Adjoint SU(5) GUT model with Modular S_4 Symmetry
aa r X i v : . [ h e p - ph ] M a r Adjoint SU(5) GUT model with Modular S Symmetry
Ya Zhao , ∗ , Hong-Hao Zhang , † School of Physics, Sun Yat-Sen University, GuangZhou, GuangDong 510275, P. R. China.
Abstract
We study the textures of SM fermion mass matrices and their mixings in a supersymmetricadjoint SU(5) Grand Unified Theory with modular S being the horizontal symmetry. The Yukawaentries of both quarks and leptons are expressed by modular forms with lower weights. Neutrinosector has an adjoint SU(5) representation 24 as matter superfield, which is a triplet of S . Theeffective light neutrino masses is generated through Type-III and Type-I seesaw mechanism. Theonly common complex parameter in both charged fermion and neutrino sectors is modulus τ .Down-type quarks and charged leptons have the same joint effective operators with adjoint scalarin them, and their mass discrepancy in the same generation depends on Clebsch-Gordan factor.Especially for the first two generations the respective Clebsch-Gordan factors made the doubleYukawa ratio y d y µ /y e y s = 12, in excellent agreement with the experimental result. We reproduceproper CKM mixing parameters and all nine Yukawa eigenvalues of quarks and charged leptons.Neutrino masses and MNS parameters are also produced properly with normal ordering is preferred. ∗ E-mail address : [email protected] † E-mail address : [email protected]
Introduction
Despite of great success, especially the discovery of Higgs boson [1,2], the standard model (SM) stillhas some problems unsolved in particle physics. One of the problems is the origin of flavor structurefor the SM fermions, which mainly refers to the enormous mass difference among different generations,and distinct mixing patterns between lepton and quark sector.The masses of charged fermions, from lightest electron of MeV level to heaviest top quark being173GeV, span almost 5 orders of magnitude. The situation becomes even worse when the neutrino sectoris contained, since the neutrino masses of sub-eV are nearly 7 orders of magnitude smaller than thatof electron. Besides the absolute mass scale and mass ordering are still yet to be determined by futurehigh-precision neutrino experiments. Apart from the large mass hierarchy, quark CKM parametersmanifest a mixing pattern of small angles [3]. However the situation is completely different in leptonMNS mixing matrix. The precise neutrino oscillation data has provided us a picture of two large angles θ and θ , and one small angle θ , which is comparable with the quark Cabibbo angle θ C .It is an unsolved puzzle to interpret the observed flavor structure in quark and lepton sector. Theflavor parameters, include all the fermion masses, mixing angles as well as CP violating phases arisefrom the dependence of Yukawa on flavor. Since symmetry plays an important role in physics, it isworth to constrain the Yukawa interactions with the supervision of flavor symmetry, hence the masshierarchies and mixing patterns of SM fermions could be interpreted by the additional symmetry beyondthe gauge symmetry of the SM. But note that flavor symmetry is not the unique top-down scenario tounderstand the flavor structure. The other possible top-down approaches include, such as anarchy [4],extra dimension [5, 6] and string theory [7–9].For the last two decades, the precise measurement of flavor parameters, especially that of leptonmixing angles, has motivated the model building by using discrete symmetry group to elucidate thedifferent flavor structures between quarks and leptons. Non-Abelian discrete groups, such as lower ordergroups A , A , S and other ones with higher order, are wildly used in such kind of works. And it isindeed easy to reproduce at least the two large leptonic mixing angles. For reviews see Refs. [10–15].In the models apart from the essential Higgs, some extra scalar fields called flavons are introduced.They are singlets under SM gauge group but have nontrivial representations under flavor group, whosevacuum orientation in flavor space can induce specific Yukawa textures for fermions. In neutrino sectorthe famous Tri-Bimaximal mixing [16, 17] is ubiquitous in many realistic models.Non-Abelian discrete groups as flavor symmetries are success in explaining the leptonic mixingpattern with large angles, with the price of introducing a number of gauge singlets called flavon fieldsin conventional studies. The flavor symmetry is broken when the scalar flavons acquiring vacuumexpectation values (VEV) by non-trivial dynamical way. Such VEVs with specific configurations controlthe flavor textures of fermions and thus a few of free coupling parameters appear in the Yukawa entries.Besides the values of VEVs themselves are determined by some other parameters. The more parametersin the traditional flavor models, the less predictive power they have. And the fermion masses are stilloften acquired by tuning the coupling parameters, even though the VEVs of the flavons can be nearlyfixed such that all the couplings can be naturally of the same order, e.g., of order one.Recently the modular symmetry provides another possibility to interpret the flavor issues [18]. Thesimplest modular symmetry implementation only demands one complex field, called modulus τ , as theunique source of modular symmetry breaking when it develops a VEV, hence the vacuum configurationproblem is greatly simplified. In such a simplest framework the Yukawa couplings are just modularforms which are the holomorphic functions of modulus τ , then the flavon fields are not indispensableingredient for model building, and thus tremendously reduce the amount of particle content and thefree parameters of the theory.The modular invariance models are based on the level N finite modular group Γ N , e.g., from N = 22o 5, Γ ≃ S [19–22], Γ ≃ A [18, 23–34], Γ ≃ S [35–43] and Γ ≃ A [41, 44, 45], which are allinhomogeneous modular groups. In such models the Yukawa couplings are weight k modular forms with k being even numbers. Most of the studies focus on the lepton flavor issues, while few of them includequark sector [20, 24, 25, 31]. The unified quark and lepton models can be implemented in the contextof SU(5) grand unified theories (GUT) combined with the modular symmetries [21, 32]. Besides thetopics on the fermion flavor structures, the related phenomenological issues has been discussed in thoseworks, such as the dark matter models [27], baryon number violation [26] and leptogenesis [22, 33, 34].On the other side, the double covering of finite modular group which is homogeneous, has been used asflavor symmetry as well [46–51].Motivated by the phenomenological viable mass ratios between quarks and leptons, and the ideaof Yukawa couplings can be modular forms, in the study we combine the modular flavor symmetryΓ ≃ S with the supersymmetric adjoint SU(5) GUT [52] to forge the flavor textures of quarks andleptons simultaneously. In refs. [21, 32] the neutrino masses are generated via type-I [53–56] seesaw byadding at least two gauge singlets, i.e., right-handed heavy neutrinos. However there exist another twopossibilities in SU(5) GUT to produce light neutrino masses: first is type-II seesaw mechanism [57–59]by adding an extra Higgs H , and second is Type-I plus Type-III [60] seesaw by adding fermionic fieldin the dimensional representation. The two cases in SU(5) GUT have not been explored yet in therecent modular flavor models. In this paper we explore for the first time the second scheme to produceeffective light neutrino masses. Meanwhile we would like to give rise to a realistic Yukawa ratios betweencharged leptons and down-quarks. The Yukawa ratio in each generation is directly derived from thenovel Clebsch-Gordan (CG) factor, which can be comparable to the phenomenological values at GUTscale. For the modular flavor model built on SU(5) GUT, the flavon-free model is such that the doubleYukawa eigenvalue ratio y µ y d /y e y s equal to 12 for the first and second families of charged leptons anddown quarks. The other viable cases which can generate the acceptable ratios, however, still requirethe flavon or so called weighton to compensate the loss of mass dimension.There is no flavon but a modulus τ in the modular symmetry breaking sector, meanwhile the gaugesymmetry is broken by one adjoint scalar H . We aim at minimizing the amount of free couplingparameters in the unified model. For higher weight modular forms, it will bring more free couplingparameters, and thus decrease the predictive power of the model. Thus we use lower weight modularforms to give the modular invariant operators.In order to achieve the above goal, we should strictly contrain the representations and weightsunder S for the fields. To be specific, all the -dimensional matter superfields are S singlets andhave distinct weights. The three families of ¯ s are divided into an doublet and a singlet but have thesame weight. At last the fermionic superfields are sealed in a triplet. For the scalar sector, the H , H and H are just singlets with distinct weights. Please see Table. 1 for details.In quark and charged lepton sector the Yukawa matrices are very sparse with several texture zeros.The adjoint scalar H couples to the matter superfields in down quark sector, then it induces the novelratio of CG factors 1/2 and 6 for the first two families of leptons and down quarks, and 3/2 for the thirdone. In neutrino sector we introduce the dimensional matter field rather than a gauge singlet toproduce the neutrino masses through Type-I and Type-III seesaw mechanism. Since we introduce theadjoint matter fields rather than gauge singlets to produce the effective masses of light neutrinos, theYukawa matrices are slightly different for the heavy ρ and ρ in the . And the same for Majoranamass matrices, since the mass terms include two nontrivial couplings: the pure mass term which is thesame for ρ and ρ , and the new interaction between and H , which splits the masses of ρ and ρ .Such new interaction is of course absent in the models which gauge singlets are responsible for seesawmechanism.The layout of the paper is arranged as follows. In Sec. 2 we introduce the framework for the modelbuilding work, including the brief review on modular group, especially the one with level N = 4,Γ ≃ S and the modular forms of weight 2 and 4. Then we give the basic aspects of adjoint SU(5)3UT. In Sec. 3 we present the flavor model based on adjoint SU(5) GUT combined with modular S flavor symmetry. We show that the GUT flavor model can be built without introducing a gauge singletscalar. The entries of Yukawa matrices of quarks and charged leptons have only modular forms in them.In Sec. 4, we first give the convention for Yukawa matrices and the SUSY threshold corrections, thenwe present the data to be used and perform the numerical fit to the Yukawa matrices with thresholdeffects included. Sec. 5 devotes to the summary of the study. In the section we shall briefly discuss the framework and environment used for the construction ofmodel. First we give a brief review on the basic concepts of modular symmetry with lower level N and modular forms of weight k . The finite modular group Γ is used for our model building work, sowe give the modular forms with weight k = 2 ,
4. Secondly we introduce the basic aspects of adjointSU(5) grand unified theory, including the matter multiplets and scalar Higgs together with the vacuumconfigurations of the scalars.
The modular group Γ implies the linear fractional transformations γ that act on the complex τ inthe upper-half complex plane τ → γτ = aτ + bcτ + d , a, b, c, d ∈ Z , ad − bc = 1 , Im τ > , (1)The generators S and T are the two transformations satisfying S = ( ST ) = I , (2)with the representation matrices as S = (cid:18) − (cid:19) , T = (cid:18) (cid:19) , (3)lead to S : τ → − τ , T : τ → τ + 1 . (4)We introduce the series of infinite normal subgroups Γ( N ), N = 1 , , , · · · of SL (2 , Z )Γ( N ) = ( (cid:18) a bc d (cid:19) ∈ SL (2 , Z ) , (cid:18) a bc d (cid:19) = (cid:18) (cid:19) (mod N ) ) . (5)For N = 1 , N ) = Γ( N ) / { I , − I } and for N > N ) = Γ( N ). Taking the quotientΓ N ≡ Γ / Γ( N ), one can obtain a finite subgroup called finite modular group. Especially for N ≤ N are isomorphic to the permutation groups Γ ≃ S , Γ ≃ A , Γ ≃ S and Γ ≃ A , whichare ubiquitous in the construction of flavor models.Modular forms f ( τ ) of weight k and level N are holomorphic functions transforming under thegroups Γ( N ) in the way as f ( γτ ) = ( cτ + d ) k f ( τ ) , γ ∈ Γ( N ) . (6)4ere k is even number, and N is natural. Given the weight k and level N , the modular forms span alinear space of dimension equals to k + 1. The basis in the linear space can be chosed such that themodular form f i ( τ ) in a multiplet transforms according to a unitary representation ρ of the group Γ N : f i ( γτ ) = ( cτ + d ) k ρ ( γ ) ij f j ( τ ) , γ ∈ Γ N . (7)In the study we take N = 4 as the case of interest and construct an explicit grand unified flavor model toelucidate the fermion masses and mixings. In the case of lowest weight 2, there are 5 linear independentmodular forms. These modular forms are explicitly expressed in terms of Dedekind eta-function η ( τ ): η ( τ ) = q / ∞ Y n =1 (1 − q n ) , q = e i πτ . (8)To be specific, the modular forms are defined as the functions of η ( τ ) and its derivatives of the form Y ( c , · · · , c | τ ) ≡ c η ′ ( τ + ) η ( τ + ) + c η ′ (4 τ ) η (4 τ ) + c η ′ ( τ ) η ( τ ) + c η ′ ( τ +14 ) η ( τ +14 ) + c η ′ ( τ +24 ) η ( τ +24 ) + c η ′ ( τ +34 ) η ( τ +34 ) , (9)with the coefficients c + · · · + c = 0. For the case of weight 2, the basis is comprised of five modularforms as follow: Y ( τ ) ≡ Y (1 , , ω, ω , ω, ω | τ ) , (10) Y ( τ ) ≡ Y (1 , , ω , ω, ω , ω | τ ) , (11) Y ( τ ) ≡ Y (1 , − , − , − , , | τ ) , (12) Y ( τ ) ≡ Y (1 , − , − ω , − ω, ω , ω | τ ) , (13) Y ( τ ) ≡ Y (1 , − , − ω, − ω , ω, ω | τ ) , (14)with ω ≡ e πi/ . The modular forms ( Y , Y ) T and ( Y , Y , Y ) T transform as an doublet and triplet of S respectively, i.e., they are denoted as Y ( τ ) ≡ (cid:18) Y ( τ ) Y ( τ ) (cid:19) , Y ′ ( τ ) ≡ Y ( τ ) Y ( τ ) Y ( τ ) . (15)One can get modular forms of higher weights from the tensor productions of weight 2 modular forms,and thus obtain different irreps of S . Taking weight k = 4 for example, we then have 9 independentmodular forms, and they are arranged into the following singlets and multiplets irreps of S Y (4) = Y Y , Y (4) = (cid:18) Y Y (cid:19) ,Y (4) = Y Y − Y Y Y Y − Y Y Y Y − Y Y , Y (4) ′ = Y Y + Y Y Y Y + Y Y Y Y + Y Y . (16)We use the above modular forms of weight 2 and 4 to construct our modular invariant model in Sec. 3.For higher weight modular forms, one may refer to Ref. [36] for further information.5 .2 Basic aspects of Adjoint SU(5) In this part we shall give the fermion sector and the scalars used for the model setup. According tothe distinct representations under the gauge symmetry, the matter fields are divided into the followingthree parts: anti-fundamental ¯ , anti-symmetrical tensor and the adjoint . We assume the usualthree generations of ¯ and , which have the decomposition under the SM gauge group SU (3) C ⊗ SU (2) L ⊗ U (1) Y . The theory of renormalizable adjoint SU(5) can be seen in ref. [61]. Concerningour model, we promote the matter fields to superfields in Minimal Supersymmetry Standard Model(MSSM) and embed them into SU(5) representations ¯ and . For ¯ we denote F i = ( d cR d cB d cG e − ν ) i = d ci ⊕ ℓ i (17)and for T i = 1 √ − u cG u cB − u R − d R u cG − u cR − u B − d B − u cB u cR − u G − d G u R u B u G − e c d R d B d G e c i = u ci ⊕ q i ⊕ e ci , (18)where i = 1 , , R, B, G stand for the colorindices. The adjoint matter field reads = 1 √ √ λ . ρ − q ρ ρ (3 , ρ (¯3 ,
2) 1 √ σ . ρ + q ρ , (19)in which λ i are the Gell-Mann matrices. The scalar fields contain the following Higgs fields H = T (3 , ⊕ H , H = S ⊕ S (¯6 , ⊕ S (3 , ⊕ S (¯3 , ⊕ S (3 , ⊕ S (¯3 , ⊕ H , H = Σ ⊕ Σ ⊕ Σ (3 , ⊕ Σ (¯3 , ⊕ Σ . (20)The VEVs of the scalars are listed as the follwing form h H i = v √ , , , , T , (21) h H i = v diag(1 , , , − / , − / , (22) h H i i j = v √ , , , − , ij , h H i inj = 0 , (23)where i, j = 1 , ..., n = 1 , ..., v = p | v | + 24 | v | = 246GeV. Now let us present the supersymmetric adjoint SU(5) GUT flavor model in details. First we assignthe representations and weights of chiral supermultiplets. For sake of simplicity we shall impose certainconstraints on the assignments. For matter fields, three generations of dimensional representations,denoted by T , , , are all S singlets, i.e., ρ T i ∼ , ( i = 1 , , ¯5 dimensional representations, named as F , , are assigned to be a doublet6f S , i.e., F = ( F , F ), while the third generation F is a S singlet. Nevertheless the three ¯5 s havethe same weight. In addition to the - and ¯ -dimensional representations in SU(5) GUTs, the adjoint dimensional matter superfields A have also three families, and they are assigned to be an triplet of S , denoted by A = ( A , A , A ).For the Higgs sector H , H , H ¯5 and H are responsible for the electroweak symmetry broken,while the GUT gauge group is broken by an adjoint scalar field H . We shall not intend to explain themass hierarchy of charged fermions, which is beyond the power of modular symmetry. And in orderto made the coupling terms minimality, we do not introduce any flavon fields like conventional modelsdid. Instead all the Yukawa couplings are modular forms with specific weights to ensure the modularinvariance. The only modular symmetry breaking source arises from the modulus τ developing its VEV.For simplicity the modular weights are assigned such that the Yukawa couplings have as lower weightsas possible. The representations and weights of the chiral supermultiplets are listed in the Table 1.Table 1: Field content and their representation assignments under the gauge group SU(5), modular S and their weights k I in the model.Fields T T T F = ( F , F ) F A H H ¯5 H H H SU (5)
10 10 10 ¯5 ¯5 24 5 ¯5 45 45 24 Γ ≡ S ′ ′ ′ k I − − − − − For the purpose of our model construction, we start with SU(5) breaking superpotential. Since wehave only one adjoint scalar field H in the model, the gauge symmetry broken into SM gauge groupis realized by the adjoint scalar acquiring the vacuum expectation value of H . The SU(5) breakingsuperpotential is simply as W = M Y (4) Tr H + λY (6) ′ Tr H , (24)then one can get the VEV of H , i.e., h H i in eq. (22) with v = 4 M Y (4) λ Y (6) ′ . (25)Besides causing the GUT breaking, the adjoint scalar field H would also couple to the matter fieldsleading to novel Clebsch-Gordan factors, and especially the exotic Yukawa coupling ratios betweendown quark sector and charged lepton sector at GUT scale. See the next section for details. In this section we will present the Yukawa couplings for charged fermions. Since the flavons are notan essential part of the model, the relevant Yukawa coupling terms in up-type quark sector manifestin renormalisable level, while those in down-type quarks and charged lepton sector show in effectiveoperators. Since the quarks have enormous mass hierarchies but small mixing angles, the resultingYukawa matrices should be controllable in the entries. A practical viable scheme is to induce texturezeros for Yukawa matrices. And another keypoint is the GUT-scale Yukawa ratios between leptons anddown-quarks have to fulfil certain phenomenological constraints from experiments as well.7he Yukawa coupling terms for up quarks in superpotential involve two Higgs fields, i.e., H and H . Because of the symmetry constraints only the following couplings are allowed W u = α u T T H + β u T T H Y (4) + γ u T T H + g u T T H , (26)which leads to a block diagonal mass matrix M u = α u v β u Y Y v g u v − g u v γ u v . (27)The operators of dimension 4, such as those in Eq. (26), are expressed by the diagram in Figure. 1 ,in which the dashed line stands for scalar Higgs and solid lines are matter multiplets. In Table 2 wegive the list of dimension 4 operators corresponding to Figure. 1 including those in the next sections. φ φ φ Figure 1:
The supergraph for dimension 4 operators.
Field 1 2 3 4 5 6 7 8 φ T T T T T F F A φ T T T T F A A A φ H H H H H H H H Table 2:
The list for dimension 4 operators corresponding to Figure. 1.
In SU(5) GUTs, the left and right handed components of up-type quarks reside in the same rep-resentations T i , that is why the Yukawa couplings are of the form T i T j H s ( s = 5 ,
45) which made theYukawa elements either symmetric or antisymmetric. However those of down-type quarks live in dif-ferent representations: F i includes the right handed component and T i has left handed quark doublet.And vice verse for charged leptons. Then the interactions of both down quarks and charged leptonsare written by the same joint operators of the form T i F j H ¯ s . Accordingly the corresponding Yukawacouplings, Y d and Y e , are just mutual transposed relation up to O (1) CG factors. Therefore the eigen-values have to satisfy certain phenomenological GUT relations. For the first two families, the followingdouble Yukawa ratio [64] y µ y s ( y e y d ) − = 10 . +1 . − . , (28)is a strong restriction on model building. Besides the Yukawa eigenvalues in both up and down quarkssectors, the quark mixing CKM parameters have to be fulfil as well. All the requirements imply thatonly certain CG factors can be realistic in GUT flavor models. To be specific, in our model the effectivesuperpotential in down quarks sector and charged leptons sector is written by the operators W d = α d Λ [ T H ] [ F H ] Y + β d Λ [ T H ] [ F H ¯5 ] Y + γ d Λ [ T H ¯5 ] [ F H ] ¯5 + g d T F H , (29) The supergraphs were drawn with
JaxoDraw [62, 63]. XY ] R of fields X and Y denotes a tensor in the representation R . Herethe adjoint scalar H is crucial to form the dimension five operators which result the new Yukawa ratiosbetween leptons and quarks. After the GUT gauge symmetry is broken when H develops its VEValong the hypercharge direction, the Clebsch-Gordan factors emerge in the entries of lepton and quarkYukawa matrices. These SU(5) tensor contractions are realized by integrating out heavy messengers.The effective operators in the superpotential are then generated, see Figure. 2. For the model to work,we list the messenger fields and their representations as well as weights in Table 3.We define the quantity ǫ = h H i / Λ, then the mass matrix of down-type quarks reads M d = α d v ǫY − α d v ǫY g d v β d v ¯5 ǫY β d v ¯5 ǫY
00 0 γ d v ¯5 ǫ , (30)and that of charged leptons is simply the transposed M d up to the CG coefficients of Yukawa couplings M e = C α d v ǫY C β d v ¯5 ǫY − C α d v ǫY C β d v ¯5 ǫY C g d v C γ d v ¯5 ǫ , (31)The CG coefficients in the model are C = − / C = 6, C = − / C = −
3. Note thatthe fourth CG factor is the famous Georgi-Jarlskog relation [65]. The first three ones are the mainpredictions for the mass relations between quarks and leptons, which can be realized in conventionalmodels, such as [66, 67]. We can check that the double ratio in Eq. (28) is satisfied for the presentchoice. The set of the CG coefficients, C = − / C = 6, is in fact the only one that can berealized in GUT without flavons. The other two possibilities in realistic flavor GUT models, e.g., (A) C = 4 / , C = 9 / C = − / , C = − , who lives in the SU(5) adjoint representation. Then theleptonic and down-type quark-like components of Γ and Γ obtain different masses split by CG factors.After integrating out the heavy messenger fields, the CG factors inversely enter in the Yukawa matrixentries of charged leptons and down-quarks [70]. We will briefly build two toy models to elucidate thetwo cases. For sake of simplicity we assume only one flavon field φ (or weighton [31]) which is a singletunder a modular symmetry (not necessary Γ ), and the appropriate weight is assigned. And we justshow the operators which result in the diagonal entries of the mass matrix. For case (A) we may writethe superpotential for the first two families as W Ad = y d T F H ¯5 φ H Y ( k ) r ,a + y d Λ [ T H ] ¯5 [ F H ] Y ( k ′ ) r ′ ,a ′ , (32)in which Y ( k ) r ,a and Y ( k ′ ) r ′ ,a ′ denote the components of modular form multiplets. For case (B) the superpo-tential is W Bd = y d T F H ¯5 φ H Y ( k ) r ,b + y d T H F Y ( k ′ ) r ′ ,b ′ . (33)The effective operators of the superpotentials are generated after integrate out the heavy messengerfields. We show in Figure. 3(a) and Figure. 3(b) the supergraphs corresponding to case (A) and case(B) respectively. The details for building such models are beyond the scope of this work. In this section we shall give the neutrino interactions which induce light neutrino masses. We assumeneutrinos to be Majorana type, and the light masses are generated by seesaw mechanism. In most of9 φ φ φ Γ ¯Γ
Figure 2:
The supergraph for dimension 5 operators in the superpotential of down quarks.
Field φ φ φ φ Γ1 T H F H Υ T H F H ¯5 Υ T H ¯5 F H Υ Field Υ Υ Υ SU(5)
45 10 ¯ Γ ≡ S ′ ′ k I − − Table 3: Left panel: The dimension 5 operators corresponding to Figure. 2, Right panel: The chargesof messenger fields appear in the dimension 5 operatorstypical flavor models, including the flavor GUT models, the light neutrino masses can be producedthrough type-I seesaw mechanism which demands at least two superheavy right-handed Majorananeutrinos to suppress the Yukawa couplings. In the implementation ways of seesaw mechanisms inSU(5) GUTs, however, there are another two ways to do the same thing, the first scheme is type-IIseesaw [57–59] by adding an extra Higgs H , and the second one is Type-I plus Type-III seesaw [60]by introducing the fermionic fields in the dimensional representation. In our model we assume thesecond scheme as the unique origin of neutrino masses and no more extra matter fields are involved. Thematter chiral superfield A lives in SU(5) adjoint representation and is also an S triplet. Accordingto the symmetry constraints in , the neutrino Yukawa interactions reads W Y A = y ν Λ F AH Y ′ + y ν Λ F AH Y ′ . (34)Note that the second term would vanish if ( ρ F , ρ A ) = ( , ) or ( ′ , ′ ). If we made the choice, the thirdcolumn of Yukawa texture will be vanishing which features a zero determinant of neutrino mass matrix,no matter what structure of Majorana mass matrix is. We drop the case at present. The Yukawamatrix is then Y ρ = y ν Y Y Y Y Y Y + y ν Y Y Y Y ρ = r Y ρ . (35)If we introduce SU(5) singlet , e.g., an N c , as right-handed Majorana neutrinos, the pure mass termof the form M N c N c is the only interaction. However in adjoint SU(5) there is an extra interaction form T F H ¯5 H φ H φX X X X T F H H X X (a) T H ¯5 H φ H H φ φ F Ω Ω Ω Ω Ω Ω T F H (b) Figure 3:
The supergraphs for the operators in Eq. (32) (left) and Eq. (33) (right). dimensional matter superfield responsible for the generationof neutrino masses, the new interactions between and H have to be considered. Now we give allthe mass terms of as W A = M AAY + M ′ AAY ′ + λ AAH Y (4) + λ AAH Y (4) ′ + λ AAH Y (4) . (36)The second and the fifth terms vanish because the tensor productions of ′ ⊗ ′ have to be antisymmetric(see Appendix) to form an invariant with modular forms. Appling the decomposition to eq. (36), thefermionic singlet ρ and triplet ρ have the mass matrices of the form M ρ = M Y Y Y Y Y Y − v ′ √ n λ − Y Y − Y Y Y − Y + λ Y (4) ′ , − Y (4) ′ , − Y (4) ′ , − Y (4) ′ , Y (4) ′ , − Y (4) ′ , − Y (4) ′ , − Y (4) ′ , Y (4) ′ , o ,M ρ = (37) M Y Y Y Y Y Y − v ′ √ n λ − Y Y − Y Y Y − Y + λ Y (4) ′ , − Y (4) ′ , − Y (4) ′ , − Y (4) ′ , Y (4) ′ , − Y (4) ′ , − Y (4) ′ , − Y (4) ′ , Y (4) ′ , o , where the abbreviation Y (4) ′ ,i ( i = 1 , ,
3) denote the components of weight 4 modular form Y (4) ′ for theexpression more compact, e.g., Y (4) ′ , = Y Y + Y Y and else, see Eq. (16).The neutrino masses are generated by Type-I and Type-III seesaw, which are realized by integratingout ρ and ρ , respectively. We write the effective light neutrino mass matrix as the sum of thecontributions from Type-I and Type-III seesaw M SSν = − ( Y Tρ M − ρ Y ρ + Y Tρ M − ρ Y ρ ) v u , (38)where v u = v sin β with v = p v u + v d =174GeV and tan β = v u /v d , as usual defined in MSSM. The superpotential in the matter sector is simply given by W matt = W u + W d + W Y A + W A , (39)in which each term is given by Eqs. (26), (29), (34) and (36), respectively. As the GUT and modularsymmetry breaking we write the Yukawa matrices of fermions in the following convention W Y = ( Y u ) ij q i ∗ H u u cj + ( Y d ) ij q i ∗ H d d cj + ( Y e ) ij ℓ i ∗ H d e cj +( Y ρ ) ij ℓ Ti iσ ( ρ ) j H u + ( Y ρ ) ij ℓ Ti iσ ( ρ ) j H u + ( M ρ ) ij Tr(( ρ ) i ( ρ ) j ) + ( M ρ ) ij ( ρ ) i ( ρ ) j , (40)where H u and H d are the two Higgs doublets in MSSM. The neutrino Yukawa matrices and massmatrices have been given by Eqs. (35) and (37) in Sec. 3.3. The Yukawa matrices of charged fermion s are then read as Y u = α u β u Y Y g u − g u γ u ≡ u u u − u u , (41)11or up quarks, and Y d = α d Y − α d Y g d β d Y β d Y
00 0 γ d ≡ d − d Y /Y d d Y /Y d
00 0 d , (42)for down quarks. Observing the Yukawa matrices Y u and Y d are very sparse with some texture zeros,one can conclude the CKM Cabibbo angle θ q is totally generated by the mixing in the down sector, andof course the same for the θ q as well as CP phase δ q . The mixing in up sector completely determinesthe angle θ q .In SU(5) GUT the Yukawa matrix of charged leptons is the transpose of that of down quarks, i.e., Y e ∼ Y Td , up to the order one CG coefficients, Y e = − d d Y /Y − d Y /Y d − d − d . (43)It is obvious that the down quarks and charged leptons follow the following Yukawa ratios y τ y b ≈ , y µ y s ≈ , y e y d ≈ , (44)in which y ℓ ( ℓ = e, µ, τ ) and y d i ( i = d, s, b ) are the eigenvalues of Y e and Y d , respectively. Thus the CGfactors C = 1 / C = 6 made the double Yukawa ratio y d y µ /y e y s = 12 which is in good agreementwith the data in Eq. (28). The Yukawa couplings and mixing observables are defined at superhigh GUT scale in our model,therefore the values from low energy experiments must run up to the ones at GUT scale. Moreoverthe SUSY radiative threshold corrections are the requisite factor for matching the MSSM at the SUSYscale M SUSY to the SM [71–74]. The running of MSSM Yukawa parameters from M Z to M GUT hasbeen analysed in [64], where the tan β enhanced 1-loop SUSY threshold effects are discussed in detail.The matching relations between the eigenvalues of the MSSM and the SM Yukawa coupling matricesare parameterized as y MSSM u,c,t = y SM u,c,t sin ¯ β , (45) y MSSM d,s = y SM d,s (1 + ¯ η q ) cos ¯ β , y MSSM b = y SM b (1 + ¯ η b ) cos ¯ β , (46) y MSSM e,µ = y SM e,µ (1 + ¯ η ℓ ) cos ¯ β , y MSSM τ = y SM τ cos ¯ β , (47)and the quark CKM parameters are also corrected by θ q, MSSM i = 1 + ¯ η b η q θ q, SM i , θ q, MSSM12 = θ q, SM12 , δ q, MSSM = δ q, SM . (48)One can notice, to a good approximation, the threshold corrections have no impact on θ q and δ q andthe running of Yukawa couplings Y MSSM f depends only on ¯ η b and tan ¯ β . Especially in the limit thatthe threshold corrections to charged leptons are neglected, i.e., ¯ η ℓ = 0, then tan ¯ β reduces to the usual12an β . We will adopt the scenario in the model. Nevertheless the ¯ η q can not be dropped. The reasonis that the ratio y µ /y s at GUT scale is approximately 4.5 without SUSY threshold corrections, but thelarge CG factor 6 in (43) needs a compensation from ¯ η q which is approximately +0.33 [75]. The two CGfactors C = 6 and C = − / Y e in Eq. (43) require a relative largetan β to generate substantial threshold corrections. In fact for the large Yukawa coupling ratios in (44),both large tan β and large threshold corrections are required (cf. [76]). Accordingly we set tan β = 35,¯ η b = 0 . η q = 0 .
3. We notice that the threshold parameters can be free, but the fixed valuesare enough to reproduce correct observables.
The Yukawa matrices for neutrinos, up- and down- quarks and leptons are presented in Eqs. (38),(41), (42) and (43), respectively. The only common parameter τ among them has been bounded in theupper half complex plane. Modular symmetry itself, however, is enable to give rise to the hierarchicalfermion masses, and mixing parameters. The free parameters appear in the mass matrices are P i = { τ, α u , β u , γ u , g u , α d , β d , γ d , g d , y ν , y ν , M, λ , λ , } , (49)and the physical observables Q obs in the GUT model include Q q = { y u , y c , y t , y d , y s , y b , θ q , θ q , θ q , δ q } ,Q ℓ = { y e , y µ , y τ , ∆ M , ∆ M , θ ℓ , θ ℓ , θ ℓ , δ ℓ } . (50)We shall construct a global χ function for all the observable quantities when fitting the couplingmatrices in Eqs.(41), (42) and (43) for charged fermions and Eqs. (35) and (37) for neutrinos. The χ function to be minimized is defined as χ = X i (cid:16) Q i ( P j ) − Q obs i σ i (cid:17) , (51)where Q i ( P j ) denote the model predicted values for observables and Q obs = Q q + Q ℓ are the centralvalues with σ i the 1 σ errors.Before performing the fit, we would like to elucidate the data used in the minimization. As stressedin Sec. 4.1, the model is defined at the high energy scale, the observable quantities should be set atthe GUT scale. The quantities at GUT energy scale can be achieved from the low energy scale wherethe experimental values are determined by the renormalization group equations (RGEs). Here in ourwork we adopt the values of Yukawa couplings at the GUT scale M X = 2 × GeV, assuming minimalSUSY breaking M SUSY =1TeV with a large tan β = 35 [64]. The Yukawa values y f give rise to thefermion masses as m f = y f v H with v H =174GeV. Meanwhile the CKM mixing angles and CP phaseare also taken as the values at the GUT scale. All the Yukawa values and CKM observables as well astheir 1 σ errors are listed in the left panel of Table 4. For the neutrino sector, the lepton mixing angles,CP phase and neutrino mass squared differences we adopted are taken from NuFit 5.0 [77]. We showtheir center values and 1 σ errors in the right panel of Table 4. The above input data are used for theestimation of our χ .Since the quark sector and neutrino sector have the modulus τ as the common parameter whichdetermines the modular forms, the remaining free parameters are just Yukawa coupling coefficients. Itis equivalent and more convenient to fit the entries of Yukawa matrices rather than Yukawa coefficientsthemselves. Therefore we just fit the entries of quark Yukawa matrices, u ij s in Eq. (41) and d ij sin Eq. (42). In neutrino sector, we define the coupling parameter ratios r = y ν y ν , r = λ v M , r = λ v M , (52)13nd an overall mass scale y ν v u /M are the parameters to be fitted. So, instead of fitting the primitivefree parameters showed in Eq. (49), we take the following equivalent parameter set P i = { τ, u , u , u , u , d , d , d , d , r , r , r , y ν v u /M } . (53)In Table 5 we show the model best fit input parameters in quark Yukawa and neutrino mass matriceswhich minimizes the χ q and χ ℓ , respectively. We present our fit results for all the Yukawas and massmatrices in Table 6. In the left panel of Table 6 we give the resulting best fit values and pulls tothe ten quark observables: three quark CKM mixing angles θ qij ( ij = 12 , ,
23) and one CP violatingphase δ q , six Yukawas y q ( q = u, c, t, d, s, b ). Also the minimum χ ,q ∼ .
45 is given at the lastrow. We also list in Table 6 (right panel) the best fit values and pulls to six neutrino observables andthree charged lepton Yukawas. The minimum χ ,ℓ is just O (1). The best fit point has the total χ = χ ,q + χ ,ℓ ≃ .
6. One can see that the model favours normal ordering neutrino masses, andwe found the minimum χ ,ℓ = 8 .
478 for inverted ordering. Besides the values of absolute neutrinomasses m i , the Majorana phases ϕ and ϕ are pure theoretical predictions. The mass sum, the β -decay effective mass m β as well as the neutrinoless double beta (0 νββ ) decay amplitude parameter m ee are also given as predictions in the table. Specifically the bounds on the above mass related quantitiesare given by X i m i ≤ ,m β = h X i m i | U ei | i / < (61 ∼ , (54) m ee = (cid:12)(cid:12)(cid:12) X i U ei m i (cid:12)(cid:12)(cid:12) < . , which are taken from PLANCK [78], KamLAND-ZEN [79] and KATRIN [80], respectively. The leptonicmixing matrix elements U ei are taken from the standard parametrization [3] U = c c s c s e − iδ − s c − c s s e iδ c c − s s s e iδ c s s s − c s c e iδ − c c − s s c e iδ c c e ϕ /
00 0 e ϕ / , (55)where s ij = sin θ ij , c ij = cos θ ij , δ is lepton Dirac CP violating phase and ϕ , ϕ are two MajoranaCP phases. Comparing the model predictions in Table 6 to these bounds, we see that the predictedvalues are well below the corresponding upper bounds. For the sum of neutrino masses, our result isstill below the tightest and most rebust upper limit M ν < . χ , i.e., χ = χ / d.o.f = 0 . τ , whichis close to the boundary (right cusp) of the fundamental region. The Yukawas are in general complex,however we can always absorb most of the phases by redefinition of the fields. Specifically for the quarkYukawa inputs, the only complex parameter is d which mainly controls the magnitude of the CKMmatrix element V ub and part of the CP phase δ q . Meanwhile the other inputs, all u ij s and the rest of d ij s ( ij = 13) are real. In neutrino sector we also have reduced the total number of real parameters tobe six, in which only r and r are complex. The Yukawa ratio r and the overall mass scale y ν v u /M are all real parameters. 14bservable µ i σ i θ q / ◦ θ q / ◦ θ q / ◦ δ q / ◦ y u . × − . × − y c . × − . × − y t y d . × − . × − y s . × − . × − y b . × − y e . × − . × − y µ . × − . × − y τ . × − Observable NO IO θ / ◦ . +0 . − . . +0 . − . θ / ◦ . +0 . − . . ± . θ / ◦ . +1 . − . . +1 . − . δ/ ◦ +51 − +27 − ∆ M / − . +0 . − . . +0 . − . ∆ M / − . +0 . − . − . +0 . − . Table 4: Left panel: The Observables of charged fermions at the GUT scale for tan β = 35 are takenfrom [64]. The SUSY breaking scale are set at M SUSY =1TeV and the threshold correction parameters¯ η b = 0 . η q = 0 .
3. Right panel: The values of neutrino Observables are taken from NuFit5.0[77] without the atmospheric data from SuperKamiokande. NO (IO) denotes the Normal (Inverted)Ordering of neutrino masses.Quark Input Value u . × − u . × − u . u − . × − d . × − d . × − d . × − d . × − e − . iπ Neutrino Input Value r = λ v /M − . . ir = λ v /M . . ir = y ν /y ν y ν v u /M (eV) 0.00362Common Input Value Re τ . Im τ . u ij and d ij include the original Yukawa coupling coefficients and the modular forms. Upper Right panel: The bestfit input parameters in neutrino sector. We list the ratios of neutrino couplings in eqs. (34) and (36).Lower Right panel: The only common input parameter to both sectors is the modulus τ .15uark output Value pull θ q / ◦ − . θ q / ◦ θ q / ◦ δ q / ◦ y u . × − y c . × − y t y d . × − y s . × − − . y b χ ,q θ / ◦ θ / ◦ − . θ / ◦ δ/ ◦ M / − M / − y e . × − − . y µ . × − y τ − . χ ,ℓ m / meV 11.728 m / meV 14.585 m / meV 51.493 P i m i / meV 77.809 m β / meV 14.674 m ee / meV 11.040 ϕ /π ϕ /π σ interval. Rightpanel: The output results of lepton sector. The absolute values of three light neutrino masses and theordering as well as the two Majorana phases are pure model predictions. We have found a minimum χ ,ℓ = 8 .
478 for inverted ordering. 16
Summary
In the study we explored an supersymmetric adjoint SU(5) GUT flavor model based on modularΓ ≃ S symmetry. We have shown the model can produce correct masses and mixing parameters ofboth quarks and leptons simultaneously. No flavons are introduced to the model, only the complexmodulus τ is responsible for the breaking of modular symmetry. By assigning suitable representationsand weights to chiral superfields, only finite coupling terms are presented in the effective operators inquark sector and lepton sector. We obtained very sparse Yukawa matrices of up- and down-quarks(and of course charged leptons) with some texture zeros. Also in neutrino sector we have only twoYukawa coupling terms, and three terms in Majorana mass terms. The effective light neutrino massesare generated through Type-I plus Type-III seesaw mechanism. The modulus τ is the only commonfield appeared in both quark and lepton sector as spurion.For simplicity we have used modular forms with lower weight in the model, since higher weightwould bring more free parameters. The assignments of representations under S and weights for thechiral superfields are also highly constrained such that the Yukawa coupling terms are uniquely fixedby the modular forms. The dimensional matter fields are all S singlets, while ¯ s are divided into a S doublet for the first two generations and a singlet for third one. Meanwhile the adjoint superfields are collected in the triplet of S . All the scalars of course transform as singlets of S . We alsoassigned distinct weights for the superfields such that the number of free Yukawa parameters is as littleas possible.With the delicate representation assignments of field content, the superpotentials of the model isrelatively simple. Unlike the models in [21,32] which have more parameters than observables, our modelhas less free parameters and thus more predictive than the above two works. The resulting operators ineach sector have limit coupling parameters. The model predicts that down quarks and charged leptonshave a rigid Yukawa coupling ratios generated by CG factors which arise from the SU(5) contractions ofthe effective 5D operators. The double Yukawa ratio y µ y d /y e y s equal to 12 for the first two families ofcharged leptons and down quarks. The model has only 17 real parameters in total, and 19 observablesto be fit, thus the degree of freedom (d.o.f) is 2. We obtained the reduced chi-square χ / d.o.f ≃ . χ ∼
7. Moreover we obtained the absolute valuesof three light neutrino masses, the effective masses of β -decay and 0 νββ decay as well as the MajoranaCP phases as model predictions.At last we give a outlook for the model. Since the masses of the fields live in the adjoint representa-tion are splited by the adjoint scalar H , the lightest one is ρ . It is crucial to realize the baryogenesisvia leptogenesis [83, 84] in the context. In the case the net asymmetry of B − L can be generated inthe out of equilibrium decays of ρ and ρ as well as their superpartners in the adjoint representation.So it is worth studying further the phenomenology according to the model. Acknowledgements
We would like to thank Dr. Shu-Jun Rong for useful discussions. This work is supported by theNational Natural Science Foundation of China (NSFC) under Grant No.11875327, the FundamentalResearch Funds for the Central Universities, China, and the Sun Yat-Sen University Science Foundation.17
PPENDIX: S GROUP
The discrete group S who has 24 elements is the permutation group of four objects. The twogenerators S and T in different irreducible presentations are given as follows : S = 1 , T = 1 (A1) ′ : S = − , T = 1 (A2) : S = (cid:18) (cid:19) , T = (cid:18) ω ω (cid:19) (A3) : S = 13 − ω ω ω ω − ω − ω , T = ω
00 0 ω (A4) ′ : S = 13 − ω − ω − ω − ω − ω − ω , T = ω
00 0 ω (A5)where ω = e πi/ = ( i √ − /
2. In the basis we can obtain the decomposition of the product repre-sentations and the Clebsch-Gordan factors. The product rules of S group, with a i , b i as the elementsof multiplet in the product, are given by following ⋆ ⊗ r = r ∼ ab i (A6) ⋆ ′ ⊗ ′ = ∼ ab (A7) ⋆ ′ ⊗ = ∼ (cid:18) ab − ab (cid:19) (A8) ⋆ ′ ⊗ = ′ ∼ ab ab ab (A9) ⋆ ′ ⊗ ′ = ∼ ab ab ab (A10)The product rules with two-dimensional representation are given by: ⋆ ⊗ = ⊕ ′ ⊕ ∼ a b + a b , ′ ∼ a b − a b ∼ (cid:18) a b a b (cid:19) (A11) ⋆ ⊗ = ⊕ ′ ∼ a b + a b a b + a b a b + a b , ′ ∼ a b − a b a b − a b a b − a b (A12) ⋆ ⊗ ′ = ⊕ ′ ∼ a b − a b a b − a b a b − a b , ′ ∼ a b + a b a b + a b a b + a b (A13)18nd the three-dimensional representations have the following product rules ⋆ ⊗ = ′ ⊗ ′ = ⊕ ⊕ ⊕ ′ ∼ a b + a b + a b ∼ (cid:18) a b + a b + a b a b + a b + a b (cid:19) ∼ a b − a b − a b a b − a b − a b a b − a b − a b , ′ ∼ a b − a b a b − a b a b − a b (A14) ⋆ ⊗ ′ = ′ ⊕ ⊕ ⊕ ′ ′ ∼ a b + a b + a b ∼ (cid:18) a b + a b + a b − a b − a b − a b (cid:19) ∼ a b − a b a b − a b a b − a b , ′ ∼ a b − a b − a b a b − a b − a b a b − a b − a b (A15) References [1]
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