Embedding A 4 into SU (3) × U (1) flavor symmetry: large neutrino mixing and fermion mass hierarchy in the SO (10) GUT

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  We present a common explanation of the fermion mass hierarchy and the large lepton mixing angles in the context of a grand unified flavor and gauge theory (GUTF). Our starting point is a SU(3)xU(1) flavor symmetry and a SO(10) GUT, a basic ingredient
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    a  r   X   i  v  :   0   8   0   2 .   1   6   9   3  v   1   [   h  e  p  -  p   h   ]   1   2   F  e   b   2   0   0   8 Embedding  A 4  into  SU  (3) × U  (1)  flavor symmetry:Large neutrino mixing and fermion mass hierarchy in  SO (10)  GUT F. Bazzocchi 1 , S. Morisi 1 , M. Picariello 2 , E. Torrente-Lujan 3 1 Instituto de F´ısica Corpuscular – C.S.I.C./Universitat de Val`encia E–46071 Val`encia, Spain  2 Dipartimento di Fisica - Universit`a del Salento  and  INFN Via Arnesano, ex collegio Fiorini, I–73100 Lecce, Italy and  3 University of Murcia - 30100 Murcia - Spain  e-mails: bazzocchi@ific.uv.es, morisi@ific.uv.es, Marco.Picariello@le.infn.it, torrente@cern.chWe present a common explanation of the fermion mass hierarchy and the large lepton mixingangles in the context of a grand unified flavor and gauge theory (GUTF). Our starting point is a( SU  (3) × U  (1)) F  flavor symmetry and a  SO (10) GUT, a basic ingredient of our theory which playsa major role is that two different breaking pattern of the flavor symmetry are at work. On one side,the dynamical breaking of ( SU  (3) × U  (1)) F  flavor symmetry into ( U  (2) × Z  3 ) F  explains why onefamily is much heavier than the others. On the other side, an explicit symmetry breaking of   SU  (3) F  into a discrete flavor symmetry leads to the observed tribimaximal mixing for the leptons. We writean explicit model where this discrete symmetry group is  A 4 . Naturalness of the charged fermionmass hierarchy appears as a consequence of the continuous  SU  (3) F  symmetry. Moreover, the samediscrete  A 4 -GUT invariant operators are the root of the large lepton mixing, small Cabibbo angle,and neutrino masses. I. INTRODUCTION Grand Unified Theory (GUT) [1, 2] are natural extensions of the Standard Model (SM) Indications toward GUT are the tendency to unify for the gauge couplings, and the possibility to explain charge quantization and anomalycancellation. One of the main features of GUT is its potentiality to unify the particle representations and thefundamental parameters in a hopefully predictive framework.  SO (10) is the smallest simple Lie group for which asingle anomaly-free irreducible representation (namely the spinor 16 representation) can accommodate the entire SMfermion content of each generation.Flavor physics appears as new extra horizontal symmetries. After the recent experimental evidences about neutrinophysics [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], within the experimental errors, the neutrino mixing matrix is compatible with the so called tri-bimaximal matrix [15] U  TB  =  − 2 / √  6 1 / √  3 01 / √  6 1 / √  3 1 / √  21 / √  6 1 / √  3  − 1 / √  2  .  (1)At this stage the parameters both the quark [16] and lepton [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28] sectors are known to a comparable level.To explain at the same moment the charged fermion mass hierarchy and the lepton-quark mixing angle hierarchy isan unsolved problem, this is the flavor puzzle. The problem of the mass hierarchy is often addressed by introducingcontinuous flavor symmetries [29, 30]. On the other hand, discrete flavor symmetry such as 2-3 [31, 32, 33],  S  3[34, 35, 36, 37],  A 4  [38, 39, 40, 41], or other symmetries [42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53], where introduced to explain large lepton mixing angles, but in that case mass hierarchy remains unexplained.A milestone in these studies has been the discovery that mass hierarchies and mixing angles can be not directlycorrelated among them in the flavor symmetry breaking [36, 54]. Fundamental steps in the realization of these ideas are given in [38, 39]. These new ingredients allow us to escape from the no-go theorem [55] that seems to indicate  2that a maximal mixing angle  θ 23  can never arise in the symmetric limit of whatever flavor symmetry (global or local,continuous or discrete), provided that such a symmetry also explains the hierarchy among the fermion masses and isonly broken by small effects, as we expect for a meaningful symmetry.In fact, in our theory, the mass hierarchy and large mixing angle are not srcinated at the same step in the symmetrybreaking pattern.Our final aim would be the construction of a grand unified  SO (10)-like model where masses and mixing angles aregenerated by the flavor and gauge symmetry breaking.We presented a viable  SO (10) model with discrete flavor symmetry in [38]. There we generated the observed leptonmixing but we fitted the fermion masses by assuming the group  A 4  as flavor symmetry and the “constrain” of assigningright and left-handed fermion fields to the same representations. Indeed, we showed in [38] that the assignment of both left-handed and right-handed SM fields to triplets of   A 4 , that is therefore compatible with  SO (10), can lead tothe charged fermion textures proposed in [56] and given by M  f   =  h f  0  h f  1  h f  2 h f  2  h f  0  h f  1 h f  1  h f  2  h f  0  ,  (2)with  h f  0 , h f  1  and  h f  2  distinct parameters. In [38], in order to obtain a mass matrix of the form of   M  f   in eq. (2) withoutspoiling the predictions of the neutrino sector, we introduced higher order operators containing simultaneously a setof   SO (10) representations  45 . The lepton mixing was naturally generated by the breaking pattern of   A 4 , while thefermion masses were obtained with a possible tuning in the flavor parameters not constrained by the symmetries.We addressed the problem of the fine tuning in [39] where the  A F  4  flavor discrete symmetry is embedded into( SO (3) L  × SO (3) R ) F  . In that way we explicitly disentangled the mixing problem from the hierarchy one. We brokethe continuous flavor ( SO (3) L  ×  SO (3) R ) F  symmetry both dynamically and explicitly. The two breaking termsproduced the charged fermion hierarchies on one hand and solved the leptonic mixing problem on the other hand. Inthis way not only a tribimaximal neutrino mixing was naturally generated but also the charged fermion hierarchiesby dynamically breaking of the continuous left-right flavor symmetry. Finally the Cabibbo angle was obtained bytaking into account higher order operators. However the left-right flavor group symmetry ( SO (3) L  × SO (3) R ) F  of [39] is not compatible with a grand unified gauge group, like  SO (10), with all the fermions of one family in the samerepresentation, because in left-right flavor symmetries the fermions of one family belong to different representationsof the flavor group.In this paper we merge all these ingredients together and we are able to construct a non renormalizable model withgrand unified gauge group  SO (10) and with an extended flavor symmetry ( SU  (3) × U  (1)) F  . In this new model boththe tribimaximal lepton mixing matrix and the hierarchy among the mass of the 3rd and the other fermion familiesnaturally appear from the symmetry breaking pattern. Our model is non renormalizable, however a renormalizableversion of it can be easily constructed because the particular structure of the operators introduced here. For thispurpose viable methods are well known, i.e. by integrated out given heavy extra fields [57].Our effective  SO (10) invariant Lagrangian is L  =  L SU  (3) F   + δ  L A 4  ,  (3)where L SU  (3) F   is  SO (10) × ( SU  (3) × U  (1)) F  invariant and  δ  L A 4  is the explicit breaking term of the  SU  (3) F  symmetrythat, at this level, leaves  SO (10) unbroken. The charge assignment of the fields is such that the  SU  (3) F  invariantoperator with lowest mass dimensions is only [38] L SU  (3) F   =  h 0 161045 A 45 B  16 Φ ,  (4)where Φ, singlet of   SO (10), transforms as  6  with respect to  SU  (3) F  . The scalar fields  10 ,  45 A  and  45 B  are singletsof   SU  (3) F  . As noticed in [38], thanks to the two  45 s scalar fields, the operator in eq. (4) can give no contribution  3to the neutrino sector for some set of the  45 s vev (i.e. in the explicit model of  [38] one vev is proportional to theright-handed isospin  T  3 R  and the other one to the hypercharge  Y  , here the directions  A  and  B  are different but stillwith the same property).When the scalar field Φ develops a vev in the direction   Φ ij  = 1( ∀ i,j ) we obtain a democratic mass matrices forall the charged fermions, that gives a massive 3rd family and two massless families. The democratic structure of thecharged fermion mass matrices avoids the fine tuning needed to explain the mass hierarchy between the 3rd and theother two families that is usually needed in presence of charged fermion mass matrices of the form of eq. (2). Thedemocratic mass matrices preserves the ( U  (2) × Z  3 ) F  subgroup of ( SU  (3) × U  (1)) F  that leaves invariant the (1 , 1 , 1)vector in the flavor space. Therefore at this stage only one mixing angle can be generated.The neutrino mass matrix and the first and second families masses arise when we switch on the explicitly breakingterms of   SU  (3) F  into  A 4 . If we neglect the ordering problem of the  45 s and the possibility to have more than oneflavon for each operator, the most general Lagrangian invariant under the flavor structure of the theory is δ  L A 4  =  h ijk  φ k 16 i 10 45 A  45 B  45 C   45 D  16 j + h ′ ijk  φ k 16 i 45 C   45 D  10 45 A  45 B  16 j + (5) h ′′ ijk ˜ φ k 16 i 10 45 C   45 D  16 j + g  16 i 126 16 i ζ  S   + g ′ ijk  16 i 126 16 j ζ  kT  where the indices  { i,j,k,l }  are  A 4 , subgroup of   SU  (3), indeces and the sum over the gauge indices is understood.The scalar field  126  is a singlet  1 ′  of   A 4 , while the  45 C  , and  45 D  are other scalars that transform as  45  of   SO (10),and are singlets of   A 4 . The flavon fields  φ , ˜ φ ,  ζ  T   are triplets under  A 4 , while  ϕ  and  ζ  S   are singlets.As found in [38] the terms in the second line of   δ  L A 4  generates the light neutrino mass matrix. The terms in thefirst two lines in  δ  L A 4  gives a contribution to the mass matrices that has the nice properties to commute with theleading order term obtained from eq. (3).After the breaking of   A 4 , it generates the first and second family masses and fix the mixing matrix in the leptonsector to be tribimaximal.The plan of the paper is as follow. First, in sec.  II  we introduce the basic ingredient of the model, i.e. thegeneral structure of the symmetry breaking, all the involved fields and how they transform under the gauge and flavorsymmetries. Then in sec.  III  we show how the 3rd family masses are generated via the breaking ( SU  (3) × U  (1)) F  into ( U  (2) × Z  3 ) F  , how the 1st and 2nd family masses are generated together with maximal mixings in the leptonsector, and how the neutrino masses are generated with a resulting tribimaximal mixing matrix in the lepton sector.Finally we show how the Cabibbo angle is naturally generated without the introduction of new operators. Finally insec.  V  we report our conclusions. II. BASIC INGREDIENTS Let us first investigate the field content of the theory and the flavor charges. We report the field content of ourmodel in Table (I). With our charge assignment, the only allowed operators of lower mass dimensions are given ineqs. (4-5), if we neglect the ordering problem of the  45  and the possibility to have more than one flavon for eachoperator. In this sense our Lagrangian is the most general one invariant under the flavor structure of the theory.Moreover, independently from the fact that nature prefer a dominant seesaw of type I (i.e. heavy Majorana right-handed neutrino mass and intermediate Dirac neutrino mass) or of type II (i.e. light Majorana left-handed neutrinomass) or a mixed scenario, the transformation properties of the  ζ  S   must be assumed to be  1 ′ , as we will explain insec.  IVB .In our opinion, the ordering problem can be related to a deeper structure of the theory, for example its version as arenormalizable model, and we will not investigate further it here. However the fact that will not be possible to expressthe directions  A  and  B  as rational combinations of   C   and  D , together with the fact that  45  appears only as couples  4( 45 A , 45 B ) and ( 45 C  , 45 D ) seems to us to indicate that the right representations to introduce are the irreducible partof the  2025  that can get a vev diagonal over the  16  matter fields with charges  AB  and  CD . If this is the case we arereally including all the allowed operator and there is not any more an ordering problem.After symmetry breaking, once the Higgs acquire vevs, the quadratic part for the fermions of the Lagrangian ineqs. (4-5) can be rewritten in a compact form, i.e. with an abuse of notation in the  SO (10) contractions, as L Dirac  =  h 0  ( 16 1 16 ′ 1  + 16 2 16 ′ 2  + 16 3 16 ′ 3 )  v 10  + (6a)+  h 1  ( 16 1 16 ′′′ 2  + 16 2 16 ′′′ 3  + 16 3 16 ′′′ 1  ) + h 2  ( 16 1 16 ′′′ 3  + 16 2 16 ′′′ 1  + 16 3 16 ′′′ 2  )  v 10  v φ  (6b)+  h ′ 1  ( 16 ′′ 1 16 ′ 2  + 16 ′′ 2 16 ′ 3  + 16 ′′ 3 16 ′ 1 ) + h ′ 2  ( 16 ′′ 1 16 ′ 3  + 16 ′′ 2 16 ′ 1  + 16 ′′ 3 16 ′ 2 )  v 10  v φ  (6c)+  h ′′ 1  ( 16 1 16 ′′ 2  + 16 2 16 ′′ 3  + 16 3 16 ′′ 1 ) + h ′′ 2  ( 16 1 16 ′′ 3  + 16 2 16 ′′ 1  + 16 3 16 ′′ 2 )  v 10  v ˜ φ  (6d)+  g ( 16 1 16 1  + 16 2 16 2  + 16 3 16 3 )  v 126  v ζ  S  +  g ′ 1 16 1 16 2  + g ′ 2 16 2 16 1  v 126  v ζ  T   (6e)where we have assumed that the two  A 4 - 3 plets  φ  and ˜ φ  acquire vev in the (1 , 1 , 1) direction of   A 4 , while the  ζ  T   vevis in the direction (0 , 0 , 1). In eqs. (6) we introduced 16 ′′′ i  ≡  v 45 A  v 45 B  v 45 C  v 45 D  16 i ,  16 ′′ i  ≡  v 45 C  v 45 D  16 i ,  (7) 16 ′ i  ≡ v 45 A  v 45 B  16 i ,  with  i  = 1 , 2 , 3 . We obtain the following expression by absorbing the vevs of the  45 s into the coupling constants 16 ′  =  x ′ Q  Q, x ′ U   U  c , x ′ D  D c , x ′ L  L, x ′ E   E  c , x ′ N   N  c  ,  (8a) 16 ′′  =  x ′′ Q  Q, x ′′ U   U  c , x ′′ D  D c , x ′′ L  L, x ′′ E   E  c , x ′′ N   N  c  ,  (8b) 16 ′′′  =  x ′′′ Q  Q, x ′′′ U   U  c , x ′′′ D  D c , x ′′′ L  L, x ′′′ E   E  c , x ′′′ N   N  c   (8c)where  x ′ f  ,  x ′′ f  , and  x ′′′ f   are the quantum numbers respectively of the product of the charges  A  and  B , of the productof the charge  C   and  D , and of the product of the charges  A ,  B ,  C  , and  D . In particular we notice that x ′′′ f   =  x ′′ f  x ′ f   .  (9)We report the charges of each fermion in Table (II). III. DYNAMICAL BREAKINGA.  ( SU  (3) × U  (1)) F  → ( U  (2) × Z  3 ) F  gives the charged fermion 3rd family masses We assume that the Φ  SU  (3) F  - 6 plet field acquire a vev   Φ ij   =  v Φ  ( ∀ i,j ). In this case the charged fermion massmatrix obtained is the so-called democratic mass matrix [58] given by M  0 f   =  m f  3 3  1 1 11 1 11 1 1  ,  (10)where m U  3  =  v Φ ( x ′ U   + x ′ Q ) v U  10 h 0 ,  (11a) m D 3  =  v Φ ( x ′ D  + x ′ Q ) v D 10 h 0 ,  (11b) m N  3  =  v Φ ( x ′ N   + x ′ L ) v U  10 h 0 ,  (11c) m E  3  =  v Φ ( x ′ E   + x ′ L ) v D 10 h 0 .  (11d)  5This matrix has only one eigenvalue different from zero,  m f  3 , and can be assumed to be the mass of the 3rd family.To avoid any non diagonal contribution to the Dirac neutrino mass matrix we impose x ′ N   + x ′ L  = 0 .  (12a)To have the bottom-tau unification, we must impose also x ′ L  + x ′ E   =  x ′ Q  + x ′ D  .  (12b)The unitary matrix U   that diagonalizes the symmetric matrix  M  0 f   has one angle and the three phases undeterminated.One possible parametrization is given by [39] U   = 1 √  3  √  2cos θe iα  √  2sin θe i ( β + γ  ) 1 − e iα ( cos θ √    2  +   32  sin θe − iγ  )  e iβ (   32  cos θ −  1 √  2  sin θe iγ  ) 1 − e iα ( cos θ √    2  −   32  sin θe − iγ  )  − e iβ (   32  cos θ  +  1 √  2  sin θe iγ  ) 1  .  (13)The freedom in the  U   matrix shows the remaining flavor symmetry  U  (2) F  . The unknow angle and phases are fixedonly after breaking the democratic structure of   M  0 f  , i.e. the  U  (2) F  flavor symmetry, with a small perturbation  δM  f  ,i.e. M  f   =  M  0 f   + δM  f   . The effect of   δM  f   is to give a small mass to the first and second family and to fix the mixing angles. To have thatthe mixing matrix diagonalizing the full  M  f   belongs to the families of matrix of eq. (13), we must require that  δM  f  commute with  M  0 f  . This has the nice consequence that we have automatically the selection of the breaking patternof   A 4  into  Z  3 . IV. EXPLICITLY BREAKING  SU  (3) F  → A 4 We will assume the presence of an hidden scalar sector that breaks spontaneously the continuous  SU  (3) F  into thediscrete  A 4 . Under this hypothesis it is quite natural to assume that the explicit breaking terms to be added to theLagrangian are small. A.  A 4  → Z  3  generates the charged fermions 1st and 2nd family masses and mixing When the  φ  and ˜ φ A 4 - 3 plets take vev as   ˜ φ  ∝  φ   =  v φ  (1 , 1 , 1) we have new contributions to the mass matrices.I.e., for the charged leptons we get the operator δ  E ijk  ǫ αβ  H  αd  L βi  E  j  φ k  →  ǫ αβ  H  αd  δ  E  1  ( L β 2 E  3  + L β 3 E  1  + L β 1 E  2 ) + δ  E  2  ( L β 3 E  2  + L β 1 E  3  + L β 2 E  1 )  v φ  ,  (14)where the two  δ  E i  arise by the two different contractions of   A 4 . The value of   δ  E  s can be read from the Lagrangian ineq. (6) and is δ  E  1  = ( h 1 x ′′′ E   + h 2 x ′′′ L ) + ( h ′ 1 x ′′ L x ′ E   + h ′ 2 x ′ L x ′′ E  ) + ( h ′′ 1 x ′′ E   + h ′′ 2 x ′′ L ) ,  (15) δ  E  2  = ( h 2 x ′′′ E   + h 1 x ′′′ L ) + ( h ′ 2 x ′′ L x ′ E   + h ′ 1 x ′ L x ′′ E  ) + ( h ′′ 2 x ′′ E   + h ′′ 1 x ′′ L ) .  (16)Because the  SO (10) unification, the operators in the up, down and neutrino sectors have similar expressions. Inparticular for the contributions to the Dirac neutrino mass matrix we have δ  N  1  = ( h 1 x ′′′ N   +  h 2 x ′′′ L ) + ( h ′ 1 x ′′ L x ′ N   + h ′ 2 x ′ L x ′′ N  ) + ( h ′′ 1 x ′′ N   + h ′′ 2 x ′′ L ) ,  (17) δ  N  2  = ( h 2 x ′′′ N   +  h 1 x ′′′ L ) + ( h ′ 2 x ′′ L x ′ N   + h ′ 1 x ′ L x ′′ N  ) + ( h ′′ 2 x ′′ N   + h ′′ 1 x ′′ L ) .  (18)
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