Universal Texture of Quark and Lepton Mass Matrices and a Discrete Symmetry Z3

PHYSICAL REVIEW D 66, 093006 ͑2002͒ Universal texture of quark and lepton mass matrices and a discrete symmetry Z3 Yoshio Koide* Department of Physics, University of Shizuoka, 52-1 Yada, Shizuoka, 422-8526, Japan Hiroyuki Nishiura Department of General Education, Junior College of Osaka Institute of Technology, Asahi-ku, Osaka, 535-8585, Japan Koichi Matsuda, Tatsuru Kikuchi, and Takeshi Fukuyama Department of Physics, Ritsumeikan University, Kusatsu, Shiga, 525-8577, Japan ͑Received 27 September 2002; published 19 November 2002͒ Recent neutrino data have been favorable to a nearly bimaximal mixing, which suggests a simple form of the neutrino mass matrix. Stimulated by this matrix form, the possibility that all the mass matrices of quarks and leptons have the same form as in neutrinos is investigated. The mass matrix form is constrained by a discrete symmetry Z3 and a permutation symmetry S2. The model, of course, leads to a nearly bimaximal mixing for the lepton sectors, while, for the quark sectors, it can lead to reasonable values of the CKM mixing matrix and masses. DOI: 10.1103/PhysRevD.66.093006 PACS number͑s͒: 12.15.Ff, 11.30.Hv, 14.60.Pq I. INTRODUCTION for the neutrino mass matrix in Refs. ͓2–7͔, using the basis where the charged-lepton mass matrix is diagonal, motivated Recent neutrino oscillation experiments ͓1͔ have highly by the experimental finding of maximal ␯␮Ϫ␯␶ mixing ͓1͔. 2 ␪ ϳ 2 ␪ suggested a nearly bimaximal mixing (sin 2 12 1,sin 2 23 In this paper, we consider that this structure is fundamental Ӎ ϵ⌬ 2 ⌬ 2 ϳ Ϫ2 1) together with a small ratio R m12/ m23 10 . for both quarks and leptons, although it was speculated from This can be explained by assuming a neutrino mass matrix the neutrino sector. Therefore, we assume that all the mass form ͓2–7͔ with a permutation symmetry between second matrices have this structure. and third generations. We think that quarks and leptons Let us look at the universal characters of the model. Here- should be unified. It is therefore interesting to investigate the after, for brevity, we will omit the flavor index. The eigen- ͑ ͒ possibility that all the mass matrices of the quarks and lep- masses mi of Eq. 1.2 are given by tons have the same matrix form, which leads to a nearly ϭ 1 bimaximal mixing and U13 0 in the neutrino sector, against Ϫm ϭ BϩCϪͱ8A2ϩ͑BϩC͒2 , ͑1.3͒ the conventional picture that the mass matrix forms in the 1 2 „ … quark sectors will take somewhat different structures from those in the lepton sectors. In the present paper, we will 1 m ϭ BϩCϩͱ8A2ϩ͑BϩC͒2 , ͑1.4͒ assume that the mass matrix form is invariant under a dis- 2 2 „ … crete symmetry Z3 and a permutation symmetry S2. Phenomenologically, our mass matrices M u, M d, M ␯ and m ϭBϪC. ͑1.5͒ ͓ 3 M e mass matrices of up quarks (u,c,t), down quarks ␯ ␯ ␯ ␮ ␶ (d,s,b), neutrinos ( e , ␮ , ␶) and charged leptons (e, , ), The texture’s components of Mˆ are expressed in terms of respectively͔ are given as follows: eigenmasses mi as ϭ † ˆ ͑ ͒ M f PLfM f PRf , 1.1 m2m1 Aϭͱ , 2 with 1 m Ϫm 0 A A ϭ ͩ ϩ 2 1 ͪ ͑ ͒ f f B m 1 , 1.6 2 3 m ˆ ϭ ͑ ϭ ␯ ͒ ͑ ͒ 3 M f ͩ A f B f C f ͪ f u,d, ,e , 1.2 A C B 1 m Ϫm f f f ϭϪ ͩ Ϫ 2 1 ͪ C m3 1 . 2 m3 where PLf and PRf are the diagonal phase matrices and A f , B f , and C f are real parameters. Namely the components are That is, Mˆ is diagonalized by an orthogonal matrix O as ˆ different in M f , but their mutual relations are the same. This Ϫ structure of mass matrix was previously suggested and used m1 00 T O Mˆ Oϭͩ 0 m2 0 ͪ , ͑1.7͒ *On leave at CERN, Geneva, Switzerland. 00m3 0556-2821/2002/66͑9͒/093006͑11͒/$20.0066 093006-1 ©2002 The American Physical Society KOIDE et al. PHYSICAL REVIEW D 66, 093006 ͑2002͒ with 1 ␻2 ␻2 ␻2 ␻␻ cs0 ͩ ͪ , ͑2.2͒ ␻2 ␻␻ s c 1 Ϫ Ϫ ͱ ͱ ͱ Oϵͩ 2 2 2 ͪ . ͑1.8͒ where s c 1 Ϫ u ␯ ͱ ͱ ͱ ϭͩ L ͪ ᐉ ϭͩ L ͪ ͑ ͒ 2 2 2 qL , L Ϫ . 2.3 dL eL Here c and s are defined by ͑ ͒ Therefore, if we assume two SU 2 doublet Higgs scalars H1 and H2, which are transformed as m2 m1 cϭͱ , sϭͱ . ͑1.9͒ !␻ !␻2 ͑ ͒ ϩ ϩ H1 H1 , H2 H2 , 2.4 m2 m1 m2 m1 the Yukawa interactions are given as follows: It should be noted that the elements of O are independent of ˆ m3 because of the above structure of M. ϭ ͑ u ¯ ˜ ϩ d ¯ ͒ Hint ͚ Y (A)ijqLiHAuRj Y (A)ijqLiHAdRj The zeros in this mass matrix are constrained by the dis- Aϭ1,2 crete symmetry that is discussed in the next section, defined ͑ at a unification scale the scale does not always mean ‘‘grand ϩ ͑ ␯ ¯ᐉ ˜ ␯ ϩ e ¯ᐉ ͒ ͒ ͚ Y (A)ij LiHA Rj Y (A)ij LiHAeRj unification scale’’ . This discrete symmetry is broken below Aϭ1,2 ␮ϭ M R , at which the right-handed neutrinos acquire heavy ϩ͑ R ¯␯c ⌽˜ 0␯ ϩ R ¯␯c ⌽0␯ ͒ϩ Majorana masses, as we discuss in Sec. IV. Therefore, the Y (1)ij Ri Rj Y (2)ij Ri Rj H.c., ͑ ͒ matrix form 1.1 will, in general, be changed by renormal- ͑ ͒ ization group equation ͑RGE͒ effects. Nevertheless, we 2.5 ͑ ͒ would like to emphasize that we can use the expression 1.1 where with ͑1.2͒ for the predictions of the physical quantities in the low-energy region. This will be discussed in the Appendix. ϩ ¯ 0 HA HA This article is organized as follows. In Sec. II we discuss H ϭͩ ͪ , H˜ ϭͩ ͪ , ͑2.6͒ A H0 A Ϫ Ϫ the symmetry property of our model. Our model is realized A HA when we consider two Higgs doublets in each up-type and so that down-type quark ͑lepton͒ mass matrices. The quark mixing matrix in the present model is argued in Sec. III. In Sec. IV, 000 the lepton mixing matrix is analyzed. Section V is devoted to a summary. u d ␯ e R ϭ 0** Y (1) ,Y (2) ,Y (1) ,Y (2) ,Y (2) ͩ ͪ , 0** II. Z SYMMETRY AND MASS MATRIX FORM 3 ͑2.7͒ We assume a permutation symmetry between second and 0** ␯ third generations, except for the phase factors. However, the Y u ,Y d ,Y ,Y e ,Y R ϭͩ *00ͪ . (2) (1) (2) (1) (1) ˆ ϭ condition (M f )11 0 cannot be derived from such a symme- *00 try. Therefore, in addition to the 2$3 symmetry, we assume a discrete symmetry Z3, under which symmetry the quark In Eq. ͑2.7͒, the symbol * denotes nonzero quantities. Here, ␺ ¯ and lepton fields L , which belong to 10L ,5L and 1L of in order to give heavy Majorana masses of the right-handed ͑ ͒ ϭ␯c ␯ ͑ ͒ SU 5 (1L ), are transformed as neutrinos R , we have assumed an SU 2 singlet Higgs sca- R ⌽0 lar , which is transformed as H1. ␺ !␺ f 1L 1L , In the present model, the phase difference arg(Y (1) ϩ f Ϫ f ϩ f Y (2))21 arg(Y (1) Y (2))31 plays an essential role. There- ␺ !␻␺ ͑ ͒ fore, for the permutation symmetry S , we set the following 2L 2L , 2.1 2 assumption: the permutation symmetry can be applied to ␺ !␻␺ only the special basis that all Yukawa coupling constants are 3L 3L , ͑ real. Of course, for the Z3 symmetry, such an assumption is not required.͒ We consider that the phase factors are caused where ␻3ϭϩ1. ͓Although we use a terminology of SU͑5͒, ͑ ͒ ͔ by an additional mechanism after the requirement of the per- at present, we do not consider the SU 5 grand unification. ͑ mutation symmetry S2 after the manifestation of the linear Then, the bilinear terms ¯q u , ¯q d , ¯ᐉ ␯ , ¯ᐉ e and ϩ Li Rj Li Rj Li Rj Li Rj combination Y (1) Y (2)). In the present paper, we consider ¯␯c ␯ ͓␯c ϭ ␯ cϭ ¯␯ T ¯␯c ϭ¯␯c ͔ Ri Rj R ( R) C R and R ( R) are transformed that although the Z3 symmetry is rigorously defined for the ͑ ͒ as follows: fields by Eq. 2.1 , the permutation symmetry S2 is a rather 093006-2 UNIVERSAL TEXTURE OF QUARK AND LEPTON MASS... PHYSICAL REVIEW D 66, 093006 ͑2002͒ phenomenological one ͑i.e., ansatz͒ for the mass matrix are diagonalized by the biunitary transformation shape. Then, under this S2 symmetry, the general forms of ϵ f ϩ f ϭ † ͑ ͒ Y f Y (1) Y (2) are given by D f ULfM f URf , 3.2 ϭ † ˆ ϵ † ϵ Y f PLfY f PRf where ULf P f O f , URf P f O f , and Od (Ou) is given by Eq. ͑1.8͒. Then, the Cabibbo-Kobayashi-Maskawa ͑CKM͒ Ϫi(␦ f Ϫ␦ f ) Ϫi(␦ f Ϫ␦ f ) 0 ae L1 R2 ae L1 R3 ͓10͔ quark mixing matrix ͑V͒ is given by Ϫi(␦ f Ϫ␦ f ) Ϫi(␦ f Ϫ␦ f ) Ϫi(␦ f Ϫ␦ f ) ϭͩ ae L2 R1 be L2 R2 ce L2 R3 ͪ .

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