Family Unification with SO(10)
aa r X i v : . [ h e p - ph ] O c t BA-07-36OSU-HEP-07-06
Family Unification with SO(10)
K.S. Babu a , S.M. Barr b and Ilia Gogoladze b, a Department of Physics, Oklahoma State University, Stillwater, OK 74078, USA b Bartol Research Institute, Department of Physics and Astronomy,University of Delaware, Newark, DE 19716, USA
Abstract
Unification based on the group SO (10) × S is studied. Each family has its own SO (10)group, and the S permutes the three families and SO (10) factors. This is the maximallocal symmetry for the known fermions. Family unification is achieved in the sense thatall known fermions are in a single irreducible multiplet of the symmetry. The symmetrysuppresses SUSY flavor changing effects by making all squarks and sleptons degenerate inthe symmetry limit. Doublet-triplet splitting can arise simply, and non-trivial structureof the quark and lepton masses emerges from the gauge symmetry, including the “doublylopsided” form. On leave of absence from: Andronikashvili Institute of Physics, GAS, 380077 Tbilisi, Georgia. n this paper we propose the idea of family unification based on the group SO (10) × SO (10) × SO (10) × S , where each family of quarks and leptons transforms as a spinor under its own SO (10), and where the S permutes the three families and the three SO (10) factors.This structure has several interesting features. First, it may be the only way to achievefamily unification [1] satisfactorily in four space-time dimensions. (Attempts to unify familiesin complex spinors of SO (4 n +2) groups have not resulted in realistic models, since these spinorscontain families and mirror families when decomposed to the Standard Model symmetry [1].Family unification can occur in higher dimensions, as in heterotic string theory [2]. For familyunification in five, six and other dimensions, see [3].) Here we define family unification tomean that all three families of quarks and leptons including their right-handed neutrinos arecontained in a single irreducible representation of the unification group, and that there is asingle gauge coupling constant at high scales. (This definition is broader than that often used,in which the three families form an irreducible representation of a simple group.) It shouldbe noted that in the scheme we propose here the families are in a reducible representation of SO (10) , namely { (16 , ,
1) + (1 , ,
1) + (1 , , } , but these form an irreducible multiplet ofthe full unification group that includes the S factor.Second, the group SO (10) × SO (10) × SO (10) × S is the largest that can be gauged with48 fermions forming a complex but anomaly-free set of representations. In that sense, it isthe “maximal local symmetry” of the known quarks and leptons, including the right-handedneutrinos. (In [4], the definition of maximal local symmetry also included the condition thatthe group be simple. By that more restrictive definition, the maximal local symmetry of 48fermions would be just SO (10).) It is easy to see that the S factor is anomaly free. S makes the gauge couplings of all SO (10) groups equal. As a result, the instanton effects ofthe three SO (10) groups will also be S –invariant, proving that it is anomaly free. (The cyclicpermutation group Z , which is often considered in such contexts is a subgroup of this S .Incidently, in the model we present it might appear that it is possible to gauge additional U (1)factors, where under these U (1)’s the fermions are rotated into themselves by a phase factorand the gauge bosons are invariant. However, the only anomaly free part of these U (1)’s arethe Z centers of the SO (10) groups. It is interesting that the S does not commute with these Z , and other SO (10), transformations.)Third, the unification of all the known quarks and leptons in a single irreducible multipletof a local group suppresses SUSY flavor changing processes by making all squark and sleptonmasses exactly degenerate at the unification scale.Fourth, the problem of “doublet-triplet splitting” can be solved more easily with thisgroup than in ordinary SO (10) unification by means of the Dimopoulos-Wilczek mechanism15] also known as the “missing VEV mechanism”. That the Dimopoulos-Wiczek mechanism isstraightforward to implement in product groups like SU (5) × SU (5) and SO (10) × SO (10) waspointed out in [6, 7, 8]. The stability of the VEV structure however is nontrivial to achieve in SO (10) × SO (10) models, as that requires rank reduction of the diagonal SO (10) [9]. As willbe shown, the present framework resolves this issue very neatly.And, finally, the “vertical” group SO (10) is also in a sense a “family group”, since thethree families transform differently under any one of the SO (10) groups. As will be seenlater, highly non-trivial patterns emerge in the mass matrices of the quarks and leptons dueprimarily to the symmetry SO (10) , though S also plays a role. Some of the patterns thatemerge almost automatically in this framework have already been proposed in the literature onpurely phenomenological grounds, such as the “doubly lopsided” structure [10].We will describe a supersymmetric SO (10) × S model that illustrates some of the possi-bilities of the idea. The quarks and leptons are in the representation { (16 , ,
1) + (1 , ,
1) +(1 , , } , which we shall denote (16 , ,
1) + cyclic for short. The Higgs doublets of the Stan-dard Model are contained in the “fundamental” Higgs multiplet (10 , ,
1) + cyclic . Two kindsof Higgs multiplets are needed to do breaking of SU (3) × S to the Standard Model and givesuperheavy mass to the right-handed neutrinos. We take these to be the “bifundamental” Higgsmultiplet (10 , ,
1) + cyclic and the “bispinor” Higgs multiplet (16 , ,
1) + cyclic (plus theconjugate bispinor multiplet (16 , , SO (10)” of the three factor SO (10) groups. Under this diagonal SO (10) subgroup, the bifundamentals contain the repre-sentations + + , while the bispinors contain + + . It is well-known that aHiggs field in the bifundamental representation of a group G × G can break it to the diagonalsubgroup G . Similarly, a set of bifundamentals can break G × G × G to the diagonal subgroupof the three factor groups. In our model, there is a minimum of the scalar potential where thevacuum expectation values (VEVs) of the bifundamentals break SO (10) all the way down toa diagonal SU (3) c × SU (2) L × U (1) × U (1), as will be seen. The bispinors, whose VEVs givemass to the right-handed neutrinos, break the extra U (1) to give the Standard Model group.The quark and lepton multiplets will be denoted ψ a , a = 1 , ,
3, where ψ ≡ (16 , , ψ ≡ (1 , , ψ ≡ (1 , , H a , where H ≡ (10 , , ab , where Ω ≡ (10 , , ′ ab . And thebispinors will be denoted ∆ ab and ∆ ab , where ∆ ≡ (16 , ,
1) and ∆ ≡ (16 , , a and b are not SO (10) indices (which we suppress) but merely labels that indicatewhich SO (10) groups the multiplets transform non-trivially under. These labels are permuted2nder the S group.Only two renormalizable terms are allowed by the gauge symmetry in the Yukawa superpo-tential of the quarks and leptons, namely W Y uk = Y ( ψ ψ H + ψ ψ H + ψ ψ H ) + Y ′ ( ψ ψ ∆ + ψ ψ ∆ + ψ ψ ∆ ) . (1)The Higgs superpotential can be written in an obvious notation as W Higgs = W H + W Ω + W ∆ + W H Ω + W Ω∆ + W H ∆ . Let us focus first on W Ω . If there is only one set of bifundamentalsΩ ab , then the most general renormalizable form of W Ω consistent with symmetry is W Ω = 12 M ( tr Ω Ω + tr Ω Ω + tr Ω Ω ) + λ ( tr Ω Ω Ω ) . (2)The order of labels ab on Ω ab is significant. If (Ω ab ) ij is the ij element of the 10 ×
10 matrix Ω ab ,then the row index i belongs to the group SO (10) a and the column index j belongs to the group SO (10) b . This superpotential gives rise to the equations of motion Ω = ( λ/M )Ω Ω , Ω =( λ/M )Ω Ω , and Ω = ( λ/M )Ω Ω , which, of course, are S permuted versions of eachother. Doublet-triplet splitting by means of the Dimopoulos-Wilczek mechanism would requirethat the VEV of at least one of these bifundamentals had the form h Ω i = diag ( a a a ⊗ iτ (corresponding to the generator B − L of the diagonal SO (10) and the generator diag ( a a a U (5)). However, it is easily seen from the three equations of motion, that ifany one of the Ω ab has a VEV of this form, all three of them would have a similar form (insome basis), i.e. would vanish in the lower 4 × SO (4) =( SU (2) L × SU (2) R ) would remain unbroken by the bifundamentals. And even after breakingby the bispinor Higgs VEVs there would still remain an unbroken ( SU (2) L ) . Thus the form inEq. (2) is too simple to give the bifundamental VEVs the Dimopoulos-Wilczek form and alsobreak the symmetry down to the Standard Model group at low energies.Satisfactory breaking can happen if there are two sets of bifundamentals, Ω ab and Ω ′ ab , whereΩ ′ ab is odd under a Z parity and Ω ab is even. Then the most general renormalizable form of W Ω is W Ω = M ( tr Ω Ω + tr Ω Ω + tr Ω Ω ) + M ′ ( tr Ω ′ Ω ′ + tr Ω ′ Ω ′ + tr Ω ′ Ω ′ )+ λ ( tr Ω Ω Ω ) + λ ′ ( tr Ω Ω ′ Ω ′ + tr Ω ′ Ω Ω ′ + tr Ω ′ Ω ′ Ω ) (3)The equations of motion then become: Ω ba = ( λ/M )Ω bc Ω ca +( λ ′ /M )Ω ′ bc Ω ′ ca and Ω ′ ba = ( λ ′ /M ′ ) × (Ω ′ bc Ω ca + Ω bc Ω ′ ca ). These equations have many interesting solutions. The one that seemsphenomenologically most interesting has VEVs of the following form:3 = (cid:18) A B (cid:19) , Ω = Ω = (cid:18) A
00 0 (cid:19) , Ω ′ = (cid:18) A ′
00 0 (cid:19) , Ω ′ = Ω ′ = (cid:18) A ′ B ′ (cid:19) , (4)where A and A ′ are 6 × B and B ′ are 4 × A = aI , A ′ = a ′ I ⊗ iτ , B = bI , B ′ = b ′ I ⊗ iτ , (5)and where a = M ′ λ ′ , a ′ = q MM ′ λ ′ + λM ′ λ ′ , b = − M ′ λ ′ , and b ′ = √− MM ′ λ ′ . At this minimum, thebifundamentals break SO (10) down to a diagonal SU (3) × SU (2) × U (1) × U (1).To illustrate some of the possibilities, we mention a few other solutions of the many thatexist: (i) One can have the same form as in Eqs. (4) and (5) but with A ′ = a ′ I , a ′ = q MM ′ λ ′ − λM ′ λ ′ . This breaks SO (10) down to SU (4) × SU (2) × U (1) × U (1). (ii) One can havethe same form as in Eqs. (4) and (5) but with B ′ = b ′ I , b ′ = √ + MM ′ λ ′ . This breaks down to SU (3) × SU (2) × SU (2) × U (1). (iii) One can have the same form as in Eqs. (4) and (5) butwith both the substitutions of cases (i) and (ii). This would break the group only down to thePati-Salam group SU (4) × SU (2) × SU (2). (iv) There are solutions where all three of the Ω ab have the form that Ω has in Eq. (4) and all three of the Ω ′ ab have the form that Ω ′ has inEq. (4). (v) There are solutions where the matrices have different forms than shown in Eq.(4), for example having different rank than 4, 6, or 10. Some of the unbroken groups that canresult are SU (5) × U (1), SU (4) × U (1) × U (1), SO (8) × SO (2). A complete analysis of all theminima would be rather lengthy.The form in Eq. (4) is a useful one for the purposes of doublet-triplet splitting, as it canlead to a single pair of Higgs doublets being light. To see this, consider W H Ω , whose mostgeneral renormalizable form consistent with symmetry is W H Ω = λ H Ω ( H Ω H + H Ω H + H Ω H ) . (6)Similar terms with Ω ′ ab are ruled out by the Z parity. If the explicit mass term ( H H + H H + H H ) is forbidden (or suppressed to be of order the weak scale) by symmetry (for example asoftly broken discrete symmetry or an R symmetry) then the mass matrix of the color-tripletsand weak-doublets in H a have the form M = a aa aa a , M = b b . (7)4t is apparent that only a single pair of doublet Higgs fields (namely those in H ) remainlight, as needed for gauge-coupling unification. Therefore, only H will get a weak-interaction-breaking VEV, and because of the form of the Yukawa superpotential given in Eq. (1) only thethird family of quarks and leptons will get mass. Below it will be shown that higher-dimensionoperators can generate other entries in the quark and lepton mass matrices, allowing non-zero(but presumably smaller) masses for the other families and CKM mixing. It is interesting thatthe SO (10) × S symmetry and a choice of minimum consistent with a single pair of lightHiggs doublets leads to a natural hierarchy wherein one family is heavier than the others.In order to give mass to the right-handed neutrinos the bispinors ∆ ab must receive non-vanishing VEVs such that both spinors of the bispinor point in the Standard-Model-singletdirection. One possible superpotential which achieves this is given below. W ∆ = λ ∆ { (∆ ∆ − M ) S + (∆ ∆ − M ) S + (∆ ∆ − M ) S } , (8)where S ab are SO (10) singlets. This superpotential admits a solution where all three ∆ ab ’sand ∆ ab ’s have equal VEVs along their respective Standard Model singlet directions. Higherdimensional operators of the form (∆∆) /M ∗ will have to be introduced to give masses to allpseudo-Goldston bosons from these fields.An interesting feature of this VEV structure is that from Eq. (1), it will generate a Majoranaright–handed neutrino mass matrix which has equal entries in the off-diagonals, and zero entriesalong the diagonals (as in M of Eq. (7)). That will result in two degnerate ν c fields, whichmay be relevant for resonant leptogenesis.There are only two renormalizable terms allowed in W Ω∆ by symmetry, namely (∆ ab ∆ ab Ω ab + cyclic ) and (∆ ab ∆ ab Ω ab + cyclic ). These terms give sufficient coupling between the bifundamen-tal Higgs sector and the bispinor Higgs sector to prevent any uneaten goldstone bosons. (Withinsufficient coupling between two kinds of Higgs, goldstone bosons can arise that correspondto relative rotations of their VEVs.) It is in this regard that the present model fares betterthan the usual SO (10) models, where new mechanisms should come in to stabilize the VEVstructure [11].Turning to higher-dimension operators, one finds that there are only a few quartic op-erators allowed by the SO (10) × S symmetry in the Higgs superpotential. Some of these(such as ( H a Ω bc + cyclic )) can be constructed by multiplying pairs of the invariant quadraticoperators H a , Ω ab , and ∆ ab ∆ ab (or by taking such products and contracting the gauge in-dices differently). In addition, there are the following five types of invariant quartic operators: O ≡ ( H a H b Ω ca Ω cb + cyclic ); O ≡ ( H a H b ∆ ab + cyclic ) and O ≡ ( H a H b ∆ ab + cyclic ); O ≡ (Ω ab Ω ac ∆ bc + cyclic ) and O ≡ (Ω ab Ω ac ∆ bc + cyclic ); O ≡ (∆ ab + cyclic ) and O ≡ (∆ ab + cyclic );and O ≡ (∆ ab ∆ bc ∆ ca H b + cyclic ) and O ≡ (∆ ab ∆ bc ∆ ca H b + cyclic ). The operators of type O ,5 , and O can exist with the product of bifundamentals being ΩΩ, ΩΩ ′ , or Ω ′ Ω ′ , as far as thesymmetry SO (10) × S is concerned; which of these is actually allowed in the superpotentialdepends on the Z parity assignments of the fields.There are only two types of quartic operators allowed by SO (10) × S in the Yukawasuperpotential, namely O Y ≡ ( ψ a ψ a H b Ω ab + cyclic ), O Y ≡ ( ψ a ψ b Ω ab ∆ ab + cyclic ). Again, inboth cases SO (10) × S permits such operators with either Ω or Ω ′ , whereas some will not beallowed by Z parity.These quartic operators, which may be induced either by Planck-scale physics or by in-tegrating out fields at the unification scale, have interesting consequences. Consider first theoperator O , which written out is ∆ ∆ ∆ H + ∆ ∆ ∆ H + ∆ ∆ ∆ H . The secondterm, which involves H , is interesting because it induces weak-scale SU (2) L -breaking VEVs inboth ∆ and ∆ .This happens as follows. If we write out this term in SO (10) notation, it has the form∆ ∆ ∆ H = (1 , , , , , , , , SU (5) × U (1)] notation [12],the first factor (∆ ) has a superlarge VEV in the (1 , , ) direction, the second factor (∆ )has a superlarge VEV in the (1 , , ) direction, the third factor (∆ ) has a superlargeVEV in the (1 − , − , ) direction, and the last factor ( H ) has a weak-scale VEV in the(1 , , − ) direction. Therefore, there is effectively a linear term for the (1 , , − ) componentof ∆ that arises from this product: (1 , , − ) · h (1 , , ) ih (1 − , − , ) ih (1 , , − ) i . Itis easy to see that this will induce a weak-scale VEV in this component. So we may write h ∆ (1 , , − ) i ∼ M W . Similarly there is a linear term for the (1 , , − ) component of the∆ coming from the product h (1 , , ) i · (1 , , − ) · h (1 − , − , ) ih (1 , , − ) i . So wemay write h ∆ (1 , , − ) i ∼ M W .These weak-scale VEVs in ∆ and ∆ are interesting, in turn, because they can contributeto quark and lepton masses if the operator O Y is present (in either its Ω or its Ω ′ form).If one examines the term ψ ψ Ω ′ ∆ = (1 , , , , , , , , , − , ) (1 , , ) h (1 , − , ) ih (1 , , − ) i . In SU (5) language, this is acontribution to a term of the form 5 h H i , i.e. the 10 of the third family times the 5 of thesecond family. Therefore this operator gives a 23 element of the charged-lepton mass matrix M L and a 32 element of the down-quark mass matrix M D . It does not contribute to any othercomponents of these matrices, and it does not contribute to the up-quark mass matrix. Thisis exactly the kind of entry that is needed in the so-called “lopsided” mass matrix models [13].If such entries are comparable to the 33 elements of M D and M L , then they explain the factthat the 2-3 mixing angle is large for the left-handed leptons (i.e. U µ ∼
1) but small for theleft-handed quarks (i.e. V cb ≪ M GUT rather than M P ℓ , there is no reason that these lopsided mass matrixelements necessarily have to be smaller than the 33 elements, even though the latter arise fromcubic terms. (Even with quartic terms generating the 23 and 13 entries, they may be comparableto the 33 entry if tan β is small.) It should be noted that in most published lopsided modelsthe operator that gives the lopsided entries ( M D ) and ( M L ) is such that these entries areequal in magnitude. It is important that they be at least approximately equal to reproduce thewell-known prediction that at the unification scale m b = m τ . Here, the operator ψ ψ Ω ′ ∆ does not make these entries equal but gives them a ratio ( M D ) / ( M L ) = a ′ /b ′ , whose valuedepends on the parameters in the Higgs superpotential. (See Eq. (5).) (If, instead of Ω ′ in thisoperator there were Ω, then the contribution to M L would vanish.)In the same way, it is easy to see that the weak-scale VEV of ∆ can generate contributionsto the 31 element of M D and the 13 element of M L . If these too are comparable to themagnitudes of the 33 elements, then a so-called “doubly lopsided” model results [10]. As hasbeen explained in the literature, such models can account for the so-called “bi-large” patternof neutrino mixing in a very simple way, and have other attractive features as well.One might expect the transposes of these lopsided mass matrix elements also to be inducedby these quartic terms (e.g. the 23 element of M D in addition to the 32 element, etc.). However,they are not. Nor are any off-diagonal elements of the up quark-mass matrices, induced untilone takes into account terms higher-order than quartic. This may well be related to the strongermass hierarchy observed among the up-type quarks. Indeed, in lopsided models, it is preciselythe absence of large lopsided terms in M U that is responsible for this. It is noteworthy thatin the lopsided models published in the literature the placement of the lopsided entries (forexample that they appear in the 32 elements but not the 12 elements, say) is to some extentcontrived with an eye to reproducing the observed pattern of masses and mixings. Here, it islargely dictated by the SO (10) × Z symmetry of the theory (and by the requirement thatonly one pair of Higgs doublets remains light).The operators O Y can also play an important role. The term ψ ψ H Ω ′ and the term ψ ψ H Ω ′ give 11 and 22 elements respectively to all the quark and lepton mass matrices.Note that the two terms ψ ψ H Ω ′ and ψ ψ H Ω ′ are related to each other by S . However,if S is broken spontaneously, either completely or down to a Z subgroup, the equality of the11 and 22 elements need not hold.The following shows at what level various elements in the quark and lepton mass matricesarise. A “3” means that such an element can arise even if only terms cubic and lower exist in W ; a “4” means that such an element can arise only if quartic (or higher) terms are present,and a “5” means that such an element can arise only if quintic (or higher) terms are present.7 U = , M D = , M L = . (9)The entries that are labelled “5” arise in a somewhat non-trivial way. Consider, for instance,the 12 and 21 elements of the mass matrices. The quintic term ∆ ∆ ∆ Ω H induces aweak-scale VEV in the component ∆ (5 − , , ) through the product (5 − , , ) ·h (1 , , ) ih (1 , − − ) ih (5 − , , ) ih (1 , , − ) i . This VEV then gives ( M D ) and ( M L ) via the quar-tic Yukawa operator ψ ψ Ω ∆ as follows: (10 , , )(1 , − , ) h (5 , − , ) ih (5 − , , ) i .The transposed elements ( M D ) and ( M L ) arise in a completely analogous way. (Just inter-change the labels 1 and 2 everywhere in the preceding discussion.) The 12 and 21 elements ofthe up quark mass matrix M U can come from a quintic Yukawa operator ψ ψ Ω Ω ∆ fromthe term (10 , , )(1 , , ) h (5 − , , ) ih (5 − , , ) ih (5 , − , ) i .The same quintic term ∆ ∆ ∆ Ω H , induces a weak-scale VEV for ∆ (5 − , , )through the product h (1 , , ) i · (5 − , , ) · h (1 , − , − ) ih (5 − , , ) ih (1 , , − ) i . ThisVEV then induces ( M D ) and ( M L ) through the quartic term that was mentioned beforeas giving ( M D ) and ( M L ) , namely ψ ψ Ω ′ ∆ . In particular, this contains the product(10 , , )(1 , , − ) h (5 , , − ) ih (5 − , , ) i . The entries ( M D ) and ( M L ) arise in asimilar way. (Just interchange the indices 1 and 2 in the foregoing discussion.)What remains is to show that the 13, 31, 23, and 32 elements of M U can arise from quinticterms. The elements ( M U ) and ( M U ) arise from the quintic term ψ ψ Ω Ω ∆ , whichcontains the product (10 , , )(1 , , ) h (5 , , − ) ih (5 , , − ) ih (1 − , , ) i . The ele-ments ( M U ) and ( M U ) arise in a similar way.One sees, then, that the requirements of SO (10) × S symmetry imply that the quark andlepton mass matrices have a non-trivial structure that contains several promising features: (i) ahierarchy among the mass matrix elements, (ii) only one family obtaining mass at lowest order,(iii) a qualitative difference between the up quark mass matrix and the other mass matrices (inparticular some of the elements of M U arise at higher order than the corresponding elements of M D and M L , which is perhaps related to the stronger hierarchy observed among the up-typequarks); (iv) relatively large off-diagonal elements in the third row of M D and third column of M L , i.e. the “doubly lopsided” pattern that is known to explain in a simple way the bilargepattern of neutrino mixing; (v) “Clebsches” in certain elements of M D and M L that may allowan explanation of the well-known Georgi-Jarlskog relations. Still, the construction of a completemodel with fully realistic quark and lepton mass matrices has not been attempted here.There are several issues that would have to be faced in constructing a fully realistic modelbased on SO (10) × S . The most difficult would be proton decay via the d = 5 operators8hat arise from the exchange of colored Higgsinos. The simple structure in Eq. (7) leads tono suppression of such decay amplitudes. It seems likely, however, that with more than twotypes of bifundamental Higgs fields adequate suppression may be achieved. Another issue is theexistence of Landau poles above the unification scale (i.e. the SO (10) × S scale) due to thelarge number of fields in the bispinor and bifundamental Higgs mutiplets. These issues requirefurther study. Acknowledgments
This work is supported in part by the DOE Grant
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