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 ͑ ͒ ϭc ͑ ͒ 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 ͪ . ϭ † ϭ T † V ULuULd Ou PuPdOd Ϫi(␦ f Ϫ␦ f ) Ϫi(␦ f Ϫ␦ f ) Ϫi(␦ f Ϫ␦ f ) ae L3 R1 ce L3 R2 be L3 R3 ϩ Ϫ Ϫ cucd susd cusd sucd su ͑2.8͒ Ϫ ϩ ϭͩ sucd cusd susd cucd cu ͪ , ͑3.3͒ Ϫ We have already assumed that L sd cd ϭ c c c c ( ,eL ,dR ;uL ,dL ,uR ,eR ; R) have the same transforma- ͑ ͒ ¯ c where and are defined by tion 2.1 under the discrete symmetry Z3, so that Li Lj are ͑ ͒ transformed as Eq. 2.2 . From this analogy, we assume that ␦ Ϫ␦ ␦ ϩ␦ 1 ␦ ␦ 3 2 3 2 the phase matrices PLf and PRf come from the replacement ϭ ͑ei 3ϩei 2͒ϭcos exp iͩ ͪ , ͑3.4͒ → 2 2 2 L P f L , i.e.
c † † c ¯ Y →¯ P Yˆ P . ͑2.9͒ 1 ␦ ␦ L f L L f f f L ϭ ͑ei 3Ϫei 2͒ 2 However, differently from the transformation ͑2.1͒,wedo ␦ Ϫ␦ ␦ ϩ␦ not assume in Eq. ͑2.9͒ that all the phase matrices P are 3 2 3 2 f ϭsin exp iͩ ϩ ͪ . ͑3.5͒ identical, but we assume that they are flavor dependent. This 2 2 2 explains the assumption ␦ ␦ ␦ ϵ †ϵ i 1 i 2 i 3 Here we have set P PuPd diag(e ,e ,e ), and we ␦ f ϭϪ␦ f ϵ␦ f , ͑2.10͒ ␦ ϭ Li Ri i have taken 1 0 without loss of generality. Then, the explicit magnitudes of the components of V are ͑ ͒ ͓ ͑ ͒ in the expression 2.8 . However, this assumption 2.10 is expressed as not essential for the numerical predictions in the present pa- per, because the predictions of the physical quantities depend ␦ Ϫ␦ 3 2 on only the phases ␦ f .͔ sin Li 2 Since the present model has two Higgs doublets horizon- ͉V ͉ϭ͉͉c ϭ , ͑3.6͒ cb u ͱ ϩ tally, in general, flavor-changing neutral currents ͑FCNCs͒ 1 mu /mc are caused by the exchange of Higgs scalars. However, this ␦ Ϫ␦ FCNC problem is a common subject to be overcome not 3 2 only in the present model but also in most models with two sin 2 mu Higgs doublets. The conventional mass matrix models based ͉V ͉ϭ͉͉s ϭ ͱ , ͑3.7͒ ub u ͱ ϩ m on a grand unified theory ͑GUT͒ scenario cannot give real- 1 mu /mc c istic mass matrices without assuming more than two Higgs ␦ Ϫ␦ scalars ͓8͔. Besides, if we admit that two such scalars remain 3 2 sin until the low energy scale, the well-known beautiful coinci- 2 dence of the gauge coupling constants at ϳ1016 GeV will ͉V ͉ϭ͉͉c ϭ , ͑3.8͒ ts d ͱ ϩ be spoiled. Although the present model is not based on a 1 md /ms GUT scenario, as are the conventional mass matrix models, ␦ Ϫ␦ for the FCNC problem, we optimistically consider that only 3 2 sin one component of the linear combinations among those 2 md ϭ ͉V ͉ϭ͉͉s ϭ ͱ , ͑3.9͒ Higgs scalars survives at the low energy scale mZ , while td d ͱ ϩ m Ͻ ͓ ͔ 1 md /ms s the other component is decoupled at M X 9 . The study of the renormalization group experiment ͑RGE͒ effects given s c in the Appendix will be based on such an ‘‘effective’’ one- ͉ ͉ϭ ͯ Ϫ u dͯ Vus cusd 1 Higgs scalar scenario. cu sd m m III. QUARK MIXING MATRIX ϭͱ c ͱ d ϩ ϩ mc mu ms md The quark mass matrices ␦ Ϫ␦ ␦ ϩ␦ m m ϫͫ Ϫ 3 2 3 2ͱ u s ϭ † ˆ †͑ ϭ ͒ ͑ ͒ 1 2 cos cos M f P f M f P f f u,d , 3.1 2 2 mcmd
093006-3 KOIDE et al. PHYSICAL REVIEW D 66, 093006 ͑2002͒
␦ Ϫ␦ 1/2 The heavy-quark-mass-independent predictions ͑3.13͒ 3 2 mums ϩcos2 ͩ ͪͬ , ͑3.10͒ and ͑3.14͒ have first been derived from a special ansatz for 2 m m c d quark mixings by Branco and Lavoura ͓13͔, and later, a simi- lar formulation has also been given by Fritzsch and Xing s c ͓ ͔ ͉ ͉ϭ ͯϪ u dͯ 14 . For example, the CKM matrix V is given by the form Vcd cusd ϭ T cu sd V R12( u)R23( Q , Q)R12( d) in the Fritzsch-Xing ansatz, and their rotation R ( , ) with a phase corresponds ␦ Ϫ␦ 23 Q Q Q mc md 3 2 Ϫ † T Ϫ ϭͱ ͱ ͫ 2 to R23( /4)PuPdR23( /4) in the present model, be- ϩ ϩ cos mc mu ms md 2 cause the present rotation given in Eq. ͑1.8͒ is expressed as O ϭR (Ϫ/4)R ( ). However, we would like to empha- ␦ Ϫ␦ ␦ ϩ␦ f 23 12 f 3 2 3 2 mums ↔ Ϫ2 cos cos ͱ size that the 2 3 mixing in V comes from only the relative ␦ Ϫ␦ 2 2 mcmd phase difference ( 2 3), and it is independent of the forms ͑ ͒ 1/2 of the up- and down-mixing matrices 1.8 . The present mass mums ϩͩ ͪͬ . ͑3.11͒ matrix texture is completely different from theirs. The red- mcmd erivation of Eqs. ͑3.13͒ and ͑3.14͒ in the present model will illuminate the farsighted instates by Branco and Lavoura. ␦ ␦ It should be noted that the elements of V are independent of Next let us fix the parameters 3 and 2. When we use the ͑ ͒ ͑ ͒ ϭ ␦ mt and mb . The independent parameters in the expression expressions 3.6 – 3.11 at mZ , the parameters 2 and ͉ ͉ ϭ Ϫ1 ϭ Ϫ1 ␦ ␦ Vij are u tan (mu /mc), d tan (md /ms), 3, and 3 do not mean the phases that are evolved from those at ␦ ϭ ␦ ␦ 2. Among them, the two parameters u and d are already M X . Hereafter, we use the parameters 2 and 3 as phe- fixed by the quark masses of the first and second generations. nomenological parameters that approximately satisfy the re- ͑ ͒ ͑ ͒ ϭ Therefore, the present model has two adjustable parameters lations 3.6 – 3.11 at mZ . In order to fix the value of ␦ ␦ ␦ Ϫ␦ ͑ ͒ 3 and 2 to reproduce the observed Cabibbo-Kobayashi- 3 2, we use the relation 3.6 , which leads to Maskawa ͑CKM͒ matrix parameters ͓11͔: ␦ Ϫ␦ m 3 2ϭͱ ϩ u͉ ͉ ϭ Ϯ ͉ ͉ ϭ Ϯ sin 1 Vcb exp 0.0401 0.0018, Vus exp 0.2196 0.0026, 2 mc ͑3.16͒ ͉ ͉ ϭ Ϯ ͑ ͒ Vcb exp 0.0412 0.0020, 3.12 ␦ Ϫ␦ ϭ Ϯ ͑ ͒ 3 2 4.59° 0.21°. 3.17 ͉ ͉ ϭ͑ Ϯ ͒ϫ Ϫ3 Vub exp 3.6 0.7 10 . Then, we obtain
It should be noted that the predictions m ͉ ͉ϭͱ u͉ ͉ ϭ Ϯ ͑ ͒ Vub Vcb exp 0.00234 0.00028, 3.18 mc ͉ ͉ Vub su mu 2.33 ϭ ϭͱ ϭͱ ϭ0.0586Ϯ0.0064, ͉V ͉ c m 677 mu cb u c 1ϩ ͑3.13͒ m ͉ ͉ϭ c ͉ ͉ ϭ Ϯ Vts ͱ Vcb exp 0.0391 0.0018, md ͉ ͉ 1ϩ Vtd sd md 4.69 ϭ ϭͱ ϭͱ ϭ Ϯ ms ͉ ͉ 0.224 0.014 ͑ ͒ Vts cd ms 93.4 3.19 ͑3.14͒ mu 1ϩ are almost independent of the RGE effects, because they do m m ͉ ͉ϭ c ͱ d͉ ͉ not contain the phase difference, (␦ Ϫ␦ ), which is highly Vtd ͱ Vcb exp 3 2 md ms dependent on the energy scale as we discuss in the Appendix 1ϩ ͓ ͑ ͔͒ m see Eq. A9 and we know that the ratios mu /mc and s md /ms are almost independent of the RGE effects. In the ϭ0.00880Ϯ0.00094, ͑3.20͒ numerical results of Eqs. ͑3.13͒ and ͑3.14͒, we have used the ϭ ͓ ͔ running quark mass at mZ 12 : which are consistent with the present experimental data. Therefore, the value ͑3.17͒ is acceptable as reasonable. Then, ͑ ͒ϭ ϩ0.42 ͑ ͒ϭ ϩ56 ͑ ͒ ͑ ͒ mu mZ 2.33Ϫ0.45 MeV, mc mZ 677Ϫ61 MeV, by using the value 3.17 and the expression 3.10 ,wecan ␦ ϩ␦ ͑3.15͒ obtain the remaining parameter ( 3 2): ͑ ͒ϭ ϩ0.60 ͑ ͒ϭ ϩ11.8 md mZ 4.69Ϫ0.66 MeV, ms mZ 93.4Ϫ13.0 MeV. ␦ ϩ␦ ϭ Ϯ Ϫ Ϯ ͑ ͒ 3 2 93° 22° or 80° 22°. 3.21 ͑ ͒ ␦ Ϫ␦ Ӎ ␦ ϩ␦ Ӎ The predicted value 3.13 is somewhat small with respect to Since sin( 3 2)/2 0.04 and cos( 3 2)/2 0.2, the ͉ ͉ ͉ ͉ϭ Ϯ the present experimental value Vub / Vcb 0.08 0.02, but present model also predicts the following approximated rela- it is within the error. tions:
093006-4 UNIVERSAL TEXTURE OF QUARK AND LEPTON MASS... PHYSICAL REVIEW D 66, 093006 ͑2002͒
s c m m ͉ ͉ϭ ͯ Ϫ u dͯӍͱ d ͑ ͒ ͉ ͉ϭ͉͉ ϭͱ͉ ͉2ϩ͉ ͉2ͱ d Vus cusd 1 , 3.22 Vtd sd Vcb Vub ϩ cu sd ms ms md Ӎ͉ ͉ ͉ ͉ ͑ ͒ Vcb • Vus . 3.24
su cd md Using the rephasing of the up-type and down-type quarks, ͉V ͉ϭc s ͯϪ ͯӍͱ , ͑3.23͒ ͑ ͒ cd u d c s m Eq. 3.3 is changed to the standard representation of the u d s CKM quark mixing matrix
␣u ␣u ␣u ␣d ␣d ␣d ϭ 1 2 2͒ 1 2 2͒ Vstd diag͑e ,e ,e V diag͑e ,e ,e Ϫi␦ c13c12 c13s12 s13e Ϫ Ϫ i␦ Ϫ i␦ ϭͩ c23s12 s23c12s13e c23c12 s23s12s13e s23c13 ͪ . ͑3.25͒ Ϫ i␦ Ϫ Ϫ i␦ s23s12 c23c12s13e s23c12 c23s12s13e c23c13
␣q ϭ 2 ␦ Here, i comes from the rephasing in the quark fields to J c13s13c12s12c23s23sin make the choice of phase convention. The CP-violating ␦ ͑ ͒ ͉ ͉͉ ͉͉ ͉͉ ͉͉ ͉ phase in the representation 3.25 is expressed with the Vud Vus Vub Vcb Vtb ͑ ͒ ϭ sin ␦ expression V in Eq. 3.3 by Ϫ͉ ͉2 1 Vub Ӎ͉V ͉͉V ͉͉V ͉sin ␦. ͑3.30͒ ͉ ͉2 us ub cb V12V22* V12 ␦ϭargͫͩ ͪ ϩ ͬ, ͑3.26͒ Ϫ͉ ͉2 V13V23* 1 V13 Comparing Eq. ͑3.29͒ with Eq. ͑3.30͒, we obtain so that we obtain
␦ ϩ␦ 3 2 ␦ϭϮ͑80°Ϯ22°͒. ͑3.27͒ sin␦Ӎsin . ͑3.31͒ 2 It is interesting that nearly maximal ͉sin ␦͉ is realized in the present model. By using the numerical results ͑3.17͒–͑3.21͒, we obtain The rephasing invariant Jarlskog parameter J ͓15͔ is de- ϭ fined by J Im(VusVcs* Vub* Vcb). In the present model with Eqs. ͑3.6͒–͑3.11͒, the parameter J is given by ͉J͉ϭ͑1.91Ϯ0.38͒ϫ10Ϫ5. ͑3.32͒ ␦ ϩ␦ ϭ͉͉2͉͉ 3 2 J cusucdsdsin 2 IV. LEPTON MIXING MATRIX ͉ ͉ ͉ ͉͉ ͉͉ ͉ ␦ ϩ␦ Vub Vtd Vts Vtb 3 2 ϭ sin . ͑3.28͒ Let us discuss the lepton sectors. We assume that the neu- ͉V ͉ ϩ͉ ͉2 2 cb 1 Vub /Vcb trino masses are generated via the seesaw mechanism ͓16͔:
͉ ͉Ӎ͉ ͉͉ ͉ ͑ ͒ Using the relation Vtd Vcb Vus in Eq. 3.24 , and the ͉ ͉2ӷ͉ ͉2ӷ͉ ͉2 ͉ ͉Ӎ͉ ͉ experimental findings Vus Vcb Vub , Vts Vcb , ϭϪ Ϫ1 T ͑ ͒ ͉ ͉Ӎ M M DM R M D . 4.1 and Vtb 1, we obtain
␦ ϩ␦ 3 2 Here M D and M R are the Dirac neutrino and the right- JӍ͉V ͉͉V ͉͉V ͉sin . ͑3.29͒ ub cb us 2 handed Majorana neutrino mass matrices, which are defined ¯ ¯c by LM D R and RM R R , respectively. Since M D ϭ † ˆ † ϭ † ˆ † On the other hand, in the standard expression of V, Eq. PM DP and M R PM RP according to the assumption ͑3.25͒, J is given by ͑2.9͒, we obtain
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ϭϪ † ˆ ˆ Ϫ1 ˆ T † M PM DM R M DP m m m m 0 ͱ 2 1 ͱ 2 1 2 2 m m 1 m Ϫm 1 m Ϫm ϭ † ͱ 2 1 ͩ ϩ 2 1 ͪ Ϫ ͩ Ϫ 2 1 ͪ † ͑ ͒ P m3 1 m3 1 P . 4.2 ͩ 2 2 m3 2 m3 ͪ m m 1 m Ϫm 1 m Ϫm ͱ 2 1 Ϫ ͩ Ϫ 2 1 ͪ ͩ ϩ 2 1 ͪ m3 1 m3 1 2 2 m3 2 m3
Here and hereafter, m1 , m2 and m3 denote neutrino masses unless they are specifically mentioned. In the last expression, we have used the fact1 that the product of ABϪ1A of the matrices A and B with the texture ͑1.1͒ with Eq. ͑1.2͒ again becomes a matrix with the texture ͑1.1͒ with ͑1.2͒. On the other hand, the charged lepton mass matrix M e is given by
mm mm 0 ͱ e ͱ e 2 2
mm 1 mϪm 1 mϪm ϭ † ͱ e ͩ ϩ e ͪ Ϫ ͩ Ϫ e ͪ † ͑ ͒ M e Pe m 1 m 1 Pe , 4.3 ͩ 2 2 m 2 m ͪ mm 1 mϪm 1 mϪm ͱ e Ϫ ͩ Ϫ e ͪ ͩ ϩ e ͪ m 1 m 1 2 2 m 2 m
where m , m and m are charged lepton masses. ϭ T e U Oe PO Those mass matrices M e and M are diagonalized as † † ϭ † † ϭ c cϩs s c sϪs c Ϫs (Pe Oe) M e(PeOe) De and (PO) M (PO) D , re- e e e e e spectively, where Ϫ ϩ ϭͩ sec ces ses cec ce ͪ , ͑4.5͒ Ϫs c
ce se 0 ␦ ␦ ␦ ϵ †ϵ i 1 i 2 i 3 where P PeP diag(e ,e ,e ). Hereafter we take se ce 1 ␦ ϭ Ϫ Ϫ 1 0 without loss of generality. ϭ ͱ ͱ ͱ Oe 2 2 2 , The explicit forms of absolute magnitudes of the compo- ͩ ͪ ͑ ͒ nents of U are given by expressions similar to Eqs. 3.4 – se ce 1 ͑ ͒ ͉ ͉ Ϫ 3.12 , where Vij ,(mu ,mc ,mt), and (md ,ms ,mb) are re- ͱ ͱ ͱ ͉ ͉ 2 2 2 placed by Uij ,(m1 ,m2 ,m3), and (me ,m ,m), ͑4.4͒ respectively. It should again be noted that the elements of U are independent of m and m3. The independent parameters c s 0 ϭ Ϫ1 of the unitary matrix U are e tan (me /m), Ϫ1 ϭtan (m /m ), ␦ , and ␦ . Among them, is given by s c 1 1 2 3 2 e Ϫ Ϫ charged-lepton masses of the first and second generations. ϭ ͱ ͱ ͱ O ͩ 2 2 2 ͪ . Therefore, the model has the three adjustable parameters ␦ , ␦ , and m /m to reproduce the experimental values s c 1 3 2 1 2 Ϫ ͓11͔. ͱ2 ͱ2 ͱ2 ␦ ␦ Let us estimate the values , 3 and 2 by fitting the experimental data. In the following discussions we consider the normal mass hierarchy ⌬m2 ϭm2Ϫm2Ͼ0 for the neu- Here c and s are obtained from Eq. ͑1.9͒ by replacing m 23 3 2 e e 1 trino mass. The case of the inverse mass hierarchy ⌬m2 and m in it by m and m ; c and s are also obtained by 23 2 e Ͻ ͓ ͔ taking the neutrino masses m . Therefore, the Maki- 0 is quite similar to it. It follows from the CHOOZ 19 , i ͓ ͔ ͓ ͔ Nakagawa-Sakata-Pontecorv ͑MNSP͒ lepton mixing matrix solar 20 , and atmospheric neutrino experiments 1 that ͓18͔ U can be written as ͉ ͉2 Ͻ ͑ ͒ U13 exp 0.03. 4.6
1The seesaw invariant texture form was discussed systematically From the global analysis of the SNO solar neutrino experi- in ͓17͔. ment ͓20͔,
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⌬m2 ϭm2Ϫm2ϭ⌬m2 ϭ5.0ϫ10Ϫ5 eV2, ͑4.7͒ 1 m 12 2 1 sol ͉ ͉2ϭ e ϭ U13 ϩ 0.00236, 2 m me tan2 ϭtan2 ϭ0.34, ͑4.8͒ ͑ ͒ 12 sol 2 ϭ 4.15 or sin 2 13 0.00942. 2 ϭ ͑ ͒ with min/DOF 57.0/72, for the large mixing angle LMA This value is consistent with the present experimental con- Mikheyev-Smirnov-Wolfenstein ͑MSW͒ solution. From the straints ͑4.6͒ and can be checked in neutrino factories ͓21͔, 2 atmospheric neutrino experiment ͓1͔, we also have which have sensitivity to sin 2 13 for 2 у Ϫ5 ͑ ͒ ⌬ 2 ϭ 2Ϫ 2Ӎ⌬ 2 ϭ ϫ Ϫ3 2 ͑ ͒ sin 2 13 10 . 4.16 m23 m3 m2 matm 2.5 10 eV , 4.9 The mixing angle sol in the present model is given by 2 Ӎ 2 ϭ ͑ ͒ sin 2 23 sin 2 atm 1.0, 4.10 2 ϵ ͉ ͉2͉ ͉2 sin 2 sol 4 U11 U12 2 ϭ ␦ ϩ␦ with min/DOF 163.2/170. 4m2m1 3 2 ␦ ␦ Ӎ ͫ 1Ϫͱ2 cos Independently of the parameters 3 and 2, the model ͑ ϩ ͒2 2 predicts the following two ratios: m2 m1 m m 1 m m ͉ ͉ ϫͱ e 2 ϩ ͩ e 2 ͪͬ U13 se me 0.487 ϭ ϭͱ ϭͱ ϭ ͑ ͒ mm 2 mm ͉ ͉ 0.0688, 4.11 1 1 U23 ce m 103 4m /m Ӎ 1 2 ͑ ͒ ͉ ͉ , 4.17 U31 s m1 ͑1ϩm /m ͒2 ϭ ϭͱ . ͑4.12͒ 1 2 ͉U ͉ c m 32 2 which leads to Here we have used the running charged-lepton mass at m1 ϭ ͓ ͔ ϭ Ϯ Ӎtan2 ϭ0.34, ͑4.18͒ mZ 12 : me(mZ) 0.48684727 0.00000014 MeV, and m sol ϭ Ϯ 2 m(mZ) 102.75138 0.00033 MeV. The neutrino mixing ͉⌬ 2 ͉ӷ͉⌬ 2 ͉ where we have used the best fit value ͑4.8͒. This value ͑4.18͒ angle atm under the constraint m23 m12 is given by guarantees the validity of the approximation ͑4.17͒, because 2 ϵ ͉ ͉2͉ ͉2 ͱ Ӎ sin 2 atm 4 U23 U33 of (me /m)/(m1 /m2) 0.12. Then, we can obtain the neu- trino masses 2 2 2 ϭ4͉͉ ͉͉ c e ϭ m1 0.0026 eV, m ϭ 2͑␦ Ϫ␦ ͒ͱ ͑ ͒ ϭ ͑ ͒ sin 3 2 ϩ . 4.13 m2 0.0075 eV, 4.19 m me ϭ m3 0.050 eV, 2 The observed fact sin 2 Ӎ1.0 highly suggests ␦ Ϫ␦ atm 3 2 ⌬ 2 Ӎ/2. Hereafter, for simplicity, we take where we have used the observed best fit values of msol and ⌬ 2 ͑ ͒ ͑ ͒ matm , Eqs. 4.7 and 4.9 , respectively. Next let us discuss the CP violation phases in the lepton ␦ Ϫ␦ ϭ ͑ ͒ 3 2 . 4.14 mixing matrix. The Majorana neutrino fields do not have the 2 freedom of rephasing invariance, so that we can use only the ͑ ͒ rephasing freedom of M e to transform Eq. 4.5 to the stan- Under the constraint ͑4.13͒, the model predicts dard form
i i(␥Ϫ␦) c13c12 c13s12e s13e ϵ ͑Ϫ Ϫ i␦͒ Ϫi Ϫ i␦ i(␥Ϫ) ͑ ͒ Ustd ͩ c23s12 s23c23s13e e c23c12 s23s12s13e s23c13e ͪ , 4.20 Ϫ i␦͒ Ϫi␥ Ϫ Ϫ i␦͒ Ϫi(␥Ϫ) ͑s23s12 c23c12s13e e ͑ s23c12 c23s12s13e e c23c13 as
␣e ␣e ␣e Ϯ ϭ i 1 i 2 i 2͒ i /2 ͒ ͑ ͒ Ustd diag͑e ,e ,e U diag͑e ,1,1 . 4.21 ␣e Here, i comes from the rephasing in the charged lepton fields to make the choice of phase convention, and the specific phase Ϯ /2 is added on the right-hand side of U in order to change the neutrino eigenmass m1 to a positive quantity. Similarly to the quark sector, the CP-violating phase ␦ in the representation ͑4.20͒ is expressed as
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͉ ͉2 The rephasing-invariant parameter J in the lepton sector is U12U22* U12 ␦ϭargͫ ϩ ͬ defined by JϭIm(U U* U* U ), which is explicitly given Ϫ͉ ͉2 12 22 13 23 U13U23* 1 U13 by
U12U22* ␦ ϩ␦ Ӎargͩ ͪ ϭ͉ ͉2͉ ͉ 3 2 J cscesesin U13U23* 2 Ӎ ϩ ͉ ͉ ͉ ͉͉ ͉͉ ͉ ␦ ϩ␦ arg * U13 U31 U32 U33 3 2 ϭ sin ͉ ͉ 2 ␦ ϩ␦ U23 1ϩ͉U /U ͉ 2 3 2 13 23 ϭϪ ϩ. ͑4.22͒ 2 ͉ ͉ ͉ ͉͉ ͉͉ ͉ U13 U31 U32 U33 р . ͑4.26͒ ͉ ͉ 2 U23 1ϩ͉U /U ͉ Though the lepton mixing matrix includes the additional Ma- 13 23  ␥ ͓ ͔ jorana phase factors and 22,23 , the number of param- The upper bound is described in terms of the ratio m1 /m2, eters which will become experimentally available in the near so that we obtain future is practically four, as in the Dirac case. The additional phase parameters are determined as Jр0.019. ͑4.27͒
It should be noted that if we again assume the maximal CP Ustd12 U12 violation in the lepton sector, the magnitude of the rephasing ϭargͩ ͪ ϭargͩ ͪ Ӎ0ϯ , ͑4.23͒ Ϯi invariant ͉J͉ can be considerably larger than in the quark Ustd11 U e 2 11 ͉ ͉Ӎ ϫ Ϫ5 sector, Jquark 2 10 .
and V. CONCLUSION In conclusion, stimulated by recent neutrino data, which Ustd13 ␥ϭargͩ ei␦ ͪ suggest a nearly bimaximal mixing, we have investigated the Ustd11 possibility that all the mass matrices of quarks and leptons have the same texture as the neutrino mass matrix. We have U13 ϭargͩ ei␦ ͪ assumed that the mass matrix form is constrained by a dis- Ϯi U11e crete symmetry Z3 and a permutation symmetry S2, i.e. that the texture is given by the form ͑1.1͒ with Eq. ͑1.2͒. The most important feature of the present model is that the tex- Ӎarg͑Ϫ͒ϩ␦ϯ 2 tures ͑1.1͒, ͑1.2͒ are practically applicable to the predictions at the low energy scale ͑the electroweak scale͒, although we Ӎ ϯ , ͑4.24͒ assume that the textures are exactly given at a unification 2 2 scale. It is well known that the matrix form ͑1.1͒ leads to a bimaximal mixing in the neutrino sector. In the present by using the relations m Ӷm and (␦ Ϫ␦ )/2Ӎ/4. e 3 2 model, the mixing angle f between the first and second Hence, we can also predict the averaged neutrino mass ͗m͘ 12 ͓23͔, which appears in the neutrinoless double beta decay, as generations is given by follows: f ϭͱ f f ͑ ͒ tan 12 m1/m2 , 5.1 ͗ ͘ϵ͉Ϫ 2 ϩ 2 ϩ 2 ͉ f f m m1U11 m2U12 m3U13 where m1 and m2 are the first and second generation fermion masses. This leads to a large mixing in the lepton mixing ͱ 2 2 ϭ͉Ϫ2c s m m ϩs ͑m Ϫm ͒ ͑ ͒ ϳ ͑ e e e 1 2 e 2 1 matrix MNSP matrix U with m1 m2 neglecting tan 12 ϭͱ ͒ ϩm s2͉. ͑4.25͒ me /m in the charge-lepton sector , and it also leads to 3 e ͓ ͔ ͉ ͉Ӎͱ the famous formula 24 Vus md /ms in the quark mixing matrix ͑CKM matrix͒ V ͑neglecting tanu ϭͱm /m in the ͑ ͒ 12 u c The value of Eq. 4.25 is highly sensitive to the value of up-quark sector͒. In the present model the mixing angle f ␦ ϩ␦ 23 ( 3 2)/2, which is unknown at present, because the val- between the second and third generation is fixed as f ϭͱ Ӎ ͱ Ӎ 23 ues se /ce me /m 0.070 and m1m2/m3 0.088 are in ϭ/4. However, the ͑2,3͒ component of the quark mixing ␦ ϩ␦ ϭ the same order. For ( 3 2)/2 0, /2 and , we obtain matrix V ͑and also the lepton mixing matrix U) is highly ͗ ͘ϭ the numerical results m 0.00018 eV, 0.00049 eV and dependent on the phase difference ␦ Ϫ␦ , as follows: 0.00069 eV, respectively. However, these values should not 3 2 be taken strictly because the value m1 /m2 is also sensitive to 1 ␦ Ϫ␦ 2 ϭ 3 2 ͑ ͒ the observed value of tan sol . In any case, the predicted V23 sin , 5.2 Ϫ3 ͱ ϩ u u 2 value of ͗m͘ will be less than the order of 10 eV. 1 m1/m2
093006-8 UNIVERSAL TEXTURE OF QUARK AND LEPTON MASS... PHYSICAL REVIEW D 66, 093006 ͑2002͒
␦ ϭ␦uϪ␦d where i i i . Replacing the arguments by their lep- the present appendix, we demonstrate this for the quark mass tonic counterparts, we have the same form for U23 . We have matrices M u and M d . understood the observed values V and U by taking (␦ It is well known ͓25͔ that the energy scale dependences 23 23 3 ϭ Ϫ␦ )/2 as a small value for the quark sectors and as /2 for R(A) A( )/A(M X) for observable quantities A approxi- 2 ͉ ͉ Ӎ ͉ ͉ Ӎ ͉ ͉ the lepton sectors, respectively. As predictions, which are mately satisfy the relations R( Vub ) R( Vcb ) R( Vtd ) Ӎ ͉ ͉ Ӎ Ӎ independent of such phase parameters, there are two rela- R( Vts ) R(md /mb) R(ms /mb), and that the ratios ͉ ͉ ͉ ͉ tions R( Vus ), R( Vcd ), R(md /ms) and R(mu /mc) are approxi- mately constant. This is caused by the fact that the Yukawa coupling constant y of the top quark is extremely large with ͉ ͉ ͉ ͉ t Vub mu Vtd md ϭͱ , ϭͱ ͑5.3͒ respect to other coupling constants. The above relations on R ͉ ͉ ͉ ͉ 2ӷ 2 2 Vcb mc Vts ms are well explained by the approximation y t y b ,y c ,... . Therefore, we will also use approximation below. ͑ ͑ ͒ and the similar relations for U). The relations 5.3 are in The one-loop RGE for the Yukawa coupling constants Y ͉ ͉ f good agreement with experiments. The relation U13 /U23 ( f ϭu,d) has the form ϭͱ ͉ ͉2Ӎ me /m in the lepton sectors leads to U13 me/2m ϭ 2 ϭ 0.0024 if we accept sin 2 atm 1.0. This value will be test- dYf 1 able in the near future. ϭ ͑C 1ϩC Y Y †ϩC Y Y † ͒Y , ͑A1͒ ͑ 2 f ff f f ffЈ f Ј f Ј f Since, in the present model, each mass matrix M f i.e. the dt 16 Yukawa coupling Y ) takes different values of A , B , and f f f ϭ ϭ ϭ ϭ so on, the present model cannot be embedded into a GUT where f Ј d ( f Ј u) for f u ( f d), and the coefficients scenario. In spite of such a demerit, however, it is worth- C f , C ff and C ffЈ are energy scale dependent factors which while noting that it can give a unified description of quark are calculated from the one-loop Feynman diagrams. We and lepton mass matrices with the same texture. start from the Yukawa coupling constants Y f (M X), corre- sponding to the mass matrix form ͑1.1͓͒with Eq. ͑1.2͔͒. † ACKNOWLEDGMENTS Since the matrix Y uY u is approximately given by One of the authors ͑Y.K.͒ wishes to acknowledge the hos- 2 00 0 pitality of the Theory group at CERN, where this work was mt ϩ ␦ ͑ ͒ †͑ ͒Ӎ Ϫ i u Y u M X Y M X ͩ 01 e ͪ , completed. Y.K. also thanks Z. Z. Xing for informing him of u 2 v Ϫ ␦ many helpful references. u 0 Ϫe i u 1 ͑A2͒ APPENDIX ␦ ϭ␦uϪ␦u ͱ ϭ͗ 0͘ where u 3 2 , vu / 2 Hu , and we have used the ͑ ͒ ͑ ͒ ͑ ͒ 2ӷ 2 2 The mass matrix texture 1.1 with Eq. 1.2 , which is relations 1.6 and the approximation y t y b ,y c ,..., the ϭ defined at the unification energy scale M X , is applicable up-quark Yukawa coupling constant Y u( ) in the neighbor- ϭ ϭ to the phenomenology at the electroweak scale mZ .In hood of M X is given by the form
00 0 ϩ ␦ ͑͒Ӎ ͑͒ ϩ ͑͒ Ϫ i u ͑ ͒ Y u ru ͫ 1 u ͩ 01 e ͪͬ Y u M X Ϫ ␦ 0 Ϫe i u 1
Ϫi␦u Ϫi␦u 0 Aue 2 Aue 3 ͑͒ ru Ϫ ␦u Ϫ ␦u Ϫ ␦uϩ␦u Ӎ ͑ ϩ Ϫ ͒ i 2 ͑ ϩ Ϫ ͒ 2i 2 ͑ ϩ Ϫ ͒ i( 2 3) ͩ Au 1 u u e Bu 1 u uCu /Bu e Cu 1 u uBu /Cu e ͪ . v /ͱ2 u Ϫ ␦u Ϫ ␦uϩ␦u Ϫ ␦u ϩ Ϫ ͒ i 3 ϩ Ϫ ͒ i( 2 3) ϩ Ϫ ͒ 2i 3 Au͑1 u u e Cu͑1 u uBu /Cu e Bu͑1 u uCu /Bu e ͑A3͒
Ӎ Although this form is one in M X , since the texture keeps the same form under the small change of energy scale, as a result, ͑ ͒ ͑ ͒ the texture of Y u( ) given by Eq. A3 holds at any energy scale . Therefore, we can obtain the expression 1.1 at an ͓ ͑ ͒ ϭ † ͑ ͒ arbitrary energy scale . The demonstration A3 has been done for the case PR PL mentioned in Eq. 2.10 . However, the conclusion does not depend on this choice.͔ On the other hand, the evolution of the down-quark Yukawa coupling constant Y d( ) is somewhat complicated. By a way similar to Eq. ͑A3͒, we obtain
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10 0 ϩ ␦ ͑͒Ӎ ͑͒ ϩ Ϫ i u ͑ ͒ Y d rd ͩ 01d de ͪ Y d M X Ϫ ␦ Ϫ i u ϩ 0 de 1 d 0 A A ͒ d d rd͑ ϩ ␦ Ϫ␦ ϩ ␦ Ϫ␦ ϩ ␦ Ϫ␦ † ͑ ϩ Ϫ i( u d)͒ ͑ ϩ Ϫ i( u d) ͒ ͑ ϩ Ϫ i( u d) ͒ † Ӎ P ͩ Ad 1 d de Bd 1 d de Cd /Bd Cd 1 d de Bd /Cd ͪ P , d d v /ͱ2 Ϫ ␦ Ϫ␦ Ϫ ␦ Ϫ␦ Ϫ ␦ Ϫ␦ d ϩ Ϫ i( u d)͒ ϩ Ϫ i( u d) ͒ ϩ Ϫ i( u d) ͒ Ad͑1 d de Cd͑1 d de Bd /Cd Bd͑1 d de Cd /Bd ͑A4͒ ␦ Ϫ␦ ϭ ␦uϪ␦u Ϫ ␦dϪ␦d ϭ␦ Ϫ␦ † † where u d ( 3 2) ( 3 2) 3 2. Note that the part that is sandwiched between Pd and Pd includes imaginary † ˆ † parts and those phase factors cannot be removed by an additional phase matrix Pd( ) into the form Pd( )Y d( )Pd( ). † However, the quantity that has the physical meaning is Y dY d . When we define Ϫ ␣ Ϫ ␦ Ϫ␦ ϩ  ϩ ␦ Ϫ␦ i ϭ ϩ Ϫ i( u d)͒ i ϭ ϩ ϩ i( u d)͒ ͑ ͒ e 1 d͑1 e , e 1 d͑1 e , A5 we obtain 2 ͑ ϩ ͒ ͑ ϩ ͒ Ad Ad Bd Cd Ad Bd Cd r2͑͒ ͑͒ †͑͒Ӎ d † ͑ ϩ ͒ 22ϩ͑ 2ϩ 2͒2 22 2i(␣Ϫ)ϩ 2 † ͑ ͒ Y d Y d PdPͩ Ad Bd Cd Ad Bd Cd Ad e 2BdCd ͪ PPd , A6 v2/2 d ͑ ϩ ͒ 22 2i(␣ϩ)ϩ 2 22ϩ͑ 2ϩ 2͒2 Ad Bd Cd Ad e 2BdCd Ad Bd Cd where
i Ϫi Pϭdiag͑1,e ,e ͒, ͑A7͒ so that we can obtain a real matrix for the part which is sandwiched by the phase matrix PdP under the approximation 2 ͉ ͉Ӎ Ad/ BdCd 0. This means that we can practically write
0 Ad Ad r ͑͒ ˆ ͑͒Ӎ d ͑ ͒ Y d ͩ Ad Bd Cd ͪ , A8 ͱ vd / 2 Ad Cd Bd with
† Ϫ ␦dϪ Ϫ ␦dϩ ͑͒ϭ ͑ i( 2 ) i( 3 )͒ ͑ ͒ Pd diag 1,e ,e , A9 ␦d→␦dϪ ␦d→␦dϩ at an arbitrary energy scale . It should be noted that the changes of the phases 2 2 and 3 3 do not come from ␦d ␦d the evolution of the phases 2( ) and 3( ), but they are brought effectively by absorbing the unfactorizable phase parts in ͑ ͒ Y d( ). Thus, we can again use the texture 1.1 at an arbitrary energy scale from a practical point of view. In the Yukawa coupling constants Y e and Y of the leptons, the RGE effects are not so large as in the quark sectors. In the 2ӷ 2 ӷ 2 ͑ ͒ charged lepton sector, since m m me , we can again demonstrate that the expression 1.1 is applicable at an arbitrary energy scale in a way similar to the quark sectors. For the neutrino Yukawa coupling constant Y (), the evolution equation is different from Eq. ͑A1͒. We must use the RGE for the seesaw operator ͓26͔. However, the calculation and result are 2ӷ 2Ͼ 2 essentially the same as those in Y u( ), Y d( ) and Y e( ), because m3 m2 m1 in the present model. ϭ Finally, we would like to add that these conclusions on the evolution of the mass matrices M f ( f u,d,e, ) are exactly confirmed by numerical study, without approximation.
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