or some other specific mass spectrum as in the focus-point region [35].
In this section, we have imposed A 0 = 0. Generally, one might expect that the A-terms are comparable in magnitude to the other soft supersymmetry-breaking parameters, m 1/2 and m 0 , at the GUT scale 1 .As we see below, large values of A 0 tend to increase the calculated Higgs mass, and thus would require a somewhat less massive supersymmetry spectrum.While one can motivate A 0 from, e.g., anomaly mediation [50], our primary motivation lies in studying the CMSSM focus-point region, which contains much of the phenomenologically viable parameter space but disappears when A 0 m 0 .
The principal dark matter strips in the CMSSM are those where the focus-point mechanism [35] is operative, or where stop [5,46,48] and/or neutralino/chargino coannihilation [47,48] is dominant, or where resonant exchanges of the neutral heavy Higgs bosons H/A in the s-channel [1,49] suppress the relic LSP density.In some parameter planes there are also important regions of the dark matter strips where a single mechanism is insufficient to lower the relic density to its cosmological value, but this is made possible by combinations of these mechanisms that are in play simultaneously.
The panels of Fig. 1 display the results of performing a Markov chain Monte Carlo (MCMC) scan of m 1/2 , m 0 parameter space with A 0 = 0 and µ > 0 for the indicated fixed values of tan β.The MCMC scan determines the coordinates (m 1/2 , m 0 ) for which Ω χ h 2 = 0.12 ± 0.0036, which is the 3σ range as determined by Planck [12].The MCMC code we run employs mainly a Metropolis-Hastings algorithm [51].The basic criterion we use to run the MCMC code is to find points in the parameter space that are compatible with the Planck 3σ range for Ω χ h 2 .This is an efficient way to delineate the dark matter strips in the parameter space.We stress that the MCMC method we employ does carry any statistical weight, but is merely a scanning code.Some of the gaps seen in the strips are due to incompleteness of the scanning process.
Information on the mechanism (or mechanisms) responsible for sufficient annihilation prior to freeze-out is coded by the color of the point.Black points lie along the focus-point strip where µ ∼ 1 TeV and the LSP is predominantly a Higgsino of mass 1.1 TeV.At slightly larger values of m 0 for each point on this strip, µ 2 is driven to negative values and radiative electroweak symmetry breaking is no longer possible.Below the strip, the LSP is predominantly a bino, with some exceptions noted below.In some cases, at values of m 0 below the focus-point region the bino LSP acquires non-negligible Higgsino components, in which case the neutralino becomes 'well-tempered' [52].These points are coloured green.
The purple points at lower m 0 and large tan β lie along the funnel where the LSP mass is close to half the heavy Higgs masses and the dominant dark matter mechanism is rapid LSP annihilation via these s-channel resonances.However, as m 1/2 increases, this s-channel annihilation is suppressed by the heavy Higgs widths and the the annihilation cross section is insufficient to reduce the relic density to the required value, which is why the purple points terminate at an endpoint.At larger m 0 , in addition to s-channel Higgs pole contributions, the LSP becomes well-tempered and the combination of these contributions leads to a low relic density.These points are colored blue.Coannihilations of the bino-like state, the two Higgsinos and chargino also play a role in determining the final relic density.In addition, as we will see, there is level crossing between the neutralino states in some areas of parameter space.Here the neutralino may be well-tempered and coannihilations may also be important.These points will be colored green.We describe in the following how each of these possibilities arises for different values of tan β.
The top left panel of Fig. 1 shows the (m 1/2 , m 0 ) plane for tan β = 5, A 0 = 0 and µ > 0. It features a single dark matter strip where the focus-point mechanism is operative, which extends beyond the range of m 1/2 ≤ 10 TeV that we display.The (m 1/2 , m 0 ) plane for tan β 20, A 0 = 0 and µ > 0 is similar, as seen in the top right panel of Fig. 1.This one is truncated at m 1/2 = 8 TeV, because the extensions of this strip to larger m 1/2 is not compatible with the LHC measurement of m h , as we see later.
In the tan β = 40 case as seen in the middle left panel of Fig. 1, in addition to the black focus point strip, a closely-spaced pair of green strips appears when m 1/2 > 7 TeV.Here, the neutralino (mostly bino) is well-tempered and the non-negligible Higgsino components in the LSP lead to enhanced annihilations that bring the relic density into the allowed range.On either side (higher and lower m 0 ) of the pair of strips, the relic density is too high, and between the strips the relic density is suppressed.However, as we will see these points are excluded by both m h and the upper limit on spin-independent dark matter scattering.
As aids to understanding the origins of the strips, we plot slices across the (m 1/2 , m 0 ) plane showing the relic density, masses, and neutralino composition as functions of m 0 for fixed values of m 1/2 .As a first example for tan β = 40, A 0 = 0 and µ > 0, we fix m 1/2 = 5 TeV in the left panels of Fig. 2 and in m 1/2 = 9 TeV in the right panels.The top panels illustrate how the relic density varies with m 0 .We see that along both slices the relic density decreases as the boundary of electroweak symmetry breaking is approached when m 0 11 (17) TeV, corresponding to the focus-point strip in the middle left panel of Fig. 1.The dark matter density is generally larger than allowed by Planck and other measurements at lower m 0 .However, there are dips in the relic density at m 0 ∼ 10 ( 14) TeV for m 1/2 = 5.0 (9.0) TeV, including a pair of m 0 values where the Planck relic density is found for m 1/2 = 9 TeV.These values lie on the pair of green lines seen in the middle left panel of Fig. 1.
The middle panels of Fig. 2 display the masses of the lightest neutralino, m χ 1 (black line), the next two neutralino masses, m χ 2 (purple line) and m χ 3 (blue line) and half the mass of the pseudoscalar Higgs boson, m A /2 (red line).The three mass eigenstates are either mostly pure or mixed states of the bino and two Higgsinos, as the wino is always heavy and does not enter in the discussion of the parameter ranges discussed here.The bino-like state (whether the LSP or not) has a relatively constant mass, while the two Higgsino masses decrease with increasing m 0 as µ decreases over the displayed range of m 0 , as a result of the Higgs potential minimization conditions.Coannihilation is the most important dark matter mechanism when two or more of the neutralinos are close in mass.Comparing the middle and top panels, one sees that the dips in the relic density coincide with the degeneracy in the neutralino masses leading to coannihilation, whereas s-channel annihilation via heavy Higgs poles is less important in this case because m χ is quite different from m A /2.The bottom panels of Fig. 2 show that the dips in the relic density at relatively low m 0 are correlated with crossovers in the neutralino mass eigenstates and corresponding variations in the LSP composition, with the LSP acquiring a significant Higgsino component near the crossovers, leading to a well-tempered neutralino.Around this cross-over the lightest eigenstate transitions from a bino-like state with Higgsino components Z 13 1/2 and Z 14 −1/2 to a Higgsino-like state with Z 33 1/2 and Z 34 −1/2 (with Z 11 1/ √ 2 in both cases).As one can see, these states are maximally well-tempered.We display the bino component (Z 11 ) and the two Higgsino components (Z 13 and Z 14 ) of the bino-like state, which is not necessarily the LSP.This state is the LSP for m 0 below the dip.When the LSP eigenstate flips, the Higgsino components of the bino-like state are now Z 13 −1/2 and Z 14 1/2 still with Z 11 1/ √ 2. At larger m 0 , this state is again mostly a pure bino, though it is no longer the LSP.As seen more clearly in the middle right panel of Fig. 2, the crossover is reflected in a bump in the LSP mass where coannihilation with χ 2 and the lighter chargino is enhanced, suppressing the relic density around the local minimum seen in the top right panel of Fig. 2. 2The middle right panel of Fig. 1 shows the (m 1/2 , m 0 ) plane for tan β = 50, where the focus-point strip (black points) is again supplemented by additional features at lower m 0 that resemble the second feature that appeared for tan β = 40, but extend to lower m 1/2 and exhibit a richer structure.As in the case of tan β = 40, there are a pair of strips where the neutralino is well-tempered.These are represented by the green points which extend down to m 1/2 ≈ 4 TeV.At larger m 1/2 , an additional pair of strips appear where the LSP is somewhat less well-tempered, but sits near the heavy Higgs pole.For these points, it is the combination of the s-channel annihilations with the bino-Higgsino mixing contributions that lead to a cross section that is large enough to obtain the correct relic density.
The interplay of different dark matter mechanisms for tan β = 50 is illustrated in Fig. 3, where m 0 varies along slices across the (m 1/2 , m 0 ) plane at fixed m 1/2 = 5 TeV (left panels) and 7.8 TeV (right panels).We see in the top left panel that the Planck value of the relic density is attained for three values of m 03 .These correspond to the focus-point strip at m 0 ≈ 11.3 TeV and a pair of values of m 0 around 10 TeV.The latter correspond to the green strips seen in Fig. 1 where the well-tempered nature of the LSP is sufficient for reducing the relic density into the cosmological range.In contrast, at higher m 1/2 , as seen in the right panels, there are five values of m 0 where the desired relic density is attained.Once again the highest value at m 0 ≈ 15.4 TeV corresponds to the focus-point strip.The cluster of points around m 0 13 TeV contains 4 values of m 0 where the cosmological value Ωh 2 = 0.12 is crossed.These 4 values arise from the interplay of rapid s-channel annihilation via the A/H resonances (which is most important where the red and black lines cross in the middle panels, which does not occur for tan β = 40) and LSP coannihilation with the Higgsinos and lighter chargino (which is most important where the blue and black lines approach each other in the lower panels).Within this cluster of points, those with lower m 0 require both s-channel annihilations and a well-tempered neutralino, whereas at higher m 0 , the neutralino level crossing is also sufficient to suppress the relic density.These correspond to the blue and green strips seen in Fig. 1.At higher m 0 , the Higgsino becomes the LSP, and the annihilation cross section enhancements disappear until m 0 is sufficiently large that m χ ≈ µ ≈ 1.1 TeV, and the relic density falls back to 0.12 due to neutralino/chargino coannihilations among the Higgsinos.Finally, the boundary of the electroweak symmetry breaking region is reached when m 0 ≈ 15.5 TeV.The (m 1/2 , m 0 ) plane for tan β = 55 shown in the bottom left panel of Fig. 1 exhibits additional features.We note that the focus-point strip terminates because the renormalizationgroup running of the top Yukawa coupling diverges at larger m 1/2 .The purple strip corresponds to a funnel region starting at low m 0 where s-channel annihilation through the heavy Higgs bosons dominates.This terminates at about m 1/2 ∼ 1.6 TeV when, due to the increasing heavy-Higgs width, the annihilation cross section becomes too small to reduce the relic density into the range allowed by Planck and other measurements.In the region above and to the right of this funnel, the relic density is too high.For fixed m 1/2 , e.g., m 1/2 = 2 TeV, as m 0 is increased, the heavy Higgs masses increase and 2m χ < m A , until at still higher m 0 , µ begins to decrease.This has the effect of decreasing m A and moving the LSP back on the Higgs pole, lowering the Higgsino/chargino masses and increasing the Higgsino component of the LSP.This results in the pair of blue strips that form a second funnel region, where the annihilation cross section is augmented by an increasing neutralino coannihilation contribution sufficiently to lower the relic density to an acceptable value.Further increasing m 0 moves the LSP off the pole and the relic density is again too big until m 0 is sufficiently large that µ has dropped enough for the LSP to become mostly Higgsino and the focus-point strip is reached.At higher m 1/2 , e.g., m 1/2 = 2.8 TeV, the funnel is not able to bring the relic density sufficiently low, even though enhanced by coannihilation.However, at larger m 0 ∼ 7 TeV a large Higgsino component in the LSP increases the cross section producing the green pair of strips below the black focus-point strip.
Fig. 4 shows slices through the (m 1/2 , m 0 ) plane for tan β = 55 with m 1/2 = 2 TeV (left panels) and 2.8 TeV (right panels).As we see in the top panels, for both values of m 1/2 the relic density exceeds the Planck range at low m 0 , namely for m 0 < 5000 and 7000 GeV, respectively.For m 1/2 = 2000 GeV, when m 0 < 1.22 TeV, the lighter stau is the LSP.At higher m 0 , the bino becomes the LSP and its mass is close to half the heavy Higgs mass, but it is past the endpoint of the funnel region and, although the relic density is low, it is still above the range allowed by cosmology 4 .At slightly higher m 0 , m A/H drops, LSP annihilation via the heavy Higgs poles is suppressed and the relic density is high.However, as one sees in the middle left panel of Fig. 4, m H/A rises, LSP annihilation via the heavy Higgs poles becomes possible, and the relic density decreases leading to the dip seen in the upper left panel at m 0 ∼ 2 TeV.However, it does not fall sufficiently before the LSP mass moves away from half the pole mass, and begins to rise for m 0 2 TeV.At m 0 4 TeV, µ and the Higgsino masses begin to drop.The Higgs pseudoscalar mass also drops and heavy-Higgs pole annihilation becomes possible again, though this time with an increased Higgsino component and a smaller mass splitting with the chargino.Then, at m 0 ≈ 5 TeV the relic density is sufficiently low to become compatible with Planck and other data.The "valley" around 5200 GeV is characteristic of s-channel annihilation via the A/H bosons and explains the funnel-like feature of the blue strips in the bottom left panel of Fig. 1.As seen there, this strip terminates when m 1/2 ∼ 2400 GeV.Above m 0 5.2 TeV, the LSP moves off the pole again and the relic density begins to rise again until m 0 ≈ 5.7 TeV, where the Higgsino becomes the LSP and the focus-point strip is encountered.
Similar behaviour is seen in the right panels of Fig. 4 for m 1/2 = 2.8 TeV.While the relic density shows a minimum around m 0 ≈ 5.5 TeV, we see that the funnel alone is not sufficient in obtain the Planck value for relic density.As the LSP moves off the pole at higher m 0 , the relic density begins to rise until the Higgsino component of the LSP increases sufficiently to raise the cross section and hence reduce the relic density.The relic density hits a local minimum and then rises as the LSP becomes a pure Higgsino, subsequently falling through   the Planck value when the Higgsino mass falls to ∼1 TeV.All these features are present also in the bottom right panel of Fig. 1, which displays the corresponding (m 1/2 , m 0 ) plane for tan β = 56.As for tan β = 55, the focus-point strip terminates because the renormalization-group running of the top Yukawa coupling diverges at larger m 1/2 .Fig. 5 shows slices across the (m 1/2 , m 0 ) plane for tan β = 56 for the fixed values of m 1/2 = 950 (upper panels) and 2500 GeV (lower panels).In the first case we see clearly that the Planck value of the relic density is attained at four different values of m 0 , corre-sponding to the funnel, two strips corresponding to the coannihilation/funnel combination and finally the focus-point region.In the left panel we see again that the LSP is near the heavy-Higgs pole at both low and high m 0 .On the other hand, for m 1/2 = 2500 GeV (right panels of Fig. 5) we see that the Planck density is attained only for two very similar values of m 0 6700 GeV that are in well-tempered region and at higher m 0 ∼ 6900 GeV when the focus point strip is reached.

Sparticle Searches
The headline sparticle searches at the LHC are those for squarks and gluinos, both of which are sensitive to masses 2 TeV in simplified models, e.g., assuming that some specific decay mode has a branching ratio of 100% and the LSP mass is negligible.The squark limit would correspond to m 0 2 TeV for small m 1/2 , and the gluino limit would correspond to m 1/2 800 GeV.These constraints do not impinge on the A 0 = 0 regions of interest in Fig. 1.
Indeed the mass spectrum we find is quite heavy.In Fig. 6, we show the lighter stop mass as a function of m 1/2 along the dark matter strips.As one can see, the stop masses are similar to (though slightly smaller than) m 0 .Similarly in Fig. 7, we show the gluino mass along the strips.The gluino mass is typically 2 − 3m 1/2 .On the other hand, as we discuss in Section 6, for A 0 = 3 m 0 searches for light stops with a compressed spectrum exclude masses just below the region of the stop coannihilation strip allowed by the FeynHiggs 2.18.1 calculation of m h at the 1-σ level.The stop masses in this case are nearly degenerate with the LSP mass.The gluino mass is again roughly 2 − 3m 1/2 .

Higgs Mass Constraint
The world average of the Higgs boson mass is 125.25 ± 0.17 GeV [53].In order to calculate m h in the CMSSM we use the FeynHiggs 2.18.1 code [15], which returns an uncertainty estimate ∆m h as well as the best estimate of m h for each choice of model parameters.The uncertainty estimate is quite variable being, e.g., significantly smaller for A 0 = 0 than for A 0 = 3 m 0 .Conservatively, for A 0 = 0 we assume that ∆m h = 1.5 GeV, which corresponds to somewhat more than twice the typical numerical estimate of ∆m h , commenting in passing on the prospective implications for ∆m h ∼ 0.5 GeV.On the other hand, the typical FeynHiggs 2.18.1 uncertainty estimate for A 0 = 3 m 0 is much larger, ∆m h 3 GeV (and tan β dependent), which does not exclude any portions of the stop coannihilation strips for The panels in Fig. 8 illustrate how the m h constraint restricts the allowed ranges of the dark matter strips for tan β = 5, 20, 40, 50, 55 and 56 when A 0 = 0 and µ > 0. For tan β = 5 (top left panel), m 1/2 7 TeV is allowed for ∆m h = 1.5 GeV, but only a limited range of m 1/2 9 TeV if we assume ∆m h 0.5 GeV.(Note that the focus-point strip continues to larger values of m 1/2 than those displayed.The apparent endpoint is due to the finite range of our MCMC scan.)For tan β = 20 (top right panel), the range 2.4 TeV m 1/2 6.6 TeV is allowed for ∆m h = 1.5 GeV, whereas only the range 3.6 TeV m 1/2 5 TeV would be favoured for ∆m h 0.5 GeV.Similar ranges of m 1/2 are allowed (favoured) for tan β = 40, as seen in the middle left panel of Fig. 8.We see now that the well-tempered (green) strips at m 1/2 > 8 TeV are excluded as the calculated Higgs mass is too large even when conservative allowance is made for the uncertainty in the calculation.When tan β = 50 (middle right panel), a portion of the well-tempered strip is allowed, extending to m 1/2 7 TeV for ∆m h = 1.5 GeV.However, the additional strips at large m 1/2 where s-channel annihilations contribute (blue points) lead to values of m h that are   too large.When tan β = 55 (bottom left panel of Fig. 8), only a portion of the focus-point strip with m 1/2 2.5 TeV is allowed for ∆m h = 1.5 GeV, increasing to m 1/2 3 TeV for ∆m h 0.5 GeV.We see also that parts of the coannihilation/well-tempered (green) strips for tan β = 55 also fall within the experimental range for the Higgs mass.In contrast, in the funnel/coannihilation/well-tempered strips (blue) the Higgs mass is always too small, as it is in the funnel region (purple).Finally, we see that for tan β = 56 (bottom right panel) the correct value of m h is only attained when m 1/2 2.5 TeV at the tip of the black and green strips. 5 Constraints on Dark Matter Interactions

Spin-Independent Dark Matter Scattering
Many experiments have used massive nuclear targets to probe coherent spin-independent LSP-nucleon scattering.The most recent such experiment is LUX-ZEPLIN (LZ) [11], which uses a xenon target to establish a stronger upper limit on the spin-independent LSP-nucleon scattering cross section than the previous XENON1T [8], PandaX-4T [10], LUX [9] and DEAP-3600 [54] experiments.
The computation of the spin-independent cross section that we use was described in detail in [55].As is well known, the spin-independent cross section is very sensitive to the quark matrix elements, N |qq|N .These can be expressed in terms of σ terms, notably the π − N σ term Σ πN and σ s (see [55] for more detail).Here we have used Σ πN = 46 ± 11 MeV and σ s = 35 ± 16 MeV, respectively.The corresponding typical uncertainties in the elastic cross sections are of order 10-15 %.
We see in the top and middle panels of Fig. 9 that the spin-independent LSP-nucleon scattering constraint allows limited regions of the dark matter focus-point strips where m χ ∼ 1.0−1.1 TeV for tan β = 5, 20, 40 and 50.Along the near vertical strips shown here, although m 1/2 (and m 0 ) changes, the LSP mass is nearly constant.For tan β = 40, we see that the direct detection experiments are (like the Higgs mass constraint) able to exclude the welltempered LSP points (green).We also see in the middle right panel of Fig. 9 that only a small fraction of the well-tempered and funnel points (green and blue) with larger values of m χ are allowed.We see in the bottom panels of Fig. 9 for tan β = 55 and 56 that points along the focus-point strips and in the well-tempered regions (green) are excluded, and only a small fraction of well-tempered/funnel points (blue) are still allowed.These points have m χ 1 TeV and, as we have discussed previously, correspond to Higgs masses that are unacceptably small.Some ranges of the funnel strips (purple) are also still allowed by the spin-independent dark matter scattering constraint, and have also shifted to lower values of m χ 800 GeV.However, these points also correspond to unacceptably low Higgs masses.
These figures show the strong impact of the spin-independent scattering limits when A 0 and µ > 0 and illustrate their potential for even more stringent constraints in the future    as their sensitivities reach down towards the neutrino 'floor' [36].We find spin-independent scattering cross sections that lie above the 'floor' for all values of tan β when A 0 = 0, for both signs of µ.

Spin-Dependent Dark Matter Scattering
Spin-dependent LSP-nucleon scattering on a nucleus is not coherent, but is proportional to the nuclear spin, which is mainly carried by an odd nucleon.The LZ experiment also set new upper limits on spin-dependent LSP scattering on both the neutron and the proton.The LZ limit on spin-dependent LSP-neutron scattering exploits the fact that two common xenon isotopes have non-zero nuclear spins that are carried mainly by unpaired neutrons: 129 Xe (abundance 26.4%, spin 1/2) and 131 Xe (21.2% abundance, spin 3/2).The LZ limit on spindependent LSP-proton scattering exploits configuration mixing between proton and neutron spin states in these isotopes, but has larger uncertainties and is significantly weaker than their limit on spin-dependent LSP-neutron scattering. 6The spin-dependent LSP-neutron and -proton scattering cross sections are calculated to be quite similar, so we use here only the LZ limit on spin-dependent LSP-neutron scattering, which provides the most stringent constraint on spin-dependent LSP-nucleon scattering.
We see in Fig. 10 that all the dark matter strips for tan β ≤ 50 are compatible with the spin-dependent scattering constraint from the LZ experiment.However, there are regions of the funnel strips (purple) for tan β = 55 and 56 with m χ 100 GeV that are excluded by this constraint, as seen in the bottom panels of Fig. 10.We note in addition that there are some well-tempered/funnel points (blue) with LSP masses ∼ 200 GeV that are marginally excluded by the LZ spin-dependent scattering constraint.

Limits on γ Fluxes from Dwarf Spheroidal Satellite Galaxies
We first examine dSphs as possible sources of γ rays from DM annihilations in the framework of the CMSSM, before revisiting the possible GC constraint on DM annihilations.The annihilation of neutralinos can yield monochromatic photons via the one-loop processes χχ → γγ and χχ → γZ, and also a continuous spectrum of photons via the decays of π 0 s and other hadrons produced by the fragmentation and hadronization of primary annihilation products.The integrated γ-ray signal flux, φ s (typically measured in photons/cm 2 /s), expected from the annihilations of DM particles with a density distribution ρ DM (r) is whose components we analyze in the following.
The first term of ( 1), Φ P P , depends on the particle physics properties.In particular, this term is dependent on the thermal annihilation cross section συ , the mass of the dark matter particle, m DM , and the differential γ ray yield per annihilation, dN γ /dE γ , integrated over the experimental energy range.The differential yield dN γ /dE γ is a sum over specific final states where B f is the branching fraction into a given final state.Branching fractions for χ pair annihilation in the CMSSM were studied in [27].As described in detail there, we have used PYTHIA tool [66] in order to simulate the gamma fluxes produced in χ pair annihilations.
The second term in ( 1), the J factor, is the integral along the light of sight (l.o.s.) [67] through the DM density profile, ρ DM , which is integrated over a solid angle, ∆Ω.The DM density profile may be given by the general expression where R is the characteristic length scale and a, b, c are parameters.In our analysis we assume that the DM distribution is a cuspy Navarro-Frank-White (NFW) [68] profile with a = c = 1.0 and β = 3.0, i.e.
We note that both the NFW and Burkert profiles [69] were studied in [21], whereas only the NFW profile was used in a subsequent Fermi-LAT study [23], where it was found that the J factors for dSphs were relatively insensitive to the choice of DM distribution.
In our analysis, we have used the information on 25 Milky Way dSphs given in [23], including the python code provided by Fermi Tools for calculating the binned likelihood with energy dispersion.In order to use this code, we used the corresponding spacecraft data files for the dSphs and a background model, which are also provided by Fermi Tools.We used the energy range from 500 MeV to 500 GeV, divided into 30 logarithmic bins, and verified that our results are insensitive to various choices of the energy range and number of bins within the limits provided by Fermi-LAT.The times of the events we used range from 01/01/2009 to 01/01/2022.The window radius for searching these data is 15 degrees and the galactic coordinates for each dSph are taken from [23].The selection of data sets is in accordance with the suggestion by the Fermi-LAT Collaboration.
After completing the analysis using Fermi Tools, we have evaluated the signals in the framework of the CMSSM, following the approach used previously in a study of the prospective γ signal from the GC [27].We calculate the χ 2 likelihood function as follows: where n bins = 30 is the number of logarithmic bins in the energy range analyzed, d i is the number of data counts per bin, b i is the expected background provided by Fermi Tools [23], and s i is the signal due to DM annihilations in each bin.The quantity σ i is defined from where σ ea is the systematic uncertainty in the effective area [27].We evaluate the χ 2 contribution for each dSph separately.Fig. 11 displays our evaluations of the contribution to the global likelihood function, ∆χ 2 , from the aggregated Fermi-LAT data on γ-rays from dSphs for the same values of tan β as in previous Figures.We see that ∆χ 2 0.1 for tan β ≤ 40, and hence is negligible in these cases.However, the situation is quite different for tan β ≥ 50, as seen in the middle right and bottom panels of Fig. 11.Values of ∆χ 2 ∼ 10 are reached in the well-tempered/funnel regions (blue) where m χ 4 TeV for tan β = 50, 1.1 TeV for tan β = 55 and m χ 0.9 TeV for tan β = 56, excluding points in these regions, where the LSP annihilates mainly into bb final states through direct-channel H/A resonances.However, these points are already excluded by the calculation of m h , as seen in Fig. 8.We note also that when tan β = 55 and 56, ∆χ 2 reaches ∼ 2 for funnel points with m χ 650 GeV.This is significant though not sufficient to exclude these points, which are in any case excluded by the constraint on m h .

Limit on γ Flux from the Galactic Centre
The production of γ rays near the GC was studied as a possible signature of the CMSSM in [27], whose analysis we follow here.Modelling the DM density near the GC entails significant uncertainties.The NFW model is frequently taken as a default, but other possibilities have been considered for comparison.For example, the H.E.S.S. Collaboration considered recently [28] an Einasto profile that would strengthen the prospective flux limit on annihilations of an LSP weighing 1 TeV by a factor > 2, and [70] considered a FIRE simulation that would strengthen the limit by a factor > 4. In a spirit of conservatism, here we use an NFW profile for setting constraints from data on the γ flux from near the GC.
Assuming the NFW profile, the recent H.E.S.S. analysis [28] finds a 95% CL upper limit on the cross section for LSP-LSP annihilation to W + W − that is stronger than that derived from the Fermi-LAT data on dSphs for m LSP 0.4 TeV, which is the mass range of most interest for our analysis.However, in the CMSSM the branching fraction for W + W − final states is < 100%, so other annihilation modes must be taken into account, which we do when comparing the sensitivities of the Fermi-LAT dSph and H.E.S.S. GC data in our analysis.Fig. 12 displays the contributions, ∆χ 2 , to the global likelihood function provided by H.E.S.S. searches for γ-rays from near the GC in the framework of the CMSSM for A 0 = 0, µ > 0 and our standard choices of tan β.The general features are quite similar the those of the corresponding plots for the Fermi-LAT dSph analysis shown in Fig. 11.The values of ∆χ 2 provided by the H.E.S.S. data are larger than for the Fermi-LAT dSph data for tan β ≤ 40, though still small.There are non-negligible peaks in ∆χ 2 for tan β ≥ 50, which are again larger than those provided by the Fermi-LAT dSph data, but the two sets of data yield similar constraints away from the peaks.Once again, the points excluded by the H.E.S.S. GC γ-ray constraint also correspond to values of m h that are too small, as seen in Fig. 8. 7

Limit on Energetic Neutrinos from the Sun
The principle of the search for energetic neutrinos from the Sun [31] is that an LSP may scatter during passage through the Sun, losing energy and becoming gravitational bound in an elliptical orbit with perihelion smaller than the solar radius.The LSP will in general scatter again during subsequent passages through the Sun, losing more energy each time and eventually settling into a thermal distribution inside the Sun.This distribution eventually equilibrates with a density that balances the LSP capture rate, C, with the loss of LSPs via annihilation, 2Γ A where Γ A is the annihilation rate.The annihilation rate in equilibrium is then determined by the capture rate: We note that this relation is not universal, as equilibrium may not be reached if C is too small.However, this is not an issue for the CMSSM scenarios we consider here, for which Eq. ( 7) is a good approximation [72].
Since nuclear matter in the Sun is largely composed of individual protons, the capture rate receives an important contribution from spin-dependent LSP-proton scattering, and limits on energetic solar neutrinos are occasionally interpreted as upper limits on spin-dependent LSP-proton scattering.However, the Sun also contains a significant fraction by mass of 4 He, as well as trace amounts of heavier nuclei, so that spin-independent LSP-nucleon scattering is also potentially important.Both spin-independent and spin-dependent scattering are included in our calculation of the capture rate.
The calculation of the fluxes of the different neutrino species is straightforward, and we follow the analysis in [72].The experimental observable is the flux of muons, which may be produced by either muon or tau neutrinos reaching Earth.We use the IceCube [32][33][34] upper limit on the flux of energetic muons from the Sun to constrain CMSSM parameters, taking into account the branching fractions for annihilations into different final states, e.g., W + W − , τ + τ − , ... that are predicted at each point in the parameter space.
For the calculation of the corresponding ∆χ 2 we are employing the nulike software [73], following the likelihood estimation described in [32,73].In addition, for the calculation 7 The projected increase in sensitivity offered by CTA [71] would probe a larger range of funnel points.  of the likelihood we are taking into account the 79-string IceCube data, that is known to provide improved limits on WIMP dark matter searches [74].As the basis for the ∆χ 2 we are plotting, we are using the p-value output from the nulike code.
The results of our analysis of the IceCube muon flux limits for tan β = 5,20,40,50,55 and 56 are shown in Fig. 13.We see that for m χ 900 GeV and tan β ≤ 50 the IceCube constraints contribute 2 ∆χ 2 1, not negligible, but insufficient to exclude any range of parameters.The tan β ≤ 50 points with larger LSP masses make negligible contributions to the χ 2 function.For tan β = 55 there are points with 900 GeV m χ 1400 GeV that have ∆χ 2  1, but there is again no exclusion.The only points excluded by the IceCube muon flux limits are focus points for tan β = 56 with 250 GeV m χ 850 GeV and some well-tempered points with larger m χ .However, these points were already excluded because they correspond to values of m h that are too low, as seen in Fig. 8, and by spin-independent cross sections that are too high as seen in Fig. 9.

Combined Results
Figs. 14 and 15 display for A 0 = 0, µ > 0 and our standard set of tan β values the results that we obtain by combining all the individual constraints discussed above, allowing uncertainties in the calculation of the Higgs mass ∆m h = 1.5 GeV and 0.5 GeV, respectively.In the case of tan β = 5 and ∆m h = 1.5 GeV (top left panel of Fig. 14) the m h and spin-independent scattering constraints combine to exclude values of m 1/ 27 TeV, increasing to ∼ 9 TeV for ∆m h ∼ 0.5 GeV (top left panel of Fig. 15). 8For tan β = 5, the Higgs mass constraint is dominant.There are analogous exclusions for tan β = 20 and 40, but extending only to lower values of m 1/2 4 TeV, where the lower limit is determined by the spin-independent scattering constraint.In the case of tan β = 50, these constraints combine to exclude the funnel region and also the part of the focus-point strip with m 1/2 3.5 TeV.There is also a small region of the focus-point strip with m 1/2 6.5 TeV that is excluded by the LHC measurement of m h .We also see that most of the well-tempered strip allowed by m h is excluded by σ SI , leaving only a small set of points near m 1/2 7 TeV.Finally, we find that all of the strips for tan β = 55 and 56 are excluded: the former by a combination of m h and spinindependent scattering, and the latter by m h alone.The other constraints considered also exclude independently parts of the regions excluded by m h and spin-independent scattering, but no supplementary regions.6 The Case A 0 = 3 m 0 , µ > 0 For A 0 = 0 and any fixed value of m 1/2 , radiative electroweak symmetry breaking, i.e. a solution for the Higgs vevs by minimizing the Higgs potential, is no longer possible for sufficiently large m 0 .This boundary is adjacent to the focus-point region where µ is driven to zero.This boundary moves to higher values of m 0 as A 0 /m 0 increases.In addition, when A 0 /m 0 is increased, there is increased splitting in the squark sector, and most notably, one of the stop masses becomes relatively light and comparable to the LSP mass allowing for the possibility that LSP-stop coannihilations determine the relic density [5,46,48].When this occurs, there is again a thin dark matter strip adjacent to the boundary of the region where the light stop becomes the LSP.At still higher m 0 , the lighter stop becomes tachyonic.calculation of m h is much larger for A 0 = 3 m 0 than for A 0 = 0, with ∆m h 3 GeV.This should be borne in mind when interpreting the top right panels of Figs.16 and 17.In particular, it implies that none of the displayed portions of the stop-coannihilation strips can be excluded on the basis of m h .When tan β = 5, the 1-σ range of m h along the dark matter strip extends down to m 1/2 ∼ 3 TeV, well above the current reach of the LHC.On the other hand, the 1-σ range of m h along the dark matter strip for tan β = 20 extends down to m 1/2 ∼ 1.4 TeV, where m χ 630 GeV and m t1 700 GeV, coinciding with the current reach of LHC stop searches for the case of a compressed spectrum [75], and we find similar results for larger tan β.However, the constraints from direct and indirect searches for astrophysical dark matter are not relevant along the stop coannihilation strip, as seen in the bottom 4 panels of Figs.16 and 17.We note that the ∆χ 2 related to neutrino searches at IceCube is significantly suppressed in this case.This is due to the fact that in the stop coannihilation region both the scalar and the spin-dependent direct cross-sections are almost five orders of magnitude smaller than in the cosmologically acceptable higgsino region for the case A 0 = 0.This results in an annihilation rate in the Sun which is about eleven orders of magnitude smaller and the same applies to the muon fluxes.Thus, the resulting ∆χ 2 is suppressed by almost nineteen orders of magnitude.
7 The Case µ < 0 The case A 0 = 0 and µ < 0 is also simpler than A 0 = 0 and µ > 0. As seen in the illustrative examples in Fig. 18 for tan β = 5 and Fig. 19 for tan β = 40, there is only a single dark matter strip close to the boundary of electroweak symmetry breaking.Overall, the results are very similar to those for µ > 0 and the same values of tan β, apart from the absence of the green well-tempered/funnel strip that appears below the focus-point strip in the (m 1/2 , m 0 ) plane when µ > 0 and tan β = 40.As for µ > 0, m h again provides the strongest constraint for A 0 = 0 and µ < 0, as seen in the top right panels of Figs.18 and Fig. 19.The direct search for spin-independent dark matter scattering also imposes an interesting constraint, as seen in the middle left panels of Figs.18 and Fig. 19, excluding a range of LSP masses 1 TeV that are, however, also excluded by m h .The searches for γ-rays from dSphs and the Galactic Centre are not constraining, as seen in the middle right and bottom left panels of these figures.Finally, we note that a range of neutralino masses ∼ 1 TeV that are excluded by m h comes under some pressure from the IceCube search for energetic solar neutrinos, as seen in the bottom right panels of Figs.18 and Fig. 19.The portions of the dark matter strips for tan β = 5 and 40 that are allowed by all the constraints, assuming an uncertainty of 1.5 (0.5) GeV in the calculation of m h , are shown in the upper (lower) pair of panels of Fig. 20.

Conclusions
We have analyzed in this paper the viability of the CMSSM in light of the available phenomenological constraints, assuming that R-parity is conserved and the lightest supersymmetric particle is a neutralino that provides the cosmological dark matter.The CMSSM framework assumes universality of all scalar masses, gaugino masses and A-terms separately at the GUT scale.We also assume a standard thermal freeze-out scenario for the relic density.Relaxing these assumptions would correspondingly relax the constraints derived here, and enlarge the regions of parameters where supersymmetry can survive.However, these assumptions restrict the allowed CMSSM parameter space to narrow strips where the relic dark matter density is compatible with the narrow range permitted by Planck and other measurements.As we have discussed in detail in Section 2, there are several such strips when the ratio of Higgs vevs tan β ≥ 40, the trilinear supersymmetry-breaking parameter A 0 = 0, and the Higgsino mixing parameter µ > 0. On the other hand, when tan β < 40 or µ < 0 there is a single CMSSM dark matter strip running close to the boundary of the   ent mechanisms may come into play simultaneously to bring the dark matter density into the allowed cosmological range, including focus-point and well-tempered neutralino effects, s-channel annihilations via heavy Higgs bosons.In contrast, the dominant dark matter mechanism when A 0 = 3 m 0 is LSP coannihilation with stop squarks.In Section 3 we have discussed the LHC constraints on parameter points along the CMSSM dark matter strips.We have found that direct LHC searches for sparticles do not constrain the CMSSM as strongly as the indirect constraint provided by the LHC measurement of the Higgs mass and calculations of m h using FeynHiggs 2.18.1.These favour relatively large values of the soft supersymmetry-breaking parameters.
In Section 4 we have discussed the constraints on the CMSSM provided by searches for astrophysical dark matter.These include the direct searches for dark matter scattering on nuclei via spin-independent and -dependent interactions.As we have shown, the non-detection of spin-independent dark matter scattering imposes a stronger constraint.It excludes sig-nificant parts of the dark matter strips, while allowed parts are in general incompatible with the m h constraint.We have also considered the constraints provided by searches for γ-rays from dwarf spheroidal galaxies and the Galactic Centre.These both exclude portions of the funnel strips for large tan β, A 0 = 0 and µ > 0, though these are also excluded by m h constraint.The IceCube search for muons produced by energetic neutrinos generated by the annihilations of dark matter particles trapped inside the Sun excludes a range of CMSSM parameters at large tan β, A 0 = 0 and µ > 0 that are, however, also excluded by the m h constraint.
The overall result of our analysis is that, whilst the searches for astrophysical dark matter do exclude portions of the CMSSM dark matter strips, these are mostly also covered by the m h constraint, which is the most powerful when A 0 = 0.In this case, as seen in Figs.14, 15 and 20, only very restricted portions of the CMSSM dark matter strips are allowed by m h .The lengths of these allowed portions are sensitive primarily to the assumed uncertainty in the calculation of m h , which is typically ∼ 0.5 GeV when A 0 = 0.This is significantly larger than the measurement uncertainty, which is between 0.1 and 0.2 GeV.We emphasize that the uncertainty in the m h calculation is much greater when A 0 = 3 m 0 , 3 GeV, rendering it an ineffective constraint for A 0 = 3 m 0 , and note that the direct and indirect searches for astrophysical dark matter are also ineffective for this value of A 0 .
For the moment the CMSSM survives Planck, the LHC, LUX-ZEPLIN, Fermi-LAT, H.E.S.S. and IceCube.However, our constraints push the supersymmetry breaking mass scale to O (10) TeV, larger than the scale originally associated with supersymmetry when proposed as a solution to the hierarchy problem.Nevertheless, supersymmetry at this scale still alleviates the hierarchy problem to some extent.In the future, the most immediate prospect for putting pressure on the CMSSM may be provided by future direct searches for spin-independent dark matter scattering, which could probe essentially all the strips for A 0 = 0 and either sign of µ if their sensitivity reaches down to the neutrino 'floor' [36].In the longer run, searches at FCC-hh or SppC offer good prospects for discovering supersymmetry within the CMSSM framework also when A 0 = 0 [76].On the other hand, the astrophysical searches for dark matter are relatively unpromising for A 0 = 3m 0 , and also have lacunae when µ < 0. The CMSSM may survive a while yet.

Figure 8 :
Figure 8: The yellow shading indicates the impacts of the LHC measurement of m h , as calculated using FeynHiggs 2.18.1 and allowing an uncertainty of 1.5 GeV, on the dark matter strips for tan β = 5, 20, 40, 50, 55 and 56 with A 0 = 0 and µ > 0.

Figure 9 :
Figure 9: Calculations of spin-independent WIMP-nucleon scattering for points along the dark matter strips for tan β = 5, 20, 40, 50, 55 and 56 with A 0 = 0 and µ > 0 compared with the LUX-ZEPLIN upper limit (solid black line).The green and yellow bands are the 1σ and 2σ LZ sensitivities.

Figure 10 :
Figure 10: Calculations of spin-dependent WIMP-nucleon scattering for points along the dark matter strips for tan β = 5, 20, 40, 50, 55 and 56 with A 0 = 0 and µ > 0 compared with the LUX-ZEPLIN upper limit on WIMP-neutron scattering (solid black line).The green and yellow bands are the 1σ and 2σ LZ sensitivities.

Figure 11 :
Figure 11: Contributions to the global χ 2 likelihood function from a comparison of the Fermi-LAT upper limit on the flux of γ-rays from dwarf spheroidal satellite galaxies with calculations for points along the dark matter strips for tan β = 5, 20, 40, 50, 55 and 56 with A 0 = 0 and µ > 0.

Figure 12 :
Figure 12: Contributions to the global χ 2 likelihood function from a comparison of the H.E.S.S. upper limit on the flux of γ-rays from the Galactic Centre with calculations for points along the dark matter strips for tan β = 5, 20, 40, 50, 55 and 56 with A 0 = 0 and µ > 0.

Figure 14 :
Figure 14:  The portions of the dark matter strips for tan β = 5, 20, 40 and 50, calculated assuming A 0 = 0 and µ > 0, that are allowed by all the constraints, assuming an uncertainty of 1.5 GeV in the calculation of m h .There are no allowed regions for tan β = 55 or 56.

Figure 15 :
Figure 15: As in Fig. 14, but assuming an uncertainty of 0.5 GeV in the calculation of m h .

Figure 20 :
Figure 20:  The portions of the dark matter strips for tan β = 5 and 40, calculated assuming A 0 = 0 and µ < 0, that are allowed by all the constraints, assuming an uncertainty of 1.5 (0.5) GeV in the calculation of m h in the upper (lower) pair of panels.