Likelihood Analysis of Supersymmetric SU(5) GUTs
We perform a likelihood analysis of the constraints from accelerator experiments and astrophysical observations on supersymmetric (SUSY) models with SU(5) boundary conditions on soft SUSY-breaking parameters at the GUT scale. The parameter space of the models studied has 7 parameters: a universal gaugino mass , distinct masses for the scalar partners of matter fermions in five- and ten-dimensional representations of SU(5), and , and for the and Higgs representations and , a universal trilinear soft SUSY-breaking parameter , and the ratio of Higgs vevs . In addition to previous constraints from direct sparticle searches, low-energy and flavour observables, we incorporate constraints based on preliminary results from 13 TeV LHC searches for jets + events and long-lived particles, as well as the latest PandaX-II and LUX searches for direct Dark Matter detection. In addition to previously-identified mechanisms for bringing the supersymmetric relic density into the range allowed by cosmology, we identify a novel coannihilation mechanism that appears in the supersymmetric SU(5) GUT model and discuss the role of coannihilation. We find complementarity between the prospects for direct Dark Matter detection and SUSY searches at the LHC.
KCL-PH-TH/2016-57, CERN-PH-TH/2016-217, DESY 16-156, IFT-UAM/CSIC-16-105
FTPI-MINN-16/29, UMN-TH-3609/16, FERMILAB-PUB-16-453-CMS, IPPP/16/97
In the absence so far of any experimental indications of supersymmetry (SUSY) [1, 2, 3, 4, 5], nor any clear theoretical guidance how SUSY may be broken, the building of models and the exploration of phenomenological constraints on them [6, 7, 8, 9, 10, 11] have adopted a range of assumptions. One point of view has been to consider the simple parametrization of soft SUSY breaking in which the gaugino and scalar masses, as well as the trilinear soft SUSY-breaking parameters, are all constrained to be universal at the SUSY GUT scale (the CMSSM [12, 13, 14, 15, 16, 6, 7, 17]). An alternative point of view has been to discard all universality assumptions, and treat the soft SUSY-breaking parameters as all independent phenomenological quantities (the pMSSM [18, 9, 19]), imposing diagonal mass matrices and the minimal flavour violation (MFV) criterion. Intermediate between these extremes, models with one or two non-universal soft SUSY-breaking contributions to Higgs masses (the NUHM1 [20, 21, 22, 23, 6, 7] and NUHM2 [24, 25, 21, 22, 23, 8]) have also been considered.
It is interesting to explore also models that are less simplified than the CMSSM, but not as agnostic as the pMSSM, in that they incorporate a limited number of simplifying assumptions. GUTs motivate the assumption that the gaugino masses are universal, and constraints on flavour-changing neutral interactions suggest that the soft SUSY-breaking masses for scalars with identical quantum numbers are also universal. However, there is no compelling phenomenological reason why the soft SUSY-breaking masses for scalars with different quantum numbers should be universal.
Specific GUTs may also provide some guidance in this respect. For example, in an SO(10) GUT the scalar masses of all particles in a given generation belonging to a single representation of SO(10) would be universal, as would those for the and SU(5) Higgs representations that belong to a single of SO(10) and break electroweak symmetry, as in the NUHM1. In contrast, the SU(5) framework is less restrictive, allowing different masses for scalars in and representations , and also for the and Higgs representations. Thus it is a 1-parameter extension of the NUHM2. In this paper we explore the theoretical, phenomenological, experimental and cosmological constraints on this SU(5)-based SUSY GUT model.
This relaxation of universality is relevant for the evaluation of several different constraints from both the LHC and elsewhere. For example, the most powerful LHC constraints on the CMSSM, NUHM1 and NUHM2 are those from the classic searches [1, 4]. These constrain principally the right-handed squarks, whose decays are dominated by the channel that maximizes the signature. On the other hand, the decay chains of left-handed squarks are more complicated, typically involving the , resulting in a dilution of the signature and more importance for final states including leptons. In a SUSY SU(5) GUT, the left-handed squarks and the right-handed up-type squarks appear in representations whereas the right-handed down-type squarks appear in representations, with independent soft SUSY-breaking masses. Hence the impacts of the LHC and other constraints need to be re-evaluated.
The possible difference between the soft SUSY-breaking contributions to the masses of the squarks appearing in a of SU(5), i.e., up-type squarks and left-handed down-type squarks, and those appearing in a of SU(5), i.e., right-handed down-type squarks, offers a new avenue for compressing the stop spectrum. Also, as we shall see, with there is the possibility that are much smaller than the other squark masses, leading to another type of compressed spectrum 111This possibility has also been noted in a supersymmetric SO(10) GUT framework ..
In principle, the constraints from flavour observables may also act differently when . For example, the soft SUSY-breaking masses of the left- and right-handed charge +2/3 quarks are independent, and flavour observables such as BR() and depend on both of them, in general.
Another experimental constraint whose interpretation may be affected by the non-universality of scalar masses is . A priori, a SUSY explanation of the discrepancy between the Standard Model (SM) prediction and the experimental measurement of requires relatively light smuons, either right- and/or left-handed, which are in and representations, respectively. It is interesting to investigate to what extent the tension between a SUSY interpretation of and the LHC constraints on squarks that is present in more constrained SUSY models could be alleviated by the extra degree of freedom afforded by the disconnect in SU(5).
Finally, we recall that in large parts of the regions of the CMSSM, NUHM1 and NUHM2 parameter spaces favoured at the 68% CL the relic density is brought into the range allowed by Planck  and other data via coannihilation with the stau and other sleptons [29, 30]. In an SU(5) GUT, the left- and right-handed sleptons are in different representations, and , respectively. Hence they have different masses, in general, providing more flexibility in the realization of coannihilation. Specifically, as mentioned above, the freedom to have allows the possibility that the right-handed up- and charm-flavour squarks, and , are much lighter than the other squarks, opening up the novel possibility of coannihilation, as we discuss below.
Our analysis of the available experimental constraints largely follows those in our previous studies of other variants of the MSSM [6, 7, 8, 9, 10, 11], the main new feature being that we incorporate the constraints based on the preliminary results from LHC searches for jets + events with /fb of data at 13 TeV . For this purpose, we recast available results for simplified models with the mass hierarchies and vice versa. We also include the preliminary constraints from LHC searches in 13-TeV data for the heavy MSSM Higgs bosons and long-lived charged particles, and incorporate in combination the recent PandaX  and LUX  data.
The SUSY SU(5) GUT model we study is set up in Section 2, and our implementations of constraints and analysis procedure are summarized in Section 3. Section 4 describes how we characterize different Dark Matter (DM) mechanisms, including the novel coannihilation mechanism, coannihilation and a hybrid possibility. Section 5 contains our results in several model parameter planes, and Section 6 describes various one-dimensional likelihood functions including those for several sparticle masses, and various other observables. Higgs boson branching ratios (BRs) are presented in Section 7, followed by a comparison of the SU(5) with the NUHM2 results in Section 8. The possibility of a long-lived is discussed in Section 9, and the prospects for direct DM detection are discussed in Section 10. Finally, Section 11 presents a summary and some conclusions.
2 Supersymmetric SU(5) GUT Model
We assume a universal, SU(5)-invariant gaugino mass parameter , which is input at the GUT scale, as are the other SUSY-breaking parameters listed below.
We assume the conventional multiplet assignments of matter fields in the minimal superymmetric GUT:
where the subscript is a generation index. The only relevant Yukawa couplings are those of the third generation, particularly that of the quark (and possibly the quark and the lepton) that may play an important role in generating electroweak symmetry breaking. In our discussion of flavour constraints, we assume the MFV scenario in which generation mixing is described by the Cabibbo-Kobayashi-Maskawa (CKM) model. This is motivated by phenomenological constraints on low-energy flavour-changing neutral interactions, as is our assumption that the soft SUSY-breaking scalar masses for the different and representations are universal in generation space, and are denoted by and , respectively. In contrast to the CMSSM, NUHM1 and NUHM2, we allow . We assume a universal soft trilinear SUSY-breaking parameter .
We assume the existence of two Higgs doublets and in and representations that break electroweak symmetry and give masses to the charge +2/3 and charge -1/3 and -1 matter fields, respectively. It is well known that this assumption gives a (reasonably) successful relation between the masses of the quark and the lepton , but not for the lighter charge -1/3 quarks and charged leptons. We assume that whatever physics resolves this issue is irrelevant for our analysis, as would be the case, for instance, if corrections to the naive SU(5) mass relations were generated by higher-dimensional superpotential terms . In the absence of any phenomenological constraints, we allow the soft SUSY-breaking contributions to the and masses, and , to be different from each other, as in the NUHM2, as well as from and . As in the CMSSM, NUHM1 and NUHM2, we allow the ratio of Higgs vacuum expectation values, , to be a free parameter.
In addition to these electroweak Higgs representations, we require one or more Higgs representations to break the SU(5) GUT symmetry. The minimal possibility is a single representation , but we do not commit ourselves to this minimal scenario. It is well known that this scenario has problems with rapid proton decay 222We note that this problem becomes less severe for supersymmetry-breaking scales beyond a TeV . and GUT threshold effects on gauge coupling unification. We assume that these issues are resolved by the appearance of additional fields at or around the GUT scale that are otherwise irrelevant for TeV-scale phenomenology. The effective low-energy Higgsino mixing coupling is a combination of an input bilinear coupling and possible trilinear and higher-order couplings to GUT-scale Higgs multiplets such as . We assume that these combine to yield and positive, without entering into the possibility of some dynamical mechanism, and commenting below only briefly on the case .
3 Implementations of Constraints and Analysis Procedure
Our treatments in this paper of many of the relevant constraints follow very closely the implementations in our previous analyses of other supersymmetric models [6, 7, 8, 9, 10]. For the convenience of the reader, we summarise the constraints in Table 1. In the following subsections we review our implementations, highlighting new constraints and instances where we implement constraints differently from our previous work.
3.1 Electroweak and Flavour Constraints
We treat as Gaussian constraints all electroweak precision observables, all -physics and -physics observables except for . The contribution from , combined here in the quantity , is calculated using a combination of the CMS  and LHCb  results described in  with the more recent result from ATLAS . We extract the corresponding contribution in Table 1 by applying to the 2-dimensional likelihood provided by the combination of these experiments the minimal flavour violation (MFV) assumption that applies in the SU(5) model. We calculate the elements of the CKM matrix using only experimental observables that are not included in our set of flavour constraints.
We have updated our implementations of all the flavour constraints, and now use the current world average value of . These and all other constraints whose implementations have been changed are indicated by arrows and boldface in Table 1.
|[GeV]|| /[41, 42]|
|[nb]|| /[41, 42]|
| /[41, 42]|
| /[41, 42]|
| /[41, 42]||0.1465 0.0032|
| /[41, 42]||0.21629 0.00066|
| /[41, 42]||0.1721 0.0030|
| /[41, 42]||0.0992 0.0016|
| /[41, 42]||0.0707 0.0035|
| /[41, 42]||0.923 0.020|
| /[41, 42]||0.670 0.027|
| /[41, 42]||0.1513 0.0021|
| /[41, 42]||0.2324 0.0012|
|[GeV]|| /[41, 42]|
|[GeV]||[46, 47] / |
|/[37, 38]||2D likelihood, MFV|
|[54, 55] /|
|[58, 54] /|
|[58, 54] /|
|[58, 54] /|
|Heavy stable charged particles||||Fast simulation based on [61, 62]|
|||limits in the planes|
|[63, 64, 65]||2D likelihood, limit|
3.2 Higgs Constraints
We use the combination of ATLAS and CMS measurements of the mass of the Higgs boson: . We employ the FeynHiggs 2.11.2 code [46, 47] to evaluate the constraint this imposes on the parameter space, assuming a one- theoretical uncertainty of 333We use a modified version of FeynHiggs 2.11.2 that includes two-loop QCD corrections in the evaluation of the running top mass and an improved evalution of the top mass in the -on-shell conversion for the scalar tops..
The contributions of 85 Higgs search channels from the LHC and the Tevatron are evaluated using HiggsSignals, see , where a complete list of references can be found. The contributions from the limits from searches for the heavy neutral MSSM Higgs bosons in the channels are evaluated using the code HiggsBounds [68, 64], which incorporates the results of CMS searches [63, 64] with of 8 TeV data. The contributions from the two possible production modes, and , are combined in a consistent manner, depending on the MSSM parameters. The results from HiggsBounds have been compared with the published CMS analysis, and are in very good agreement . The corresponding contribution is labelled as “2D likelihood” in Table 1. For the corresponding constraint with fb of 13 TeV data, we implement an approximate treatment of the contribution using the preliminary result of ATLAS , as we describe in more detail below. Limits from other Higgs boson searches are not relevant for the investigation in this paper and are therefore not included.
3.3 Lhc constraints at 13 TeV
ATLAS and CMS have recently announced preliminary results from searches with /fb of data at 13 TeV, using simplified models for gluino and squark pair production [3, 5]. These searches assume and , respectively, and 100% BRs for the decays () and , respectively, which maximize the possible corresponding signatures. Neither of these assumptions is valid in the SUSY SU(5) GUT model: as we will see in more detail later, the and masses are quite similar in much of the favoured region of parameter space 444An exception is provided by the and , which may be much lighter than the gluino and other squarks in some regions of parameter space. We will discuss this possibility in detail below., and in general other decay modes dilute the signature, although larger-multiplicity final states may compensate through an increase in transverse energy . These other decay modes populate other search channels including leptons, which we do not consider in this paper as they were of limited importance in our previous analyses of the CMSSM, NUHM1 and NUHM2, having impact only for relatively large squark masses and small .
Fig. 1 displays the ratios of the cross section (left panel) and the cross section (right panel) that we find in ranges of and that are representative of those favoured in our analysis before implementing the LHC 13-TeV constraint, relative to the cross sections found in the simplified models with and , respectively. We have used NLL-fast-3.1  to obtain the cross section at NLO + NLL level. In both plots a large area at higher squark masses is visible, as well as a thin strip at . The latter corresponds to lighter and discussed below. We see that the cross section (left panel) is generally smaller than in the corresponding simplified model by a factor due to the destructive interference between the -channel gluon exchange diagram and the -channel squark exchange diagram in , thus weakening the LHC constraints as discussed below. On the other hand, the cross section (right panel) is generally a factor larger than in the simplified model, except in the coannihilation strip at small and , to which we return later. The enhancement of the squark cross-section is due to the fact that in the squark-neutralino simplified model there is no production mode with total baryon number , , because gluinos are assumed to be absent. On the other hand, in our model , and (with -channel exchange) becomes the dominant squark production mode in the large region, due to the valence quark-parton dominance in the proton in the large regime.
Fig. 2 displays the CMS 95% confidence limits in the plane from a hadronic jets plus search  within a simplified model assuming that the decay mode occurs with 100% BR (solid black lines). These limits are compared with the best-fit points (green stars) and the regions in the fits that are preferred at and (red and blue contours, respectively). Here and in the following analogous parameter planes, we use the and contours as proxies for the boundaries of the 68% and 95% CL regions in the fit.
In addition, within the 95% CL region in Fig. 2 we have indicated the dominant (%) decays found in our analysis. We note that many model points do not have any decay mode with BR % within the 95% CL region and that, for those that do, the dominant decays are two-body modes that were not considered in . Because of this and the fact that the cross section is always smaller than in the gluino simplified model by a factor (see the left panel of Fig. 1), the LHC 13-TeV constraint from the gluino simplified model has only negligible impact. Our LHC 13-TeV constraint on the gluino mass actually comes indirectly from the squark mass constraint estimated using the squark simplified model discussed below, since the squark and gluino masses are related via renormalization group evolution in the SU(5) model. The left panel in Fig. 2 was obtained before implementing the LHC 13-TeV 95% confidence limit on gluino and squark pair-production, while in the right panel this constraint is included. We note that the simplified model exclusion in this analysis extended to , below the gluino mass at the pre-LHC 13 TeV best-fit point, and barely reaching the 68% CL contour (solid red line).
Fig. 3 contains an analogous set of planes for CMS searches for squarks, where the CMS limit assuming a simplified model with heavy gluino and 100% BRs for is displayed (black lines): the solid lines assume that all the squarks of the first two generations are degenerate, the dashed lines assume two degenerate squarks, and the dotted lines assume just one squark. The planes in the upper panels display and the masses of the first- and second-generation right-handed up-type squarks (here commonly denoted as ), while the planes in the lower panels are for the down-type squarks (here commonly denoted as ). The main decay modes of the (upper) and the (lower) are indicated over much of the preferred parameter space, and we note that the dominant (%) decay modes of both right-handed up- and down-type squarks are indeed into the corresponding quark flavour + for nearly the whole 68% CL regions, as assumed in the squark simplified-model search. This is, however, not the case for the left-handed up- and down-type squarks (not shown), whose dominant decays are into and electroweak doublet partner quark flavours. Furthermore, within the displayed 95% CL regions there are also large areas where decays into gluinos, not considered in the simplified model, are dominant.
Because the decays are important, and also because the cross section in our sample is much larger than that found at large for in the simplified model with , as seen in the right panel of Fig. 1, we have implemented a recast of this search in our global analysis 555The coannihilation strip visible in the upper panels of Fig. 3 at is the subject of a later dedicated discussion., and the comparison between the left panels (without this contribution) and the right panels (with this contribution) in Fig. 3 shows the importance of this constraint.
Our implementation of the LHC 13-TeV constraint is based on . In this analysis, the CMS Collaboration provides a map of the 95% CL cross-section upper limit as a function of and assuming and 100% BR for . This is indeed the dominant production and decay mode in most parts of the 68% CL regions of the considered model, as can be seen in Figs. 1 and 3. For each point we compare our calculation of with the CMS 95% CL upper limit on the cross section: . We model the penalty as
so that the CMS 95% CL upper limit corresponds to and scales as the square of the number of signal events, , which gives the right scaling. We have checked that our implementation (2) reproduces the band in the 2-dimensional exclusion limit provided by CMS , with a discrepancy that is much smaller than the width of the band.
The aforementioned CMS analysis  also looks at three simplified gluino models assuming 100% BR for with , respectively, and provides corresponding cross-section upper limit maps as a function of and . We implement these constraints by defining by analogy with Eq. (2).
We also consider the process, treating it as follows. This process is only relevant when . In this regime, if (), () tends to decay into (), radiating soft jets. If these soft jets are ignored, we are left with the () system. In this approximation, the impact of can therefore be estimated by adding an extra contribution () to (). In general, SUSY searches are designed to look for high objects, and one loses a small amount of sensitivity by ignoring soft jets. We therefore believe that our implementation of the process is conservative.
Finally, we estimate the total penalty from the LHC 13-TeV constraint to be .666One could be concerned that summing up the contributions from different simplified model limits would overestimate the exclusion limit, since these signal regions are not necessarily independent. This would be indeed the case if the same event sample were confronted with multiple overlapping signal regions. In our case, however, the signal sample is divided into statistically independent sub-samples, corresponding to the simplified model topologies , etc., and these sub-samples are confronted with the corresponding simplified model limits only once. In such a case the estimate (2) provides a conservative limit when there is no significant excess in the data.
3.4 Constraints on long-lived charged particles
We also include in our analysis LHC constraints from searches for heavy long-lived charged particles (HLCP) that are, in general, relevant to coannihilation regions where the mass difference between the lightest SUSY particle (LSP) and the next-to-lightest SUSY particle (NLSP) may be small and the NLSP may therefore be long-lived. As we discuss below, important roles are played in our analysis by , and coannihilation, but only in the case is the NLSP - LSP mass difference small enough to offer the possibility of a long-lived charged particle. We implement in our global analysis the preliminary CMS 13-TeV result  using tracking and time-of-flight measurements, based on the recipe and the efficiency map as a function of the pseudo-rapidity and velocity of the HLCP given in . We use Pythia 8  and Atom  to generate and analyse the events, and assume that the efficiencies for detecting slow-moving s are similar at 8 and 13 TeV. 777A similar recasting method was used in . See also  for another approach using simplified model topologies. The efficiency contains a lifetime-dependent factor , where is a distance m that depends on the pseudorapidity, and and are the mass, momentum and lifetime of the long-lived particle. This factor drops rapidly for particles with lifetimes ps, corresponding to .
3.5 Constraints on heavy neutral Higgs bosons from Run II
Concerning the production of heavy neutral Higgs bosons, in addition to the TeV constraints on provided by HiggsBounds, we also take into account the preliminary exclusion limits obtained by ATLAS from searches for generic spin-0 bosons in the final state with an integrated luminosity of 13.3 fb at 13 TeV that were presented at the ICHEP 2016 conference and described in  (see also the CMS results in ). Upper bounds on are reported for each separately for the gluon fusion production channel and for production in association with a pair assuming there is no contamination between the modes, assuming a single resonance. We compute the cross sections and the BRs in the MSSM using FeynHiggs, adding the contributions for and , using the average of the two masses, which are degenerate within the experimental resolution. This result is compared with the upper limit from the corresponding channel neglecting contamination. This approach leads to a conservative limit since we underestimate the signal yield in each channel by neglecting the contamination (the events from the other production mode). As in Eq. (2), the penalties are modelled as
where , is the production mode, is the corresponding search channel and is the 95% CL upper limit evaluated at by ATLAS . Finally we take the stronger rather than combining them, in order to be on the conservative side 888 A more conservative approach would be to choose the strongest search channel based on the expected sensitivity rather than that observed. However, the expected limit shown in  is similar to that observed, so we do not expect that this approach would lead to a significant change. : .
3.6 Other constraints
The most important other constraint update is that on spin-independent DM scattering. We incorporate in our global fit the recent result published by the PandaX-II experiment , which we combine with the new result from the LUX Collaboration , as discussed in more detail in Section 8.
For the electroweak observables we use FeynWZ , and for the flavour constraints we use SuFla . For the Higgs observables, we use FeynHiggs 2.11.2 [46, 47] (including the updates discussed in Sect. 3.2), HiggsBounds 4.3.1  and HiggsSignals 1.4.0 . We calculate the sparticle spectrum using SoftSusy 3.3.10  and sparticle decays using SDECAY 1.3b  and StauDecay 0.1. The DM density and scattering rate are calculated using micrOMEGAs 3.2  and SSARD , respectively. Finally, we use SLHALib 2.2  to interface the different codes.
3.7 Sampling procedure
As discussed in the previous Section, the SUSY SU(5) GUT model we study has 7 parameters: , , , , , and . The ranges of these parameters that we scan in our analysis are listed in Table 2. The quoted negative values actually correspond to negative values of and : for convenience, we use the notation . The negative values of and that are included in the scans may be compatible with early-Universe cosmology , and yield acceptable tachyon-free spectra. In the portions of the scans with negative values of and , although the effect of the top quark Yukawa coupling in the renormalization group equations is important, it may not be the mechanism responsible for generating electroweak symmetry breaking, since and are negative already at the input scale.
|( 0 , 4)||2|
|( - 2.6 , 8)||2|
|( - 1.3 , 4)||3|
|( -7 , 7)||3|
|( -7 , 7)||3|
|( -8 , 8)||1|
|( 2 , 68)||1|
|Total number of boxes||108|
We sample this parameter space using MultiNest v2.18 , dividing the 7-dimensional parameter space into 108 boxes, as also described in Table 2. This has two advantages: it enables us to run MasterCode on many nodes in parallel, and it enables us to probe more efficiently for local features in the likelihood function. For each box, we choose a prior such that 80% of the sample has a flat distribution within the nominal range, while 20% of the sample is in normally-distributed tails outside the box. Our resultant total sample overlaps smoothly between boxes, avoiding any spurious features at the box boundaries. The total number of points in our sample is , of which have .
4 Dark Matter Mechanisms
The relic density of the LSP, assumed here to be the lightest neutralino, , which is stable in supersymmetric SU(5) because of -parity, may be brought into the narrow range allowed by the Planck satellite and other measurements  via a combination of different mechanisms. It was emphasized previously  in studies of the CMSSM, NUHM1 and NUHM2 that simple annihilations of pairs of LSPs into conventional particles would not have been sufficient to bring the relic density down into the Planck range for values of compatible with the LHC search limits and other constraints on these models. Instead, there has to be some extra mechanism for suppressing the LSP density. Examples include enhanced, rapid annihilation through direct-channel resonances such as . Another possibility is coannihilation with some other, almost-degenerate sparticle species [12, 80]: candidates for the coannihilating species identified in previous studies include the and .
We introduced in  measures on the sparticle mass parameters that quantify the mass degeneracies relevant to the above-mentioned coannihilation and rapid annihilation processes, of which the following are relevant to our analysis of the SUSY SU(5) GUT model 999We note that the focus-point mechanism  does not play a role in the SU(5) model.:
We also indicate above the colour codes used in subsequent figures to identify regions where each of these degeneracy conditions applies. We have verified in a previous study  that CMSSM, NUHM1 and NUHM2 points that satisfy the DM density constraint fulfill one or more of the mass-degeneracy conditions, and that they identify correctly the mechanisms that yield the largest fractions of final states, which are usually % [8, 82].
In much of the region satisfying the degeneracy criterion above, the has a similar mass, and can contribute to coannihilation . We highlight the parts of the sample where sneutrino coannihilation is important by introducing a shading for regions where the is the next-to-lightest sparticle (NLSP), and obeys the degeneracy condition
We discuss later the importance of this supplementary DM mechanism.
As we discuss in this paper, a novel possibility in the SU(5) SUSY GUT is coannihilation with right-handed up-type squarks, and , which may be much lighter than the other squarks in this model, as a consequence of the freedom to have . We quantify the relevant mass degeneracy criterion by
As we shall see in the subsequent figures, this novel degeneracy condition can play an important role when . The existence of this new coannihilation region was verified using SSARD , an independent code for calculating the supersymmetric spectrum and relic density.
We also distinguish in this analysis ‘hybrid’ regions where the coannihilation and funnel mechanisms may be relevant simultaneously:
also with the indicated colour code.
5.1 Parameter Planes
We display in Fig. 4 features of the global function for the SUSY SU(5) GUT model in the plane (left panel) and the plane (right panel), profiled over the other model parameters101010We have used Matplotlib  and PySLHA  to plot the results of our analysis.. Here and in subsequent parameter planes, the best-fit point is shown as a green star, the 68% CL regions are surrounded by red contours, and the 95% CL regions are surrounded by blue contours (as mentioned above, we use the and contours as proxies for the boundaries of the 68% and 95% CL regions in the fit). The regions inside the 95% CL contours are shaded according to the dominant DM mechanisms discussed in the previous Section, see the criteria (4, 6, 7). In the (relatively limited) unshaded regions there is no single dominant DM mechanism.
As we see in Fig. 4, the best-fit point is at relatively small values of and , close to the lower limit on , whereas the 68% CL region extends to much larger values of and . The values of the model parameters at the best-fit point are listed in Table 3 111111The SLHA files for the best-fit point and other supplementary material can be found in .. The upper row of numbers are the results from the current fit including the latest LHC 13-TeV and PandaX-II/LUX constraints, and the numbers in parentheses in the bottom row were obtained using instead the previous LHC 8-TeV and XENON100 constraints, but the same implementations of the other constraints. The most significant effect of the new LHC data has been to increase the best-fit value of by : the changes in the other fit parameters are not significant, in view of the uncertainties. As we discuss in more detail later, the favoured fit regions are driven by the constraint towards the boundary of the region excluded by the constraint. Away from this boundary, the global function is quite flat.
The best-fit point and much of the 68% CL region lie within the pink shaded region where coannihilation is the dominant DM mechanism. At larger values of and we encounter a blue shaded region where rapid annihilation via direct-channel poles is dominant. We also see darker shaded hybrid regions where and annihilation are important simultaneously. At larger values of , in the green shaded regions, the dominant DM mechanism is coannihilation. There is also a band in the plane with and , allowed at the 95% CL, where coannihilation is important.
We also note the appearance within the 95% CL region at , and of the novel coannihilation region (shaded yellow). To understand the origin of this novelty, consider the one-loop renormalization-group equations for the states in the representations of SU(5), namely , above the highest MSSM particle mass (all masses are understood to be scalar fermion masses, and we suppress subscripts ):
where with the renormalization scale and some reference scale,
where the trace in sums over the generations. The coannihilation mechanism becomes important in a region of the SUSY SU(5) GUT parameter space where is very large and positive ( TeV), is small and negative ( TeV), is very large and negative ( TeV), and is very large and positive ( TeV). In this region, therefore, is very large and negative ( TeV), and are suppressed because of small Yukawa couplings ( is not large in this region), and is also very large and negative ( TeV), since is large and negative and vanishes at the GUT scale. Inspection shows that the terms in (8) and (9) drive the stop and sbottom masses upwards, and the terms in (8) and (10) drive the left-handed squark and right-handed slectron masses upwards. On the other hand, the term in (9) drives the right-handed squark masses downwards. Since there are no counteracting terms for the and , these have lower masses than the other sfermions, opening the way to a coannihilation region.121212 An SLHA file corresponding to the coannihilation region can be found in .
As discussed in more detail later, we used the Atom  simulation code for a dedicated verification that points in this region escape all the relevant LHC constraints. These points avoid exclusion by the LHC constraints through a combination of a strong mass degeneracy, , leading to strong suppression of the standard signature, and the reduction of the production rate compared to the simplified model that assumes mass degeneracy of all 8 light flavour squarks (see Fig. 1). These effects are clearly visible in Fig. 18 of .
Fig. 5 displays the corresponding information in the plane of the SUSY SU(5) GUT model. As already reported in Table 3, here we see directly that the best-fit point has very small (and slightly negative) , and that is somewhat larger, exploiting the possibility that that is offered in this model. We also see again that the 68% CL region extends to values of and beyond the coannihilation region. We also note that in most of the rest of this plane coannihilation is dominant, with only scattered regions where rapid annihilation is important, even in combination with coannihilation.
Projections of our results in the and planes are shown in Fig. 6. We see that values of are allowed at the 95% CL, that the range is favoured at the 68% CL, and that there is no phenomenological upper limit on at the 95% CL131313The RGE evolution of the Yukawa couplings blows up for .. The best-fit point has , as also reported in Table 3.
The pink coannihilation region is very prominent in the