Likelihood analysis of the minimal AMSB model

We perform a likelihood analysis of the minimal anomaly-mediated supersymmetry-breaking (mAMSB) model using constraints from cosmology and accelerator experiments. We find that either a wino-like or a Higgsino-like neutralino LSP, χ~10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\chi }^0_{1}$$\end{document}, may provide the cold dark matter (DM), both with similar likelihoods. The upper limit on the DM density from Planck and other experiments enforces mχ~10≲3TeV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_{\tilde{\chi }^0_{1}} \lesssim 3 \,\, \mathrm {TeV}$$\end{document} after the inclusion of Sommerfeld enhancement in its annihilations. If most of the cold DM density is provided by the χ~10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\chi }^0_{1}$$\end{document}, the measured value of the Higgs mass favours a limited range of tanβ∼5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tan \beta \sim 5$$\end{document} (and also for tanβ∼45\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tan \beta \sim 45$$\end{document} if 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}) but the scalar mass m0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_0$$\end{document} is poorly constrained. In the wino-LSP case, m3/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_{3/2}$$\end{document} is constrained to about 900TeV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$900\,\, \mathrm {TeV}$$\end{document} and mχ~10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_{\tilde{\chi }^0_{1}}$$\end{document} to 2.9±0.1TeV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2.9\pm 0.1\,\, \mathrm {TeV}$$\end{document}, whereas in the Higgsino-LSP case m3/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_{3/2}$$\end{document} has just a lower limit ≳650TeV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gtrsim 650\,\, \mathrm {TeV}$$\end{document} (≳480TeV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gtrsim 480\,\, \mathrm {TeV}$$\end{document}) and mχ~10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_{\tilde{\chi }^0_{1}}$$\end{document} is constrained to 1.12(1.13)±0.02TeV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1.12 ~(1.13) \pm 0.02\,\, \mathrm {TeV}$$\end{document} in the 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} (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}) scenario. In neither case can the anomalous magnetic moment of the muon, (g-2)μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(g-2)_\mu $$\end{document}, be improved significantly relative to its Standard Model (SM) value, nor do flavour measurements constrain the model significantly, and there are poor prospects for discovering supersymmetric particles at the LHC, though there are some prospects for direct DM detection. On the other hand, if the χ~10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\chi }^0_{1}$$\end{document} contributes only a fraction of the cold DM density, future LHC -based searches for gluinos, squarks and heavier chargino and neutralino states as well as disappearing track searches in the wino-like LSP region will be relevant, and interference effects enable BR(Bs,d→μ+μ-)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{BR}(B_{s, d} \rightarrow \mu ^+\mu ^-)$$\end{document} to agree with the data better than in the SM in the case of wino-like DM with 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}.

3 TeV after the inclusion of Sommerfeld enhancement in its annihilations. If most of the cold DM density is provided by theχ 0 1 , the measured value of the Higgs mass favours a limited range of tan β ∼ 5 (and also for tan β ∼ 45 if μ > 0) but the scalar mass m 0 is poorly constrained. In the wino-LSP case, m 3/2 is constrained to about 900 TeV and mχ0 1 to 2.9 ± 0.1 TeV, whereas in the Higgsino-LSP case m 3/2 has just a lower limit 650 TeV ( 480 TeV) and mχ0 1 is constrained to 1.12 (1.13) ± 0.02 TeV in the μ > 0 (μ < 0) scenario. In neither case can the anomalous magnetic moment of the muon, (g − 2) μ , be improved significantly relative to its Standard Model (SM) value, nor do flavour measurements constrain the model significantly, a e-mail: martino.borsato@cern.ch and there are poor prospects for discovering supersymmetric particles at the LHC, though there are some prospects for direct DM detection. On the other hand, if theχ 0 1 contributes only a fraction of the cold DM density, future LHC / E T -based searches for gluinos, squarks and heavier chargino and neutralino states as well as disappearing track searches in the wino-like LSP region will be relevant, and interference effects enable BR(B s,d → μ + μ − ) to agree with the data better than in the SM in the case of wino-like DM with μ > 0.
In the first part of our likelihood analysis of the mAMSB parameter space, we combine the assumption that the LSP is the dominant source of CDM with other measurements, notably of the mass of the Higgs boson, M h = 125.09 ± 0.24 GeV [79] (including the relevant theory uncertainties [80]) and its production and decay rates [81,82]. In addition to solutions in which theχ 0 1 is wino-or Higgsino-like, we also find less-favoured solutions in which theχ 0 1 is a 1 In pure gravity-mediated models [70][71][72], m 0 is constrained to be equal to the gravitino mass, resulting in a two-parameter model in which tan β is strongly constrained to a value near 2. mixed wino-Higgsino state. In the wino case, whereas m 3/2 and hence mχ0 1 are relatively well determined, as is the value of tan β, the value of m 0 is quite poorly determined, and there is little difference between the values of the global likelihood functions for the two signs of μ. On the other hand, in the case of a Higgsino-likeχ 0 1 , while tan β has values around 5, m 0 and m 3/2 are only constrained to be larger than 20 TeV and 600 TeV, respectively, in the positive μ case. For negative μ, the m 0 and m 3/2 constraints are lowered to 18 TeV and 500 TeV, respectively.
If there is some other contribution to the CDM, so that χ 0 1 < CDM , the SUSY-breaking mass scale m 3/2 can be reduced, and hence also mχ0 1 , although the value of M h still imposes a significant lower limit. In this case, some direct searches for sparticles at the LHC also become relevant, notably / E T -based searches for gluinos, squarks and heavier chargino and neutralino states as well as disappearing track searches for the next-to-LSP charged wino. We discuss the prospects for sparticle searches at the LHC in this case and at the 100 TeV FCC-hh collider, and also find that some deviations from Standard Model (SM) predictions for flavour observables may become important, notably BR(b → sγ ) and BR(B s,d → μ + μ − ).
Using the minimum value of the χ 2 likelihood function and the number of effective degrees of freedom (excluding the constraint from HiggsSignals [81,82], as was done in [4][5][6]) leads to an estimate of ∼11% for the χ 2 probability of the mAMSB model if most of the CDM is due to thẽ χ 0 1 , for both signs of μ in both the wino-and Higgsino-like cases. When this CDM condition is relaxed, the χ 2 probability is unchanged if μ < 0, but increases to 18% in the wino-like LSP case if μ > 0 thanks to improved consistency with the experimental measurement of BR(B s,d → μ + μ − ). 2 These χ 2 probabilities for the mAMSB model cannot be compared directly with those found previously for the CMSSM [4], the NUHM1 [4], the NUHM2 [5] and the pMSSM10 [6], since those models were studied with a different dataset that included an older set of LHC data.
The outline of this paper is as follows. In Sect. 2 we review briefly the specification of the mAMSB model. In Sect. 3 we review our implementations of the relevant theoretical, phenomenological, experimental, astrophysical and cosmological constraints, including those from the flavour and Higgs sectors, and from LHC and dark matter searches (see [6,8] for details of our other LHC search implementations). In the case of dark matter we describe in detail our implementation of Sommerfeld enhancement in the calculation of the relic CDM density. Section 4 reviews the MasterCode framework. Section 5 then presents our results, first under the assumption that the lightest neutralinoχ 0 1 is the dominant form of CDM, and then in the more general case when other forms of CDM may dominate. This Section is concluded by the presentation and discussion of the χ 2 likelihood functions for observables of interest. Finally, we present our conclusions in Sect. 6.
The following are some characteristic features of mAMSB: near mass-degeneracy of the left and right sleptons: ml R ≈ ml L , and of the lightest chargino and neutralino, mχ± 1 ≈ mχ0 1 . The mass hierarchy between sleptons and gauginos is dependent on the numerical values of the input parameters, and the squark masses are typically very heavy, because they contain a term proportional to g 4 3 m 2 3/2 . In addition, the measured Higgs mass and the relatively low values of the trilinears A i together imply that the stop masses must also be relatively high. The LSP composition may be wino-, Higgsino-like or mixed, as we discuss in more detail below.

Implementations of constraints
Our treatments in this paper of many of the relevant constraints follow very closely the implementations in our previous analyses which were recently summarized in [8].
In the following subsections we review the implementations, highlighting new constraints and instances where we implement constraints differently from our previous work.

Flavour, electroweak and higgs constraints
Constraints from B-physics and K -physics observables are the same as in [8]. 3 In particular, we include the recent ATLAS result in our global combination of measurements of BR(B s,d → μ + μ − ) [84][85][86][87]. In contrast to our previous studies [4][5][6][7][8], in this study we do not evaluate independently the constraints from EWPOs, since for SUSYbreaking parameters in the multi-TeV range they are indistinguishable from the Standard Model values within the current experimental uncertainties, as we have checked using FeynWZ [88,89]. The only exception is the mass of the W boson, M W , which is evaluated using FeynHiggs. 4 For the other EWPOs we use the theoretical and experimental values given in the review [82]. We use the combination of ATLAS and CMS measurements of the mass of the Higgs boson: M h = 125.09 ± 0.24 GeV [79]. We use a betaversion of the FeynHiggs 2.12.1 code [80,[90][91][92][93][94] to evaluate the constraint this imposes on the mAMSB parameter space. It improves on the FeynHiggs versions used for previous analyses [4][5][6][7] by including two-loop QCD corrections in the evaluation of the DR running top mass and an improved evaluation of the top mass in the DR-on-shell conversion for the scalar tops. At low values of mt 1 , we use, as previously, a one-σ theoretical uncertainty of 1.5 GeV. In view of the larger theoretical uncertainty at large input parameter values, this uncertainty is smoothly inflated up to 3.0 GeV at mt 1 > 7.5 TeV, as a conservative estimate. The χ 2 contributions of 85 Higgs search channels from LHC and Tevatron are evaluated using HiggsSignals [81,82] and HiggsBounds [95][96][97][98] as detailed in our previous paper [8].

LHC constraints
If the entire CDM relic density is provided by the lightest neutralino, all sparticles are heavy, and the current results of the direct sparticle searches at the LHC have no impact on our global fit, though there is some impact from H/A searches [99,100]. On the other hand, ifχ 0 1 accounts only for a fraction of the relic CDM density, some sparticles can be light enough to be produced at the LHC. However, as we discuss in more detail later, even for this case we find that the sleptons, the first two generations of squarks and the thirdgeneration squarks are heavier than 0.7, 3.5 and 2.5 TeV at the 2 σ level, respectively, well beyond the current LHC sensitivities [101][102][103]. On the other hand, gluinos and winos can be as light as 2.5 and 0.5 TeV, respectively, at the 2 σ level, so we have considered in more detail the constraints from searches at the LHC. Currently they do not impact the 68 and 95% CL ranges we find for the mAMSB, but some impact can be expected for future LHC runs, as we discuss in Sect. 5.4.

Sommerfeld enhancement in the wino dark matter region
For a wino-like dark matter particle, the non-perturbative Sommerfeld effect [78] needs to be taken into account in the calculation of the thermal relic abundance. Dedicated studies have been performed in the literature [74][75][76][77], with the result that the correct relic abundance is obtained for mχ0 1 3.1 TeV (with a spread in numerical results of a few %, which may be taken as an estimate of the uncertainty) after inclusion of Sommerfeld enhancement in the thermally averaged coannihilation cross sections, compared to mχ0 1

TeV at tree level.
Because of the large number of points in our mAMSB sample, we seek a computationally efficient implementation of the Sommerfeld enhancement. We discuss this now, and consider its implications in the following subsections.
It is sufficient for our χ 2 likelihood analysis to use a phenomenological fit for the Sommerfeld enhancement that is applicable near 3.1 TeV. One reason is that, away from ∼3.1 TeV, the χ 2 price rises rapidly due to the very small uncertainty in the Planck result for CDM h 2 . Another reason is that the enhancement factor depends very little on the particle spectrum and mostly on mχ0 1 . Therefore, we extract the Sommerfeld factor by using a function to fit the 'non-perturbative' curve in the right panel of Fig. 2 in [74]. One can see that the curve has a dip at ∼2.4 TeV, due to the appearance of a loosely bound state. The calculated relic abundance near the dip is much smaller than the Planck value, so it gives a very large χ 2 , and therefore we do not bother to fit the dip. Considering that the Yukawa potential approaches the Coulomb limit for mχ0 1 M W , and that only the electromagnetic force is relevant for mχ0 1 M W , we fit the annihilation cross section using, 5 where a eff is the effective s-wave coannihilation cross section (including the Sommerfeld enhancement) for the wino system including the wino-like LSP,χ 0 1 , and the corresponding chargino,χ ± 1 , and a eff SE=0 is the effective s-wave coannihilation cross section calculated ignoring the enhancement. The latter is defined as where 1 /T ) expressed as functions of the temperature, T , at which the coannihilations take place. The indices refer toχ 0 1 ,χ + 1 andχ − 1 , and g i is the number of degrees of freedom, which is 2 for each of the three particles, , a i j is the total s-wave (co)annihilation cross section for the processes with incoming particles i and j, and c pm is the fraction of the contribution of theχ + 1χ − 1 s-wave cross section in a eff SE=0 , namely, In practice, since mχ+ /a eff SE=0 . In Eq. (1), S α em and S α 2 are the thermally averaged s-wave Sommerfeld enhancement factors for attractive Coulomb potentials with couplings α em and α 2 , respectively. We use the function given in Eq. (11) of [104] for these quantities, namely where y ≡ α x π mχ0 1 /T .
Because the curve in [74] is obtained by taking the massless limit of the SM particles in a i j , we do the same for our fit to obtain the fitting parameter κ. We find that a κ = O(1) can give a good fit for the curve, and that the fit is not sensitive to the exact value of κ. We choose κ = 6 in our calculation, which gives a good fit around mχ0 ). The perturbative p-wave contribution is also included. We note that, whereas the Sommerfeld enhancement depends almost entirely on mχ0 1 , the values of a eff SE=0 and c pm depend on the details of the supersymmetric particle spectrum. In particular, due to a cancellation between s-and t-channel contributions in processes with SM fermion anti-fermion pairs in the final states, a eff SE=0 becomes smaller when the sfermion masses are closer to mχ0 For a small subset of our mAMSB parameter sample, we have compared results obtained from our approximate implementation of the Sommerfeld enhancement in the case of wino dark matter with more precise results obtained with SSARD. As seen in the left panel of Fig. 1, our implementation (red line) yields results for the relic density that are very similar to those of complete calculations (black dots). In the right panel we plot the ratio of the relic density calculated using our simplified Sommerfeld implementation for the subsample of mAMSB points to SSARD results, connecting the points at different mχ0 1 by a continuous blue line. We see that our Sommerfeld implementation agrees with the exact results at the 2% level (in particular when mχ0 1 ∼ 3 TeV), an accuracy that is comparable to the current experimental uncertainty from the Planck data. We conclude that our simplified Sommerfeld implementation is adequate for our general study of the mAMSB parameter space. 6 Figure 2 illustrates the significance of the Sommerfeld enhancement via a dedicated scan of the (m 0 , m 3/2 ) plane for tan β = 5 using SSARD. The pink triangular region at large m 0 and relatively small m 3/2 is excluded because there are no consistent solutions to the electroweak vacuum conditions in that region. The border of that region corresponds to the line where μ 2 = 0, like that often encountered in the CMSSM at low m 1/2 and large m 0 near the so-called focus-point region [105][106][107][108][109][110]. The dark blue strips indicate where the calculated χ 0 1 density falls within the 3-σ CDM density range preferred by the Planck data [73], and the red dashed lines are contours of M h (labelled in GeV) calculated using FeynHiggs 6 As stated above, a full point-by-point calculation of the relic density would be impractical for our large sample of mAMSB parameters.
2.11.3. 7 The Sommerfeld enhancement is omitted in the left panel and included in the right panel of Fig. 2. We see that the Sommerfeld enhancement increases the values of m 3/2 along the prominent near-horizontal band (where the LSP is predominantly wino) by ∼200 TeV, which is much larger than the uncertainties associated with the CDM density range and our approximate implementation of the Sommerfeld enhancement. We stress that any value of m 3/2 below this band would also be allowed if theχ 0 1 provides only a fraction of the total CDM density.

Interpolation between the wino and Higgsino dark matter regions
We note also the presence in both panels of a very narrow V-shaped diagonal strip running close to the electroweak vacuum boundary, where theχ 0 1 LSP has a large Higgsino component as mentioned previously. As this Higgsino strip is rather difficult to see, we show in Fig. 3 a blowup of the Higgsino region for μ > 0 (the corresponding region for μ < 0 is similar), where we have thickened artificially the Higgsino strips by shading dark blue regions with As the nearly horizontal wino strip approaches the electroweak symmetry-breaking boundary, the blue strip deviates downward to a point, and then tracks the boundary back up to higher m 0 and m 3/2 , forming a slanted V shape.
The origin of these two strips can be understood as follows. In most of the triangular region beneath the relatively h 2 ≤ 0.2 so as to thicken the slanted V-shaped Higgsino LSP strip. Towards the upper part of the Higgsino strip, there is a thin brown shaded strip that is excluded because the LSP is a chargino. Contours of M h calculated (labelled in GeV) using FeynHiggs 2.11.3 (see text) are shown asred dashed lines thick horizontal strip, the LSP is a wino with mass below 3 TeV, and the relic density is below the value preferred by the Planck data. For fixed m 3/2 , as m 0 is increased, μ drops so that, eventually, the Higgsino mass becomes comparable to the wino mass. When μ > 1 TeV, the crossover to a Higgsino LSP (which occurs when μ M 2 ) yields a relic density that reaches and then exceeds the Planck relic density, producing the left arm of the slanted V-shape strip near the focus-point boundary where coannihilations between the wino and Higgsino are important. As one approaches closer to the focus point, μ continues to fall and, when μ 1 TeV, the LSP becomes mainly a Higgsino and its relic density returns to the Planck range, thus producing right arm of the slanted V-shape strip corresponding to the focus-point strip in the CMSSM. In the right panel of Fig. 2 Fig. 3 the tip of the V where these narrow dark matter strips merge occurs when m 0 ∼ 1.8 × 10 4 GeV.

and in
In the analysis below, we model the transition region by using Micromegas 3.2 [111] to calculate the relic density, with a correction in the form of an analytic approximation to the Sommerfeld enhancement given by SSARD that takes into account the varying wino and Higgsino fractions in the composition of the LSP. In this way we interpolate between the wino approximation based on SSARD discussed above for winos, and Micromegas 3.2 for Higgsinos.
Comparing the narrowness of the strips in Figs. 2 and 3 with the thickness of the near-horizontal wino strip, it is clear that they are relatively finely tuned. We also note in Fig. 3 a thin brown shaded region towards the upper part of the V-shaped Higgsino strip that is excluded because the LSP is a chargino.
We also display in these (m 0 , m 3/2 ) planes contours of M h (labelled in GeV) as calculated using FeynHiggs 2.11.3 (see above). Bearing in mind the estimated uncertainty in the theoretical calculation of M h [80], all the broad near-horizontal band and the narrow diagonal strips are compatible with the measured value of M h , both with and without the inclusion of the Sommerfeld enhancement.

Dark matter detection
We implement direct constraints on the spin-independent dark matter proton scattering cross section, σ SI p , using the SSARD code (Information about this code is available from K. A. Olive: it contains important contributions from J. Evans, T. Falk, A. Ferstl, G. Ganis, F. Luo, A. Mustafayev, J. McDonald, K. A. Olive, P. Sandick, Y. Santoso, V. Spanos, and M. Srednicki.), as reviewed previously [4][5][6][7][8]. As discussed there and in Sect. 5.5, σ SI p inherits considerable uncertainty from the poorly constrained p|ss| p matrix element and other hadronic uncertainties, which are larger than those associated with the uncertainty in the local CDM halo density.
We note also that the relatively large annihilation cross section of wino dark matter is in tension with gamma-ray observations of the Galactic centre, dwarf spheroidals and satellites of the Milky Way made by the Fermi-LAT and H.E.S.S. telescopes [112][113][114][115][116][117]. However, there are still considerable ambiguities in the dark matter profiles near the Galactic centre and in these other objects. Including these indirect constraints on dark matter annihilation in our likelihood analysis would require estimates of these underlying astrophysical uncertainties [118], which are beyond the scope of the present work.
In general, we treat the observables as Gaussian constraints, combining in quadrature the experimental and applicable SM and supersymmetric theory errors, which are enumerated in Table 1 of [8]. The exceptions are BR(B s,d → μ + μ − ) and σ SI p , for which we construct full likelihood functions combining the available data, as also described in [8].

Parameter ranges
The ranges of the mAMSB parameters that we sample are shown in Table 1. We also indicate in the right column of this Table how we divide the ranges of these parameters into segments, as we did previously for our analyses of the CMSSM, NUHM1, NUHM2, pMSSM10 and SU(5) [4][5][6][7][8]. The combinations of these segments constitute boxes, in which we sample the parameter space using the MultiNest package [131][132][133]. For each box, we choose a prior for which 80% of the sample has a flat distribution within the nominal range, and 20% of the sample is outside the box in normally distributed tails in each variable. In this way, our total sample exhibits a smooth overlap between boxes, eliminating spurious features associated with box boundaries. Since it is relatively fine-tuned, we made a dedicated supplementary 36-box scan of the Higgsino-LSP region of the mAMSB parameter space, requiring the lightest neutralino to be Higgsino-like. We have sampled a total of 11(13) × 10 6 points for μ > 0 (μ < 0).

Case I: CDM is mainly the lightest neutralino
We display in Fig. 4 the (m 0 , m 3/2 ) planes for our sampling of mAMSB parameters with μ > 0 (left panel) and μ < 0 (right panel). The coloured contours bound regions of parameter space with χ 2 = 2.30 and χ 2 = 5.99 contours, which we use as proxies for the boundaries of the 68% (red) and 95% (dark blue) CL regions. The best-fit points for the two signs of μ are indicated by green stars, closed in the case of wino-like DM, open in the case of Higgsino-like DM. The shadings in this and subsequent planes indicate the composition of the sample point with the lowest χ 2 in this projection: in general, there will also be sample points with a different composition and (possibly only slightly) larger χ 2 . Different shading colours represent the composition of theχ 0 1 LSP: a region with Higgsino fraction exceeding 90% is shaded yellow, one with wino fraction exceeding 90% is shaded light blue, while other cases are shaded orange. 8 Most of blue shading corresponds to a wino-like LSP, and in only a small fraction of cases to a mixed wino-Higgsino state. We see that in the case of a wino-like LSP, the regions favoured at the 2-σ level are bands with 900 TeV m 3/2 1000 TeV corresponding to the envelope of the nearhorizontal band in the right panel of Fig. 2 Fig. 3 that is obtained when profiling over tan β. For both signs of μ, the lower limit m 0 5 TeV is due to theτ 1 becoming the LSP.

and in
The yellow Higgsino-LSP regions correspond to the envelope of the V-shaped diagonal strips seen in Fig. 2 Fig. 3. The locations of these diagonal strips vary significantly with tan β and m t , and their extents are limited at small and large gravitino mass mainly by the Higgs mass constraint. The best-fit point for the Higgsino-LSP scenario has a total χ 2 very similar to the wino-LSP case, as is shown in Fig. 5. The χ 2 values at the best-fit points in the winoand Higgsino-like regions for both signs of μ are given in Table 2, together with more details of the fit results (see below). 3 are allowed at the 95% CL with an upper limit at 48 only in the μ > 0 case. There are regions favoured at the 68% CL with small values of tan β 10 in both the wino-and Higgsino-like cases for both signs of μ. In addition, for μ > 0 there is another 68% CL preferred region in the wino case at tan β 35, where supersymmetric contributions improve the consistency with the measurements of BR(B s,d → μ + μ − ), as discussed in more detail in Sect. 5.3. 9 The parameters of the best-fit points for μ > 0 and μ < 0 are listed in Table 2, together with their 68% CL ranges corresponding to χ 2 = 1. We see that at the 68% CL the range of tan β is restricted to low values for both µ > 0 µ < 0   Fig. 6 The (tan β, m 0 ) planes for μ > 0 (left panel) and for μ < 0 (right panel), assuming that theχ 0 1 provides all the CDM density. The colouring convention for the shadings and contours is the same as in Fig. 4, and the best-fit points for the two signs of μ are again indicated by green stars Fig. 7 The (tan β, m 3/2 ) planes for μ > 0 (left panel) and μ < 0 (right panel), assuming that theχ 0 1 provides all the CDM density. The shadings and colouring convention for the contours are the same as in Fig. 4, and the best-fit points for the two signs of μ are again indicated by green stars LSP compositions, with the exception of the μ > 0 wino-LSP case, where also larger tan β values around 45 are allowed. In the wino-LSP scenario, m 3/2 is restricted to a narrow region around 940 TeV and m 0 is required to be larger than 4 TeV. The precise location of the Higgsino-LSP region depends on the spectrum calculator employed, and also on the version used. These variations can be as large as tens of TeV for m 0 or a couple of units for tan β, and can change the χ 2 penalty coming from the Higgs mass. In our implementations, we find that m 3/2 can take masses as low as 650 TeV (480 TeV) while m 0 is required to be at least 23 TeV (18 TeV) at the 68% CL in the μ > 0 (μ < 0) case. This variability is related to the uncertainty in the exact location of the electroweak symmetry-breaking boundary, which is very sensitive to numerous corrections, in particular those related to the top quark Yukawa coupling.

and in
The minimum values of the global χ 2 function for the two signs of μ are also shown in Table 2, as are the χ 2 probability values obtained by combining these with the numbers of effective degrees of freedom. We see that all the cases studied (wino-and Higgsino-like LSP, μ > 0 and μ < 0) have similar χ 2 probabilities, around 11%, calculated taking into account the relevant numbers of degrees of freedom (including the electroweak precision observables) using the standard prescription [82].
We show in Fig. 8 the contributions to the total χ 2 of the best-fit point in the scenarios with different hypotheses on the sign of μ and the composition of CDM. In addition, we report the main χ 2 penalties in Table 3. Figure 9 shows the best-fit values (blue lines) of the particle masses and the 68 and 95% CL ranges allowed in both the wino-and Higgsino-like LSP cases for both signs of μ. More complete spectra at the best-fit points for the two signs of μ are shown in Fig. 10 in both the winoand Higgsino-LSP cases, where branching ratios exceeding 20% are indicated by dashed lines. As was apparent from the previous figures and tables, a relatively heavy spectrum is favoured in our global fits. The difference between the best-fit spectra in the Higgsino LSP case for μ > 0 and < 0 reflects the fact that the likelihood function is quite flat in the preferred region of the parameter space. In the Higgsino-LSP case, the spectra are even heavier than the other one with a wino LSP, apart from the gauginos, which are lighter. Overall, these high-mass scales, together with the minimal flavor violation assumption, implies that there are, in general, no significant departures from the SM predictions in the flavour sector or for (g − 2) μ . Figure 11 shows the (M A , tan β) planes for μ > 0 (left panel) and for μ < 0 (right panel), assuming that theχ 0 1 contributes all the CDM density. As previously, the red (blue) contours represent the 68% (95%) CL contours, and the winolike (Higgsino-like) DM regions are shaded blue (yellow), and mixed wino-Higgsino regions are shaded orange. We find that the impact of the recent LHC 13-TeV constraints on the (M A , tan β) plane is small in these plots. We see here that the large-tan β 68% CL region mentioned above corresponds to M A 6 TeV.
As anticipated in Sect. 1, the wino-LSP is almost degenerate with the lightest chargino, which acquires a mass about 170 MeV larger through radiative corrections. Therefore, because of phase-space suppression the chargino acquires a lifetime around 0.15 ns, and therefore may decay inside the ATLAS tracker. However, the ATLAS search for disappearing tracks [134] is insensitive to the large mass ∼2.9 TeV expected for the mAMSB chargino if the LSP makes up all the dark matter. In Sect. 5.2 we estimate the LHC sensitivity to the lower chargino masses that are possible if theχ 0 1 Fig. 8 All the contributions to the total χ 2 for the best-fit points for mAMSB assuming different hypotheses on the composition of the dark matter relic density and on the sign of μ as indicated in the legend contributes only a fraction of the cold dark matter density. In the Higgsino-LSP case, the chargino has a mass ∼1.1 TeV in the all-DM case, but its lifetime is very short, of the order of few ps.
The 68% CL ranges of the neutralino masses, the gluino mass, theχ ± 1 −χ 0 1 mass splitting and theχ ± 1 lifetime are reported in Table 4, assuming that theχ 0 1 accounts for all the CDM density. Each parameter is shown for both the winoand Higgsino-like LSP scenarios and for the two signs of μ. Figure 12 shows our results in the (mχ0 in the case when theχ 0 1 is required to provide all the CDM density, within the uncertainties from the Planck and other measurements. The left panel is for μ > 0 and the right panel is for μ < 0: they are quite similar, with each featuring two distinct strips. The strip where mχ0 1 ∼ 1 TeV corresponds to a Higgsino LSP near the focus-point region, and the strip where mχ0 1 ∼ 3 TeV is in the wino LSP region of the parameter space. In between these strips, the make-up of the LSP changes as the wino-and Higgsino-like neutralino states mix, and coannihilations between the three lightest neutralinos and both charginos become important. The Sommerfeld enhancement varies rapidly (we recall that it is not important in the Higgsino LSP region), causing the relic density to rise rapidly as well. We expect the gap seen in Fig. 12 to be populated by points with very specific values of m 0 .

Case II: the LSP does not provide all the cold dark matter
If the LSP is not the only component of the cold dark matter, mχ0 1 may be smaller, m 3/2 may also be lowered substantially, and some sparticles may be within reach of the LHC. The preferred regions of the (m 0 , m 3/2 ) planes for μ > 0 (left panel) and μ < 0 (right panel) in this case are shown in the upper panels of Fig. 13. 10 We see that the wino region allowed at the 95% CL extends to smaller m 3/2 for both signs of μ, and also to larger m 0 at m 3/2 300 TeV when μ < 0. We also see that the 68% CL region extends to much larger m 0 and m 3/2 when μ < 0, and the best-fit point also moves to larger masses than for μ > 0, though with smaller tan β.
The best-fit points and mass ranges for the case where the LSP relic density falls below the Planck preferred density are given in Table 5. As one can see, the best fit for μ > 0 has a somewhat lower value of χ 2 and a significantly higher value of tan β. This is because in the case Table 3 The most important contributions to the total χ 2 of the best-fit points for mAMSB assuming different hypotheses on the composition of the dark matter relic density and on the sign of μ. In the μ > 0 scenario with χ 0 1 < CDM andW -LSP, the experimental constraints from (g − 2) μ and BR(B s → μ + μ − ) can be accommodated and get a lower χ 2 penalty of positive μ there is negative interference between the mAMSB and SM contributions to the decay amplitude in this parameter-space region, reducing BR(B s,d → μ + μ − ) and allowing a better fit to the latest experimental combination of ATLAS, CMS and LHCb measurements (see Fig. 19).
On the other hand, in the negative-μ case, the interference is constructive and thus the best fit to the experimental measurement is for a SM-like branching ratio, which is predicted in a much wider region of the parameter space. As previously, the quoted χ 2 probabilities are calculated taking into account the relevant numbers of degrees of freedom (including the electroweak precision observables) using the standard prescription [82]. The lower panels of Fig. 13 show the (tan β, m 3/2 ) planes for μ > 0 (left) and for μ < 0 (right). Comparing with the corresponding planes in Fig. 7 for the case in which the LSP provides all the dark matter, we see a large expansion of the wino-like region, we note that the allowed range of m 3/2 extends down to ∼ 100 TeV, and the 68% CL region is found to extend to large values of tan β.
We display in Fig. 14 the (M A , tan β) planes in the partial-CDM case for μ > 0 (left panel) and μ < 0 (right panel). Comparing with the corresponding Fig. 11 for the all-CDM case, we see that a large region of smaller values of M A and tan β are allowed in this case. We also note that the best-fit point in the wino-like region for μ > 0 has moved to a much smaller value of M A and a much larger value of tan β, much closer to the region currently excluded by LHC searches. In this connection, we note that the fit including only the LHC 8-TeV H/A → τ + τ − constraint [99] is slightly weaker in this region than that including the 13-TeV constraint [100]. This gives hope that future improvements in the LHC H/A search may be sensitive to the preferred region of the mAMSB parameter space in the partial-CDM case.  Figure 16 shows the mass spectra allowed in the wino-like LSP case for μ > 0 (top panel) and μ < 0 (second panel), and also in the Higgsino-like LSP case for μ > 0 (third panel) and μ < 0 (bottom panel). The one-and two-σ ranges are again shown in dark and light orange respectively, and the best-fit values are represented by blue lines. We see that the spectra in the wino-like LSP case are quite different for the two signs of μ, whereas those in the Higgsino-like LSP case resemble each other more. Table 6 provides numerical values for the 68% CL ranges for the neutralino masses, the gluino mass, the mass difference between the lightest chargino and neutralino, as well as for the corresponding chargino lifetime.
Finally, Fig. 17 displays the spectra of our best-fit points in the case that the LSP contributes only a fraction of the cold dark matter density. As previously, the left panels are for μ > 0 and the right panels are for μ < 0 (note the different scales on the vertical axes). Both the wino-(upper) and the Higgsino-like LSP (lower) best-fit points are shown. In each W -LSP for µ > 0, Ωχ0 Fig. 9 The ranges of masses obtained for the wino-like LSP case with μ > 0 (top panel) and μ < 0 (second panel), and also for the Higgsino-like LSP case for μ > 0 (third panel) and μ < 0 (bottom panel), assuming that the LSP makes the dominant contribution to the cold dark matter density Fig. 10 The spectra of our best-fit points for μ > 0 (left panel) and μ < 0 (right panel), assuming that the LSP makes the dominant contribution to the cold dark matter density. Both the wino-like (upper) and the Higgsino-like LSP (lower) best-fit points are shown. In each case, we also indicate all the decay modes with branching ratios (BRs) above 20%, with darker shading for larger BRs, and the colours of the horizontal bars reflect particles electric charges µ > 0, Ωχ0 Fig. 11 The (M A , tan β) planes for μ > 0 (left panel) and for μ < 0 (right panel), assuming that theχ 0 1 contributes all the CDM density. As previously, the red (blue) contours represent the 68% (95%) CL con-tours, and the wino-like (Higgsino-like) (mixed wino-Higgsino) DM regions are shaded blue (yellow) (orange) Table 4 The 68% CL ranges for the masses of the LSPχ 0 1 and of the heavier neutralinos χ 0 2 ,χ 0 3 andχ 0 4 , as well as the mass splitting between the lighter charginoχ ± 1 and the LSP and the corresponding lifetime of theχ ± 1 , for the case in which theχ 0 1 accounts for all the CDM density. Each parameter is shown for both the wino-and Higgsino-LSP scenarios, as well as for both signs of μ  h 2 ) planes for μ > 0 (left panel) and μ < 0 (right panel) assuming that all the CDM density is provided by theχ 0 1 . The shadings are the same as in Fig. 4 case, we also indicate all the decay modes with branching ratios above 20%.

χ 2 likelihood functions for observables
We show in Fig. 18 the one-dimensional likelihoods for several sparticle masses. In all cases the solid lines correspond to the case in which the LSP accounts for all of the CDM, and the dashed lines for the case in which it may provide only a fraction of the CDM. The blue lines are for μ > 0 and the red lines are for μ < 0. It is apparent that in the all-CDM case the sparticles in the mAMSB are expected to be too heavy to be produced at the LHC: mg, mt 1 10 TeV, mq 15 TeV, mτ 1 3 TeV, mχ0 1 1 TeV. However, in the part-CDM case the sparticle masses may be much lighter, with strongly interacting sparticles possibly as light as ∼2 TeV and much lighterχ 0 3 andχ ± 2 also possible, so that some of them may become accessible at LHC energies. Indeed, as we discuss below, future LHC runs should be able to explore parts of the allowed parameter space.
As shown in Fig. 19, there are some interesting prospects for indirect searches for mAMSB effects. There are in gen-eral small departures from the SM if the LSP accounts for all of the CDM, whereas much more significant effects can arise if the CDM constraint is relaxed. In particular, we find that significant destructive interference between mAMSB effects and the SM may cause a sizeable decrease of the BR(B s,d → μ + μ − ) branching ratio in the positive μ case, which can be significant within the range of model parameters allowed at the 2-σ level and improves the agreement with the experimental measurement shown by the dotted line. This effect arises from a region of parameter space at large tan β where M A can be below 5 TeV, as seen in the bottom right panel of Fig. 18. We find that the destructive interference in BR(B s,d → μ + μ − ) is always accompanied by a constructive interference in BR(b → sγ ). There is also some possibility of positive interference in BR(B s,d → μ + μ − ) and negative interference in BR(b → sγ ) when μ < 0 and the LSP does not provide all the dark matter, though this effect is much smaller. Finally, we also note that only small effects at the level of 10 −10 can appear in (g − 2) μ , for either sign of μ. Fig. 13 The (m 0 , m 3/2 ) planes (upper panels) and the (tan β, m 3/2 ) planes (lower panels) for μ > 0 (left panels) and for μ < 0 (right panels), allowing theχ 0 1 to contribute only part of the CDM density. The shadings are the same as in Fig. 4

Table 5
Fit results for the mAMSB assuming that the LSP accounts for just a fraction of the cold dark matter density. The 68% CL ranges correspond to χ 2 = 1. We also display the values of the global χ 2 function omitting the contributions from HiggsSignals, and the corresponding χ 2 probability values calculated taking into account the relevant numbers of degrees of freedom (including the electroweak precision observables) using the standard prescription [82]. Each mass range is shown for both the wino-and Higgsino-LSP scenarios and both signs of μ

Discovery prospects at the LHC and FCC-hh
As mentioned above, current LHC searches are not sensitive to the high-mass spectrum of mAMSB, even if the LSP is not the only component of CDM. However, simple extrapolation indicates that there are better prospects for future LHC searches with 300 or 3000 fb −1 . Searches for chargino tracks disappearing in the tracker, such as that performed by ATLAS [134] start to be sensitive with 300 fb −1 of data, and become much more sensitive with 3000 fb −1 , as shown in Fig. 20. We have obtained the projected contours for the 13 TeV LHC with 13,300 and 3000 fb −1 by rescaling the Run-1 µ > 0, Ωχ0 Fig. 14 The (M A , tan β) planes for μ > 0 (left panel) and for μ < 0 (right panel), allowing theχ 0 1 to contribute only part of the CDM density. The shadings are the same as in Fig. 4 µ > 0, Ωχ0 h 2 ) planes in the mAMSB for μ > 0 (left) and μ < 0 (right), allowing theχ 0 1 to contribute only part of the CDM density. The red (blue) contours represent the 68% (95%) CL contours. The shadings are the same as in Fig. 4 sensitivity presented in [134]. In doing so, at a given lifetime we shift the Run-1 value of the 95% CL reach for the wino mass to the higher value at which the wino cross section times luminosity (13,300 and 3000 fb −1 ) at the 13-TeV LHC coincides with the reach achieved during Run-1. This method is often used and is known to give a reasonable estimate [135]. We find that the disappearing-track search is more sensitive in the μ > 0 case than in the μ < 0 case because, in order to accommodate the reduced BR(B s → μ + μ − ), the wino-LSP solution is preferred over the Higgsino-LSP one.
The large mass reach of a 100 TeV pp collider would extend these sensitivities greatly. Reference [136,137] studied the capability of exploring the wino LSP scenario at a 100-TeV collider and found that the sensitivity reaches around mχ± 1 ∼ 3 TeV at 3000 fb −1 . We therefore expect that at a 100-TeV collider with 3000 fb −1 almost the entire 68% CL region and a part of the 95% CL with τχ± 1 > 0.1 ns will be explored for μ > 0 and < 0, respectively. If improvements on the detector and the analysis allow the sensitivity to be extended to mχ± 1 3 TeV, the wino-like dark matter region in the scenario with χ 0 1 = CDM can also be probed. Coloured sparticle searches that will become sensitive in future LHC runs are shown in Figs. 21,22, and 23 in the (mg, mq R ), (mq R , mχ0 1 ) and (mg, mχ0 1 ) planes, respectively, the latter being the most promising one. In these figures, exclusion contours from the Run-1 (orange) and the 13 TeV with 13 fb −1 (purple) data are taken from ATLAS analyses [138,139], respectively. Also superimposed are the projected 95% CL contours at the 14 TeV LHC with 300 (green dashed) and 3000 (green dotted) fb −1 estimated by ATLAS [140,141]. We also show, by grey dotted contours, the sensitivity at a 100 TeV collider with 3000 fb −1 taken from [142]. These contours assume simplified models with BR(q → qχ 0 1 ) = BR(g → qqχ 0 1 ) = 100% for Figs. 22 and 23. For Fig. 21, in addition to these decays the heavier of the gluino and squark can also decay into the lighter one with an appropriate branching ratio. In Fig. 22 the projected LHC contours are estimated postulating mg = 4.5 W -LSP for µ > 0, Ωχ0 16 The ranges of masses obtained for the wino-like LSP case with μ > 0 (top panel) and μ < 0 (second panel), and also for the Higgsino-like LSP case for μ > 0 (third panel) and μ < 0 (bottom panel), relaxing the assumption that the LSP contributes all the cold dark matter density. The one-and two-σ CL regions are shown in dark and light orange respectively, and the best-fit values are represented by blue lines TeV, which is the right ball-park in our scenario. We see that with 3000 fb −1 the LHC could nip the tip of the 95% CL region in these planes, whereas with the same luminosity a 100-TeV collider would explore a sizeable region of the parameter space. In particular, the best-fit point and the 68% CL region are within this sensitivity for the μ > 0 case.  The spectra of our best-fit points for μ > 0 (left panel) and μ < 0 (right panel), allowing the LSP to contribute only part of the cold dark matter density. Both the wino-(upper) and the Higgsino-like LSP (lower) best-fit points are shown. In each case, we also indicate all the decay modes with branching ratios (BRs) above 20%, with darker shading for larger BRs, and the colours of the horizontal bars reflect particles electric charges. The range of masses shown for theW -LSP μ > 0 best-fit point (top-left panel) is smaller than the others, since its mass spectrum is considerably lighter Allowing χ 0 1 < CDM , we found in our sample very light winos as well as Higgsinos. If both of them are light but with a sufficiently large mass hierarchy between them, the LHC and a 100-TeV collider may be able to detect the production of a heavier state decaying subsequently into the lightest state by emitting the heavy bosons W ± , Z and h. In Fig. 24 we plot the current and future LHC reaches as well as the sensitivity expected at a 100-TeV collider with the same luminosity assumptions as in the previous figures. The current limit (purple) and projected sensitivity (green)  [141] and assume wino-like chargino and neutralino production and a 100% rate for decay into the W ± Z + / E T final state. As can be seen in Fig. 24, the region that can be explored is mainly the Higgsino-like LSP region, whereas we are interested in the wino-like chargino and neutralino production. However, unlike the simplified model assumption employed by ATLAS and CMS, the charged wino decays into neutral or charged Higgsinos emitting W ± , Z and h with 50, 25 and 25% branching ratio, respectively [144][145][146]. Similarly, the branching ratios of the neutral wino are 50, 25 and 25% for decays into W ± , Z and h, respectively. In total, only 25% of the associated charged and neutral wino production events contribute to the W ± Z + / E T channel. The LHC contours shown in Fig. 24 should be considered with this caveat. Also shown by the grey dotted line is the sensitivity expected at a ) plane allowed in the χ 0 1 ≤ CDM case for μ > 0 (left) and μ < 0 (right). The orange solid line represents the limit from the ATLAS 8-TeV search for disappearing tracks [134]. The magenta solid, green dashed and green dotted lines represents the projection of this limit to 13-TeV data with 13, 300 and 3000 f b −1 , respectively. The shadings are the same as in Fig. 4 µ > 0, Ωχ0 Fig. 21 The region of the (mq R , mg) plane allowed in the χ 0 1 ≤ CDM case for μ > 0 (left) and μ < 0 (right). The orange solid line represents the LHC 8-TeV 95% CL exclusion [138]. The green dashed and dotted lines show the projection estimated by ATLAS [140] for 14-TeV data with 300 and 3000 f b −1 , respectively. The grey dotted line is the 95% CL sensitivity expected at a 100 TeV pp collider with a 3000 f b −1 integrated luminosity obtained from [142]. All contours assume masslessχ 0 1 . The shadings are the same as in Fig. 4 µ > 0, Ωχ0 Fig. 22 The region of the (mq R , mχ0 1 ) plane allowed in the χ 0 1 ≤ CDM case for μ > 0 (left) and μ < 0 (right). The purple solid line represents the ATLAS 13-TeV 95% CL exclusion using 13 f b −1 of data [139]. The green dashed and dotted lines show the projected 95% CL sensitivity estimated by ATLAS [141] for 14-TeV data with integrated luminosities of 300 and 3000 f b −1 , respectively. The grey dotted line is the 95% CL sensitivity expected at a 100 TeV pp collider with a 3000 f b −1 integrated luminosity obtained from [142]. All contours assume a simplified model with 100% BR forq → qχ 0 1 . The current limit and 100 TeV projection assumes decoupled gluino, while the projection to the higher luminosity LHC assumes a 4.5-TeV gluino. The shadings are the same as in Fig. 4 100-TeV collider with 3000 fb −1 luminosity studied in [144] (see also [146]) assuming a Higgsino-like LSP and wino-like chargino and neutralino production, taking into account the correct branching ratios mentioned above. As can be seen, a 100-TeV collider is sensitive up to mχ0 3 ∼ 2 TeV, and a large part of the 95% CL region would be within reach, and also a substantial portion of the 68% CL region if μ < 0, though not the best-fit point for either sign of μ.
Finally, in Fig. 25 we show the (mg, mq R ) plane for the scenario with χ 0 1 = CDM . We found that a small part of the wino-like dark matter region and a good amount of the Higgsino-like dark matter region are within the 95% CL sensitivity region at a 100-TeV collider with 3000 fb −1 . In particular, the sensitivity contour reaches the best-fit point for the μ > 0 case.

Prospects for direct detection of dark matter
While the heavy spectra of mAMSB models may lie beyond the reach of the LHC constraints, future direct DM search experiments may be capable of detecting the interaction of a mAMSB neutralino, even if it does not provide all the CDM density [147][148][149][150]. Figure 26 displays the cross section for spin-independent scattering on a proton, σ SI p , versus the neu- Fig. 23 The region of the (mg, mχ0 1 ) plane allowed in the χ 0 1 ≤ CDM case for μ > 0 (left) and μ < 0 (right). The purple solid line represents the ATLAS 13-TeV 95% CL exclusion with the data with 13 f b −1 [139]. The green dashed and dotted lines show the projected 95% CL sensitivity estimated by ATLAS [141] for 14-TeV data with integrated luminosities of 300 and 3000 f b −1 , respectively. The grey dotted line is the 95% CL sensitivity expected at a 100 TeV pp collider with a 3000 f b −1 integrated luminosity obtained from [142]. All contours assume a simplified model with 100% BR forg → qqχ 0 1 . The shadings are the same as in Fig. 4 µ > 0, Ωχ0 line shows the projected sensitivity for 14-TeV data with an integrated luminosity of 300 f b −1 (3000 f b −1 ) estimated by ATLAS [141]. The grey dotted line is the 95% CL sensitivity expected at a 100 TeV pp collider with a 3000 f b −1 integrated luminosity obtained from [144].
The shadings are the same as in Fig. 4 tralino mass. As previously, the left plane is for μ > 0, the right plane is for μ < 0, the 1 and 2 σ CL contours are shown as red and blue lines, and the wino-and Higgsino-LSP regions are shaded in pale blue and yellow. The palegreen-shaded region represents the range of σ SI p excluded at the 95% CL by our combination of the latest PandaX and LUX results [151,152], while the purple and blue lines show the prospective sensitivities of the LUX-Zeplin (LZ), XENON1T and XENONnT experiments [153][154][155]. Also shown, as a dashed orange line, is the neutrino 'floor', below which astrophysical neutrino backgrounds would dominate any DM signal [156] (grey region). The mAMSB region allowed at the 2 σ level includes points where σ SI p is nominally larger than that excluded by LUX and PandaX at the 95% CL, which become allowed when the large theoretical uncertainty in σ SI p is taken into account. This uncertainty stems largely from the uncertainty in the strangeness contribution to the nucleon, which receives contributions from two sources. The strange scalar density can be written as y = 1 − σ 0 / π N where σ 0 is the change in the nucleon mass due to the non-zero u, d quark masses, and is estimated from octet baryon mass differences to be σ 0 = 36 ± 7 MeV [157][158][159]. This is the dominant source of error in the computed cross section. In addition, the πµ > 0, Ωχ0 1 = Ω CDM µ < 0, Ωχ0 1 = Ω CDM Fig. 25 The region of the (mq R , mg) plane allowed in the χ 0 1 = CDM case for μ > 0 (left) and μ < 0 (right). The orange solid line represents the LHC 8-TeV 95% CL exclusion [138]. The green dashed and dotted lines show the projection estimated by ATLAS [140] for 14-TeV data with 300 and 3000 f b −1 , respectively. The grey dotted line is the 95% CL sensitivity expected at a 100 TeV pp collider with a 3000 f b −1 integrated luminosity obtained from [142]. All contours assume masslessχ 0 1 . The shadings are the same as in Fig. 4 µ > 0, Ωχ0

Fig. 26
The (mχ0 1 , σ SI p ) planes in the mAMSB for μ > 0 (left) and μ < 0 (right) in the case when the LSP accounts for the whole DM density. The red and blue solid lines are the 1 and 2 σ CL contours, and the solid purple and blue lines show the projected 95% exclusion sensitivities of the LUX-Zeplin (LZ) [153] and XENON1T/nT experiments [154,155], respectively. The green line and shaded region show the combined limit from the LUX and PandaX experiments [151,152], and the dashed orange line shows the astrophysical neutrino 'floor' [156], below which astrophysical neutrino backgrounds dominate (grey region). The blue, orange and yellow shadings are the same as in Fig. 4 nucleon term is taken as 50 ± 8 MeV. Another nonnegligible source of error comes from the uncertainty in quark masses. The resulting 68% CL uncertainty in the calculated value of σ SI p is more than 50%. However, the current data already put pressure on the mAMSB when μ > 0, in both the wino-and Higgsino-like LSP cases, corresponding to the left and right vertical strips in Fig. 26. 11 The Higgsino-likeχ 0 1 with this sign of μ could be explored completely with the prospective LZ sensitivity, while the wino-like LSP may have σ SI p below the neutrino 'floor'. The wino-like LSP region lying below the LZ sensi- 11 The arches between them, where the LSP has a mixed wino/Higgsino composition, are under severe pressure from LUX and PandaX. tivity could be partly accessible to a 20-tonne DM experiment such as Darwin [160]. When μ < 0, σ SI p may be lower than for positive μ, possibly lying below the LZ sensitivity in the Higgsino case and far below the neutrino 'floor' in the wino case. Figure 27 extends the analysis to the case in which the LSP is allowed to contribute only a fraction of the CDM density. In this case we weight the model value σ SI p by the ratio χ 0 1 / CDM , since this would be the fraction of the galactic halo provided by the LSP in this case. There are still reasonably good prospects for future DM direct detection experiments when μ > 0, with only a small fraction of the parameter space lying below the neutrino 'floor'. However, µ > 0, Ωχ0 Fig. 27 The (mχ0 1 , σ SI p ) planes in the mAMSB for μ > 0 (left) and μ < 0 (right) in the case when the LSP only accounts for a fraction of the CDM density. The legends, line styles and shadings are the same as in Fig. 26 when μ < 0 σ SI p may fall considerably below the 'floor', because of cancellations [161][162][163][164][165] in the scattering matrix element.

Summary
Using the MasterCode framework, we have constructed in this paper a global likelihood function for the minimal AMSB model and explored the constraints imposed by the available data on flavour, electroweak and Higgs observables, as well as by LHC searches for gluinos via / E T signatures. We have also included the constraint imposed by the cosmological cold dark matter density, which we interpret as either a measurement or an upper limit on the relic LSP density, and searches for dark matter scattering.
In the all-CDM case, we find that the spectrum is relatively heavy, with strongly interacting sparticles weighing 10 TeV, but much smaller masses are possible if the LSP contributes only a fraction of the overall CDM density. In the all-CDM case, the LSP composition may be either winoor Higgsino-like with almost equal likelihood, weighing ∼3 TeV and ∼1 TeV, respectively. On the other hand, in the part-CDM case much lighter LSP masses are allowed at the 68% CL, as are intermediate LSP masses.
Because of the high masses in the all-CDM case, the prospects for discovering sparticles at the LHC are small, and there are limited prospects for observing significant deviations for the SM predictions for flavour observables. However, in the part-CDM case some sparticles may well be within reach of the LHC, and there are more interesting possibilities for observing mAMSB effects on flavour observables, e.g., BR(B s,d → μ + μ − ). In both cases, wide ranges of dark matter scattering cross sections, σ SI p are allowed: σ SI p may be very close to the upper limits established recently by the PandaX and LUX experiments, or it may be within reach of the planned LUX-Zeplin experiment, or it may even be far below the neutrino 'floor'.
The mAMSB scenario discussed in this paper clearly presents different challenges from the models with GUTscale unification of (at least some of) the soft SUSY-breaking parameters that we have studied previously [1][2][3][4][5]7,8], and it does not share the flexibility of pMSSM models [6,166]. As such, the mAMSB serves as a useful reminder that SUSY phenomenology may differ significantly from what is preferred in these other scenarios.