Modular invariant flavor model of A 4 and hierarchical structures at nearby fixed points
AAPCTP Pre2020-024
Modular invariant flavor model of A andhierarchical structures at nearby fixed points Hiroshi Okada a,b ∗ and Morimitsu Tanimoto c † a Asia Pacific Center for Theoretical Physics, Pohang 37673, Republic of Korea b Department of Physics, Pohang University of Science and Technology, Pohang 37673,Republic of Korea c Department of Physics, Niigata University, Niigata 950-2181, Japan
Abstract
In the modular invariant flavor model of A , we study the hierarchical structure of lepton/quarkflavors at nearby fixed points of τ = i and τ = ω of the modulus, which are in the fundamentaldomain of PSL(2 , Z ). These fixed points correspond to the residual symmetries Z S = { I, S } and Z S = { I, ST, ( ST ) } of A , where S and T are generators of the A group. The infinite τ = i ∞ alsopreserves the residual symmetry of the subgroup Z T = { I, T, T } of A . We study typical two-typemass matrices for charged leptons and quarks in terms of modular forms of weights 2, 4 and 6while the neutrino mass matrix with the modular forms of weight 4 through the Weinberg operator.Linear modular forms are obtained approximately by performing Taylor expansion of modular formsaround fixed points. By using them, the flavor structure of the lepton and quark mass matrices areexamined at nearby fixed points. The hierarchical structure of these mass matrices is clearly shownin the diagonal base of S , T and ST . The observed PMNS and CKM mixing matrices can bereproduced at nearby fixed points in some cases of mass matrices. By scanning model parametersnumerically at nearby fixed points, our discussion are confirmed for both the normal hierarchy andinverted one of neutrino masses. Predictions are given for the sum of neutrino masses and the CPviolating Dirac phase of leptons at each nearby fixed point. ∗ E-mail address: [email protected] † E-mail address: [email protected] a r X i v : . [ h e p - ph ] S e p Introduction
In spite of the remarkable success of the standard model (SM), the origin of the flavor of quarks andleptons is still a challenging issue. Indeed, a lot of works have been presented by using the discretegroups for flavors to understand the flavor structures of quarks and leptons. In the early models ofquark masses and mixing angles, the S symmetry was used [1, 2]. It was also discussed to understandthe large mixing angle [3] in the oscillation of atmospheric neutrinos [4]. For the last twenty years,the discrete symmetries of flavors have been developed, that is motivated by the precise observation offlavor mixing angles of leptons [5–14].Many models have been proposed by using the non-Abelian discrete groups S , A , S , A and othergroups with larger orders to explain the large neutrino mixing angles. Among them, the A flavor modelis attractive one because the A group is the minimal one including a triplet irreducible representation,which allows for a natural explanation of the existence of three families of leptons [15–21]. However,variety of models is so wide that it is difficult to show a clear evidence of the A flavor symmetry.Recently, a new approach to the lepton flavor problem appeared based on the invariance of themodular group [22], where the model of the finite modular group Γ (cid:39) A has been presented. Thiswork inspired further studies of the modular invariance to the lepton flavor problem. The modulargroup includes the finite groups S , A , S , and A [23]. Therefore, an interesting framework for theconstruction of flavor models has been put forward based on the Γ (cid:39) A modular group [22], andfurther, based on Γ (cid:39) S [24]. The flavor models have been proposed by using modular symmetriesΓ (cid:39) S [25] and Γ (cid:39) A [26]. Phenomenological discussions of the neutrino flavor mixing havebeen done based on A [27–29], S [30–32] and A [33]. A clear prediction of the neutrino mixingangles and the CP violating phase was presented in the simple lepton mass matrices with A modularsymmetry [28]. The Double Covering groups T (cid:48) [34, 35] and S (cid:48) [36, 37] have also obtained from themodular symmetry.The A modular symmetry has been also applied to the leptogenesis [38–40], on the other hand,it is discussed in the SU(5) grand unified theory (GUT) of quarks and leptons [41, 42]. The residualsymmetry of the A modular symmetry has presented the interesting phenomenology [43]. Furthermore,modular forms for ∆(96) and ∆(384) were constructed [44], and the extension of the traditional flavorgroup is discussed with modular symmetries [45]. The level 7 finite modular group Γ (cid:39) PSL(2 , Z ) isalso presented for the lepton mixing [46]. Moreover, multiple modular symmetries are proposed as theorigin of flavor [47]. The modular invariance has been also studied combining with the CP symmetriesfor theories of flavors [48, 49]. The quark mass matrix has been discussed in the S and A modularsymmetries as well [50–52]. Besides mass matrices of quarks and leptons, related topics have beendiscussed in the baryon number violation [50], the dark matter [53, 54] and the modular symmetryanomaly [55]. Furthere phenomenology has been developed in many works [56–74] while theoreticalinvestigations are also proceeded [75–79].As well known, in non-Abelian discrete symmetries of flavors, residual symmetries provides inter-esting phenomenology of flavors. They arise whenever the modulus τ breaks the modular group onlypartially. In this work, we study the hierarchical flavor structure of leptons and quarks in context withthe residual symmetry, in which the modulus τ is at fixed points. We examine the flavor structure ofmass matrices of leptons and quarks at nearby fixed points of the modulus τ in the framework of themodular invariant flavor model of A . It is challenging to reproduce the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing angles [83,84] and the CP violating Dirac phase of leptons, which is expected tobe observed at T2K and NO ν A experiments [85, 86], as well as observed Cabibbo-Kobayashi-Maskawa1CKM) matrix elements at nearby fixed points.We have already discussed numerically both mass matrices of leptons and quarks in the A modularsymmetry [52, 82], where modular forms of weights 2, 4 and 6 are used. In the same framework, wediscuss the flavor structure of the lepton and quark mass matrices forcusing on nearby fixed points.For this purpose, we give linear forms of Y ( τ ), Y ( τ ) and Y ( τ ) approximately by performing Taylorexpansion of modular forms around fixed points of the modulus τ in the A modular symmetry.The paper is organized as follows. In section 2, we give a brief review on the modular symmetryand modular forms of weights 2, 4 and 6. In section 3, we discuss the residual symmetry of A andmodular forms at fixed points. In section 4, we present modular forms at nearby fixed points. In section5 and 6, we discuss flavor mixing angles at nearby fixed points in lepton mass matrices and quark massmatrices, respectively. In section 7, the numerical results and predictions are presented. Section 8 isdevoted to a summary. In Appendix A, the tensor product of the A group is presented. In AppendixB, the transformation of mass matrices are discussed in the arbitrary bases of S and T . In AppendixC, the modular forms are given at nearby fixed points. In Appendix D, we present how to obtain DiracCP phase, Majorana phases and the effective mass of the 0 νββ decay. , , The modular group ¯Γ is the group of linear fractional transformation γ acting on the modulus τ ,belonging to the upper-half complex plane as: τ −→ γτ = aτ + bcτ + d , where a, b, c, d ∈ Z and ad − bc = 1 , Im[ τ ] > , (1)which is isomorphic to PSL(2 , Z ) = SL(2 , Z ) / { I , − I } transformation. This modular transformation isgenerated by S and T , S : τ −→ − τ , T : τ −→ τ + 1 , (2)which satisfy the following algebraic relations, S = I , ( ST ) = I . (3)We introduce the series of groups Γ( N ) ( N = 1 , , , . . . ), called principal congruence subgroups,defined by Γ( N ) = (cid:26)(cid:18) a bc d (cid:19) ∈ SL(2 , Z ) , (cid:18) a bc d (cid:19) = (cid:18) (cid:19) (mod N ) (cid:27) . (4)For N = 2, we define ¯Γ(2) ≡ Γ(2) / { I , − I } . Since the element − I does not belong to Γ( N ) for N > N ) = Γ( N ). The quotient groups defined as Γ N ≡ ¯Γ / ¯Γ( N ) are finite modular groups. Inthis finite groups Γ N , T N = I is imposed. The groups Γ N with N = 2 , , , , A ,S and A , respectively [23].Modular forms of level N are holomorphic functions f ( τ ) transforming under the action of Γ( N ) as: f ( γτ ) = ( cτ + d ) k f ( τ ) , γ ∈ Γ( N ) , (5)2here k is the so-called as the modular weight.Superstring theory on the torus T or orbifold T / Z N has the modular symmetry [87–92]. Its lowenergy effective field theory is described in terms of supergravity theory, and string-derived supergravitytheory has also the modular symmetry. Under the modular transformation of Eq.(1), chiral superfields φ ( I ) transform as [93], φ ( I ) → ( cτ + d ) − k I ρ ( I ) ( γ ) φ ( I ) , (6)where − k I is the modular weight and ρ ( I ) ( γ ) denotes an unitary representation matrix of γ ∈ Γ N .In this paper, we study global supersymmetric models, e.g., minimal supersymmetric extensions ofthe Standard Model (MSSM). The superpotential which is built from matter fields and modular formsis assumed to be modular invariant, i.e., to have a vanishing modular weight. For given modular formsthis can be achieved by assigning appropriate weights to the matter superfields.The kinetic terms are derived from a K¨ahler potential. The K¨ahler potential of chiral matter fields φ ( I ) with the modular weight − k I is given simply by K matter = 1[ i (¯ τ − τ )] k I | φ ( I ) | , (7)where the superfield and its scalar component are denoted by the same letter, and ¯ τ = τ ∗ after takingthe vacuum expectation value (VEV). Therefore, the canonical form of the kinetic terms is obtainedby changing the normalization of parameters [28]. The general K¨ahler potential consistent with themodular symmetry possibly contains additional terms [94]. However, we consider only the simplestform of the K¨ahler potential.For Γ (cid:39) A , the dimension of the linear space M k (Γ ) of modular forms of weight k is k + 1 [95–97],i.e., there are three linearly independent modular forms of the lowest non-trivial weight 2. These formshave been explicitly obtained [22] in terms of the Dedekind eta-function η ( τ ): η ( τ ) = q / ∞ (cid:89) n =1 (1 − q n ) , q = exp ( i πτ ) , (8)where η ( τ ) is a so called modular form of weight 1 /
2. In what follows we will use the following basis ofthe A generators S and T in the triplet representation:S = 13 − − − , T = ω
00 0 ω , (9)where ω = exp( i π ) . The modular forms of weight 2, Y ( ) = ( Y ( τ ) , Y ( τ ) , Y ( τ )) T transforming as atriplet of A can be written in terms of η ( τ ) and its derivative [22]: Y ( τ ) = i π (cid:18) η (cid:48) ( τ / η ( τ /
3) + η (cid:48) (( τ + 1) / η (( τ + 1) /
3) + η (cid:48) (( τ + 2) / η (( τ + 2) / − η (cid:48) (3 τ ) η (3 τ ) (cid:19) ,Y ( τ ) = − iπ (cid:18) η (cid:48) ( τ / η ( τ /
3) + ω η (cid:48) (( τ + 1) / η (( τ + 1) /
3) + ω η (cid:48) (( τ + 2) / η (( τ + 2) / (cid:19) , (10) Y ( τ ) = − iπ (cid:18) η (cid:48) ( τ / η ( τ /
3) + ω η (cid:48) (( τ + 1) / η (( τ + 1) /
3) + ω η (cid:48) (( τ + 2) / η (( τ + 2) / (cid:19) . q -expansions: Y (2) = Y ( τ ) Y ( τ ) Y ( τ ) = q + 36 q + 12 q + . . . − q / (1 + 7 q + 8 q + . . . ) − q / (1 + 2 q + 5 q + . . . ) . (11)They satisfy also the constraint [22]: ( Y ( τ )) + 2 Y ( τ ) Y ( τ ) = 0 . (12)The modular forms of the higher weight, k , can be obtained by the A tensor products of the modularforms with weight 2, Y (2) , as given in Appendix A. For k = 4, there are five modular forms by thetensor product of ⊗ as: Y (4) = Y + 2 Y Y , Y (4) (cid:48) = Y + 2 Y Y , Y (4) (cid:48)(cid:48) = Y + 2 Y Y = 0 , Y (4) = Y (4)1 Y (4)2 Y (4)3 = Y − Y Y Y − Y Y Y − Y Y , (13)where Y (4) (cid:48)(cid:48) vanishes due to the constraint of Eq. (12). For k = 6, there are seven modular forms by thetensor products of A as: Y (6) = Y + Y + Y − Y Y Y , Y (6) ≡ Y (6)1 Y (6)2 Y (6)3 = Y + 2 Y Y Y Y Y + 2 Y Y Y Y + 2 Y Y , Y (6) (cid:48) ≡ Y (cid:48) (6)1 Y (cid:48) (6)2 Y (cid:48) (6)3 = Y + 2 Y Y Y Y Y + 2 Y Y Y Y + 2 Y Y . (14)By using these modular forms of weights 2 , ,
6, we discuss lepton and quark mass matrices. A at fixed points Residual symmetries arise whenever the VEV of the modulus τ breaks the modular group Γ onlypartially. Fixed points of modulus are the case. There are only 2 inequivalent finite points in thefundamental domain of Γ, namely, τ = i and τ = ω = − / i √ /
2. The first point is invariant underthe S transformation τ = − /τ . In the case of A symmetry, the subgroup Z S = { I, S } is preservedat τ = i . The second point is the left cusp in the fundamental domain of the modular group, whichis invariant under the ST transformation τ = − / ( τ + 1). Indeed, Z ST = { I, ST, ( ST ) } is one ofsubgroups of A group. The right cusp at τ = − ω = 1 / i √ / τ = ω by the T transformation. There is also infinite point τ = i ∞ , in which the subgroup Z T = { I, T, T } of A ispreserved. 4 r τ = i τ = ω τ = i ∞ Y (1 , − √ , − √ Y (1 , ω, − ω ) Y (1 , , (6 − √ Y (1 , , Y (1 , − ω, ω ) Y (1 , , { , (cid:48) } Y { √ − , − √ } { , Y ω } { Y , } Y ( − √ , − √ , − √
3) 0 Y (1 , , (cid:48) Y ( −
12 + 7 √ , − √ , − √ Y ( − , ω, ω ) 0 Y Y Y Y ( i ) = 1 . ... Y ( ω ) = 0 . ... Y ( i ∞ ) = 1Table 1: Modular forms of weight k = 2, k = 4 and k = 6 at fixed points of τ .It is possible to calculate the values of the A triplet modular forms of weight 2, 4 and 6 at τ = i , τ = ω and τ = i ∞ . The results are summarized in Table 1.If a residual symmetry of A is preserved in mass matrices of leptons and quarks, we have commu-tation relations between the mass matrices and the generator G ≡ S, T, ST as:[ M † RL M RL , G ] = 0 , [ M LL , G ] = 0 , (15)where M RL denotes the mass matrix of charged leptons and quarks, M E and M q , on the other hand, M LL denotes the left-handed Majorana neutrino mass matrix M ν .Therefore, the mass matrices M † E M E , M † q M q and M ν could be diagonal in the diagonal basis ofG at the fixed points. The hierarchical structures of flavor mixing are easily realized near those fixedpoints. However, we should be careful with the generator S , in which two eigenvalues are degenerate.At τ = i , one (2 ×
2) submatrix of the mass matrix respecting S are not diagonal in general sincetwo eigenvalues of S are degenerate such as ( − , − , S symmetry provides us anadvantage to reproduce the large mixing angle of neutrinos as discussed in section 5. S and ST S The modular forms of Eq. (10) is obtained in the base of Eq. (9) for S and T . In order to present themass matrices in the diagonal base of S , we move to the diagonal base of S as follows: V S S V † S = − − , V S S V † S = − − , V S S V † S = − − , (16)where V Si ≡ P i √ − √ − √ √ √ √ − √ √ , P = , P = , P = . (17)5hen, the generator T is not anymore diagonal.If there is a residual symmetry of A in the Dirac mass matrix M RL , for example, Z S = { I, S } , thegenerator S commutes with M † RL M RL , (cid:104) M † RL M RL , S (cid:105) = 0 . (18)Therefore, the mass matrix is expected to be diagonal in the diagonal base of S . However, the eigenvalue − S is degenerated, and so one pair among off diagonal terms of M † RL M RL is not necessary to vanishdepending on V i of Eq. (17). For diagonal matrices S = ( − , , − , − , −
1) and ( − , − , M † RL M RL = × × × ×
00 0 × , × × × × × , × × × × × , (19)respectively, where ” × ” denotes non-vanishing entry. Thus, one flavor mixing angle appears even ifthere exists the Z S = { I, S } symmetry.. ST and T If there exists the residual symmetries of the A group Z ST = { I, ST, ( ST ) } or Z S = { I, T, T } , wehave (cid:104) M † RL M RL , ST (cid:105) = 0 , (cid:104) M † RL M RL , T (cid:105) = 0 , (20)respectively, which lead to the diagonal M † RL M RL because ST and T have three different eigenvalues.The generator T is already diagonal in the original base of Eq. (9). On the other hand, we can moveto the diagonal base of ST by the unitary transformation V ST as follows: V ST i
ST V † ST i = P i ω ω
00 0 1 P Ti , (21)where V ST i = 13 P i − ω − ω − ω ω ω − ω , P = , P = , (22)and P i ( i = 1 , ,
3) are given in Eq. (17). The order of eigenvalues of ST depends on P i . We haveeigenvalues { ω, ω , } for P , { ω , , ω } for P , { , ω, ω } for P and { , ω , ω } for P , respectively.In the diagonal bases of S and ST , the Dirac mass matrix ˆ M RL is given by the unitary transformationas (See Appendix B): ˆ M RL = M RL V † Si , ˆ M RL = M RL V † ST i , (23)respectively. On the other hand, the Majorana mass matrix M LL is given as:ˆ M LL = V Si M LL V † Si , ˆ M LL = V ST i M LL V † ST i , (24)respectively. We will discuss the lepton and quark mass matrices in the diagonal base of the generatorsby using these transformations. 6 Modular forms at nearby fixed points
The mass matrices of leptons and quarks have simple flavor structures due to simple modular forms atfixed points. At τ = i , those mass matrices have one flavor mixing angle because the representation of S for the A triplet has two degenerate eigenvalues. On the other hand, at τ = ω and τ = i ∞ , the squareof the mass matrix is diagonal one because ST and T of the A triplet have three different eigenvalues.Therefore, the modulus τ should deviate from the fixed point to reproduce the observed PMNS andCKM matrix elements. We present the explicit modular forms by performing Taylor expansion aroundfixed points. τ = i Let us discuss the behavior of modular forms at nearby τ = i . We consider linear approximation of themodular forms Y ( τ ), Y ( τ ) and Y ( τ ) by performing Taylor expansion around τ = i . We parametrize τ as: τ = i + (cid:15) , (25)where | (cid:15) | is supposed as | (cid:15) | (cid:28)
1. We obtain the ratios of the modular forms approximately as: Y ( τ ) Y ( τ ) (cid:39) (1 + (cid:15) ) (1 − √ , Y ( τ ) Y ( τ ) (cid:39) (1 + (cid:15) ) ( − √ , (cid:15) = 12 (cid:15) = 2 . i (cid:15) . (26)These approximate forms are agreement with exact numerical values within 0 . | (cid:15) | ≤ .
05. Detailsare given in Appendix C.1. The higher weight modular forms Y ( k ) i in Eqs. (13) and (14) are also givenin terms of (cid:15) and (cid:15) in Appendix C.1. τ = ω We perform linear approximation of the modular forms Y ( τ ), Y ( τ ) and Y ( τ ) by performing Taylorexpansion around τ = ω . We parametrize τ as: τ = ω + (cid:15) = −
12 + √ i + (cid:15) , (27)where we suppose | (cid:15) | (cid:28)
1. We obtain the ratios of modular forms approximately as: Y ( τ ) Y ( τ ) (cid:39) ω (1 + (cid:15) ) , Y ( τ ) Y ( τ ) (cid:39) − ω (1 + (cid:15) ) , (cid:15) = 12 (cid:15) = 2 . i (cid:15) . (28)These approximate forms are agreement with exact numerical values within 1 % for | (cid:15) | ≤ .
05. Detailsare given in Appendix C.2.The higher weight modular forms Y ( k ) i in Eqs. (13) and (14) are also given in terms of (cid:15) and (cid:15) inAppendix C.2. 7 .3 Modular forms towards τ = i ∞ We show the behavior of modular forms at large Im τ , where the magnitude of q = exp (2 πiτ ) issuppressed. Taking leading terms of Eq. (11), we can express modular forms approximately as: Y ( τ ) (cid:39) p (cid:15) , Y ( τ ) (cid:39) − p (cid:15) , Y ( τ ) (cid:39) − p (cid:15) , p = e πi Re τ , (cid:15) = e − π Im τ . (29)Indeed, we obtain (cid:15) = 3 . × − for Im τ = 2. The leading correction is (cid:15) = 0 . Y ( τ ) whileother corrections of (cid:15) and (cid:15) is negligibly small. Then, | Y (2 i ) | (cid:39) . , | Y (2 i ) | (cid:39) . , | Y (2 i ) | (cid:39) . , (30)which agree with exact values within 0 . Y ( k ) i in Eqs. (13) and (14) arealso given in terms of p and (cid:15) approximately in Appendix C.3. A modular invariance Let us discuss models of the lepton mass matrices. There are freedoms for the assignments of irreduciblerepresentations of A and modular weights to charged leptons and Higgs doublets. The simplest as-signment has been given in the conventional A model [17, 18], in which three left-handed leptons arecomponents of the triplet of the A group, but three right-handed charged leptons, ( e c , µ c , τ c ) are threedifferent singlets ( , (cid:48)(cid:48) , (cid:48) ) of A , respectively.Supposing neutrinos to be Majorana particles, we present the neutrino mass matrix through theWeinberg operator. The simplest one is given by assigning the A triplet and weight − singlets with weight 0. On the other hand, thecharged lepton mass matrix depends on the assignment of weights for the right-handed charged leptons.If the weight is 0 for all right-handed charged leptons, the charged lepton mass matrix are given interms of only the weight 2 modular forms of Eq. (10). That is the simplest one.Alternatively, we also consider weight 4 and 6 modular forms of Eqs. (13) and (14) in addition toweight 2 modular forms. The assignment is summarized in Table 2. L ( e c , µ c , τ c ) H u H d Y (6) , Y (6) (cid:48) Y (4) , Y (4) , Y ( )1 (cid:48) Y (2) SU (2) A (cid:48)(cid:48) , 1 (cid:48) ) 1 1 3 3 , (cid:48) − k I − , ,
0) 0 0 k = 6 k = 4 k = 2II: ( − , − , − k I for MSSM fields and modular forms. Let us begin with discussing the neutrino mass matrix. The superpotential of the neutrino mass term, w ν is given as: w ν = −
1Λ ( H u H u LL Y (k) r ) , (31)8here L is the left-handed A triplet leptons, H u is the Higgs doublet, and Λ is a relevant cut off scale.Since the left-handed lepton doublet has weight −
2, the superpotential is given in terms of modularforms of weight 4, Y (4) , Y (4) and Y (4) (cid:48) . By putting the VEV of the neutral component of H u , v u andtaking ( ν e , ν µ , ν τ ) for left-handed neutrinos, we have w ν = v u Λ ν e ν e − ν µ ν τ − ν τ ν µ ν τ ν τ − ν e ν µ − ν µ ν τ ν µ ν µ − ν τ ν e − ν e ν τ ⊗ Y (4) + ( ν e ν e + ν µ ν τ + ν τ ν µ ) ⊗ g ν Y (4) + ( ν e ν τ + ν µ ν µ + ν τ ν e ) ⊗ g ν Y (4) (cid:48) (cid:105) = v u Λ (cid:104) (2 ν e ν e − ν µ ν τ − ν τ ν µ ) Y (4)1 + (2 ν τ ν τ − ν e ν µ − ν µ ν e ) Y (4)3 + (2 ν µ ν µ − ν τ ν e − ν e ν τ ) Y (4)2 + ( ν e ν e + ν µ ν τ + ν τ ν µ ) g ν Y (4) + ( ν e ν τ + ν µ ν µ + ν τ ν e ) g ν Y (4) (cid:48) (cid:105) , (32)where Y (4) , Y (4) and Y (4) (cid:48) are given in Eq. (13), and g ν , g ν are complex parameters. The neutrinomass matrix is written as follows: M ν = v u Λ Y (4)1 − Y (4)3 − Y (4)2 − Y (4)3 Y (4)2 − Y (4)1 − Y (4)2 − Y (4)1 Y (4)3 + g ν Y (4) + g ν Y (4) (cid:48) LL . (33) The relevant superpotentials of the charged lepton masses are given for two cases as follows:I : w E = α e e c H d Y (2) L + β e µ c H d Y (2) L + γ e τ c H d Y (2) L , (34)II : w E = α e e c H d Y (6) L + α (cid:48) e e c H d Y (6) (cid:48) L + β e µ c H d Y (4) L + γ e τ c H d Y (2) L , (35)where L is the left-handed A triplet leptons, H d is the Higgs doublet.The charged lepton mass matrices M E are given as:I : M E = v d α e β e
00 0 γ e Y (2)1 Y (2)3 Y (2)2 Y (2)2 Y (2)1 Y (2)3 Y (2)3 Y (2)2 Y (2)1 RL , (36)(37)II : M E = v d α e β e
00 0 γ e Y (6)1 + g e Y (cid:48) (6)1 Y (6)3 + g e Y (cid:48) (6)3 Y (6)2 + g e Y (cid:48) (6)2 Y (4)2 Y (4)1 Y (4)3 Y (2)3 Y (2)2 Y (2)1 RL , (38)respectively, where coefficients α e , β e and γ e are real parameters while g e is complex one, and v d is VEVof the neutral component of H d .Model parameters of leptons are α e , β e , γ e , ( g e ), g ν and g ν in addition to the modulus τ . Weexamine these mass matrices around the fixed points.9 .2 Lepton mass matrix at τ = i τ = i We get the neutrino mass matrix at τ = i by putting modular forms in Table 1 into Eq. (33) as: M ν = v u Λ (6 − √ Y − − − − − + g + g , (39)where g = 6 √ − − √ g ν = √ g ν , g = 9 − √ − √ g ν = −√ g ν . (40)We move to the disgonal basis of S . By using the unitary transformation of Eq. (22), V S , the massmatrix is transformed as:ˆ M ν ≡ V ∗ S M ν V † S = v u Λ Y g + g g − g √ g √ g − g + g . (41)Off diagonal entries of (2,3) and (3,2) are remained as discussed in Eq. (19). At the limit of vanishing g and g , the lightest neutrino mass is zero and other ones are degenerated.In order to discuss the flavor mixing angle, we show ˆ M † ν ˆ M ν as M ν ≡ ˆ M † ν ˆ M ν = (cid:18) v u Λ Y (cid:19) | g + g | G ν + 6Re[ g ] − g ] √ (6 Re[ g ] + 2 i Im[ g ∗ g ])0 √ (6 Re[ g ] − i Im[ g ∗ g ]) G ν − g ] + 3Re[ g ] , (42)where G ν = 9 + | g | + | g | − Re[ g ∗ g ] . (43)The imaginary part of this matrix is factored out by using a phase matrix P ν as: (cid:18) v u Λ Y (cid:19) P ν | g + g | G ν + 6Re[ g ] − g ] √ (cid:112) g ]) + (Im[ g ∗ g ]) √ (cid:112) g ]) + (Im[ g ∗ g ]) G ν − g ] + 3Re[ g ] P ∗ ν , (44)where P ν = e − iφ ν , (45)with tan φ ν = Im[ g ∗ g ]3 Re[ g ] . (46)10n the other hand, mass eigenvalues m , m and m satisfy: m = | g + g | , m + m = 18 + 2( | g | + | g | ) − g ∗ g ) , m m = | − g − g + g g | , (47)in the unit of ( v u / Λ) Y . The mixing angle between the 2nd- and 3rd-family, θ ν is given as:tan 2 θ ν = 1 √ (cid:112) g ]) + (Im[ g ∗ g ]) Re[ g ] − g ] . (48)If we put Re[ g ] = 2Re[ g ], we obtain the maximal mixing angle θ ν = 45 ◦ . Thus, the large mixingangle is easily obtained by choosing relevant parameters g and g . It is also noticed that θ ν vanishesfor g = 0. Thus, θ ν could be 0–45 ◦ depending on g and g . τ = i As discussed in the previous subsubsection, the large θ ν is easily reproduced at τ = i . The large flavormixing angle between the 1st- and 2nd-family, θ ν is also realized at nearby τ = i . Mass matrix ofneutrinos in Eq. (33), M ν are corrected due to the deviation from the fixed point of τ = i . Puttingmodular forms of Eq. (26) (see also Appendix C.1) into M ν , the corrections to Eq. (42) are given byonly a small variable (cid:15) in Eq. (26) in the diagonal base of S . In the 1st order approximation of (cid:15) , thecorrection M ν is given as: M ν = (cid:18) v u Λ Y (cid:19) δ ν δ ν δ ∗ ν δ ν δ ν δ ∗ ν δ ∗ ν δ ν , (49)where δ νi are given in terms of (cid:15) , g and g . Due to the 1st order perturbation of (cid:15) , we can obtainthe mixing angle θ ν , which vanishes in the 0th order of perturbation. In order to estimate the flavormixing angles, we present relevant δ νi explicitly as: δ ν = − √ { ( g ∗ + g ∗ )[(1 + √ (cid:15) + (cid:15) ] + (cid:15) ∗ [(3 + g )(1 + √ − g ] + (cid:15) ∗ [(3 + g ) + (1 − √ g ] }(cid:39) − . g ∗ + g ∗ ) (cid:15) − (10 .
04 + 3 . g − . g ) (cid:15) ∗ ,δ ν = 1 √ { ( g ∗ + g ∗ )[(3 − √ (cid:15) + (2 √ − (cid:15) ] + (cid:15) ∗ [(3 − √ − g ) − √ g ]+ (cid:15) ∗ [(2 √ − − g ) − (3 − √ g ] } (cid:39) . g ∗ + g ∗ ) (cid:15) + (2 . − . g − . g ) (cid:15) ∗ , (50)where (cid:15) = 2 . i(cid:15) , and (cid:15) = 2 (cid:15) in Eq. (26) is used in last approximate equalities.Let us estimate the mixing angles, θ ν and θ ν in terms of δ ν and δ ν . The eigenvectors of the lowestorder in M ν is given, u (0) ν = , u (0) ν = θ ν − sin θ ν e − iφ ν , u (0) ν = θ ν cos θ ν e − iφ ν , (51)for eigenvalues m , m and m of Eq. (47), respectively.11e can calculate corrections of eigenvectors in the 1st order of (cid:15) . In order to estimate the non-vanishing mixing angle between the 1st- and 2nd-family, we calculate the eigenvoctor of 1st order, u (1) ν ,which is given u (1) ν = C ν u (0) ν + C ν u (0) ν , (52)where C νji = (cid:104) u (0) νj |M ν | u (0) νi (cid:105) m j − m i . (53)Therefore, the non-vanishing (1-2) mixing appears at the first component of u (1) ν as: u (1) ν [1 ,
1] = C ν = δ ∗ ν cos θ ν − δ ∗ ν sin θ ν e iφ ν m − m . (54)Here, we take 2 g = g , which leads to the maximal mixing θ ν = 45 ◦ as seen in Eq. (48). Then, themass squares are given from Eq. (47) as: m = 9 | g | , m = 3 (cid:16) | g | − √ | Re g | (cid:17) , m = 3 (cid:16) | g | + 2 √ | Re g | (cid:17) , (55)in the unit of ( v u / Λ) Y . Supposing NH of neutrino masses, we take the observed ratio of ∆ m / ∆ m =34 .
2, which leads to g = 0 .
61 by neglecting the imaginary part of g . Then, δ ∗ ν and δ ∗ ν are given interms of (cid:15) by using (cid:15) = 2 . i (cid:15) in Eq. (26) as follows: δ ∗ ν = − . i (cid:15) − . i (cid:15) ∗ , δ ∗ ν = − . i (cid:15) − . i (cid:15) ∗ . (56)Neglecting δ ν because of | δ ∗ ν | (cid:29) | δ ∗ ν | , we have u (1) ν [1 , (cid:39) δ ∗ ν cos θ ν m − m = − i . (cid:15) + 12 . (cid:15) ∗ . √ , (57)where θ ν = 45 ◦ is put. We obtain u (1) ν [1 , (cid:39) .
55 ( θ ν (cid:39) ◦ ) by putting (cid:15) = 0 . i . Thus, the large(1-2) mixing angle could be reproduced by the correction terms in the neutrino mass matrix due to thesmall deviation from τ = i . It is remarked that the sum of three neutrino masses is around 110 meVtaking 2 g = g = 1 . u (1) ν [1 ,
1] = C ν = δ ∗ ν sin θ ν + δ ∗ ν cos θ ν e iφ ν m − m . (58)Since ( m − m ) is 30 times larger than ( m − m ), u (1) ν [1 ,
1] is suppressed compared with u (1) ν [1 , O (0 . θ ∼ .
15 of the PMNS matrixshould be derived from the charged lepton sector. It is noted that the correction to the (2-3) mixing isalso O (0 .
01) because u (1) ν [2 ,
1] is suppressed due to the large ( m − m ).We can also discuss the case of IH of the neutrino masses by taking ∆ m / ∆ m = − .
2. Thelarge mixing angles θ ν and θ ν are obtained if we take g = g / − .
45. The sum of three neutrinomasses is around 90 meV.Thus, our neutrino mass matrix is attractive one at nearby τ = i . Therefore, we should examinethe contribution from the charged lepton sector carefully for both NH and IH of neutrinos.12 .2.3 Charged lepton mass matrix I at τ = i The charged lepton mass matrix I is the simplest one, which is given by using only weight 2 modularforms. It is given at fixed points of τ = i in the base of S of Eq. (9) as follows: M E = v d α e β e
00 0 γ e Y Y Y Y Y Y Y Y Y = ˜ α e β e
00 0 ˜ γ e − √ − √ − √ − √ − √ − √ , (59)where ˜ α e = v d Y α e , ˜ β e = v d Y β e and ˜ γ e = v d Y γ e . We move to the diagonal basis of S . By using theunitary transformation of Eq. (17), the mass matrix is transformed as presented in Eq. (23). Then, wehave: M E ≡ V S M † E M E V † S = 32 α e + 2(2 − √
3) ˜ β e + (7 − √ γ e − (2 − √ α e − β e + ˜ γ e )0 − (2 − √ α e − β e + ˜ γ e ) (7 − √
3) ˜ α e + 2(2 − √
3) ˜ β e + ˜ γ e , (60)which is a real matrix with rank 2.Since the lightest charged lepton is massless at τ = i , the small deviation from τ = i is required toobtain the electron mass. It is remarked that the flavor mixing between 2nd- and 3rd-family appearsat the fixed point τ = i as seen in Eq. (60). It is given as:tan 2 θ e = − − √
3) ( ˜ α e − β e + ˜ γ e )2(2 √ −
3) (˜ γ e − ˜ α e ) = − √ α e − β e + ˜ γ e ˜ γ e − ˜ α e , (61)which leads to θ e (cid:39) ◦ for ˜ α e (cid:29) ˜ β e , ˜ γ e , θ e (cid:39) − ◦ for ˜ γ e (cid:29) ˜ β e , ˜ α e , θ e (cid:39) ◦ for ˜ β e (cid:29) ˜ α e (cid:29) ˜ γ e and θ e (cid:39) − ◦ for ˜ β e (cid:29) ˜ γ e (cid:29) ˜ α e , respectively. This mixing angle leads to θ of the PMNS matrix bycooperating with the neutrino mixing angle θ ν in Eq. (48). τ = i In order to obtain the electron mass, τ should be deviated a little bit from the fixed point τ = i . Byusing modular forms at nearby τ = i in Eq. (26), we obtain the additional contribution M E to M E in Eq. (60) of order (cid:15) as: M E (cid:39) δ e δ e δ ∗ e δ e δ e δ ∗ e δ ∗ e δ e , (62)where δ ei are given in terms of (cid:15) , ˜ α e , ˜ β e and ˜ γ e . In order to estimate the flavor mixing angles, wepresent relevant δ ei as: δ e = 1 √ { [( √ − (cid:15) ∗ + ( √ − (cid:15) ∗ ] ˜ α e + [(4 − √ (cid:15) ∗ + (3 √ − (cid:15) ∗ ] ˜ β e + [(3 √ − (cid:15) ∗ + (7 − √ (cid:15) ∗ ]˜ γ e } (cid:39) √ (cid:15) ∗ [(3 √ −
5) ˜ α e + 2(2 √ −
3) ˜ β e + (9 − √ γ e ] , (63) δ e = 1 √ { [(9 − √ (cid:15) ∗ + (7 √ − (cid:15) ∗ ] ˜ α e + [(4 √ − (cid:15) ∗ + (9 − √ (cid:15) ∗ ] ˜ β e + [( √ − (cid:15) ∗ + (3 − √ (cid:15) ∗ ]˜ γ e } (cid:39) √ (cid:15) ∗ [(3 √ −
5) ˜ α e + 2(2 − √
3) ˜ β e + (1 − √ γ e ] , (64)13here (cid:15) = 2 (cid:15) in Eq. (26) is used in the last approximate equalities. The mixing angle of 1st- and2nd-family as:tan 2 θ e = 2 | δ e | [ ˜ α e + 2(2 − √
3) ˜ β e + (7 − √ γ e ] (cid:39) √ − √ − √ | (cid:15) ∗ | (cid:39) √ √ | (cid:15) ∗ | (cid:39) . | (cid:15) ∗ | , (65)where the denominator comes from the (2 ,
2) element of Eq. (60). In the last approximate equality, wetake ˜ γ e (cid:29) ˜ α e , ˜ β e , which is the case in the numerical fits of section 7. We estimate θ e to be 0 .
22 at | (cid:15) | = | . i (cid:15) | = 0 .
1. This magnitude of θ e leads to θ (cid:39) .
15 of the PMNS matrix by cooperatingwith the neutrino mixing angle θ ν in Eq. (48). The mixing angle between 1st- and 3rd-family θ e isfound to be much smaller than θ e in the similar calculation.In conclusion, the charged lepton mass matrix I combined with the neutrino mass matrix of Eq. (33)is expected to be consistent with the observed three PMNS mixing angles at nearby τ = i . Indeed, thiscase works well for both NH and IH as seen in numerical results of section 7. The output of the DiracCP violating phase and the sum of neutrino masses will tested in the future experiments. τ = i We discuss another charged lepton mass matrix II at τ = i , which is : M E = v d α e β e
00 0 γ e Y (6)1 + g e Y (cid:48) (6)1 Y (6)3 + g e Y (cid:48) (6)3 Y (6)2 + g e Y (cid:48) (6)2 Y (4)2 Y (4)1 Y (4)3 Y (2)3 Y (2)2 Y (2)1 = v q ˜ α e β e
00 0 ˜ γ e √ − g e (7 √ −
12) 12 − √ g e (9 − √
3) 5 √ − g e (3 − √ − √ − √ , (66)where ˜ α e = 3 v d Y α e , ˜ β e = (6 − √ v d Y β e and ˜ γ e = v d Y γ e .We move to the diagonal basis of S . The mass matrix M † E M E is transformed by the unitarytransformation V S as: M E ≡ V S M † E M E V † S = 32 β e A ˜ γ e + 3( A + B e + | g e | C ) ˜ α e − D ˜ γ e − B e + Ag e + Cg ∗ e ) ˜ α e )0 − D ˜ γ e − B e + Ag ∗ e + Cg e ) ˜ α e ) ˜ γ e + 3( C + B e + | g e | A ) ˜ α e , (67)where A = 7 − √ , B = 26 − √ , C = 97 − √ , D = 2 − √ ,B e = B ( g e + g ∗ e ) = 2 B Re[ g e ] , B e = B (1 + | g e | ) , A = C , D = A , A + C = 4 B . (68)The flavor mixing between the 2nd- and 3rd-family appears at the τ = i as well as the charged leptonmass matrix I. 14he mass eigenvalues satisfy m e = 3 ˜ β e , m e m e = 81(97 − √
3) ˜ α e ˜ γ e ,m e + m e = 6(2 − √ γ e + 3(78 − √ g e ] + | g e | ) ˜ α e . (69)The imaginary part of the matrix in Eq. (67) is factored out by using a phase matrix P e as:32 P e β e A ˜ γ e + 3( A + B e + | g e | C ) ˜ α e − (cid:112) [ D ˜ γ e + 3( B e + E e ) ˜ α e )] + F e ˜ α e − (cid:112) [ D ˜ γ e + 3( B e + E e ) ˜ α e )] + F e ˜ α e ˜ γ e + 3( C + B e + | g e | A ) ˜ α e P ∗ e , (70)where E e = ( A + C )Re[ g e ] , F e = ( A − C )Im[ g e ] , (71)and P e = e − iφ e , (72)with tan φ e = F e ˜ α e D ˜ γ e + 3( B + E e ) ˜ α e . (73)The mixing angle θ e is given as:tan 2 θ e = − (cid:112) [ D ˜ γ e + 3( B e + E e ) ˜ α e )] + F e ˜ α e (2 √ − γ e + 3(45 − √ − | g e | ) ˜ α e . (74)Neglecting the imaginary part of g ( g = Re[ g ]), it is given simply as:tan 2 θ e = − √ γ e + 3(7 − √ g e + g e ) ˜ α e ˜ γ e − − √ − g e ) ˜ α e . (75)We take ˜ β e (cid:28) ˜ α e , ˜ γ e due to the mass hierarchy of the charged lepton masses. There are two possiblechoices of ˜ α e (cid:28) ˜ γ e and ˜ γ e (cid:28) ˜ α e .In the case of ˜ α e (cid:28) ˜ γ e , tan 2 θ e (cid:39) − √ − √ g e ) ˜ α e ˜ γ e ] . (76)At the limit of ˜ α e / ˜ γ e = 0, we obtain θ e = − ◦ .On the other hand, in the case of α e (cid:29) γ e , Eq. (75) turns totan 2 θ e (cid:39) √ g e + g e − g e , (77)which gives | θ e | = 0–45 ◦ by choosing relevant g e . Thus, the large θ e is obtained easily.15 .2.6 Charged lepton mass matrix II at nearby τ = i The mass matrix of the charged lepton in Eq. (66), M E is corrected due to the deviation from the fixedpoint of τ = i . In the 1st order approximation of (cid:15) , the correction M E to M E of Eq. (67) is givenby the following matrix: M E = δ e δ e δ e δ ∗ e δ e δ e δ ∗ e δ ∗ e δ e , (78)where δ ei are given in terms of (cid:15) , g e , ˜ α e , ˜ β e and ˜ γ e . By the 1st order perturbation of (cid:15) , we can obtainthe mixing angle θ e , which vanishes in the 0th order of perturbation. In order to estimate the flavormixing angles, we present relevant δ ei as: δ e = 3 √ α e ( g ∗ e − { [ (11 √ −
19) + (41 √ − g e ] (cid:15) ∗ − [ (15 √ −
26) + (56 √ − g e ] (cid:15) ∗ } + 1 √ γ e [ (3 √ − (cid:15) ∗ + (7 − √ (cid:15) ∗ ] (cid:39) (0 .
193 + 0 . g e ) ˜ α e ( g ∗ e − (cid:15) ∗ + 0 . γ e (cid:15) ∗ ,δ e = 1 √ α e ( g ∗ e − { [ 3(71 √ − √ − g e ] (cid:15) ∗ − [ 3(97 √ − √ − g e ] (cid:15) ∗ } + 1 √ γ e [ (1 − √ (cid:15) ∗ + ( √ − (cid:15) ∗ ] (cid:39) − (0 .
052 + 0 . g e ) ˜ α e ( g ∗ e − (cid:15) ∗ − . γ e (cid:15) ∗ , (79)where O ( ˜ β e ) is neglected and (cid:15) = 2 (cid:15) of Eq. (26) is taken in last approximate equalities.Let us discuss the mixing angles of θ e and θ e of the charged lepton flavors, which vanish in theleading terms of the mass matrix. As seen in Eq. (79), both δ e and δ e are of O ( ˜ α e , ˜ γ e ) × (cid:15) for g e = O (1).Suppose ˜ γ e (cid:28) ˜ α e to realize the hierarchy of charged lepton masses in Eq. (69) . Then, we have masseigenvalues from Eq. (69) as: m e = 3 ˜ β e , m e (cid:39) − √ g e ] + | g e | ˜ γ e , m e (cid:39) − √ g e ] + | g e | ) ˜ α e , (80)which lead to m e m e (cid:39) √ g e ] + | g e | ) ˜ γ e ˜ α e . (81)The mixing angles between 1st- and 2nd-family θ e and between 1st- and 3rd-family θ e are givenapproximately as: θ e (cid:39) (cid:12)(cid:12)(cid:12)(cid:12) δ e m e (cid:12)(cid:12)(cid:12)(cid:12) , θ e (cid:39) (cid:12)(cid:12)(cid:12)(cid:12) δ e m e (cid:12)(cid:12)(cid:12)(cid:12) , (82)where δ e (cid:39) (0 .
193 + 0 . g e ) ˜ α e ( g ∗ e − (cid:15) ∗ , δ e (cid:39) − (0 .
052 + 0 . g e ) ˜ α e ( g ∗ e − (cid:15) ∗ , (83) Indeed, a successful numerical result is obtained for ˜ γ e (cid:28) ˜ α e in section 7. θ e and θ e . The mixing angle of θ e is given as: θ e (cid:39) (cid:12)(cid:12)(cid:12)(cid:12) (0 .
193 + 0 . g e )( g ∗ e − − √
3) (2 + 2Re[ g e ] + | g e | ) ˜ α e ˜ γ e (cid:15) ∗ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (0 .
193 + 0 . g e )( g ∗ e − √ g e ] + | g e | ) m e m e (cid:15) ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:39) . | (0 .
193 + 0 . g e )( g ∗ e − | g e ] + | g e | | (cid:15) ∗ | × , (84)where the mass ration of Eq. (81) is used to remove the ratio ˜ γ e / ˜ α e . In the last equality, observedmasses of the tauon and the muon are input. Let us suppose the magnitude of | (cid:15) ∗ | to be 0 .
02 as atypical value. As seen in Eq. (84), θ e depends on g e . Indeed, θ e vanishes at g e = 1 or − .
62 while itis of order one if | g e | (cid:28) | g e | (cid:29)
1. On the othe hand, θ e is suppressed due to the factor of 1 /m e as seen Eq. (82).In conclusion, the charged lepton mass matrix II combined with the neutrino mass matrix of Eq. (33)is expected to be consistent with the observed three PMNS mixing angles at nearby τ = i as well ascharged lepton mass matrix I. Indeed, this case works well for NH, but it leads to the sum of neutrinomasses larger than 120 meV for IH as seen in numerical results of section 7. τ = ω τ = ω Let us consider the neutrino mass matrix at τ = ω , where there exists the residual symmetry of the A group Z ST = { I, ST, ( ST ) } . By putting the modular forms in Table 1 into Eq. (33), the neutrino massmatrix is written as: M ν = v u Λ Y − ω ω − ω − ω − ω − ω + 94 ωg ν , (85)where the g ν term of Eq. (33) disappears because of Y (4) = 0 at τ = ω . We move to the diagonalbasis of ST . By using the unitary transformation of Eq. (22), V ST or V ST , the neutrino mass matrixis transformed as: M ν ≡ V ST ˆ M † ν ˆ M ν V † ST = (cid:18) v u Λ Y (cid:19) | g ν | | − g ν |
00 0 | − g ν | . (86)The neutrino mass matrix is diagonal and two neutrinos are degenerated at τ = ω . Three neutrinomasses are degenerate if g ν = − .
5. Then, large flavor mixing angles are possibly reproduced if smalloff disgonal elements are generated by the deviation from τ = ω . τ = ω Neutrino mass matrix in Eq. (33), M ν is corrected due to the deviation from the fixed point of τ = ω .After putting modular forms of Eq. (28) and moving to the diagonal base of ST by V ST , the corrections17o Eq. (86) are given by only a small variable (cid:15) of in Eq. (28). In the 1st order approximation of (cid:15) , thecorrection M ν to M ν of Eq. (86) is given by the following matrix: M ν = (cid:18) v u Λ Y (cid:19) δ ν δ ν δ ν δ ∗ ν δ ν δ ν δ ∗ ν δ ∗ ν δ ν , (87)where δ νi are given in terms of (cid:15) , g ν and g ν . By the 1st order perturbation of (cid:15) , we can obtain themixing angle θ ν , which vanishes in the 0th order of perturbation. In order to estimate the flavor mixingangles, we present off diagonal elements, δ ν , δ ν and δ ν as: δ ν = 34 (2 + g ∗ ν ) (cid:15) + 38 (1 + 6 g ∗ ν )(1 − g ν ) (cid:15) ∗ −
218 (2 + g ∗ ν ) (cid:15) −
34 (4 − g ∗ ν )(1 − g ν ) (cid:15) ∗ (cid:39) −
92 (2 + g ∗ ν ) (cid:15) −
98 (5 − g ∗ ν )(1 − g ν ) (cid:15) ∗ ,δ ν = 38 (1 + 6 g ν )(2 + g ∗ ν ) (cid:15) + 34 (1 − g ν ) (cid:15) ∗ −
34 (4 − g ν )(2 + g ∗ ν ) (cid:15) −
218 (1 − g ν ) (cid:15) ∗ (cid:39) −
98 (5 − g ν )(2 + g ∗ ν ) (cid:15) −
92 (1 − g ν ) (cid:15) ∗ ,δ ν = 34 (1 − g ∗ ν )(1 − g ν ) (cid:15) ∗ −
32 (1 − g ∗ ν ) (cid:15) −
34 (8 + 3 g ∗ ν )(1 − g ν ) (cid:15) ∗ + 214 (1 − g ∗ ν ) (cid:15) (cid:39) −
94 (5 + 3 g ∗ ν )(1 − g ν ) (cid:15) ∗ + 9(1 − g ∗ ν ) (cid:15) , (88)where (cid:15) = 2 . i (cid:15) , and (cid:15) = 2 (cid:15) of Eq. (28) is used for last approximate equalities. If we move to thediagonal base of ST by using V ST instead of V ST , we obtain the corrections by exchanging the aboveresults as: δ ν ↔ δ ν , δ ν ↔ δ ∗ ν . (89)Indeed, we move to the diagonal base of ST by using V ST for the charged lepton mass matrix II insection 5.3.5.It is noticed that the off-diagonal elements are enhanced by large coefficients in front of (cid:15) and (cid:15) ∗ .For example, | δ ν | could be comparable to diagonal element if | (cid:15) | = 0 . g ν and g ν .The (1–3) mixing angle is relatively small due to the fixed mass square difference ∆ m . On the otherhand, the sum of neutrino masses may increase if mass eigenvalues become quasi-degenerate. Then, itscosmological upper-bound provides a crucial test for the lepton mass matrices. Therefore, we shouldexamine the contribution from the charged lepton sector carefully for both NH and IH of neutrinos tojudge it working well or not. Indeed, we will see in section 7 that the model of the charged leptonmass matrix I is excluded by the sum of neutrino masses while the model with the charged lepton massmatrix II is consistent with it for both NH and IH of neutrino masses.18 .3.3 Charged lepton mass matrix I at τ = ω We discusses the charged lepton mass matrix I at the fixed point τ = ω by using modular forms inTable 1. In the base of S and T of Eq. (9), the charged lepton mass matrix I in Eq. (36) is given as: M E = ˜ α e β e
00 0 ˜ γ e − ω ωω − ω − ω ω , (90)where ˜ α e = v d Y α e , ˜ β e = v d Y β e and ˜ γ e = v d Y γ e . By using the unitary transformation of Eq. (22), V ST , like the case of the neutrino mass matrix, M † E M E is transformed as: M E ≡ V ST M † E M E V † ST = 94 ˜ α e γ e
00 0 ˜ β e . (91)It is remarked that it is diagonal one as well as the neutrino mass matrix in Eq. (86). τ = ω The charged lepton mass matrix in Eq. (90), M E is corrected due to the deviation from the fixed pointof τ = ω . After putting modular forms of Eq. (28) and moving to the diagonal base of ST by V ST , thecorrection M ν to M ν of Eq. (91) is given in the 1st order approximation of (cid:15) as: M E = δ e δ e δ e δ ∗ e δ e δ e δ ∗ e δ ∗ e δ e , (92)where δ e = i ˜ α e ( (cid:15) − (cid:15) ) + 12 i ˜ γ e ( (cid:15) ∗ + (cid:15) ∗ ) = 32 i ˜ γ e (cid:15) ∗ , (93) δ e = 12 i ˜ α e ( (cid:15) + (cid:15) ) + i ˜ β e ( (cid:15) ∗ − (cid:15) ∗ ) = 32 i ˜ α e (cid:15) , (94) δ e = − i ˜ γ e ( (cid:15) − (cid:15) ) − i ˜ β e ( (cid:15) ∗ + (cid:15) ∗ ) = − i ˜ β e (cid:15) ∗ , (95)where (cid:15) = 2 (cid:15) of Eq. (28) is used for last equalities. Due to ˜ β e (cid:29) ˜ γ e (cid:29) ˜ α e , mixing angles θ eij is easilyobtained by using (cid:15) = 2 . i(cid:15) as follows: θ e (cid:39) θ e (cid:39) | (cid:15) | (cid:39) . | (cid:15) | , (96)which are smaller than 0 .
1, moreover, θ e is highly suppressed due to the factor ˜ α e / ˜ β e . Thus, theflavor mixing angles of the charged lepton is very small at nearby the fixed point τ = ω . The PMNSmixing angles come from mainly the neutrino sector in this case. Therefore, the increase of the sum ofneutrino masses is unavoidable since mass eigenvalues become quasi-degenerate in order to reproducelarge mixing angles. 19 .3.5 Charged lepton mass matrix II at τ = ω We discusses the charged lepton mass matrix II at the fixed point τ = ω by using modular forms inTable 1. The charged lepton mass matrix II in Eq. (38) is given as: M E = ˜ α e β e
00 0 ˜ γ e g e − ω g e − ωg e − ω ω − ω ω , (97)where ˜ α e = (9 / v d Y α d , ˜ β e = (3 / v d Y β q and ˜ γ e = v d Y γ e . By using the unitary transformationof Eq. (22), V ST , which is different from the case of the charged lepton mass matrix I, M † E M E istransformed as: M E ≡ V ST M † E M E V † ST = 94 α e | g e | + ˜ β e + ˜ γ e , (98)which gives two massless charged leptons. τ = ω The charged lepton mass matrix in Eq. (97), M E is corrected due to the deviation from the fixed pointof τ = ω . After putting modular forms of Eq. (28) and moving to the diagonal base of ST by V ST , thecorrection M ν to M ν of Eq. (98) is given as: M E = δ e δ e δ ∗ e δ ∗ e δ e (99)where δ ei are given in terms of (cid:15) , g e , ˜ α e , ˜ β e and ˜ γ e . By the 1st order perturbation of (cid:15) , we can obtainthe mixing angles θ e and θ e , which vanish in the 0th order of perturbation. In order to estimate theflavor mixing angles, we present δ e and δ e as: δ e = − α e g e (2 + g ∗ e )( (cid:15) ∗ + (cid:15) ∗ ) + 16 ˜ β e ( (cid:15) ∗ − (cid:15) ∗ ) + 12 i ˜ γ e ( (cid:15) ∗ + (cid:15) ∗ ) (cid:39) [ − α e g e (2 + g ∗ e ) −
52 ˜ β e + 32 i ˜ γ e ] (cid:15) ∗ ,δ e = ˜ α e | g e | ( − (cid:15) ∗ + 2 (cid:15) ∗ ) + ˜ β e ( 13 (cid:15) ∗ − (cid:15) ∗ ) + i ˜ γ e ( (cid:15) ∗ − (cid:15) ∗ ) (cid:39) − β e (cid:15) ∗ , (100)where (cid:15) = 2 (cid:15) of Eq. (28) is used in last approximate equalities. If ˜ β e (cid:29) ˜ α e | g e | , ˜ γ e , mixing angles θ e and θ e are given : θ e (cid:39) | (cid:15) | (cid:39) | (cid:15) | , θ e (cid:39) | (cid:15) | (cid:39) | (cid:15) | , (101)where (cid:15) = 2 . i (cid:15) in Eq. (28) is taken. Therefore, these mixing angles are at most 0 .
1. It is noticed that θ e vanishes. 20n the other hand, if ˜ α e | g e | (cid:29) ˜ β e , ˜ γ e mixing angles θ e is given : θ e (cid:39) (cid:12)(cid:12)(cid:12)(cid:12) g ∗ e g e (cid:15) (cid:12)(cid:12)(cid:12)(cid:12) (cid:39) . (cid:12)(cid:12)(cid:12)(cid:12) g e (cid:15) (cid:12)(cid:12)(cid:12)(cid:12) , (102)where | g e | is supposed to be much smaller than 1 in the last equality. Therefore, θ e is enhanced bytaking | g e | (cid:39) .
1. It could be of order 1 if | (cid:15) | = 0 .
05. Thus, the flavor mixing θ e contribute significantlyto the PMNS mixing angle θ .Indeed, we obtain the allowed region of | (cid:15) | (cid:39) . | g e | (cid:39) . | (cid:15) | (cid:39) .
15 is obtained with large | g e | = 5–10. τ = i ∞ τ = i ∞ Let us consider the neutrino mass matrix at τ = i ∞ , where there exists the residual symmetries of theA group Z S = { I, T, T } . By putting the modular forms in Table 1 into Eq. (33), the neutrino massmatrix is written as: M ν = v u Λ Y − − + g ν , (103)where the g ν term of Eq. (33) disappears because of Y (4) (cid:48) = 0 at τ = i ∞ . Since T is already in thediagonal base as seen in Eq. (9), we can write down M † ν M ν straightforward as follows: M ν ≡ M † ν M ν = (cid:18) v u Λ Y (cid:19) | g ν | | − g ν |
00 0 | − g ν | , (104)which is a diagonal matrix as well as the neutrino mass matrix at τ = ω in Eq. (86). Three neutrinomasses are degenerate if g ν = − .
5. Then, large flavor mixing angles are possibly reproduced if smalloff diagonal elements are generated due to finite effect of τ . τ = i ∞ Neutrino mass matrix in Eq. (33), M ν is given from the finite correction of τ = i ∞ . Taking account ofmodular forms of Eq. (29), the corrections to Eq. (104) are given by only a small variable (cid:15) of in Eq. (29).In the 1st order approximation of (cid:15) , the correction M ν to M ν of Eq. (104) is given in terms of δ = − e πi Re τ e − π Im τ . (105)It is given by the following matrix: M ν (cid:39) (cid:18) v u Λ Y (cid:19) − δ ∗ (1 − g ν )(1 + 2 g ∗ ν ) δ (2 + g ∗ ν )(1 + 2 g ν ) − δ (1 − g ∗ ν )(1 + 2 g ν ) 0 2 δ ∗ (1 − g ν )(1 − g ∗ ν ) δ ∗ (2 + g ν )(1 + 2 g ∗ ν ) 2 δ (1 − g ∗ ν )(1 − g ν ) 0 . (106)21f we take Im τ = 1 .
6, we get | δ | (cid:39) .
21, which is derived in Eq. (105). Thus, the large (2–3) mixingangle is easily obtained since 2nd and 3rd eigenvalues are degenerated as seen in Eq. (104). The large(1–2) mixing angle is also possible by choosing relevant g ν and g ν . The (1–3) mixing angle is expectedrelatively small due to the fixed mass square difference ∆ m . Then, the cosmological upper-bound ofthe sum of neutrino masses is a crucial criterion to test neutrino mass matrices. In section 7, we willsee that both charged lepton mass matrix I and II satisfy the sum of neutrino masses less than thecosmological upper-bound 120 meV for NH of neutrinos, but they do not satisfy it for IH. τ = i ∞ The charged lepton mass matrices of I and II in Eqs. (36) and (38) are simple at τ = i ∞ since themodular forms of weight 2, 4 and 6 are given in the T diagonal base. Putting them of Table 1 into thecharged lepton mass matrices in Eqs. (36) and (38), we obtain M E = ˜ α e β e
00 0 ˜ γ e , (107)where ˜ α e = v d Y α e , ˜ β e = v d Y β e and ˜ γ e = v d Y γ e for the case I and ˜ α e = v d Y α e , ˜ β e = v d Y β e and˜ γ e = v d Y γ e for the case II. The mass matrix M † E M E is given as: M E ≡ M † E M E = ˜ α e β e
00 0 ˜ γ e . (108)The flavor mixing appears through the finite effect of Im [ τ ]. τ = i ∞ The charged lepton mass matrices of I and II in Eqs. (36) and (38) is given from the finite correctionof τ = i ∞ . By using modular forms of Eq. (29), the corrections to Eq. (108) are given by only a smallvariable (cid:15) of in Eq. (29). In the 1st order approximation of (cid:15) , the corrections M E to M E of Eq. (108)is given in terms of δ of Eq. (105) as: M E (cid:39) δ ∗ ˜ β e δ ˜ α e δ ˜ β e δ ∗ ˜ γ e δ ∗ ˜ α e δ ˜ γ e , (109)for the charged lepton mass matrix I. On the other hand, for the charged lepton mass matrix II, it is: M E (cid:39) − δ ∗ ˜ β e (1 + 2 g e ) δ ˜ α e − δ ˜ β e δ ∗ ˜ γ e (1 + 2 g ∗ e ) δ ˜ α e δ ˜ γ e . (110)In both charged lepton mass matrices I and II, (1–2) and (2–3) families mixing angles θ e , θ e , aregiven as: θ e (cid:39) | δ ∗ | ˜ β e ˜ β e (cid:39) | δ | , θ e (cid:39) | δ ∗ | ˜ γ e ˜ γ e = | δ | , (111)22espectively, where ˜ γ e (cid:29) ˜ β e (cid:29) ˜ α e . If we take Im τ = 1 .
6, the magnitude of θ e (cid:39) | δ | (cid:39) .
21. Thismagnitude of θ e contributes significantly to the PMNS mixing angle θ . On the other hand, the mixingangle θ e between 1st- and 3rd-family is highly suppressed due to the factor ˜ α e / ˜ γ e .It is remarked that the mass matrix of Eq. (110) is agreement with Eq.(109) in the case of | g e | (cid:28) θ and θ togetheralthough the charged lepton mass matrix I is successful to reproduce the observed PMNS angles.Alternatively, the observed PMNS mixing angles can be reproduced in the charged lepton massmatrix II if a large mixing angle for θ e is obtained by taking | g e | (cid:29) α e (cid:29) ˜ β e , ˜ γ e . This case isshown numerically in section 7. A modular invariance If flavors of quarks and leptons are originated from a same two-dimensional compact space, the leptonsand quarks have same flavor symmetry and the same value of the modulus τ . Therefore, the modularsymmetry provides a new approach towards the unification of quark and lepton flavors. In order toinvestigate the possibility of the quark/lepton unification, we discuss a A modular invariant flavormodel for quarks together with the lepton sector. We take the assignments of A irreducible representations and modular weights for quarks like thecharged leptons. That is, three left-handed quarks are components of the triplet of the A group,but three right-handed quarks, ( u c , c c , t c ) and ( d c , s c , b c ) are three different singlets ( , (cid:48)(cid:48) , (cid:48) ) of A ,respectively. Quark mass matrices depend on modular weights of the left-handed and the right-handedquarks since the sum of their weight including modular forms should vanish. Let us fix the weights ofleft-handed quarks to be − −
4. Therefore, the up-type quark mass matrix is given in terms of the weight6 modular forms, in which two different triplet modular forms are available. This model has alreadydiscussed in Ref. [52] numerically. We reexamine the flavor structure of these quark mass matrices atnearby fixed point explicitly, and then we can understand why this model works well.Alternatively, another quark mass matrix is also considered as the case II. In this case, weightsof the right-handed up-type quarks and the down-type ones are same ones, which are also discussednumerically in Ref. [82]. The modular forms of weight 6 join only in the 1st-family.23 ( u c , c c , t c ), ( d c , s c , b c ) H q Y (6) , Y (6) (cid:48) Y (4) Y (2) SU (2) A (cid:48)(cid:48) , 1 (cid:48) ) 1 3 3 3 − k I − − , − , − , ,
0) 0 k = 6 k = 4 k = 2II :( − , − , − , − , − k I for MSSM fields and modular forms.The relevant superpotentials of the quark sector are given for two cases as follows:I : w u = α u u c H u Y (6) Q + α (cid:48) u u c H u Y (6) (cid:48) Q + β u c c H u Y (6) Q + β (cid:48) u c c H u Y (6) (cid:48) Q + γ u t c H u Y (6) Q + γ (cid:48) u t c H q Y (6) (cid:48) Q ,w d = α d d c H d Y (2) Q + β d s c H d Y (2) Q + γ d b c H d Y (2) Q , (112)II : w q = α q q c H q Y (6) Q + α (cid:48) q q c H q Y (6) (cid:48) Q + β q q c H q Y (4) Q + γ q q c H q Y (2) Q , (113)where q = u, d , and the argument τ in the modular forms Y i ( τ ) is omitted. Couplings α q , α (cid:48) q , β q , β (cid:48) q , γ q and γ (cid:48) q can be adjusted to the observed quark masses.The quark mass matrix is written as:I : M u = v u α u β u
00 0 γ u Y (6)1 Y (6)3 Y (6)2 Y (6)2 Y (6)1 Y (6)3 Y (6)3 Y (6)2 Y (6)1 + g u g u
00 0 g u Y (cid:48) (6)1 Y (cid:48) (6)3 Y (cid:48) (6)2 Y (cid:48) (6)2 Y (cid:48) (6)1 Y (cid:48) (6)3 Y (cid:48) (6)3 Y (cid:48) (6)2 Y (cid:48) (6)1 RL ,M d = v d α d β d
00 0 γ d Y Y Y Y Y Y Y Y Y RL , (114)II : M q = v q α q β q
00 0 γ q Y (6)1 + g q Y (cid:48) (6)1 Y (6)3 + g q Y (cid:48) (6)3 Y (6)2 + g q Y (cid:48) (6)2 Y (4)2 Y (4)1 Y (4)3 Y (2)3 Y (2)2 Y (2)1 RL , (115)where g u = α (cid:48) u /α u , g u = β (cid:48) u /β u , g u = γ (cid:48) u /γ u and g q ≡ α (cid:48) q /α q . The VEV of the Higgs field H q isdenoted by v q . Parameters α q , β q , γ q can be taken to be real, on the other hand, g u , g u , g u , g u and g d are complex parameters.These mass matrices turn to the simple ones at the fixed points, τ = i , τ = ω and τ = i ∞ . Wediscuss them in the diagonal bases of S , ST and T , respectively.24 .2 Quark mass matrix at the fixed point of τ = i τ = i The quark matrix I is given by using modular forms in Table 1 at fixed point τ = i in the base of S ofEq. (9) as follows: M u = ˜ α u β u
00 0 ˜ γ u × √ − g u (7 √ −
12) 12 − √ g u (9 − √
3) 5 √ − g u (3 − √ √ − g u (3 − √
3) 2 √ − g u (7 √ −
12) 12 − √ g u (9 − √ − √ g u (9 − √
3) 5 √ − g u (3 − √
3) 2 √ − g u (7 √ − ,M d = ˜ α d β d
00 0 ˜ γ d − √ − √ − √ − √ − √ − √ , (116)where ˜ α u = 3 v u Y α u , ˜ β u = 3 v u Y β u , ˜ γ u = 3 v u Y γ u , ˜ α d = (6 − √ v d Y α d , ˜ β d = (6 − √ v d Y β d and˜ γ q = (6 − √ v d Y γ d .We move the quark mass matrix to the diagonal base of S . By using the unitary transformation ofEq. (17), V S , the mass matrix M † u M u is transformed as: M u ≡ V S M † q M u V † S = 92 a ˜ α u + b ˜ β u + c ˜ γ u a ˜ α u + b ˜ β u + c ˜ γ u a ∗ ˜ α u + b ∗ ˜ β u + c ∗ ˜ γ u a ˜ α u + b ˜ β u + c ˜ γ u . (117)Each coefficient is given as: a = A + 2 B Re[ g u ] + C | g u | , b = 2 B + 2( A − B )Re[ g u ] + A | g u | ,c = C + 2( C − B )Re[ g u ] + 2 B | g u | , a = − B − Ag u − Cg ∗ u − B | g u | ,b = 2 B + ( C − B ) g u + ( A − B ) g ∗ u − B | g u | , c = − B + ( C − B ) g u + ( A − B ) g ∗ u + 2 B | g u | ,a = C + 2 B Re[ g u ] + A | g u | , b = 2 B + 2( C − B )Re[ g u ] + C | g u | ,c = A + 2( A − B )Re[ g u ] + 2 B | g u | , (118)where A , B and C are given in Eq. (68). On the other hand, the mass matrix M † d M d is transformed as: M d ≡ V S M † d M d V † S = 32 α d + 2 D ˜ β d + A ˜ γ d − D ˜( α d − β d + ˜ γ d )0 − D ˜( α d − β d + ˜ γ d ) A ˜ α d + 2 D ˜ β d + ˜ γ d . (119)It is remarked that the lightest quarks are massless for both up-type and down-type quarks at τ = i .Therefore, the small deviation from τ = i is required to avoid the massless quark. There exists a non-vanishing flavor mixing angle θ u at τ = i as discussed in Eq. (19). Supposing ˜ γ q (cid:29) ˜ β q , ˜ α q , the mixingangle θ u is given from Eq. (117) as:tan 2 θ u (cid:39) | − B + ( C − B ) g u + ( A − B ) g ∗ u + 2 B | g u | | ( A − C )(1 + 2Re[ g u ])= 2 (cid:112) [ − B + 2 B Re[ g u ] + 2 B | g u | ] + [( C − A )Im g u ] √ B (1 + 2Re[ g u ]) (cid:39) √ (cid:12)(cid:12)(cid:12)(cid:12) g u + 2 g u −
11 + 2 g u (cid:12)(cid:12)(cid:12)(cid:12) , (120)25here A + C = 4 B is used and the imaginary part of g q is neglected in the last equation ( g u = Re[ g u ]).In this case, tan 2 θ u vanishes at g u = ( − ± √ /
2, while θ u = 15 ◦ at g u = 0.On the other hand, the mixing angle θ d is simply given from Eq. (119) as:tan 2 θ d (cid:39) D − A = 1 √ , (121)which leads to θ d = 15 ◦ . Since the observed small CKM mixing angle θ CKM23 (around 2 ◦ ) is given by thedifference ( θ d − θ u ), the magnitude of g u should be small in order to realize the enough cancellationbetween θ d and θ u . Indeed, | g u | is in [0 , , .
07] in our numerical results of section 7. τ = i By using the approximate modular forms of weight 2 and 6 in Eqs. (183) and (185) of Appendix C.1, wepresent the deviation from M u and M d in Eqs. (117) and (119). Then, the additional contribution M u to M u of Eq. (117) of order (cid:15) is given in terms of A , B and C in Eq. (68) as follows: M u (cid:39) δ u δ u δ ∗ u δ u δ u δ ∗ u δ ∗ u δ u , (122)where δ u = 3 √ { [ ( A − B + ( B − C ) g u ) (cid:15) ∗ + ( B + Cg u ) (cid:15) ∗ ]( g ∗ u −
1) ˜ α u + ( − B + ( B − A ) g u ) (cid:15) ∗ + ( C − B − Bg u ) (cid:15) ∗ ]( g ∗ u −
1) ˜ β u + ( C − B + 2 Bg u ) (cid:15) ∗ + ( − C + ( B − C ) g u ) (cid:15) ∗ ]( g ∗ u − γ u }(cid:39) √ (cid:15) ∗ { [ ( A + B ) + ( B + C ) g u ]( g ∗ u −
1) ˜ α u + [ 2( C − B ) − ( A + B ) g u ]( g ∗ u −
1) ˜ β u + [ − ( B + C ) + 2(2 B − C ) g u ]( g ∗ u − γ u } , (123) δ u = 3 √ { [ ( C − B − ( A − B ) g u ) (cid:15) ∗ − ( C + Bg u ) (cid:15) ∗ ]( g ∗ u −
1) ˜ α u + ( − B + ( B − C ) g u ) (cid:15) ∗ + ( C − B − Cg u ) (cid:15) ∗ ]( g ∗ u −
1) ˜ β u + ( A − B + 2 Bg u ) (cid:15) ∗ + ( B + ( B − C ) g u ) (cid:15) ∗ ]( g ∗ u − γ u }(cid:39) √ (cid:15) ∗ {− [ ( C + B ) + ( A + B ) g u ]( g ∗ u −
1) ˜ α u + [ 2( C − B ) + ( B + C ) g u ]( g ∗ u −
1) ˜ β u + [ A + B + 2(2 B − C ) g u ]( g ∗ u − γ u } . (124)In the approximate equalities, (cid:15) = 2 (cid:15) in Eq. (26) is put. In order to estimate the Cabibbo angle, wecalculate the mixing angle of the 1st- and 2nd-family as:tan 2 θ u = 2 | δ u | ( a ˜ α u + b ˜ β u + c ˜ γ u ) (cid:39) √ B + CC | (cid:15) ∗ | (cid:39) √ √ | (cid:15) ∗ | (cid:39) . | (cid:15) ∗ | , (125)where the denominator comes from Eq. (117). In the second approximate equality, ˜ γ u (cid:29) ˜ α u , ˜ β u and | g u | (cid:28) c is given in Eq. (118). 26he additional contribution M d to M d of Eq. (119) of order (cid:15) is: M d (cid:39) δ d δ d δ ∗ d δ d δ d δ ∗ d δ ∗ d δ d , (126)where δ d = 1 √ { [( √ − (cid:15) ∗ + ( √ − (cid:15) ∗ ] ˜ α d + [(4 − √ (cid:15) ∗ + (3 √ − (cid:15) ∗ ] ˜ β d + [(3 √ − (cid:15) ∗ + (7 − √ (cid:15) ∗ ]˜ γ d } (cid:39) √ (cid:15) ∗ [(3 √ −
5) ˜ α d + 2(2 √ −
3) ˜ β d + (9 − √ γ d ] , (127) δ d = 1 √ { [(9 − √ (cid:15) ∗ + (7 √ − (cid:15) ∗ ] ˜ α d + [(4 √ − (cid:15) ∗ + (9 − √ (cid:15) ∗ ] ˜ β d + [( √ − (cid:15) ∗ + (3 − √ (cid:15) ∗ ]˜ γ d } (cid:39) √ (cid:15) ∗ [(3 √ −
5) ˜ α d + 2(2 − √
3) ˜ β d + (1 − √ γ d ] . (128)In the last approximate equalities, (cid:15) = 2 (cid:15) in Eq. (26) is put. The mixing angle of the 1st- and2nd-family as:tan 2 θ d = 2 | δ d | ( ˜ α d + 2 D ˜ β d + A ˜ γ d ) (cid:39) √ − √ A | (cid:15) ∗ | (cid:39) √ √ | (cid:15) ∗ | (cid:39) . | (cid:15) ∗ | , (129)where the denominator comes from Eq. (119). In the second approximate equality, ˜ γ d (cid:29) ˜ α d , ˜ β d is taken.Since the magnitudes of θ u and θ d in Eqs. (125) and (129) are almost same, the phase of (cid:15) is importantto reproduce the Cabibbo angle. If we take | (cid:15) | = 0 . τ = i + (cid:15) and (cid:15) = 2 . i (cid:15) in Eq. (26)), both θ u ( d )12 are approximately 0 .
22. Thus, the magnitude of Cabibbo angle is easily reproduced by taking therelevant phase of (cid:15) . Indeed, the observed CKM elements are reproduced at τ (cid:39) i + (0 . . e iφ withrelevant φ as numerically discussed in section 7. τ = i Let us discuss the quark mass matrix II in Eq. (115) at fixed points of τ by using modular forms inTable 1. At τ = i , both up-type and down-type quark mass matrices are given in the base of S ofEq. (9) as: M q = ˜ α q β q
00 0 ˜ γ q × √ − g q (7 √ −
12) 12 − √ g q (9 − √
3) 5 √ − g q (3 − √ − √ − √ , (130)where ˜ α q = 3 v q Y α q , ˜ β q = (6 − √ v q Y β q and ˜ γ q = v q Y γ q ( q = u, d ).27et us move them to the diagonal base of S . By using the unitary transformation of Eq. (17), V S ,the matrix M † q M q is transformed as ( M q V S ) † M q V S . Then, we have M q ≡ V S M † q M q V † S = 32 A ˜ γ q + 3( A + B q + | g q | C ) ˜ α q − [ D ˜ γ q + 3( B q + Ag q + Cg ∗ q ) ˜ α q )] 0 − [ D ˜ γ q + 3( B q + Ag ∗ q + Cg q ) ˜ α q )] ˜ γ q + 3( C + B q + | g q | A ) ˜ α q
00 0 2 ˜ β , (131)with A = 7 − √ , B = 26 − √ , C = 97 − √ , D = 2 − √ ,B q = B ( g q + g ∗ q ) = 2 B Re[ g q ] , B q = B (1 + | g q | ) , A = C , D = A , A + C = 4 B , (132)where A , B , C and D in Eq. (68) are again presented for convenience. The mass eigenvalues satisfy: m q m q = 81 C ˜ α q ˜ γ q , m q + m q = 6 D ˜ γ q + 9 B (2 + 2Re[ g q ] + | g q | ) ˜ α q , m q = 3 ˜ β q . (133)The mixing angle between 1st- and 2nd-family, θ q , is given as:tan 2 θ q = − (cid:113) [ D ˜ γ q + 3( B + E q ) ˜ α q )] + 9 F q ˜ α q (2 √ − γ q + 3(45 − √ − | g q | ) ˜ α q , (134)where E q = ( A + C )Re[ g q ] = (104 − √ g q ] , F q = ( A − C ) Im[ g q ] = (52 √ −
90) Im[ g q ] . (135)Neglecting the imaginary part of g ( g = Re[ g ]), it is simply given as:tan 2 θ q = − √ γ q + 3(7 − √ g q + g q ) ˜ α q ˜ γ q − − √ − g q ) ˜ α q . (136)where | g q | is supposed to be O (1). We take ˜ α q , ˜ γ q (cid:28) ˜ β q due to the mass hierarchy of quark masses.There are two possible choices of ˜ α q (cid:28) ˜ γ q and ˜ γ q (cid:28) ˜ α q .In the case of ˜ α q (cid:28) ˜ γ q ,tan 2 θ q (cid:39) − √ − √ g q ) ˜ α q ˜ γ q ] (cid:39) − √ , (137)which gives θ q = − ◦ at the limit of ˜ α q / ˜ γ q = 0. This is common for both up-quark and down-quark mass matrices because it is independent of g q . Then, the flavor mixing (CKM) between 1st- and2nd-family vanishes due to the cancellation between up-quarks and down-quarks.On the other hand, in the case of ˜ γ q (cid:28) ˜ α q , we obtaintan 2 θ q (cid:39) √ g q + g q − g q , (138)where the imaginary part of g q and terms of ˜ γ q are neglected. The Cabibbo angle could be reproducedby choosing relevant values of g d and g u of order one. However, the CKM matrix elements V cb and V ub vanish at τ = i . In order to obtain desirable CKM matrix, τ should be deviated from i a little bit.28 .2.4 Quark mass matrix II at nearby τ = i By using modular forms of weight 2, 4 and 6 in Appendix C.1, we obtain the deviation from M q inEq. (131). Then, the additional contribution M Q to M Q of Eq.(131) of order (cid:15) is: M Q (cid:39) O ( ˜ α q , ˜ γ q , (cid:15) , (cid:15) ) O ( ˜ α q , ˜ γ q , , (cid:15) , (cid:15) ) ˜ β q √ [( √ − (cid:15) ∗ + (2 − √ (cid:15) ∗ ] O ( ˜ α q , ˜ γ q , (cid:15) , (cid:15) ) O ( ˜ α q , ˜ γ q , (cid:15) , (cid:15) ) ˜ β q √ [(3 + √ (cid:15) ∗ + √ (cid:15) ∗ ] ˜ β q √ [( √ − (cid:15) + (2 − √ (cid:15) ] ˜ β q √ [(3 + √ (cid:15) + √ (cid:15) ] ˜ β q [4Re( (cid:15) ) + 2(2 − √ (cid:15) )] , (139)where O ( ˜ α q , ˜ γ q , (cid:15) , (cid:15) ) terms are highly suppressed compared with elements (1,3), (3,1), (2,3), (3,2), (3.3)due to ˜ β q (cid:29) ˜ α q , ˜ γ q . Therefore, the 2nd- and 3rd-family mixing angle θ q is given as: θ q (cid:39) √ ˜ β q | (3 + √ (cid:15) ∗ + √ (cid:15) ∗ | β q = 3 + √ √ | (cid:15) ∗ | (cid:39) . | (cid:15) ∗ | , (140)and the 1st- and 3rd-family mixing angle θ q is: θ q (cid:39) √ ˜ β q | ( √ − (cid:15) ∗ + (2 − √ (cid:15) ∗ | β q = 3 − √ √ | (cid:15) ∗ | (cid:39) . | (cid:15) ∗ | , (141)where 3 ˜ β q in the denominators is the (3,3) element of Eq. (131), and (cid:15) = 2 (cid:15) = 4 . i (cid:15) of Eq. (26) isused. The ratio θ q /θ q (cid:39) .
27 is rather large compared with observed CKM ratio | V ub /V cb | (cid:39) . θ q spoils to reproduce observed CKM elements V cb and V ub at the nearby fixed point τ = i . τ = ω τ = ω In the quark mass matrix I of Eq. (114), the up-type and down-type mass matrices are given at τ = ω by using modular forms in Table 1: M u = − g u ˜ α q − g u ˜ β q
00 0 − g u ˜ γ q − ω − ω − ω − ω − ω − ω ,M d = ˜ α d β d
00 0 ˜ γ d − ω ωω − ω − ω ω , (142)where ˜ α u = (9 / v u Y α q , ˜ β u = (9 / v u Y β q , ˜ γ u = (9 / v u Y γ q for up-type quarks, and ˜ α d = v d Y α d ,˜ β d = v d Y β d , ˜ γ d = v d Y γ d for down-type quarks, respectively. By using the unitary transformation ofEq. (22), V ST , the mass matrix M † u M u is transformed as: M u ≡ V ST M † q M q V † ST = 9 | g u | ˜ β u | g u | ˜ α u
00 0 | g u | ˜ γ u . (143)29he mass matrix M † d M d is transformed as: M d ≡ V ST M † d M d V † ST = 94 ˜ α d γ d
00 0 ˜ β d . (144)It is remarked that both are diagonal ones. τ = ω Quark mass matrix I in Eq. (142) is corrected due to the deviation from the fixed point of τ = ω . Byusing modular forms of weight 2, 4 and 6 in Appendix C.2, we obtain the deviation from M u and M d in Eqs. (143) and (144). In the diagonal base of ST , the corrections are given by only a smallvariable (cid:15) as seen in Eq. (27). In the 1st order perturbation of (cid:15) , the corrections M u and M d aregiven as: M u = δ u δ u δ u δ ∗ u δ u δ u δ u ∗ δ ∗ u δ u , M d = δ d δ d δ d δ ∗ d δ d δ d δ d ∗ δ ∗ d δ d , (145)where off diagonal elements δ q δ q and δ q are: δ u = 2 ˜ β u | g u | (2 (cid:15) − (cid:15) ) − α u (2 + g ∗ u ) g u ( (cid:15) ∗ + (cid:15) ∗ ) = − g ∗ u ) g u (cid:15) ∗ ˜ α u , (146) δ u = 2 ˜ β u (2 + g u ) g ∗ u ( (cid:15) + (cid:15) ) + 2˜ γ u | g u | ( − (cid:15) ∗ + (cid:15) ∗ ) = 6(2 + g u ) g ∗ u (cid:15) ˜ β u , (147) δ u = 2˜ γ u (2 + g ∗ u ) g u ( (cid:15) ∗ + (cid:15) ∗ ) + 2 ˜ α u | g u | ( − (cid:15) + (cid:15) ) = 6(2 + g ∗ u ) g u (cid:15) ∗ ˜ γ u , (148) δ d = i ˜ α d ( (cid:15) − (cid:15) ) + 12 i ˜ γ d ( (cid:15) ∗ + (cid:15) ∗ ) = 32 i (cid:15) ∗ ˜ γ d , (149) δ d = 12 i ˜ α d ( (cid:15) + (cid:15) ) + i ˜ β d ( (cid:15) ∗ − (cid:15) ∗ ) = 32 i (cid:15) ˜ α d , (150) δ d = − i ˜ β d ( (cid:15) ∗ + (cid:15) ∗ ) − i ˜ γ d ( (cid:15) − (cid:15) ) = − i (cid:15) ∗ ˜ β d . (151)In last equalities, (cid:15) = 2 (cid:15) of Eq. (28) is used.Taking account of ˜ γ u (cid:29) ˜ α u (cid:29) ˜ β u and ˜ β d (cid:29) ˜ γ d (cid:29) ˜ α d as seen in Eqs. (143) and (144), mixing angles θ q and θ q are given as: θ u (cid:39) | (2 + g ∗ u ) g u (cid:15) ∗ | , θ u (cid:39) | (2 + g ∗ u ) g u (cid:15) ∗ | , θ d (cid:39) θ d (cid:39) | (cid:15) ∗ | , (152)respectively, while both θ q ( q = u, d ) are highly suppressed.Since up-type quark mixing angles depend on the magnitudes of g u and g u , the magnitudes ofCKM matrix elements V us and V cb could be reproduced by choosing relevant g u and g u . For example,we can take θ u ∼ λ and θ u ∼ θ d ∼ θ d ∼ λ , where λ (cid:39) . | V us | , | V cb | and | V ub | . However, this scheme leads to | V td | ∼ λ , which is much smaller than the observed one.Indeed, the observed | V td | is not reproduced at nearby τ = ω in section 7.30 .3.3 Quark mass matrix II at τ = ω We discuss the quark mass matrix II at the fixed point τ = ω by using modular forms in Table 1. Inthe base of S and T of Eq. (9), it is given at the fixed point τ = ω : M q = − g q ˜ α q β q
00 0 ˜ γ q − ω − ω − ω ω − ω ω , (153)where ˜ α q = (9 / v q Y α q , ˜ β q = v q Y β q and ˜ γ q = v q Y γ q . By using the unitary transformation ofEq.(22), V ST , the mass matrix M † q M q is transformed as: M q ≡ V ST M † q M q V † ST = 94 g q ˜ α q + ˜ β q + ˜ γ q , (154)which gives two massless quarks. Therefore, it seems very difficult to reproduce observed quark massesand CKM elements even if we shift τ from τ = ω a little bit and choose relevant g q . τ = ω Quark mass matrix II in Eq. (153) is corrected due to the deviation from the fixed point of τ = ω .By using modular forms of weight 2, 4 and 6 in Appendix C.2, we obtain the deviation from M q in Eq. (154). In the diagonal base of ST , the correction is given by only a small variable (cid:15) as seen inEq. (27). In the 1st order approximation of (cid:15) i , the correction M q is given as: M q = δ q δ q δ ∗ q δ ∗ q δ q , (155)where δ qi are given in terms of (cid:15) , g q , ˜ α q , ˜ β q and ˜ γ q . In order to estimate the flavor mixing anles, wepresent relevant δ qi as: δ q = − α q g q (2 + g ∗ q )( (cid:15) ∗ + (cid:15) ∗ ) + 16 ˜ β q ( (cid:15) ∗ − (cid:15) ∗ ) + 12 i ˜ γ q ( (cid:15) ∗ + (cid:15) ∗ ) (cid:39) − α q g q (2 + g ∗ q ) (cid:15) ∗ −
52 ˜ β q (cid:15) ∗ + 32 i ˜ γ q (cid:15) ∗ ,δ q = ˜ α q | g q | ( − (cid:15) ∗ + 2 (cid:15) ∗ ) + ˜ β q ( 13 (cid:15) ∗ − (cid:15) ∗ ) + i ˜ γ q ( (cid:15) ∗ − (cid:15) ∗ ) (cid:39) − β q (cid:15) ∗ , (156)where (cid:15) = 2 (cid:15) of Eq. (28) is used in last approximate equalities. By using Eqs. (154) and (155), weobtain Det[ M Q + M Q ] = 0. Therefore, it is impossible to reproduce observed quark masses atnearby τ = ω in the 1st order perturbation of (cid:15) . Indeed, this model cannot reproduce the observedCKM elements at nearby τ = ω in section 7. 31 .4 Quark mass matrix at τ = i ∞ τ = i ∞ The mass matrices of I and II in Eqs. (115) and (114) are simply given by using modular forms in Table1 at τ = i ∞ since the modular forms of weight 2, 4 and 6 are same. Those are both diagonal ones asfollows: M q = ˜ α q β q
00 0 ˜ γ q , (157)where ˜ α u = v u Y α q , ˜ β u = v u Y β u , ˜ γ u = v u Y γ u , ˜ α d = v d Y α d , ˜ β d = v d Y β d , ˜ γ d = v d Y γ d for quark massmatrix I and ˜ α q = v q Y α q , ˜ β q = v q Y β q and ˜ γ q = v q Y γ q for quark mass matrix II.In the diagonal base of T of Eq. (9), the mass matrix M † q M q is given as: M q ≡ M † q M q = ˜ α q β q
00 0 ˜ γ q . (158)Mixing angles appear through the finite effect of Im [ τ ]. τ = i ∞ Quark mass matrix I in Eq. (157) is corrected due to the finite effect of τ = i ∞ . By using modular formsof Eqs. (192), (193) and (194) in Appendix C.3, we obtain the deviation from M q in Eq. (158) for thequark mass matrix I. We present the first order corrections M q for up-type quarks and down-typequarks to M q of Eq. (158), respectively : M u (cid:39) g ∗ u ) ˜ β u δ ∗ (1 + 2 g u ) ˜ α u δ (1 + 2 g u ) ˜ β u δ g ∗ u ) ˜ γ u δ ∗ (1 + 2 g ∗ u ) ˜ α u δ ∗ (1 + 2 g u ) ˜ γ u δ , M d (cid:39) β d δ ∗ ˜ α d δ ˜ β d δ γ d δ ∗ ˜ α d δ ∗ ˜ γ d δ , (159)where δ is given in Eq.(105). We obtain mixing angles as: θ u (cid:39) | (1 + 2 g ∗ u ) δ ∗ | , θ u (cid:39) | (1 + 2 g ∗ u ) δ ∗ | , θ d (cid:39) θ d (cid:39) | δ ∗ | , (160)respectively. The 1st- and 3rd-family mixing angle θ q is suppressed due to the factor ˜ α q / ˜ γ q for bothup- and down-type quarks. Since θ u and θ u depend on the magnitudes of g u and g u , the CKM matrixelements V us and V cb could be reproduced by choosing relevant g u and g u . For example, we can take θ u ∼ λ and θ u ∼ θ d ∼ θ d ∼ λ , where λ (cid:39) . | V us | , | V cb | and | V ub | . However,this scheme leads to | V td | ∼ λ , which is much smaller than the observed one. Indeed, the successfulCKM matrix elements are not reproduced at large Im τ in the numerical results of section 7.32 .4.3 Quark mass matrix II towards τ = i ∞ Quark mass matrix II in Eq. (157) is corrected due to the finite effect of τ = i ∞ . By using modularforms of Eqs .(192), (193) and (194) in Appendix C.3, we obtain the deviation from M q in Eq. (158)for the quark mass matrix II. The first order correction M q to M q of Eq. (158) is given as : M q (cid:39) − δ ∗ ˜ β q (1 + 2 g q ) δ ∗ ˜ α q − δ ˜ β q δ ∗ ˜ γ q (1 + 2 g ∗ q ) δ ˜ α q δ ˜ γ q , (161)where ˜ α q (cid:28) ˜ β q (cid:28) ˜ γ q . Therefore, the mixing angles θ q and θ q , are given as: θ q (cid:39) | δ ∗ | ˜ β q ˜ β q (cid:39) | δ ∗ | , θ q (cid:39) | δ ∗ | ˜ γ q ˜ γ q = | δ ∗ | , (162)respectively. On the other hand, 1st- and 3rd-family mixing angle θ q is highly suppressed due to thefactor ˜ α q / ˜ γ q . Since θ q and θ q are the same magnitude for both up-type and down-type quarks, it isimpossible to reproduce observed CKM mixing angles.In conclusion of section 6, it is found that the only quark mass matrix I works well at nearby τ = i . We have presented analytical discussions of lepton and quark mass matrices at nearby fixed points ofmodulus. In this section, we show numerical results at the nearby fixed points of τ = i , τ = ω and τ = i ∞ to confirm above discussions and present predictions. In order to calculate the left-handed flavor mixing of leptons numerically, we generate random numberfor model parameters. The modulus τ is scanned around fixed points τ = i and τ = ω . It is also scannedIm τ ≥ . τ = i ∞ . We keep the parameter sets, in which the neutrino experimental data andcharged lepton masses are reproduced within 3 σ interval of error-bars. We continue this procedure toobtain enough points for plotting allowed region.As input of the neutrino data, we take three mixing angles of the PMNS matrix and the observedneutrino mass ratio ∆ m / ∆ m with 3 σ , which are given by NuFit 4.1 in Table 4 [99]. Since there aretwo possible spectrum of neutrinos masses m i , which are the normal hierarchy (NH), m > m > m ,and the inverted hierarchy (IH), m > m > m , we investigate both cases. We also take account of thesum of three neutrino masses (cid:80) m i since it is constrained by the recent cosmological data [100–102].We impose the constraint of the upper-bound (cid:80) m i ≤
120 meV.Since the modulus τ obtains the expectation value by the breaking of the modular invariance at thehigh mass scale, the observed masses and lepton mixing angles should be taken at the GUT scale by therenormalization group equations (RGEs). However, we have not included the RGE effects in the leptonmixing angles and neutrino mass ratio ∆ m / ∆ m in our numerical calculations. We suppose thatthose corrections are very small between the electroweak and GUT scales. This assumption is confirmedwell in the case of tan β ≤ × GeV withtan β = 5 in the framework of the minimal SUSY breaking scenarios [103, 104]: y e = (1 . ± . × − , y µ = (4 . ± . × − , y τ = (7 . ± . × − , (163)where lepton masses are given by m (cid:96) = y (cid:96) v H with v H = 174 GeV.observable 3 σ range for NH 3 σ range for IH∆ m (2 . . × − eV − (2 . . × − eV ∆ m (6 . . × − eV (6 . . × − eV sin θ . .
609 0 . . θ . .
350 0 . . θ . . . . σ ranges of neutrino parameters from NuFIT 4.1 for NH and IH [99].For the quark sector, we also adopt numerical values of Yukawa couplings of quarks at the GUTscale 2 × GeV with tan β = 5 in the framework of the minimal SUSY breaking scenarios [103, 104]: y d = (4 . ± . × − , y s = (9 . ± . × − , y b = (6 . ± . × − ,y u = (2 . ± . × − , y c = (1 . ± . × − , y t = 0 . ± . , (164)which give quark masses as m q = y q v H with v H = 174 GeV.We also use the following CKM mixing angles at the GUT scale 2 × GeV with tan β = 5 [103,104]: θ CKM12 = 13 . ◦ ± . ◦ , θ CKM23 = 2 . ◦ ± . ◦ , θ CKM13 = 0 . ◦ ± . ◦ . (165)Here θ CKM ij is given in the PDG notation of the CKM matrix V CKM [102]. In addition, we impose therecent data of LHCb [102]: (cid:12)(cid:12)(cid:12)(cid:12) V ub V cb (cid:12)(cid:12)(cid:12)(cid:12) = 0 . ± . , (166)where V ij ’s are CKM matrix elements. This ratio is stable against radiative corrections. The observedCP violating phase is given at the GUT scale as: δ CKMCP = 69 . ◦ ± . ◦ , (167)which is also in the PDG notation. The error intervals in Eqs. (164), (165), (166) and (167) represent1 σ interval. τ at nearby fixed points We have examined eighteen cases of leptons and quarks in above framework numerically as shown inTable 5. In this Table, the successful cases for the mass matrix I and II at nearby fixed points are34 odulus nearby τ = i nearby τ = ω towards τ = i ∞ Lepton/Quark Lepton Quark Lepton Quark Lepton QuarkNeutrino mass hierarchy NH IH NH IH NH IHmass matrix I for M E and M q (cid:13) (cid:13) (cid:13) (cid:78) × × (cid:13) × × mass matrix II for M E and M q (cid:13) (cid:78) × (cid:13) (cid:13) × (cid:13) (cid:78) × Table 5: The successful cases for the mass matrix I and II at nearby fixed points are denoted by (cid:13) .On the other hand, × denotes a failure to reproduce observed mixing angles, and (cid:78) denotes the casein which observed mixing angles are reproduced, but (cid:80) m i ≥
120 meV.denoted by (cid:13) . On the other hand, × denotes a failure to reproduce observed mixing angles, and (cid:78) denotes the case in which observed PMNS mixing angles are reproduced, but (cid:80) m i ≥
120 meV.Among eighteen cases, seven cases of leptons and one case of quarks are consistent with recentobserved data. It is emphasized that the all cases of the mass matrix I work well at nearby τ = i . Theseresults confirm our previous discussions.We show allowed regions of τ at nearby τ = i , τ = ω and towards τ = i ∞ for eleven cases inFigs. 1, 2 and 3, respectively. In these figures, green points denote allowed ones by inputting masses andmixing angles with the constraint (cid:80) m i ≤
120 meV for leptons, but blue points denote the regions inwhich the sum of neutrino masses (cid:80) m i is larger than 120 meV. It is noted that blue points are hiddenunder green points in the case of the charged lepton II (NH) of Fig. 2 and charged lepton I (NH) ofFig. 3. Green points for quarks denote allowed region of τ by inputting masses, mixing angles and CPviolating phase δ CKMCP .As seen in Fig. 1, the constraint (cid:80) m i ≤
120 meV excludes the charged lepton II with IH of neutrinos.The allowed regions of τ (green points) deviate from the fixed point τ = i in magnitude of 5–10%, whichconfirm the discussions in section 5. It is reasonable that the allowed points appear frequently at nearby τ = i since one flavor mixing angle is generated even at the fixed point τ = i as discussed in section5.2. In the quark sector, the mass matrix I works well, but the matrix II does not because the mixingangles are canceled out each other in the same type mass matrices of up-type and down-type quarks.It is emphasized that there is the common region of τ between charged lepton I (NH) and quark I. Thiscommon region has already discussed in context with the quark-lepton unification in Ref. [52].As seen in Fig. 2, at nearby τ = ω , the charged lepton mass matrix I with NH is excluded bythe constraint of (cid:80) m i ≤
120 meV. In the charged lepton mass matrix I with IH, the PMNS mixingangles are not reproduced. On the other hand, the allowed regions are marginal in the charged leptonII. Indeed, the green points are 0 . .
15 for IH away from τ = ω , respectively. Theperturbative discussion of this IH case is possibly broken. Moreover, we cannot find allowed region ofquarks at nearby τ = ω . That is expected in the discussion in section 6.3.As seen in Fg. 3, towards τ = i ∞ , both charged lepton mass matrix I and II reproduce the observedPMNS mixing angles for NH of neutrinos. In the charged lepton mass matrix I with IH, the PMNSmixing angles are not reproduced. Although the charged lepton mass matrix II with IH reproduce threePMNS mixing angles, it is excluded by the constraint of (cid:80) m i ≤
120 meV. We cannot find allowedregion for quarks. These results are also consistent with discussions of section 5.4 and 6.4.35 uark mass matrix IINo allowed region around τ = i Figure 1: Allowed regions of τ at nearby τ = i are shown by green points for charged lepton massmatrices I and II with NH and IH of neutrinos, and quark mass matrices I, respectively. Blue pointsdenote regions in which the sum of neutrino masses (cid:80) m i is larger than 120 meV.36 harged lepton mass matrix I IH for neutrino masses
No allowed region around τ = ω quark mass matrix INo allowed region around τ = ω quark mass matrix II No allowed region around τ = ω Figure 2: Allowed regions of τ at nearby τ = ω are shown by green points for the charged lepton massmatrix I and II with NH and IH of neutrinos, respectively. Blue points denote regions in which the sumof neutrino masses (cid:80) m i is larger than 120 meV. 37 harged lepton mass matrix I IH for neutrino masses
No allowed region around τ = i∞ quark mass matrix INo allowed region towards τ = i∞ quark mass matrix IINo allowed region towards τ = i∞ Figure 3: Allowed regions of τ towards τ = i ∞ are shown by green points for charged lepton massmatrices I and II with NH and IH of neutrinos, respectively. Blue points denote regions in which thesum of neutrino masses (cid:80) m i is larger than 120 meV.38 .3 Predictions of CP violation and masses of neutrinos We predict the leptonic CP violating phase δ (cid:96) CP , the sum of neutrino masses (cid:80) m i and the effectivemass for the 0 νββ decay |(cid:104) m ee (cid:105)| for each case of leptons since we input four observed quantities ofneutrinos (three mixing angles of leptons and observed neutrino mass ratio ∆ m / ∆ m ) and threecharged lepton masses. For quark sector, there is no prediction because ten observed quantities (quarkmasses and CKM elements) are put to obtain the region of the modulus τ .In Table 6, the predicted ranges of the effective mass for the 0 νββ decay, (cid:104) m ee (cid:105) are presented foreach case. We also summarize magnitudes of parameters g ν , g ν , g e for leptons and g u , g u , g u forquarks. Their phases are broad. We add hierarchies of ˜ α e , ˜ β e , ˜ γ e and ˜ α q , ˜ β q , ˜ γ q . |(cid:104) m ee (cid:105)| [meV] | g ν | | g ν | | g e | ˜ α e , ˜ β e , ˜ γ e NH, charged lepton I, τ (cid:39) i . . γ e (cid:29) ˜ α e (cid:29) ˜ β e IH, charged lepton I, τ (cid:39) i . . . . γ e (cid:29) ˜ α e (cid:29) ˜ β e NH, charged lepton II, τ (cid:39) i . .
53– 7 . . . . . α e (cid:29) ˜ γ e (cid:29) ˜ β e NH, charged lepton II, τ (cid:39) ω . . . .
05 0 . .
65 0 . .
28 ˜ α e (cid:29) ˜ β e (cid:29) ˜ γ e IH, charged lepton II, τ (cid:39) ω . . . . . . α e (cid:29) ˜ β e (cid:29) ˜ γ e NH, charged lepton I, τ (cid:39) i ∞ . .
53 1 . . γ e (cid:29) ˜ β e (cid:29) ˜ α e NH, charged lepton II, τ (cid:39) i ∞ . . .
33 0 . .
87 3 . . α e (cid:29) ˜ γ e (cid:29) ˜ β e | g u | | g u | | g u | ˜ α q , ˜ β q , ˜ γ q quark mass matrices I, τ (cid:39) i — 0 . .
86 0 . .
29 0 . .
07 ˜ γ u (cid:29) ˜ β u (cid:29) ˜ α u ˜ γ d (cid:29) ˜ α d (cid:29) ˜ β d Table 6: Magnitudes of parameters g ν , g ν , g e for leptons and g u , g u , g u for quarks are shown.Predicted ranges of the effective mass for the 0 νββ decay, (cid:104) m ee (cid:105) [meV] are also given. In addition,hierarchies of ˜ α e , ˜ β e , ˜ γ e and ˜ α q , ˜ β q , ˜ γ q are presented.We present numerical predictions on (cid:80) m i – δ (cid:96) CP and δ (cid:96) CP –sin θ planes for successful seven cases inFigs. 4–10. In Fig. 4, we show them at nearby τ = i for the charged lepton mass matrix I with NH ofneutrinos. The predicted range of the sum of neutrino masses is (cid:80) m i = 86–120 meV. The predicted δ (cid:96) CP depends on (cid:80) m i . A crucial test will be presented in the near future by cosmological observations.The correlation between sin θ and δ (cid:96) CP is also helpful to test this case.In Fig. 5, we show them at nearby τ = i for the charged lepton mass matrix I with IH of neutrinos.Th predicted range of the sum of neutrino masses is (cid:80) m i = 90–120 meV. The prediction of δ (cid:96) CP isclearly given versus (cid:80) m i . On the other hand, sin θ is predicted to be smaller than 0 .
52. Crucial testwill be available by cosmological observations and neutrino oscillation experiments in the near future.In Fig. 6, we show them at nearby τ = i for the charged lepton mass matrix II with NH of neutrinos.The predicted range of the sum of neutrino masses is (cid:80) m i = 58–83 meV while δ (cid:96) CP is allowed in [ − π, π ].There is no correlation between sin θ and δ (cid:96) CP . The rather small value of the sum of neutrino massesis a characteristic prediction in this case. 39igure 4: Allowed regions on (cid:80) m i – δ (cid:96) CP and δ (cid:96) CP –sin θ planes at nearby τ = i for the charged leptonmass matrix I with NH of neutrinos. The solid black line denotes observed best-fit value of sin θ , andred dashed-lines denote its upper(lower)-bound of 3 σ interval.Figure 5: Allowed regions on (cid:80) m i – δ (cid:96) CP and δ (cid:96) CP –sin θ planes at nearby τ = i for the charged leptonmass matrix I with IH of neutrinos.Figure 6: Allowed regions on (cid:80) m i – δ (cid:96) CP and δ (cid:96) CP –sin θ planes at nearby τ = i for the charged leptonmass matrix II with NH of neutrinos. 40igure 7: Allowed regions on (cid:80) m i – δ (cid:96) CP and δ (cid:96) CP –sin θ planes at nearby τ = ω for the charged leptonmass matrix II with NH of neutrinos.Figure 8: Allowed regions on (cid:80) m i – δ (cid:96) CP and δ (cid:96) CP –sin θ planes at nearby τ = ω for the charged leptonmass matrix II with IH of neutrinos.Figure 9: Allowed regions on (cid:80) m i – δ (cid:96) CP and δ (cid:96) CP –sin θ planes towards τ = i ∞ for the charged leptonmass matrix I with NH of neutrinos. 41igure 10: Allowed regions on (cid:80) m i – δ (cid:96) CP and δ (cid:96) CP –sin θ planes towards τ = i ∞ for the charged leptonmass matrix II with NH of neutrinos.Let us give our predictions on (cid:80) m i – δ (cid:96) CP and δ (cid:96) CP –sin θ planes at nearby τ = ω . In Fig. 7, weshow them for the charged lepton mass matrix II with NH of neutrinos. The predicted range of thesum of neutrino masses is (cid:80) m i = 65–71 meV. The ranges of δ (cid:96) CP is clearly given in [110 ◦ ,180 ◦ ] and[ − ◦ , − ◦ ]. On the other hand, sin θ is predicted in both 1st- and 2nd-octant.In Fig. 8, we show them for the charged lepton mass matrix II with IH of neutrinos at nearby τ = ω . The predicted range of the sum of neutrino masses is (cid:80) m i = 112–120 meV, which may beexcluded in the near future due to the cosmological observations. The predicted CP violating phase is δ (cid:96) CP = [ − ◦ , − ◦ ] and [110 ◦ , ◦ ]. There is no clear correlation between sin θ and δ (cid:96) CP .It is noticed that the predicted CP violating phase δ (cid:96) CP is asymmetric for plus and minus signs inboth Figs. 7 and 8. That is due to excluding the tau region at nearby τ = ω outside the fundamentaldomain of PSL(2 , Z ). Indeed, the excluded region corresponds to the other region inside at nearby thefixed point τ = − ω , where we obtain δ (cid:96) CP with the reversed sign of Figs. 7 and 8.Finally, we show predictions on (cid:80) m i – δ (cid:96) CP and δ (cid:96) CP –sin θ planes towards τ = i ∞ . In Fig. 9, weshow them for the charged lepton mass matrix I with NH of neutrinos. The predicted range of the sumof neutrino masses is in the narrow range of (cid:80) m i = 94–120 meV. The predicted δ (cid:96) CP is close to ± π/ θ is predicted to be smaller than 0 .
45. The predicted CP violation is favoredby the T2K experiment [85], however the predicted sin θ may be excluded in the near future since itis far from the best fit value.In Fig. 10, we show them for the charged lepton mass matrix II with NH of neutrinos. The predictedrange of the sum of neutrino masses is in (cid:80) m i = 105–120 meV. The predicted δ (cid:96) CP is is clearly givenin ± (100 ◦ –180 ◦ ). On the other hand, sin θ is allowed in full range of 3 σ error-bar. Crucial test willbe available by cosmological observations and CP violation experiments of neutrinos in the future.Thus, lepton mass matrices at nearby fixed points provide characteristic predictions for (cid:80) m i and δ (cid:96) CP . On the other hand, there is no prediction for the quark sector. In the modular invariant flavor model of A , we have studied the hierarchical structure of lepton/quarkflavors at the nearby fixed points of the modulus. There are only two inequivalent fixed points in thefundamental domain of PSL(2 , Z ), τ = i and τ = ω . These fixed points correspond to the residual42ymmetries Z S = { I, S } and Z S = { I, ST, ( ST ) } of A , respectively. There is also infinite point τ = i ∞ , in which the subgroup Z T = { I, T, T } of A is preserved. We have examined typical two-typemass matrices for charged leptons and quarks by using modular forms of weights 2, 4 and 6 whilethe neutrino mass matrix with the modular forms of weight 4 through the Weinberg operator. Byperforming Taylor expansion of modular forms around fixed points, we have obtained linear modularforms in good approximations. By using those explicit modular forms, we have found the hierarchicalstructure of these mass matrices in the diagonal base of S , T and ST , in which the flavor mixingangles are easily estimated. The observed PMNS mixing angles are reproduced at the nearby fixedpoint in ten cases of lepton mass matrices. Among them, seven cases satisfy the cosmological bound (cid:80) m i ≤
120 meV. On the other hand, only one case of quark mass matrices is consistent with theobserved CKM matrix. Our results have been confirmed by scanning model parameters numerically asseen in τ regions of Figs. 1, 2 and 3.We have also presented predictions for (cid:80) m i and δ (cid:96) CP for seven cases. Some cases will be testedin the near future. Although there is no prediction for the quark sector, the obtained τ provides aninteresting subject, the possibility of the common τ between quarks and leptons.We have worked by using two-type specific mass matrices for charged leptons and quarks while oneMajorana neutrino mass matrix in order to clarify the behavior at nearby fixed points. More studiesincluding other mass matrices are necessary to understand the phenomenology of fixed points completely.The modular symmetry provides a good outlook for the flavor structure of leptons and quarks at nearbyfixed points. We also should pay attention to the recent theoretical work: the spontaneous CP violationin Type IIB string theory is possibly realized at nearby fixed points, where the moduli stabilization isperformed in a controlled way [80, 81]. Thus, the modular symmetry at nearby fixed points gives us anattractive approarch to flavors. Acknowledgments
This research was supported by an appointment to the JRG Program at the APCTP through theScience and Technology Promotion Fund and Lottery Fund of the Korean Government. This was alsosupported by the Korean Local Governments - Gyeongsangbuk-do Province and Pohang City (H.O.).H. O. is sincerely grateful for the KIAS member.
AppendixA Tensor product of A group We take the generators of A group for the triplet as follows: S = 13 − − − , T = ω
00 0 ω , (168)43here ω = e i π for a triplet. In this base, the multiplication rule is a a a ⊗ b b b = ( a b + a b + a b ) ⊕ ( a b + a b + a b ) (cid:48) ⊕ ( a b + a b + a b ) (cid:48)(cid:48) ⊕ 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 , ⊗ = , (cid:48) ⊗ (cid:48) = (cid:48)(cid:48) , (cid:48)(cid:48) ⊗ (cid:48)(cid:48) = (cid:48) , (cid:48) ⊗ (cid:48)(cid:48) = , (169)where T ( (cid:48) ) = ω , T ( (cid:48)(cid:48) ) = ω . (170)More details are shown in the review [6, 7]. B Mass matrix in arbitrary bases of S and T Define the new basis of generators, ˆ S and ˆ T by a unirary transformation as:ˆ S = U SU † , ˆ T = U T U † , (171)where ˆ S , S , ˆ T , T and U are 3 × triplet transforms under the S ( T ) transfor-mation as: a a a → S ( T ) a a a = U † ˆ S ( ˆ T ) U a a a . (172)Thus, in the new basis, the A triplet transforms as: ˆ a ˆ a ˆ a → ˆ S ( ˆ T ) ˆ a ˆ a ˆ a , (173)where ˆ a ˆ a ˆ a = U a a a . (174)Let us rewrite the Dirac mass matrix M RL in the new base ( ˆ S , ˆ T ) of the triplet left-handed fields.Denoting L and ˆ L to be triplets of the left-handed fields in the basse of S and ˆ S , respectively, and R to be right-handed singlets, the Dirac mass matrix is written as:¯ RM RL L = ¯ RM RL U † ˆ L (175)44here ˆ L = U L . (176)Then, the Dirac mass matrix ˆ M RL in the new base is given as:ˆ M RL = M RL U † . (177)On the other hand, the Majorana mass matrix M LL in the new base ( ˆ S , ˆ T ) is written as L c M LL L = ˆ L c U M LL U † L . (178)Therefore, the Majorana mass matrix ˆ M LL is given as:ˆ M LL = U M LL U † . (179) C Modular forms at nearby fixed points
C.1 Modular forms at nearby τ = i Let us present the behavior of modular forms at nearby τ = i . We obtain approximate linear forms of Y ( τ ), Y ( τ ) and Y ( τ ) by performing Taylor expansion of modular forms around τ = i . We parametrize τ as: τ = i + (cid:15) , with (cid:15) = (cid:15) R + i (cid:15) I , (180)where | (cid:15) | is supposed to be enough small | (cid:15) | (cid:28)
1. For the case of the pure imaginary number of (cid:15) , thatis (cid:15) = i (cid:15) I ( (cid:15) I is real), we obtain the linear fit of (cid:15) by Y ( τ ) Y ( τ ) (cid:39) (1 − . (cid:15) I ) (1 − √ , Y ( τ ) Y ( τ ) (cid:39) (1 − . (cid:15) I ) ( − √ , (181)where coefficients are obtained by numerical fittings. These ratios decrease linearly for (cid:15) I ≥ (cid:15) , that is (cid:15) = (cid:15) R , ( (cid:15) R is real), we obtain as:Re Y ( τ ) Y ( τ ) (cid:39) (1 − . (cid:15) R ) (1 − √ , Re Y ( τ ) Y ( τ ) (cid:39) (1 − (cid:15) R ) ( − √ , Im Y ( τ ) Y ( τ ) (cid:39) . (cid:15) R (1 − √ , Im Y ( τ ) Y ( τ ) (cid:39) . (cid:15) R ( − √ , (182)where the liner terms of (cid:15) disappear in the real parts. Finally, after neglecting O ( (cid:15) R ), we obtainapproximately Y ( τ ) Y ( τ ) (cid:39) (1 + (cid:15) ) (1 − √ , Y ( τ ) Y ( τ ) (cid:39) (1 + (cid:15) ) ( − √ , (cid:15) = 12 (cid:15) = 2 . i (cid:15) . (183)These approximate forms are agreement with exact numerical values within 0 . | (cid:15) | ≤ . Y ( k ) i in Eqs. (13) and (14) in terms of (cid:15) and (cid:15) . Forweight 4, they are Y (4)1 ( τ ) Y ( τ ) (cid:39) − √ − √ (cid:15) + (cid:15) ) , Y (4)2 ( τ ) Y ( τ ) (cid:39) − √ √ − (cid:15) + (14 − √ (cid:15) ,Y (4)3 ( τ ) Y ( τ ) (cid:39) − √ − √ (cid:15) + (2 − √ (cid:15) , (184) Y (4) ( τ ) Y ( τ ) (cid:39) − √ √ − (cid:15) + (cid:15) ) , Y (4) (cid:48) ( τ ) Y ( τ ) (cid:39) − √ − √ (cid:15) + (14 − √ (cid:15) . For weight 6, they are Y (6)1 ( τ )3 Y ( τ ) (cid:39) √ − (cid:18) √ − (cid:19) ( (cid:15) + (cid:15) ) ,Y (6)2 ( τ )3 Y ( τ ) (cid:39) √ − (cid:18) √ − (cid:19) (cid:15) + (cid:18) √ − (cid:19) (cid:15) ,Y (6)3 ( τ )3 Y ( τ ) (cid:39) − √ (cid:18) − √ (cid:19) (cid:15) + (cid:18) − √ (cid:19) (cid:15) ,Y (cid:48) (6)1 ( τ )3 Y ( τ ) (cid:39) √ −
12 + (cid:18) √ − (cid:19) (cid:15) + (cid:18) √ − (cid:19) (cid:15) ,Y (cid:48) (6)2 ( τ )3 Y ( τ ) (cid:39) − √ (cid:18) − √ (cid:19) (cid:15) + (cid:18) − √ − (cid:19) (cid:15) ,Y (cid:48) (6)3 ( τ )3 Y ( τ ) (cid:39) − √ (cid:18) − √ (cid:19) (cid:15) + (cid:18) − √ (cid:19) (cid:15) ,Y (6) ( τ )3 Y ( τ ) (cid:39) (15 − √ (cid:15) + (12 √ − (cid:15) . (185) C.2 Modular forms at nearby τ = ω Let us present the behavior of modular forms at nearby τ = ω . We perform linear approximation of themodular forms Y ( τ ), Y ( τ ) and Y ( τ ) by performing Taylor expansion around τ = ω . We parametrize τ as: τ = ω + (cid:15) , with (cid:15) = (cid:15) R + i (cid:15) I , (186)where we suppose | (cid:15) | (cid:28)
1. For the case of (cid:15) = i (cid:15) I , which is a pure imaginary number, we obtain thelinear fit of (cid:15) as: Y ( τ ) Y ( τ ) (cid:39) ω (1 − . (cid:15) I ) , Y ( τ ) Y ( τ ) (cid:39) − ω (1 − . (cid:15) I ) , (187)46here coefficients are obtained by numerical fittings. These ratios decrease linearly for (cid:15) I ≥
0. On theother hand, for the case of (cid:15) = (cid:15) R , which is a real number, we obtain as:Re Y ( τ ) Y ( τ ) (cid:39) ω (1 − (cid:15) R ) , Re Y ( τ ) Y ( τ ) (cid:39) − ω (1 − (cid:15) R ) . Im Y ( τ ) Y ( τ ) (cid:39) ω (2 . (cid:15) R ) , Im Y ( τ ) Y ( τ ) (cid:39) − ω (4 . (cid:15) R ) , (188)where the linear terms of (cid:15) disappear in the real parts. After neglecting O ( (cid:15) R ), we obtain approximately Y ( τ ) Y ( τ ) (cid:39) ω (1 + (cid:15) ) , Y ( τ ) Y ( τ ) (cid:39) − ω (1 + (cid:15) ) , (cid:15) = 12 (cid:15) = 2 . i (cid:15) , (189)where | (cid:15) | (cid:28)
1. These approximate forms are agreement with exact numerical values within 1 % for | (cid:15) | ≤ . Y ( k ) i in Eqs. (13) and (14) in terms of (cid:15) and (cid:15) . Forweight 4, they are Y (4)1 ( τ ) Y ( τ ) (cid:39)
32 (1 + (cid:15) + (cid:15) ) , Y (4)2 ( τ ) Y ( τ ) (cid:39) − ω (cid:18)
12 + 23 (cid:15) + 16 (cid:15) (cid:19) , Y (4)3 ( τ ) Y ( τ ) (cid:39) ω (cid:18) − (cid:15) − (cid:15) (cid:19) ,Y (4) ( τ ) Y ( τ ) (cid:39) − ( (cid:15) + (cid:15) ) , Y (4) (cid:48) ( τ ) Y ( τ ) (cid:39) ω (cid:18) (cid:15) + 29 (cid:15) (cid:19) . (190)For weight 6, they are Y (6)1 ( τ ) Y ( τ ) (cid:39) − ( (cid:15) + (cid:15) ) ,Y (6)2 ( τ ) Y ( τ ) (cid:39) − ω ( (cid:15) + (cid:15) ) ,Y (6)3 ( τ ) Y ( τ ) (cid:39) ω ( (cid:15) + (cid:15) ) ,Y (cid:48) (6)1 ( τ ) Y ( τ ) (cid:39) − (cid:18) (cid:15) + 119 (cid:15) (cid:19) ,Y (cid:48) (6)2 ( τ ) Y ( τ ) (cid:39) ω (cid:18) (cid:15) + 29 (cid:15) (cid:19) ,Y (cid:48) (6)3 ( τ ) Y ( τ ) (cid:39) ω (cid:18) (cid:15) + 29 (cid:15) (cid:19) ,Y (6) ( τ ) Y ( τ ) (cid:39) (cid:18) (cid:15) + 13 (cid:15) (cid:19) . (191) C.3 Modular forms towards τ = i ∞ We show the behavior of modular forms at large Im τ , where q = exp (2 πiτ ) is suppressed. Takingleading terms of Eq. (11), we can express modular forms approximately as: Y ( τ ) (cid:39) p (cid:15) , Y ( τ ) (cid:39) − p (cid:15) , Y ( τ ) (cid:39) − p (cid:15) , p = e πi Re τ , (cid:15) = e − π Im τ . (192)47igher weight modular forms Y ( k ) i in Eqs. (13) and (14) are obtained in terms of p and (cid:15) approxi-mately. For weight 4, they are Y (4)1 ( τ ) (cid:39) − p (cid:15) , Y (4)2 ( τ ) (cid:39) p (cid:15) , Y (4)3 ( τ ) (cid:39) p (cid:15) ,Y (4) ( τ ) (cid:39) p (cid:15) , Y (4) (cid:48) ( τ ) (cid:39) − p (cid:15) . (193)Weight 6 modular forms are given: Y (6)1 ( τ ) (cid:39) p (cid:15) , Y (6)2 ( τ ) (cid:39) − p (cid:15) , Y (6)3 ( τ ) (cid:39) − p (cid:15) ,Y (cid:48) (6)1 ( τ ) (cid:39) p (cid:15) , Y (cid:48) (6)2 ( τ ) (cid:39) − p (cid:15) , Y (cid:48) (6)3 ( τ ) (cid:39) p (cid:15) ,Y (6) ( τ ) (cid:39) − p (cid:15) . (194) D Ma jorana and Dirac phases and (cid:104) m ee (cid:105) in ν ββ decay Supposing neutrinos to be Majorana particles, the PMNS matrix U PMNS [83, 84] is parametrized interms of the three mixing angles θ ij ( i, j = 1 , , i < j ), one CP violating Dirac phase δ (cid:96) CP and twoMajorana phases α , α as follows: U PMNS = c c s c s e − iδ (cid:96) CP − s c − c s s e iδ (cid:96) CP c c − s s s e iδ (cid:96) CP s c s s − c c s e iδ (cid:96) CP − c s − s c s e iδ (cid:96) CP c c e i α
00 0 e i α , (195)where c ij and s ij denote cos θ ij and sin θ ij , respectively.The rephasing invariant CP violating measure of leptons [105, 106] is defined by the PMNS matrixelements U αi . It is written in terms of the mixing angles and the CP violating phase as: J CP = Im (cid:2) U e U µ U ∗ e U ∗ µ (cid:3) = s c s c s c sin δ (cid:96) CP , (196)where U αi denotes the each component of the PMNS matrix.There are also other invariants I and I associated with Majorana phases I = Im [ U ∗ e U e ] = c s c sin (cid:16) α (cid:17) , I = Im [ U ∗ e U e ] = c s c sin (cid:16) α − δ (cid:96) CP (cid:17) . (197)We can calculate δ (cid:96) CP , α and α with these relations by taking account ofcos δ (cid:96) CP = | U τ | − s s − c c s c s c s s , Re [ U ∗ e U e ] = c s c cos (cid:16) α (cid:17) , Re [ U ∗ e U e ] = c s c cos (cid:16) α − δ (cid:96) CP (cid:17) . 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