Unification of SUSY breaking and GUT breaking

We build explicit supersymmetric unification models where grand unified gauge symmetry breaking and supersymmetry (SUSY) breaking are caused by the same sector. Besides, the SM-charged particles are also predicted by the symmetry breaking sector, and they give the soft SUSY breaking terms through the so-called gauge mediation. We investigate the mass spectrums in an explicit model with SU(5) and additional gauge groups, and discuss its phenomenological aspects. Especially, nonzero A-term and B-term are generated at one-loop level according to the mediation via the vector superfields, so that the electro-weak symmetry breaking and 125 GeV Higgs mass may be achieved by the large B-term and A-term even if the stop mass is around 1 TeV.


I. INTRODUCTION
realized.
The breaking sector consists of one SU(5) F adjoint plus singlet filed (Φ) and SU(5) F fundamental and anti-fundamental fields (φ, φ). The vector-like pairs (φ, φ) are also charged under SU(2) × U(1) φ . As discussed in Ref. [6], this type of gauge theory causes SUSY breaking along with the gauge symmetry breaking. In our model, SU (5)  breaking effect through the gauge coupling with Φ, and play a crucial role in generating the non-zero A-term and B-term as discussed in Refs. [7,8]. It is well-known that SUSY-scale A-term could shift the upper bound on the lightest Higgs mass in the MSSM, even if squark is light, and the SUSY-scale B-term is required to realize the EW symmetry breaking. Our A-term and B-term are given at one-loop level, so that they are the same order as the squark masses and gaugino masses. In fact, we will see that Higgs mass could be around 125 GeV, even if Λ SU SY is less than O(1) TeV, and the B-term is consistent with the EW symmetry breaking, if we allow O(1)-factor difference.
In Sec. II, we introduce the SUSY and GUT breaking sector in generic SU(N F ) F × SU(N) × U(1) φ gauge theory. There, we discuss not only the symmetry breaking, but also the behavior of the gauge couplings and soft SUSY breaking terms according to the gauge mediation with the mediators of the chiral superfields and the vector superfields. In Sec. III, we apply the breaking sector to the SU(5) F ×SU(2) ×U(1) φ gauge theory. As we mentioned above, an interesting aspect of this model is the improvement of the consistency with the EW symmetry breaking and Higgs mass in the case with low-scale SUSY. We investigate the soft SUSY breaking terms, and discuss how well it is achieved in our scenario. In Sec.
IV, we give a comment on the possibility that the breaking sector is applied to other GUT models. Sec. V is devoted to the summary. In Appendix A, we give the mass spectrum in the SUSY breaking sector. In Appendix B, we show examples of mass spectrums in the MSSM sector.
In this section, we introduce the model which causes SUSY breaking together with gauge symmetry breaking, based on Ref. [6].
We consider SU(N F ) F ×SU(N) ×U(1) φ gauge theory with N F > N. The matter content is shown in Table I: Φ is the SU(N F ) F adjoint plus singlet filed and (φ, φ) pair is the vector- The superpotential is given by assigning U(1) R symmetry: the R-charge of Φ is 2 and the R-charge of (φ, φ) is vanishing.
However, there would be an issue about how to break R-symmetry and how to avoid the massless particle according the U(1) R symmetry breaking. Let us introduce explicit U(1) R breaking terms, and discuss the superpotential as W SB = W R +W / R . In Ref. [6], W R is generated, considering the dual side of SU(N F ) F × SU(N + N F ) gauge theory with the N F vector-like pairs (q d , q d ) of SU(N + N F ) gauge group. Φ is interpreted as the composite operator as Φ ≡ q d q d , and hΛ G Tr N F (Φ) in W R corresponds to the mass term of the (q d , q d ).
Some ideas to induce W / R have been proposed in Ref. [9], where the small wave-function factor of Φ suppresses Φ 2 and Φ 3 terms according to the strong dynamics or the profile in the extra dimension. In Ref. [10], the effect of the explicit R-symmetry breaking terms is well studied. Here, we simply start the discussion from the superpotential W SB assuming such a mechanism like Ref. [9], and study the symmetry breaking.
In the global SUSY with canonical Kähler potential, the scalar potential is given by V = In this model, ∂ Φ W SB is given by and all elements cannot be vanishing, because N F ×N F matrix (φ φ) has the rank N (< N F ).
This means that SUSY is broken by the F-components of (N F − N) elements in Φ and SU(N F ) F would be also broken.
Following Ref. [6], we decompose Φ and (φ,φ) as whereŶ ,χ andχ are N × N matrices,X is an This solution also satisfies the D-flat conditions. v X is a flat direction in global SUSY. If we consider gravity and one-loop corrections, it would be stabilized at the nonzero value [6,11].
The nonzero VEVs breaks

A. gauge bosons
After the symmetry breaking, massive gauge bosons appear according to the Higgs mechanism. Let us decompose the vector field (V µ F ) for SU(N F ) F as is defined. W µ F and G µ are the adjoint representations of the subgroups of SU(N F ) F : SU(N) F and SU( N ). X µ is the anti-fundamental and fundamental The nonzero VEVs generate the following mass terms, where ∆v = v X − v Y is defined. W ′Aµ and Z ′µ are given by the linear combinations of W Aµ F and SU(N) gauge boson (W Aµ N ), and B ′µ and where cos θ Y and cos θ is defined as G µ , W µ , and B µ are the gauge bosons for SU( N ) × SU(N) D × U(1) Y gauge symmetry, and their gauge couplings are given by B. SM-charged fields from symmetry breaking sector According to the decomposition in Eqs. (4) and (5), we introduce the charge assignment of (Z, Z), (ρ, ρ), Y , (χ, χ), and X in Table II. Y , (χ, χ), and X are the adjoint parts ofŶ , (χ,ˆ χ), andX. The singlet parts are not charged under the SM, and they are not so relevant to our analysis. The mass matrices are studied in Appendix A.
These fields obtain masses according to the nonzero VEVs, v χ , v Y and v X as we see in the Appendix A. They decouple at some scales above the EW scale. In the next subsection, we investigate the RG flows of the gauge couplings including the threshold corrections and discuss the soft SUSY breaking terms mediated by the heavy fields.

C. RG flows of the gauge couplings
In this model, two kinds of symmetry breaking actually happen: one is SU( is caused by ∆v, and the later is by v χ . We consider a simple scenario assuming ∆v ≫ v χ . According to the one-loop RG equations, the gauge couplings at the EW scale (M Z ) are evaluated as follows: SU(N) F , SU(N) and SU(N) D gauge couplings (α F N , α ′ N , α N ) are Λ is the cut-off scale and T i , , and ρ i decouple respecitvely. According to the mass spectrums at each scale The factor in front of each intermediate scale describes the freedom of the particles decoupling at the scale: T ex and ∆b J ex may be required to achieve realistic mass spectrums and the doublet-triplet mass splitting in explicit GUT models.
We also study the soft SUSY breaking terms of sfermions in the next subsection. Let us also introduce the wave function renormalization factor (Z q ) for SU(N) F -charged field (q). The one-loop renormalization group for Z q can be integrated analytically, if the Yukawa coupling is negligible, and c q i are the second Casimir of the field q, corresponding to the gauge groups. The masses squared of sfermions can be derived by the v X -dependence in Z q . v X appears in the gauge couplings, so that v X -dependence on the gauge couplings is only relevant to the sfermion masses [12]. [12]. In our model, massive gauge bosons and the fermionnic partners also works as the mediators to generate the soft SUSY breaking terms [7,8,13,14].

D. Soft SUSY breaking terms
In Eqs. (17) , (20), and (21), the only intermediate scales, where ξ N , ξ N , and ξ 1 describe the v X dependence on the masses of the extra particles.
Let us consider the soft SUSY breaking terms corresponding to the trilinear (A-term) and bilinear couplings (B-term) of the scalar components of the SU(N F ) F -charged fields (q I ). They could be relevant to the v X -dependence of the wave renormalization factor. For instance, the A-terms corresponding to the Yukawa couplings y IJK q I q J q K in the superpotential is given by A I is obtained from Eq. (31), The masses squared (m 2 q ) of q could be also estimated by the Eq. (31), seeing the |v X | 2dependence of Z q [12]. As discussed in Ref. [13], the gauge mediation with gauge messengers may contribute to the masses squared at the one-loop level, if the gauge symmetry breaking and SUSY breaking are caused by the VEVs and F-components of several fields. In our case, we simply assume v χ ≪ ∆v, so that the gauge symmetry breaking and SUSY breaking are caused by only ∆v and the F-component of ∆v. ‡ The one-loop correction is strongly suppressed by (v χ /∆v) 2 according to Ref. [13], so that we have to investigate the two-loop corrections, as discussed in Refs. [7,12].
Following Refs. [7,12], m 2 q could be written as . ‡ ∆v corresponds to the VEV of one adjoint field.
In the next section, we discuss one explicit model, where SU( N ) × SU(N) D × U(1) Y is the SM gauge groups corresponding to (N F , N) = (5,2). In the explicit model, we see that a few parameters control all soft SUSY breaking terms according to this analysis. Then, Λ SU SY is roughly given by (α G /(4π)) × (F X /|∆v|), and A-term and B-term could also be of O(Λ SU SY ), which we could expect that are consistent with the condition for the EW symmetry breaking. We study the compatibility with the EW condition and the Higgs mass, in Sec. III D.
In this section, we consider a SU (5) where 5 k and 10 l are defined as the matter fields. As well-known,ŷ u kl andŷ d kl may require Φ and (φ, φ) dependences in order to generate realistic mass matrices at the EW scale according to the higher-dimensional operators. Here, we simply assume that the contributions to the soft SUSY breaking terms are enough small.
One serious problem in the SU(5) GUT is how to generate the mass splitting between the colored Higgs and the MSSM Higgs doublet. The mass of colored Higgs should be around the GUT scale to avoid the too short life time of proton: m Hc 10 16 GeV×(1TeV/Λ SUSY ) [15]. In our SU(5) F × SU(2) × U(1) φ model, the relevant terms to the Higgs masses is written as After the symmetry breaking, the colored Higgs mass and MSSM Higgs mass are given by If v Y = m φ /h is the GUT scale, µ should be also around the GUT scale and then the fine-tuning between µ and λv Y is required: On the other hand, the colored-Higgs mass is , so that ξ for the colored Higgs in soft SUSY breaking terms is approximately estimated as ξ ≈ sign(λ H ). The one-loop correction of H c to m 2 q would be suppressed, because the m Hc -dependence appears in Z q as ln(|m Hc + λ H F X θ 2 | 2 ) according to the study in Ref. [12]. We could apply our analysis in Sec. II D to this scenario.  Fig. 1 shows the allowed region for T X , which may not be far from O(Λ SU SY ). Fig. 2 shows the gauge couplings, (α F 2 , α 2 , α F 1 , α φ ) at the SUSY breaking scale. Fig. 3 describes RG flows of the gauge couplings (α 3 , α 2 (α F 2 ), α 1 (α F 1 )), when T X = 10 7 GeV, T χ = 3.8 × 10 10 GeV, T ρ = 7.9 × 10 11 GeV, and T GU T = 2 × 10 16 GeV. § T χ1 = T χ2 , T ρ1 = T ρ3 , and T G = T 1 = T 2 = T 3 are assumed.

B. soft SUSY breaking terms
We qualitatively evaluate the soft SUSY breaking terms in this scenario. According to the analysis in Sec. II D, the gaugino masses at µ < T χ are written as so that we could derive the following mass relation: The masses are almost degenerate, and this may be a specific feature of the gauge messenger model [7,16]. ¶ If all intermediate scales are close to the GUT scale, the fine-tuning of µ term may be drastically reduced, as discussed in Ref. [18]. Fig. 2 tells us that the extra SU(3)-adjoint field reside in the low-scale, so that the condition for the small µ-term would be modified. The one-loop running correction of m 2 Hu with T X = 10 7 GeV from T χ to M Z is estimated as where the ellipsis denotes the terms including A-term and scalar masses and those are not important when they are comparable to the gluino mass. This leads that the condition to ¶ The gaugino masses are degenerate in the TeV-scale mirage mediation scenario, too [17].
cancel the large contribution of gluino is M 2 /M 3 (M Z ) ≈ 1.23, which suggests the almost degenerate mass spectrum. However, we have a large A-term contribution to ∆m 2 Hu in our model, so that it may be difficult to avoid a certain fine-tuning even if the gaugino masses are degenerate.
According to Eqs. (36) and (35), the masses squared of superpartners and A-term are evaluated explicitly. Setting T G = T Hc > T ρ > T χ and ξ = 1, stop masses at T χ are given by As we see, large stop masses are generated by the large second casimir (c t 2 = 18/5), but they might be driven to the tachyonic if T χ and T ρ are close to the GUT scale. The SUSY scale (Λ SU SY ) from the gauge mediation is defined as 16 GeV, so that Λ SU SY might be compatible with m 3/2 .
If the correction from the gravity mediation is estimated as O(m 3/2 ), it is the same order as the one from the gauge mediation and it may make it difficult to control flavors. We assume that the gravity effect is sub-dominant and discuss the SUSY mass spectrum without the gravity mediation. * * A t , which is the trilinear coupling of stops ( t) as y t A t t L H u t R is given by and the B-term, which is the bilinear coupling of two Higgs µBH u H d , is estimated as As we see, the A-term and B-term might be large as O(10)Λ SU SY . This may be good to achieve the EW symmetry breaking, but too large A-term makes the stop masses tachyonic because of the running correction such as (50) * * The gauge-mediation contributions are large in our model, compared with the gravitino mass, as we see in Table IV. In this case, we could expect the gravity-mediation effect is sub-dominant. If the gravity mediation is comparable to the gauge mediation, we have to consider the dynamics to suppress the contribution of the gravity mediation above the GUT scale, as discussed in Refs. [19].
In our model, the gluino mass M 3 is relatively small as wee see in Eq. (40), so ∆m 2 U (M Z ) becomes easily negative and stop mass becomes tachyonic even if the positive m 2 U is generated at the SUSY breaking scale T χ . In order to avoid the tachyonic stop masses, we add an extra contribution to the gluino mass, as we see below.

C. Shift of the gluino mass
We consider an extra term, which contributes to the gluino mass, There are several ways to introduce this term, such as gravity effect. Here, we simply assume that N extra extra heavy SU(5) vector-like pairs (ψ, ψ) with the masses ψ(Λ 0 + λ X Φ)ψ induce this term, integrating out them at the scale Λ 0 . After the SU(5) breaking, the gauge coupling would have the extra v X dependence as This additional coupling could shift the gluino mass as where N ef f may not be N extra because of the scale difference between Λ 0 and the GUT scale.
Including N ef f , the gluino mass becomes so N ef f should be bigger than 2 in order to shift M 3 . In fact, we discuss N ef f = 3, 4 cases and find that N ef f enables us to evade the negative squared masses. Λ SU SY O(10) TeV in the simple scenarios as discussed in Ref. [3]. O(10)-TeV SUSY scale would require 0.01% fine-tuning against µ without any cancellation in m 2 Hu . As pointed out in Refs. [20,21], it is known that a special relation between A t and squark mass relaxes the fine-tuning, maximizing the loop corrections in the Higgs mass in the MSSM. This relation is so-called "maximal mixing" and described as X t /m stop = √ 6, where X t = A t − µ/ tan β and  Fig. 5 shows that small tan β is consistent with the EW symmetry breaking. B EW is the value to realize the EW symmetry breaking, and B is our prediction via the gauge mediation. Unfortunately, at least tan β 4 is necessary to achieve 125 GeV Higgs mass, so that we could conclude that an additional contribution to B, which is the same order as the contribution of the gauge mediation, is required by the one-loop calculation. Table IV  We introduce SU(3) vector-like fields (H 3 , H 3 ) and assign Z 3 symmetry to the fields as in Table III. Z 3 symmetry is broken by the VEV of S. The superpotential for the Higgs sector is given by H u and H d correspond to the Higgs SU(2) L doublets in MSSM, and they could get the supersymmetric mass term according to the nonzero VEV of S. This type product GUT has been studied in Ref. [23].
In order to avoid the bound from the proton decay caused by the five dimensional oper- F X is given by −hv 2 χ , so that very tiny h is necessary to achieve the low-scale SUSY. When v χ ≈ 10 16 GeV and Λ SUSY = 1 TeV are set, h should be around O(10 −10 ), because of We conclude that high-scale SUSY is favored to avoid such an extremely small h.
We can consider the applications of our symmetry breaking models to the other BSMs, such as We would study such patterns elsewhere [24]. In these models, all of chiral superfields appear as adjoint representations and bi-fundamental representations. Such models can be constructed in D-brane models, e.g. intersecting/magnetized D-brane models (see for a review [25,26] and references therein). Thus, the above models are interesting from the viewpoint of superstring theory.

V. SUMMARY
The MSSM is one of the attractive BSMs to solve the hierarchy problem in the SM and it may be expected to be found near future. One big issue in the MSSM is how to control the SUSY breaking parameters, so that many ideas and works on spontaneous SUSY breaking and mediation mechanisms of the SUSY breaking effects have been discussed so far. In this Our light SUSY particles are wino, bino, and gravitino, and the mass difference is not so big. The lightest particle is bino, and wino is slightly heavier than bino. The mass difference is 0.06 × m 3/2 GeV. This might be one specific feature of the gauge messenger scenario in SU(5) GUT, as discussed in Ref. [16]. We investigate the mass matrices for the remnant fields in the symmetry breaking sector.
First, let us discuss (Z, Z) and (ρ, ρ) components. We define Z ± and ρ ± as The fermion masses are given by where the mass matrices (M f ± ) are and λ ± are the linear combinations of the gauginos (X (+) ) which are the suparpartners of X µ , The masses for the bosonic superpartners are where the mass matrices (M 2 ± ) are given by The F-term F X is F X = −h 2 v 2 χ , so that M 2 + includes the Goldstone mode. The fermion masses for the other particles are also generated by the VEVs: