Unification of SUSY breaking and GUT breaking
aa r X i v : . [ h e p - ph ] F e b EPHOU-14-018
Unification of SUSY breaking and GUT breaking
Tatsuo Kobayashi and Yuji Omura Department of Physics, Hokkaido University, Sapporo 060-0810, Japan Department of Physics, Nagoya University, Nagoya 464-8602, Japan (Dated: July 7, 2018)
Abstract
We build explicit supersymmetric unification models where grand unified gauge symmetry break-ing and supersymmetry (SUSY) breaking are caused by the same sector. Besides, the SM-chargedparticles are also predicted by the symmetry breaking sector, and they give the soft SUSY breakingterms through the so-called gauge mediation. We investigate the mass spectrums in an explicitmodel with SU (5) and additional gauge groups, and discuss its phenomenological aspects. Espe-cially, nonzero A-term and B-term are generated at one-loop level according to the mediation viathe vector superfields, so that the electro-weak symmetry breaking and 125 GeV Higgs mass maybe achieved by the large B-term and A-term even if the stop mass is around 1 TeV. . INTRODUCTION As well-known, the Standard Model (SM) is very successful in describing our nature, andit is firmly established by the Higgs discovery at the LHC [1]. There are still some ambiguitiesin not only the signal strength of the Higgs particle but also the other observations such asflavor physics, but it would be getting more difficult to consider new-physics effects in anysignals.On the other hand, we are sure that the SM remains several mysteries about our nature:the origin of the fermion generations, the hyper-charge assignment, the Higgs mass, andso on. Many Beyond Standard Models (BSM) were proposed so far motivated by thosemysteries, and some of them are expected to be found near future. One of the candidates isthe supersymmetric grand unified theory (GUT), which reveals the origin of the Higgs massand the fermion charges. There are some issues in Yukawa couplings, for instance, how togenerate realistic Yukawa couplings and heavy colored Higgs, but it succeeds in the chargequantization ( | Q e + Q p | < − [2]) and naturally deriving the electro-weak (EW) scale, ifthe supersymmetry (SUSY) scale (Λ SUSY ) is close to the EW scale. The supersymmetricGUT scenario is constrained by the observation of the proton decay, the direct search ofSUSY particles, and the SM measurements. Especially, the Higgs discovery around 125GeV may require high-scale SUSY (Λ
SUSY ≫ O (1)TeV) [3], which may discard the strongmotivation of SUSY, that is, the natural explanation of the EW scale. Furthermore, thegauge coupling unification of supersymmetric SU (5) GUT might be lost in high-scale SUSY,depending on the mass spectrum of the SUSY particles. The supersymmetric models couldhave so many parameters in the bottom-up approach, so that we could have some solutionsfor the Higgs mass and the gauge coupling unification. However, it is very important to findhow to derive such a specific SUSY mass spectrum.In this paper, we propose an explicit supersymmetric GUT with SU (5) F × SU (2) × U (1) φ gauge groups. We discard the miracle of the gauge coupling unification in the MinimalSupersymmetric SM (MSSM), but SUSY breaking and GUT breaking sectors are unified. ∗ The SM-charged particles also appear after the symmetry breaking, so the messenger fieldsfor the gauge mediation is also introduced by the breaking sector in our model. † The SMfields are only charged under the SU (5) F gauge group, so that the charge quantization is ∗ This type of scenario has been proposed in Refs. [4, 5]. † The messenger sector and SUSY breaking sector are unified, for instance, in Refs. [6]. SU (5) F adjoint plus singlet filed (Φ) and SU (5) F fundamental and anti-fundamental fields ( φ, e φ ). The vector-like pairs ( φ, e φ ) are also chargedunder SU (2) × U (1) φ . As discussed in Ref. [7], this type of gauge theory causes SUSYbreaking along with the gauge symmetry breaking. In our model, SU (5) F × SU (2) × U (1) φ symmetry breaks down to the SM gauge groups, SU (3) c × SU (2) L × U (1) Y , where SU (3) c is from the subgroup of SU (5) F , and SU (2) L × U (1) Y are the linear combinations of thesubgroup of SU (5) F and SU (2) × U (1) φ . SUSY is broken by the F-component of the part ofΦ. After the symmetry breaking, SM-charged particles are generated by the fluctuation ofΦ and ( φ, e φ ) around the vacuum expectation values (VEVs). One interesting point is thatthe massive gauge boson of SU (5) F and the fermionic partners could mediate the SUSYbreaking effect through the gauge coupling with Φ, and play a crucial role in generating thenon-zero A-term and B-term as discussed in Refs.[8, 9]. It is well-known that SUSY-scaleA-term could shift the upper bound on the lightest Higgs mass in the MSSM, even if squarkis light, and the SUSY-scale B-term is required to realize the EW symmetry breaking. OurA-term and B-term are given at one-loop level, so that they are the same order as the squarkmasses and gaugino masses. In fact, we will see that Higgs mass could be around 125 GeV,even if Λ SUSY is less than O (1) TeV, and the B-term could be consistent with the EWsymmetry breaking.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 alsothe behavior of the gauge couplings and soft SUSY breaking terms according to the gaugemediation 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 mentionedabove, an interesting aspect of this model is the improvement of the consistency with theEW symmetry breaking and Higgs mass in the case with low-scale SUSY. We investigatethe 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 GUTmodels. Sec. V is devoted to the summary. In Appendix A, we give the mass spectrumin the SUSY breaking sector. In Appendix B, we show examples of mass spectrums in theMSSM sector. 3 I. SU ( N ) × SU ( N F ) F × U (1) φ GAUGE THEORY
In this section, we introduce the model which causes SUSY breaking together with gaugesymmetry breaking, based on Ref. [7].We consider SU ( N F ) F × SU ( N ) × U (1) φ gauge theory with N F > N . The matter contentis shown in Table I: Φ is the SU ( N F ) F adjoint plus singlet field and ( φ, e φ ) pair is the vector-like under SU ( N F ) F × SU ( N ) × U (1) φ gauge group. φ e φ Φ SU ( N F ) F N F N F adj N F + SU ( N ) N N 1 U (1) φ Q φ − Q φ SU ( N F ) F × SU ( N ) × U (1) φ gauge theory The superpotential is given by W R = − hT r N ( e φ Φ φ ) + h Λ G Tr N F (Φ) , (1)assigning U (1) R symmetry: the R-charge of Φ is 2 and the R-charge of ( φ, e φ ) is vanishing.However, there would be an issue about how to break R-symmetry and how to avoid themassless particle according the U (1) R symmetry breaking. Let us introduce explicit U (1) R breaking terms, W /R = m φ T r N ( e φφ ) + c, (2)and discuss the superpotential as W SB = W R + W /R . In Ref. [7], W R is generated, consideringthe dual side of SU ( N F ) F × SU ( N + N F ) gauge theory with the N F vector-like pairs ( q d , e q d )of SU ( N + N F ) gauge group. Φ is interpreted as the composite operator as Φ ≡ e q d q d , and h Λ G Tr N F (Φ) in W R corresponds to the mass term of the ( q d , e q d ).Some ideas to induce W /R have been proposed in Ref. [10], where the small wave-functionfactor of Φ suppresses Φ and Φ terms according to the strong dynamics or the profile inthe extra dimension. In Ref.[11], the effect of the explicit R-symmetry breaking terms is wellstudied. Here, we simply start the discussion from the superpotential W SB assuming thatsuch a mechanism, as discussed in Ref. [10], works in underlying theories above the GUT,and study the symmetry breaking. In the global SUSY with canonical K¨ahler potential, the4calar potential is given by V = | ∂ Φ W SB | + | ∂ φ W SB | + | ∂ e φ W SB | , and SUSY vacua satisfy ∂ Φ W SB = ∂ φ W SB = ∂ e φ W SB = 0. In this model, ∂ Φ W SB is given by ∂ Φ ji W SB = − h ( φ e φ ) ij + h Λ G δ ij , (3)and all elements cannot be vanishing, because N F × N F matrix ( φ e 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. [7], we decompose Φ and ( φ, ˜ φ ) asΦ = ( v Y ) N + ˆ Y e ZZ ( v X ) e N + ˆ X , (4) φ = ( v χ ) N + ˆ χρ , e φ T = ( v χ ) N + ˆ e χ T e ρ T , (5)where ˆ Y , ˆ χ and ˆ˜ χ are N × N matrices, ˆ X is an e N × e N matrix ( e N = N F − N ), Z and ρ ( e Z and e ρ ) are N × e N matrices ( e N × N matrices). The VEVs, v Y and v χ , are fixed by thestationary conditions v Y = m φ h , (6) v χ = Λ G . (7)This solution also satisfies the D-flat conditions. v X is a flat direction in global SUSY. If weconsider gravity and one-loop corrections, it would be stabilized at the nonzero value [7, 12].The nonzero VEVs break SU ( N ) × SU ( N F ) F × U (1) φ gauge symmetry to SU ( e N ) × SU ( N ) D × U (1) Y . SU ( N ) D and U (1) Y are the linear combinations of the subgroups of SU ( N F ) F and SU ( N ) × U (1) φ . A. gauge bosons
After the symmetry breaking, massive gauge bosons appear according to the Higgs mech-anism. Let us decompose the vector field ( V µF ) for SU ( N F ) F as V µF = W µF − aB ′ µ √ ( X µ ) † √ X µ G µ + N e N aB ′ µ , (8)5here a = √ e N √ N ( N + e N ) is defined. W µF and G µ are the adjoint representations of the sub-groups of SU ( N F ) F : SU ( N ) F and SU ( e N ). X µ is the anti-fundamental and fundamentalrepresentations of SU ( N ) F × SU ( e N ), and B ′ µ is the U (1) F vector field, where U (1) F is from SU ( N F ) F .The nonzero VEVs generate the following mass terms, L g = M X X † µ X µ + 12 M W ′ W ′ Aµ W ′ Aµ + 12 M Z ′ Z ′ µ Z ′ µ , (9) M X = g F ( v χ + ∆ v ) , (10) M W ′ = 2( g F + g ′ N ) v χ , (11) M Z ′ = 4 N ( Q φ g φ + a g F ) v χ , (12)where ∆ v = v X − v Y is defined. W ′ Aµ and Z ′ µ are given by the linear combinations of W AµF and SU ( N ) gauge boson ( W AµN ), and B ′ µ and U (1) φ gauge boson ( A µφ ) respectively: B ′ µ A µφ = cos θ Y sin θ Y − sin θ Y cos θ Y B µ Z ′ µ , (13) W AµF W AµN = cos θ − sin θ sin θ cos θ W Aµ W ′ Aµ , (14)where cos θ Y and cos θ are defined ascos θ Y = Q φ g φ q Q φ g φ + a g F , cos θ = g ′ N p g ′ N + g F . (15) G µ , W µ , and B µ are the gauge bosons for SU ( e N ) × SU ( N ) D × U (1) Y gauge symmetry, andtheir gauge couplings are given by g N = g F cos θ, g e N = g F , g ′ = aN g = aN g F cos θ Y . (16) B. SM-charged fields from symmetry breaking sector
According to the decomposition in Eqs. (4) and (5), we introduce the charge assignmentof ( Z, e Z ), ( ρ, e ρ ), Y , ( χ, e χ ), and X in Table II. Y , ( χ, e χ ), and X are the adjoint parts of ˆ Y ,( ˆ χ, ˆ e χ ), and ˆ X . The singlet parts are not charged under the SM, and they are not so relevantto our analysis. The mass matrices are studied in Appendix A.6 e Z ρ e ρ Y χ e χ XSU ( e N ) e N e N e N e N 1 1 1 adj e N SU ( N ) D N N N N adj N adj N adj N U (1) Y N + e NN e N − N − e NN e N N + e NN e N − N − e NN e N SU ( e N ) × SU ( N ) D × U (1) Y . These fields obtain masses according to the nonzero VEVs, v χ , v Y and v X as we see inthe 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 anddiscuss 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 ( N F ) F break-ing, SU ( N F ) F → SU ( e N ) × SU ( N ) F × U (1) F , and the other is SU ( N ) F × U (1) F × SU ( N ) × U (1) φ breaking: SU ( N ) F × SU ( N ) → SU ( N ) D and U (1) F × U (1) φ → U (1) Y . The formeris caused by ∆ v , and the later is by v χ . We consider a simple scenario assuming ∆ v ≫ v χ .As we see in Appendix A, there will be several intermediate scales, where heavy particlesin the symmetry breaking sector are decoupled and the RG flow of gauge couplings is modify.According to the one-loop RG equations, the gauge couplings at the EW scale ( M Z ) areevaluated as follows: SU ( N ) F , SU ( N ) and SU ( N ) D gauge couplings ( α F N , α ′ N , α N ) are4 πα − N ( M Z ) = 4 πα − F N ( T χ N ) + 4 πα ′− N ( T χ N ) + b N ln (cid:18) M Z T χ N (cid:19) + ∆ b N ex (cid:18) T Λ (cid:19) , (17)4 πα − F N ( T χ N ) = 4 πα − G (Λ) + ∆ b N ln (cid:18) T N Λ (cid:19) + ( b N − N ) ln (cid:18) T χ N Λ (cid:19) , (18)4 πα ′− N ( T χ N ) = 4 πα ′− N (Λ) + ∆ b ρ N ln (cid:18) T ρ N Λ (cid:19) − N ln (cid:18) T χ N Λ (cid:19) . (19) SU ( e N ) gauge coupling ( α e N ) is4 πα − e N ( M Z ) = 4 πα − G (Λ) + b e N ln (cid:18) M Z Λ (cid:19) + ∆ b e N ln T e N Λ ! + ∆ b e N ex ln (cid:18) T Λ (cid:19) + ∆ b ρ e N ln T ρ e N Λ ! + ∆ b X ln T χ e N Λ ! . (20)7 (1) F , U (1) φ , and U (1) Y gauge couplings ( α F , α φ , α ) are4 πα − ( M Z ) = 4 πα − F ( T χ ) + 4 πa Q φ α − φ ( T χ ) + b ln (cid:18) M Z T χ (cid:19) + ∆ b ln (cid:18) T Λ (cid:19) , (21)4 πα − F ( T χ ) = 4 πα − G (Λ) + ∆ b ln (cid:18) T Λ (cid:19) + ∆ b ρ ln (cid:18) T ρ Λ (cid:19) + ( b + ∆ b χ ) ln (cid:18) T χ Λ (cid:19) , (22)4 πα − φ ( T χ ) = 4 πα − φ (Λ) + ∆ b ρ φ ln (cid:18) T ρ Λ (cid:19) + ∆ b χ φ ln (cid:18) T χ Λ (cid:19) . (23)Λ is the cut-off scale and T i , T χ i and T ρ i ( i = N, e N ,
1) are the intermediate scales where X µ , χ i ( χ e N ≡ X ), and ρ i decouple respecitvely. According to the mass spectrums at each scalein Appendix A, T i , T χ i and T ρ i ( i = N, e N ,
1) are estimated as( T N , T ρ N , T χ N ) = ( M X , h ∆ v, √ hM G ′ ) , (24)( T e N , T ρ e N , T χ e N ) = ( M X , h ∆ v, m X ) , (25)( T , T ρ , T χ ) = ( M X , h ∆ v, √ hM Z ′ ) . (26)The factor in front of each intermediate scale describes the freedom of the particles decou-pling at the scale: (∆ b N , ∆ b ρ N ) = (2( N F − N ) , − e N ) , (27)(∆ b e N , ∆ b ρ e N , ∆ b X ) = (2( N F − e N ) , − N, − e N ) , (28)(∆ b , ∆ b ρ , ∆ b χ ) = (cid:18) N F , − a N N e N , − a N (cid:19) , (29)(∆ b ρ φ , ∆ b χ φ ) = ( − N e N Q φ , − N Q φ ) . (30)We may also have to introduce additional particles charged under the gauge symmetry, inorder to achieve realistic mass spectrums. For instance, colored Higgs would be necessaryto derive the MSSM Higgs doublet at the low scale in Sec. III, and it is charged under SU ( e N ) × U (1) F in our explicit model. Such an extra intermediate scale and the coefficientis defined as T ex and ∆ b J ex ( J = e N , Z q ) for SU ( N ) F -charged field( q ). The one-loop renormalization group for Z q can be integrated analytically, if the Yukawa8oupling is negligible,ln Z q ( M Z ) = ln Z q (Λ) + 2 c qG b G ln (cid:18) α G (Λ) α G ( T i ) (cid:19) + 2 c qi b G − ∆ b i ln (cid:18) α F i ( T i ) α F i ( T ex ) (cid:19) + 2 c qi b G − ∆ b i − ∆ b i ex ln (cid:18) α F i ( T ex ) α F i ( T ρ i ) (cid:19) + 2 c qi b G − ∆ b i − g ∆ b ρ i − ∆ b i ex ln (cid:18) α F i ( T ρ i ) α F i ( T χ i ) (cid:19) + 2 c qi b i ln (cid:18) α i ( T χ i ) α i ( M Z ) (cid:19) , (31)where ( ] ∆ b ρ N , ] ∆ b ρ e N , ] ∆ b ρ ) = (0 , ∆ b ρ e N , ∆ b ρ ) is defined and T i ≥ T ex ≥ T ρ i is assumed. c qG and c qi are the second Casimir of the field q , corresponding to the gauge groups. The massessquared of sfermions can be derived by the v X -dependence in Z q . v X appears in the gaugecouplings, so that v X -dependence on the gauge couplings is only relevant to the sfermionmasses [13]. D. Soft SUSY breaking terms
Based on the above results, we investigate soft SUSY breaking terms which relate to par-ticles charged under the gauge symmetry. Soft SUSY breaking terms in SU ( e N ) × SU ( N ) D × U (1) are calculated by substituting v X + θ F X for v X in the gauge couplings [13]. Comparedto typical gauge mediation, where messengers are only chiral superfields, massive gaugebosons and the fermionic partners also work as the mediators to generate the soft SUSYbreaking terms, in our models [8, 9, 14, 15].In Eqs. (17) , (20), and (21), the only intermediate scales, T i , T ρ i , and T ex depend on v X . This leads the masses ( M e N , M N , M ) of the gauginos, which are the superpartner of SU ( N ) D × SU ( e N ) × U (1) gauge bosons, as follows: M N ( µ ) = − (∆ b N + ∆ b ρ N + ∆ b N ex ξ N ) α N ( µ )4 π F X | ∆ v | , (32) M e N ( µ ) = − (∆ b e N + ∆ b ρ e N + ∆ b e N ex ξ e N ) α e N ( µ )4 π F X | ∆ v | , (33) M ( µ ) = − ∆ b + ∆ b ρ + a Q φ ∆ b ρ φ + ∆ b ξ ! α ( µ )4 π F X | ∆ v | . (34) ξ N , ξ e N , and ξ describe the v X dependence on the mass scale of extra particles, T ex . Forexample, the holomorphic mass of extra particles may be given by m ex + λ ex ( v X + θ F X ),where m ex and λ ex are a supersymmetric mass term and Yukawa coupling involving theextra particles. That is, the gaugino mass contribution of ln( T ex ) would be proportional9o λ ex F X /m ex , if m ex is larger than λ ex v X . In this case, ξ i is approximately given by ξ i = λ ex | ∆ v | /m ex .Let us consider the soft SUSY breaking terms corresponding to the trilinear (A-term) andbilinear couplings (B-term) of the scalar components of the SU ( N F ) F -charged fields ( q I ).They are 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 aregiven by A IJK = A I + A J + A K , where A I = ∂ ln Z I ∂ ln v X is defined and the trilinear coupling isdescribed as y IJK A IJK q I q J q K .Eventually, A I is obtained from Eq. (31), A I = ( c IG α G ( T i )4 π − b G c Ii b G − ∆ b i α F i ( T i )4 π + (cid:18) c Ii b G − ∆ b i − c Ii b G − ∆ b i − ∆ b i ex (cid:19) ( b G ξ + ∆ b i (1 − ξ )) α F i ( T ex )4 π + c Ii b G − ∆ b i − ∆ b i ex − c Ii b G − ∆ b i − ∆ b i ex − g ∆ b ρ i ! ( b G − ∆ b i ex (1 − ξ )) α F i ( T ρ i )4 π + b i + ∆ b i ex ξ + g ∆ b ρ i ) c Ii b G − ∆ b i − ∆ b i ex − g ∆ b ρ i ! α F i ( T χ i )4 π − b i + ∆ b i ex ξ + ∆ b ′ ρ i ) c Ii b i ! α i ( T χ i )4 π + b i + ∆ b i ex ξ + ∆ b ′ ρ i ) c Ii b i ! α i ( µ )4 π ) F X | ∆ v | , (35)assuming ξ N = ξ e N = ξ = ξ . α F e N ≡ α e N is defined.The masses squared ( m q ) of q could be also estimated by the Eq. (31), seeing the | v X | -dependence of Z q [13]. As discussed in Ref. [14], the gauge mediation with gauge messengersmay contribute to the masses squared at the one-loop level, if the gauge symmetry breakingand SUSY breaking are caused by the VEVs and F-components of several fields. In ourcase, we simply assume v χ ≪ ∆ v , so that the gauge symmetry breaking and SUSY breakingare caused by only ∆ v and the F-component of ∆ v . ‡ The one-loop correction is stronglysuppressed by ( v χ / ∆ v ) according to Ref. [14], so that we have to investigate the two-loopcorrections, as discussed in Refs. [8, 13]. ‡ ∆ v corresponds to the VEV of one adjoint field. m q could be written as m q ( µ ) = ( b G c qG α G ( T i )(4 π ) − b G c qi b G − ∆ b i α F i ( T i )(4 π ) + (cid:18) c qi b G − ∆ b i − c qi b G − ∆ b i − ∆ b i ex (cid:19) ( b G ξ + ∆ b i (1 − ξ )) α F i ( T ex )(4 π ) + c qi b G − ∆ b i − ∆ b i ex − c qi b G − ∆ b i − ∆ b i ex − g ∆ b ρ i ! ( b G − ∆ b i ex (1 − ξ )) α F i ( T ρ i )(4 π ) + b i + ∆ b i ex ξ + g ∆ b ρ i ) c qi b G − ∆ b i − ∆ b i ex − g ∆ b ρ i ! α F i ( T χ i )(4 π ) − (cid:18) b i + ∆ b i ex ξ + ∆ b ′ ρ i ) c qi b i (cid:19) α i ( T χ i )(4 π ) + (cid:18) b i + ∆ b i ex ξ + ∆ b ′ ρ i ) c qi b i (cid:19) α i ( µ )(4 π ) ) F X | ∆ v | , (36)where ∆ b ′ ρ i is (∆ b ′ ρ , ∆ b ′ ρ , ∆ b ′ ρ ) = (∆ b ρ , ∆ b ρ , ∆ b ρ + a ∆ b ρ φ /Q φ ).In the next section, we discuss one explicit model, where SU ( e N ) × SU ( N ) D × U (1) Y is theSM gauge groups corresponding to ( N F , N ) = (5 , SUSY isroughly given by ( α G / (4 π )) × ( F X / | ∆ v | ), and A-term and B-term are of O (Λ SUSY ), whichwe could expect that are consistent with the condition for the EW symmetry breaking. Westudy the compatibility with the EW condition and the Higgs mass, in Sec. III D.
III. SU (5) F × SU (2) × U (1) φ GAUGE THEORY: ( N F , N ) = (5 , In this section, we consider a SU (5) F × SU (2) × U (1) φ gauge symmetric model, whichcorrespond to the ( N F , N ) = (5 ,
2) case. We expect that the MSSM fields are embedded into and representation as in the Georgi-Glashow SU (5) GUT. Involving -representationHiggs ( H, H ), the superpotential for the Yukawa couplings in the visible sector is W vis = ˆ y ukl H k l + ˆ y dkl H k l , (37)where k and l are defined as the matter fields. As well-known, ˆ y ukl and ˆ y dkl may require Φand ( φ, e φ ) dependences in order to generate realistic mass matrices at the EW scale accordingto the higher-dimensional operators. Here, we simply assume that the contributions to thesoft SUSY breaking terms are enough small.One serious problem in the SU (5) GUT is how to generate the mass splitting between thecolored Higgs and the MSSM Higgs doublet. The mass of colored Higgs should be around11he GUT scale to avoid the too short life time of proton: m H c & GeV × (1TeV / Λ SUSY )[16]. In our SU (5) F × SU (2) × U (1) φ model, the relevant terms to the Higgs masses iswritten as W H = µHH + λ H H Φ H. (38)After the symmetry breaking, the colored Higgs mass and MSSM Higgs mass are given by µ + λv X and µ + λ H v Y . If v Y = m φ /h is the GUT scale, µ should be also around the GUTscale and then the fine-tuning between µ and λv Y is required: µ + λ H v Y ≈ O ( M Z ). On theother hand, we could expect that the colored Higgs is enough heavy because of µ , if thereis no cancellation between µ and λ H v X . Let us also consider the case that v X is the GUTscale. In this case, the MSSM Higgs mass could be light if µ and v Y are around the weakscale, and the colored Higgs is heavy: m H c ≈ λ H v X .In both cases, the colored Higgs couples with v X + F X θ , so it mediates the SUSYbreaking effect to the soft SUSY breaking terms. The supersymmetric mass for SU (2) L Higgs doublet is µ = µ + λ H v Y ≈ O ( M Z ). On the other hand, the colored-Higgs mass is m H c = µ + λ H v X ≈ λ H ( v Y − v X ), so that ξ for the colored Higgs in soft SUSY breaking termsis approximately estimated as ξ ≈ sign ( λ H ). The one-loop correction of H c to m q would besuppressed, because the m H c -dependence appears in Z q as ln( | m H c + λ H F X θ | ) accordingto the study in Ref. [13]. We could apply our analysis in Sec. II D to this scenario. A. gauge couplings
In this model, SU (5) F × SU (2) × U (1) φ breaks down to the SM gauge group, SU (3) c × SU (2) L × U (1) Y . SU (3) c is the subgroup of SU (5) F and SU (2) L × U (1) Y are the linearcombination of SU (2) F × U (1) F and SU (2) × U (1) φ .On the other hand, there are several intermediate scales: ( T G , T ρ , T χ , T X ). § T G is theGUT scale, where X µ decouples, and T ρ is the messenger scale fixed by the parameter h and the GUT scale. T χ is interpreted as the SUSY breaking scale, because T χ ≈ √ F X = p M p m / , so that it is almost fixed around O (10 ) GeV when m / = O (100) GeV. T X isfixed by the mass scale of X ( m X ), which is massless at the tree-level. X could be expectedto be O (Λ SUSY ), because the one-loop corrections shift the mass, but it may be difficult toclearly fix the masses of bosonic and fermonic X in our model. Let us simply treat m X as § T χ = T χ , T ρ = T ρ , and T G = T = T = T are assumed. l og ( T X / G e V ) log (m /GeV) FIG. 1. Gravitino mass ( m / ) and the scale T X with 10 GeV ≤ T G ≤ M p . T X should be smallto rase the GUT scale above 10 GeV. T χ is T χ ≈ p M p m / . The constraints, T ρ > T χ and T X > m / , are also assigned. All gauge couplings and Yukawa couplings satisfy the perturbativebounds as α F i < π. α ’ ( T χ ) α F2 (T χ ) 0.01 0.1 1 0.01 0.1 1 α φ ( T χ ) α F1 (T χ ) FIG. 2. α F vs. α ′ and α F vs. α φ at the symmetry breaking scale, T χ . the free parameter, and Fig. 1 shows the allowed region for T X , which may not be far from O (Λ SUSY ). Fig. 2 shows the gauge couplings, ( α F , α , α F , α φ ) at the SUSY breaking scale.Fig. 3 describes RG flows of the gauge couplings ( α , α ( α F ) , α ( α F )), when T X = 10 GeV, T χ = 3 . × GeV, T ρ = 7 . × GeV, and T GUT = 2 × GeV.13 - Α - Α - H Μ (cid:144) M z L Α - FIG. 3. RG flows of the subgroups of SU (5) F , with T X = 10 GeV, T χ = 3 . × GeV, T ρ = 7 . × GeV, and T GUT = 2 × GeV. The green, blue, and red lines correspond to thegauge couplings of U (1) Y × SU (2) L × SU (3) c below T χ and U (1) F × SU (2) F × SU (3) c above T χ respectively. The input parameters for the couplings are in Eq. (42). B. soft SUSY breaking terms
We qualitatively evaluate the soft SUSY breaking terms in this scenario. According tothe analysis in Sec. II D, the gaugino masses at µ < T χ are written as M ( µ ) = − α ( µ )4 π F X | ∆ v | , (39) M ( µ ) = − (2 − ξ ) α ( µ )4 π F X | ∆ v | , (40) M ( µ ) = − (cid:18) − ξ (cid:19) α ( µ )4 π F X | ∆ v | . (41)Let us consider the case with ξ = 1 and the gaugino masses at the EW scale. The gaugecouplings at the EW scale are [2] α ( M Z ) ≈ . , α ( M Z ) ≈ . , α ( M Z ) ≈ . , (42)so that we could derive the following mass relation: M ( M Z ) M ( M Z ) ≈ . , M ( M Z ) M ( M Z ) ≈ . . (43)The masses are almost degenerate, and this may be a specific feature of the gauge messengermodel [8, 17]. ¶ If all intermediate scales are close to the GUT scale, the fine-tuning of µ ¶ The gaugino masses are degenerate in the TeV-scale mirage mediation scenario, too [18]. SU (3)-adjoint field reside in the low-scale, so that the condition for the small µ -term wouldbe modified. The one-loop running correction of m H u with T X = 10 GeV from T χ to M Z isestimated as∆ m H u ≈ − . M ( M Z ) − . M ( M Z ) M ( M Z ) + 0 . M ( M Z ) + . . . , (44)where the ellipsis denotes the terms including A-term and scalar masses and those are notimportant when they are comparable to the gluino mass. This leads that the condition tocancel the large contribution of gluino is M /M ( M Z ) ≈ .
23, which suggests the almostdegenerate mass spectrum. However, we have a large A-term contribution to ∆ m H u in ourmodel, so that it may be difficult to avoid a certain fine-tuning even if the gaugino massesare degenerate.According to Eqs. (36) and (35), the masses squared of superpartners and A-term areevaluated explicitly. Setting T G = T H c > T ρ > T χ and ξ = 1, stop masses at T χ are given by m Q ( T χ ) ≈ (cid:18) . − . α ( T ρ ) α G − . α F ( T χ ) α G − . α F ( T χ ) α G (cid:19) Λ SUSY , (45) m U ( T χ ) ≈ (cid:18) . − . α ( T ρ ) α G − . α F ( T χ ) α G − . α F ( T ρ ) α G (cid:19) Λ SUSY . (46)As we see, large stop masses are generated by the large second casimir ( c t = 18 / T χ and T ρ are close to the GUT scale. The SUSY scale(Λ SUSY ) from the gauge mediation is defined asΛ
SUSY = α G (4 π ) F X | ∆ v | ≈ α G (4 π ) M p T G m / . (47) α G is of O (0 .
1) when T G is around 10 GeV, so that Λ
SUSY might be compatible with m / .If T G is smaller, the situation, Λ SUSY ≫ m / , is achieved but suffers from the constraint fromproton decay. The correction from the gravity mediation is naively estimated as O ( m / ). Itis almost the same order as the one from the gauge mediation in our model, and it may makeit difficult to control flavors. In fact, the gauge-mediation contributions are typically at least5 times as large as the gravitino mass in our model, as we see in Table IV. In this case, wecould expect the gravity-mediation effect is sub-dominant, and the SUSY scale is governedby the gauge-mediation. However, the gravity-mediation contribution should be O (10 − )times suppressed, if it contributes to the sparticles masses squared flavor-universally [20].15n order to realize such a suppression and control flavor in the MSSM, we have to considerflavor symmetry or some dynamics above the GUT scale, as discussed in Refs. [21]. ∗∗ Indeed, explicit contributions on soft masses through the gravity mediation depend on theUV completion of our model. In this letter, one of our main motivations is to achieve125 GeV Higgs mass and realistic EW symmetry breaking, which may be independent ofthis issue about the constraint from flavor physics, so that we will discuss our SUSY massspectrums assuming that the gauge-mediation is dominant. The underlying theory abovethe GUT scale will be studied in Ref. [26]. A t , which is the trilinear coupling of stops ( e t ) as y t A t e t L H u e t R is given by A t ( T χ ) ≈ (cid:18) . − . α ( T ρ ) α G − . α F ( T χ ) α G − . α F ( T χ ) α G + 0 . α F ( T ρ ) α G (cid:19) Λ SUSY , (48)and the B-term, which is the bilinear coupling of two Higgs µBH u H d , is estimated as B ( T χ ) ≈ (cid:18) . − . α F ( T χ ) α G − . α F ( T χ ) α G + 0 . α F ( T ρ ) α G (cid:19) Λ SUSY . (49)As we see, the A-term and B-term might be large as O (10)Λ SUSY . This may be good toachieve the EW symmetry breaking, but too large A-term makes the stop masses tachyonicbecause of the running correction such as∆ m U ( M Z ) ≈ − . A t ( T χ ) + 1 . M ( T χ ) − . A t ( T χ ) M ( T χ ) . (50)In our model, the gluino mass M is relatively small as wee see in Eq. (40), so ∆ m U ( M Z )becomes easily negative and stop mass becomes tachyonic even if the positive m U is generatedat the SUSY breaking scale T χ . In order to avoid the tachyonic stop masses, we add an extracontribution 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, W = 1Λ T r (Φ W W ) . (51)There are several ways to introduce this term, such as gravity effect. Here, we simply assumethat N extra extra heavy SU (5) vector-like pairs ( ψ, ψ ) with the masses ψ (Λ + λ X Φ) ψ induce ∗∗ In fact, such strong dynamics has been proposed not only to suppress flavor changing currents but alsoto realize the superpotential W SB in Sec. II [10]. . After the SU (5) breaking, the gauge couplingwould have the extra v X dependence as α − → α − − N extra π ln (cid:18) ( | Λ + λ X ( v X + F X θ ) | )Λ (cid:19) . (52)This additional coupling could shift the gluino mass as M → M − α N eff π F X | ∆ v | , (53)where N eff may not be N extra because of the scale difference between Λ and the GUT scale.Including N eff , the gluino mass becomes M ( µ ) = − (1 − N eff ) α ( µ )4 π F X | ∆ v | , (54)so N eff should be bigger than 2 in order to shift M . In fact, we discuss large N eff casesand find that N eff enables us to evade the negative squared masses and achieve the largeSM Higgs mass. D. Consistency with the Higgs mass and the EW symmetry breaking
One issue in supersymmetric models is how to realize the µ and B terms which areconsistent with the EW scale. Especially, µ relates to the lightest Higgs mass, becauseof the upper bound in MSSM, so that the recent Higgs discovery with the mass 125 GeVmay impose unnatural SUSY scenarios on us. In fact, 125 GeV Higgs mass may requireΛ SUSY & O (10) TeV in the simple scenarios as discussed in Ref. [3]. O (10)-TeV SUSY scalewould require 0 .
01% fine-tuning against µ without any cancellation in m H u . As pointed outin Refs. [22, 23], it is known that a special relation between A t and squark mass relaxes thefine-tuning, maximizing the loop corrections in the Higgs mass in the MSSM. This relationis so-called “maximal mixing” and described as X t /m stop = √
6, where X t = A t − µ/ tan β and m stop = q m Q m U are defined. If this relation is satisfied, the 125 GeV Higgs masscould be achieved even if the stop is light. We can see our prediction on X t and the upperbound on the Higgs mass in the case with 0 ≤ N eff ≤ ≤ N eff ≤ T G , T X ) are fixed at (2 × GeV , GeV). We find that our A-term is too large to realize X t /m stop = √
6, but the maximal mixing could be achieved, if we allow large N eff , andenhance the Higgs mass, even if m stop is around 1 TeV.17 t (cid:144) m s t op = £ N eff £ £ N eff £ X t (cid:144) m stop m s t op @ G e V D £ N eff £ £ N eff £ CMS Β m h @ G e V D FIG. 4. X t /m stop vs. m stop and tan β vs. the lightest Higgs mass in the case with ( T GUT , T X ) =(2 × GeV , GeV) and 0 ≤ N eff ≤ ≤ N eff ≤ X t /m stop = √
6. In the right figure, m h is calculated at the two-loop level using m t = 172 . m stop is lighter than 2 TeV. The green band is the CMS result on Higgs massfrom h → γγ , ZZ channels [24]. On the other hand, we notice that there is no special cancellation in m H u and m H d , aswe see in Fig. 5. Large m stop corresponds to large µ , so that 1-TeV squark mass requires 1%fine-tuning against µ . The right figure in Fig. 5 shows that small tan β is consistent withthe EW symmetry breaking. B EW is the value to realize the EW symmetry breaking, B EW ≡ − µ { ( m H d − m H u ) tan 2 β + M Z sin 2 β } , (55)and B is our prediction via the gauge mediation. It seems that 2 . tan β . β region may be inconsistent with the one requiredby 125 GeV Higgs (tan β &
4) with m stop ≤ m h ≈
125 GeV and | B EW /B | ≈
1. There, m stop and | µ | are around 3 TeV, and O (0 .
1) % fine-tuning is required against µ term. IV. SU (5) F × SU (3) × U (1) φ GAUGE THEORY: ( N F , N ) = (5 , Our symmetry breaking model could be embed into other type GUT model. One simpleexample would be the SU (5) F × SU (3) × U (1) φ gauge symmetric model, and we couldconsider the same setup as in the SU (5) F × SU (2) × U (1) φ gauge theory. The visible sectoris given by Eq. (37). However, the modification of the Higgs sector may be required because18
100 1000 600 800 1000 1200 1400 1600 1800 2000 µ [ G e V ] m stop [GeV] 0.1 1 2 4 6 8 10 12 14 16 18 20 B E W / B tan β FIG. 5. m stop vs. µ and tan β vs. B-term in the case with ( T GUT , T X ) = (2 × GeV , GeV)and 0 ≤ N eff ≤ (blue), 6 ≤ N eff ≤ m stop is lighter than 2 TeV. Thedashed line is consistent with the condition for the EW symmetry breaking. λH Φ H term gives the very large B-term, λF X H u H d . There may be a solution to realizethe EW symmetry breaking, but the serious fine-tuning may be required. Here, we consideranother solution to shift the colored Higgs mass which maybe favor high-scale SUSY.We introduce SU (3) vector-like fields ( H , H ) and assign Z symmetry to the fields asin Table III. Z symmetry is broken by the VEV of S . The superpotential for the Higgssector is given by W H = λ S SHH + λ φ HH φ + λ e φ e φH H + λ S . (56) i i H H H H S φ e φ Φ SU (5) F + SU (3) U (1) φ Q φ − Q φ Q φ − Q φ Z ω ω ω ω ω ω ω TABLE III. Chiral superfields in SU (5) F × SU (3) × U (1) gauge theory After the GUT symmetry breaking, (
H, H ) are decomposed as (( H u , H ′ ) , ( H d , H ′ )) andthe mass terms for ( H , H ′ ) and ( H ′ , H ) pairs appear as W effH = λ φ v χ H ′ H + λ e φ v χ H H ′ . (57)19 u and H d correspond to the Higgs SU (2) L doublets in MSSM, and they could get thesupersymmetric mass term according to the nonzero VEV of S . In Refs. [25], we can seenot only the SU (5) F × SU (2) × U (1) φ -type but also this type of product-GUT.In order to avoid the bound from the proton decay caused by the five dimensional oper-ators, v χ should be large as v χ & GeV × (cid:18) SUSY (cid:19) . (58) F X is given by − hv χ , so that very tiny h is necessary to achieve the low-scale SUSY. When v χ ≈ GeV and Λ
SUSY = 1 TeV are set, h should be around O (10 − ), because of h = 4 πα G ∆ vv χ Λ SUSY ≈ − × (cid:18) GeV v χ (cid:19) (cid:18) Λ SUSY (cid:19) . − × (cid:18) Λ SUSY (cid:19) . (59)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 • SU (3) c × SU (2) L × SU (2) R × U (1) B − L → SU (3) c × SU (2) L × U (1) Y , • SU (4) × SU (2) L × U (1) → SU (3) c × SU (2) L × U (1) Y . We would study such patterns elsewhere [26]. In these models, all of chiral superfieldsappear as adjoint representations and bi-fundamental representations. Such models canbe constructed in D-brane models, e.g. intersecting/magnetized D-brane models (see for areview [27, 28] and references therein). Thus, the above models are interesting from theviewpoint of superstring theory.
V. SUMMARY
The MSSM is one of the attractive BSMs to solve the hierarchy problem in the SM and itmay be expected to be found near future. One big issue in the MSSM is how to control theSUSY breaking parameters, so that many ideas and works on spontaneous SUSY breakingand mediation mechanisms of the SUSY breaking effects have been discussed so far. In thispaper, we proposed an explicit and simple supersymmetric model, where the spontaneousSUSY breaking and GUT breaking are achieved by the same sector. The origin of the hyper-charge assignment in the MSSM is also explained by the analogy with the Georgi-Glashow20 U (5) GUT. The SM-charged particles are also introduced by the breaking sector, so that wecould also predict the soft SUSY breaking terms via the gauge mediation with the gauge andchiral messenger superfields. The crucial role of the gauge-messenger mediation is to inducelarge A-terms and B-terms at the one-loop level. We investigated the scenario with lightsuperpartners that such a large A-term realizes the maximal mixing and shift the lightestHiggs mass. In fact, we have to introduce additional contribution to the gluino mass, but125 GeV Higgs mass could be achieved, even if stop is light. m stop should be as light aspossible to relax the fine-tuning of µ parameter. On the other hand, the one-loop B-termcould be also consistent with the EW symmetry breaking, if tan β is within 2 . tan β . β may require large stop mass, as we see in Figs. 4 and 5. In fact, we seethat about 3 TeV m stop can achieve 125 GeV Higgs mass and the EW symmetry breakingin Table IV.Our light SUSY particles are wino, bino, and gravitino, and the mass difference is notso big. The lightest particle is bino, and wino is heavier than bino. The mass difference is O (0 . × m / GeV. This might be one specific feature of the gauge messenger scenario in SU (5) GUT, as discussed in Ref. [17]. ACKNOWLEDGMENTS
We are grateful to Hiroyuki Abe for useful discussions and comments. This work issupported by Grant-in-Aid for Scientific research from the Ministry of Education, Science,Sports, and Culture (MEXT), Japan, N0. 25400252 (T.K.) and No. 23104011 (Y.O.).
Appendix A: mass spectrums of the particles in the symmetry breaking sector
We investigate the mass matrices for the remnant fields in the symmetry breaking sector.First, let us discuss ( Z, e Z ) and ( ρ, e ρ ) components. We define Z ± and ρ ± as Z ± = e Z ± Z † √ , ρ ± = ρ ± e ρ † √ . (A1)21he fermion masses are given by L f = − (cid:16) λ − Z + ρ + (cid:17) M f + λ − Z + ρ + − (cid:16) λ + Z − ρ − (cid:17) M f − λ + Z − ρ − , (A2)where the mass matrices ( M f ± ) are M f + = − g ∆ v gv χ − g ∆ v − hv χ gv χ − hv χ − h ∆ v , M f − = − g ∆ v gv χ − g ∆ v hv χ gv χ hv χ h ∆ v , (A3)and λ ± are the linear combinations of the gauginos ( X (+) ) which are the suparpartners of X µ , λ ± = X + ± X √ . (A4)The masses for the bosonic superpartners are L B = − (cid:16) Z † + ρ † + (cid:17) M Z + ρ + − (cid:16) Z †− ρ †− (cid:17) M − Z − ρ − , (A5)where the mass matrices ( M ± ) are given by M = h v χ − h v χ ∆ v − h v χ ∆ v h ( v χ + ∆ v ) + F X , (A6) M − = h v χ + g ∆ v − ( h + g )∆ vv χ − ( h + g )∆ vv χ h ( v χ + ∆ v ) + g v χ − F X . (A7)The F-term F X is F X = − h v χ , so that M includes the Goldstone mode.The fermion masses for the other particles are also generated by the VEVs: L Y = − (cid:16)f W A Y A (cid:17) M Y χ A e χ A − (cid:16) χ A e χ A (cid:17) M TY f W A Y A + h.c., (A8)where f W is the superpartner of W ′ and M Y are defined as M Y = − √ M W ′ √ M W ′ − hv χ − hv χ . (A9)The eigenvalues are M W ′ , M W ′ , √ hv χ , √ hv χ and the bosonic masses are given by thesame mass spectrum. The imaginary part of χ − e χ corresponds to the Goldstone boson,22nd the real part has the mass, M W ′ , according to the D-term. The other masses, √ hv χ ,correspond to the ones of χ + e χ and Y .The singlet components ( Y , χ , e χ ) of ˆ Y and ( ˆ χ, ˆ e χ ) also get masses, according to thenonzero v χ . The fermionic mass matrix is L Y = − (cid:16) e Z ′ Y (cid:17) M Y χ f χ − (cid:16) χ f χ (cid:17) M TY e Z ′ Y + h.c., (A10)where M Y are defined as M Y = − √ M Z ′ √ M Z ′ − hv χ − hv χ . (A11)The mass spectrums are given, relplacing M W ′ with M Z ′ . Appendix B: Concrete parameter set
The parameter sets which predict m h ≈
125 GeV are in Table IV. The Higgs mass iscalculated by FeynHiggs [23, 29]. m e t , are the stop masses in the mass eigenstate. m e Q L , m e d R , m e l L and m e e R are the soft SUSY breaking terms of the squarks ( e Q L , e d R ) and sleptons( e l L , e e R ). 23 eff = 6 N eff = 6 N eff = 6 . N eff = 7 . m / .
84 GeV 741 .
31 GeV 495 .
79 GeV 245 .
02 GeV T ρ . × GeV 5 . × GeV 5 . × GeV 2 . × GeV T X . × GeV 1 . × GeV 1 . × GeV 1 . × GeVtan β .
69 3 .
93 3 .
43 4 . m h .
20 GeV 125 .
89 GeV 124 .
65 GeV 124 .
03 GeV m stop .
05 TeV 3 .
61 TeV 2 .
93 TeV 1 .
90 TeV X t . × m stop . × m stop . × m stop . × m stop | µ | .
72 TeV 4 .
38 TeV 3 .
27 TeV 1 .
93 TeV | B | .
21 TeV 4 .
72 TeV 3 .
22 TeV 1 .
97 TeV | B EW /B | .
92 1 .
00 0 .
91 0 . | M | . × m / . × m / . × m / . × m / | M | . × m / . × m / . × m / . × m / | M | . × m / . × m / . × m / . × m / m e t .
52 TeV 4 .
12 TeV 3 .
31 TeV 2 .
15 TeV m e t .
62 TeV 3 .
17 TeV 2 .
57 TeV 1 .
65 TeV m e Q L .
72 TeV .
56 TeV .
36 TeV .
24 TeV m e d R .
97 TeV .
52 TeV .
60 TeV .
40 TeV m e l L .
78 TeV .
93 TeV .
44 TeV .
18 TeV m e e R .
42 TeV .
75 TeV .
81 TeV .
31 TeV TABLE IV. SUSY mass spectrums and parameters with Λ
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