Confronting Grand Unification with Lepton Flavour Violation, Dark Matter and LHC Data

We explore possible signatures for charged lepton flavour violation (LFV), sparticle discovery at the LHC and dark matter (DM) searches in grand unified theories (GUTs) based on SU(5), flipped SU(5) (FSU(5)) and SU(4)$_c \times $SU(2)$_L \times $SU(2)$_R$ (4-2-2). We assume that soft supersymmetry-breaking terms preserve the group symmetry at some high input scale, and focus on the non-universal effects on different matter representations generated by gauge interactions at lower scales, as well as the charged LFV induced in Type-1 see-saw models of neutrino masses. We identify the different mechanisms that control the relic DM density in the various GUT models, and contrast their LFV and LHC signatures. The SU(5) and 4-2-2 models offer good detection prospects both at the LHC and in LFV searches, though with different LSP compositions, and the SU(5) and FSU(5) models offer LFV within the current reach. The 4-2-2 model allows chargino and gluino coannihilations with neutralinos, and the former offer good detection prospects for both the LHC and LFV, while gluino coannihilations lead to lower LFV rates. Our results indicate that LFV is a powerful tool that complements LHC and DM searches, providing significant insights into the sparticle spectra and neutrino mass parameters in different models.


Introduction
Experimental and theoretical considerations both require extending the Standard Model (SM) of particle physics, which can neither accommodate massive neutrinos, nor explain the observed baryon asymmetry of the universe, nor provide dark matter [1,2,3,4]. Nevertheless, the data from the LHC [5,6,7,8] and dark matter searches [9,10,11,12,13] have not yet yielded any positive signature of physics beyond the SM. On the contrary, severe constraints have been derived for the simplest extensions of the SM that address these issues, including the most simplified versions of supersymmetric unified theories. However, supersymmetry (SUSY) continues to have strong theoretical attraction and, among other features, provides a natural candidate for dark matter (DM) [14] and facilitates the construction of grand unified theories (GUTs) [15]. It is therefore premature to exclude SUSY before studying in more detail non-simplified models that have not yet been explored.
In doing so, flavour physics inevitably plays a crucial role, since it also provides severe bounds on extensions of the SM that would have resulted in exotic manifestations of flavour violation that have not yet been observed. In particular, supersymmetric theories have several possible sources of lepton flavour violation (LFV), which would yield unacceptably large effects unless off-diagonal entries in the sfermion mass matrices were small at some high scale. Even in this case, however, quantum corrections during the running from high scales to low energies would modify this simple structure. This effect is particularly significant in see-saw models for neutrino masses, where the Dirac neutrino Yukawa couplings cannot be diagonalised simultaneously with the charged (s)lepton Yukawa couplings [16]. In this case, the large mixing of neutrino families required by the data also implies that charged LFV may occur at enhanced rates for sufficiently small soft supersymmetry-breaking masses. This can occur in rare decays and conversions (e.g., µ → eγ and τ → µγ, µ → 3e, τ → eγ and µ − e conversion [17,18]), but also in other processes such as sparticle production at the LHC [19,20,21,22,23,24,25,26] and slepton pair production at a Linear Collider (LC) [27,28,29,30,31,32,33,34], particularly in the decays of the second-lightest neutralino.
In this paper we study the possibilities for LFV, LHC and dark matter searches in models with various grand unification groups. We pay particular attention to comparisons between their respective signatures, and to ways to differentiate between the various schemes in present and future searches. Since the SO(10) GUT model is now severely constrained by the data, we focus on GUTs based on SU(5), flipped SU(5) (FSU (5)) and SU(4) c ×SU(2) L ×SU(2) R (4-2-2) [35,36,37]. Similarly to previous works within various GUT models [38,39,40,41,42,43,44], we assume that at the unification scale the soft SUSY-breaking terms preserve the group symmetry, and focus on the non-universal effects on different matter representations due to the gauge structures of the groups. We also include the charged LFV induced in these models via Type-1 see-saw models of neutrino masses with right-handed neutrinos at some high intermediate mass scale.
Different mechanisms that control the relic DM density in the various GUT models, lead to contrasting LFV and LHC signatures. Although the results are sensitive to the scale of the right-handed neutrinos (with larger scales being linked to larger couplings and thus larger flavour violation due to quantum corrections), for similar see-saw parameters, detailed comparisons between different unification schemes can be made. In all cases, coannihilations result to higher LFV rates. We find that the SU(5) and 4-2-2 models have different LSP compositions, but both offer good prospects for detection at both the LHC and in LFV searches. The FSU(5) model also offers LFV within the current reach, for instance in stop and stau coannihilation scenarios. The 4-2-2 model admits novel DM mechanisms such as chargino and gluino coannihilations with neutralinos. The former offers good detection prospects for both the LHC and LFV, whereas gluino coannihilations lead to lower LFV rates, and Higgsino DM models do not predict detectable LFV. In addition, we derive specific correlations between the respective sparticle spectra, providing further input on the experimental signatures that can be expected in each scheme, commenting on the prospects for direct detection of DM as well as LHC and LFV searches.
In Section 2 we summarise the basic non-universal features of the GUTs we study that are relevant for our discussion. In Section 3 we discuss different mechanisms for determining the relic density of dark matter (DM) in the presence of non-universal SUSY-breaking mass terms. In Section 4 we discuss lepton flavour mixing effects in the presence of see-saw neutrinos. In Section 5 we look at the branching ratios for rare LFV decays in different DM scenarios, also taking LHC and direct DM searches into account. Finally, our main results and future detection prospects are summarised in Section 6.

Non-universal soft supersymmetry breaking in GUT models
In our analysis, we assume that SUSY breaking occurs in a hidden sector at some scale M X > M GU T , via a mechanism that generates flavour-blind soft terms in the visible sector. Between the scales M X and M GU T , although the theory still preserves the GUT symmetry, quantum corrections may induce non-universalities for soft terms that belong to different GUT representations, while particles that belong to the same representation have common soft masses.
The soft SUSY-breaking scalar terms for the fields in an irreducible representation r of the unification group are parametrised as multiples of a common scale m 0 : while the trilinear terms are defined as: where Y r is the Yukawa coupling associated with the representation r. We use the standard parametrization with a 0 a dimensionless factor, which we assume to be representation-independent. Since the two Higgs fields of the MSSM arise from different SU(5) representations, they have in general different soft masses. The situation in the different GUT groups is then as follows: • SU(5): In this case, the multiplet assignments are as follows: We assume that the soft terms are the same for all the members of the same representation at the GUT scale, but may be different for the 10 and 5 representations. Here we use as reference the common soft SUSY-breaking masses for the fields of the 10, denoted by m 10 . The masses for the other representations are then defined as: and the A terms are specified via a common mass scale: • Flipped SU(5) (FSU(5)): Since the particle assignments are now different, namely: and the parametrisation changes to where x R refers to the SU(2)-singlet fields. As previously, the A terms are specified as universal: • 4-2-2 symmetry: In this case, a significant modification arises already in the correlation of gaugino masses, since the embedding of the hypercharge generator in the 4-2-2 group implies: yielding gluino coannihilations that are absent in models based on other groups [38]. Sfermions are accommodated in 16-dimensional spinor representations, with their common soft mass parameter being m 16 . The electroweak MSSM doublets lie in the 10-dimensional representation with D-term contributions that split their soft masses: m 2 H u,d = m 2 10 ± 2M 2 D . In our notation: with x u < x d . In the left-right asymmetric 4-2-2 model, a new parameter is introduced: where m L is the mass of the left-handed sfermions (that preserve the definition of m 16 = m 0 ), and m R is the mass of the corresponding right-handed ones.

Relic density mechanisms and GUT mass relations
We assume the following relic density constraint [4]: with a (fixed) theoretical uncertainty of τ = 0.012, (following Refs. [45]) to account for numerical uncertainties in the relic density calculation. This narrow range on the relic density imposes a strong constraint on the DM candidate and the mechanisms that determine its density. It is well known that particular mass relations must be present in the supersymmetric spectrum if the required amount of relic dark matter is provided by neutralinos. In addition to mass relations, we use the neutralino composition to classify the relevant points of the supersymmetric parameter space. The higgsino 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. Thus, we classify the points that pass the relic density constraint discussed above according to the following criteria: Higgsino DM: The first condition in (13) ensures that the lightest neutralino is higgsino-like and, as we discuss later, the lightest chargino χ ± 1 is almost degenerate in mass with 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 DM abundance. Clearly, coannihilation channels involving χ ± 1 and χ 0 2 also contribute. The second condition in (13) implies that the DM density is not controlled by rapid annihilation through s-channel resonances.
A/H resonances: This condition ensures that the correct value of the relic DM abundance is achieved thanks to s-channel annihilation, enhanced by the resonant heavy neutral Higgs (A and H) propagators. The thermal average σ ann v spreads out 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 −χ 0 1 coannihilations: The first condition in (15) ensures that the neutralino is bino-like, in which case annihilation into leptons through t-channel slepton exchange is suppressed, and when the second condition in (15) is satisfied coannihilations involving the nearly-degenerateτ 1 enhance the thermal-averaged effective cross section.
This is similar to the previous case, but also the ντ is nearly degenerate in mass with theτ 1 .
In this case thet 1 is light and nearly degenerate with the bino-like neutralino. These coannihilations are present in the flipped SU(5) model and in the 4-2-2 model, but not SU(5).
In previous work, we had found that the 4-2-2 model may be distinguished clearly from the other GUT groups, due to the appearance of three additional modes of coannihilation that are not present in other groups: In this case the Higgsino component in the LSP is small, but the lightest chargino is light and nearly degenerate with the bino-like neutralino.
g−χ 0 1 coannihilations: In this case the gluino can be relatively light and nearly degenerate with the bino-like neutralino.
b−χ 0 1 coannihilations: in this case, in the presence of the LR asymmetry, theb can be light and nearly degenerate with a bino-like neutralino [46].

Lepton-flavour mixing effects and see-saw neutrino masses
In what follows, we supplement the previous framework with a see-saw mechanism, so as to incorporate also neutrino masses. Specifically, we implement the type-I see-saw [47], via the following superpotential: where W MSSM is the MSSM superpotential and the N c i are additional superfields that contain the three singlet (right-handed) neutrinos, ν Ri , and their scalar partners,ν Ri , and M ij N denotes the 3 × 3 Majorana mass matrix for the heavy right-handed neutrinos. The full set of soft SUSY-breaking terms is given by where L soft contains the MSSM soft SUSY-breaking masses, and (m 2 ν ) i j , A ij ν and B ij ν are the new soft SUSY-breaking parameters in the see-saw sector.
The see-saw mechanism yields three heavy neutral mass eigenstates which are mainly right-handed and decouple at a high energy scale, with masses that we denote as M N . Below this scale the effective theory contains the MSSM plus a higher-dimensional operator that provides masses for the light neutrinos, which are mainly left-handed: As the right-handed neutrinos decouple at their respective mass scales, at low energy we have the same particle content and mass matrices as in the MSSM. This framework naturally explains neutrino oscillations that are consistent with experimental data [48]. At the electroweak scale an effective Majorana mass matrix for light neutrinos, arises from Dirac neutrino Yukawa coupling matrix Y ν (with entries that can be assumed to be of the same order of magnitude as the charged-lepton and quark Yukawa couplings), and the heavy Majorana masses M N . We observe from (21 that we can rotate the fields L i and N c i in such a way that the matrices of the lepton Yukawa couplings, Y ij l , and of the right-handed neutrinos, M ij N , become diagonal. However, in this basis, the neutrino Yukawa couplings Y ij ν are not in general diagonal, giving rise to lepton-flavourviolating (LFV) effects [49,50,51,52,53,54,55]. It is important to note here that lepton-flavour conservation is not a consequence of the SM gauge symmetry, even in the absence of the right-handed neutrinos. Consequently, slepton mass terms can violate lepton-flavour conservation in a manner consistent with the gauge symmetry. Indeed, the scale of LFV can be identified with the EW scale, much lower than the right-handed neutrino scale, M R , which we assume to be common, for simplicity. In the basis where the charged-lepton Yukawa matrix Y is diagonal, the soft slepton mass matrix acquires corrections that contain off-diagonal contributions from the RGE running from M GUT down to M R , which are of the following form in the leading-log approximation [56]: SUSY parameters Table 1: Parameter ranges sampled in our scan of the parameter spaces of the GUT models we study.
Below M R , the off-diagonal contributions remain almost unchanged. Their depends on the structure of Y ν at M R , in a basis where Y l and M N are diagonal. Using the approach of [54,57] a generic form for Y ν that contains all neutrino experimental information can be obtained: where O R is a general orthogonal matrix and M δ N and m δ ν denote the diagonalized heavy and light Majorana neutrino mass matrices, respectively. In this basis the matrix U can be identified with the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix, U PMNS : In order to determine the slepton mixing parameters, we need a specific form of the product Y † ν Y ν , shown in (25). Assuming hierarchical light neutrinos and a common scale for the right-handed neutrinos provides a simple benchmark. In this case, using (26) we find Under the assumption of common masses for the heavy Majorana Neutrinos, LFV effects are independent of the matrix O R .

LFV, Dark Matter and the LHC
We perform parameter space scans similar to those in [38,44,43], 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 ) = 0.1187 The MultiNest [65] algorithm is used to scan the parameter space and identify regions compatible with the data. We have scanned the parameter spaces of the three GUT groups over the broad ranges of parameters shown in Table 1, including soft SUSY-breaking terms up to 10 TeV, with the results that we now discuss. In addition to the dark matter density constraint mentioned above (11), we impose the following constraints: which includes an allowance for the theoretical uncertainty in the calculation of m h in the CMSSM, which is computed using [61]. We extract the following B-physics constraints from [66]: which accommodates the range allowed experimentally at the 2-σ level, which also covers the 2-σ experimental range, and which covers the 3-σ experimental range. These constraints are implemented as described in [60].

BR(l i → l j γ)
The processes l i → l j γ with i = j are allowed at potentially observable levels in SUSY models with flavour mixing among leptons and their scalar partners. In the CMSSM this mixing does not occur, due to the assumption of universal soft terms at the GUT scale. However, this simple SUSY extension of the SM cannot explain neutrino flavour oscillations and, when the model is supplemented with a mechanism to account for them, flavour oscillations of charged leptons also occur. In the MSSM supplemented by a a type-I see-saw as described in the previous Section, which is compatible with the available neutrino data, the uncertainties in the latter may lead to LFV predictions that can differ by several orders of magnitude. Our target in this work, therefore, is not only to analyze the possibility of observing LFV in current experiments, but also to understand the impact of the bounds on BR(µ → eγ) on the perspectives for LHC data. In the simplified see-saw scenario presented in Section 4, we must still specify the following parameters: • The right-handed neutrino scale, which we assume to be common for all generations.
• Lepton-slepton mixings parametrized by a matrix similar to the PMNS matrix at the GUT scale.
We fix the entries assuming that they are real and that their values are such that the neutrino observables are predicted at their experimental central values. In this simple scheme, the product Y † ν Y ν is defined by the PNMS matrix as in (28).   (5), the FSU(5) and 4-2-2 models, in scenarios where sfermions coannihilate with the LSP. We use the same notation for the DM models as in Fig. 1. Models with parameters not detectable at the LHC are marked in grey, while excluded models are marked with purple dots. We assume M N = 2.5 · 10 12 GeV in all three cases.
• SUSY soft masses are flavour-independent at the GUT scale. However, we allow the sfermions belonging to different representations of the unification group to have different soft masses.
The first two points are discussed in this Section, as illustrated in Fig. 1, whereas the third point requires a more elaborate treatment, and we dedicate the two subsequent Sections to it.
For fixed light neutrino masses, Eq. (24) links the ratio of the square of the Yukawa couplings to the right-handed neutrino masses. We see that higher right-handed neutrino masses imply, in general, higher Yukawa couplings and larger mixings of the scalar sleptons in Eq. (25), and hence larger LFV branching ratios (BRs). Fig. 1 compares model predictions with the experimental upper limits on BR(τ → µγ) and BR(µ → eγ). Comparing the upper and lower panels, we can understand how an increase in M R by a factor of 4 would imply the exclusion of many models by the current bound on BR(µ → eγ). For the rest of our analysis, we use M R = 2.5 · 10 12 GeV. In this case, most of the points that can be explored at the LHC will predict BR(µ → eγ) between the current upper limit and a possible future sensitivity one order of magnitude lower.
The correlation between BR(τ → µγ) and BR(µ → eγ) is almost linear, with the prediction of the first being larger than the second by a factor of 10, while the experimental bounds are five orders of magnitude apart. In our study we fixed Y † ν Y ν from the PMNS matrix, requiring common righthanded neutrino masses. Although this cannot be considered general, the values of LFV tau decays are maximized by large 2-3 mixing in the PNMS matrix. Furthermore, in SU(5) group symmetries can relate the PMNS and Cabibbo-Kobayashi-Maskawa (CKM) matrix, leading to large mixing in the 2-3 sector. It is nevertheless possible to find particular textures for Y ν and M N for which the ratios of µ and τ decays are simultaneously closer to the experimental bounds. These cases will, however, typically imply smaller values for the Dirac Yukawa couplings of the first and second generations, predicting less restrictive BRs. Our study can be considered as targeting the kinds of textures that predict large charged LFV.

Combining µ → eγ and LHC bounds
The scale M R = 2.5·10 12 GeV was chosen as representative. It also turns out that no points are excluded by τ → µγ, since µ → eγ is more restrictive. This bound, in combination with large mixing for solar and atmospheric neutrinos, also excludes models with rare τ decays at the levels of the experimental limits in almost all natural textures. Keeping this in mind, we proceed to analyze the predictions for this decay in different unification schemes, studying all kinds of DM models. Since the signal largely depends on the SUSY particle spectroscopy, we combine our analysis with consideration of the LHC data for the specific unified SUSY models under consideration. For this purpose we follow a similar procedure as that applied in Refs. [44,43]. Each model can be associated to a particular set of particle mass hierarchies and decays, which are then compared with the generic analyses provided by the ATLAS and CMS collaborations [67,68]. These comparisons are made with the help of Simplified Model Spectra (SMS) which can be defined by a set of hypothetical SUSY particle masses and a sequence of decay patterns that have to be compared with those expected in any specific model. An individual check has to be done for every model, while, due to mismatches between the theoretical predictions and the experimental analyses, it is not possible to provide contour plots where one can easily see which mass ranges are excluded. This task is simplified by using public packages like Smodels-v1.2.2 [69], which provides a powerful tool for performing a fast analysis of a large number of models [70,71]. Using this package, the theoretical models are mapped onto SMS and can be compared with the existing LHC bounds if there is a match in the respective topologies. In each model the mass spectrum is generated using SoftSusy and the corresponding decay ratios are calculated using SUSY-HIT [72]. The cross-section information is then inserted in Smodels-v1.2.2 through a call to Pythia 8.2 [73].
We classify the models as follows, according to their LHC prospects: (i) Those that are excluded by the current LHC bounds; (ii) Those that can be compared with the LHC data and are not excluded; (iii) Those that cannot be tested at the LHC, i.e., points that predict either processes with very low cross sections or topologies that are not tested at the LHC.
In Fig. 2 and following figures, we denote points of the same DM class with the same symbol as in Fig. 1 but changing the colour according to the LHC prospects of the model: Points of categories (i) and (ii) retain the same colour as in Fig. 1, adding a magenta dot for the excluded ones, whereas points of category (iii) are drawn in grey.
We see in Fig. 2 that the µ → eγ bound may be violated in DM models with coannihilations, whereas models with resonant annihilations and higgsino DM are not affected by this bound, as seen in Fig. 3. This can be attributed to the lighter masses of the sleptons in the coannihilation scenarios.
The LHC and charged LFV predictions of models populating classes of points with different DM mechanisms can be compared in the different unification scenarios: These mechanisms are particularly interesting, since they both predict LFV and LHC signals within experimental reach. In theτ − χ scenario, the lighter stau is determined by left-right mixing, and theτ −ν − χ is the limiting case where theτ 1 is mainly left-handed. We should also take into account the fact that LFV is induced mainly in the leftleft sector of the slepton mass matrix, due to the see-saw mechanism, therefore models with larger left-stau composition and smaller masses tend to have larger LFV decay rates. As seen in Fig. 3, this scenario is very interesting in SU(5) and FSU(5), since these models predict both LFV and LHC signals within experimental reach. Models with LSP masses above 400 GeV are not excluded in either scenario, but the different representation assignments and hence soft masses change the slepton compositions, with manifest implications for the LFV predictions, which are specific for each group. For instance, whereas in SU(5) most of the points with τ −ν − χ coannihilations violate the experimental bound, in FSU(5) they are still allowed. In the case of 4-2-2 models, there is a left-right splitting of the sfermion soft masses, implying that points with stau coannihilations are more difficult to find than in SU(5), as can be seen in the corresponding panel of Fig.2. Moreover, due to gaugino mass relations, the charginos and neutralinos can be heavier than in SU(5) models, leading to lower BR( µ → eγ). in 4-2-2 models the LSP mass can be larger, with BR(µ → eγ) two orders of magnitude below the experimental limit.
A/H resonances: As can be seen in Fig. 3, the predictions for LFV decays are below the current limits. However, there are some differences between the three GUTs for the points with good prospects for both the LHC and BR(µ → eγ), which are easier to find in 4-2-2 and SU(5) than in FSU(5).
Higgsino DM: Fig. 3 shows that this class of points does not predict charged LFV of experimental interest, due to the heavy SUSY masses in the three GUT schemes; these models are also out the LHC reach in all SU(5) cases. However, the LSP composition is different in the three schemes; for instance, in the 4-2-2 model the LSP is almost a pure Higgsino and, even if BR( µ → eγ) is low, some model points can be tested at the LHC.
χ + − χ andg − χ coannihilations: These DM classes appear only in the 4-2-2 case, due to its GUT relation on gaugino masses. As can be seen in Fig. 4, models withχ + − χ coannihilations have good detection prospects for both LFV decays and at the LHC. Points withg − χ coannihilation are still within the LHC reach, while the BR(µ → eγ) predictions are low.

LFV signals, SUSY spectroscopy and DM detection
In this Section we discuss the LHC prospects for discovering SUSY combined with a possible charged LFV signal. The results are shown in Figs. 5 and 6, which plot SUSY particle masses vs. m χ , in order to compare directly the range of SUSY masses to which the LHC is sensitive with those that give rise to detectable LFV signatures. In the case of SU(5) and FSU(5), each panel contains all classes of points, while in the 4-2-2 case the different classes are shown in two panels, for clarity of presentation. We follow the same notation as in the previous Section, with purple dots denoting points excluded by the LHC. In addition to the symbols introduced in the previous Sections, we introduce two more, to show the impact of the LFV predictions on the SUSY spectrum: -Indigo crosses mark points excluded by the current bound on BR(µ → eγ), and -Green crosses mark points with predictions for BR(µ → eγ) between the present bound and a factor of 10 below this value. In addition, the solid red lines are obtained by combining the simplified model bounds from LHC searches. Since these bounds often do not apply directly to our particular cases, this boundary should not be considered as an exclusion line, though excluded points would lie within at least one of these contours. Nevertheless, it is useful to include this line for illustrative purposes, since it gives an idea of the range of masses explored at the LHC for every SUSY particle.
The upper panels in Fig 5 and 6 display LHC and LFV results on mg − m χ contour plots. Since in SU(5) and FSU(5) we assume universal gaugino masses at the GUT scale, all except the Higgsino Figure 5: LHC prospects for the SU5 and FSU5 models. The points follow the notation of Figs. 2, 3 and 4. The meanings of the solid red lines are explained in the text. Indigo crosses indicate points excluded by the limit on BR(µ → eγ), whereas the green crosses mark points that lie between the current bound and one order of magnitude below it.  Fig. 5. For clarity of presentation, in the left panels we display predictions for models with sfermion coannihilations, whereas in the right panels we display the remaining cases.
DM models lie on the proportionality lines obtained from the GUT relations. Among other relations, the neutralino mass is in general proportional to that of the gluino, something that does not hold in 4-2-2 where, due to its different group structure, the distribution of models (shown in Fig. 6), follows different patterns. The sfermion coannihilation cases (left panel) do not show any correlation in the mg − m χ plane. The same holds for models with Higgsino DM and with A/H resonances (right panel), which deviate from the proportionality line. Chargino and gluino coannihiliations, on the other hand, display the pattern of mass correlations described in Ref. [43]. We have checked that the excluded points inside the red contour in mg − m χ plots in Figs. 5 and 6 violate the constraint from the 0-lepton + jets + E T channel [74,75]. This bound affects all the models excluded by the LHC in SU (5), and most of the models excluded in the other two scenarios.
Although the superposition of models on Figs. 5 and 6 does not by itself allow a clear distinction among different DM scenarios, we can associate the excluded points to specific models by confronting these figures with the LFV predictions of Figs. 2, 3 and 4. We see that models with sfermion coannihilations in SU(5) and FSU(5) are more affected by the LHC bounds than in the 4-2-2 case, especially fort − χ coannihilations. In all scenarios, LFV enables exploring a range of mg far beyond the LHC bounds (up to about 4 TeV in SU(5) and FSU (5), and even larger values in 4-2-2 in the chargino coannihilation scenario).
The analysis of excluded models shown in the mt − m χ plots (middle panels of Figs. 5 and 6) indicates that they are affected by the bound due to searches for stop decays into t−χ ± [76,77,78,79]. We see that the exclusion bound in the SU(5) and FSU(5) panels contains many points with slepton coannihilations, while points witht − χ coannihilations escape this bound. In the case of the 4-2-2 models shown in the left middle panel of Fig. 6, we see that this bound is less effective for the same kind of models than in the other GUTs. In the middle right panel of the same Figure, we see that the bound excludes many models withχ ± − χ coannihilations with m χ below 300 GeV. Regarding LFV, we see that the models present good detection prospects up to stop masses above 3 TeV (and even further in 4-2-2 models). However, in the case of 4-2-2, only models withχ + − χ coannihilation predict LFV within one order of magnitude of the current bound.
The mχ± − m χ plane (bottom panels of Figs. 5 and 6) shows that it is possible to see models excluded by electroweak searches through the ATLAS multi-leptons + E T channel [80]. This channel is particularly important in models withχ ± − χ coannihilation in the 4-2-2 scenario, where it can exclude models allowed by 0-lepton +jets + E T . We see that most of the models with chargino mass m χ ± 300 GeV are excluded by these searches. Regarding LFV in SU(5) and FSU(5), we see that the models give rise to good detection prospects for chargino masses up to 1.5 TeV whilst, in the case of 4-2-2, only models with chargino coannihilation present better prospects for LFV detection. In these cases, the masses reach the maximum value of 1 TeV within our data range. The impact is weaker for sbottom searches, as was shown in [44,43]. This is due to the fact that in our scenarios the sbottom squarks are heavy and outside the area covered by the LHC; the same happens with signals involving squarks of the lighter generations.
Finally, we display in Fig. 7 the spin-independent (SI) neutralino-nucleon cross section as a function Figure 7: SI neutralino-nucleon cross section versus m χ in SU (5), FSU(5) and the 4-2-2 escenarios. The solid lines corresponds to the Xenon-1T bound [12], and the dashed and dot-dashed lines correspond to the projected sensitivities of the LZ [81] and DARWIN [82] experiments.
of the neutralino mass in the different GUT models, and we see that the predictions depend on the unification scenario. We note in particular that the FSU(5) model predicts a lower SI cross section than the SU(5) model, in general, while the 4-2-2 model may yield a relatively large SI cross section even for large neutralino masses > 1 TeV. The current bound from the Xenon-1T experiment [12] already excludes many models where the neutralino has a large higgsino component, and the projected sensitivities of the LZ [81] and DARWIN [82] experiments will be able to cover most of the models studied on this work. In particular, only models withχ ± − χ coannihilations may escape the projected DARWIN sensitivity. Comparing the sensitivities of the LFV, LHC and SI DM searches, we see that the latter are potentially very promising probes of SUSY models. On the other hand, as in [43], we find in each model that the spin-dependent (SD) neutralino-neutron cross section is below the projected limit from the LZ [81] experiment. .

Conclusions
In previous work, we studied the predictions of different unified theories for DM and the LHC. We investigated several GUT scenarios, comparing the areas allowed within different symmetry schemes. We considered scenarios with gaugino unification, such as SU(5) and FSU (5), and models where it can be relaxed, such as 4-2-2 models. Among others, we had reached the following conclusions: • Models based on SO(10) are very restricted by data, as can be seen in Refs. [42,43,44]. In contrast, SU(5) models contain several areas of interest for higgsino dark matter, resonant annihilations and coannihilations. However, due to its multiplet structure, SU(5) models not allow stop-neutralino coannihilations, in contrast to the other groups.
• Flipped SU(5) models can be clearly distinguished from SU(5), and have several additional features, including stop-neutralino coannihilations.
• Models based on 4-2-2 not only give rise to stop-neutralino and sbottom-neutralino coannihilations, they also allow novel DM mechanisms, including gluino and chargino coannihilations, as a direct consequence of the distinctive gauge structure.
Here we have combined these analyses with the study of LFV, which turns out to be particularly relevant, using updated LHC data. The large mixing for solar and atmospheric neutrinos implies strong correlations between different rare decays. Since the limits for µ → eγ are significantly stronger, it made sense to focus mostly on this mode and comment on τ → µγ where relevant. We have found the following: • The three groups have distinctive LFV signatures, making it possible to link specific signatures in rare decays and colliders to the gauge and multiplet structure of the theory.
• The results are naturally sensitive to the scale of the right-handed neutrinos, M R . The see-saw mechanism implies that larger scales are linked to larger couplings and thus larger quantum corrections that violate flavour. For smaller values of M R the available parameter space is significantly enhanced: indeed, a change of M R by a factor of 4 is sufficient to exclude or allow a large number of models.
• In all three groups, coannihilations lead to higher rates for LFV, while resonant annihilations and higgsino dark matter are mostly not affected. Overall, in SU(5) and 4-2-2 it is easier to find annihilation models with good detection prospects both at the LHC and in LFV searches. Higgsino DM models do not predict detectable LFV. Still, it is interesting to note that the LSP composition is different in each scheme, yielding an almost pure Higgsino spectrum in the 4-2-2 model.
• Since the see-saw mechanism introduces LFV only in the LL sector, stau coannihilations with smaller masses and larger left-stau components lead to LFV within the current reach. This is particularly relevant for SU(5) and flipped SU(5), since in 4-2-2 models the left-right splitting of the soft fermion masses makes stau coannihilations more difficult to find. However, the two groups can be clearly distinguished, since SU(5) is more restrictive than flipped SU(5).
• Stop-neutralino coannihilations appear only in flipped SU(5) and 4-2-2 models but, once more, with distinct signatures in each case. In 4-2-2 models the LSP can be heavier, and significantly smaller LFV rates are to be expected.
• The 4-2-2 model also allows chargino and gluino coannihilations with neutralinos, due to the different GUT relations for gaugino masses. Chargino-neutralino coannihilations have good detection prospects for both the LHC and LFV, while gluino coannihilations lead to lower LFV rates.
• There are specific correlations between the sparticle masses, leading to interesting signatures. In flipped SU(5), gaugino mass universality results in a proportionality between the gluino and neutralino masses for most of the models under study (corresponding to Higgsino DM and resonant annihilations). Larger masses have good LFV detection prospects, even when they are out of the LHC reach. This is also true for stop-neutralino coannihilations, as well as for models with compressed spectra, such as stau coannihilations.
• In 4-2-2 models, a proportionality relation is found only in chargino-neutralino coannihilations, again due to the GUT relation when the chargino is mostly a Wino. This class of models provides good prospects for both LFV and the LHC, while in other scenarios LFV is significant only for neutralino masses below 500 GeV. This is an additional feature that enables detailed tests of neutralino-chargino coannihilations versus alternative possibilities.
• The experimental advances in direct LSP DM detection are already reaching the sensitivity needed to provide a verdict on many models, specially on the SU(5) and 4-2-2 GUTs. Also, the projected sensitivities of the LZ and DARWIN experiments will provide probes of models that are complementary to LFV searches, even models that cannot be explored at the LHC.
Overall, our results indicate that LFV is a powerful tool that complements LHC and DM searches, and provides valuable information that can help identify optimal modes for future LHC searches. Moreover, not only does it distinguish clearly between various GUTs via the observability of different channels, but it can also provide significant insight into the respective sparticle spectra and neutrino mass parameters.