Scenarios for Gluino Coannihilation

We study supersymmetric scenarios in which the gluino is the next-to-lightest supersymmetric particle (NLSP), with a mass sufficiently close to that of the lightest supersymmetric particle (LSP) that gluino coannihilation becomes important. One of these scenarios is the MSSM with soft supersymmetry-breaking squark and slepton masses that are universal at an input GUT renormalization scale, but with non-universal gaugino masses. The other scenario is an extension of the MSSM to include vector-like supermultiplets. In both scenarios, we identify the regions of parameter space where gluino coannihilation is important, and discuss their relations to other regions of parameter space where other mechanisms bring the dark matter density into the range allowed by cosmology. In the case of the non-universal MSSM scenario, we find that the allowed range of parameter space is constrained by the requirement of electroweak symmetry breaking, the avoidance of a charged LSP and the measured mass of the Higgs boson, in particular, as well as the appearance of other dark matter (co)annihilation processes. Nevertheless, LSP masses $m_\chi \lesssim 8$~TeV with the correct dark matter density are quite possible. In the case of pure gravity mediation with additional vector-like supermultiplets, changes to the anomaly-mediated gluino mass and the threshold effects associated with these states can make the gluino almost degenerate with the LSP, and we find a similar upper bound.


Introduction
The absence of supersymmetry so far, at LHC Run I [1,2] and elsewhere, raises the question where, if anywhere, is it hiding. There are scenarios for which, at least, some supersymmetric particles were produced in LHC Run I, but have been overlooked. Examples include models where R-parity is violated [3], or the spectra are compressed [4,5]. Alternatively, sparticles might be too heavy to have been detected at LHC Run I, but might be within range of future LHC runs [6]. It is also possible that supersymmetric particles may lie beyond the reach of the LHC altogether, and require a future higher-energy pp collider for its detection.
If one assumes that R-parity is conserved, the lightest supersymmetric particle (LSP) must be stable, and hence make at least a contribution to the cosmological cold dark matter density [7]. The total density of cold dark matter is very tightly constrained by measurements of the cosmic microwave background radiation [8]. It is clear, therefore, that the parameters of generic models are constrained in very specific ways in order to realize the correct dark matter density [9,10,11,12]. Moreover, this parameter space with the correct density is likely to be found in a region of parameter space where the density varies rapidly with the parameters. In these cases, regions where the LSP contributes only a fraction of the cold dark matter density will have parameters similar to those regions yielding the correct total density. This sensitivity of the dark matter density to parameters are particularly relevant for models with compressed and/or very heavy spectra that have survived LHC searches. Examples of specific choices of heavy spectra that yield the correct cosmological dark matter density include scenarios in which the LSP, χ, would have annihilated with itself through a direct-channel boson such as the heavier neutral Higgs bosons A and H [13]. Alternatively, there might be one or more heavier supersymmetric particles that are nearly degenerate with the LSP, χ, and would have coannihilated with it in the early Universe [14]. There are several examples of possible coannihilating sparticles, including the lighter stau, or possibly some other slepton [15], the lighter stop squark [16,17,18], the lighter chargino [19,17] and the gluino [20,21,22,23,24,25,18,26].
In most cases, coannihilation with a sparticle having stronger interactions extends the allowed mass range of the LSP. The possibility of gluino coannihilation is therefore particularly interesting since it interacts strongly suggesting it can accommodate a heavier LSP than is possible from coannihilation with a stau or slepton. In fact, it has been shown that a dark matter density realized by an LSP coannihilating with the gluino could lie well beyond the reach of the LHC, with a mass as heavy as m χ 8 TeV [26].
The possibility of gluino coannihilation does not arise in the minimal supersymmetric extension of the Standard Model (MSSM) with the soft supersymmetry-breaking parameters constrained to be universal at the input GUT scale (the CMSSM) [13,27,28,29,9,30], nor in related models with non-universal Higgs masses [31,28,32,33,34]. However, as we discuss in this paper, gluino coannihilation can become important in variants of the MSSM with non-universal gaugino masses, and in variations of pure gravity mediation (PGM) with non-minimal matter content such as additional vector-like supermultiplets [22,23].
The layout of this paper is as follows. In Section 2 we set out the coupled set of Boltzmann equations that we use to calculate the relic LSP density, discussing the circumstances under which the analysis can be reduced to a single Boltzmann equation for a particular combination of sparticle abundances [26]. Then, in Section 3 we discuss various scenarios with non-universal gaugino masses in which gluino coannihilation can become important, delineating the corresponding strips in parameter space and comparing their extents with the results of [26]. We find that this scenario is constrained by the requirement of consistent electroweak symmetry breaking (EWSB), by the measurement of m H , and by avoidance of a stop or chargino LSP. We give examples showing that the correct dark matter density is possible with LSP masses as large as 8 TeV. Section 4 contains a similar analysis of PGM models with vector-like supermultiplets, focusing on an example with a single extra pair of 10 and 10 representations of SU (5). Because the anomaly-mediated contribution to the gluino mass is zero in this case, threshold effects due to these additional states generate almost the entire gluino mass. This suppresses the gluino mass relative to those of the other gauginos, leading to near-degeneracy between the gluino and the LSP. Neutralino dark matter candidates with similarly large values of m χ are again possible. Finally, Section 5 summarizes our conclusions and discusses the prospects for discovering supersymmetry in these gluino coannihilation scenarios.

Calculations of Gluino Coannihilation
In this section, we present general formulae for calculating the dark matter thermal relic density, and then specialize it to the case of the gluino coannihilation scenarios we consider in this paper, taking into account the effects of gluino-gluino bound states. ratios: n i /n 1 = n eq i /n eq 1 for (i = 1, ..., l). For any of the N species, the evolution of its number density is governed by the Boltzmann where σv ij→SM is the product of the thermally-averaged relative velocity with the total cross section for the channels of i and j (co)annihilating into Standard Model particles, and Γ i→j is the sum of all the thermally-averaged decay and conversion rates for decay and conversion processes for which there is one particle i in the initial state and one particle j in the final state, with all other particles involved in these processes being Standard Model particles. The relations between the thermally-averaged forward and backward reactions are used in (1), and we assume for all the Standard Model particles involved that n SM = n eq SM , so that, for example, Γ i→j n eq i = Γ j→i n eq j .
Written in terms of the yields, and g * s and g * are the total numbers of effectively massless degrees of freedom associated with the entropy density and the energy density, respectively.
Using Y i /Ỹ = Y eq i /Ỹ eq for i = 1, ..., l and summing over (3) for the first l species, we find while for each of the species k (l < k ≤ N), we get where We note that in the case N = l + 1, the final term in (7) does not appear. In the case N = l, (6) does not have the two terms in the squared bracket, and reverts to the familiar form for coannihilations when all the N species are sufficiently coupled to the LSP. We now specialize the above general formulae to the gluino coannihilation scenarios we consider in this paper. First of all, following the discussion in [26], the effect of gluinogluino bound states on the calculation of the dark matter relic density can be taken into account simply by modifying the Boltzmann equation by including the Sommerfeld-enhanced thermal-averaged velocity-weighted gluino pair annihilation cross section [24], which includes gluino-pair annihilation to two gluons and to all the quark anti-quark pair channels: where σv bsf , Γ R and Γ dis are the thermally-averaged formation cross section, decay rate and dissociation rate for the bound stateR, respectively. The details of these quantities and the derivation of eq. (11) can be found in Section 3, 5 and Appendix B of [26].
When the rate for interconverting the neutralino LSP and the gluino is sufficiently large, compared to the Hubble rate, so that to a good approximation the relation Yg(T )/Y 1 (T ) = Y eq g (T )/Y eq 1 (T ) holds at all temperatures during which the sum of Yg(T ) and Y 1 (T ) changes non-negligibly, we can use a single Boltzmann equation to solve for the dark matter relic abundance, including the gluino species inỸ and σ ef f v and using (6) without the two terms in the squared bracket. Otherwise, one should use a coupled set of Boltzmann equations, namely (6) and (7), to solve for the dark matter relic abundance. For the scenarios considered in this paper, any of the R-odd species apart from the gluino is either sufficiently coupled to the LSP by having a Standard Model particle in the propagator of a tree-level Feynman diagram describing its interconversion with the LSP, or is so heavy compared to the LSP that it is effectively not participating coannihilations. Therefore, when using a coupled set of Boltzmann equations, we have N = l + 1 in (6) and (7), and the index k is for the gluino.
We end this Section by emphasizing that, in principle, the coupled set of Boltzmann equations can always be used to solve for the dark matter relic density, whether the rate for interconverting the gluino and the LSP is sufficiently large compared to the Hubble rate or not. However, for the former case, solving a single Boltzmann equation is usually easier than solving the coupled ones and requires less computing time.

The Non-Universal MSSM Scenario
It was assumed in [26] that the squarks were all degenerate with a common mass mq, and the effects of sparticles with only electroweak interactions were neglected. It was found in [26] that in the presence of gluino coannihilation, a Bino LSP, χ, could be the dark matter of the universe if it weighed 8 TeV, the exact value depending on the ratio mq/m χ , with smaller values of m χ being found for mq/m χ 5 and 100. Here we make a more complete study in a variant of the MSSM with universal soft supersymmetry-breaking scalar masses m 0 and trilinear couplings A 0 , allowing a restricted form of non-universality in the gaugino sector with M 1 = M 2 = M 3 at the input GUT scale. The results therefore depend on M 1 /M 3 as well as the usual CMSSM parameters m 0 , A 0 and tan β (the ratio of MSSM Higgs vev's).
This is therefore a one-parameter extension of the CMSSM (with the new free parameter being M 3 ) as is the NUHM1 (with the soft Higgs masses m 1 = m 2 = m 0 ) [33,34]. We this case, the gluino is the LSP in the shaded region. Because of the scale of the plot, it is difficult to discern the relic density strip, which lies very close to the boundary of this region. However, we note that it lies to the left of the red shaded region only when M 3 is between ∼ 400 and ∼ 1200 GeV, as shown in the right panel of Fig 500 GeV for A 0 /m 0 = 1 and no consistent solutions across the plane at somewhat lower A 0 /m 0 . This is also the case for tan β = 10 and A 0 /m 0 = 1.5, for which m H ∼ 134 GeV. Larger values of A 0 /m 0 also give values that tend to increase m H and, if increased too much, the stop becomes the LSP and eventually tachyonic.
Next we consider some sample (M 1 , M 3 ) planes with m 0 = 20 TeV and tan β = 5, corresponding to the lower end of the m χ plateau found in [26]. Fig. 3 is for the case A 0 /m 0 = 1, where we see in the left panel that electroweak symmetry breaking is possible up to values of M 1 14 TeV. There is a gluino coannihilation strip close to the colored LSP boundary for M 1 9 TeV. This is terminated by a spur extending to large M 3 when 9 TeV M 1 10 TeV, where the lighter chargino is the LSP. There is no chargino coannihilation strip along the boundary of this region at large M 3 , because the relic density is too high: for these values of M 1 and M 3 , the Higgsino mass is too large and other coannihilations are not sufficient to bring the relic density down. At larger values, 10 TeV M 1 11 TeV, there is a Higgsino-gluino coannihilation strip, which is followed at larger M 1 by a focuspoint strip [36] hugging the electroweak symmetry breaking boundary where the neutralino is well-tempered [37]. In this case we see both the 124 and 125 GeV Higgs mass contours and, as in the previous example, m H is compatible with experiment whenever the dark matter density falls within the allowed range.
Because the relic density strip is a multi-valued function of M 3 , the structure of the gluinoneutralino mass difference ∆M (blue curve) and the neutralino mass (red curve) shown in the right panel of Fig. 3 are more complicated than in the previous cases. After growing to a local maximum ∼ 170 GeV when M 3 ∼ 1 TeV, ∆M starts to fall at larger M 3 . We then see a change in behaviour at M 3 ≈ 1800 GeV along the gluino coannihilation strip. Here, the neutralino becomes Higgsino-like and, as M 1 is increased, the coannihilation strip tends toward lower M 3 with an increasing mass difference, as seen in the lower branch of the blue curve. A Higgsino LSP emerges for larger M 1 because it gives a positive contribution to the up Higgs soft mass from renormalization group running. As the up Higgs soft mass goes to zero so does µ and the Higgsino becomes the LSP. Once µ is small enough, the Higgsino can be a thermal relic without any assistance in setting the relic density from other particles. In this focus-point-like region, the mass difference increases beyond the range displayed. This behaviour is correlated with the value of m χ (red curve), which increases monotonically to ∼ 4 TeV. When the bino/Higgsino transition occurs at M 3 ≈ 1800 GeV, m χ doubles back down to M 3 ≈ 500 GeV. Then, on the focus-point branch of the relic density strip, the LSP is mostly Higgsino, the value of M 3 grows, and the lightest neutralino mass takes the characteristic value m χ ∼ 1 TeV.
In Fig. 4  region has disappeared, due to the large A 0 driving the Higgs mass to large negative values.
The end-point of the gluino coannihilation strip is clearly seen in the right panel of Fig. 4, where ∆M → 0 (blue curve) at M 3 ≃ 3300 GeV. Qualitatively, this case is similar to that shown in Fig. 2, rather than to Fig. 3 with its truncated gluino coannihilation strip. In this case, the LSP mass (red curve) rises monotonically to m χ ∼ 7.5 TeV at the end-point of the strip. In the left panel of Fig. 5 we display the (M 1 , M 3 ) plane for m 0 = 20 TeV and A 0 /m 0 = 2. We see again a gluino coannihilation strip, but extending only to M 1 ∼ 7.5 TeV. It is terminated by a stop LSP region that extends to larger values of M 3 than those displayed. In principle, one might have expected to see a stop coannihilation strip running up along the boundary of the stop LSP region. However, in this case the relic density is too high along the boundary shown in this figure: as in the chargino case mentioned earlier, the would-be end-point of the stop coannihilation strip lies within the gluino LSP region. The value of m H is generally higher than in the previous case, though compatible with experiment along all the dark matter strip. In the right panel of Fig. 5, the curves for ∆M and m χ terminate when the stop becomes the LSP, with m χ 3.5 TeV.  [26]. Fig. 7 is for tan β = 10 and A 0 /m 0 = 1 and displays a truncated gluino coannihilation strip extending to M 1 ∼ 4 TeV, followed by a Higgsino coannihilation strip extending to M 1 ∼ 5 TeV, and then a focus-point strip extending beyond the limits of the plot. We find that m H is always compatible with the experimental measurement. This example resembles that of Fig. 3 for m 0 = 20 TeV and A 0 /m 0 = 1, the main difference being that the chargino spur has disappeared: we see instead a chargino LSP island at M 1 ∼ 6 TeV and M 3 2 TeV. As in Fig. 3, we see in the right panel that the gluino-neutralino mass difference (blue curve) has a multivalued form, and becomes larger as the neutralino becomes more Higgsino-like along the gluino coannihilation strip. At larger M 1 , M 3 drops along the Higgsino-gluino coannihilation strip and the mass difference increases slightly as M 3 decreases, increasing beyond the displayed range as one moves on to the focus point strip. Similarly, the neutralino mass rises as M 1 is increased, then falls back to about 1.1 TeV when the LSP is mostly a Higgsino.  Gluino coannihilation is also prominent for m 0 = 20 TeV and tan β = 5, but the focus point may also be important (it becomes dominant when tan β = 10), as is Higgsino coannihilation.

Pure Gravity Mediation with Vector Multiplets
Another possible way of realizing a spectrum that can lead to gluino coannihilation is in models with pure gravity mediation [38,39,40] of supersymmetry breaking with additional vector multiplets [22,23]. The model we consider here is based on that in [23], whose setup we briefly review here. The effective potential is similar to that for the CMSSM: where W (3) corresponds to the trilinear terms of the superpotential, W (2) contains the bilinear terms of the superpotential, and the φ i signify the MSSM fields, with The scalar masses are generated through gravity mediation with a minimal Kähler potential, in an identical manner to mSUGRA [41], and hence are equal to m 3/2 at the GUT scale. In general, as seen in (12), the form of W (3) dictates the pattern of the trilinear supersymmetrybreaking terms. However, the trilinear couplings are suppressed in PGM models, because the supersymmetry-breaking field is charged. Thus, we take A 0 = 0.
Since A 0 = 0 and the Kähler potential is minimal in this model, we have B 0 = −m 3/2 for the Higgs fields. This leaves three free parameters, two of which are determined by enforcing the electroweak symmetry-breaking conditions; and where ∆ (1) µ , ∆ µ and ∆ B are loop corrections to the relationships [42]. This scenario has a very restricted parameter space that does not, in general, realize electroweak symmetry breaking [40]. Therefore, we add a Giudice-Masiero (GM) term [43] for the Higgs fields, which modifies the GUT-scale values of both B 0 and µ: This additional degree of freedom in the EWSB sector allows us to choose tan β (in addition to m 3/2 ) as a free parameter, and one finds viable parameter space as long as tan β 3 [40].
We consider here an extension of this simplest viable version of the PGM scenario that includes an additional 10 and 10. Because these states are vector-like, the most general form of the Kähler potential will be K = |10| 2 + |10| 2 + C 10 (10 · 10) + h.c. , which includes a GM-like coupling C 10 that generates a supersymmetric mixing mass term, µ 10 , and a supersymmetry-breaking B term for the additional vector-like fields. Because of the minimal form of the kinetic terms in the limit C 10 → 0, the additional fields also have a gravity-mediated tree-level soft supersymmetry-breaking mass equal to m 3/2 . Since the 10 contains fields with the same quantum numbers as SM fields, the 10 (10) can be combined into gauge-invariant operators with H 2 (H 1 ). If we impose only gauge symmetries the most generic contribution to the superpotential is where Q ′ and U ′ are from the 10 andQ andŪ are from the 10. However, to preserve R symmetry we must take either y ′ b = 0 or y ′ t = 0. Here, we take y ′ b = 0. The interactions proportional to y ′ t contribute to the beta function of the up Higgs soft mass in a similar way to those controlled by y t . Specifying a comparable value of y ′ t helps drive radiative electroweak symmetry breaking, which in turn allows larger values of tan β > 3. The extended theory now has four parameters: m 3/2 , tan β, C 10 , and y ′ t . The gaugino masses in the models are generated by anomalies [44]. Thus because the contributions to gaugino masses are proportional to m 3/2 , scalar masses tend to be much heavier than the gaugino masses, reminiscent of split supersymmetry [45]. With the addition of the 10 and 10 the anomaly-mediated contributions to the gaugino masses are In addition, the gauginos then get rather large threshold corrections from the 10 and 10 when they are integrated out, which is in addition to the large threshold correction coming from integrating out the Higgsinos: for more details see [23]. Since the only contribution to the mass of the gluino comes from the threshold corrections, it tends to be lighter than in typical PGM models. Hence there are regions where the gluino can coannihilate with the bino, yielding the possibility of a relatively heavy dark matter candidate.
Our results for the PGM model with vector 10 and 10 multiplets can be displayed in (c H , m 3/2 ) planes for fixed values of the Yukawa coupling, y ′ t and tan β. Two examples of these planes are shown in Figs. 9 and 10. In the former, we have fixed tan β = 3 (mainly to get an acceptable value for the Higgs mass, m H , over the range of m 3/2 ≤ 600 TeV shown) and y ′2 t = 0.15. In the left panel, we see a large red shaded region at small c H where the gluino is the LSP. To the right of this boundary, we see the gluino coannihilation strip 2 . In the lower right corner, the pink shaded region is excluded because one or more of the new vector scalars becomes tachyonic. As in previous figures, the Higgs mass contours are shown as red dot-dashed curves as labelled. Within the theoretical uncertainties, the Higgs mass agrees with experiment over the part of the plane that is shown. In the right panel we see, as before, the gluino-neutralino mass difference ∆M along the gluino coannihilation strip (blue) and the neutralino mass along the strip (red). We see that the curve for ∆M has the same characteristic shape due to strong coannihilations involving the gluino and peaks at In Fig. 10, we show in the left panel corresponding (c H , m 3/2 ) plane for fixed y ′2 t = 0.65 and the same value of tan β = 3. In this case with a higher Yukawa coupling, slightly higher c H is needed to obtain a neutralino LSP. As in the previous case, we see a brick-red shaded gluino LSP for low c H and, at slightly larger c H , a gluino coannihilation strip. As previously, m H is acceptable along all the displayed portion of the strip where m 3/2 ≤ 400 TeV. In this case, the region at larger c H where one or more of the new vector scalars becomes tachyonic also extends to low c H for small m 3/2 . The right panel of Fig. 10 shows the values of ∆M and m χ along the gluino coannihilation strip. In this case, we see that ∆M is maximized at ∼ 160 GeV for m 3/2 ∼ 150 TeV. The end-point of the gluino coannihilation strip also occurs at m 3/2 ≃ 500 TeV with m χ around 8.3 TeV.
Finally, we show in Fig. 11 an example of a (y ′2 t , c H ) plane with tan β = 5 for four choices of the gravitino mass, namely m 3/2 = 30, 50, 100 and 250 TeV. The red shaded region has a gluino LSP only in the m 3/2 = 30 TeV case. In the other three cases this region would that the gluino-neutralino mass difference is almost independent of y ′2 t . We do not show the neutralino mass for these cases, as it is largely independent of y ′ t and is determined from the gravitino mass and can be read from either of the two previous figures.

Summary and Conclusions
We have presented in this paper a couple of representative MSSM scenarios in which the gluino may be nearly degenerate with the neutralino LSP χ, whose relic dark matter density is brought into the range favoured by Planck and other data by gluino coannihilation. It had been shown previously that values of m χ 8 TeV are in principle possible when gluino coannihilation is operative [26], and we have shown in this paper how such a possibility can be embedded within a scenario for non-universal soft supersymmetry breaking within the MSSM, on the one hand, and within a simple extension of the MSSM with pure gravitational mediation of soft supersymmetry breaking that includes a vector-like 10 + 10 multiplet pair, on the other hand.
In both scenarios, the upper bound on m χ depends on the details of the models. In 4 We recall also that the Higgs mass is sensitive to the choice of tan β. particular, in the non-universal MSSM scenario there is competition from other mechanisms for bringing the dark matter density into the Planck range. These, together with other phenomenological constraints such as the mass of the Higgs boson and the requirement to ensure electroweak symmetry breaking, restrict the parameter region where gluino coannihilation is dominant. In the PGM scenario with extra vector-like multiplets, the allowed range of m χ depends on the gravitino mass as well as a vector-like Yukawa coupling. In both cases, values of m χ ∼ 8 TeV are quite possible. Gluino coannihilation therefore offers the possibility that the LSP, and hence the rest of the supersymmetric spectrum, may lie in the multi-TeV range, beyond the reach of the LHC. Of course, we sincerely hope, if not expect, that supersymmetry will be discovered during future LHC runs. That said, the scenarios discussed here illustrate one way in which detection at the LHC could be evaded. An interesting and important question that lies beyond the scope of this paper is how to detect supersymmetry in a gluino coannihilation scenario with a multi-TeV LSP (see, e.g., the discussion in [46]). As we have discussed in this paper, the gluino-neutralino mass difference in such a scenario is typically O(100) GeV, resulting in a suppressed missing-energy signature whose detection at a future 100-TeV proton-proton collider might be challenging.