Non-universal gaugino mass GUT models in the light of dark matter and LHC constraints

We perform a comprehensive study of $SU(5)$, $SO(10)$ and $E(6)$ supersymmetric GUT models where the gaugino masses are generated through the F-term breaking vacuum expectation values of the non-singlet scalar fields. In these models the gauginos are non-universal at the GUT scale unlike in the mSUGRA scenario. We discuss the properties of the LSP which is stable and a viable candidate for cold dark matter. We look for the GUT scale parameter space that leads to the the lightest SM like Higgs mass in the range of 122-127 GeV compatible with the observations at ATLAS and CMS, the relic density in the allowed range of WMAP-PLANCK and compatible with other constraints from colliders and direct detection experiments. We scan universal scalar ($m_0^G$), trilinear coupling $A_0$ and $SU(3)_C$ gaugino mass ($M_3^G$) as the independent free parameters for these models. Based on the gaugino mass ratios at the GUT scale, we classify 25 SUSY GUT models and find that of these only 13 models satisfy the dark matter and collider constraints. Out of these 13 models there is only one model where there is a sizeable SUSY contribution to muon $(g-2)$.


Introduction
Supersymmetry (SUSY) is an aesthetically appealing model which provides a natural mechanism to stabilise the Higgs mass and solves the gauge hierarchy problem of the Standard Model. The general Supersymmetric Standard Model at the electroweak scale has more than a hundred parameters which make the predictability of such models questionable. An economical Supersymmetric Standard Model can be constructed which contains only a few free parameters known as the constrained Minimal Supersymmetric Standard Model (cMSSM), which relates to the high scale minimal supergravity models (mSUGRA) through renormalisation groups. In mSUGRA there are only 5 parameters: universal scalar mass m 0 , universal gaugino mass M 1/2 , tan β, sign of µ (sgn(µ)) and universal tri-linear couplings A 0 . The lightest supersymmetric particle (LSP) is mostly bino-like. But the recent LHC data is ruling out most of its parameter space for obtaining the WMAP-PLANCK measured relic density of bino as cold dark matter. But it is not necessary to have all the gauginos unified at the unification scale.
It has been noted in [1][2][3][4][5][6][7] that in Supersymmetric Grand Unified Theories (SUSY GUTs) the boundary conditions at the high scale itself can be different than that in mSUGRA. The gaugino masses can be non-universal at the GUT scale itself. The renormalisation group evolutions (RGEs) further change their ratios at the electroweak scale and thus the phenomenology of such models can be completely different compared to mSUGRA. But these non-universalities in SUSY GUT models are completely determined from the group theoretic structure of the symmetry breaking scalar fields. In [1][2][3][4][5][6][7] these non-universal gaugino mass ratios were first calculated for SU (5) group with 24-, 75-, and 200-dimensional scalar fields. Later in [8][9][10][11] the non-universal gaugino mass ratios are presented for all possible breaking patterns having all possible scalar fields for SO (10), and E(6) GUT gauge groups.
In this paper we have encapsulated the parameter space for all models (25) arising from different GUT gauge groups, like SU (5), SO (10), and E(6) and the symmetry breaking patterns from all the possible scalar representations which can break the F-term and gauge symmetries as well. These give rise to different mass ratios of the three gauginos at the GUT scale. Here we have considered only those models for which all of them are non-zero at the unification scale.
Running the masses down to the electroweak scale we get the ratios M 1 : M 2 : M 3 for different models which are quite distinct from the mSUGRA relation 1 : 2 : 6.7 at electroweak scale. Here M 1 , M 2 , M 3 1 are the gaugino masses corresponding to the U (1) Y , SU (2) L , SU (3) C gauge groups respectively. We scan the parameters M G 3 , m G 0 , A 0 , tan β and test the range of parameters for each model which give the lightest Higgs mass in the range 122 GeV < M h < 127 GeV [31], and the dark matter relic density within 3-sigma of the WMAP-PLANCK [32,33] measured band 0.112 < Ωh 2 < 0.128. In addition we have other constraints: within the allowed parameter space the contribution to the B s → X s γ [34], B s → µ + µ − [35] and the muon (g − 2) [36] must satisfy the experimental bounds. We have also set the lower limit on the gluino mass (mg) to be 1.4 TeV 2 . Once these criterion are satisfied we compute the best fit value for the SUSY contribution to muon (g − 2) within the parameter space of the models constrained by the other experimental limits.
Of the 25 models examined we find that only 13 models satisfy the collider and dark matter experimental constraints and we find however that none of these 13 models explain the experimental value of muon (g − 2) [36]. The other 12 models are mainly ruled out when we impose light Higgs mass and 3-sigma relic density constraints together. The largest contribution to muon (g − 2) comes from the the models where the gaugino mass ratio at GUT scale is M 1 : M 2 : M 3 ≡ −1/2 : −3/2 : 1 and this model has a bino like dark matter with mass 177 GeV.
There are five wino, five bino and three higssino dark matter models which give the WMAP-PLANCK relic density. Some of the models can be probed by the XENON1T [37] and Super-CDMS [38] experiments and one model is ruled out by XENON100 [39].

SUSY GUT and non-universal gauginos
Supersymmetry and Grand Unified Theory both have different motivations to be suitable theories beyond the Standard Model. Supersymmetry justifies the gauge hierarchy problem and predicts many other superpartners of SM particles. In R-parity conserving SUSY theories LSP is stable and can be a viable cold dark matter candidate. Here we will focus only on the neutralino LSPs. Within this framework SUSY is expected to explain the observed relics of the Universe. Added with these nice outcomes the extra feature of this theory in the GUT framework is very encouraging. SUSY improves the gauge coupling unification in most of the GUT models. Thus SUSY GUT models are phenomenologically interesting and motivating.
The GUT symmetry is broken when a non-singlet direction under that gauge group acquires vacuum expectation value. In SUSY GUT unified frame work most of the couplings (masses) are degenerate at the unification scale. In its minimal form all the gauginos and scalars are universal respectively. The other free parameters are tri-linear coupling (A 0 ) which is also universal, tan β (ratio of the vacuum expectation values, vev, of two Higgs doublets), and sign of µ (Higgs parameter). But we can have other possibilities, like gauginos or scalars are non-universal at the High scale themselves when we work under SUSY GUT framework. The scalars that cause the GUT symmetry breaking may develop a F-term breaking vev. Thus GUT and supersymmetry are broken via a single scalar but through the vevs in different directions. The gauge kinetic term can be recast in a much simpler form as: η M T r(F µν ΦF µν ) where η is dimensionless parameter, M = M P l / √ 8π (reduced Planck mass). As F µν transforms as adjoint of the unbroken GUT groups, Φ belongs to the symmetric product of the two adjoints.
In this paper we have worked on SU (5), SO(10), E(6) GUT groups, thus the choices of scalars are as following: where 24, 45, 78 are the dimensions of the adjoint representations of SU (5), SO(10), E(6) respectively. It has been noted earlier that these operators also change the gauge coupling unification conditions at the high scale and in many cases it improves the unifications, see for example [40][41][42][43]. As these scalars are non-singlet, their vev treat the SM gauginos in different footing. Thus the SM gauge fields, i.e. the gauge couplings are scaled differently. These types of operators can inject non-universality in the gaugino masses.
In SU (  universal gaugino mass ratios. But as the ranks of SO(10) and E(6) are larger than that of the SM there are more than one possible breaking patterns of these GUT symmetry groups. We have noted the gaugino mass ratios for the following intermediate Here we briefly mention our model identifications depending on the GUT groups, choices of scalar fields and symmetry breaking patterns, see Tables 1 and 2. Here we would like to pass a remark that while calculating these gaugino mass ratios for different models it has been assumed that all the intermediate symmetry scales are same as the unification (GUT) scale, i.e., the GUT symmetry is broken to the SM gauge group at the unification scale itself.

Results
We examine the different non-universal gaugino mass models in the light of relic density, direct detections and collider bounds. We have classified all the models in three categories depending on the compositions of the LSPs: bino-dominated, wino-dominated, and higgsino-dominated.

Branching fraction for
There is a discrepancy in anomalous muon magnetic moment, a µ ≡ (g − 2)/2, between experimental value [36] and SM prediction [46], We compute the SUSY contribution to a µ for each of the models which satisfies the other criterion listed above. We find only one model where there is a substantial SUSY contribution with a SU SY µ = 2.65 × 10 −10 .
For our analysis we use the two-loop RGE code SuSpect [47] to obtain the weak scale SUSY particle spectrum. In addition we use the MicrOMEGAs code [48] to evaluate low energy constraints like B s → µ + µ − , B s → X s γ, muon (g − 2) and relic density. The parameter scan performed in this analysis takes the following ranges of parameters : Here we define M 3 as M G 3 at GUT scale and other gaugino masses M 1 , M 2 are set by the gaugino mass ratios at that scale. We have performed our analysis for three different choices of tri-linear coupling A 0 = −1, 0, 1 TeV. We have chosen tan β = 10 unless mentioned otherwise.
We see that for large A 0 theτ mass becomes very large thereby precluding the stau-coannihilation channel and as a result the relic density which depends on the  Table 3: Input parameters at GUT scale for the benchmark point chosen for each of the 13 models. We choose the parameters such that in each case we get a maximal contribution from SUSY to muon (g − 2).
stau coannihilation becomes too large (this holds for light bino DM and applies to model 24 only). Also very large tan β leads to conflict with the B s → µ + µ − constraint since the SUSY contribution to this process goes as O(tan 6 β).
In Table 2, model 24 which has a gaugino mass ratio of −1/2 : −3/2 : 1 having a bino LSP at low scale, is compatible with all the low energy constraints considered in this work. But it is mainly dependent on the stau coannihilation channel for achieving the correct relic density which means that one has to choose m 0 such that τ mass is quasi degenerate with the LSP mass. The sign of µ is chosen to be negative as that gives the a positive contribution to (g − 2). Table 2, model 20 which has the gaugino mass ratio 5/2 : −3/2 : 1 having a higgsino dominated LSP is compatible with all the low energy constraints but only

Also in
We show the mass spectrum for wino models in Table 4, bino models in Table 5 and higgsino models in Table 6 which satisfy all the low energy constraints listed in the beginning of the section. These are models 1 − 5, 9 − 11, 18 − 20, 22 and 24 as given in Tables 1 and 2. The input parameters for each of the benchmark scenarios are shown in Table 3. The non-universal gaugino models 11 and 19 have been examined in ref [49]. For models 11, 19 and 24 the parameter space which satisfies all the constraints is restricted in the neighbourhood of the values shown in the benchmark table.     Table 3 for each of the wino models which satisfy all the low energy constraints. In addition we also mention the Higgs mass and the relic density in each case. All masses are in GeV.
Of all the wino models only model 8(77/5 : 1 : 1) does not have any valid parameter space for the region that we scan. Here, for M G We have noted that if we allow the larger parameter space for M G 3 , models 1 and 4 allows some parameter space which is consistent with the constraints that we have imposed in our study. It is interesting to mention that for these models to be compatible with the correct relic density, M G 3 needs to be more than 2 TeV in both cases, see as they already qualify to be allowed models for smaller ranges of parameters.

Bino DM
There are three models which have bino LSP as the DM but with very different benchmark spectrum. In model 10 (9/5 : 1 : 1) the DM is a 934 GeV bino LSP. The chargino mass is close to the LSP mass and chargino coannihilation processes, In model 24 (−1/2 : −3/2 : 1) the LSP is a bino of mass 178 GeV and the main annihilation channel is the stau coannihilationχ 0 1τ → Aτ ;ττ → ττ , AA;χ 0 1τ → Zτ which are all an order of magnitude larger than the annihilation channelχ 0 1χ 0 1 → ττ . The stau coannihilation channels are boosted up by taking the stau mass 184.5 GeV close to the LSP mass. In addition the models 18 and 22 also show a very small parameter space in the stau coannihilation region. These two models in particular require that theτ 1 mass be taken very close to the LSP mass (within 5 GeV) and in that sense are more fine tuned than the rest of the successful models.
The bino models which do not work in our parameter scan are models 14,15,16,17,18,21,22,23 and 25 with their ratios as given in Table 2. For these models     Table 3 for each of the bino models which satisfy all the low energy constraints. In addition we also mention the Higgs mass and the relic density in each case. All masses are in GeV. in stau-coannihilation region of the parameter space.

Higgsino DM
In model 9 (10 : 2 : 1) the LSP is a higgsino and the relic density is via the chargino coannihilation processesχ 0 1χ + 1 → ud, cs. The NLSP mass is close to the LSP mass and the NLSP coannihilationχ 0 2χ + 1 → ud, cs also contributes to the relic density. In model 11 (−1/5 : 3 : 1) the LSP is a higgsino with mass 1015 GeV and the relic density is via the same chargino coannihilation processes as in model 9 including the NLSP coannihilation contribution.
In model 20 (5/2 : −3/2 : 1) the LSP is a higgsino of mass 1507 GeV and the contributions to the relic density are due to the chargino coannihilationχ 0 1χ + 1 → tb; χ − 1χ + 1 → tt, bb in addition to the main annihilation channelχ 0 1χ 0 1 → bb, tt. The NLSP mass is close to the LSP mass and the NLSP coannihilationχ 0 2χ + 1 → tb also contributes to the relic density. This model gives the correct relic density for A 0 ∼ −1 TeV.
The failed higgsino models are models 6(122/5 : 1 : 1), 7(−101/10 : −3/2 : 1), 12(1 : 35/9 : 1) and 13(1 : −5 : 1). All of these models fail because the spectrum is unphysical or the higgs sector is unstable. In model 6 for m 0 ≤ 1200 GeV the spectrum contains tachyonic modes, while for m 0 ≥ 1200 GeV there is no EWSB and as M 3 increases one again encounters tachyonic modes in the spectrum. In model 7 the relic density is under abundant for M G 3 < 1.3 TeV while for higher values of M G 3 there is no EWSB. Model 12 behaves very similar to model 6 and so fails for the same reasons. For model 13, there is no EWSB below a certain value of M 3 for a given m 0 , and this value increases with m 0 . Above this value of M 3 some of the scalar modes are tachyonic .  Table 3 for each of the higgsino models which satisfy all the low energy constraints. In addition we also mention the Higgs mass and the relic density in each case. All masses are in GeV.

Direct detection constraints
The elastic scattering of neutralinos with nucleons which results in spin-independent cross section is by Higgs exchange. The Higgs coupling to the lightest neutralino depends upon the product of the higgsino and the gaugino fraction of the neutralino. Pure bino DM therefore easily evade the direct detection limits from XENON100 [39]. In model 24 ( 5/2 : −3/2 : 1) with a 176 GeV bino DM evades the XENON100 bound but may be probed in Xenon 1000 as shown in  Figure 6: The direct detection spin independent proton-DM scattering cross section plotted with the constraint from XENON100 [39]. These plots show selected points for bino models satisfying all the low energy constraints considered here, except for muon (g − 2). These bino models satisfy the XENON100 constraint.
in Fig 5. These wino DM models may be within the reach of XENON1T [37] and Super-CDMS [38] experiments.

Muon (g − 2)
It has long been recognised that to explain the discrepancy between experiment and SM prediction for muon anomalous magnetic moment from a SUSY contribution would require a light mass spectrum on the gauginos and the sleptons [51,52] which would put a severe restriction on the SUSY models. The SUSY contribution to muon (g − 2) for light binos is through the binosmuon loop [53,54] so the largest a SU SY µ = 2.65 × 10 −10 [36,46] comes from model 24 which has the lightest LSP (177 GeV bino) and slepton spectrum. In model 24 (M G 1 : M G 2 : M G 3 = −1/2 : −3/2 : 1) it would have been easy to adjust the smuon mass (through m G 0 ) and the bino mass through M G 3 (as M G 1 is related to M G 3 ) to get a much larger contribution to muon (g − 2). However the relic density of bino DM in model 24 depends on the stau coannihilation which has to be close to the bino DM mass of 177 GeV which is again determined by the universal scalar mass m 0 . So demanding the correct relic density results in a less than optimum contribution The direct detection spin independent proton-DM scattering cross section plotted with the constraint from XENON100 [39]. These plots show selected points for the heavy higgsino models satisfying all the low energy constraints considered here, except for muon (g−2). These heavy higgsino models satisfies the XENON100 constraint.
to the muon (g − 2) in this model. In Table 2 one can note that the gaugino mass ratio referred to here as model 24, can arise from three possible breaking patterns of SO(10), each of them through a different intermediate symmetry group. It will be interesting to see if we distinguish intermediate scale separately than the unification scale then muon (g − 2) is further improved or not. We have kept this issue for our further publication. The gaugino mass ratio of model 24 has been studied in ref. [55] in the context of Yukawa unification in SO (10), but in the benchmark models examined in [55] the SUSY contribution to muon g − 2 is an order of magnitude smaller than the benchmark parameters for model 24 shown in Table 5.
In this paper we have chosen a single non-singlet scalar for giving masses to the gauginos. By choosing a the gaugino masses to arise from more than one scalar representation like 1+24, 1+75 and 1+200 of SU (5) [14,16,56] it is possible to explain muon (g − 2) from SUSY contributions along with the Planck-WMAP relic density [57]. It has been noted [58] that in a mSUGRA model the gaugino mass ratio M 1 : M 2 : M 3 = 1 : 1 : 10 at the GUT scale gives the required muon (g − 2), but in this paper we see that this gaugino ratio does not arise from any of the GUT breaking patterns if one considers one non-singlet Higgs representation for generating the gaugino masses.
If one were to have non-universal scalar masses [27,28] it may be possible to adjust the stau mass to control the relic relic density and the smuon mass to fit muon (g − 2) using a single scalar representation for getting non-universal gaugino masses.

Conclusions
In this paper we have exhaustively analysed all possible non-universal gaugino mass models that arise from SU (5), SO(10), E(6) SUSY GUT models. The underlying assumption is that the full gauge symmetry is broken to the SM symmetry group at the GUT scale itself, i.e., the intermediate scales are same as the GUT scale. We have considered all these models in its minimal versions, i.e., we have not probed the effect of the presence of multiple non-singlet scalars. If one considers that the contribution to the effective gaugino mass ratios are outcome of the contributions from more than one scalar field with the introduction of one or more free parameters, the the unique group theoretic characteristics of the models are lost. Thus we restrict ourselves to the minimal versions (from the point of number of free parameters) of the non-universal gaugino models. We have shown different models predict different kind of LSP compositions. Thus the contributions to the relic density from such models are discriminated. We have performed a comparative study among such models using the collider constraints, lightest Higgs mass and the relic density. We also emphasise the importance of muon (g − 2) and briefly argue why model 24 (M G 1 : M G 2 : M G 3 = −1/2 : −3/2 : 1) is the best candidate among other models in the context of muon (g − 2) contribution. We also check the status of bino-, wino-, and higgisno-dominated models in the context of Direct detection constraints. The model 19 (−5 : 3 : 1) is ruled out by XENON100 [39]. The three models 2(−3 : 1 : 1), 3(−13/5 : 1 : 1) and 5(41/5 : 1 : 1) where the dark matter is a TeV scale wino can be probed in upcoming direct detection experiments like XENON1T [37] and Super-CDMS [38].
Finally we would like to comment on the impact of the insertions of the intermediate scales. In supersymmetric grand unified theories in case of one step breaking the usual trend of the intermediate scale is to lie around the unification scale, see [43]. Thus we expect that the ratios at the GUT scale will not change visibly by the new set of RGEs from intermediate scale to the unification scale. But in case of two step symmetry breaking the second intermediate scale can as low as 100 TeV [43] within a proper unification frame work. If the second intermediate scale is low enough then a new set of RGEs will change the gaugino mass ratios at the GUT scale widely. We are looking into this issue in detail and postpone and will present the results in a future publication.