Solving the muon g-2 anomaly in CMSSM extension with non-universal gaugino masses

We propose to generate non-universal gaugino masses in SU(5) Grand Unified Theory (GUT) with the generalized Planck-scale mediation SUSY breaking mechanism, in which the non-universality arises from proper wavefunction normalization with lowest component VEVs of various high dimensional representations of the Higgs fields of SU(5) and an unique F-term VEV by the singlet. Different predictions on gaugino mass ratios with respect to widely studied scenarios are given. The gluino-SUGRA-like scenario, where gluinos are much heavier than winos, bino and universal scalar masses, can be easily realized with appropriate combinations of such high-representation Higgs fields. With six GUT-scale free parameters in our scenario, we can solve elegantly the tension between mSUGRA and the present experimental results, including the muon g-2, the dark matter (DM) relic density and the direct sparticle search bounds from the LHC. Taking into account the current constraints in our numerical scan, we have the following observations: (i) The large-tan β (≳35) samples with a moderate M3 (∼5 TeV), a small |A0/M3| (≲0.4) and a small mA (≲4 TeV) are favoured to generate a 125 GeV SM-like Higgs and predict a large muon g-2, while the stop mass and μ parameter, mainly determined by |M3| (≫ M0, |M1|, |M2|), can be about 6 TeV; (ii) The moderate-tan β (35 ∼ 40) samples with a negative M3 can have a light smuon (250 ∼ 450 GeV) but a heavy stau (≳1 TeV), which predict a large muon g-2 but a small Br(Bs → μ+μ−); (iii) To obtain the right DM relic density, the annihilation mechanisms should be stau exchange, stau coannihilation, chargino coannihilation, slepton annihilation and the combination of two or three of them; (iv) To obtain the right DM relic density, the spin-independent DM-nucleon cross section is typically much smaller than the present limits of XENON1T 2018 and also an order of magnitude lower than the future detection sensitivity of LZ and XENONnT experiments.


Introduction
Low energy supersymmetry (SUSY), which is well motivated to solve the hierarchy problem, is one of the most appealing new physics candidates beyond the Standard Model (SM). The gauge coupling unification, which cannot be realized in the SM, can be successfully realized in the framework of weak scale SUSY. Besides, assuming R-parity conservation, the lightest SUSY particle (LSP) can be a promising dark matter (DM) candidate with the right DM relic density.
However, there are over 100 physical free parameters in the minimal SUSY model (MSSM), including the soft masses, phases and mixing angles that cannot be rotated away by redefining the phases and flavor basis for the quark and lepton supermultiplets. In practice, some universalities of certain soft SUSY breaking parameters as high scale inputs are usually adopted. In the constrained MSSM (CMSSM), the gaugino masses, the sfermion masses and the trilinear couplings are all assumed to be universal at the GUT scale, respectively. Thus, CMSSM only has five free parameters, i.e., tan β, M 0 , A 0 , M 1/2 and the sign of µ. All the low energy soft SUSY breaking parameters can be determined by these five inputs through the renormalization group equations (RGEs) running from the GUT scale to the EW scale.
So far the null search results of the gluino and the first two generations of squarks together with the 125 GeV Higgs discovery [1,2] at the LHC have severely constrained the parameter space of CMSSM. For example, to provide a 125 GeV SM-like Higgs, the stop masses should be around 10 TeV or the trilinear stop mixing parameter A t should be quite large. Besides, in order for the gluino to escape the LHC bounds, the universal gaugino mass at the GUT scale |M 1/2 | should be larger than about 1 TeV (mg 2|M 1/2 | ), and thus the bino-like neutralino is bounded to be higher than about 400 GeV. All JHEP12(2018)041 some UV-completed models that can predict the low energy soft SUSY breaking parameters with few input parameters.
Some UV-completed models of MSSM, e.g., the gravity mediated SUSY breaking scenarios with the simplest Kahler potential, predict universal gaugino masses at the GUT scale. After RGE running to EW scale, the approximate ratio for gaugino masses 1 is M 1 : M 2 : M 3 ≈ 1 : 2 : 6. We know that the latest LHC results have pushed the gluino mass to about 2 TeV, and thus the neutral electroweakinos are also heavy and cannot solve the muon g − 2 anomaly. Actually, the gaugino masses at the GUT scale are not necessarily universal. In realistic SUSY GUT models, certain high dimensional GUT group representations of Higgs fields may play an essential role in solving the doublet-triplet splitting problem or generating realistic fermion ratios if Yukawa unification is further assumed. With such high dimensional Higgs fields, the universal soft SUSY breaking masses can receive additional non-universal parts. For example, the scenarios with non-universal gaugino masses have been studied in [27,[44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60][61][62] and references therein. General results of soft SUSY breaking parameters in the generalized SUGRA [63][64][65] for SU (5), SO (10) and E6 GUT models involving various high dimensional Higgs fields with different symmetry breaking chains have been discussed in [63,64,66]. Some applications have been also studied [67][68][69][70].
The gaugino masses can always be given by the following non-renomalizable superpotential terms with Λ being a typical UV energy scale (say the Planck scale M P l ) upon the GUT scale. The chiral superfield T is a GUT group singlet and Φ ab is a chiral superfield lying in any of the irreducible representations within the symmetric group production decomposition of adjoint representations. For example, in the framework of SU(5) GUT, Φ ab can belong to After Φ ab or T acquiring an F-term VEVs, soft SUSY breaking gaugino masses will be predicted. For example, the term proportional to a 1 will generate universal gaugino masses with non-vanishing F T , while the term proportional to b 1 will generate non-universal gaugino parts with non-vanishing F Φ ab . In most of the previous studies, non-vanishing Fterm VEVs of the GUT non-singlet Higgs field Φ ab are necessarily present to generate nonuniversal gaugino masses. In principle, the soft sfermion masses or trilinear couplings may also receive additional non-universal contributions from such high dimensional operators.
Although it is indeed possible to realize SUSY breaking with a F-term VEV for a gauge non-singlet superfield through model buildings, for example in typical dynamical SUSY breaking models or ISS-type models, it is more natural to realize SUSY breaking with a gauge singlet F-term VEV. We propose a new approach in which the leading nonuniversality of gaugino masses comes from the wavefunction normalization with a F-term 1 Such a ratio at the EW scale is also predicted by the minimal gauge mediated SUSY breaking mechanism.

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VEV for a gauge singlet only. 2 Although simple, this possibility, which leads to different predictions on gaugino hierarchy, has not been emphasized and discussed in previous nonuniversal gaugino masses literatures.
We can consider the most general combination involving the 24, 75 and 200 representations of Higgs fields of SU(5) GUT group and the gauge singlet T In previous studies on non-universal gaugino masses, as noted previously, non-vanishing F Hr are necessarily present to generate non-universal gaugino masses with (almost) trivial kinetic terms for gauginos. It is however possible that only the GUT group singlet T acquires both F-term VEV F T and lowest component VEV T while all other high dimensional representation Higgs fields acquire only the lowest component VEVs that still with M r ab being the group factor for the representation r and v U the GUT breaking scale which is assumed to be independent of the SU(5) representation and universal for all H r ab . The VEVs of the Higgs field Φ 24 in the adjoint representation can be expressed as a 5 × 5 matrix while the VEVs of the Higgs field Φ 75 can be expressed as a 10 × 10 matrix Similarly, the VEVs of the Higgs field Φ 200 can be expressed as a 15 × 15 matrix As T 0 is a GUT group singlet, the VEV T 0 can be of order Λ without spoiling GUT. The kinetic term after substituting the lowest component VEV will take the form As v U Λ and T 0 Λ, the term proportional to δ ab will be the leading normalization factor. If this term nearly vanishes by choosing a proper a 1 , the second term, which is nonuniversal for three gauge couplings, will generate a different wavefunction normalization 2 We should note that it is possible to generate subleading non-universal gaugino masses from the c1 term of eq. (2.1) with a suppression factor Λ −2 in comparsion with the leading universal gaugino mass part which is suppressed by Λ −1 .

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factor. On the other hand, substituting the F-term VEV F T will generate universal gaugino masses for non-canonical gauge fields. After normalizing the gauge kinetic term to canonical form, non-universal gaugino masses will be generated. The prediction in this scenario is different from previous studies. In table 1, we list our prediction for gaugino mass ratios in different SU (5) representations, in comparison with previous studies (e.g., ref. [62]). For example, if only the 24 representation Higgs is present other than T , the gaugino ratio is given by at the GUT scale which will predict the gaugino ratio 3 : 2 : −9 at the EW scale. So the wino is the lightest gaugino and possibly be the DM candidate in contrary to the widely studied scenarios with bino as the lightest gaugino for F 24 = 0. Another example, although we adopt the most general form of combinations, gluino SUGRA can in fact be realized with only one 200 or 75 representation, in which the gluino can be much (almost 5∼10 times) heavier than wino and bino at EW scale. For the most general combinations involving all 24, 75 and 200 Higgs fields, we can obtain the gaugino ratio M 1 : at the GUT scale. So we can see that one can get an arbitrary gaugino ratio at GUT scale with different choices of c i . This result is different from the widely studied scenarios in which both the high-representation Higgs fields H ab and T acquire universal F-term VEVs F U with trivial kinetic terms, which gives the gaugino mass ratio M 1 : M 2 : M 3 at the GUT scale as (2.11) So, an arbitrary gaugino ratio of M 1 : M 2 : M 3 at the GUT scale can be obtained with properly chosen coefficients c 1 , c 2 , c 3 or c 0 , c 1 , c 2 , c 3 . Besides, the new and old approaches will in general lead to different predictions on the nature of lightest gaugino.
An interesting region will appear if M 3 M 2 , M 1 . In this region, the colored sfermions are heavy even for a very small M 0 (which is the universal sfermion mass parameter) because of the loop corrections involving a heavy M 3 . The non-colored sfermions will, however, still be light if the GUT scale mass M 1,2 M 3 . This region, which is called the gluino-SUGRA region [43], is well motivated to solve the muon g − 2 anomaly [71] and at the same time be consistent with the LHC predictions. In our new approach, the gluino-SUGRA region is easily realized if the denominators of the third term within eq. (2.10) nearly vanishes while the denominators of the first two terms are non-vanishing. In the widely studied approach which is given by eq. (2.11), to realize the gluino-SUGRA region, the first two terms within the second line of eq. (2.11) need to vanish approximately while the third term should be non-vanishing. Solving for c 1 , c 2 in terms of c 0 and c 3 gives

The scan and constraints
In order to illustrate the salient features of our scenarios, we scan the six dimensional parameter space considering all current experimental constraints. The package NMSSM-Tools [72,73] is used in our numerical scan to obtain the low energy SUSY spectrum. We know that in case λ ∼ κ → 0 and A λ is small, the singlet superfield within the NMSSM will decouple from other superfields and the NMSSM will reduce to the MSSM plus a decoupled heavy singlet scalar and singlino. So the MSSM spectrum can be calculated with NMSSM-Tools. In our scan, we use the program NMSPEC MCMC [74] in NMSSMTools 5.2.0 [72,73]. The ranges of parameters in our scan are where we choose a large |M 3 | to escape the LHC bounds on colored sparticles and a large |A 0 | to generate the 125 GeV Higgs mass. Small M 0 , |M 1 |, |M 2 | and a large tan β are chosen to give large SUSY contributions to the muon g-2 and a low mass for dark matter particle. In our scan, we consider the following constraints: (1) The theoretical constraints of vacuum stability, and no Landau pole below M GUT [72][73][74]. charginos/neutralinos, and gauge bosons at leading logs 1 loop level. 3 Its production rates should fit the LHC data globally [123,124] with the method in refs. [125,126].
(3) The searches for low mass and high mass resonances at the LEP, Tevatron, and LHC, which constrained the production rates of heavy Higgs bosons. We implement these constraints by the package HiggsBounds-5.1.1beta [127].

Numerical results and discussions
, a little larger than that in refs. [140,141]. And the theoretical uncertainty is added linearly to ∆aµ, totally required to satisfy a ex µ − a SM µ at 2σ level.

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Then we have where v = 174 GeV is the Higgs VEV in the SM, and X t ≡ A t − µ/ tan β. The SM-like Higgs mass with one-loop correction of stops is given by  We can see these characteristics jointly from the right plane of figure 1, the left and middle planes of figure 2. In SUSY models, we have the following equation for B s → µ + µ − branch ratio In figure 3, we project surviving samples in the A 0 versus M 0 (left), and tan β versus the lighter smuon mass mμ 1 (middle and right) planes, with colors indicating mμ 1 (left), SUSY contributions to muon g-2 ∆a µ (middle), and the lighter stau mass mτ 1 (right), respectively. In these three 2-dimension planes, larger-∆a µ samples are also shown on top of smaller ones. From the middle plane in figure 3, we can see that the muon g-2 anomaly can be solved in our scenario. In fact, light smuon and large tan β can give a sizable contribution to ∆a µ with positive µ in MSSM. Combined with figure 2, we can see that the moderate-tan β (35 ∼ 40) samples with negative-M 3 and predicting small Br(B s → µ + µ − ) can contribute sizably to ∆a µ for lightμ 1 (250 ∼ 450 GeV), but with heavyτ 1 ( 1 TeV) because of the exotic tuning among GUT parameters. This is also a new finding in this work. From the right plane we can know the light-μ 1 and moderatetan β (10 ∼ 45) regions with positive M 3 are missed mainly because of the lower bounds of stau mass mτ 1 . The confusing missed part was shown in a figure in refs. [27,55] but without an explanation, while we give a clear interpretation here in this work. While the wino neutralino always coannihilates with the wino charginos, the relic density is always too small to account for full abundance. We checked that for bino LSP, all the decay modes of wino neutralino χ 0 2 and chargino χ ± 1 contain a τ orτ 1 final state. Thus, from the bottom right plane, we can see the searches for EW gauginos at the LHC set important constraints to the model.  From the bottom middle and right planes, we can also glimpse the annihilation mechanisms of bino-like LSP in our model. We checked that for samples predicting the right relic density (0.107 < Ωh 2 < 0.131), there are mainly five single mechanisms 5 and several combined ones: stau exchange (χ 0

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, other slepton annihilation (˜ /ν ˜ /ν → XY ) . Thus we sort our surviving samples into six classes by 5 Notice that unlike that in CMSSM, we do not have surviving samples with stop as the next-to-lightest SUSY particle (NLSP), and stop coannihilation mechanism in our NUGM extension, which is because in our scenario we can have lighter bino-like neutralino and wino-like charginos for NUGM, and lighter sleptons to solve the muon g-2.
µν e → eν µ , 11.3% eν µ → ν e µ, 11.3% µν e → ν e µ, 4.7% eν µ → eν µ , 4.7% judging if it is a single or a combined mechanism: τ 1 exchange : mτ 1 < 200 GeV, For the hybrid2 samples, the dominated mechanism is a combined one byτ 1 coannihilation and χ ± 1 coannihilation; while for stau hybrid3, it is combined byτ 1 exchange,τ 1 coannihilation andτ 1 annihilation, and the heavierτ 1 , the more annihilation and the less exchange; but whenτ 1 are heavier than 400 GeV, the dominated mechanism becomes other sleptons coannihilation, which is very complex in income and outcome particles. In table 2, we give the detail annihilation information for 7 benchmark points. For each point, we list its main JHEP12(2018)041 annihilation channels and the relative contributions (> 1.5%) to σv . For completeness, we list the other information for the benchmark points in table 3 in appendix B. In this work, we show in detail the various annihilation mechanisms of DM, which is not done in refs. [27,55]. And our findings is some different from these in refs. [8,149] for NUGM version of pMSSM. We do not have annihilation mechanisms of A/H funnel, focus point, and stop coannihilation, since our H/A, µ,t 1 are much heavier, but we have stau exchange which may be omitted in refs. [8,149].
In figure 5, we show the six classes of samples with sufficient relic density (0.107 < Ωh 2 < 0.131) on the plane of SI DM-nucleon cross section σ SI (original values without being rescaled by Ω/Ω 0 ) versus LSP mass m χ 0 1 . We can see that, most of the samples predict small σ SI , which are over one order of magnitude lower than the future detection accuracy of LZ and XENONnT experiments. The σ SI are smaller than these in refs. [27,55], because we required sufficient relic density for these samples, and we checked that σ SI can be larger for samples with insufficient relic density. However, a few samples corresponding toτ 1 coannihilation, χ ± 1 coannihilation and hybrid2 can be covered by the two detectors, with the LSP mass at about 200-400 GeV. It is because these samples have large percentages of coannihilation channels contributing to the DM relic density, which is also a new finding in this work.

JHEP12(2018)041 5 Summary and conclusions
We propose to generate non-universal gaugino masses in SU(5) GUT with the generalized Planck-scale mediation SUSY breaking mechanism, in which the non-universality arises from proper wavefunction normalization with lowest component VEVs of various high dimensional representations of the Higgs fields of SU(5) and an unique F-term VEV by the singlet. Different predictions on gaugino mass ratios with respect to widely studied scenarios are given. The gluino-SUGRA-like scenarios, where gluinos are much heavier than winos, bino and universal scalar masses, can be easily realized with appropriate combinations of such high-representation Higgs fields. With six GUT-scale free parameters in our scenario, we can solve elegantly the tension in mSUGRA between the muon g-2 and other constraints including the dark matter relic density and the direct sparticle search bounds from the LHC.
Taking into account the current constraints, we performed a scan and obtained the following observations: • The large-tan β ( 35) samples with a moderate M 3 (∼ 5 TeV), a small |A 0 /M 3 | ( 0.4) and a small m A ( 4 TeV) are favoured to generate a 125 GeV SM-like Higgs and predict a large muon g-2, while the stops mass and µ parameter, which are mainly determined by |M 3 | ( M 0 , |M 1 |, |M 2 |), can be about 6 TeV.
• The lightest neutralino can be as light as 100 GeV, which can predict a right relic abundance if it is bino-like and a much smaller relic density if it is wino-like.
• To obtain the right DM relic density, the annihilation mechanisms should be stau exchange, stau coannihilation, chargino coannihilation, slepton annihilation and the combination of two or three of them; • To obtain the right DM relic density, the spin-independent DM-nucleon cross section is typically much smaller than the present bounds of XENON1T 2018, and an order of magnitude lower than the future detection sensitivity of LZ and XENONnT experiments.
For the linear-correlation parameters we calculate the coefficients by For the quadratic-correlation parameters we calculate the coefficients by which can be written in a 5×5 triangular-matrix for each parameter. In the following equations, we list the coefficients for the benchmark point P4 in Class C and D in eq. (4.1), and P7 in Class A and B in eq. (4.1). We checked that these coefficients coincide approximately with our parameter-running results in NMSSMTools-5.2.0. Most of these equations (except M 2 H d and m 2 A , for example) can be generalized roughly to other surviving samples in their represented classes, because all of them satisfy tan β 1. However, most coefficients will change a lot if one change the SUSY scale too much, e.g., to M SUSY = 400 GeV as in ref. [28]. These equations are given as follows.
For Benchmark Point P4 (with tan β = 20.8 fixed and on behalf of Class C and D),

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