Sizable D-term contribution as a signature of $E_6 \times SU(2)_F \times U(1)_A$ SUSY GUT model

We show that the sizable $D$-term contributions to sfermion mass spectrum can be signatures of certain GUT, $E_6\times SU(2)_F\times U(1)_A$ GUT. Note that these $D$-term contributions deviate the degenerate sfermion masses among different generations in this model. This is different from the previous works which have argued the $D$-term contributions, which deviate masses only between sfermions with different quantum charges, as a signature of GUT with larger rank unification group. Such $D$-terms are strongly constrained by the FCNC processes if the SUSY breaking scale is the weak scale. However, in $E_6\times SU(2)_F\times U(1)_A$, natural SUSY type sfermion mass spectrum is obtained, and if the masses of ${\bf 10}_3$ sfermions are larger than $O(1{\rm TeV})$ to realize 126 GeV Higgs and the other sfermion masses are $O(10{\rm TeV})$, then sizable $D$-term contribution is allowed. If the deviations by these $D$-terms can be observed in future experiments like 100 TeV proton collider or muon collider, we may confirm the $E_6\times SU(2)_F\times U(1)_A$ GUT.


Introduction
Grand unified theory (GUT) [1] is one of the most promising extensions of the standard model (SM). It unifies not only three gauge interactions in the SM into a single gauge interaction, but also, e.g., quarks and leptons into a few multiplets, 10 and5 of SU (5). Moreover, there is experimental support for both unifications. For the unification of forces, the measured values of three gauge couplings are quantitatively consistent with the unification of gauge interactions in the minimal supersymmetric (SUSY) SM (MSSM). For the unification of matters in SU (5) GUT, if we assume that the 10 matter fields induce stronger hierarchies of Yukawa couplings than the5 matter fields, the various measured hierarchies of quark and lepton masses and mixings can be explained qualitatively at the same time [2][3][4][5][6][7][8][9].
In E 6 unification [10][11][12][13][14][15][16][17], the above assumption for the origin of the hierarchies can be derived [9]. As a result, we can obtain various realistic hierarchies of Yukawa couplings from one basic Yukawa hierarchy that realizes the hierarchy of up-type quarks. Moreover, if the family symmetry [18][19][20][21][22][23][24][25][26][27], SU (3) F or SU (2) F , is introduced, we can obtain a model in which all three generations of quarks and leptons can be unified into a single multiplet or two multiplets, and after breaking the family symmetry and E 6 unified symmetry, realistic quark and lepton masses and mixings can be realized [28][29][30][31]. Such models predict a peculiar sfermion mass spectrum in which all sfermions except the third generation of the 10 matter 10 3 have universal sfermion masses. This is called modified universal sfermion masses (MUSM). When the mass of 10 3 is smaller than the other universal sfermion masses, the mass spectrum is nothing but the natural SUSY-type sfermion mass spectrum [32,33], in which the SUSY flavor-changing neutral current (FCNC) processes are suppressed due to large sfermion masses while the weak scale is stabilized.
The most difficult problem in the SUSY GUT scenario is the doublet-triplet splitting problem (for a review, see Ref. [34]). One pair of Higgs doublets in the MSSM can be included in 5 H and5 H with one pair of triplet (colored) Higgs. The mass of the triplet Higgs must be larger than the GUT scale to stabilize the nucleon, while the mass of the doublet Higgs must be around the weak scale. It is difficult to realize such a splitting without fine-tuning. Several ideas to solve this problem have been discussed in various models in the literature. Unfortunately, in most of the models, very small parameters are required or the terms that are allowed by the symmetry are dropped just by hand. Such a feature is, in a sense, fine-tuning.
If the anomalous U (1) A gauge symmetry [35][36][37][38] is introduced, the doublet-triplet splitting problem can be solved in a natural assumption that all the interactions that are allowed by the symmetry are introduced with O(1) coefficients. Note that all higher-dimensional interactions that are allowed by the symmetry are introduced. Because of this natural assumption, we call the GUT scenario with the anomalous U (1) A gauge symmetry "natural GUT" [4][5][6][7][8]39]. Note that in natural GUT the vacuum expectation value (VEV) of an operator O i that is a singlet under gauge groups, except U (1) A , can be determined by its where λ is determined from the Fayet-Iliopoulos parameter ξ as λ ≡ ξ/ . In this paper, we take λ ∼ 0.22 and adopt the unit in which the cutoff = 1. This feature is important in solving the doublet-triplet splitting problem. If we consider the E 6 GUT with family symmetry and the anomalous U (1) A gauge symmetry at the same time, more attractive GUT model with E 6 × SU (2) F × U (1) A gauge symmetry can be obtained. Since the μ problem is also solved in the natural GUT [39,40], we can discuss the SUSY CP problem. Actually, by imposing the CP symmetry and considering the spontaneous CP violation, we can solve not only the usual SUSY CP problem but also the new SUSY CP problem on the chromoelectric dipole moment (CEDM) [41][42][43], which is more serious in the natural SUSY-type sfermion mass spectrum [44][45][46][47].
How can this interesting SUSY GUT scenario be tested? Since the unification scale is so large that it is difficult to produce GUT particles directly, it is important to examine various indirect searches. The most promising candidate for the indirect search is to find the nucleon decay. In the natural GUT, the nucleon decay via dimension-6 operators is enhanced while the nucleon decay via dimension-5 operators is suppressed [4][5][6][7][8]. We have proposed how to identify the unification group in the natural GUT by observing the decay modes of nucleons in Refs. [48,49].
Recently, one of the authors has pointed out that if the gravitino mass is O(100 TeV) to solve the gravitino problem and the other SUSY breaking parameters are O(1 TeV) for the gauge hierarchy problem, the little hierarchy problem becomes less severe (O(%) tuning is realized) [50]. In this scenario, a sizable anomaly mediation [51,52] contribution cancels the renormalization group (RG) effects of the gravity mediation. As a result, if the mirage scale, at which three gaugino masses meet, is O(TeV), we can observe directly the gravity contribution at the GUT scale by low-energy experiments at the TeV scale, except for the stop, the right-handed stau, and the up-type Higgs masses. (In the usual mirage mediation scenario, all sfermion masses can converge at the mirage scale by taking the special boundary conditions [53][54][55][56][57], while in the scenario without the special conditions the stop masses do not become the gravity contribution at the GUT scale because of the large top Yukawa couplings [50].) Note that the sfermions' masses other than the stops and the right-handed stau do not have to be universal in the arguments [50]. Therefore, we can test the GUT scenario by measuring the sfermion mass spectrum if some signatures of the GUT appear in the sfermion masses. For example, if the rank of the unification group is larger than 4, the non-vanishing D-term contributions, which are usually flavor blind, can be a signature of the GUT scenario [58][59][60][61]. The MUSM can be a signature of E 6 × SU (2) F GUT, since the most serious CEDM constraints for the natural SUSY-type mass spectrum can be avoided in the scenario by spontaneous CP violation [44][45][46]. One more interesting test for the E 6 × SU (2) F GUT scenario is to observe the non-vanishing D-term contributions of the E 6 and SU (2) F gauge symmetry to sfermion masses. This is interesting because they spoil the universality of the sfermion masses. Before the LHC found the 126 GeV Higgs [62,63], these D-term contributions were strongly constrained to be small from the various FCNC processes [64]. However, the stop mass must be larger than 1 TeV in order to realize the 126 GeV Higgs, and therefore the other sfermion masses can be O(10 TeV). Since FCNC constraints become much weaker when the SUSY breaking scale is O(10 TeV), a sizable D-term contribution may be allowed.
In this paper, we clarify the D-term contributions of the E 6 × SU (2) F × U (1) A GUT model and discuss the FCNC constraint from of K 0K 0 mixing because it is the strongest. We will conclude that a sizable D-term contribution is possible. If the D-term contributions are sufficiently large and observed by future experiments, e.g., by the SuperLHC, then we can obtain precious evidence of the GUT scenario.

E 6 × SU(2) F × U(1) A SUSY GUT model
In this section we give a short review of the Refs. [10][11][12][13][14][15][16][17] for a more detailed explanation of the model. The notation for the GUT model in this paper is almost the same as that for the model in Ref. [49].

Yukawa matrices for quarks and charged leptons
The contents of matters and Higgs and their charge assignments are shown in Table 1, though this model is just an example. In the model we introduce three 27 dimensional (fundamental) fields of E 6 as matters. The 27 is decomposed in the E 6 ⊃ SO(10) × U (1) V notation (and in the [SO(10) ⊃ SU (5) × U (1) V ] notation) as The 27 of E 6 includes not only spinor 16 but also vector 10 of SO (10). These 10s of SO(10) play an important role in obtaining realistic quark and lepton masses and mixings. The spinor and vector of SO (10) Matter fields 27 i (i = 1, 2, 3) include six5s of SU (5). Three of the six5s become superheavy by developing the VEVs , which breaks E 6 into SO (10), and C , which breaks SO(10) into SU (5), through the superpotential where a, b, c, f , and , and , and it contributes to 1 2 . The other three5s become the SM5 0 i whose main components become . This is a critical observation in calculating the D-term contribution to the sfermion masses. After where a, b, c, d q , d 5 , d l , f , g, and β H are real O(1) coefficients, ρ and δ are O(1) phases, and λ ∼ 0.22 is taken to be the Cabibbo angle. In this paper, we begin our arguments from these Yukawa matrices that have only 9 real parameters and 2 CP phases.

Mass spectrum of sfermions
The sfermion mass matrices can be obtained mainly from the SUSY breaking potential where the terms in the last line give the mass terms between5 and5 after developing the VEVs , ¯ , C , and C . These mass parameters are considered to be the SUSY breaking scale O(1-10 TeV). The D-term contributions are written as    where the contribution of the term m 2 † a † a to |16 a | 2 is included in m 2 0 by redefinition of m 2 0 . Then, the sfermion mass matrix for SM5 fields, which are represented as (5 0 1 ,5 0 2 ,5 0 3 ) ∼ (5 1 ,5 1 ,5 2 ), becomesm Moreover, the contributions from the sub-leading components of5 0 i become These sfermion mass matrices give interesting predictions of E 6 × SU (2) F × U (1) A GUT, though the terms that are suppressed by the power of λ are strongly dependent on the explicit model. In the next section, we discuss how to obtain GUT information from the sfermion mass spectrum.

Signatures of E 6 × SU(2) F × U(1) A GUT from the sfermion mass spectrum
Suppose that, in the future, all sfermion and gaugino masses are measured by experiments, and three gaugino masses meet at the GUT scale or at a mirage scale. Then, in principle, we can calculate the sfermion mass spectrum at the GUT scale or the mirage scale from the measured values by RG equations. If the mass spectrum respects the SU (5) symmetry, this can be a signature for GUT, though in the generalized mirage mediation scenario, sfermions that have large Yukawa couplings like top Yukawa coupling do not have masses that are consistent with the SU (5) GUT symmetry generically at the mirage scale [50]. In this section, we discuss the signatures of GUT in the sfermion mass spectrum at the GUT scale. The constraints from the FCNC processes will be discussed in the next section.
If the observed sfermion mass spectrum at the GUT scale or at the mirage scale is the MUSM as The third generation of 10 of SU (5) may not respect SU (5) in generalized mirage mediation because of large top Yukawa coupling [50].) Of course, the MUSM is nothing but a natural SUSY-type sfermion mass spectrum, which is predicted by a lot of models. However, the generically natural SUSY-type sfermion mass spectrum suffers from the CEDM problem [41][42][43], and there are few models in which the problem can be solved in a natural way. We would like to emphasize that the CEDM problem can be solved in the E 6 × SU (2) F × U (1) A GUT by spontaneous CP violation in a non-trivial way [44][45][46]. 6 In order to obtain more specific signatures of the E 6 × SU (2) F × U (1) A GUT, we study the Dterm contributions. For a while, we neglect the terms that are suppressed by the power of λ. We will discuss these terms later. Then, the mass matrices ofm 2 10 andm 2 5 0 are rewritten as where 1 3×3 is a 3 × 3 unit matrix. A non-trivial prediction of this model is m 2 10,2 = m 2 5,3 . If this relation is observed, we obtain strong evidence for this model and can know the D F . The D 6 and D 10 can be determined if m 2 10,0 − m 2 5 0 ,0 and m 2 5,2 are observed. If these small modifications from the MUSM and m 2 10,2 = m 2 5,3 are observed, we think that the What size D-terms are allowed? If these D-terms are very small, it may become difficult to measure them, and if these D-terms are large, the FCNC constraints cannot be satisfied. In the next section, we study the constraints to the D-terms from the FCNC processes, especially from the parameter in K 0K 0 mixing, which gives the strongest constraints.

FCNC constraints to D-terms
In this section, we focus on the natural SUSY-type sfermion masses, i.e., m 0 m 3 , because the FCNC constraints become weaker and a sizable D-term may be allowed. Therefore, we fix m 2 10,3 = m 2 0 . To obtain the 126 GeV Higgs, m 3 must be larger than 1 TeV. Since the smaller m 3 tends to be more natural, we take m 3 ∼ O(1 TeV). In the literature, the upper bound for the ratio m 0 /m 3 has been studied; it is derived from the requirement of the positivity of the stop mass square to be roughly 5 through a two-loop RG contribution [32,33]. Therefore, we expect that m 0 is O(10 TeV). In this paper, we do not argue the upper bound of m 0 /m 3 explicitly, because a larger stop mass can always satisfy the positivity and the upper bound is dependent on the explicit models between the GUT scale and the SUSY breaking scale.
If the D-term contributions can be negligible, the contributions to the FCNC processes from the MUSM become sufficiently small to satisfy the experimental bounds [28][29][30][31], though the CEDM constraint is quite severe, which will be discussed later. When the D-terms become sizable, the strongest constraints can be given from the CP-violating parameter in K 0K 0 mixing. Basically, if these constraints are satisfied, the other FCNC constraints are automatically satisfied. Therefore, we consider here the constraints from the CP-violating parameter in K 0K 0 mixing. Since we calculate constraints from the FCNC processes with the mass eigenstates of quarks and leptons, we need diagonalizing matrices that make Yukawa matrices diagonal as where ψ is a flavor eigenstate, ψ is a mass eigenstate, and Y D ψ is a diagonalized matrix of ψ. We summarize the detailed expression of these diagonalizing matrices with the explicit O(1) coefficients in Appendix A. Here we roughly show the diagonalizing matrices for up-type quark, down-type quark, and charged lepton without O(1) coefficients as We have two types of diagonalizing matrices for 10 of SU (5) sfermions and for5 sfermions as where a i j and b i j are generically complex O(1) coefficients, respectively. The mass insertion parameters defined as where mψ is the averaged mass of ψ = u, d, e, ν and is taken as m 0 in many cases in this paper, can be calculated as · · · a * 21 λ m 2 10,2 + a * 31 a 32 λ 5 m 2 10,3 (a * 21 a 23 m 2 10 2 + a * 31 m 2 10,3 )λ 3 · · · · · · (a 23 m 2 10,2 + a * 32 m 2 10,3 )λ 2 · · · · · · · · · ⎞ ⎟ ⎠ m 2ψ , for 10 fields and5 fields, respectively. In Appendix B, we show each mass insertion in this model with explicit O(1) coefficients. Let us calculate the constraints from the parameter in K 0K 0 mixing. We use the constraints for which are obtained in Refs. [65,66] by including the SM contribution and next-to-leading order calculation of QCD. These parameters can roughly be calculated as By taking m 2 10,3 = m 2 0 = m 2d , we can obtain the allowed region in | m 2 5,2 | md , | m 2 10,2 | md = | m 2 5,3 | md space, which is shown in Fig. 1. Note that m 2 10,2 = m 2 5,3 is one of the predictions in the  In the above arguments, we have neglected the contributions to the sfermion masses that are suppressed by the power of λ in Eqs. (13) and (14). All these terms except the λ 2 m 2 term in Eq. (14) can be neglected in the above arguments. However, the λ 2 m 2 term gives non-vanishing | m 2 5,2 | md , which becomes O(λ) if m ∼ m 0 . The FCNC constraints in this situation can be easily extracted from Fig. 1.
At the end of this section, we comment about the CEDM constraints. As noted in Refs. [44][45][46], the CEDM constraints from the neutron (Hg) are very severe, especially for the models with natural SUSY-type sfermion masses like the MUSM as Since we usually take the complex Yukawa couplings to obtain the sizable Kobayashi-Maskawa (KM) phase, (δ u 13 ) L L and (δ u 31 ) R R become complex generically. If (δ u 13 ) L L (δ u 31 ) R R has O(1) complex phase, the above constraints cannot be satisfied because |(δ u 13 ) L L (δ u 31 ) R R | ∼ λ 6 ∼ 10 −4 in this model. Note that models with the natural SUSY-type sfermion mass spectrum are severely constrained by the CEDM generically. However, in E 6 × SU (2) F × U (1) A with spontaneous CP violation, the up-type Yukawa matrix becomes real as in Eq. (6) and therefore L u and R u are also real as in Eq. (20). As a result, (δ u 13 ) L L and (δ u 31 ) R R become real, and the above severe constraints can be satisfied in a non-trivial way.

Discussions and summary
We have shown that the sizable D-term contributions to the sfermion mass spectrum can be signatures of a certain GUT, E 6 × SU (2) F × U (1) A GUT. Note that these D-term contributions destroy 10/18 the degeneracy of sfermion masses among different generations in this model. This point is one large difference between our work and the previous works, which have argued the D-term contributions [58][59][60][61], which destroy the degeneracy of masses only between sfermions with different quantum charges, as a signature of GUT with larger rank unification group. Such D-terms are strongly constrained by the FCNC processes if the SUSY breaking scale is the weak scale. However, in E 6 × SU (2) F × U (1) A , a natural SUSY-type sfermion mass spectrum is obtained, and if the masses of 10 3 sfermions are larger than O(1 TeV) to realize the 126 GeV Higgs and the other sfermion masses are O(10 TeV), then a sizable D-term contribution is allowed. A novel relationm 2 5 3 −m 2 5 1 = m 2 10 2 −m 2 10 1 is predicted in this model. If these D-terms can be observed in future experiments like the 100 TeV proton collider or muon collider, we may confirm the E 6 × SU (2) F × U (1) A GUT.
Since we have in mind the generalized mirage mediation scenario in which the mirage scale is the weak scale [50], we have not considered the RG effects in estimating the FCNC constraints in this paper. However, for the other SUSY breaking scenario, we have to consider the renormalization group (RG) effects in the estimation generically. It is possible that m 3 is much smaller than 1 TeV, while a sufficiently large stop mass for the 126 GeV Higgs can be obtained from the RG effects via the gluino. In such a situation, the lepton flavor violation processes can be sizable [28][29][30][31]. However, the constraint is quite weak as m 3 > 200 GeV.
Since the GUT scale is much larger than the TeV scale that we can reach by experiments, it is important to consider how to test the GUT scenario. We have discussed the D-term contributions that are dependent on generations, and they can be a promising signature of the E 6 × SU (2) F × U (1) A GUT scenario.

Appendix A. The coefficients of diagonalizing matrices (in leading order)
In Appendix A in Ref. [49], we showed how to diagonalize the 3 × 3 matrix Y i j . Here we show the diagonalizing matrices for up-type quark, down-type quark, and charged lepton. The diagonalizing matrices L ψ and R ψ come from mixing angles s