Dark Matter in a Constrained $E_6$ Inspired SUSY Model

We investigate dark matter in a constrained $E_6$ inspired supersymmetric model with an exact custodial symmetry and compare with the CMSSM. The breakdown of $E_6$ leads to an additional $U(1)_N$ symmetry and a discrete matter parity. The custodial and matter symmetries imply there are two stable dark matter candidates, though one may be extremely light and contribute negligibly to the relic density. We demonstrate that a predominantly Higgsino, or mixed bino-Higgsino, neutralino can account for all of the relic abundance of dark matter, while fitting a 125 GeV SM-like Higgs and evading LHC limits on new states. However we show that the recent LUX 2016 limit on direct detection places severe constraints on the mixed bino-Higgsino scenarios that explain all of the dark matter. Nonetheless we still reveal interesting scenarios where the gluino, neutralino and chargino are light and discoverable at the LHC, but the full relic abundance is not accounted for. At the same time we also show that there is a huge volume of parameter space, with a predominantly Higgsino dark matter candidate that explains all the relic abundance, that will be discoverable with XENON1T. Finally we demonstrate that for the $E_6$ inspired model the exotic leptoquarks could still be light and within range of future LHC searches.


Introduction
A plethora of astrophysical and cosmological observations provide strong evidence for the presence of non-baryonic, non-luminous matter, so called dark matter (DM), that constitutes about 25% of the energy density of the Universe [1]. So far its microscopic composition remains unknown. However it is clear that dark matter can not consist of any standard model (SM) particles. Therefore its existence represents the strongest piece of evidence for physics beyond the SM.
Models with softly broken supersymmetry (SUSY) are currently the best motivated extensions of the SM. Within these models the quadratic divergences, which give rise to the destabilization of the electroweak (EW) scale, get cancelled [2][3][4][5]. Models with softly broken SUSY also provide an attractive framework for the incorporation of the gravitational interactions. Indeed, a partial unification of the SM gauge interactions with gravity can be attained within models based on the (N = 1) local SUSY (supergravity). Nevertheless (N = 1) supergravity (SUGRA) is a nonrenormalizable theory. The (N = 1) SUGRA models can arise from ten dimensional E 8 × E 8 heterotic string theory [6]. The compactification of the extra dimensions in this theory results in breaking E 8 → E 6 [7][8][9]. The remaining E 8 constitutes a hidden sector that gives rise to spontaneous breakdown of local SUSY. The hidden sector and visible sectors interact only gravitationally, which allows for the breaking of local SUSY in the hidden sector to be communicated to the visible sector and results in a set of soft SUSY breaking interactions.
When R-parity is conserved the lightest SUSY particle (LSP) in the models with softly broken SUSY is stable and therefore can play the role of dark matter [10]. Moreover in the simplest SUSY extension of the SM, i.e., the minimal supersymmetric standard model (MSSM), the SM gauge couplings extrapolated to high energies using the renormalization group (RG) equations (RGEs) converge to a common value at some high energy scale M X ∼ 10 16 GeV [11][12][13][14]. This permits to embed the SM gauge group into Grand Unified Theories (GUTs) [15] based on E 6 or its subgroups such as SU (5) and SO (10).
In this context it is especially important to explore the implications for dark matter and collider phenomenology within well motivated E 6 inspired SUSY extensions of the SM. The breakdown of E 6 may lead to a variety of SUSY models at low energies. In particular, a set of the simplest E 6 inspired SUSY extensions of the SM includes supersymmetric models based on the SM gauge group, like the MSSM, as well as extensions of the MSSM with an extra U (1) gauge symmetry. Within the class of the E 6 inspired U (1) extensions of the MSSM, there is a unique choice of Abelian U (1) N gauge symmetry that allows zero charges for right-handed neutri-nos and this is the U (1) that appears in the exceptional supersymmetric standard model (E 6 SSM) [16,17]. This choice ensures that the right-handed neutrinos can be superheavy, so that a high scale see-saw mechanism can be used to generate the mass hierarchy in the lepton sector, providing a comprehensive understanding of the neutrino oscillations data. Successful leptogenesis is also a distinctive feature of the E 6 SSM because the heavy Majorana right-handed neutrinos may decay into final states with lepton number L = ±1, creating a lepton asymmetry in the early Universe [18,19]. Since sphalerons violate B + L but conserve B − L, this lepton asymmetry gets converted into the observed baryon asymmetry of the Universe through the EW phase transition. In this case substantial values of the CPasymmetries can be generated even for the lightest right-handed neutrino masses M 1 ∼ 10 6 GeV so that successful thermal leptogenesis may be achieved without encountering a gravitino problem [19].
To ensure anomaly cancellation the matter content of the E 6 SSM is extended to include three 27 representations of E 6 . In addition the low energy spectrum can be supplemented by a SU (2) W doublet L 4 and anti-doublet L 4 from extra 27 and 27 to preserve the unification of the SM gauge couplings at high energies [20]. Thus the E 6 SSM contains extra exotic matter beyond the MSSM. Over the last ten years, several variants of the E 6 SSM have been proposed [16,17,[21][22][23][24][25][26][27][28][29][30][31]. The E 6 inspired SUSY models with an extra U (1) N gauge symmetry have been thoroughly investigated as well. For example, the possibility of mixing between doublet and singlet neutrinos [32], the effects of Z − Z mixing [33], the neutralino sector [33][34][35], the implications of the exotic states for the dark matter [36], the renormalization group flow [20,34] and EW symmetry breaking (EWSB) in the model [34,37,38] have all been studied. More recently, the RG flow of the Yukawa couplings and the theoretical upper bound on the lightest Higgs boson mass were explored in the vicinity of the quasi-fixed point [39,40] that appears as a result of the intersection of the invariant and quasi-fixed lines [41]. Detailed studies of the E 6 SSM have established that the additional exotic matter and Z in the model would lead to distinctive LHC signatures [16,17,22,25,[42][43][44][45][46][47], as well as result in non-standard Higgs decays for sufficiently light exotics [30,40,[48][49][50][51][52][53]. In this SUSY model the particle spectrum has been examined in Refs. [54][55][56][57], including the effects of threshold corrections from heavy states [58]. The renormalization of the vacuum expectation values (VEVs) that lead to EWSB in the model has also been calculated [59,60], and the fine tuning in the model has been studied [61,62].
Although the presence of exotic matter in the E 6 SSM may lead to spectacular collider signatures it also gives rise to non-diagonal flavor transitions and rapid proton decay. In principle, an approximate Z H 2 symmetry can be imposed to suppress flavor changing processes in these U (1) extensions of the MSSM while the most dangerous baryon and lepton number violating operators can be forbidden by another exact Z 2 symmetry which plays a similar role to the R-parity in the MSSM [16,17].
Using the method proposed in [63][64][65] it was shown that the LSP and next-tolightest SUSY particle (NLSP) in the E 6 SSM have masses below 60 − 65 GeV [49]. As a consequence these states can give rise to unacceptably large branching ratios of the exotic decays of the SM-like Higgs boson into the LSP and NLSP. In order to suppress such exotic Higgs decays and to prevent the decays of the lightest MSSMlike neutralino into the LSP and NLSP in models with approximate Z H 2 symmetry an additional Z S 2 symmetry needs to be postulated [26]. All discrete symmetries mentioned above do not commute with E 6 and the imposition of such symmetries to ameliorate phenomenological problems, which generically arise because of the presence of the exotic matter at low energies, is an undesirable feature of the models under consideration.
Here we focus on the investigation of the U (1) N extension of the MSSM (SE 6 SSM) in which a single discreteZ H 2 symmetry forbids tree-level flavor-changing transitions and the most dangerous operators that violate baryon and lepton numbers [28,30,39]. In a recent letter [66] we specified a set of benchmark points representing scenarios with a 125 GeV SM-like Higgs, which are consistent with the LHC limits on SUSY particles and measured dark matter abundance, within the constrained version of the above SE 6 SSM (CSE 6 SSM). As in any other constrained SUSY model, the soft SUSY-breaking scalar masses, gaugino masses, the trilinear and bilinear scalar couplings in the CSE 6 SSM are each assumed to be universal at the scale M X , where all gauge couplings coincide, i.e., m 2 i (M X ) = m 2 0 , M i (M X ) = M 1/2 , A i (M X ) = A 0 and B i (M X ) = B. The benchmark scenarios presented in Ref. [66] lead to large spin-independent (SI) dark matter-nucleon scattering cross section observable soon at XENON1T experiment and new physics signatures that may be observable at the 13 TeV LHC. These new signatures should allow to distinguish the SUSY model under consideration from the simplest SUSY extensions of the SM. At the same time in this letter we did not examine the CSE 6 SSM parameter space thoroughly and did not provide full details of our calculations. We also did not include the full set of the two-loop RGEs which were used in our analysis.
In this article we present the results of the comprehensive analysis of the CSE 6 SSM parameter space which is consistent with the 125 GeV SM-like Higgs, measured dark matter density and present LHC limits on sparticle masses. As in the MSSM the matter parity in the SE 6 SSM is preserved. Therefore in both models the lightest R-parity odd state, i.e., LSP, is absolutely stable. In most scenarios that have been explored within the MSSM and its extensions the LSP is the lightest neutralino. In the CMSSM the lightest neutralino state is predominantly a linear superposition of the Higgsino and bino. Since the lightest neutralinos are heavy weakly interacting massive particles (WIMPs) they explain well the large scale structure of the Universe [67] and can provide the correct relic abundance of dark matter as long as the mass of the lightest neutralino is below the TeV scale [10]. The conservation of Z H 2 symmetry and matter parity in the SE 6 SSM results in the lightest neutralino as well as the lightest exotic state being stable. In the simplest phenomenologically viable scenarios the lightest exotic states have masses substantially lower than 1 eV forming hot dark matter in the Universe. The results of our analysis indicate that in this case the lightest neutralino in the CSE 6 SSM, which is either mostly Higgsino or a mixed bino-Higgsino state, can account for all or some of the observed cold dark matter relic density.
We perform a scan of the parameter space of the CSE 6 SSM enforcing successful EW symmetry breaking and imposing theoretical and low energy experimental constraints mentioned above. We also compute the dark matter density and SI neutralino-nucleon scattering cross section as well as examine their dependence on the parameters of the CSE 6 SSM. The obtained results are compared with the corresponding ones in the CMSSM. We show that present LUX bounds set sufficiently stringent constraints on the mixing between bino and Higgsino states, for cases where they give a substantial contribution to the observed dark matter density. We therefore find that if the relic density is to be explained with Higgsino dark matter in either the CMSSM or CSE 6 SSM, then the lightest neutralino must be a relatively pure Higgsino state with a highly restricted level of bino mixing, and this is what we find in most of the allowed parameter space. As a consequence the observed dark matter abundance can be reproduced only if the mass of lightest neutralino is relatively close to 1 TeV. In this scenario all sparticles are so heavy that it won't be possible to discover these states at the LHC. If the lightest neutralino is considerably lighter than 1 TeV then this state can account for only a small fraction of the measured dark matter density in the allowed part of parameter space within both the CSE 6 SSM and CMSSM. At the same time we argue that the scenarios with relatively small masses of lightest neutralino and low relic dark matter abundance can still lead to the spectrum of SUSY particles that may be observed at the 13 TeV LHC. In the CSE 6 SSM the set of states detectable at the LHC includes gluino, chargino and neutralino states as well as exotic fermions. In the most part of the allowed CSE 6 SSM parameter space the lightest neutralino has sufficiently large direct detection cross section which should be observable soon at the XENON1T experiment.
The paper is organized as follows. In the next section we briefly review the E 6 inspired U (1) N extension of the MSSM with exact custodialZ H 2 symmetry and define the CSE 6 SSM. In Section 3 we consider the breakdown of gauge symmetry within this SUSY model. In Section 4 the analytical expressions for the mass matrices and masses of all new states that appear in the SE 6 SSM are specified. In Section 5 we discuss the implications of the SUSY model under consideration for dark matter and summarize the results of our studies. Section 6 is reserved for our conclusions. Appendix A contains the complete system of the two-loop RGEs that we use in our analysis.

The SE 6 SSM
In orbifold GUT models the breakdown of E 6 gauge symmetry can lead to the SM gauge group along with two additional U (1) factors [28], i.e., where U (1) ψ and U (1) χ are associated with the subgroups E 6 ⊃ SO(10) × U (1) ψ ⊃ SU (5) × U (1) χ × U (1) ψ . Further symmetry breaking can then result in a low-energy model with a single additional U (1) that is a linear combination of U (1) ψ and U (1) χ , In this case, the value of the mixing angle θ E 6 characterizes the resulting U (1) at low-energies, and several choices of symmetry breaking pattern have been considered (for reviews, see for example Refs. [68][69][70]). In U (1) extensions with E 6 inspired charges, gauge anomalies automatically cancel provided that the low-energy matter content fills in complete representations of E 6 . The SM particle content can be accommodated if each generation is embedded within a fundamental 27-plet of E 6 , which requires the introduction of extra matter to form complete multiplets. In addition to the SM fermions, each of these 27-plets (27 i , i = 1, 2, 3) contains a pair of SU (2) L doublets, H u , H d , a pair of color triplets, D i , D i , a right-handed neutrino, N c i , and a SM singlet, S i . In general, both N c i and S i carry non-zero U (1) charges. The doublets H u i and H d i may be identified as Higgs or inert Higgs doublets, the distinction being that the latter do not develop VEVs. The states D i and D i have electric charge ±1/3 and carry B − L charge twice that of ordinary quarks, and therefore may either be diquarks or leptoquarks.
The potential for interesting phenomenology associated with these exotic states, along with at least one Z boson, has provided substantial motivation for studying E 6 inspired models [71][72][73][74][75][76][77][78][79][80]. Possible signatures of the exotic states at colliders have been studied [81], as well as limits on the Z mass [82]. In addition to observing these exotic states, an underlying E 6 GUT might leave identifiable fingerprints on the ordinary MSSM mass spectrum, such as in the pattern of first and second generation sfermion masses [83]. Further motivation for studying this class of models has come from the fact that they are able to address several weaknesses of the MSSM. The extended gauge sector and the presence of additional singlets, some of which may get VEVs, allows for the solution of the MSSM µ-problem [84] in a way similar to in the next-to-MSSM (NMSSM) [85,86]. These same features also lead to a theoretical upper bound on the lightest Higgs boson mass that is larger than can be achieved in the MSSM, and indeed in the NMSSM [38,43,87,88]. The accompanying enlarged Higgs [16,87,88] and neutralino [33][34][35][88][89][90][91][92][93][94][95][96] sectors have been extensively studied. It has been proposed that the extra D-terms could also solve the tachyon problems encountered in anomaly mediated SUSY breaking scenarios [97], while the inclusion of appropriate family symmetries could provide an explanation for the hierarchy of fermion masses and mixings [23,24,98,99]. Many further implications of these models have also been considered, including for EWSB [34,37,38,[100][101][102][103], neutrino physics [32,104], leptogenesis [18,19] and EW baryogenesis [105,106], the muon anomalous magnetic moment [107,108], electric dipole moments [89,90], lepton flavor violating processes [91] and the possibility of CP-violation in the extended Higgs sector [109].
As was noted above, in the rank-5 models described by Eq. (2) both the singlets S i and the right-handed neutrinos N c i are charged under the additional U (1) in general. However, for the choice of θ E 6 = arctan √ 15, the right-handed neutrinos are uncharged under the resulting U (1) , denoted U (1) N . In this case, a large Majorana mass is allowed for the N c i and a see-saw mechanism can be used to explain the observed neutrino masses, while also allowing for an explanation of the baryon asymmetry via leptogenesis [18,19].
In this article we study a U (1) N extension of the MSSM in which tree-level flavorchanging transitions and the most dangerous baryon and lepton number violating operators are forbidden by a single discreteZ H 2 symmetry [28,30,39]. In the SUSY models under consideration [28], E 6 is assumed to be broken directly to SU (3) C × SU (2) L × U (1) Y × U (1) χ × U (1) ψ at or near the GUT scale, M X , which can be achieved in 5 or 6 dimensional orbifold GUT models. The additional U (1) χ × U (1) ψ is then broken near M X to U (1) N × Z M 2 , where the matter parity Z M 2 is defined by Below M X , the three complete 27-plets of E 6 are taken to be accompanied by a set of pairs of multiplets M l , M l , coming from incomplete 27 and 27 representations, respectively. Note that anomalies still cancel, since the fields from M l and M l carry opposite U (1) charges. A single exactZ H 2 , commuting with E 6 , may then be imposed under which all components of the 27-plets are odd, thereby forbidding both interactions that generate large flavor changing neutral currents (FCNCs) and those that would lead to rapid proton decay. Doing so precludes any of the components of the 27-plets from getting VEVs to break EW symmetry, so that, for example, all of the 27-plet Higgs states H u i , H d i are inert and cannot be identified with the usual MSSM Higgs doublets. But, at the same time the multiplets M l and M l may be either even or odd underZ H 2 , allowing some of them to get VEVs for spontaneous symmetry breaking. In the model considered here we include two pairs of SU (2) L doublets, H u and H u , H d and H d , as well as a pair of singlets S and S. The fields H u , H d , S and S are postulated to be even underZ H 2 symmetry and are responsible for the breaking of SU (2) L × U (1) Y × U (1) N → U (1) em at the TeV scale 1 . The doublets H u and H d are odd underZ H 2 , so that they can mix with a combination of the 27-plet states, defined to be the third generation H u 3 , H d 3 . In this case they may form vectorlike states with masses of order M X , and so may be integrated out of the low-energy spectrum.
With only this set of multiplets, the imposedZ H 2 would forbid any renormalizable operators allowing the exotic quarks to decay. Such long-lived exotics would be produced in the early Universe and would lead to estimated concentrations [110,111] in excess of the observed limits on heavy isotopes [112][113][114]. To avoid this, a pair ofZ H 2 even SU (2) L doublets L 4 and L 4 with the quantum numbers of leptons are also included at the TeV scale that couple to the exotic D i , D i and allow the exotic quarks to decay. This choice also implies that D i and D i are leptoquarks in this scenario.
In addition to the above sets of multiplets, in the model considered here we also include a pure singlet superfieldφ in the spectrum below the GUT scale, which is uncharged under all of the gauge symmetries [30]. This superfield is likewise taken to be even underZ H 2 so that the superpotential may contain a term proportional toφŜŜ, to stabilize the scalar potential, and the scalar component ofφ is allowed to develop a non-zero VEV. The fields H u , H d , S, S andφ are all expected to get masses at or below the TeV scale. Thus after integrating out superheavy states the low-energy matter content in this model, which we refer to as the SE 6 SSM, consists of the superfields shown in Table 1. At low-energies and neglecting suppressed non-renormalizable interactions, the superpotential can then be written We denote superfields with hats, and adopt the conventionÂ ·B ≡ αβÂ αBβ = A 2B1 −Â 1B2 for the SU (2) dot product. The exactZ H 2 symmetry forbids all terms of the form 27 × 27 × 27, so that the allowed trilinear interactions involving nonsinglet fields are of the form 27 × 27 × 27 or 27 × 27 × 27. By making appropriate rotations of the superfields (Ĥ d α ,Ĥ u α ) and (D i ,D i ), the trilinear couplingsλ αβ and κ ij are chosen to be flavor diagonal, while the other new couplings aref iα , f iα , g D ij and h E iα are not, in general. The superpotential also contains several bilinear terms, such as those of the form 27 × 27 . The corresponding couplings, for example µ L , may be generated 2 through the Giudice-Masiero mechanism [116].
As well as being invariant under the single imposedZ H 2 symmetry, the superpotential is also invariant under the residual Z M 2 symmetry resulting from the break- The presence of multiple Z 2 symmetries suggests that it is not unreasonable to expect multiple stable states that may play the role of DM. For our analysis, it is convenient to define a combination of these two The transformation properties of each field under this Z E 2 symmetry are also shown in Table 1, and henceforth we shall refer to states that are odd under Z E 2 as exotics. Since the Lagrangian is separately invariant underZ H 2 and Z M 2 , it is also the case that transformations under Z E 2 leave the Lagrangian invariant. In particular, this means that the lightest Z E 2 -odd, exotic state is absolutely stable and so can potentially be a DM candidate. The automatically conserved matter parity Z M 2 , meanwhile, is equivalent to R-parity and also implies the existence of a stable state, as in the MSSM. Examination of the possible cases shows that these two states are in fact distinct, so that the model contains two DM candidates. In the case that the stable, lightest Z E 2 odd state is R-parity even 3 , then the lightest R-parity odd state must be stable, as usual. Conversely, if the lightest Z E 2 odd state is also the lightest R-parity odd state, then either the lightest R-parity even, Z E 2 odd state or the lightest R-parity odd, Z E 2 even state (depending on which is lighter) is absolutely stable.
By applying the method described in Ref. [63][64][65], it has previously been found that the lightest inert neutralinos can have masses no larger than 60 − 65 GeV [49][50][51]. These states then tend to be the lightest exotic states in the spectrum, and are predominantly combinations of the fermionic components of the inert singlet superfieldsŜ i . Substantial masses for these inert singlinos, of more than ∼ 1 eV, are ruled out by measurements of the SM-like Higgs branching ratios and the DM relic density. The simplest viable solution is instead for the inert singlino masses to be much lighter than 1 eV, which can be achieved provided that the couplings f iα , f iα 10 −6 . This results in the inert singlinos forming hot dark matter, giving a negligible contribution to the observed relic density 4 .
In this case, the second DM candidate should account fully or partially for the DM density, with the latter possibility requiring either additional DM candidates or a non-standard thermal history of the Universe to be consistent with measurements. The sub-eV inert singlinos are both the lightest exotic and lightest R-parity odd states in the spectrum. This implies that the lightest R-parity even exotic state or the lightest R-parity odd, Z E 2 even state is a possible second DM candidate. As can be read from Table 1, the possible exotic candidates are the exotic squarks arising from the superfields (D i ,D i ), the inert Higgs scalars coming from the mixing of (Ŝ i ,Ĥ u α ,Ĥ d α ), or the fermionic components of (L 4 ,L 4 ). The masses of these states are required to be sufficiently heavy to have evaded detection to date. In particular, for large values of the SUSY breaking scale M S the scalars receive large soft SUSY breaking masses and can be of similar mass to the ordinary squarks. The fermionic components of (L 4 ,L 4 ), meanwhile, receive a supersymmetric mass contribution from the superpotential bilinear term µ LL4 ·L 4 , which is not constrained by the requirement of successful EWSB and need not be small 5 . In the model studied here this means that the lightest R-parity odd, Z E 2 even state tends to be the stable state, corresponding to the lightest neutralino with Z E 2 = +1. Depending on the composition of this state, it may then account for some or all of the DM relic density, as in the MSSM. In the following we shall focus on cases where the lightest neutralino is a mixed bino-Higgsino state, or pure Higgsino; we will find that this leads to a DM candidate that is also MSSM-like in its interactions and predictions for the DM relic density.
As usual in low-energy SUSY models, the relevant masses and mixings of interest in the neutralino sector are governed both by the superpotential interactions in Eq. (4) as well as a subset of the soft SUSY breaking interactions. Including the standard set of soft scalar masses, soft trilinears, and soft gaugino masses, the full set of soft SUSY breaking terms that we consider is The general soft SUSY breaking Lagrangian, in which all of the soft parameters are treated as independent, introduces a large number of additional free parameters on top of the extra couplings already present in the superpotential. The number of free parameters can be much reduced by considering a constrained model in which certain relations are assumed to hold between the soft parameters at some high scale. The CSE 6 SSM is defined by imposing boundary conditions at the GUT scale M X where all gauge couplings coincide. In the SE 6 SSM, since all of the low-energy matter content can be placed in complete SU (5) multiplets with the exception of the doubletsL 4 andL 4 , gauge coupling unification still occurs at the two-loop level for any value of α 3 (M Z ), the strong coupling evaluated at the scale M Z , consistent with the measured value [20,28]. Therefore, at the GUT scale M X we take where g 1 , g 1 , g 2 and g 3 are the GUT-normalized U (1) Y , U (1) N , SU (2) L and SU (3) C gauge couplings, respectively. This allows for the U (1) N gauge coupling g 1 to be fixed. The presence of multiple U (1) symmetries implies the possibility of kinetic mixing between the U (1) field strengths [118,119]. In practice, this mixing can be handled by working in a rotated basis for the U (1) gauge fields where the mixing leads instead to non-zero off-diagonal gauge couplings, i.e., in covariant derivatives one finds terms of the form This field redefinition is also responsible for the appearance of the mixed gaugino soft mass, M 11 , in the last bracketed term of Eq. (5). It is natural to expect that at M X , the kinetic mixing should vanish so that g 11 (M X ) = 0, M 11 (M X ) = 0. However, even if this holds at M X , in general non-zero mixing terms will be generated at lowenergies by RG running [120,121]. Previous analyses [20,122] suggest that in this particular model, provided that the off-diagonal gauge coupling vanishes at M X , it remains very small at all scales below M X as well, g 11 ∼ 0.02 g 1 , g 1 . Therefore in our analysis we neglect the effects of gauge kinetic mixing, setting g 11 (M X ) = 0, M 11 (M X ) = 0 and taking them to vanish at scales below this. Nevertheless, it is important to note that in general the effects of this kinetic mixing can be nonnegligible [122][123][124]; it is small here as the only non-vanishing contribution to the mixing comes from the (L 4 ,L 4 ) multiplet pair.
The remaining soft masses satisfy high-scale relations analogous to those applied in the CMSSM. The soft scalar masses squared are taken to be flavor diagonal with diagonal elements set to the common value m 2 0 at M X , and similarly the gaugino masses (with the exception of M 11 , as noted above) are assumed to unify to the value M 1/2 at this scale. The values of the soft breaking trilinears are related to a single common trilinear parameter A 0 by Similarly, the soft breaking bilinears are assumed to unify, The parameter B 0 is taken to be independent of A 0 ; to do so we assume that these soft terms are also generated via a Giudice-Masiero term, as used to produce the superpotential bilinears. The soft breaking tadpole Λ S is not required to be related to other soft parameters by the high-scale boundary condition.
With this choice of boundary conditions, the remaining unfixed parameters in the CSE 6 SSM consist of the new superpotential couplings, namely λ(M X ), σ(M X ), , and the soft breaking parameters m 0 , M 1/2 , A 0 , B 0 and Λ S . To simplify our analysis, in the following we assume that all of these parameters are real. Once these high-scale parameters, together with the MSSM gauge and Yukawa couplings are specified, the model at low-energies is studied by integrating the RGEs given in Appendix A from M X to the EWSB scale.

Gauge Symmetry Breaking
The Higgs fields H u , H d , S, S and φ develop non-zero VEVs breaking SU The relevant part of the scalar potential reads whereḡ 2 = g 2 2 +3g 2 1 /5 and ∆V contains the loop corrections to the effective potential. We denote by Q Φ the U (1) N charge of the field Φ. In the presence of kinetic mixing these charges should be replaced by effective U (1) N charges [16].
At the physical minimum of this potential, the VEVs of the Higgs fields are taken to be of the form The corresponding conditions for these non-zero VEVs to be a stationary point of the potential are 6 , Of the 14 degrees of freedom associated with this set of Higgs fields, after EWSB four massless Goldstone modes are swallowed to generate masses for the physical W ± , Z and Z bosons. The masses of the charged gauge bosons remain the same as in the MSSM. The neutral gauge boson masses are rather different, since the fields H 0 u and H 0 d are charged under both U (1) groups and therefore there is Z − Z mixing even when gauge kinetic mixing is neglected. It is convenient to define the combinations of the VEVs, The tree-level masses M Z 1 , M Z 2 of the physical Z and Z bosons are then found by diagonalizing the squared mass matrix where M 2 Z =ḡ 2 v 2 /4 and The mixing between the two gauge bosons is strongly constrained by EW precision measurements [125], while LHC searches currently place lower bounds on the mass of the extra Z in U (1) N models of M Z 2 3.4 TeV [126]. The physical Z mass can be made acceptably large provided that the combination of the SM singlet VEVs is large, s 9 TeV. This leads to negligible mixing between the physical states Z 1 and Z 2 , with a mixing angle 10 −4 , so that the light state Z 1 is approximately the SM Z boson with M Z 1 ≈ M Z =ḡv/2 and v ≈ 246 GeV, while the heavier gauge boson has its mass set by the singlet VEVs with M Z 2 ≈ M Z ≈ g 1 Q S s. The presence of the singlet fields involved in EWSB means that the set of EWSB conditions, Eq. (11), is somewhat larger than in the MSSM. In the MSSM, there are two such conditions, which read Imposing the EWSB conditions allows for a subset of the model parameters to be fixed. Conventionally in the CMSSM, the two parameters fixed by Eq. (14) are chosen to be µ and B. However, this choice is not unique, nor is it always the most convenient. In particular, when studying scenarios for dark matter, it is ideal to be able to vary µ directly, as this controls the Higgsino masses and therefore permits the composition of the lightest neutralino to be directly chosen. In all of our results below, in both models we allow µ (eff) to remain free and instead fix m 0 using the EWSB conditions. This can be done by expressing the soft masses in terms of the GUT scale parameters using semi-analytic solutions to the RGEs, as detailed in Section 5 below. In the MSSM, the remaining EWSB condition can be used to fix B 0 , while in the SE 6 SSM there are still four conditions available.
In this paper we primarily examine the part of the parameter space where all SUSY particles are considerably lighter than M Z . This corresponds to s 1 , s 2 and ϕ being much larger than the SUSY breaking scale M S . These VEVs are fixed using two of the EWSB conditions to determine tan θ and ϕ, with the value of s being a free input parameter. The remaining two conditions can be used to fix the GUT scale parameters Λ F (M X ) and Λ S (M X ). The appropriate stationary points of the scalar potential in Eq. (9) arise if Λ F M 2 S and Λ S M 3 S . In this case the structure of the potential is further simplified if the dimensionless couplings κ φ and σ are small. Then in the leading approximation the quartic part of the scalar potential in Eq. (9) is just given by so that in the limit | S |, | S | → ∞ the SM singlet VEVs tend to lie approximately along the D-flat direction s 1 ≈ s 2 . The inclusion of non-zero couplings σ and κ φ stabilize the potential along this direction resulting in large SM singlet VEVs, i.e., For the ratio of the SM singlet VEVs s 2 /s 1 one can obtain a more accurate estimate using the minimization conditions Eq. (11c) and Eq. (11d), If the VEVs of the SM singlets ϕ, s 1 and s 2 are rather large due to the large values of parameters Λ F and Λ S then M Z M S and from Eq. (17) it follows that tan θ 1. This is in marked difference to the situation in the simplest variants of the E 6 SSM where the EWSB conditions imply that M Z ∼ M S , forcing the SUSY spectrum to be substantially heavier than is required, for example, in the MSSM by collider searches, due to the large lower bound on M Z . In our numerical studies we take advantage of this behavior to search for solutions with a heavy Z with a mass well above current limits and a somewhat lighter SUSY scale than could be achieved in the simplest E 6 inspired extensions of the MSSM.
After fixing the parameters m 0 , tan θ, ϕ, Λ F (M X ) and Λ S (M X ), the remaining parameters listed after Eq. (8) are still free, up to the constraint of requiring a viable mass spectrum. In our analysis we mostly focus on the scenarios with small Yukawa couplings λ, σ,λ αβ , κ ij ,f iα and f iα that can lead to a set of relatively light exotic fermions which might be discovered at the LHC. Consequently for the high-scale boundary condition m 2 S (M X ) = m 2 S (M X ) = m 2 0 , the running of m 2 S and m 2 S is such that at the EWSB scale m 2 S m 2 S . Thus the value of tan θ is always extremely close to unity.

Particle Spectrum
The extension of the Higgs sector responsible for the breaking of U (1) N and EW symmetry also modifies the masses of the physical states in the spectrum compared to those found in the simplest variants of the E 6 SSM. The masses of the MSSM sfermions are almost unchanged. The smallness of the first and second generation Yukawa couplings leads to negligible mixing between the left-and right-handed states, so that their masses may be summarized as [56] m DR for which the one-loop corrections ∆g can be quite large, of up to 20%-30%. Pair production of gluinos would lead to a significant enhancement in p p → qqqq+E miss T + X, with X denoting any number of light quark or gluon jets [56]. This signature can be used to discover the model when mg is within the LHC reach, or exclude regions of SE 6 SSM parameter space where this is the case. As the SE 6 SSM contains the same colored states as in the E 6 SSM, the form of these radiative corrections ∆g is unchanged between the two models.

The Chargino and Neutralino Sector
On the other hand, the predictions for the masses of some other remaining states, that is, the neutralino sector, the exotic states and the Higgs sector, are rather different in the SE 6 SSM compared to the E 6 SSM. At the same time because the supermultiplet of the Z boson and the additional singlet superfields in the Higgs sector are electrically neutral, the fermion components of these superfields do not mix with chargino states,χ ± 1,2 . Therefore the tree-level chargino mass matrix and its eigenvalues are almost identical to the ones in the MSSM, the only difference being that µ → µ eff , where By contrast, the neutral fermion components ofĤ u ,Ĥ d ,Ŝ,Ŝ andφ as well as the neutral gauginos may all mix, leading to a Z E 2 = +1 neutralino sector that is twice as large as the MSSM neutralino sector. The neutralino mass eigenstates,χ 0 i , i = 1, . . . , 8, are linear combinations of the neutral Higgsino and singlino fieldsH 0 u , H 0 d ,S,S,φ, the binoB, the neutral SU (2) L gauginoW 3 , and the U (1) N gauginõ B , and are obtained by diagonalizing the mass matrix The 8×8 tree-level mass matrix in the basis (H 0 d ,H 0 u ,W 3 ,B,B ,S cos θ−S sin θ,S sin θ+ S cos θ,φ) can be written in terms of 4 × 4 sub-matrices as As noted above, we neglect the mixed gaugino soft mass M 11 arising from U (1) mixing.
For general values of the parameters and VEVs, the neutralino mass matrix of the SE 6 SSM is clearly more complicated than its counterpart in the MSSM. In the parameter space that we consider here, however, the mass matrix has a rather simple structure so that the MSSM-like neutralinos and the states beyond the MSSM tend not to mix. Inspection of Eq. (34) shows that two of the neutralinos, those that are a mixture ofB andS cos θ −S sin θ, have their masses set by the large value of M Z . For large values of the singlet VEV, the value of µ eff would be similarly large unless λ is taken to be sufficiently small. For large values of µ eff the states that are superpositions ofH 0 u andH 0 d become very heavy, leading to two very heavy pure Higgsino neutralinos that cannot account for the relic dark matter density. Therefore we restrict our attention to small values of λ so that µ eff 1 TeV. When λ σ while σ is rather small and M Z M S as implied by Eq. (16), the aforementioned states with masses set by M Z become very heavy and decouple from the rest of the spectrum. For very large s, Eq. (17) implies that tan θ ≈ 1 to high precision, and we find this is indeed the case in our numerical results below, allowing us to express the masses of these states approximately as When M S M Z and λ is small, the mixing of the remaining extra states, which are a mixture ofS sin θ +S cos θ andφ, and the MSSM-like neutralinos is also highly suppressed. The masses of these states are then approximately given by For large values of M S 1 TeV, these states will be heavy and, due to the lack of significant mixing, can also be ignored in the first approximation as far as determining the mass of the DM candidate goes. Provided this is the case, the neutralino DM candidate is expected to be predominantly MSSM-like, that is, a mixture ofH d , H u ,W 3 andB, with mass given by the lightest eigenvalue of the 4 × 4 sub-matrix in Eq. (33). In particular, since this matrix is identical to the MSSM neutralino mass matrix (with µ → µ eff ), when M S M Z the masses of the four lightest neutralinos are determined by µ eff , M 1 and M 2 as they are in the MSSM. In the CSE 6 SSM, the condition of universal gaugino masses at M X further implies that so that the MSSM-like neutralino sector in our case depends only on the two parameters µ eff and M 1/2 . These values can also be compared to the relations found in the CMSSM, which are quite different due to the modified RG flow in the SE 6 SSM.

The Exotic Sector
The states that are odd under Z E 2 do not mix with the ordinary MSSM states or the Higgs fields, forming a separate sector containing the second DM candidate as well as additional exotic states, some of which may generate spectacular collider signals. As discussed above, the DM candidate in this sector is expected to be an almost massless inert singlino, which is the lightest of the inert neutralinos. The inert neutralino sector is formed by the fermion components (S i ,H u α andH d α ) of the superfieldsŜ i ,Ĥ u α andĤ d α . The scalar components of the corresponding superfields also mix to form a set of inert charged and neutral Higgs scalars. The general inert neutralino and neutral inert Higgs mass matrices are 7 × 7 matrices. In the basis ((H d,0 , the inert neutralino mass matrix is of the form where contains the tree-level masses of the inert Higgsinos, µH0 Iα =λ αα s cos θ/ √ 2, in the absence of mixing with the inert singlinos, while the mixing is given by the 3 × 4 sub-matrix C I with elements The couplings of the inert singlinos are required to satisfy f iα ,f iα 10 −6 to yield almost massless hot DM candidates. Then, provided thatλ αβ 10 −6 , the mixing between the inert Higgsinos and the inert singlinos is entirely negligible, and the inert neutralinos correspond to two degenerate pairs of inert Higgsinos with treelevel masses given by Eq. (41) and three almost massless inert singlinos. The inert charginos similarly have tree-level masses given by µH± When the couplings f iα ,f iα are negligibly small, the mass matrix associated with the scalar components of the superfieldsŜ i ,Ĥ u α andĤ d α also simplifies in a similar fashion. In this case, the mixing between the neutral inert Higgs scalars (H u α and H d α ) and the inert singlets S i can be ignored and the corresponding mass matrix decomposes into a 3 × 3 singlet mass matrix and a 4 × 4 mass matrix for the inert Higgs scalars 7 . The family-diagonal structure of the couplingsλ αβ , as well as the fact that the off-diagonal soft scalar masses vanish at the GUT scale, ensures that the mixing between generations is very small. Thus the mass matrix for the inert singlets is approximately diagonal, with the tree-level masses for the inert singlet scalars given by For tan θ ≈ 1, the inert singlet masses are therefore ∼ M S , and so are somewhat lighter than M Z . In the absence of generation mixing, the inert Higgs mass matrix can be written as two 2 × 2 matrices, yielding the tree-level masses through off-shell W and Z bosons. They then decay into an inert singlino and an on-shell W or Z boson, or a Z E 2 even Higgs boson, through the mixing induced by the f iα andf iα superpotential couplings. When both of the produced states decay into gauge bosons it is expected that they should lead to enhancements in the rates of p p → Z Z + E miss T + X, p p → W Z + E miss T + X and p p → W W + E miss T + X. The choice of flavor diagonal couplings κ ij also means that there is no substantial mixing between generations of the exotic leptoquarks, D i and D i . The 6 × 6 mass matrix for the scalar leptoquarks reduces to three 2×2 matrices, giving the tree-level masses where (λv 2 sin 2β + 2σϕs sin θ) and the corresponding spin-1/2 leptoquark masses are µ D i = κ ii s cos θ/ √ 2. The same potentially dangerous contribution to the mixing that occurs in the inert Higgs mass matrices is also present here. To ensure that this does not lead to an instability of the physical vacuum, we require the couplings κ ij to be small as well, κ ij ∼ 10 −3 . As is the case for the inert Higgs states, this leads to the scalar leptoquarkD i being heavier, with masses of the order of M S , while the exotic fermions D i can be light. These exotic fermion states are colored and, once past threshold, can be pair produced at the 13 TeV LHC. They subsequently decay with missing energy via a decay chain involving an initial decay into an ordinary squark (quark) and an exotic L 4 fermion (scalar) component, through the couplings g D ij . This is followed by a decay involving the couplings h E iα of the exotic L 4 state into a lepton and inert Higgs or singlet (inert neutralino). If a hierarchy exists in the sizes of the couplings g D ij and h E iα as is present in the SM Yukawas, then such a process leads to an enhancement in signals with third generation final states, namely in p p → tt τ + τ − + E miss t + X and p p → bb τ + τ − + E miss T + X. For the branching ratio of these leptoquark decays to be significant, and also for the lifetimes of the exotic leptoquarks to be sufficiently short, the states associated withL 4 andL 4 should not be too heavy. The fermion and scalar components ofL 4 andL 4 form a set of exotic lepton and slepton states that do not mix with the other exotic fields. The fermion components lead to a pair of charged and neutral states L ± 4 andL 0 4,1 ,L 0 4,2 with degenerate tree-level masses given by The tree-level masses of the neutral exotic sleptons are given by where the mixing parameter is while those for the charged exotic sleptons read By tuning the above mixing parameter, the exotic sleptons could be made light enough so that the exotic D fermions decay rapidly enough. Alternatively, these states are allowed to be heavier than the spin-1/2 leptoquarks provided that the couplings g D and h E are taken to be sufficiently large. In the numerical results below, we find that taking values for these couplings of ∼ 10 −2 lead to lifetimes of the exotic fermions short enough to be consistent with constraints from Big Bang nucleosynthesis. At the same time, the impact of the couplings g D and h E on the mass spectrum and DM predictions is negligible for these small values of the couplings. Consequently they may be safely varied in this range without having any substantial impact on the other sectors.

The Higgs Sector
The Higgs sector of the SE 6 SSM is substantially different from the simplest version of the E 6 SSM, for which the spectrum of the Higgs bosons was explored in Ref. [16].
In the simplest case the sector responsible for the breakdown of the SU (2) L × U (1) Y × U (1) N gauge symmetry includes just H u , H d and S resulting in three CP-even, one CP-odd and two charged states. One CP-even Higgs state, which is predominantly SM singlet field, is always almost degenerate with the Z gauge boson. The qualitative pattern of the Higgs spectrum in the simplest variant of the E 6 SSM depends on the coupling λ which is a coupling of the SM singlet superfield S to the Higgs doubletsĤ u andĤ d , i.e., λŜĤ uĤd , as in the SE 6 SSM. If λ < g 1 the singlet dominated CP-even state is very heavy and decouples which makes the rest of the Higgs spectrum indistinguishable from the one in the MSSM. When λ g 1 the spectrum of the Higgs bosons has a very hierarchical structure, which is similar to the one that appears in the NMSSM with the approximate Peccei-Quinn (PQ) symmetry [127][128][129][130][131]. As a result the mass matrix of the CP-even Higgs sector can be diagonalized using the perturbation theory [131][132][133][134]. In this case the heaviest CP-even, CP-odd and charged states are almost degenerate and lie beyond the multi-TeV range whereas the mass of the second lightest CP-even Higgs state is set by the Z boson mass.
As was mentioned before in the SE 6 SSM the sector responsible for the breakdown of gauge symmetry involves five multiplets of scalar fields H u , H d , S, S and φ that give rise to ten physical degrees of freedom in the Higgs sector which form a set of charged and neutral Higgs bosons. The unbroken U (1) em symmetry ensures that the charged components of H u and H d do not mix with the other Higgs and singlet fields. Two massive charged Higgs states are formed by the linear combination with a mass given by The linear combination orthogonal to Eq. (51) constitutes the longitudinal degrees of freedom of the W ± bosons. In the absence of CP-violation in the Higgs sector, the real and imaginary parts of the neutral components of the Higgs and singlets fields do not mix, leading to three physical CP-odd Higgs bosons and five CP-even states. The Goldstone states that are absorbed by the Z and Z bosons are mixtures of the imaginary parts of where For phenomenologically viable scenarios with s v, tan γ goes to zero. Expressed in terms of the field basis (P 1 , P 2 , P 3 ), where the pseudoscalar mass matrixM 2 has elements (56) In the parameter space of interest here, the structure of the full 3 × 3 matrix is such that it can be approximately diagonalized analytically. Because M Z , M S M Z and we restrict our attention to small values of λ, the mixings between P 1 and P 2 , P 3 are rather small and may be safely neglected. In this approximation, the mass of one CP-odd state is set byM 11 . Thus it has almost the same mass as the charged Higgs states. The masses of two other CP-odd states are set bym + andm − which are given by As follows from Eq. (57) in some casesm − can be rather small so that the lightest CP-odd state A 1 becomes the lightest particle in the spectrum. This happens, for example, in the limit κ φ , µ φ , Λ F , Λ S → 0, when m DR A 1 vanishes and the superpotential possesses a global U (1) P Q PQ symmetry which is spontaneously broken by the VEVs s 1 , s 2 and ϕ. For small but non-vanishing U (1) P Q violating couplings, the state A 1 is a light pseudo-Goldstone boson of the approximate PQ symmetry and can be lighter than the SM-like Higgs. In this case, the decay h 1 → A 1 A 1 is kinematically allowed and can in principle lead to non-negligible branching fractions for non-standard decays of the SM Higgs [30]. Even for larger values of the couplings κ φ , µ φ , Λ F and Λ S , m A 1 may be small provided that the remaining parameters in Eq. (57) are tuned so thatm − → 0. It is important to note that in either case, the vanishing of m DR A 1 ≈m − does not also require that the lightest neutralino mass becomes small, as occurs for example in the PQ-symmetric NMSSM. Indeed, from Eq. (36) and Eq. (37) it is clear that the singlino dominated states should remain heavy, while mχ0 1 is governed by the values of the gaugino masses and µ eff . This means that by varying the other Lagrangian parameters for fixed M 1/2 and µ eff , the value of m A 1 can be chosen independently of mχ0 1 . In particular, for a given mχ0 1 this allows for the possibility of resonant annihilationsχ 0 1 , leading to regions of parameter space in which the well-known A-funnel mechanism is responsible for setting the DM relic density [135][136][137].
The real parts of H 0 d , H 0 u , S, S and φ form five physical CP-even Higgs states, h i , related by the unitary transformation where U h diagonalizes the CP-even Higgs mass matrix, M 2 . In the basis ( and using the EWSB conditions Eq. (11) to eliminate the soft Higgs masses, this has elements With the exceptions of M 2 45 , M 2 54 and M 2 55 , the size of the mass matrix elements is determined by the singlet VEVs s and ϕ. For small values of λ such that λs ∼ σs ∼ σϕ ∼ M S , it is therefore expected that all but the lightest state have masses of the order of the SUSY scale or heavier. In particular, for λ ∼ σ → 0 the element M 2 11 ∼ M Z M S , while all other matrix elements are substantially smaller. Thus the mass of the heaviest CP-even state is approximately degenerate with the Z mass. After neglecting all terms which are proportional to λv in Eqs. (60) it is easy to see that in the limit M S M Z the mass of another CP-even state is set by M 44 , i.e., this state is almost degenerate with the charged Higgs states, while the masses of two other CP-even states are determined by m + and m − , The mass of the lightest state, on the other hand, is bounded from above by the smallest element M 2 55 , i.e., the tree-level lightest CP-even Higgs mass satisfies Consequently h 1 ≈ S 5 is always light, and for M S M Z is SM-like in its interactions. While the upper bound Eq. (62) is larger than in the MSSM, it is still the case that radiative corrections are important for reaching m h 1 ≈ 125 GeV. Moreover, the by-now very precise measurement of the Higgs mass, m exp. h 1 = 125.09 ± 0.21 ± 0.11 GeV [138], strongly constrains the parameter space of SUSY models and necessitates a reliable calculation of the physical Higgs mass. In principle, the physical Higgs masses can be determined from the poles in the propagator after including the one-loop self-energies by solving is the tree-level Higgs mass matrix, evaluated here at M S , and Σ h (p 2 ) denotes the self-energies. The required one-loop self-energies are automatically included in the tools used in our numerical studies, described below. However, for large values of M S M Z this strategy leads to large logarithmic contributions to the Higgs masses due to heavy states, which should be resummed to get an accurate estimate for the Higgs mass. In our case, the discussion above indicates that the SUSY spectrum is split, containing many heavy scalars, notably the MSSM sfermions and the exotic scalars, as well as light neutralinos and exotic fermions. Such scenarios are well handled by an effective field theory (EFT) approach to calculate the lightest Higgs mass, in which the large logarithms are resummed. In the MSSM, the largest of these contributions is usually associated with the third generation sfermions, and in particular the stops. In the SE 6 SSM, there are also contributions from the heavy exotic scalars that should be accounted for. Because the exotic Yukawa couplingsλ αβ and κ ij are very small in the models we consider, these logarithmic corrections to the Higgs mass are very small and can be neglected compared to the contributions from the stops and other MSSM sfermions 8 . In our results below, to obtain the light CP-even Higgs mass we therefore make use of the known EFT calculation in the MSSM, which includes the dominant contributions to the Higgs mass. While a complete EFT calculation including the exotic states would be more accurate 9 , we expect that in this case the accuracy of our calculation should not be significantly reduced due to the small size of the exotic contributions.

Scan Procedure
To study scenarios in the CSE 6 SSM that are able to account for the observed relic DM density with a MSSM-like DM candidate, a dedicated CSE 6 SSM spectrum generator was created using FlexibleSUSY-1.1.0 [140,141] and SARAH-4.5.6 [142][143][144][145]. The generated code 10 , which internally also relies on some routines from SOFTSUSY [146,147], provides a precise determination of the mass spectrum by making use of the full two-loop RGEs and one-loop self-energies for all of the masses. Leading two-loop contributions to the CP-odd and CP-even Higgs masses taken from the known NMSSM [148] and MSSM [149][150][151][152][153] expressions were initially also included 11 , since the additional contributions from new states are expected to be small by virtue of their small couplings. However, as noted above, for the solutions presented below this fixed order Higgs mass calculation suffers from the effects of large logarithmic contributions that are not resummed and so a MSSM EFT calculation is employed to predict m h 1 instead. To do so, at the SUSY scale defined by M S = m DR t 1 m DR t 2 , with m DR tα given by Eq. (26), we performed a simple tree-level matching to the MSSM. In this simple matching procedure, the DR MSSM soft scalar masses are set at M S to their values obtained in the CSE 6 SSM after running from M X . The MSSM µ parameter is set to its effective value at M S , Eq. (30), while an effective MSSM pseudoscalar mass, (m A ) eff , is obtained from the effective soft bilinear The lightest CP-even Higgs mass was then calculated using SUSYHD-1.0.2 [156] to obtain a more accurate estimate for the SM-like Higgs mass. The remaining heavy CP-even Higgs masses were computed using the ordinary fixed order approach. As mentioned above, for the purposes of studying the MSSM-like DM candidate it is most convenient to directly vary the parameters M 1/2 and µ eff . For this reason, we implemented a solver algorithm in FlexibleSUSY that makes use of the semianalytic solutions to the RGEs. A similar algorithm has previously been used in studies of the constrained E 6 SSM, where it was described in Ref. [56]. The main advantage of this algorithm over the standard two-scale fixed point iteration is that by expanding all of the soft parameters at low-energies using the semi-analytic solutions, the EWSB conditions can be used to fix a subset of the input high-scale parameters in terms of the remaining input parameters. In particular, the low-energy soft Higgs and singlet masses can be written in the form for Φ = H d , H u , S, S, φ. Imposing the EWSB conditions Eq. (11) then allows m 0 to be fixed, as desired, along with tan θ, ϕ, Λ F (M X ) and Λ S (M X ). The parameters λ and M 1/2 remain free parameters that can be varied to set the mass and composition ofχ 0 1 .
To satisfy the limits on the Z mass, we take advantage of the mechanism described below Eq. (16) to set M Z well above the current limits, and so we set M Z ≈ 240 TeV. This requires a very large value of s = 650 TeV at the SUSY scale. time our numerical study was done.
Acceptably small values of µ eff 1 TeV for reproducing the DM relic density are then achieved for very small |λ|, though µ eff is still large enough to evade limits from LEP. In this study we focus on scenarios in which the LSP is either a mixed bino-Higgsino or pure Higgsino dark matter candidate. To do so, we considered |λ(M X )| = 9.15181 × 10 −4 and |λ(M X )| = 2.4 × 10 −3 , for both λ < 0 and λ > 0. Because tan θ ≈ 1 for such large values of s, this corresponds to |µ eff (M X )| ≈ 347 GeV and |µ eff (M X )| ≈ 898 GeV, giving values at the SUSY scale of |µ eff (M S )| ≈ 417 GeV and |µ eff (M S )| ≈ 1046 GeV, respectively 12 .
To prevent tachyonic states in the exotic sector, the exotic couplings cannot be too large, and for our scans we chose fixed values satisfyingλ αβ (M X ), κ ij (M X ) ≤ 3 × 10 −3 . Additionally, to simplify our analysis we took these couplings to be family universal withλ αβ (M X ) =λ 0 δ αβ and κ ij (M X ) = κ 0 δ ij . A SUSY scale somewhat below M Z was obtained by choosing small σ(M X ) = 2 × 10 −2 . Light inert singlinos in the spectrum were ensured by choosing extremely small values for the couplings f iα and f iα , while for simplicity we set the couplingsσ(M X ), µ φ (M X ), g D ij (M X ) and h E iα (M X ) to zero. We stress that the impact of the latter two sets of couplings on the quantities we investigate is numerically negligible. We have checked that their values could also be increased to satisfy constraints on the exotic lifetimes without altering our results. We also chose κ φ (M X ) = 10 −2 , and µ L (M X ) = 10 TeV. While the above fixed couplings impact the mass spectrum, they do not play a significant role in the predictions for dark matter, for the scenarios considered here in which the dark matter candidate is the lightest MSSM-like neutralino, and hence we do not scan over them.
For each parameter point in the scan, the GUT scale M X at which these values are set is defined to be the scale at which g 1 (M X ) = g 2 (M X ). This condition is solved iteratively, as described in Ref. [140]. We do not require that g 3 (M X ) is also unified, but this will be approximately fulfilled due to the inclusion of theL 4 and L 4 states. This is similar to what occurs in the E 6 SSM [20].
For λ ḡ, the tree-level upper bound on the SM-like Higgs mass is maximized for large tan β. We took tan β(M Z ) = 10 to saturate this limit. As in the CMSSM, We use this symmetry to fix M 1/2 ≥ 0. Setting B 0 = 0, we 12 The values of |µ eff | given are the mean values over all of the obtained valid solutions. The exact values of |µ eff (M S )| and |µ eff (M X )| vary over the parameter space scanned, since tan θ varies slightly over the scanned region, as it is an EWSB output parameter, and the RG evolution also changes slightly due to sparticle threshold corrections.   TeV, 20 TeV], respectively, to find solutions with the correct Higgs mass and an allowed DM relic density. The relic density and direct detection cross section were calculated numerically with micrOMEGAs-4.1.8 [157][158][159][160][161][162][163], using CalcHEP [164] model files automatically generated with SARAH. The values of the CSE 6 SSM parameters used are summarized in Table 2. 20,20] [−20, 20] tan β(M Z ) 10 10  For this choice of parameters the lightest neutralino is expected to be MSSMlike in its composition and couplings. At the same time, the spectrum and the RG flow of couplings in the CSE 6 SSM is very different to that in the CMSSM. While the two models may in this limit make very similar predictions concerning DM, the ranges of parameter space in which this occurs and their collider signatures can therefore be quite distinct. This makes it interesting to compare the CSE 6 SSM and CMSSM directly. To do this comparison, we also generated a CMSSM spectrum generator using FlexibleSUSY and SARAH as described above, and modified it to make use of the semi-analytic solver algorithm. The MSSM EWSB conditions were used to fix the common soft scalar mass m 0 and soft breaking bilinear B 0 at the GUT scale, and M 1/2 and A 0 were scanned over the same ranges as in the CSE 6 SSM. This was done for values of µ(M S ) fixed to the mean values obtained in the CSE 6 SSM, that is, |µ(M S )| = 417 GeV and |µ(M S )| = 1046 GeV, respectively.
The same fixed value of tan β(M Z ) = 10 was used. In this way we are able to present a more direct comparison of the two models, in which analogous parameters are approximately matched between the two 13 . The CMSSM solutions that we obtained have a heavy SUSY scale as well, so that we again used SUSYHD to compute the lightest Higgs mass. The predicted DM relic density and direct detection cross section were calculated in micrOMEGAs using model files generated by SARAH 14 .
In both models, valid points were selected by imposing the theoretical constraints that the point should have a valid spectrum with correct EWSB and no tachyonic states. We required that all couplings remain perturbative up to the GUT scale. Since we perform only a naïve matching to the MSSM in the EFT calculation, we allowed for an uncertainty of ±3 GeV in the result for m h 1 , which is somewhat larger than is reported by SUSYHD. For the CSE 6 SSM we accepted points with calculated light Higgs masses satisfying 122 GeV ≤ m h 1 ≤ 128 GeV, and for comparison we allowed the same range of Higgs masses in the CMSSM. A point predicting a relic density Ωh 2 greater than that determined by Planck observations [165], is effectively ruled out if one assumes a standard cosmological history. Points with a predicted relic density that does not exceed this value are not ruled out in the same way, though in this case additional contributions to DM are required. In our scans we excluded all points that have a predicted relic density (Ωh 2 ) th. > (Ωh 2 ) exp. . To make a clear comparison of the impact of collider bounds on the CSE 6 SSM and CMSSM, model specific limits should be applied to each. However in the CSE 6 SSM the RGEs drive the sfermions to masses which are substantially larger than the gaugino masses, creating a hierarchical spectrum that persists even with the decoupling of the Z mass from the rest of the spectrum. This means that typically LHC collider limits come from the gaugino sector, especially the gluino which is produced through strong interactions. The gluino decays in an MSSM-like manner and as a result the gluino mass limit set in the CMSSM in the heavy sfermion limit should, to a reasonable approximation, apply to the gluino in the CSE 6 SSM also 15 .
So that the reader can see where current and future collider limits should constrain the models we will show explicit gluino mass contours in each model, along with contours for the physical first generation squark mass, mũ 6 . Note that this is approximately degenerate with the remaining first and second generation squark masses, i.e., mq 1,2 ≈ mũ 6 .

Mixed Bino-Higgsino Dark Matter
We first consider cases with a light Higgsino mass term of |µ (eff) (M S )| ≈ 417 GeV. The results obtained in the CSE 6 SSM and the CMSSM for this value of |µ (eff) | are compared in Figure 1 and Figure 2.
In the top row of Figure 1 we compare the mass of the SM-like Higgs in the two models. In both we find solutions consistent with m h 1 ≈ 125 GeV, but the allowed regions in the M 1/2 − m 0 plane clearly differ quite substantially. For such large values of s and small values of λ the tree-level mass of the lightest CP-even Higgs in the SE 6 SSM is approximately the same as it is in the MSSM, (m DR h 1 ) 2 ≈ M 2 Z cos 2 2β, as follows from approximately diagonalizing the mass matrix in Eq. (60). Without substantial tree-level contributions from the additional F -and D-terms, a 125 GeV Higgs is achieved with large radiative corrections in the CSE 6 SSM as well as in the CMSSM. In principle, large enough loop corrections result from either large sparticle masses, particularly stop masses, or large stop mixing. However, increasing A 0 or M 1/2 to generate large mixings for fixed µ (eff) leads to the value of m 0 increasing as needed to satisfy the EWSB conditions. As a result in the solutions we obtain m 0 > A 0 , M 1/2 and large enough radiative corrections must arise from sufficiently heavy sparticle masses instead. The effect of the Higgs mass constraint can be clearly seen in the top row of Figure 1 and   is true for µ (eff) < 0 in Figure 2, though the position of the boundary is modified, leading to the much smaller range of acceptable m 0 values in the CSE 6 SSM for this value of |µ eff |. It should be noted, however, that these results are obtained for a single value of s. It is expected that if s and λ are allowed to vary while maintaining fixed µ eff , additional solutions would be obtained, as is found in the constrained E 6 SSM [56,57]. It is important to emphasize that in the CSE 6 SSM there is still additional parameter space available, and that the constraints shown here apply only for a single value of M Z in the model.
The large values of m 0 required result in a large SUSY scale and all scalars except the SM-like Higgs h 1 , and the lightest pseudoscalar A 1 in the CSE 6 SSM, are very heavy. In the top row of Figure 1 and Figure 2 we show contours of the gluino and first and second generation squark masses. The viable solutions that we find in the CSE 6 SSM all have squark masses mq 1,2 ≥ 5.4 TeV, while in our CMSSM solutions mq 1,2 ≥ 6.5 TeV, so that these states are not observable at the LHC. On the other hand, the small exotic couplings lead to light exotic fermions. For |µ eff (M X )| ≈ 347 GeV, the choice of κ 0 = 10 −3 leads to exotic D fermion masses of ≈ 1.3 TeV. Similarly, settingλ 0 = 10 −3 leads to inert Higgsinos with masses ≈ 580 GeV. Both sets of states are therefore light enough to be produced at the LHC and would be detectable via the signatures discussed in Section 4. Given the increasingly large SUSY scale required by LHC searches in constrained models, this makes searches targeting the exotic spin-1/2 leptoquark and inert Higgsino states attractive for still being able to probe the CSE 6 SSM parameter space. Because the exotic couplings cannot be too large in the scenarios considered here, improved limits on these states would strongly constrain the solutions we have found with very small values of |µ eff |.
In addition to the restriction on the allowed values of m 0 , there is also a lower bound on M 1/2 in both models, which is determined by the relic density constraint. The behaviour in the CMSSM in this case is well understood. When M 1 is sufficiently large,χ 0 1 is a nearly pure, light Higgsino that is underabundant [170]. The opposite limit, with small M 1/2 and M 1 µ, leads to an almost pure bino LSP that is overabundant, due to its small annihilation cross section. Therefore requiring Ωh 2 ≤ 0.1188 amounts to placing a lower bound on M 1/2 for fixed µ.
Since µ (eff) is small in this case, an acceptable relic density is achieved with relatively low values of M 1/2 . The minimal allowed value of M 1/2 in the CMSSM, M 1/2 ≈ 0.85 TeV, leads to M 1 ≈ µ and the LSP is a so-called "well-tempered" highly mixed bino-Higgsino state [171] that saturates the relic density. This region is evident in the middle rows of Figure 1 and The low allowed values of M 1/2 imply that in the light µ (eff) scenario the gluino as well as the ordinary neutralino and chargino states can be light. Though the location of the well-tempered strip differs in the two models, the masses of the gluino, neutralino and charginos are rather similar. For example, in both models in this strip mχ0 1 ≈ 370 GeV. In the CMSSM, we find that mg 2.1 TeV, the minimum value occurring in the well-tempered region. A very similar result can be seen in the CSE 6 SSM, with mg 2 TeV except for a narrow line of solutions where the gluino can be as light as mg ≈ 1 TeV.
For these solutions, the bino DM candidate is viable due to the A-funnel mechanism. In the CMSSM, m A is only light enough so that m A ≈ 2mχ0 1 at large tan β 50 [136]. Because we only considered tan β(M Z ) = 10 in our scans, m A > 6 TeV is always very heavy in our CMSSM results and the A-funnel region is not accessible. In the CSE 6 SSM, for a given value of tan β and M 1/2 one can make m A 1 ≈ 2mχ0 1 light by fine tuning A 0 appropriately. This corresponds to the lower boundary of the solution region in Figure 3. Therefore even for tan β(M Z ) = 10 light bino DM can satisfy the relic density constraint in the CSE 6 SSM. This does, however, imply a substantial fine tuning; in our scans, additional points were sampled from this region to overcome this.
In either the bulk or A-funnel regions, the gluino is thus observable at run II or at the high luminosity LHC (HL-LHC); indeed, gluino masses under 2 TeV are already rather close to the limits based on the most recent √ s = 13 TeV data and so LHC searches will soon be probing this part of the parameter space. Similarly, both models also predict light neutralinos and charginos with masses of a few hundred GeV. To be precise, our CMSSM solutions satisfy 366 GeV ≤ mχ0 1 ≤ 452 GeV, 428 GeV ≤ mχ0 2 ≤ 453 GeV and 419 GeV ≤ mχ± 1 ≤ 453 GeV, while in the CSE 6 SSM the ranges are 182 GeV ≤ mχ0 1 ≤ 426 GeV, 335 GeV ≤ mχ0 2 ≤ 438 GeV, and 335 GeV ≤ mχ± 1 ≤ 431 GeV. This suggests the neutralinos and charginos could also be discoverable at the HL-LHC [172] in the small µ (eff) case. The overall picture for the solutions presented with |µ(M S )| ≈ 417 GeV is of a split spectrum, with unobservably heavy scalars but light exotic fermions and EW-inos, as well as a sufficiently light gluino. This scenario would therefore predict interesting collider phenomenology in tandem with accounting for the observed DM relic density.
However, while small values of µ (eff) permit the neutralinos and gluino to be observable at the LHC, models with a highly mixed bino-Higgsino DM candidate are strongly constrained by null results from direct detection experiments. In the bottom rows of Figure 1 and Figure 2 we show theχ 0 1 -proton SI cross section for each sign of µ (eff) . In the region where (Ωh 2 ) th. matches the observed value, the direct detection cross section peaks at ∼ 10 −45 − 10 −44 cm 2 and is above the 90% exclusion limits set by LUX [173,174]. In both the CSE 6 SSM and CMSSM, the SI cross section in this part of the parameter space is dominated by t-channel exchange of the lightest CP-even Higgs h 1 . Thus in the leading approximation the SI part of χ 0 1 -nucleon cross section takes the form where  67) is set by the h 1χ 0 1χ 0 1 coupling g h 1 χ 1 χ 1 , which is given by 17 The values of these hadronic matrix elements are the default values used in micrOMEGAs, as determined in Ref. [162] from lattice results. A review of some recent determinations of the required sigma terms σ πN and σ s has been given in Ref. [175], while an extraction of these quantities from phenomenological inputs using chiral effective field theory has been presented in Refs. [176,177].
where the neutralino mixing matrix elements N ij are defined 18 in Eq. (31) and the Higgs mixing matrix U h is defined by Eq. (58). In the CSE 6 SSM, the contributions to this coupling involving the singlet mixing components N 1j , j = 5, 6, 7, 8, are negligible in our case and can be ignored. In the highly mixed case with |µ| ≈ M 1 and N 13 N 14 , the products N 11 N 14 and N 12 N 14 that appear above are large and the SI cross section is enhanced [178]. Therefore points with a mixed bino-Higgsino DM candidate that saturates the relic abundance are excluded, for both 19 signs of µ (eff) . As M 1/2 is increased (decreased) so thatχ 0 1 has a smaller (larger) bino component, the SI cross section decreases as N 14 → 0 (N 11 , N 12 → 0). Additionally, the reduction in Ωh 2 for larger values of M 1/2 implies a reduction in the local number density of WIMPs and thereby weakens the limits from direct detection. We estimate the extent to which this occurs by rescaling the given limits by the predicted relic abundance, so that a given set of values (mχ0 where σ p,LUX SI (mχ0 1 ) is the LUX limit at the WIMP mass mχ0 1 . Thus points away from the well-tempered strip may still avoid the direct detection limits. In the CSE 6 SSM, the presence of the A-funnel region also allows for solutions with (Ωh 2 ) th. ≈ (Ωh 2 ) exp. and a predicted SI cross section below current limits for λ < 0. Nevertheless, as discussed below future limits are expected to probe a substantial portion of the remaining parameter space. Therefore scenarios with small µ (eff) and a mixed bino-Higgsinoχ 0 1 are very tightly constrained.

Pure Higgsino Dark Matter
Scenarios with a heavy, pure Higgsino DM candidate are less constrained by direct detection limits due to both the weaker limits at high WIMP masses and the suppression of the SI scattering cross section for a pure Higgsino LSP [179]. Analyses of the CMSSM parameter space that also account for limits from collider searches suggest that this part of the parameter space is favored by experimental constraints [180], though scenarios with a relatively light LSP can still fit the data [181]. To see that this is also true in the CSE 6 SSM, in Figure 4 and Figure 5 we compare the  As in the previous case with small µ (eff) , the region in which we find solutions in the CSE 6 SSM is much smaller than in the CMSSM. The upper bound on m 0 again arises from tachyonic CP-even and CP-odd Higgs states that occur as |A 0 | is increased. At the same time, the minimum value of M 1/2 that satisfies the relic density constraint is much larger. This is because a relic density consistent with Eq. (66) requiresχ 0 1 to be nearly purely Higgsino with mχ0 1 ≈ 1 TeV, which is achieved for |M 1 | |µ (eff) | ≈ 1 TeV. The condition of universal gaugino masses at M X then means that the gluino is now very heavy along with the sfermions. In the CSE 6 SSM we find solutions with mg ≥ 3.8 TeV, compared to the minimum value of mg ≥ 5.7 TeV in the CMSSM scan. The prospects for an LHC discovery in this scenario are fairly poor in the CMSSM, as the gluino and all sfermions would be out of reach at run II.
For the CSE 6 SSM points shown in Figure 4 and Figure 5 we considered slightly larger exotic couplings with κ 0 =λ 0 = 3 × 10 −3 . The couplings are required to be large enough to ensure thatχ 0 1 is still the stable second DM candidate, rather than one of the exotic sector possibilities. The exotic fermions are correspondingly heavier, with masses satisfying 3 TeV ≤ µ D i ≤ 3.3 TeV and 1.63 TeV ≤ µH0 Iα ≤ 1.67 TeV, which also makes them unlikely to be observable at run II or at the HL-LHC. Note however that, in addition to being able to vary M Z , there is also some freedom to vary the exotic couplings to obtain lighter exotic states. We illustrate this in Figure 6, where we plot the valid solutions with κ 0 = 1.4 × 10 −3 , giving D fermion masses of µ D i ∈ [1.5 TeV, 1.6 TeV], comparable with the potential exclusion reach for third generation squarks at the HL-LHC [182]. For fixed |λ(M X )| = 2.4 × 10 −3 the effect of this is to slightly increase the minimum allowed value of M 1/2 outside of the A-funnel region. This is due to an increase in the calculated (Ωh 2 ) th. , which was already rather close to the value from Planck observations. The larger value of the relic density in turn arises because of the increase in µ eff (M S ) that results for smaller values of κ 0 in the RG running; this can be seen, for example, from Eq. (A.18). A compensating small reduction in λ(M X ) can be used to maintain the low-energy value of µ eff and therefore (Ωh 2 ) th. , in which case the smaller values of M 1/2 shown in Figure 4 and Figure 5 continue to be allowed. The presence of light exotics is an important possible signature that allows the model to be discovered when the SUSY breaking scale is very heavy, as well as distinguishing the E 6 inspired model from the CMSSM. As can be seen in the middle rows of Figure 4 and Figure 5, and in Figure 6, the prediction for the relic density in the CSE 6 SSM remains similar to that in the CMSSM. In both models a Higgsino with a mass of approximately 1 TeV saturates the observed value in Eq. (66). The narrow A-funnel region at lower M 1/2 is again accessible in the CSE 6 SSM by tuning A 0 to reduce m A 1 . As large mixings are no longer required to reproduce the relic density for |µ (eff) | ≈ 1 TeV, a large fraction of the solutions found have a predicted SI cross section below the current LUX limits. Points in both models with M 1/2 where the LSP transitions from being pure bino to pure Higgsino, i.e., where M 1 ≈ µ (eff) near the lower bound on M 1/2 , present a larger cross section that is in excess of the LUX limits. Therefore even for heavy µ (eff) in the CMSSM and CSE 6 SSM constraints can be put on the parameter space by direct detection searches. At larger M 1/2 (that is, where M 1 is significantly larger than µ (eff) ) the models currently evade the SI direct detection limits, and are very unlikely to be probed by direct collider searches in the near future if the exotic fermions in the CSE 6 SSM are not light. However, this part of the CSE 6 SSM, and CMSSM, parameter space will be constrained by results from XENON1T, as we now discuss in more detail.

Impact of Current and Future Searches
In Figure 7 we show the current and future regions probed by LUX and XENON1T for |µ (eff) (M S )| ≈ 417 GeV in both models. As described above, the existing 2015 LUX limits already essentially exclude the well-tempered bino-Higgsino solution region at low mg, i.e., low M 1/2 , where the SI cross section is enhanced by large mixings. The effect of the new 2016 limit is to extend this exclusion to larger gluino masses, despite the reduction in the predicted relic density and SI cross section. This is as expected from the results of dedicated MSSM studies [183,184]. XENON1T [185] is projected to exclude (or discover) even larger values of mg. In this CMSSM scenario, XENON1T can potentially excludeg masses up to 4 − 5 TeV.
The exclusions set by direct detection searches in the CSE 6 SSM are to some extent similar to those in the CMSSM. In particular, outside of the A-funnel region in the CSE 6 SSM, the LUX limits exclude gluino masses mg 3 TeV for µ (eff) > 0 and mg 2.5 TeV for µ (eff) < 0 in both models. Similarly, XENON1T will be able to probe gluino masses up to 4 − 5 TeV in the CSE 6 SSM as well. This accounts for a large fraction of as yet unexcluded solutions in the CSE 6 SSM.
However, as can be seen from the left column of Figure 7, some points in the A-funnel region will still not be excluded by LUX or XENON1T. These points have a suppressed SI cross section or do not saturate the relic density bound, or both.  [174] LUX limits (pink and red, respectively) and points that are not currently excluded but are within the projected reach [185] of XENON1T (blue). In each case, the exclusion limit is determined according to Eq. (70). Finally, points that are not excluded by any limits but that predict a relic density that is less than 90% of the measured value are shown in yellow, while those points with 0.9 < (Ωh 2 ) th. /(Ωh 2 ) exp. ≤ 1 are shown in green. This is also true in both models for those points not excluded at large mg. Points close to the well-tempered region, where the amount of mixing is still relatively large, only escape being excluded if they lead to an extremely small relic density. If it is required that the LSP explains a substantial fraction of the observed relic abundance, for example (Ωh 2 ) th. /(Ωh 2 ) exp. > 0.1, then these points are removed. This is illustrated in Figure 8, where we show the variation in the bino fraction for points satisfying this criterion. The effect of the direct detection limits is to heavily restrict the amount of mixing allowed. The surviving points are forced to either be almost pure bino, at small M 1/2 , or almost pure Higgsino at large M 1/2 and hence having a heavy SUSY spectrum. While the A-funnel points will not be observable at XENON1T, the fact that mg 2 TeV for these solutions means that most are in reach of LHC searches targeting gluinos. This highlights the complementary nature of collider and direct detection searches; similar observations have been made for the CMSSM (see, for example, Ref. [186]). Given the similarity of the lightest Z E 2 = +1 neutralinos in the CSE 6 SSM to the ordinary MSSM neutralino sector, it is not so surprising that this continues to hold. In particular, results from XENON1T will be able to constrain the CSE 6 SSM (and CMSSM) at much higher SUSY scales than are expected to be reached at the LHC. We conclude from this that direct detection searches, if no WIMPs are observed, will be able to place indirect limits on the sparticle masses much higher than can be achieved at run II, when the neutralino does not annihilate via special mechanisms such as the A-funnel. Thus direct detection limits are a particularly strong constraint on the CSE 6 SSM parameter space.
The solutions that we find with a heavy Higgsino DM candidate lead to gluino and MSSM sfermion masses beyond the exclusion reach at run II. This is shown in Figure 9. Consequently there are effectively no constraints on this part of parameter space coming from collider limits, at least in the CMSSM. In the CSE 6 SSM, the possibility of light exotic fermions, as in Figure 6, would allow for the model to be discovered even if all MSSM-like states and exotic scalars are heavy. However, if these states are also heavy then limits from direct detection searches are much more effective at constraining the parameter space.
Prior to the most recent LUX limits, all of our solutions with heavy |µ (eff) | were consistent with direct detection limits. This is no longer true for the new 2016 LUX limits, which now exclude points with M 1 ≈ µ (eff) . Therefore the current direct detection limits are already probing the heavy |µ (eff) | parameter space. Scenarios with a highly mixed bino-Higgsinoχ 0 1 accounting for at least 10% of the relic abundance are again all excluded by the current limits. This is shown in Figure 10. Thus in the case that the LSP is relevant for addressing the DM problem, direct detection limits place stringent constraints on the allowable bino-Higgsino admixture. More extensive coverage of the valid, low mixing regions will require results from XENON1T, however.
It is clear that in the CSE 6 SSM, results from XENON1T will place very strong constraints on the parameter space, as it should be possible to cover almost all of the allowed region. As for the previous small |µ (eff) | case, the surviving regions are the A-funnel region and at very large mg. In this scenario the A-funnel region cannot be searched for directly at the LHC; from the left column of Figure 9 it can be seen that the gluino mass is always greater than ≈ 4 TeV. An interesting question is to what extent indirect DM detection experiments or results from flavor physics can GeV, after also requiring that the LSP accounts for at least 10% of the observed relic density. The scaling of the limits and the colour coding is the same as in Figure 7.
constrain the CSE 6 SSM here; we leave this for a future study. On the other hand, for very heavy spectra without light exotic fermions neither collider searches nor results from XENON1T will constrain the CSE 6 SSM or the CMSSM. Even more sensitive direct detection experiments, such as results from LZ, will be required to directly search for these scenarios.
It should be noted that the large number of solutions for which (Ωh 2 ) th. is indicated as being less than 90% of the Planck value in Figure 9 still account for a very large fraction of the observed relic abundance. Small changes in λ(M X ), or µ(M X ) in the CMSSM, are enough to closely reproduce the value in Eq. (66) without significantly changing any other results, unlike in the light Higgsino case where the DM candidate is severely underabundant assuming a standard freeze-out scenario. At large M 1/2 the relic density is still fully accounted for by the Higgsino DM candidate. Unfortunately, while these scenarios can explain the observed DM density entirely, the expected collider phenomenology is rather uninteresting as all states are too heavy to be observable.

Conclusions
We have studied dark matter and LHC phenomenology implications in both the CMSSM and a constrained version of an E 6 inspired model (CSE 6 SSM). The SE 6 SSM is a string inspired alternative to the MSSM, where the break down of the E 6 gauge group leads to a discrete R-parity and a U (1) N gauge extension surviving to the TeV scale that forbids the µ-term of the MSSM. The charges allow the standard see-saw mechanism for neutrino masses and a leptogenesis explanation of the matter-antimatter asymmetry. The model contains exotic states at low energies needed to fill three generations of complete 27-plet representations of E 6 and ensure anomaly cancellation, and can give rise to spectacular collider signatures. A single additional discrete symmetry which commutes with E 6 is imposed to forbid FCNCs and this along with R-parity leads to multiple dark matter candidates. In this paper we focused on scenarios where the lightest exotic particle is an extremely light singlino which forms hot dark matter, but contributes negligibly to the relic density. We showed that the relic density can instead be explained entirely by the lightest MSSM-like neutralino.
We have performed a detailed exploration of the parameter space of both the CMSSM and CSE 6 SSM and compared the results. We find that in both models one may fit the observed relic density with a pure Higgsino neutralino that has a mass around 1 TeV. Alternatively this can be achieved with a mixed bino-Higgsino dark matter candidate, requiring a fine tuning of M 1 and µ (eff) to obtain the well-tempered strip and this can work for lighter neutralino masses (≈ 400 GeV in our example). However recent direct detection results have placed strong limits on this mixing, placing a significant tension between fitting the observed relic density and evading direct detection limits. Indeed we find that the recent LUX 2016 direct detection limits constrain Higgsino-bino mixing such that it rules out this well-tempered strip for both models for light and heavy neutralinos.
However we also found that the CSE 6 SSM can have special A-funnel solutions where the correct relic density can be achieved for lighter M 1/2 , a scenario that is only possible in the CMSSM for a much larger tan β than is considered here. Such scenarios exist for both the heavier and lighter Higgsino masses considered. For lighter Higgsino masses this A-funnel region, which can escape direct detection limits even from the future results of XENON1T, will be probed by the LHC run II. This demonstrates an important complementarity between collider searches and experiments for the direct detection of dark matter.
Such special regions aside however it is now rather difficult to explain dark matter in the lighter scenarios. Nonetheless if one requires only that the relic density is not too large then many scenarios are still viable and have phenomenology that will be probed with run II of the LHC. Since the sfermions will still be very heavy the main signatures arise from the production of gluinos, charginos and neutralinos, with MSSM-like signatures. On the other hand the leptoquarks in the CSE 6 SSM can be light enough to detect even when the SUSY scale is very heavy. These exotic states would lead to considerable enhancement of p p → tt τ + τ − + E miss t + X and p p → bb τ + τ − + E miss T + X, where X stands for any light quark or gluon jets.
Heavier scenarios with a Higgsino dark matter candidate of around 1 TeV are also not currently constrained so much by direct detection and it is possible to fit the relic density in both the CMSSM and CSE 6 SSM for a wide range of the parameter space. These scenarios have a rather heavy spectrum which is not accessible to the LHC, however they will be probed by future direct detection experiments, such as XENON1T which will be able to probe most of the viable solutions we have found in the CSE 6 SSM. Therefore the future impact of XENON1T on these models will be very significant.

A RGEs
In our analysis, the SUSY preserving and soft SUSY breaking parameters at M S are obtained from the GUT scale boundary conditions by running them using twoloop RGEs. These RGEs were automatically derived using SARAH-4.5.6, which makes use of the general results given in Refs. [187][188][189]. For completeness, in this appendix we summarize the complete set of RGEs used to obtain our results. For a general parameter p, the RG equation for p is expressed in terms of the one-and two-loop β functions, β (1) p and β (2) p respectively, according to where t = ln Q/M X gives the scale at which p is evaluated.

A.1 Gauge Couplings
In general, kinetic mixing of the U (1) Y and U (1) N leads to a set of RGEs for the Abelian gauge couplings involving a set of off-diagonal gauge couplings. In the triangle basis of Eq. (7), these RGEs can be written where the matrix of β functions is B = β g 1 g 2 1 2g 1 g 1 β g 11 + 2g 1 g 11 β g 1 0 g 2 1 β g 1 + 2g 1 g 11 β g 11 + g 2 11 β g 1 .

(A.3)
The off-diagonal β function β g 11 is rather small, with β (1) g 11 = − √ 6/5 at one-loop. As discussed in Section 2, the effects of kinetic mixing are therefore small if g 11 vanishes at the GUT scale, and so we neglect it. When this is done, the two-loop RGEs for the diagonal Abelian gauge couplings are The β functions for the SU (2) L and SU (3) C gauge couplings are the same irrespective of whether or not the kinetic mixing is taken into account. They are

A.3 Superpotential Bilinear and Linear Couplings
The β functions of the bilinear superpotential parameters µ φ and µ L read while that for the linear superpotential parameter Λ F is

A.4 Gaugino Masses
The two-loop β functions for the soft gaugino masses are β (1) Tr Tr As mentioned above, kinetic mixing in this class of E 6 inspired models is small and so we neglect the mixed gaugino mass M 11 .

A.6 Soft-breaking Bilinear and Linear Couplings
The β functions for the soft-breaking bilinears are given by The two-loop β function for the soft-breaking linear coupling Λ S is

A.7 Soft Scalar Masses
In writing down the two-loop β functions for the soft scalar masses, the following quantities are defined, − 6κ T κ * Tr T κ * T κT − 4T κT κ * Tr Tλ * λT − 4κ T κ * Tr Tλ * Tλ T − 12g DT g D * Tr g D g D † m 2 * Q − 12g DT g D * Tr g D m 2 * D g D †  − 6 Tr g D κ † κm 2 * D g D † − 6 Tr g D κ † m 2 * D κg D † − 18 Tr g D m 2 * D g D † g D g D † − 6 Tr g D m 2 * D g D † y DT y D * − 6 Tr g D m 2 * D g D † y U T y U * − 6 Tr g D m 2 * D κ † κg D †