Stop Co-Annihilation in the Minimal Supersymmetric Standard Model Revisited

We re-examine the stop co-annihilation scenario of the Minimal Supersymmetric Standard Model, wherein a bino-like lightest supersymmetric particle has a thermal relic density set by co-annihilations with a scalar partner of the top quark in the early universe. We concentrate on the case where only the top partner sector is relevant for the cosmology, and other particles are heavy. We discuss the cosmology with focus on low energy parameters and an emphasis on the implications of the measured Higgs boson mass and its properties. We find that the irreducible direct detection signal correlated with this cosmology is generically well below projected experimental sensitivity, and in most cases lies below the neutrino background. A larger, detectable, direct detection rate is possible, but is unrelated to the co-annihilation cosmology. LHC searches for compressed spectra are crucial for probing this scenario.


I. INTRODUCTION
The Minimal Supersymmetric Standard Model (MSSM) is a leading candidate for physics beyond the Standard Model (SM). However, superpartners have remained stubbornly absent at the Large Hadron Collider (LHC). Moreover, bounds from direct detection experiments, including LUX, PandaX, and Xenon1T [1][2][3] present an increasingly strong challenge to the WIMP (weakly interacting massive particle) paradigm both in the MSSM and more broadly [4]. These experiments place the most pressure on models where dark matternucleon scattering is related to the cosmological history via a crossing symmetry. Crossing symmetry is spoiled if the dark matter coinhabits the thermal bath with another exotic state at the time of freeze-out [5]. Then processes involving this co-annihilating partner can be important for the determination of the relic density, but are unrelated (at tree level) to direct detection. These co-annihilating scenarios are therefore among the WIMP models least constrained by direct detection bounds. The MSSM realizes this scenario when the superpartner of the top quark, the stop, is light, and neutralino co-annihilations with this state determine the relic density.
In this work, we examine the stop co-annihilation scenario in terms of the low energy parameters most relevant for cosmology and direct detection. Many analyses of the stop co-annihilation parameters place an emphasis on simplified high-energy models, see e.g.
Ref. [6][7][8][9] However, because there are typically a small number of processes that dominate the cosmology involving only a handful of particles and couplings, it is illuminating to analyze these models in terms of the low energy parameters. Consistent with this approach, in previous work [10] we considered the possibility where a single top partner (perhaps the superpartner of the right-handed top) was responsible for co-annihilation, see also Refs. [11][12][13][14][15][16] for related work. In the context of the MSSM, however, a simplified model that includes a single co-annihilator, e.g. at R , may be too simple to capture the physics of both cosmology and direct detection. Mixing between the light stop and the heavy stop can impact both of these processes. Indeed, the measurement of the Higgs boson mass suggests that there may be large mixing in the stop sector. In this case, it makes sense to include the full stop sector in the simplified model. This article is organized as follows. In Sec. II we discuss the basics of the parameter space and discuss constraints unrelated to the dark matter story. Cosmology is analyzed in Sec. III and we turn to implications for direct detection in Sec. IV. Finally, Sec. V is reserved for our conclusions.

II. ORIENTATION
We consider the case of pure bino neutralino dark matter χ, with co-annihilations resulting from the presence of one or more colored states Y with masses not dissimilar to that of the neutralino. If equilibrium between the neutralino and the colored state is maintained (as is typically the case due to scatterings off the SM bath), then processes of the form (χY → SM ) or (Y Y → SM ) are relevant for setting the dark matter abundance.
A small admixture of Higgsino will not affect the cosmological history in detail. However, doping the bino with even a small Higgsino fraction can impact direct detection since the Higgs boson has a tree-level bino-Higgsino coupling. The Higgsino fraction of the neutralino is controlled by mixing suppressed by M Z /µ, with µ the Higgsino mass parameter. The tree-level direct detection cross section is well approximated by [10]: (1) Here, s 2β ≡ sin 2β where tan β is the ratio of vacuum expectation values of the two Higgs doublets. In this work, we will be interested in the case where this contribution to direct detection is subdominant. That is, we consider the case where µ is (quite) large, and we ask the question: what is the direct detection cross section induced only by the presence of a stop co-annihilation cosmology?
Because we will be interested in discussing the effects of reproducing the required Higgs boson mass, we will include the full stop sector, i.e botht R andt L . This means that the left-handed sbottomb L is necessarily in the spectrum as well. For simplicity, we assume the partner of the right-handed bottom is decoupled. The precise value of its mass has little effect on either the physical Higgs boson mass or any fine-tuning arguments. 1 The stop mass matrix is given by: where X t = (A t − µ/ tan β), m Q 3 , m u 3 are the left and right-handed soft masses, and the D-terms are: D L = M 2 Z cos 2β( 1 2 − 2 3 s 2 W ) and D R = 2 3 M 2 Z cos 2βs 2 W . Here, s W = sin θ W is the weak mixing angle. We choose sign conventions for the sign of the stop mixing angle consistent with SuSpect 2.41 [17]. We have: where we denote the stop mass eigenstates ast 1 = c ttL + s ttR andt 2 = −s ttL + c ttR and use the shorthand s t = sin θt, etc. The stop mixing angle is a function of X t rather than A t , µ or tan β directly. In the region of interest, X t ∼ A t , and the dependence on tan β is minimal.
For concreteness we fix tan β = 10 but numerically verify that results are approximately independent of tan β, as long as the values are not too extreme.
The heaviness of the right-handed sbottom ensures that theb 1 state has mass given approximately by m Q 3 . But for simplicity, we also set X b = 0 so there is no mixing in the sbottom sector, regardless of tan β. This ensures that theb 1 is completely left-handed, and its mass may be expressed in terms of the stop masses and mixing angles: For later reference, we record the couplings of the stops and the left-handed sbottom to the Higgs and the bino: where the Higgs vacuum expectation value v = 246 GeV, g Y is the hypercharge gauge coupling and P L , P R are the projection operators.
We implemented this simplified model containing the third generation squark doublet, right-handed stop and a bino neutralino in MicroOmegas [18]. As a cross-check we validated our numerics with the full MSSM implementation.

A. Indirect Constraints
Under the assumption that the MSSM is the underlying theory, important information about the superpartner spectrum can be gleaned from the Higgs boson mass. Indeed, the MSSM cannot reproduce a Higgs boson mass of 125 GeV without substantial radiative corrections from the stop sector. Because the Higgs mass constraint impacts the allowed properties of the stops, it is of interest to study whether it has phenomenological implications for the stop co-annihilation region. Nevertheless, one should bear in mind the possibility that the observed Higgs mass is generated by physics beyond the MSSM.
We therefore prefer to begin by considering additional indirect consequences of the stop sector, potentially relevant even if there are additional contributions to the mass of the Higgs boson. In particular, before discussing the impact of the Higgs mass, we discuss three sets of constraints: electroweak precision observables (EWPO), Higgs boson production and decays, and the stability of the physical vacuum due to the presence of charge and color breaking vacuua. It is a model-dependent question as to whether the above would be affected by deviations from the MSSM. At minimum, however, these constraints give an indication of cancellations that would need to occur between the stop sector and any additional contributions. Once collider bounds are imposed, we find constraints from vacuum stability are generally the strongest of the three.
Regarding EWPO, we require the stop sector contribution to δρ not exceed the 2σ bound from the measured δρ = 3.7 ± 2.3 × 10 −4 [19]. Assuming negligible mixing in the sbottom sector, the third generation squarks yield [20]: with G F the Fermi constant while the loop function F 0 is given by In the case with hierarchical stops, where the lightest stop is still mostly left-handed, it can be shown that X t ∼ mt 2 minimizes the above contribution to δρ [21]. This is due to an approximate custodial symmetry between the sbottom and the lightest stop -these two states have an approximate mass degeneracy in this regime: . Deviating too far from this limit (i.e. very large X t ), causes the constraints to become significant. The excluded region is shown in Fig. 1 (red region) in the sin θt vs. mt 2 plane for two different values of mt 1 = 600 GeV (a) and 1.5 TeV (b).
Higgs boson properties can also be influenced by the presence of a light stop. The primary impact is a potential contribution to the gluon fusion production cross section. The contribution to H → γγ, while less pronounced, should also be considered. Using the Higgs low energy theorem, the contribution oft 1,2 to the production amplitude of the Higgs boson in gluon fusion can be approximated as [22][23][24] where A SM hgg = 4 is the SM amplitude. Analogously we can write the amplitude for H → γγ as where A SM hγγ = −13 [22][23][24][25][26][27]. Again, if X t mt 2 , both gluon fusion and the diphoton rate will be impacted, but because |A SM hgg | |A SM hγγ |, the stop contribution to gluon fusion provides a larger deviation from SM expectations. The combined analysis of the 8 TeV LHC run by ATLAS and CMS constrains the signal strength of the production in gluon fusion, normalized to the SM prediction, to be µ ggF = 1.03 +0. 17 −0.15 while µ γγ = 1.16 +0.20 −0.18 [28]. Allowing for near future improvements in these measurements, we allow a deviation of up to 20% in these observables. With the increasingly powerful direct searches for the stop (which imposẽ t 1 > ∼ 500 GeV, more below), these constraints are only relevant for particularly large values of the X t parameter. The region excluded by Higgs signal strength measurements is shown in orange in Fig. 1.
Charge and color breaking vacua may appear at large values of the trilinear coupling A t and the physical vacuum can become metastable, possibly with a lifetime shorter than that of the observed universe. To ensure a metastable physical vacuum with a sufficiently long lifetime, following Ref. [29] 2 we impose the rule of thumb condition with β = −0.5, α = 3.4, γ = 60 and r ≡ m 2 u 3 /m 2 Q 3 . We see that stability of the physical vacuum roughly requires |A t | mt 2 . Because X t ≡ (A t − µ/ tan β), the precise impact of Eq. (15) on the stop sector will depend on the choice of µ and tan β. However, even for fairly large values of µ, as long as tan β is larger than a few, X t ∼ A t and the impact of the exact choice of µ and tan β is minimal. For concreteness, when imposing the vacuum stability constraint, Eq. (15), on the stop parameters, we fix µ = 3 mt 1 ∼ 3 m χ , a value large enough that the tree-level direct detection cross-section will be small, see Eq. (1). 3 In Fig. 1, the purple region denotes where the vacuum becomes unstable according to this prescription. The slight asymmetry between positive and negative values of sin θt is because of the difference between X t and A t due to our choice of µ/ tan β.
In the absence of Higgs mass information, the above constraints are responsible for limiting the maximal value of X t allowed. Because X t enters the Higgs boson couplings to the stops, this affects both cosmology and direct detection. As we will now see, the Higgs mass constraint gives comparable, and often stronger bounds on X t .

B. Higgs Mass
At tree level, the MSSM predicts a Higgs boson with mass below M Z . However, loop effects give important contributions [30][31][32]. In our analysis, we employ the Higgs mass calculation as implemented in FeynHiggs-2.13.0 [33][34][35][36][37][38]. In the stop sector, we scan over m Q 3 , m u 3 and A t . For concreteness we fix M 3 = ±1.5 TeV, X b = 0, and all other soft masses and trilinear couplings to 5 TeV. To capture potential uncertainties due to even higher order corrections and more importantly uncertainties in inputs (such as the top quark mass or the Gluino mass, which can change the Higgs mass up to 4 GeV when the Gluino mass is varied from 1 to 3 TeV [39,40]) we allow a Higgs boson mass in a window from 122 to 128 2 We have verified with the authors of this reference that this approximation remains roughly applicable for the larger values of µ which we are interested in here. 3 In discussing direct detection in Sec. IV, however, we explicitly set bino-Higgsino mixing to zero (as would be appropriate for very large µ), focusing instead on the contributions arising from the stops/sbottom. GeV. The sign of (A t M 3 ) has an asymmetric effect on the Higgs mass primarily due to the running of the top Yukawa (see Ref. [41] and references therein). However, this sign does not otherwise affect the dark matter phenomenology. Hence, we check both signs of the gluino mass, to see whether either realizes a valid Higgs mass.
Consistency with the observed Higgs boson mass in the presence of a relatively light t 1 < ∼ TeV requires a relatively heavyt 2 . Unless one wishes to allow for an extreme hierarchy between the stop masses, an appreciable mixing, |X t | ∼ mt 2 is also preferred [21,24,42,43].
In Fig. 1, the green swath denotes the region with an observationally consistent Higgs boson mass. The two bands that comprise the swath correspond to the two signs of M 3 . Fig. 1, we indicate where the sbottom mass is degenerate with mt 1 by the solid black line. The two dashed (dotted) contours show where the sbottom is 5 GeV lighter (heavier) than mt 1 . The central region enclosed by the thick black lines is then where the sbottom is lighter than the lightest stop, and hence would be the next to lightest supersymmetric particle (NLSP). Close to these lines, we expect the sbottom and stop to act as co-NLSPs. This information will be relevant when we discuss the cosmology associated with this scenario.

Also in
To summarize, the region compatible with the Higgs boson mass is mostly unaffected by the indirect constraints of the previous section. However, the largest values of X t consistent with the Higgs mass can be in tension with the stability of the vacuum.

C. LHC
Owing to its small production cross section, pure bino dark matter is difficult to probe at the LHC. The colored, nearly degenerate co-annihilation partner, on the other hand, possesses a relatively large production cross section. While small mass splittings between the states degrade the classic missing energy signature, a variety of searches targeting this region now exist. We review a few of the most relevant searches here. Other particles in the SUSY sector are also potentially accessible, i.e.t 2 andb 1 . Since they will typically have a larger splitting with the lightest supersymmetric particle (LSP), the sensitivity to these states can be enhanced. Hence, in some cases it is possible that their limits can be important, despite the smaller production cross section.
In the co-annihilation scenario the mass splitting between thet 1 and the dark matter is always less than m t , so the two-body decay of the stop (t 1 → χ 0 t) is kinematically forbidden.
Indeed, for all but the heaviest stops, the splitting is much less than M W , which also forbids the three-body decay (t 1 → χ 0 W + b), leaving only the four-body decay (t → bχ 0 ff ), or the flavor violating two-body decay (t → cχ 0 ). The relative importance of these last two decays depends on the flavor properties of the SUSY breaking sector [44,45]. Recent searches by the CMS collaboration have focused on this compressed region, and have ruled out stop masses up to roughly 500 GeV [46]. The exact mass excluded depends on the precise value of the mass splitting and the relative branching ratio of these two channels. Monojet searches, such as those undertaken at ATLAS [47] can also be relevant, but typically are not as constraining.
For mb > mt 1 the sbottom decay pattern depends strongly on the mass splitting. If kinematically accessible and if there is any left-handed component in the lightest stop, (b →tW ) dominates, which is challenging to observe. However, a dedicated search may be possible. If the splitting is less than M W , (b → bχ) robustly excludes mb < 625 GeV [48]. A phenomenological projection [49] suggests that a combined search for all the different decay modes of the sbottom could ultimately probe mb 900 GeV with 300 fb −1 of data.
Finally, the decay width of the lightest stop is suppressed if the mass splitting is small, and for ∆m 10 GeV thet 1 has a proper decay length exceeding a meter [50]. Current limits on such long-lived particles exclude stable stops with mt 1 < 1020 GeV [51]. As we shall see, a co-annihilation cosmology does not motivate such a small mass splitting for stops this light (see next section). Even with 300 fb −1 it will be challenging for these searches to probe the relevant region, where the stop mass exceeds 1.5 TeV.
At present, stops with modest mass splittings from the neutralino and mass roughly greater than 500 GeV are allowed, so in what follows we will focus on this region. The LHC will continue to push these bounds upwards as additional data are taken.

III. RELIC DENSITY
In the model studied here, co-annihilation of χ witht 1 andb 1 [6], as well as the annihilation of pairs oft 1 and/orb 1 are relevant for the precise determination of Ωh 2 = 0.11805 ± 0.0031 [52]. For some spectra, even processes involvingt 2 can be relevant. Contributions from processes with particles other than χ in the initial state must be appropriately weighted by their thermal abundance. The impact of these additional processes is accounted for via an effective annihilation cross section [5] where n eq i = g i [m i T /(2π)] 3/2 e −m i /T ; m i is the mass of the particle i, and g i counts the number of internal degrees of freedom.
Since the abundance of a heavier state i is suppressed relative to that of the LSP by Thus, co-annihilations impose a lower limit on ∆m as a function of m χ , since co-annihilations will be too effective if the mass splitting is too small. 4 A complication arises from the Sommerfeld effect, which is known to have an impact on the annihilation rate of charged non-relativistic particles [54][55][56][57]. While the Sommerfeld effect does not directly affect the annihilation of binos, it will have an impact on the annihilation rates oftt * andtt (and potentially their analogs including the sbottom). These corrections can modify the relic density significantly [10,12,[58][59][60]. Qualitatively, the Sommerfeld effect can be understood as an enhancement (or suppression) of the leading order cross section due to the presence of an attractive (repulsive) potential between the initial state particles generated by the exchange of light force carriers. In the context of color charged particles such a potential is generated by gluon exchange and can be approximated by the leading term of the static QCD potential [61], [12]: where C R and C R are the quadratic Casimirs of the color representation of the incoming particles while C Q is the quadratic Casimir of the final state. In this case the long and the short range contributions to the annihilation process factorize. The Sommerfeld corrected s-wave cross section is given by σ S ≡ S 0 σ 0 where σ 0 is the short distance contribution, and S 0 is an enhancement (suppression) factor given by Here corresponds to the velocity of the particles in the initial state. The appropriate Sommerfeld factors for higher partial waves S l can be constructed using a recursion relation starting from S 0 [62]. For a detailed description of our treatment of the Sommerfeld effect see Ref. [10].
We have implemented the Sommerfeld effect in MicrOmegas 3.3 [18] which we use to solve the relevant Boltzman equations numerically 5 .
NLO calculations of the relic density have made significant progress in recent years and indicate that perturbative corrections can have a relevant impact on the annihilation rate [64][65][66][67][68][69]. For stop coannihilation these corrections were found to change the relic density by ≈ 20% [67]. As expected the impact is even larger if Sommerfeld corrections are also included [68]. This is certainly larger than the observational uncertainty on the relic density and should be included in detailed studies of the MSSM. In the scenario at hand this correction can be absorbed via a shift of ∆m which does not have an appreciable effect of the qualitative features of the phenomenology.
It has recently been emphasized that bound state formation may have an impact on dark matter freeze-out [70][71][72][73][74][75][76]. For squarks the non-Abelian structure of QCD leads to a partial accidental cancellation in the matrix element for bound state formation [75] which somewhat diminishes its importance. Once all electroweak annihilation channels are included in the freeze-out calculation, the impact of bound states was found to be at the 10% level for all viable neutralino masses [76]. We do not consider their effects here, but note that while the formation of bound states would allow for somewhat heavier dark matter consistent with the relic density constraint, the qualitative picture would remain the same.
In Fig. 2 we show the mass difference ∆m between χ and the NLSP where co-annihilations provide a viable mechanism for reproducing the dark matter density. We have shown separately the regions where the left-handed sbottom is the NLSP and where the lightest stop is the NLSP. To give guidance for the importance of various processes in the early universe we have shown two additional curves. The first is the ∆m curve (solid black line) if only the process (ff * → gg) were active. The true ∆m is always larger, showing the importance of other annihilation channels. Another curve, labeledt R (dot-dashed black line), shows the mass splitting if the NLSP were a single pure right-handed stop. In this case, there are no X t enhanced couplings to the Higgs boson, but the pure right-handed stop has the largest 5 It has recently been emphasized that the color structure of the final state for theff * → gg process forces a separate treatment for the odd and even angular momentum states [63]. This leads to a different Sommerfeld factor for the p-wave compared to our implementation. We have checked the numerical impact of this correction and find that it changes the predicted relic density by less than 1% throughout the parameter space considered in this study. coupling to the bino [cf. Eqs. (5)- (10)]. In this case, the processes χχ → tt, χt R → tg/h, t Rt * R → tt/gg/hh are relevant [10]. The precise value of the mass splitting depends strongly on the mixing in the stop sector.
This mixing controls both interactions with the Higgs boson and the bino, see Eqs. (5)-(9).
The variation in these couplings due to different mixings explains the width of the band of consistent ∆m at each given value of the dark matter mass.
In Fig. 2, the allowed mass splitting for stop NLSPs broadens for higher dark matter masses. Near m χ ∼ 500 GeV, it populates a band from ∼ 35 − 60 GeV, but in the multi-TeV range it can range from near degeneracy to mass splittings approaching 100 GeV. The largest mass splittings require contributions from channels such as (t 1t left panel results from the need for an X t (A t ) so large as to make the vacuum unstable. The smallest ∆m obtained are somewhat smaller than the values consistent with a purely right handed stop NLSP. This is because perturbing away from the puret R case, an important effect is that thet L admixture has a much smaller hypercharge and hence the coupling to χ is reduced, see Eq. (8). This suppresses diagrams that rely on this coupling, necessitating a smaller ∆m to maintain the thermal relic abundance.
In contrast, sbottom NLSPs never have ∆m exceeding 80 GeV, irrespective of the vacuum stability constraint. This is because for the sbottom to be the NLSP, X t cannot be too large (lest thet 1 mass be driven down by level repulsion). In the sbottom NLSP case, the lightest stop will always be nearby, and for some parameter choices the heavier stop is also allowed to be degenerate. In fact the largest mass splitting obtained for the sbottom NLSP is precisely where both the stops are nearly degenerate with the sbottom, and dominant annihilation channels may includet 1,2 andb 1 ; for example (ff * → gg) or (t 2 /t 1b1 → W g). As the heavier stop decouples from the cosmology, the required mass splitting decreases and the smallest ∆m is obtained when onlyt 1 andb 1 contribute to co-annihilations.
The right panel of Fig. 2 shows the region that is carved out demanding consistency with the Higgs mass. Note that in particular the width of the sbottom NLSP region is significantly reduced. This can be understood by looking at the region in Fig. 1 where the sbottom is

IV. DIRECT DETECTION
Having determined the regions of parameter space in which thermal freeze-out can account for the observed relic density, we now turn to direct searches for dark matter. The bino dark matter candidate considered here has a vanishing direct detection cross section at tree level. A process that can generate a direct detection cross section arises from the loopinduced coupling of the dark matter with the Higgs boson. Such an effective χχh coupling is generated by triangle diagrams with tops and stops in the loop, see Figs. 3a and 3b. This effective coupling has been calculated in Ref. [77] and re-derived by us using the low energy Higgs theorem [22]. 6 In addition to the Higgs triangle-diagrams there are also box-diagrams witht 1,2 /t andb 1 /b in the loop which induce an effective coupling between the bino and the gluon content in the nucleus (see Fig. 3c as an example diagram). The full loop result for this contribution is available in the literature [79] and is already included in the MSSM implementation of MicrOmegas.
We show the direct detection cross section as a function of the neutralino mass in Fig. 4 considering different constraints. First, in red, we show all points in our scan, which include X t that would violate the vacuum stability constraint by as much as 50%. Next, in blue, we show points consistent with the observed Higgs signal strength (gluon fusion rate), EWPO (δρ), and a metastable vacuum with sufficiently long lifetime. Because all points shown have 6 Loops with thet will also contribute to the Higgs coupling to gluons. This effect is always subleading and we do not include it in the following qualitative discussion. Our numerical calculations take the effect into account. For a discussion of the full SUSY-QCD corrections to direct detection see for instance [78] •    m χ > 500 GeV, they are consistent with current stop searches [46]. In addition, we ensure that the points are consistent with the recent sbottom searches [48], which eliminates some points with mass between 500 and 625 GeV where the sbottom was nearly degenerate with the LSP. Finally, in green, we show points that also yield a Higgs mass 122 GeV < m h < 128 GeV. Once the Higgs mass is imposed, all points lie well below future experimental sensitivity, and indeed below the neutrino floor. We can understand the behavior of the direct detection in some detail by examining the relevant amplitudes.
At the nucleon level the direct detection cross section is given by: where n specifies the nucleon (i.e. the neutron or the proton) and A is the amplitude which can be separated into a Higgs exchange contribution and the gluon box contribution We discuss each of these contributions in turn, attempting to understand how large an amplitude can be realized for each.
The gluon box diagram is suppressed by the relevant fermion mass and hence can be enhanced when the sbottom is the NLSP and the mass splitting is comparable to m b . In this case, A g is well approximated by of 2 larger in magnitude than the s t > 0 case, but with a negative sign. This negative sign yields destructive interference with the subdominant gluon contribution. We now turn to the case of small, but finite mixing, (s t ∼ 1 or 0). In this case, a consistent Higgs mass is obtained when mt 1 mt 2 and |X t | ∼ mt 2 . For a predominantly right-handed light stop, we find A h tt ∼ m t /mt 1 3 − X t /mt 2 1 + 2X t /mt 2 + X t mt 1 /m 2 t 2 −1 + 2X t /mt 2 ∼ 0.35 for mt 1 = 2 TeV, mt 2 = 3 TeV and positive s t (X t < 0). As can be seen from Eqs. (24)- (27), most of the contributions to the direct detection cross section for a purely left-handed stop, close to c t ∼ 1, are suppressed by factors of 1/16 compared to the right-handed approximation owing to the smaller hypercharge of the left-handed multiplet. However, there is an unsuppressed contribution from A h t 1,2 , and as long as the mixing angle is not too small, it tends to dominate. It leads to an approximate amplitude A h tt ∼ −m t X t /mt 1 mt 2 ∼ 0.09 for negative X t and mt 1 = 2 TeV. Given these considerations, and demanding consistency with the Higgs mass, it is clear that unless the mass splitting between χ andb L is ∼ m b , which happens for m χ 2.5 TeV, the direct detection cross section for a predominantly lefthanded stop is much smaller than a right-handed stop. Due to the destructive interference of the stop contributions with the gluon contribution for s t < 0, the maximal cross section, in the absence of very small mass-splitting with the sbottom, consistent with the Higgs mass is obtained for a dominantly right-handed light stop with s t > 0. This maximal value can be roughly approximated by σ SI ∼ 10 −49 1 + 3t 1 /2t 2 2 2 TeV/t 1 2 cm 2 , a value well below the neutrino floor.
Finally, a comment regarding the direct detection prospects for a mixedb co-annihilation scenario is in order. We have set X b identically zero throughout, in part for simplicity, and in part because a mixing in the sbottom sector is not motivated by the Higgs boson mass.
Relaxing this assumption would allow the lightest sbottom to be an admixture of the right and left-handed states. In cases where the sbottom is the co-NLSP, this enhances the direct detection cross section with respect to Eq. (28) due to the larger hypercharge of theb R . To get a feel for the maximum size of this effect, we consider the limiting case of a pureb R NLSP. Theb L and the entire stop sector can be substantially heavier than the NLSP. In this case the thermal expectation for ∆m found in Ref. [76] is rather similar to the expectation for (ff * → gg) shown in Fig. 2. Note that this is substantially smaller than the expectation for a purely left-handed sbottom NLSP and hence will lead to significant enhancement of the gluon box contribution, Eq. (22). The corresponding direct detection cross section has recently been studied in Ref. [16], and for a thermal mass splitting, as read from our Fig. 2, their results indicate an expected scattering rate above the neutrino floor throughout the allowed parameter space. However, LZ is only expected to be sensitive to the very smallest mass splittings, corresponding to 1300 GeV < ∼ m χ < ∼ 1700 GeV.

V. CONCLUSION
We have revisited the stop co-annihilation scenario wherein dark matter freeze-out, stopinduced corrections to the Higgs boson mass as well as an irreducible direct detection cross section mediated by stop/sbottom loops are interrelated. After accounting for collider limits on light stops and sbottoms we find that the constraints on the stop sector from the stability of the vaccuum are more stringent than EWPOs and the Higgs signal strength measurements.
The region of stop parameters consistent with the Higgs mass is generally unconstrained by any of the above listed indirect constraints, however there may be slight tension between the largest values of X t consistent with the Higgs mass and the stability of the vacuum.
We analyze the irreducible, stop/sbottom-loop induced direct detection cross section in detail taking into account the mass splittings motivated by cosmology. Unfortunately, we find that this minimal direct detection rate generally falls well below the neutrino floor, particularly once the Higgs mass is taken into account. It should be kept in mind, however, that even a small amount of bino-Higgsino mixing can induce substantial spin-independent neutralino-nucleus scattering at tree-level. In this case the rate could be large, even going up to the current limit, but the direct connection between the cosmological origin of dark matter in the early Universe and direct detection rates would be lost.
In light of the potentially challenging direct detection situation, collider searches may be the best available probes for this scenario. The mass splitting between thet (orb) and the neutralino is small in the co-annihilation region, and naively, absent a detailed model of supersymmetry breaking it is not clear why this should be so. However, given that it produces the proper thermal dark matter relic density consistent with increasingly strong direct detection bounds, this mass spectrum is extremely well motivated. Therefore the region in the ∆m vs. m χ plane shown in Fig. 2 represents an important target. Dedicated LHC searches for such compressed spectra will be crucial in testing the co-annihilation scenario.