Dark Matter, Sparticle Spectroscopy and Muon $(g-2)$ in $SU(4)_c \times SU(2)_L \times SU(2)_R$

We explore the sparticle mass spectra including LSP dark matter within the framework of supersymmetric $SU(4)_c \times SU(2)_L \times SU(2)_R$ (422) models, taking into account the constraints from extensive LHC and cold dark matter searches. The soft supersymmetry-breaking parameters at $M_{GUT}$ can be non-universal, but consistent with the 422 symmetry. We identify a variety of coannihilation scenarios compatible with LSP dark matter, and study the implications for future supersymmetry searches and the ongoing muon g-2 experiment.


Introduction
In recent years, a large body of experimental data, including Higgs boson measurements [1,2] and cosmological observations [3][4][5][6], have provided increasingly strong constraints on new physics beyond the Standard Model (SM). Nonetheless, some new physics is required to explain, for instance, the observed solar and atmospheric neutrino oscillations, provide a plausible dark matter (DM) candidate, explain the observed baryon asymmetry in the universe, help understand electric charge quantization, etc.
Because of its gauge structure, the 422 model naturally allows one to consider nonuniversal soft SUSY breaking masses at M GU T for the gluino and scalar sectors, leading to significant differences from other GUTs (Grand Unified Theories). Note also that left-right symmetry may not hold at M GU T . We explore the implications for particle spectroscopy focusing, mostly, on the yet to be found supersymmetric partners of the SM particles, as well as LSP DM. We identify a variety of coannihilation scenarios that are compatible with the current searches at the LHC and the presence of primordial LSP DM. In addition, supersymmetric contributions to the anomalous magnetic moment of the muon (g − 2) could help explain the discrepancy between the SM prediction and the experimental value [30]. In particular, we identify models and SUSY mass relations for which the neutralino relic density is consistent with the cosmological bounds and explore how their parameter space is constrained by the LHC data. These predictions will be tested by the ongoing and future DM and LHC searches. 2 The SU (4) c × SU (2) L × SU (2) R model We start by briefly reviewing the salient features of the 422 model [25,27], which shares many features, but also shows fundamental differences from standard GUTs such as SU (5) and SO (10). The 422 gauge symmetry can be obtained from a spontaneous breaking of SO(10) by utilizing either the 54 dimensional or the 210 dimensional representation. The breaking of SO(10) with a Higgs 54-plet yields two connected components, namely the 422 subgroup and Σ 67 .422, where Σ 67 is a rotation by π in the 6-7 plane [31,32].
Instead of Σ 67 , we could alternatively use the rotation C given by C = (Σ 23 )(Σ 67 ), which is also an element of SO(10). This C-transformation interchanges the left-handed and right-handed fields and conjugates the representations. The SO(10) breaking with a Higgs 210-plet also yields the 422 symmetry, but the C-symmetry (and left-right (LR) symmetry) is explicitly broken in this case.
Previous investigations of particle spectroscopy in 422 models have relied on the presence of left-right symmetry [33], in order to keep the number of soft SUSY breaking parameters to a minimum. In this paper, we go a step further and assume that the soft scalar masses do not necessarily respect the discrete left-right symmetry. In principle, in the left-right asymmetric 422 model, the soft gaugino masses are not necessarily equal, M SU (2)L = M SU (2)R , and the SM hypercharge generator is given by where the SU (4) gaugino mass parameter M 4 will be identified with M 3 . Then, if the 422 gaugino masses are independent, this will also hold for the SM gaugino masses. With additional assumptions, the number of free parameters can be reduced. Here, we will follow the approach of Ref. [26,33] for the gaugino sector. Supplementing 422 with a discrete left-right C-symmetry, reduces the number of independent gaugino masses from three to two. Indeed, while the gaugino masses associated with SU (2) L and SU (2) R are the same, the gluino mass, associated with SU (4) c , in principle can be different. The hypercharge generator from 422 implies: Our framework is the following: we assume that SUSY breaking occurs in a hidden sector at a scale M X > M GU T , via a mechanism that generates flavour-blind soft terms in our visible sector. Between the scales M X and M GU T , while the theory still preserves the 422 symmetry, renormalisation and additional flavour symmetries may induce nonuniversalities for soft terms that belong to different representations (while particles that belong to the same representation have common soft masses).
We employ GUT relations among the soft terms derived from the unification group structure [36][37][38][39][40][41]. The soft terms for the scalar fields in an irreducible representation r of the 422 unification group are defined as multiples of a common scale m 0 : while the trilinear terms are defined as Here, Y r is the Yukawa coupling associated with the r representation and a 0 is a dimensionless factor, which is representation independent (the representation dependence is taken into account in the Yukawa couplings).
In view of the above discussion, we expect the following: • Gluino masses: We assume the relation in eq. (2.2) among gaugino masses. We will see that this relation will yield gluino coannihilation as a viable scenario [34,35], which was absent in other groups, namely SO(10), SU(5) and flipped SU(5) [36].
• Soft masses: Sfermions are accommodated in 16-dimensional spinor representations and their soft mass parameter is m 16 . The electroweak MSSM doublets lie in the 10-dimensional representation with D-term contributions that result in splitting of their soft masses. Indeed, m 2 H u,d = m 2 10 ± 2M 2 D , and, in our notation: • LR asymmetric 422: In this case there is additional freedom, as the left-right asymmetry introduces a new parameter where m L is the mass of the left-handed sfermions (that preserve the definition of m 16 = m 0 ), and m R the mass of the corresponding right-handed ones.

Exploring the model: Methodology
We perform parameter space scans similar to [36], where the initial conditions of the soft terms are determined by a unification group that breaks at M GU T (defined as the scale where the g 1 and g 2 couplings meet, while g 3 (M GU T ) is obtained by requiring α s (M Z ) = 0.187). For our analysis we use Superbayes [42][43][44], a package to perform statistical inference of SUSY models which is linked to SoftSusy [45] to compute the SUSY spectrum, to MicrOMEGAs [46] and DarkSUSY [47] to compute DM observables, SuperIso [48] to compute flavour physics and the muon g − 2, and it uses Multinest [49] for sampling the parameter space of the models. The likelihood function, which drives our exploration of regions of the parameter space where the model predictions fit the data well, is built from the following components: ln L Joint = ln L EW + ln L B(D) + ln L Ωχh 2 + ln L DD + ln L Higgs + ln L SUSY + ln L g−2 . (3.1) Here: • L EW is the part corresponding to electroweak precision observables, where constraints from LEP and Tevatron are implemented as summarised in [50,51].
• L Ωχh 2 is for measurements of the cosmological DM relic density. Assuming that the lightest neutralino is the dominant DM component, we include constraints from Planck temperature and lensing data Ω χ h 2 = 0.1186 ± 0.0031 [6], with a (fixed) theoretical uncertainty τ = 0.012, following Refs. [43,56,57], to account for numerical uncertainties.
• L DD is for constraints from direct DM detection searches; we apply data from the Xenon-1T [14] and PICO-60 [16] experiments. The likelihood is computed with the DDCalc code [58], and for the computation of the spin-independent and spindependent neutralino-nucleon cross-sections, we adopt hadronic matrix elements determined by lattice QCD [59,60].
• L Higgs implements bounds obtained from Higgs searches at LEP, Tevatron and LHC via HiggsBounds [61] and LHC Higgs-boson bounds [1,2]. For this we use HiggsSignals [62], assuming a 2 GeV theoretical uncertainty in the lightest Higgs mass computation.
• L SUSY stands for sparticle searches at colliders. The constraints from SUSY searches at LEP and Tevatron are evaluated following the prescription proposed in [63].
The MultiNest [49] algorithm is used to scan the parameter space and identify regions compatible with the data, though we do not perform any statistical interpretation of the results. Instead, we select only model points that predict the value of all observables within the 2σ interval (with σ obtained by summing in quadrature the experimental and theoretical errors); however, we go to 3σ for muon g −2. We combine the samples produced using logarithmic and linear priors of the model parameters. We finally produce scatter plots showing the correlations of pairs of parameters and/or observables in various planes.

Results of the parameter space scan
As mentioned above, we perform two scans: one with logarithmic priors that scan over a wide range of parameters as shown in Eq. 4.1 and another one with flat priors, that are appropriate for looking for correlations. In the first case, we find many points with Higgsino DM and resonances in the annihilation channels, while flat priors are more appropriate when searching for coannihilations.
The 422 non universal soft masses are parametrized using the following definitions: Here M 1 is determined by eq. (2.2). Note that M 3 and m R are allowed to be negative.
It is well known that if the required amount of relic DM is provided by neutralinos, particular mass relations must be present in the supersymmetric spectrum [64][65][66][67][68][69][70][71]. We therefore use these mass relations, together with the neutralino composition, in order to classify the points that pass the constraints discussed in Sec. 2, according to the following criteria: The Higgsino-like fraction of the lightest neutralino mass eigenstate is characterized by the quantity where the N ij are the elements of the unitary mixing matrix that correspond to the Higgsino mass states. In this case, the lightest chargino χ ± 1 and the second lightest neutralino χ 0 2 are almost degenerate in mass with the χ 0 1 . The couplings to the SM gauge bosons are not suppressed and χ 0 1 pairs have large cross sections for annihilation into W + W − and ZZ pairs, which may reproduce the observed value of the relic abundance. Clearly, coannihilation channels involving χ ± 1 and χ 0 2 also contribute.
A/H resonances: The correct value of the relic abundance is achieved thanks to s-channel annihilation, enhanced by the resonant A propagator. The thermal average σ ann v spreads out over the peak in the cross section, so that neutralino masses for which 2m χ m A is not exactly realized can also experience resonant annihilations.
τ coannihilations: The neutralino is bino-like, annihilations into leptons through t-channel slepton exchange are suppressed, and coannihilations involving the nearly-degenerateτ 1 are necessary to enhance the thermal-averaged effective cross section.
τ −ν τ coannihilations: This is similar to the previous case but, in addition, the ντ is nearly degenerate in mass withτ 1 .
Thet 1 is light and nearly degenerate with the bino-like neutralino. These coannihilations were found to be present also in the flipped SU(5) model, but not in SO (10) or SU(5) [36].
What is particularly interesting in the 422 model, which distinguishes it from the other GUT groups, is that in this case we get three additional modes of coannihilation, namely: •χ + coannihilations: The lightest chargino is light and nearly degenerate with the bino-like neutralino.
•g coannihilations: since the gluino can be relatively light and nearly degenerate with the bino-like neutralino.
•b coannihilations: since, due to the LR asymmetry, theb can be light and nearly degenerate with the bino-like neutralino [72].

GUT inputs and Planck compatible regions
In this subsection we present the phenomenological consequences of relaxing the universality of the SUSY breaking terms following the 422 pattern. Specifically, we concentrate on the differences with respect to the groups based on SO(10) and SU(5) that assume gaugino mass universality. As discussed in Sec. 2, following the 422 group structure, the gaugino masses are not universal at the GUT scale, and we also assume left-right asymmetry for the scalar soft masses at the GUT scale. In the figures that follow, we show combined points arising from the linear and logarithmic sampling of parameters. In both cases,χ + coannihations and Higgsino DM are the points found most frequently.
In Figure 1 we clearly observe that the vast majority of points lie in the upper right region. Points withχ + coannihilations have a preference for x LR > 0. We find that obtaining the correct prediction of m h imposes a correlation between the signs of M 3 and A 0 . The majority of models satisfying this constraint correspond to M 3 > 0 and A 0 < 0, however, a few models with M 3 < 0 and A 0 < 0 are also found.
We observe that most of the classes of models satisfying the Planck constraints can be found even if sfermion LR symmetry is preserved. However, points withτ −ν τ − χ (red asterisks) andb − χ (maroon down triangles) coannihilations appear only when the LR symmetry is broken (x LR < 1). Although the constraints imposed by the anomalous magnetic moment of the muon and the LHC searches will be discussed in the following sections, we find it illustrative to anticipate our results in all the plots. Therefore, we enclose in a green circle the points that explain the discrepancy of the experimental bound with respect to the SM prediction at the 3 − σ level. Similarly, points excluded by the LHC searches, according to the analysis presented in Sec. 5, are enclosed in magenta squares.  In both cases, the lightest chargino is mostly wino and x LR > 0. We also find points withχ + − χ coannihilations for different values of M 2 /M 3 , for cases whereχ + is not a pure wino. Coannihilationsτ −ν − χ where theτ is mostly left-handed, correspond to |x LR | < 1, while larger values of |x LR | allow right handed stau dominatedτ − χ, as well ast − χ coannihilations. In order to classify coannihilations of the LSP with a sparticlep, we used as a criterion a mass ratio mp/m χ = 0.1, although we can see that coannihilations among several particles are also possible. Some coannihilation points classified asτ − χ andτ −ν − χ show ratios M 2 /M 3 typical of the chargino coannihilations, indicating that coannihilationsτ −ν −χ + − χ are possible. The neutralino masses satisfying the Planck constraints are displayed in the right panel of Figure 2. We can see that points satisfying the muon (g − 2) constraints require values of m χ below 500 GeV, and all of them correspond to ratios M 2 /M 3 below 2. Higgsino like neutralino masses are in the 1 TeV range, similar to what was found in other GUTs [36]. We can see again that LR asymmetry allowsτ −ν − χ andb − χ coannihilations that are not present when the LR symmetry holds (x LR = 1). Figure 3 provides a clear picture of the constraints imposed by gaugino mass relations, as shown in eq. (2.2), along with the Planck constraints. These may imply a second condition on the gaugino masses due to relations of the LSP mass with other particles in order to fulfill the relic density requirements; this relation is more diffuse, due to RGE dependence at the low energy mass scale. In the left panel, we can see that two ratios are favored:

Higgs mass and Muon (g − 2)
Connecting the Higgs boson discovery with the lightest neutral SUSY particle of the MSSM requires a rather heavy SUSY spectrum that makes it challenging to explain the discrepancy between the experimental value of (g µ − 2) and its SM prediction, at least in the simplest SUSY models. The value δa SU SY µ = (28.7 ± 8.2)10 −10 is difficult to reach in models with universal soft terms. Even after allowing non-universalities at the GUT scale for scalar soft terms, like the models based on SU(5) and SO(10) of Ref. [36], the SUSY contribution remains below the central value. Therefore, we wish to investigate whether the pattern of soft terms introduced by the 422 symmetry can result in models with a larger contribution to muon (g − 2).
To display the relevance of the particular relation of soft terms introduced by the 422 symmetry, we present in the upper panels of Figure 4 the variation of the prediction of δa SU SY µ with the GUT gaugino mass ratios (left) and the LR asymmetry (right). We can observe that the highest values of δa SU SY µ are obtained for M 2 /M 3 ratios that favor chargino coannihilations. We also see that only M 2 /M 3 below 2 can result in a a SUSY contribution compatible with (g µ − 2). The right upper panel shows that the LR soft mass asymmetry results in some points withτ − χ coannihilations crossing the δa SU SY µ lower bound. These points include cases where the stau is mostly left-handed, so thatτ −ν − χ coannihilations take place.
In the lower panels of Figure 4, we explore the m χ values that can simultaneously explain the experimental value of the Higgs mass and the discrepancy δa µ . A SUSY contribution to δa SU SY µ above the lower bound is possible for points with chargino and stau-coannihilations for m χ < 500 GeV. Note that many of the points of [73,74] satisfying  [14] and the dotted line below, to its recent update [75].The dash and the dot-dash lines correspond to the projected sensitivities from LZ [12] and DARWIN [15]. Right panel: same plot for the SD neutralino-neutron cross section and the projected limit from LZ [12].
δa SU SY µ are now excluded because of the m h bound (and will be further constrained by the LHC, as we will discuss in section 5).

Dark Matter Searches
The current choice of soft terms allows for models where the neutralino relic density is located inside the cosmological bounds in scenarios that imply different relations among SUSY masses. In each scenario, the composition of the LSP determines its detection prospects.
In Figure 5 we display the Spin Independent (SI) and Spin Dependent (SD) neutralinonucleon cross sections as functions of the neutralino mass, comparing the theoretical predictions with updated experimental bounds, as summarised in the respective figure captions (the line corresponding to the latest announced update from Xenon-1T [75] is also included). The SI bounds are the most restrictive, and the current bounds from Xenon-1T [14] exclude many models where the LSP has a relevant Higgsino component. According to our classification of section 4, these correspond to Higgsino DM (h f > 0.1). However, points where the LSP has a smaller Higgsino component, such as A/H resonances, are on the scope of coming experiments like LZ [12]. Furthermore, even models where the LSP has a high degree of bino purity can be reached at sensitivities such as the ones expected with a multi-ton mass experiment like the DARWIN project [15]. These experiments can explore most of the models presented here, including the (g µ − 2) favored points.
Regarding the SD neutralino-nucleon cross section, bounds from experiments sensitive to neutralino-proton interactions like PICO [16] are less restrictive than the SI case. Models predicting SI cross sections on the Xenon-1T bound are below the PICO bound by two orders of magnitude. The predictions for neutralino-neutron cross sections, like the ones displayed in the right panel of Figure 5 are higher. However, we can see on the figure that the LZ prospects still favors the SI over the SD interaction sensitivity.  Figure 6 depicts the total non-relativistic LSP annihilation cross section times relative velocity as a function of the neutralino mass. Here as well, we see that for a subset of points like Higgsino DM and A/H resonances where the neutralino has an important Higgsino component and we hope that further light can be shed in the near future.

LHC searches
In previous sections we have seen how the implementation of the 422 group expands the possibilities for DM predictions with respect to more constrained models. In this section, we derive sparticle mass correlations, combining the experimental and cosmological data summarised in the previous sections with results from applying the LHC constraints. The results indicate the complementarity of DM experiments and of LHC SUSY searches for the asymmetric 422 group, similar to what was found in other GUTs [36]. The so far unsuccessful searches for SUSY particles impose severe bounds on their spectrum and interactions. However, it is not straightforward to translate these bounds to SUSY masses, because the ATLAS and CMS experiments typically show results in a model-dependent fashion. Namely, the recast of the data is done in the framework of so-called Simplified Model Spectra (SMS) that can be considered indicative rather than conclusive for real models [77,78].
Every SMS can be defined by a set of hypothetical particles and a sequence of their products and decay modes. Therefore, to confront the theoretical models against the LHC bounds, the predictions must be expressed in the SMS language. There are several tools designed for such purpose [79][80][81] and by using these packages, we can go one step further in comparing our models with LHC data, applying this procedure to a large number of models without the need of a huge computing power [82,83].
In our analysis, we compute for every model its particle mass spectrum using SoftSusy [45] and the decay branching ratios (B) using SUSY-HIT [86]. Then, we pass this information to Smodels-v1.1.1. [79] in form of a SLHA [84] file. Production cross-sections (σ) are calculated by Smodels-v1.1.1 which calls Pythia 8.2 [85]. Smodels-v1.1.1 decomposes production chains in SMS topologies that are confronted with the ones constrained by data. It cannot test all the models we provide, either because their topologies do not match any of the existing experimental results or because their masses fall outside the ranges considered by the experimental searches. These models along with models with weak signals (below 0.05 fb) are considered as beyond the scope of the LHC and classified as not tested. Besides if the mass gap between mother and daughter is small, the decays products will be too soft to trigger any signal. We use 5 GeV as the minimum required mass difference for the decay products to be visible.
In order to present our results, we distinguish among models where the SMS results apply 1 and the ones not tested 2 . For the first, we keep the notation from the previous sections, while not tested models are displayed as gray symbols (squares for Higgsino DM, circles for A/H resonances, diamonds for thet-coannihilation). Not tested models are about 50% of the total investigated models; however this percentage changes depending on the class of models. For clarity reasons we display only the cases where the number of not-tested models dominates over the analysed ones (Higgsino DM, A/H resonances and t -coannihilation). The other classes of models lie in the same areas of the graphs as the displayed points.
In Figure 7, we display µ versus M 2 at m susy = √ mt 1 mt 2 in order to infer the composition of the lightest chargino. We can see that the models with gaugino-dominant charginos are classified asχ ± − χ coannihilations; we also notice that up to a value of M 2 (m susy ) ∼ 300 GeV, many of these models are affected by the LHC exclusion bounds.
In Figure 8 we show the impact of LHC constraints in some of the mass planes. The results can be summarized as follows: • On the top left panel we show the impact of strong production through the 0-lepton + jets + E T channel where the excluded points can be compared with the current coverage by CMS [88,89] using SMS results. It is interesting to notice that there are points with gluino masses about 1.3 TeV away from the gluino-compress region, for which the 13 TeV searches should have good sensitivity, which are not excluded by the SMS results. The reason for this is that the produced gluino-pairs decay asymmetrically via, for instance, one into bb and the another one into light jets. It is also visible how there are points in the gluino-compress spectra region sensitive to monojet searches. Besides, due to the correlations of the gaugino masses induced by GUT-scale boundary conditions imposed by the model some of the points lying into the chargino-coannihilations region are also excluded by this search. These conclusion hold for the exclusion of squarks.
• Next we show the impact of the third generation squarks searches on both stop/sbottomneutralino mass planes on top right panel and bottom left panel respectively. For stops a reasonable correspondence is found between the sensitivity to the model points and those of the simplified-model decay considered in ATLAS and CMS analyses [90][91][92][93], in which the top squark was assumed to decay to t + χ ± . Beyond theses points, points not excluded typically undergo long chain decays. For the sbottoms the impact of the LHC constraints is rather weak since for light neutralinos, sbottoms are too heavy to be excluded. Points with sensitivity are well captured by a simplified model where the sbottom decays into a bottom quark and a neutralino.
• Finally, the impact of electroweak searches through the multi-leptons + E T channel [94] is shown in the chargino-neutralino mass plane in the bottom right panel. The largest impact of this channel is on the chargino-coannihilation region through the soft two-lepton channel which is sensitive to compress spectrum, specially for wino-like charginos. As it can be seen, most of the points with mχ ± 300 GeV are excluded by this search. As it is mentioned this search is complementary to the 0-lepton + jets + E T search.
The impact of this search to sleptons is however insignificant because slepton production cross sections are small. The lower values for the slepton masses are of the order of 300 GeV for the staus and 400 GeV for the other generations.
In order to compare the 422 LHC predictions with other possible signals of the 422 modes, we can consider several reference values for the LSP masses and see the correspon-dence among different plots. For instance, for values of m χ below 500 GeV we find models that satisfy the muon (g -2) 3 − σ bound. These points (in green circles) are classified as models withχ+ orτ coannihilations. Since their mass spectrum is relatively low, many of them are excluded by the current LHC bounds according to Smodels. From future experiments, only the sensitivity of DARWIN will suffice to explore this area. Their indirect detection signals are also weak since they lie two orders of magnitude below the reference line of Figure 6.
The 422 structure for the soft terms allows coannihilations at values of m χ ≈ 1 TeV and beyond, which is not possible in models with universal soft-terms like the CMSSM. The analysis with Smodels indicate that many of the classes of models produce signals that can be compared with the LHC bounds up to values of m χ about 1.5 TeV. Beyond this mass we find only models with A/H resonances andt coannihilations that are not tested with Smodels. In contrast, the predictions of these models are the most promising for indirect detection (see Figure 6) and they will be tested at Xenon-nT and LZ, before the sensitivity of DARWIN is reached as we can see in Figure 6.

Conclusions
In this work we explored the predictions of supersymmetric SU (4) c × SU (2) L × SU (2) R (422) models for supersymmetric particle spectra, taking into account the constraints from LHC and cold dark matter searches. The gauge and symmetry breaking structure of these models leads to very distinct predictions, which deviate significantly from other models.
In particular, our results are the following: • A variety of coannihilation scenarios compatible with LSP dark matter and the LHC have been identified. This clearly indicates that, despite the fact that no SUSY signal has been found so far, there is still a lot of ground to cover and several alternative possibilities to explore.
• The particular relations between the gaugino masses in 422 result in relatively light gluinos with gluino coannihilations, a feature that is very particular for these models and does not appear in other GUT schemes. Similarly, chargino coannihilations are also found in this case, and in fact, together with Higgsino DM, are the most frequently encountered scenarios.
• The fact that the soft supersymmetry breaking parameters at M GU T can be nonuniversal, while compatible with the 422 symmetry, gives rise to additional possibilities and unique features compared to other GUTs, including stop (also found in flipped SU(5), but not SO(10) or SU(5)) and sbottom coannihilations.
• We find very concrete predictions for the gaugino mass ratios that favor scenarios such as chargino-neutralino coannihilations. For sfermion-neutralino coannihilations (and particularly staus where there is no dependence on M 3 /M 1 ), the gaugino mass relations that lead to viable schemes are not as sharp.
• Overall, the LHC and dark matter searches complement each other in covering the available parameter space, and among others, accommodate solutions with prospects for reducing the muon g − 2 discrepancy via a SUSY contribution. These solutions are mostly found in the stau-neutralino and chargino-neutralino coannihilation areas.