Probing Bino-Gluino Coannihilation at the LHC

It has been widely known that bino-like dark matter in the supersymmetric (SUSY) theories in general suffers from over-production. The situation can be drastically improved if gluinos have a mass slightly heavier than the bino dark matter as they reduce the dark matter abundance through coannihilation. In this work, we consider such a bino-gluino coannihilation scenario in high-scale SUSY models, which can be actually realized when the squark-mass scale is less than 100-1000 TeV. We study the prospects for exploring this bino-gluino coannihilation scenario at the LHC. We show that the searches for long-lived colored particles with displaced vertices offer a strong tool to test this scenario in collider experiments.


Introduction
restricted by the direct search at the LHC as MW > 270 GeV [33]. The wino DM scenario is also being constrained by the indirect DM searches using gamma rays [34,35]. These experiments, as well as the DM direct detection experiments [36], can probe this scenario in future. Higgsino DM with a mass of ∼ 1 TeV can also account for the observed DM density [37]. For the recent study of the phenomenology and future prospects for this Higgsino DM scenario, see Ref. [38] and references therein.
The last possibility is bino DM. If the scalar SUSY particles and Higgsino are significantly heavy, bino DM is usually over-produced as the interactions of bino with the SM sector tend to be suppressed. To avoid the over-production and get correct dark matter abundance, we need some exceptional mechanism to reduce the bino abundance, such as coannihilation and Higgs funnel [39]. If the Higgsino mass is heavier than O(10) TeV, the remaining possibility is the coannihilation. In this case, its thermal relic agrees to the observed value if there exist some particles degenerate with the bino DM in mass. In fact, as shown in Refs. [40][41][42][43][44][45], bino DM can explain the correct DM density if wino or gluino has a mass slightly above the bino mass. After all, there are various options for DM candidates in the high-scale SUSY scenario, and therefore it is quite important to experimentally examine each possibility.
Among the possibilities mentioned above, the collider testability of the bino-gluino coannihilation is expected to be the most promising since this case requires light gluinos. As we shall see below, we expect an O(1) TeV gluino mass in this case, which can be within the reach of the LHC. This could be compared to other DM scenarios in high-scale SUSY models; for instance, if the gaugino masses follow the spectrum predicted by the anomaly mediation, wino is the LSP and it becomes the main component of DM if it has a mass of 3 TeV, as mentioned above. In this case, the gluino mass is predicted to be O(10) TeV, which is of course far above the possible reach of the LHC. In this sense, it could be much easier to look for gluinos in the bino-gluino scenario than other cases. This naive expectation, however, turns out to be questionable. The bino-gluino coannihilation scenario requires that the mass difference between bino and gluino, ∆M , be ∆M 100 GeV. Such small mass difference results in soft jet emissions, which make it extremely challenging to detect the signal of gluino production. For this reason, previous studies have concluded that it is difficult to probe this bino-gluino coannihilation scenario at the LHC if the DM mass is heavier than 1 TeV [44,46,47].
In this work, we show that this small mass difference actually helps us to probe the bino-gluino coannihilation. When ∆M 100 GeV and the sfermion masses are much heavier than the gaugino masses, the lifetime of gluinos τg can be long enough to distinguish its decay signal from that of prompt decay. As will be shown below, we expect its decay length to be cτg O(0.1 − 10) cm when the sfermion masses are O(100) TeV. A decay length of this order is in fact the main target of searches for long-lived colored particles with displaced vertices (DVs) [48]. We will find that this search technique indeed gives a stringent limit on the bino-gluino coannihilation region, and probe wide range of the parameter space in future experiments. This paper is organized as follows. In the next section, we consider the bino-gluino coannihilation scenario and show the parameter region which accomplishes the correct DM density. The lifetime of gluino predicted in this parameter region is given in Sec. 3. Then, in Sec. 4, we discuss the strategy of the long-lived gluino searches at the LHC, and present the current constraint and future prospects for the bino-gluino coannihilation scenario. Finally, Sec. 5 is devoted to conclusion and discussion.

Bino-gluino coannihilation
To begin with, let us discuss the bino-gluino coannihilation scenario [41][42][43][44][45] to clarify the target parameter space we consider in the following analysis. Throughout this paper, bino is assumed to be the LSP and be the DM in the Universe. We consider the case where the bino-gluino coannihilation is effective so that the thermal relic abundance of the bino LSP is consistent with the observed DM density Ω DM h 2 = 0.12. Thus, bino and gluino should be degenerate in mass, i.e., ∆M ≡ Mg − MB 100 GeV, with Mg and MB being the gluino and bino masses, respectively. We further assume that the typical mass of scalar SUSY particles, m, as well as the Higgsino mass MH, is as high as the gravitino mass m 3/2 . This setup is realized with a generic Kähler potential. The gaugino masses are supposed to be suppressed by a loop factor compared with m 3/2 , which occurs when the SUSY breaking superfields are non-singlet. Namely, we require MB ∼ Mg m ∼ MH ∼ m 3/2 in what follows. Moreover, we assume the wino is heavy enough not to contribute to the coannihilation process. It turns out that such a mass spectrum can be in fact realized in the high-scale SUSY models [27,44]. We will see below that the scalar mass scale m gives the significant effects on the determination of the bino DM abundance. 1 The relevant annihilation processes to the computation of the thermal relic abundance are the self-annihilation and coannihilation of bino and gluinos. Among them, gluino self-annihilation is the most effective because of the strong interaction, and this plays the dominant role in the determination of the bino relic abundance. The bino self-annihilation and bino-gluino annihilation are much smaller than the gluino self-annihilation, since these cross sections are suppressed by heavy Higgsino and sfermion masses. Hence, these annihilation processes scarcely affect the following calculation.
An important caveat here is that the bino-gluino coannihilation does not work efficiently without chemical equilibrium between bino and gluinos [45,49]. Therefore we should require that the transition rate between them should be fast enough compared to the Hubble expansion rate. The transition rate is, however, again suppressed by heavy squark masses. Thus, we obtain an upper bound on m by imposing the above condition. The transition rate of bino into gluino via quark scattering, Γ( Bq → gq), is estimated by the product of the corresponding scattering cross section, σ( Bq → gq), and the number density of initial state quarks, n q . The former is approximately given by σ( Bq → gq) ∼ T 2 / m 4 with T being the temperature of the Universe, while the latter is n q ∼ T 3 since quarks are relativistic when the transition process is active. Consequently, the transition rate is given by On the other hand, the Hubble rate H goes like H ∼ T 2 /M Pl with M Pl the Planck scale in the radiation dominated epoch. In order to sufficiently reduce the bino density through coannihilation, the condition Γ( Bq → gq) H should be satisfied until the bino DM decouples from thermal bath at the freeze-out temperature T f ∼ MB/20. This reads which then gives an upper bound on the scalar mass scale m.
TeV . ( We find that when the DM mass is O(1) TeV the upper bound on the scalar mass scale lies around O(10 (2−3) ) TeV; indeed, many high-scale SUSY models [15][16][17][18][19][20][21][22][23] predict the SUSY breaking scale to be this order, with which the 125 GeV Higgs mass is naturally accounted for. Therefore, it is quite important to take into account the constraint on m when we discuss the bino-gluino annihilation in the high-scale SUSY scenario. To make the above discussion more accurately, we perform the numerical computation by solving the Boltzmann equation to obtain the bino-gluino conversion rate and the resultant relic abundance. All of the squark masses are assumed to be equal to the universal mass m. When we evaluate the transition cross sections and (inverse) decay rate of gluino and bino, we use the effective theoretical approach to properly deal with sizable quantum corrections resulting from large difference between the gluino and squark mass scales; we first integrate out squarks to obtain a set of dimension-six operators which involve quarks, bino and gluino, and then evolve these operators down to the gluino mass scale by using the renormalization group equations, which results in a several tens percent enhancement of the transition rate, compared to the tree level calculation [50][51][52]. The loop-induced dimension-five dipole operator (gluon-bino-gluino) is found to be quite suppressed and thus its contribution is negligible in the present analysis. In addition, we include the so-called Sommerfeld effects [53] on the gluino annihilation. On top of that, p-wave contribution, finitetemperature effects, the scale dependence of the strong coupling constant in the QCD potential [43], and the bound-state effects on a pair of gluinos [45] may change the results by a factor of O(10)%. The above figure shows that the conversion rate decreases as m or ∆M is taken to be larger. In particular, if the squark mass scale m is several hundred of TeV with the DM mass being a relatively small, then the condition ΓB →g H does not hold any more when the DM abundance freezes out. In Fig. 2, we plot on the MB − m plane the mass difference ∆M with which the thermal relic abundance of bino DM explains the observed DM density Ω DM h 2 = 0.12. In the red shaded region, the squark mass is too heavy for the coannihilation process to work well and therefore the DM is overproduced. We will discuss how to probe the parameter space shown in Fig. 2 at the LHC in the subsequent section.

Gluino lifetime
Next, we study the lifetime of gluino, which plays a crucial role in the discussion of the testability of the bino-gluino coannihilation scenario at the LHC in the following section. As mentioned in the Introduction, in this scenario, a relatively light gluino mass is expected. Thus, the gluino pair production is suitable target for the hadron collider experiments like the LHC in this case. After the pair production, a gluino decays into a bino, a quark, and an anti-quark through the squark-exchange processes [50][51][52]54]. When the gluino is degenerate with the bino in mass, which is required in the bino-gluino coannihilation scenario, the decay length of the gluino, cτg, is approximately given as From this equation, we see that the decay length gets longer as the mass difference ∆M is taken to be smaller or the scalar mass scale m is set to be larger. Therefore, we expect a relatively long decay length when the bino-gluino coannihilation is achieved in the high-scale SUSY scenario.
To illustrate the gluino decay length corresponding to the bino-gluino coannihilation region, in Fig. 3, we plot contours of the gluino decay length in colored lines with the squark masses set to be m = 100 TeV, which we denote by cτ 100 TeṼ g , on the MB-∆M plane. We also show the mass difference ∆M with which the thermal relic of the bino DM agrees to Ω DM h 2 = 0.12; the black solid line shows the case where the bino-gluino chemical equilibrium is assumed, while the other black lines represent the cases of m = 100, 300 and 500 TeV. To avoid overproduction, ∆M should be below these lines. From Fig. 3, we find that the gluino decay length is scarcely dependent on the bino mass, which has been already shown in Eq. (4) implicitly. We have cτg > O(1) cm where the thermal relic abundance of the bino DM explains the observed DM density. This is a crucial observation for the strategy of exploring the bino-gluino coannihilation region at the LHC.

LHC search
If gluino decays promptly and the bino and gluino masses are almost degenerate, it is quite hard to search for the gluino at the LHC, since the small mass difference makes the missing energy and jet activities tiny. Currently the ATLAS and CMS collaborations have put limits on such a degenerate neutralino, i.e., bino in our case, with a mass of around 600 GeV [6,7]. The bounds are expected to reach ∼ 1200 GeV with the integrated luminosity of 300 fb −1 at the 14 TeV LHC [55].
These limits are in fact drastically improved once we consider the fact that in the case of the bino-gluino coannihilation scenario, the gluino lifetime is as long as cτg > O(1) cm, as we have seen in the previous section. Such a gluino has a distinct property in the collider experiments; a gluino with a decay length of cτg > O(1) cm leaves a visible DV in the detectors, which greatly helps the gluino search. At present, however, there have been no dedicated searches from this aspect so far. 2 The ATLAS collaboration has searched for DVs in the region of |z| < 30 cm and r < 30 cm in the inner detector [48], where z-axis points along the LHC beam line and r denotes the radial coordinate in the plane perpendicular to the z-axis. They use the DVs reconstructed only in the air-gap region, namely, discard the DVs reconstructed within the material layers. This leads to significant background reduction. The signal region for the DVs is defined such that the number of  Figure 4: Current constraints (red and solid lines) and future prospects (blue and dashed lines) for the gluino searches. Favored region for the DM relic abundance is also shown in black lines for the cases of m = 50, 100, 200, and 300 TeV, with ∆M chosen so that the thermal relic abundance of the bino equals to the current observed DM density. We also show the current constraint [6] and future prospect of the 14 TeV LHC run [55] from the search for the prompt-decay gluino in horizontal red solid and blue dashed lines, respectively. tracks associated with the DV is larger than four and m DV > 10 GeV, where m DV is the invariant mass of the tracks evaluated with the charged-pion mass hypothesis. Since they have observed no event in the signal region, they have given an upper limit on the long-lived gluino production cross section, which is interpreted as bound on the gluino mass in the high-scale SUSY scenario with a fixed neutralino mass of 100 GeV [48].
We re-interpret this low m DV search result in the case of the degenerate bino-gluino system, and obtain constraints on the bino-gluino coannihilation scenario, which is shown in Fig. 4. Here, the red and blue bands (from cτg = 1 mm to 1 m) show the estimated sensitivities of the DV search with the total luminosity of 20 fb −1 at the 8 TeV running and with 300 fb −1 at 14 TeV, respectively. The upper lines of these bands are for the cases where only the trigger efficiency is taken into account, which are simulated with HERWIG 6 [57] and AcerDET [58] to be 40% for 8 TeV with the threshold of the missing energy of 100 GeV, and 15% for 14 TeV with the missing energy trigger of 200 GeV. Their dependence on the mass of gluino is only a few percent level. The lower lines, on the other hand, correspond to the reconstruction efficiency for DVs that is estimated from Refs. [59,60], where the long-lived neutralino decaying to two quarks and one muon is discussed in the R-parity violating SUSY scenario. The reconstruction efficiency for the 108 GeV neutralino is about 20% of that for the 494 GeV neutralino in this case; we use this 20% for the lower lines, which gives conservative limits rather than the previous ones. In Fig. 4, we also show the favored region in terms of the DM relic abundance in black lines for the cases of m = 50, 100, 200, and 300 TeV. Here, the bino-gluino mass difference ∆M is taken such that the thermal relic of bino DM explains the correct DM density. This reads that the present LHC data have already constrained a considerable range of parameter region consistent with the bino-gluino coannihilation scenario. This constraint is in fact much stronger than the ordinary limit from the searches of promptly decaying gluinos, which are based on only jets and missing energy [6,7]. This constraint is indicated by the red and solid horizontal line in this figure. The 14 TeV LHC running can further probe this scenario and reach Mg ∼ 2.5 TeV when cτg = O(1 − 10) cm; this sensitivity is better than that by search with only jets and missing energy [55] (shown in the horizontal blue dashed line in the above figure) by almost a factor of two.
In addition, the ATLAS collaboration searches for massive charged meta-stable particles, such as R-hadrons [61]. A characteristic feature of such particles is that they are produced with relatively low velocities, β ≡ v/c < 1. This signature can be seen by means of large energy loss, dE/dx, in the Pixel detector and the late timing signal detected with the muon chambers and calorimeter. Here, we note that this analysis requires gluinos to form charged R-hadrons. Although the estimation of the charged hadronization fraction of gluinos may suffer from large theoretical uncertainty, this search offers the best sensitivity for cτg > 1 m. In Ref. [62], the result of this search is given as limits on the gluino mass in the case of ∆M = 100 GeV. We use the trigger efficiency given there for our computation for the 8 TeV case, and estimate the efficiency for the 14 TeV case by re-scaling it with a factor obtained by simulations. The red and blue solid curves in Fig. 4 show the estimated sensitivities of this search with 20 fb −1 at 8 TeV and with 300 fb −1 at 14 TeV, respectively. We find that the the searches of heavy stable charged particles give the most stringent constraints when cτg > 1 m, and are complementary to the DV searches. In particular, they are of importance when the scalar mass scale is relatively higher, say, a few hundred TeV.

Conclusion and discussion
In this paper, we study the bino-gluino coannihilation in the high-scale SUSY scenario. We have found that the squark mass scale cannot be too large for the coannihilation to work well. The upper bound on the squark mass is 200-1000 TeV for the gluino mass 1-8 TeV. Actually this mass scale is coincident with the prediction of the spectrum often called the spread or mini-split SUSY [15][16][17][18][19][20][21][22][23]. This constraint will provide a new perspective on the model-building to realize such mass spectrum.
We also discuss the LHC signatures of this scenario. Because of the small mass difference between the bino LSP and gluino, which is necessary for coannihilation, as well as heavy squark masses, the gluino decay length is considerably prolonged. Despite the small jets and missing energy activity, the DV and R-hadron searches can efficiently probe such long-lived gluinos. If the squark mass scale is higher than about 100 TeV, the current lower bound on the gluino mass is around 1.2 TeV. The 13/14 TeV LHC run2 stage is expected to be able to explore gluinos with a mass of ∼ 2 TeV.
Let us speculate possible sensitivities for much higher energy machines. For gluinos with the decay length longer than O(1) mm, a mass of 4.5 (10) TeV can be probed using a √ s = 33 (100) TeV running proton collider with the integrated luminosity of 100 fb −1 , provided that the background is sufficiently small and the detection efficiency of gluinos is the same as that of the current LHC detector. This estimation may, of course, be too naive. Further detailed studies should be dedicated to see more precise prospects for such colliders, though we expect that they can probe the most of parameter space of the bino-gluino coannihilation.
Lastly, we discuss the possibility of other gaugino coannihilation scenarios. As in the case of the current study, the small mass difference and heavier sfermion scale easily make the next LSP live long. For instance, in the case of the wino and gluino coannihilation, we may observe very exotic signatures; if the gluino lifetime is long enough, the gluino can carry the charged wino to the LHC trackers. In this case, we may observe displaced and disappearing tracks of the charged wino. The large gluino production cross section and the long-lived nature of the charged wino make it rather easy to look for this scenario in the LHC experiments. Another very interesting and plausible possibility is wino-bino coannihilation. This spectrum can be relatively easily realized even in the minimal anomaly mediation model. In this case, we may have another long-lived particle, which may play an important role at the LHC searches. A detailed analysis for this scenario will be done elsewhere [63].