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Flavored Gauge-Mediated Supersymmetry Breaking Models with Discrete Non-Abelian Symmetries

We investigate flavored gauge mediation models in which the Higgs and messenger doublets are embedded in multiplets of the discrete non-Abelian symmetry S3. In these theories, the S3 symmetry correlates the flavor structure of the quark and lepton Yukawa couplings with the structure of the messenger Yukawa couplings that contribute to the soft supersymmetry breaking mass parameters. We provide a systematic exploration of possible scenarios within this framework that can accommodate hierarchical quark and charged lepton masses, and examine the resulting phenomenological implications in each case. We find a heavier spectrum for the superpartner masses compared to flavored gauge mediation models controlled by Abelian symmetries, which can be directly traced back to the need in our scenarios for two vectorlike pairs of messenger fields for viable electroweak symmetry breaking.

the splitting of the squark and gluino masses compared to the slepton and electroweakino masses.
Further embedding of the MSSM fields in S 3 representations allows for the possibility that S 3 can play a role as part of the family symmetry that governs the SM fermion masses and mixings. A specific implementation of this idea was explored in [38], as well as in [43], in which the Higgsmessenger singlets play a dominant role in generating the third family SM fermion masses.
The purpose of this paper is to provide a comprehensive analysis of the FGM S 3 scenario, summarizing and extending our previous work. The aim is to explore other viable corners of parameter space of these theories and the subsequent effects of including nonleading corrections to the fermion masses. We identify several viable parameter regions, describe their phenomenological consequences, and compare them to the U (1) FGM benchmark scenarios in the literature. We will see that quite generally, it is not easy to generate viable fermion masses while maintaining flavor-diagonal soft terms, and we will characterize the extent to which such flavor nondiagonal terms are constrained in these theories. The examples studied here all feature very heavy squarks and gluinos, very heavy Higgs fields, and lighter sleptons, charginos, and neutralinos. As such, they provide working examples of currently allowed MSSM parameter space that will continue to be constrained at the LHC and future colliders. This paper is structured as follows. We begin with a brief overview of the flavored gauge mediation framework studied here, and describe various options for obtaining hierarchical quark and charged lepton masses. Next, we describe several concrete models, and analyze their mass spectra in detail. Finally, we present our summary and conclusions.

II. THEORETICAL BACKGROUND
As described in [37], the FGM S 3 scenario studied here assumes a specific set of Higgs-messenger fields and supersymmetry-breaking fields. The quantum numbers of these fields with respect to S 3 are given in Table I. Here the H (2) u,d are Higgs-messenger S 3 doublets, the H  S 3 singlets, and X H is a supersymmetry breaking field that also breaks the S 3 symmetry. The Td k,dk denote the SU (3) c triplets which have the appropriate quantum numbers to complete approximate 5, 5 multiplets with the messengers and X T is the supersymmetry breaking field that couples to these triplets [44]. Focusing on the Higgs-messenger fields, we can write H (2) u,d and H (1) u,d as in which H u,d are the electroweak Higgs fields of the MSSM, M u1,d1 and M u2,d2 are gauge mediation messenger doublets, and R u,d are unitary matrices whose form is governed by the couplings of the Higgs-messenger fields to X H , which obtains both a scalar and F -component vacuum expectation value (VEV). As shown in [37], consistency requirements and obtaining the needed mass hierarchy between the MSSM Higgs fields H u,d and the heavy messengers M ui,di require that R u,d are given by We turn now to the MSSM fields and their interactions with the Higgs-messenger fields. Although various possibilities exist, as discussed in [37], we make the key assumption that the three generations of SM quarks and leptons are embedded into doublet and singlet representations of S 3 , as summarized in Table II. With these S 3 charge assignments, the superpotential couplings of the  MSSM matter fields and the Higgs-messenger fields, for example for the up quarks, are given by In Eq. (3),ỹ u is a dimensionless overall factor, and the quantities β 1u , β 2u , β 3u , and β 4u are dimensionless quantities that characterize the different couplings as allowed by S 3 . (Analogous forms hold for the down quarks and the charged leptons; we will ignore the effects of neutrino masses.) In the basis given by the superpotential couplings of Eq. (3) can be expressed in matrix form as this system explicitly and examine parameter sets where viable eigenvalue hierarchies can be obtained. For example, in the up quark sector, we can follow standard procedures and consider the Hermitian combinations Y u Y † u and Y † u Y u . It is straightforward to calculate following exact results for their eigenvalues (denoted by λ 1u,2u,3u ): in which Λ u is given by Clearly, for arbitrary values of the parameters, the eigenvalues are not hierarchical. However, in looking for leading-order results in which only one eigenvalue is sizable, we can easily identify two general scenarios of interest, depending on the ordering of the mass eigenvalues. One option is that λ 1u is one of the small eigenvalues, which would have β 1u → 1, and λ 2u is the other, and hence λ 3u generically has an O(1) value. Another option is that λ 1u is the large eigenvalue, such that β 1u = 1, and both λ 2u,3u are small. We now discuss each possibility in turn.
In what follows, we will focus on the up quarks, but our default assumption will be that the down quarks and the charged leptons will take similar forms. Mixing possible options for eigenvalue hierarchies in the different charged fermion sectors will not be considered here for simplicity.
A. Case 1: λ 1u,2u λ 3u (encompassing the "singlet-dominated" and "democratic" limits) We begin with the situation that β 1u → 1, such that λ 1u is a small eigenvalue, and explore parameter regimes in which λ 2u is also small. For simplicity, we first consider the case in which both vanish, such that to this order of approximation we have one massive third-generation, and two massless generations. It is easily verified that in this regime, both eigenvalues vanish for This case includes what we call the democratic limit, in which all the β iu = 1, and thus the MSSM Yukawas take on the well-known democratic form [41]. The democratic limit was originally studied at leading order in [37], and will be studied in more detail below, including subleading corrections. This case also includes what we will call the singlet-dominated limit, which is the case in which β 4u β 1u,2u,3u , as β 4u is the parameter related to the strength of the superpotential coupling involving only S 3 singlet fields. In the singlet-dominated limit, the MSSM and messenger Yukawa couplings at leading order, in the diagonal quark mass basis, only have nonvanishing 3 − 3 entries, allowing for sizable stop mixing and consequently lighter superpartner masses than the other examples we will consider (as we will see). This limiting case was studied in some detail in [38] and [43], and will be considered below as a benchmark scenario for purposes of comparison.
For Case 1, incorporating Eq. (11) and up to possible rephasings to ensure that the fermion masses are real and positive, the diagonalization matrices U uL and U uR take the form Assuming these forms with no further rephasings, the messenger Yukawa couplings in the diagonal quark mass basis then take the form From these forms, we see that in the democratic limit, the messenger Yukawas only have nonvanishing entries in the upper 2 × 2 block, as follows: In the singlet-dominated limit, the 3 − 3 entries dominate, with Y u1,u2 = Diag(0, 0,ỹ u β 2u β 3u / √ 3).

III. MODELS
As described in the previous subsection, we have identified two cases with hierarchical quark and charged lepton masses. The first (Case 1) satisfies Eq. (11), and includes two possible scenarios at leading order: the singlet-dominated limit, in which it is the S 3 singlet couplings of the MSSM fields and the Higgs-messenger fields that dominate the superpotential, and the democratic limit, in which all the couplings of S 3 representations in the superpotential are precisely equal at leading order, resulting in an enhanced S 3L × S 3R symmetry. The singlet-dominated limit was explored in [38] and [43], and the democratic limit at leading order in [37]. The second (Case 2) is what we call the doublet-dominated limit, as in this case the dominant couplings are those involving only In what follows, we will discuss these scenarios in greater detail.

A. Case 1 models
We begin the discussion of Case 1 models with the singlet-dominated limit, which was studied in detail in [38,43]. In this scenario, sizable stop mixing can occur due to the FGM contributions to the third-generation soft trilinear scalar coupling. This in turn allows for the squarks and gluinos to be in the O(5 − 6 TeV) range, which is relatively light compared to generic parameter choices for this class of FGM models. A variety of subleading corrections to this limit can be considered, including the possibility of generating nontrivial masses for the second-generation fields and the possibility of viable quark mixing at the first subleading order. For the case described in [43], the corrections to the soft terms that result from these terms have only minimal effects on the superpartner masses. Furthermore, in this case flavor-violating contributions to the soft terms also do not result at the first subleading order in the quantities that control the lighter generation quark and lepton masses, though this is not necessarily generic. Here we will not revisit this case in detail other than as a point of comparison for the new scenarios considered in this work.
Let us now turn to the democratic limit, for which the Yukawa coupling parameters β 1i = β 2i = β 3i = β 4i = 1, where i = u, d, e. In this case, the MSSM Yukawa matrices take the form This is the well-known flavor democratic mass matrix form, which exhibits an S 3L × S 3R symmetry.
At leading order, this mass matrix has two vanishing eigenvalues, and one O(1) eigenvalue, to be identified with the third-generation. As shown in [37], the messenger Yukawa matrices have nonzero entries only in the upper 2 × 2 block in the diagonal quark mass basis.
We now address the generation of the first-and second-generation fermion masses and the effects on the sfermion masses through the messenger Yukawa corrections. Here we choose to break the S 3L × S 3R symmetry to S 2L × S 2R and then to S 1L × S 1R , which generates a nonzero mass for the first-and second-generation fermions (see e.g. [42]). This can be achieved via the following terms: in which i and σ i are real dimensionless perturbative parameters associated with symmetry breaking from S 3 to S 2 and S 2 to S 1 respectively. In our scenario, the perturbations of the up quarks (the down quarks and charged leptons have analogous structures) can be generated in superpotential at the renormalizable level by while the σ perturbations can be generated via non-renormalizable operators. These superpotential terms add corrections of the form of Eq. (21) to the Yukawa matrix for the up-type quarks, and corrections of the following form to the up-type messenger Yukawa matrices: Including these correction terms along with the leading order results, the eigenvalues λ 1u,2u,3u , are then found to be In these relations, we have identified the top quark Yukawa coupling In the diagonal quark mass basis, the messenger Yukawa matrices for the up-type quark sector are given to order in σ u / u by Analogous forms are easily obtained for the MSSM and messenger Yukawa matrices for down-type quarks and leptons in the diagonal quark mass basis with the replacements u → d,e and y t → y b,τ .
The relative strengths of the parameters u,d,e and σ u,d,e can be estimated from the fact that these parameters govern the fermion masses of the lighter generations. More precisely, up to O(1) Similarly, d ≈ 0.1, σ d ≈ 9 × 10 −3 , e ≈ 0.3, and σ e ≈ 8 × 10 −3 . These parameter values also yield hierarchical quark mixing angles of the Cabibbo-Kobayashi-Maskawa (CKM) matrix, in which the largest angle is the Cabibbo angle, sin θ c ∼ 0.17. While the quark mixing angles are not fully realistic (the Cabibbo angle is clearly too small compared to its experimentally determined value), for the purposes of this study it is a reasonable starting point for the analysis.
We now find the nonvanishing corrections to the soft supersymmetry breaking terms, assuming for simplicity that the ratio of the F terms to the scalar VEVs for the X H and X T terms are identical (both will be denoted as Λ). We provide the expressions for these correction terms in the Appendix. As expected, in the limit that the perturbation parameters are set to zero, the result is what was found in [37]. When the perturbations are added, the diagonal entries of the soft masssquared terms are corrected at second order in the parameters. This generates nonzero (but small) diagonal 3 − 3 entries. In addition, with nonzero perturbations, flavor off-diagonal contributions to the corrections to the soft terms are generated. More precisely, the u,d,e parameters introduce nonvanishing δm 2 f 23 terms at first order in , while the σ u,d,e introduce nonvanishing δm 2 f 12 and δm 2 f 21 terms. Therefore, the dominant effects are expected to be seen in the 2 − 3 sfermion mixings. Further details will be discussed in the next section.
• Ordering β 3i > β 2i . Let us first consider the case in which β 3i > β 2i , for which we have in which U iL,iR take the simple forms We see that for this ordering, the β 3i control the second-generation masses and the β 2i control the first-generation masses. The messenger Yukawas in the diagonal fermion mass basis (the SCKM basis) are given by Given that we can identify β 2i,3i with the first and second-generation masses, respectively, we can write for example for the up-type quarks (with y u,d,e = β 2u,d, y t / √ 2 and y c = β 3u y t / √ 2): From the quark and charged lepton masses, we can roughly estimate (neglecting running effects) Hence, to leading order we can neglect the effects proportional to the first-generation masses (here the β 2i ), and treat the effects due to the second-generation masses (the β 3i ) perturbatively. We thus calculate the corrections to the soft supersymmetry terms in this limit. As before, we assume for simplicity that the ratio of the F terms to the scalar vevs for the X H and X T terms are identical. The detailed forms of these soft supersymmetry breaking terms are presented in the Appendix.
We note that in this case, there are flavor off-diagonal contributions in the δm 2 Q,L 12 that are proportional to the β 3i , and thus scale with the second-generation quark and lepton masses. This is reminiscent of the Case 1 democratic limit with perturbations, though the dominant off-diagonal contributions occurred there in the 2 − 3 sector, and here they arise in the more dangerous 1 − 2 sector. We will discuss their effects in the next section.
• Ordering β 2i > β 3i . We now consider the case in which β 2i > β 3i , for which the roles of β 2i and β 3i are switched in Eq. (27). We now have The messenger Yukawa matrices in the diagonal quark mass basis are of the form As in the previous section, we can ignore effects that scale with the first-generation fermion masses and keep leading contributions involving the second-generation fermion masses. Thus, we now neglect the terms proportional to β 3i and keep leading-order terms proportional to the β 2i . We can again calculate the soft supersymmetry breaking terms, subject to the same assumptions as given for the alternate ordering. The detailed forms are included in the Appendix.
One interesting feature of this mass ordering (β 2i > β 3i ) is the corrections to the soft supersymmetry breaking mass terms are flavor-diagonal if we neglect effects proportional to the firstgeneration fermion masses. As in the alternate ordering, here we obtain contributions to δm 2 Q,L 12 that are proportional to the β 3i , but now these quantities must be much smaller since they govern the masses of the first-generation. Given the high degree of suppression of the flavor off-diagonal elements, in this case the model is clearly safe from flavor-changing neutral current constraints.

IV. RESULTS AND DISCUSSION
In this section, we analyze the mass spectra of these scenarios and their phenomenological implications. We start with Case 1, focusing solely on the democratic limit with symmetry breaking effects, and then study Case 2, the doublet-dominated limit, with both orderings of the β 2i and β 3i . The model parameters are M mess , Λ, tan β = H u / H d , the sign of µ (sgn(µ), taken here to be +1), and the relevant perturbation parameters, which depend on the scenario in question. Here we have followed standard procedures and replaced |µ| and b with tan β and the Z boson mass.

A. Case 1 models
We start with the flavor democratic limit, which was explored in [37] for the case of thirdgeneration masses only, i.e. in the absence of the small perturbations that break the S 3L × S 3R symmetry. It was shown in [37] that this scenario leads to heavy superpartner masses, which can be traced to the absence of large stop mixing in this limit. In the presence of nonvanishing perturbations, this picture generically continues except for specific small regions of parameter space where the Higgs mass constraint can be satisfied without being bolstered by very heavy squarks.
In Figure 1, we show a representative mass spectrum for an intermediate messenger mass scale of M mess = 10 12 GeV and tan β = 10, where Λ is chosen to satisfy the Higgs mass constraint [47].
As seen, the heavy Higgs particles are nearly 8 TeV, the gluino is approximately 10 TeV, and the squarks fall into three groupings: a lightest set that is close in mass to the heavy Higgs particles, a set in between, and a heavier set that is similar to the gluino mass. The sleptons are close in mass to the lightest neutralino, and the next-to-lightest superpartner (NLSP) is the lightest slepton.
The effects of nonzero σ u,d,e lead to small (O(1 GeV)) splittings in the masses ofd 1 andd 2 , and u 4 andũ 5 , which are each originally identical up to order 10 −2 GeV. The effect of u,d,e is larger, which is expected as these have larger numerical values. For nonzero u , there is a splitting of order ∼ 70 GeV in the masses ofũ 1 andũ 2 , which are also identical up to order of 10 −2 GeV in the S 3L × S 3R limit. Similar features are seen forũ 4 andũ 5 . The u corrections also introduce a small (∼ 25 GeV) mass splitting ford 1 andd 2 , which is a sign of the symmetry breaking from  [48]. We expect rather mild constraints due to the heavy sfermion and gluino masses and the suppression factors in the off-diagonal contributions to the soft terms. For the set of model parameters in Fig. 1, we obtain 2 − 3 squark and slepton mass insertion parameters of the order |(δ 23 as well as small contributions to LR mixings in the 2 − 3 sector (ranging from 10 −4 to 10 −7 . The 1 − 3 and 1 − 2 mass insertions are parametrically smaller, with limits that range from 10 −4 to 10 −12 , except for |(δ 12 l ) RR | ∼ 3 × 10 −3 . The resulting effects are small and within the allowed ranges (see e.g. [49]).
The composition of the mass eigenstates of the sfermions is shown in Fig. 2 Mass/GeV It is illustrative to compare this scenario with the singlet-dominated limit [38,43]. In this case, the dominant contributions to the soft terms arise in the diagonal third-generation (33)  We now turn to the Case 2 models, for which the superpotential couplings only involving the S 3 -doublets dominate. As described in the previous section, in this case there are two subcategories, depending on whether the β 3i or the β 2i parameters control the second-generation quark and charged lepton masses. Here we will label the mass ordering β 3i > β 2i by Case 2a, and the alternate mass ordering β 2i > β 3i as Case 2b. The soft supersymmetry breaking terms for Case 2a are given in Eq. (36), and the analogous quantities for Case 2b are given in Eq. (37). Case 2b with β 2d = 0.03, β 2u = 0.01, β 2l = 0.08, β 3u = β 3d = β 3l = 0 (right).
In Case 2 models, there is a nonvanishing trilinear scalar parameter A t that is present in the absence of the first and second-generation quark and charged lepton masses, in contrast to the Case 1 democratic limit. Hence, the Higgs and superpartner masses are lighter than their Case 1 democratic counterparts, though not as light as in the Case 1 singlet-dominated limit. In Fig. 3, Mass/GeV

C. Discussion
In comparing the mass spectra of these scenarios (Case 1: democratic, and Cases 2a and 2b: doublet-dominated, as well as the Case 1: singlet-dominated limit as studied in [37]), there are several features of interest. For fixed M mess , the mass spectra are more compressed for larger values of tan β (tan β > 10) because the contributions from the bottom and tau Yukawa couplings are more significant than in the low tan β regime. For smaller tan β values, the sparticle masses are heavier as the tree-level contribution to the light Higgs mass has decreased, requiring larger radiative corrections to boost its mass to its experimentally allowed range. The superpartner masses in this limit are thus highly split, with heavy squarks and gluinos, and lighter sleptons.
For fixed tan β (here taken to be tan β = 10), lower values of the messenger mass scale generally lead to heavier spectra, as larger values of Λ are needed to satisfy the light Higgs mass constraint.
For higher messenger scales, due to increased renormalization group running effects, the µ and b/µ terms needed to satisfy the electroweak symmetry breaking constraints are smaller, and thus the heavy charginos and neutralinos become lighter.
To further investigate the dependence of the mass spectra on M mess and tan β, we show the Higgs mass curve for fixed Λ, with the color representing the mass of the lightest squark. We show these results in Fig. 5 for the Case 1 democratic scenario, which show several excluded regions.
When u = d = e = 0, the central big "hole" appears because the mass-squared of the lightest slepton is negative. When the parameters are nonzero, the size of the holes increases, and there are also small holes that appear above the central void because A 0 becomes tachyonic in those regions. Quite generally, we see that the parameters that satisfy the Higgs mass constraint could be very different in these two cases. The lightest value for the mass of theũ 3 squark is in the  in this parameter region. A detailed resolution of these issues is beyond the scope of this paper, and is deferred to future study. Case 2b with fixed Λ = 6.3 × 10 5 GeV, β 2d = 0.03, β 2u = 0.01, β 2l = 0.08, β 3u = β 3d = β 3l = 0 (right).
For the Case 2 models, we also see excluded regions in the parameter scan in Fig. 7 is from M mess ∼ 10 6 − 10 9 GeV and tan β ∼ 5 − 50, is ruled out because the desired electroweak minimum is not present. We also see that in both cases, the viable Higgs region does not generally intersect with the region whereẽ 1 is lighter. In Fig 8, we fix tan β = 10 and show both Case 2a and Case 2b with nonzero perturbations. Note that the effects of the perturbations only slightly shift the mass curve of the NLSP upward, and have almost no effect on the viable Higgs mass region and the gluino masses, in contrast to what we have seen in the democratic limit. Case 2b with β 2d = 0.03, β 2u = 0.01, β 2l = 0.08, β 3u = β 3d = β 3l = 0 (right).
We close this section by commenting on further phenomenological aspects of this set of models.
For all cases described here (both democratic and doublet-dominated models), the superpartner masses are generally heavy and split, in a way that is reminiscent of minimal gauge mediation with N = 2. As previously discussed, the constraints of the non-Abelian Higgs-messenger symmetry have led us to include at least two messenger pairs to avoid a catastrophic µ/B µ problem. Ultimately, this means that the scenarios studied in this paper have heavier and more split spectra than what can be obtained in Abelian flavored gauge mediation models (where a judicious choice of U (1) charges can be made to avoid the µ/B µ issue seen here, without increasing the number of messenger pairs), such as in [35], or in general gauge mediation scenarios [17]. We recall that in our scenario in the singlet-dominated limit as studied in [38,43], it is also possible to minimize the splitting of the mass spectra, though not to the extent that is possible in the Abelian flavored gauge mediation models.
As a result, the discovery potential for the scenarios studied here either via direct LHC searches or indirect constraints is not as promising as it can be in Abelian flavored gauge mediation models, or even in the singlet-dominated non-Abelian scenario. For example, it is straightforward to see that the supersymmetric contribution to muon anomalous magnetic moment (MDM) in the democratic and doublet-dominated non-Abelian flavored gauge mediation scenarios studied here is generically about two orders smaller than the current experimental value [50]. This is both due to the heavy superpartner masses as described above, and that we are generally precluded from having large values of tan β in these scenarios, which usually provide the largest enhancement to the MDM.
Therefore, if new physics is required to resolve any future confirmed discrepancy between the SM prediction and the measured value of the muon anomalous magnetic moment, this set of flavored gauge mediation models would need to be extended to accommodate the experimental result.
One notable difference in the non-Abelian flavored gauge mediation scenarios studied here compared to minimal gauge mediation with N = 2 as well as the non-Abelian singlet-dominated flavored gauge mediation scenario is in regards to the NLSP composition. Here, for messenger mass scales of 10 12 GeV as displayed in Fig. 1 and Fig. 3, the NLSP is the lightest slepton, which is a right-handed smuon. This is different from the minimal GMSB scenario and the singlet- We also note that in both minimal N = 2 GMSB and our non-Abelian flavored gauge-mediation scenarios, for messenger scales of M mess = 10 12 GeV, the gravitino has a mass of O(0.1 GeV), and the NLSP is not long-lived enough to decay during or after Big Bang nucleosynthesis (BBN).
Therefore, the successful predictions of BBN will not be spoiled (see e.g. [52][53][54]). For gravitinos of this mass range, there are well-known mechanisms to ensure the desired reheating temperatures and late entropy production to avoid having the gravitinos overclose the universe, so that gravitinos can then be a plausible dark matter candidate. For lower values of the messenger scale, the situation is further improved, as the gravitinos are lighter (with masses of the order of tenths of keV for M mess = 10 6 GeV) and the NLSP decays to gravitinos much more rapidly than in the higher messenger scale case, thus avoiding the need for gravitino dilution.

V. CONCLUSIONS
In this paper, we have explored MSSM flavored gauge mediation models in which the Higgsmessenger mixing is controlled by a discrete non-Abelian symmetry, here taken for simplicity to be S 3 . Building on previous analyses [37] which showed that viable models can be constructed for an extended Higgs-messenger sector that includes both S 3 doublet and singlet fields that mix to yield one light MSSM Higgs pair and two messenger pairs, we studied various possibilities for generating plausible SM quark and charged lepton masses in the case in which the MSSM matter fields also carry S 3 quantum numbers. While additional relations beyond S 3 are generically needed to obtain the desired hierarchical SM fermion masses, we have identified two general categories of solutions that we broadly categorized as Case 1 and Case 2 models. The Case 1 models obey Eq. (11), and encompass two regimes of interest: (i) the singlet-dominated limit, in which the Yukawa couplings involving only the S 3 singlets dominate, and (ii) the democratic limit, in which the Yukawa superpotential for the MSSM fields has an enhanced S 3L × S 3R symmetry. The Case 2 models, in contrast, include the doublet-dominated limit, in which the Yukawa couplings involving only the S 3 doublet fields dominate. The singlet-dominated limit was previously investigated in [38,43] and served here as a point of comparison for a general analysis of the Case 1 democratic limit and the Case 2 doublet-dominated models. We include corrections to obtain nonvanishing masses for one or both of the lighter families, as well as for the third family. In certain cases such corrections lead to off-diagonal corrections to the soft supersymmetry breaking mass terms, but these corrections are relatively mild (a feature that is known in the literature for flavored gauge mediation models of this general type) and as a result, do not immediately lead to insurmountable problems with flavor-changing neutral current constraints.
Within Case 1 models, our analysis shows that while the singlet-dominated limit allows for examples with optimized parameter sets that yield gluino and squark masses in the 4-5 TeV range, the Case 1 democratic limit generically has significantly heavier squark and gluino masses. The Case 2 models generally also yield heavier superpartner masses, with the heavier squarks and gluino in the 7 TeV mass range. Ultimately, the fact that the squark and gluino masses cannot be made lighter than 4-5 TeV even in the singlet-dominated limit is related to the fact that this non- analyses such as this one can serve as a reminder of the rich framework of TeV-scale N = 1 supersymmetry, and the many ways in which it might still be hiding at or just above TeV energies.
As the Terascale continues to be explored in this data-rich era for high energy physics, hopefully we will know relatively soon if TeV-scale supersymmetry is indeed part of our physical world.

ACKNOWLEDGMENTS
This work is supported by the U.S. Department of Energy under the contract number DE-SC0017647.