Neutrino mixing from finite modular groups
aa r X i v : . [ h e p - ph ] J u l EPHOU-18-002
Neutrino mixing from finite modular groups
Tatsuo Kobayashi ∗ , Kentaro Tanaka † , and Takuya H. Tatsuishi ‡ Department of Physics, Hokkaido University, Sapporo 060-0810, Japan
Abstract
We study the lepton flavor models, whose flavor symmetries are finite subgroups of the mod-ular group such as S and A . In our models, couplings are also nontrivial representationsof these groups and modular functions of the modulus. We study the possibilities that thesemodels realize realistic values of neutrino masses and lepton mixing angles. ∗ E-mail address: [email protected] † E-mail address: [email protected] ‡ E-mail address: [email protected] ontents A model 6 A model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.2 Inverted ordering in A model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 S model 7 S models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95.2 Inverted ordering in S models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.2.1 Model 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.2.2 Model 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.2.3 Model 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Introduction
One of the unsolved but important mysteries in particle physics is the mystery about the flavorstructure of the quarks and leptons such as the generation number, mass hierarchy and mixingangles. Such a mystery would provide us with hints to explore physics beyond the standardmodel. Indeed, several types of scenarios have been proposed to understand quark and leptonmasses and mixing angles as well as CP phases.One of interesting ideas is to impose non-Abelian discrete flavor symmetries. Many modelshave been proposed imposing non-Abelain discrete flavor symmetries, e.g., S , A , S andother various finite groups. (See for review [1–4].) In particular, the lepton sector has beenintensively studied, because at least two of three lepton mixing angles are large compared withthe quark mixing angles and their experimental results have been improved precisely. Recently,CP symmetry and its violation were also studied [5–7].Superstring theory is a promising candidate for the unified theory of all interactions in-cluding gravity and matter fields such as quarks and leptons as well as Higgs fields. It isshown that superstring theory on certain compactifications leads to non-Abelian discrete flavorsymmetries. For example, heterotic string theory on orbifolds can lead to D , ∆(54), etc. [8].(See also [9, 10].) In addition type II magnetized and intersecting D-brane models can lead tosimilar flavor symmetries [12–16].On the other hand, string theory on tori and orbifolds has the modular symmetry. Forexample, modular symmetries were studied in heterotic orbifold models [17–19] and magnetizedD-brane models [20–22]. In general, modular transformations act nontrivially on string modesand interchange massless modes such as quarks and leptons to each other. In this sense, modularsymmetry is a non-Abelain discrete flavor symmetry. Furthermore, it is interesting that themodular symmetry includes S , A , S , A as its congruence subgroups, Γ( N ). However, thereis a difference between the modular symmetry and the usual flavor symmetries. Couplingconstants such as Yukawa couplings also transform nontrivially under the modular symmetry[17, 20–22], while coupling constants are invariant under the usual flavor symmetries, althoughflavon fields, which develop their vacuum expectation values (VEVs), transform nontriviallyunder flavor symmetries. Moreover, Yukawa couplings as well as higher order coupling constantsare modular functions of moduli [20, 23–25].By use of the above aspects, an interesting ansatz was proposed in Ref. [26], where Γ(3) ≃ A was used and leptons were assigned to triplets and singlets of A . Also coupling constants wereassigned to A triplet and singlets, which are modular functions. Then, neutrino masses andmixing angles were analyzed. Such an ansatz would be interesting in order to bridge a gapbetween underlying theory such as superstring theory and low-energy physics like neutrinophenomena. Our purpose in this paper is to study systematically the above approach. Westudy the A model following Ref. [26]. Also, we extend our analysis to S models.This paper is organized as follows. In section 2, we give a briel review on modular symmetry.In section 3, using experimental values, we write the neutrino mass matrix, which is convenient In Ref. [11], relations between enhanced gauge symmetries and non-Abelian discrete flavor symmetrieswere studied. See for their difference and relation [22].
1o our analyses. In section 4, we study systematically the A model following Ref. [26]. Insection 5, we construct S models in a way similar to the A model and study them systemat-ically. Section 6 is our conclusion and discussions. In appendix A, we briefly review modularfunctions and show modular functions corresponding to the A triplet and the S doublet. In this section, we give a brief review on the modular symmetry and its congruence subgroups.Toroidal compactification is one of simple compactifications. The two-dimensional torus T is obtained as T = R / Λ. Here, Λ is the two-dimensional lattice, which is spanned by twolattice vectors, α = 2 πR and α = 2 πRτ . We use the complex coordinate, where R is a realparameter, and τ is a complex modulus parameter.The choice of the basis vectors has some ambiguity. The same lattice can be spanned byother bases, (cid:18) α ′ α ′ (cid:19) = (cid:18) a bc d (cid:19) (cid:18) α α (cid:19) , (1)where a, b, c, d are integer with satisfying ad − bc = 1. That is the SL (2 , Z ) transformation.The modular parameter τ = α /α transforms as τ −→ τ ′ = aτ + bcτ + d , (2)under (1). Both lattice bases ( α , α ) and ( α ′ , α ′ ), and both modular parameters, τ and τ ′ ,lead to the same lattice Λ and the same T . The modular transformation is generated by the S and T transformations, S : τ −→ − τ , (3) T : τ −→ τ + 1 . (4)In addition, these satisfy the following algebraic relations, S = 1 , ( ST ) = 1 . (5)On top of that, when we impose T N = 1, the so-called congruence subgroups Γ( N ) can berealized. The congruence subgroups Γ( N ) are isomorphic to (even) permutation groups, e.g.Γ(2) ≃ S , Γ(3) ≃ A , Γ(4) ≃ S , and Γ(5) ≃ A . (For subgroups of the modular group, e.g.,see Ref. [27].)String theory on T as well as orbifolds T /Z N has the modular symmetry. Furthermore,four-dimensional low-energy effective field theory on the compactification T × X as well as( T /Z N ) × X also has the modular symmetry, where X is a four-dimensional compact space.A set of chiral superfields φ ( I ) transform under the modular transformation (2) as a multiplet[28], φ ( I ) → ( cτ + d ) − k I ρ ( I ) ( γ ) φ ( I ) , (6)2here − k I is the so-called modular weight and ρ ( I ) denotes a representation matrix. Modularinvariant kinetic terms expanded around a VEV of the modulus τ are written by | ∂ µ τ | h τ − ¯ τ i + X I | ∂ µ φ ( I ) | h τ − ¯ τ i k I . (7)Also, the superpotential should be invariant under the modular symmetry. That is, the su-perpotential should have vanishing modular weight in global supersymmetric models. Indeed,Yukawa coupling constants as well as higher-order couplings constants are modular functionsof τ [20, 23–25]. In the framework of supergravity theory, the superpotential must be invariantup to the K¨ahler transformation [28]. That implies that the superpotential of supergravitymodels with the above kinetic term should have modular weight one. In sections 4 and 5, weconsider the global supersymmetric models, and require that the superpotential has vanishingmodular weight, although it is straightforward to arrange modular weights of chiral superfieldsfor supergravity models.The Dedekind eta-function η ( τ ) is one of famous modular functions, which is written by η ( τ ) = q / ∞ Y n =1 (1 − q n ) , (8)where q = e πiτ . The η ( τ ) function behaves under S and T transformations as η ( − /τ ) = √− iτ η ( τ ) , η ( τ + 1) = e iπ/ η ( τ ) . (9)The former transformation implies that the η ( τ ) function has the modular weight 1 / Y , Y , Y ) with weight 2, which behave as an A triplet, are obtainedas Y ( τ ) = i π (cid:18) η ′ ( τ / η ( τ /
3) + η ′ (( τ + 1) / η (( τ + 1) /
3) + η ′ (( τ + 2) / η (( τ + 2) / − η ′ (3 τ ) η (3 τ ) (cid:19) ,Y ( τ ) = − iπ (cid:18) η ′ ( τ / η ( τ /
3) + ω η ′ (( τ + 1) / η (( τ + 1) /
3) + ω η ′ (( τ + 2) / η (( τ + 2) / (cid:19) , (10) Y ( τ ) = − iπ (cid:18) η ′ ( τ / η ( τ /
3) + ω η ′ (( τ + 1) / η (( τ + 1) /
3) + ω η ′ (( τ + 2) / η (( τ + 2) / (cid:19) , in Ref. [26], where ω = e πi/ . (See Appendix A.)We can obtain the modular functions with weight 2, which behave as an S doublet, Y ( τ ) = i π (cid:18) η ′ ( τ / η ( τ /
2) + η ′ (( τ + 1) / η (( τ + 1) / − η ′ (2 τ ) η (2 τ ) (cid:19) ,Y ( τ ) = √ i π (cid:18) η ′ ( τ / η ( τ / − η ′ (( τ + 1) / η (( τ + 1) / (cid:19) , (11)by a similar technique. (See Appendix A.) 3e use the following expansions: Y ( τ ) = 18 + 3 q + 3 q + 12 q + 3 q · · · ,Y ( τ ) = √ q / (1 + 4 q + 6 q + 8 q · · · ) . (12)In section 5, we fit Y /Y to experimental data. That is, we use the following expansion:1 Y = 18 ·
11 + 24 q + · · · . (13)Such an expansion would be valid for | q | . .
1, i.e.Im( τ ) & . . (14) Before studying the A and S models, here using experimental values of neutrino oscillationswe write the neutrino mass matrix, which is convenient to our analyses in Sections 4 and 5. Flavor eigenstates of neutrino ( ν e , ν µ , ν τ ) are linear combinations of mass eigenstates ( ν , ν , ν ).Their mixing matrix U , i.e. the so-called PMNS matrix can be written by U = U e U e U e U µ U µ U µ U τ U τ U τ = c s − s c c s e − iδ CP − s e iδ CP c c s − s c
00 0 1 e iα /
00 0 e iα / , (15)where c ij = cos θ ij and s ij = sin θ ij for mixing angles θ ij , δ CP is the Dirac CP phase, and α i areMajorana CP phases. The mass-squared differences are defined by δm = m − m , (16)∆ m = m − m + m , (17)where m i is the mass eigenvalue of ν i . We also define the ratio between the mass-squareddifferences as r = δm | ∆ m | . (18)Experimental values with normal ordering (NO) and inverted ordering (IO) are shown in Ta-ble 1. 4arameter Normal Ordering Inverted Ordering δm / − eV . +0 . − . . +0 . − . | ∆ m | / − eV . +0 . − . . +0 . − . sin θ / − . +0 . − . . +0 . − . sin θ / − . +0 . − . . +0 . − . sin θ / − . +0 . − . . +0 . − . ⊕ . +0 . − . δ CP /π . +0 . − . . +0 . − . r . +0 . − . × − . +0 . − . × − Table 1: The best-fit values and 1 σ -ranges in experiments with NO and IO from Ref. [29]. In sections 4 and 5, we consider the models, where the charged lepton mass matrix is diagonal,following [26]. In such models, the neutrino mass matrix m ν is written as m ν = U ∗ diag( m , m , m ) U † , (19)by use of the PMNS matrix. By using three column vectors U = ( ~U , ~U , ~U ), the matrix m ν can be decomposed into m ν = m (1) ν + e − iα m (2) ν + e − iα m (3) ν , (20)where m (1) ν ≡ m ( ~U ∗ · ~U † ) , e − iα m (2) ν ≡ m ( ~U ∗ · ~U † ) , e − iα m (3) ν ≡ m ( ~U ∗ · ~U † ) . (21)The matrices m ( i ) ν are symmetric matrices and do not depend on Majorana phases α i in thisdefinition. The overall sizes of m ( i ) ν are of O ( m i ).In the case of NO ( m ≃ m ≪ m ), m ν can be approximated by m ν ≃ m (3) ν = m s e iδ CP c s s e iδ CP c s c e iδ CP c s c c s c c , (22)up to the phase. Here, we have omitted to write explicitly some elements, because m (3) ν is thesymmetric matrix. These matrix elements have the following relations:( m ν ) ( m ν ) = ( m ν ) , ( m ν ) ( m ν ) = ( m ν ) , ( m ν ) ( m ν ) = ( m ν ) , ( m ν ) ( m ν ) = t , ( m ν ) ( m ν ) + ( m ν ) = t e iδ CP , (23)where t ij = tan θ ij . Thus, realistic models should be consistent with these rules (23).In the case of IO ( m ≪ m ≃ m ), the mass matrix m ν can be approximated by m ν ≃ m (1) ν + e − iα m (2) ν . (24)Since α has not been determined by experiments, there is one more parameter in IO than inNO. 5 A model In this section, we consider the model in Ref. [26] systematically. This model is the supersym-metric model, although we can construct a similar nonsupersymmetric model. This model hasthe flavor symmetry Γ(3) ≃ A . SU (2) L × U (1) Y A k I e cR ( , +1) k e e cR ( , +1) ′′ k e e cR ( , +1) ′ k e L ( , − / k L H u ( , +1 / k H u H d ( , − / k H d φ ( , k φ Table 2: A representations and k I in the A modelWe concentrate on the lepton sector. Table 2 shows the A representations and k I of leptonand Higgs superfields, L i , e cRi and H u,d . Recall that − k I is the modular weight. The superfield φ is a flavon field and the A triplet. We arrange k I such that the charged lepton masses arenot modular functions of τ and the flavon field does not appear in the Weinberg operator. Forexample, we take k φ = 3, k H u,d = 0, k L = 1 and k ei = − W e = β e cR H d ( Lφ ) + β e cR H d ( Lφ ) ′ + β e cR H d ( Lφ ) ′′ , (25)where the β i are constant coefficients.We assume that the flavon multiplet develops the VEV along the direction, h φ i = ( u, , v d of the neutral component of H d leads to the diagonal chargedlepton mass matrix, m e = uv d β β β . (26)By choosing proper values of couplings β i , we can realize the experimental values of the chargedlepton masses, m e,µ,τ .On the other hand, we can write the Weinberg operator in the superpotential W ν = 1Λ ( H u H u LLY ( τ )) . (27)Since k L = 1 and k H u = 0, the couplings Y = ( Y , Y , Y ) must be a modular form with modularweight 2 and an A triplet. We use the modular functions (10). Then, we obtain the mass6atrix of neutrinos written by m model ν = v u Λ Y − Y − Y − Y Y − Y − Y − Y Y , (28)where v u denotes the VEV of the neutral component of H u .Note that the charged lepton mass matrix is diagonal. Thus, mixing angles as well as theCP phases originated from the neutrino mass matrix. In the following subsections, we performnumerical analyses by using the mass matrix (28). A model Here, we study the NO case. In the case of NO, there are constraints (23) for the realistic massmatrix. By using the equations in the second line of (23), we obtain t e iδ CP = Y Y + Y = (cid:20) (cid:18) Y Y (cid:19) Y Y (cid:21) − = 14 t (1 + t ) . (29)This is the theoretical prediction in this model. By putting experimental values with ± σ , theleft- and right-hand sides of (29) become (cid:12)(cid:12) t e iδ CP (cid:12)(cid:12) ∼ . , t (1 + t ) ∼ . , (30)respectively. The theoretical prediction (29) is inconsistent with experimental data (30). Thus,this model does not reproduce the experimental results for NO. A model Here, we study the IO case. The mass matrix in this model m model ν (28) obeys the followingrelations: F ( α ; s ij , δ CP ) ≡ ( m model ν ) + 2( m model ν ) = 0 , (31) F ( α ; s ij , δ CP ) ≡ ( m model ν ) + 2( m model ν ) = 0 , (32) F ( α ; s ij , δ CP ) ≡ ( m model ν ) + 2( m model ν ) = 0 . (33)Figure 1 shows a plot of Re F . The shaded region shows the 3 σ deviation of s ij and δ CP .Figure 2 shows plots of Re F i , i = 2 ,
3. Since a realistic model must satisfy F i = 0 at the same α , this model does not reproduce the experimental results within the 3 σ range for IO. S model In this section, we construct the models with the flavor symmetry Γ(2) ≃ S in a way similarto the A model, and study them systematically.7 .0 0.5 1.0 1.5 2.00.00.20.40.60.81.0 Α (cid:144) Π R e F Figure 1: Plot of Re F with 3 σ deviation of s ij and δ CP Α (cid:144) Π R e F - - - Α (cid:144) Π R e F Figure 2: Plots of Re F (left) and Re F (right) with 3 σ deviation of s ij and δ CP SU (2) L × U (1) Y S k I e cR a ( , +1) − e cR b ( , +1) − e cR c ( , +1) ′ − L (1) ( , − / L (2) ( , − / H u ( , +1 / H d ( , − / φ (1) ( , φ (2) ( , S representations and k I in the S models.8able 3 shows the S representations and k I of lepton and Higgs superfields. The φ (1) and φ (2) fields are flavon fields, and φ (1) and φ (2) are S singlet and doublet, respectively. In orderto distinguish e R a and e R b , we assign k I different from each other. For such a purpose, we canimpose an additional symmetry, e.g. Z . We assign k I such that we can realize the diagonalcharged lepton mass matrix similar to the A model. Indeed, the superpotential terms in thecharged lepton sector can be written by W e = β a e cR a H d ( L (1) φ (1) ) + β b e cR b H d ( L (2) φ (2) ) − β c e cR c H d ( L (2) φ (2) ) ′ , (34)where the β i are constant coefficients. We assume that the flavon fields develop their VEVs as h φ (1) i = u , h φ (2) i = ( u , . (35)Then, we can realize the diagonal charged lepton mass matrix when the neutral componentof H d develops its VEV. Similar to the A model, we can realize the experimental values ofthe charged lepton masses, m e,µ,τ by choosing proper values of couplings β a . Note that theassignment of generations to e R i , i = a, b, c is not fixed yet.Modular invariant Weinberg operators in the superpotential can be written by L ν eff = 1Λ (cid:2) dHH (cid:0) L (2) L (2) (cid:1) Y (2) + aHH (cid:0) L (1) L (2) (cid:1) Y (2) + bHH (cid:0) L (1) L (1) (cid:1) Y (1) + cHH (cid:0) L (2) L (2) (cid:1) Y (1) (cid:3) , (36)where a, b, c, d ∈ C are constant coefficients. Y (1) and Y (2) are modular forms with modularweight 2, and Y (1) and Y (2) are S singlet and doublet , respectively. Note that since ′ in × = + ′ + is antisymmetric, (cid:0) L (2) L (2) (cid:1) ′ = 0. We denote Y (1) = Y and Y (2) = ( Y , Y ).There are 6 ways to assign 3 generations of lepton L i to S singlet L (1) and doublet L (2) . Thereplacement of L (2) = ( L i , L j ) → ( L j , L i ) corresponds to the replacement of Y ↔ Y and lessaffects analysis. Therefore, we study 3 models shown in table (4). The fifth column shows theneutrino mass matrix in each model. Since parameters b and c always appear in bY and cY ,respectively, we rewrite B ≡ bY and C ≡ cY . S models Here, we study the NO case in the three S models. In the model 1, the neutrino mass matrixis written by m model ν = C + dY dY aY dY C − dY aY aY aY B . (37)Since the charged lepton mass matrix is diagonal, the PMNS matrix is determined only by theneutrino mass matrix. The consistency conditions (23) are written by( C + dY )( C − dY ) = ( dY ) , B ( C + dY ) = ( aY ) , B ( C − dY ) = ( aY ) , (38) C − dY B = t , C + dY B + ( C − dY ) = e iδ CP t . (39) There are two independent modular forms with weight 2 and Γ(2) [26, 30]. Thus, there is only oneindependent modular form doublet Y (2) in (36). L (1) L (2) m model ν L ( L , L ) dY dY aY dY − dY aY aY aY + cY cY
00 0 bY L ( L , L ) dY aY dY aY aY dY aY − dY + cY bY
00 0 cY L ( L , L ) aY aY aY − dY dY aY dY dY + bY cY
00 0 cY Table 4: Three S modelsWe obtain dY = C/ t = C B , e iδ CP t = 31 + BC (40)from (39). Thus, a prediction of this model is e iδ CP t = 3 s . (41)Experimental values lead to t = O (10 − ) and 3 s = O (1). Hence this model is inconsistentwith experiments. Note that since this result (41) does not depend on Y i , we obtain the sameresult by replacing Y ↔ Y , that is, L (2) = ( L , L ) → ( L , L ).In the models 2 and 3, we can analyze in the same way as in the model 1. A prediction ofthe model 2 is e iδ CP t = 3 c , (42)and a prediction of the model 3 is t = 3 . (43)Both of these predictions are inconsistent with experiments. Hence, these models are notrealistic. S models Here, we study the IO case in the three S models. In the model 1, the mass matrix is written by m model ν = C + dY dY aY dY C − dY aY aY aY B . (44)10rom this matrix, we find( m model ν ) − ( m model ν ) m model ν ) = ( m model ν ) ( m model ν ) = Y Y , (45)and hence F model ≡ m model ν ) ( m model ν ) − ( m model ν ) (cid:2) ( m model ν ) − ( m model ν ) (cid:3) = 0 . (46)This is a prediction of this model.First, we define a function F ( α ; s ij , δ CP ) = 2( m ν ) ( m ν ) − ( m ν ) [( m ν ) − ( m ν ) ] , (47)with m ν in (24), and search a set of the values of ( α , s ij , δ CP ) satisfying F = 0 within the 3 σ experimental range. Here, we treat α as a free parameter. Figure 3 shows a plot of the F - - F I m F Figure 3: F in the model 1.on a complex plane. The blue curve is an one-parametric plot of the F ( α ) with α ∈ [0 , π )and the best-fit values of s ij and δ CP . The blue shaded region is a two-parametric plot ofthe F ( α ; δ CP ) with α ∈ [0 , π ), δ CP ∈ (3 σ -range), and the best-fit values of s ij . The origin F = 0 is the point consistent with the model prediction (46), and this point corresponds to α /π ≃ .
26 and δ CP /π ≃ . m model ν = m ν with numerical matrix m ν . Since there are six complex modelparameters and six complex equations, we can solve these equations. An approximate solutionis a = 0 . − . i, B = 0 . − . i, C = 0 . − . i,d = − . − . i, Y = 0 .
163 + 0 . i, Y = − .
468 + 0 . i. (48)At this stage, numerically expressed mass matrix m ν has degenerate eigenvalues m = m , andtherefore m ν does not reproduce the mixing angle s and mass-squared differences.11e can resolve the degeneracy m = m by changing parameters slightly. That is, we modifythe parameter B → B + ǫ B to reproduce the s and mass-squared differences by resolving thedegeneracy. When the ǫ B = 0 .
030 + 0 . i , observables are given by s = 2 . × − , s = 2 . × − , s = 5 . × − ,r = 2 . × − , sin δ CP = − . , (49)and Majorana phases are given by α /π = 0 . , α /π = 0 . . (50)These observables are consistent with experimental results up to 3 σ , and the values of Majoranaphases are predictions of the model 1.Finally, we fit the values of Y and Y in (48) with the modular functions in (11). Since theoverall coefficient of Y i ( τ ) is not fixed in (89), we fit the value of Y /Y by Y ( τ ) Y ( τ ) ≃ . × e . πi . (51)Such a ratio can be obtained for τ = 0 .
505 + 0 . i . This value is sufficiently large comparedwith Eq. (14).When we change the generation assignment as L (2) = ( L , L ) → ( L , L ), Y and Y arereplaced each other. Thus, we fit the values of Y and Y by Y ( τ ) Y ( τ ) ≃ . × e − . πi , (52)and we get the solution τ = − .
507 + 0 . i . In the model 2, we can analyze with the same way as in the model 1.The model 2 predicts F model ≡ m model ν ) ( m model ν ) − ( m model ν ) (cid:2) ( m model ν ) − ( m model ν ) (cid:3) = 0 . (53)We define F ( α ; s ij , δ CP ) = 2( m ν ) ( m ν ) − ( m ν ) [( m ν ) − ( m ν ) ] , (54)and this is shown in Figure 4 similar to the model 1. The condition F = 0 is satisfied at α /π ≃ .
16 and δ CP /π ≃ . m model ν = m ν is a = 0 . − . i, B = 0 . − . i, C = 0 . − . i,d = − . − . i, Y = − . − . i, Y = − .
523 + 0 . i. (55)12 - F I m F Figure 4: F in the model 2 - - F I m F Figure 5: F in the model 3We can tune B → B + ǫ B with ǫ B = 0 .
033 + 0 . i , and we obtain s = 2 . × − , s = 2 . × − , s = 5 . × − ,r = 2 . × − , sin δ CP = 0 . , (56)and α /π = 0 . , α /π = 0 . . (57)These observables are consistent with experimental results up to 3 σ except for the δ CP . Thevalue of δ CP is out of 3 σ deviation. The values of Majorana phases are predictions of themodel 2. In this model, we can also tune a or d instead of B .Finally, we fit the values of Y i by Y ( τ ) Y ( τ ) ≃ . × e − . πi . (58)By solving this equation, we obtain the solution τ = − .
487 + 0 . i . When we change thegeneration assignment as L (2) = ( L , L ) → ( L , L ), we find Y ( τ ) Y ( τ ) ≃ . × e . πi , (59)and we obtain the solution τ = 0 .
480 + 1 . i , The model 3 predicts F model ≡ m model ν ) ( m model ν ) − ( m model ν ) (cid:2) ( m model ν ) − ( m model ν ) (cid:3) = 0 . (60)13e define F ( α ; s ij , δ CP ) = 2( m ν ) ( m ν ) − ( m ν ) [( m ν ) − ( m ν ) ] , (61)and this is shown in Figure 5 similar to the model 1. The condition F = 0 is satisfied at α /π ≃ .
14 and δ CP /π ≃ . m model ν = m ν is a = 0 .
000 + 0 . i, B = 0 . − . i, C = 0 . − . i,d = 0 . − . i, Y = − .
203 + 0 . i, Y = 0 . − . i. (62)We can tune B → B + ǫ B with ǫ B = 0 .
001 + 0 . i , and we obtain s = 3 . × − , s = 2 . × − , s = 5 . × − ,r = 2 . × − , sin δ CP = 0 . , (63)and α /π = − . , α /π = − . . (64)These observables are consistent with experimental results up to 1 σ except for the δ CP . Thevalue of δ CP is out of 3 σ deviation. The values of Majorana phases are predictions of themodel 3. In this model, we can also tune C or d instead of B .Finally, we fit the values of Y i by Y ( τ ) Y ( τ ) ≃ . × e − . πi . (65)By solving this equation, we obtain the solution τ = 1 .
000 + 1 . i . When we change thegeneration assignment as L (2) = ( L , L ) → ( L , L ), we get Y ( τ ) Y ( τ ) ≃ . × e . πi , (66)This equation has no solution. We have studied neutrino mixing in the models with A and S discrete flavor symmetries. Inour models, couplings are also nontrivial representations under the discrete flavor symmetries,and they are modular functions. In the A model, following [26], we assigned the three gen-erations of leptons to the triplet of A . In this case, the form of the neutrino mass matrix isstrongly restricted as (28), and there are no realistic solution. In the S models, we assignedthe three generations of leptons to a singlet and a doublet of S . In these cases, there arefive model parameters except for an overall coefficient in Table 4. It may be easier to fit theexperimental data by increasing the number of parameters. However, neutrino mass matricesin our models have restricted forms, and are written by modular functions. Thus, it is non-trivial to realize the experimental values by many parameters. Indeed, there are no solution in14he case of normal ordering. In the case of inverted ordering, we can reconstruct experimentalresults except for the δ CP within the 3 σ -range in all three models. Additionally, we can fit themass matrix by using modular functions in all three models. Also, we have predictions on theMajorana CP phases.It would be interesting to study other assignments of leptons in A and S models such thatthe charged lepton mass matrix is not diagonal and depend on modular functions of τ . Also,it would be interesting to extend our analyses to other congruence subgroups, e.g. Γ(4) ≃ S and Γ(5) ≃ A . Acknowledgement
The authors would like to thank Y.Takano for useful discussions. T. K. was is supported inpart by MEXT KAKENHI Grant Number JP17H05395 and JSP KAKENHI Grant NumberJP26247042.
A Modular funcitons
Here, following [26], we derive modular functions with modular weight 2, which behave as an A triplet and an S doublet.Suppose that the function f i ( τ ) has modular weight k i . That is, it transforms under themodular transformation (2), f i ( τ ) → ( cτ + d ) k i f i ( τ ) . (67)Then, it is found that ddτ X i log f i ( τ ) → ( cτ + d ) ddτ X i log f i ( τ ) + c ( cτ + d ) X i k i . (68)Thus, ddτ P i log f i ( τ ) is a modular function with the weight 2 if X i k i = 0 . (69)We find the following transformation behaviors under T , η (3 τ ) → e iπ/ η (3 τ ) ,η ( τ / → η (( τ + 1) / ,η (( τ + 1) / → η (( τ + 2) / , (70) η (( τ + 2) / → e iπ/ η ( τ / , S , η (3 τ ) → r − iτ η ( τ / ,η ( τ / → √− i τ η (3 τ ) ,η (( τ + 1) / → e − iπ/ √− iτ η (( τ + 2) / , (71) η (( τ + 2) / → e iπ/ √− iτ η (( τ + 1) / . Using them, we can construct the modular functions with weight 2 by Y ( α, β, γ, δ | τ ) = ddτ ( α log η ( τ /
3) + β log η (( τ + 1) /
3) + γ log η (( τ + 2) /
3) + δ log η (3 τ )) , (72)with α + β + γ + δ = 0 because of Eq.(69). These functions transform under S and T as S : Y ( α, β, γ, δ | τ ) → τ Y ( δ, γ, β, α | τ ) ,T : Y ( α, β, γ, δ | τ ) → Y ( γ, α, β, δ | τ ) . (73)Now let us construct an A triplet by the modular functions Y ( α, β, γ, δ | τ ). We use the(3 ×
3) matrix presentations of S and T as ρ ( S ) = 13 − − − , ρ ( T ) = ω
00 0 ω , (74)where ω = e πi/ . They satisfy( ρ ( S )) = I , ( ρ ( S ) ρ ( T )) = I , ( ρ ( T )) = I , (75)that is, Γ(3) ≃ A . Using these matrices and Y ( α, β, γ, δ | τ ), we search an A triplet, whichsatisfy, Y ( − /τ ) Y ( − /τ ) Y ( − /τ ) = τ ρ ( S ) Y ( τ ) Y ( τ ) Y ( τ ) , Y ( τ + 1) Y ( τ + 1) Y ( τ + 1) = ρ ( T ) Y ( τ ) Y ( τ ) Y ( τ ) . (76)Their solutions are written by Y ( τ ) = 3 cY (1 , , , − | τ ) , Y ( τ ) = − cY (1 , ω , ω, | τ ) , Y ( τ ) = − cY (1 , ω, ω , | τ ) , (77)up to the constant c . They are explicitly written by use of eta-function as Y ( τ ) = i π (cid:18) η ′ ( τ / η ( τ /
3) + η ′ (( τ + 1) / η (( τ + 1) /
3) + η ′ (( τ + 2) / η (( τ + 2) / − η ′ (3 τ ) η (3 τ ) (cid:19) ,Y ( τ ) = − iπ (cid:18) η ′ ( τ / η ( τ /
3) + ω η ′ (( τ + 1) / η (( τ + 1) /
3) + ω η ′ (( τ + 2) / η (( τ + 2) / (cid:19) , (78) Y ( τ ) = − iπ (cid:18) η ′ ( τ / η ( τ /
3) + ω η ′ (( τ + 1) / η (( τ + 1) /
3) + ω η ′ (( τ + 2) / η (( τ + 2) / (cid:19) , c = i/ (2 π ). They can be expanded as Y ( τ ) = 1 + 12 q + 36 q + 12 q + · · · ,Y ( τ ) = − q / (1 + 7 q + 8 q + · · · ) , (79) Y ( τ ) = − q / (1 + 2 q + 5 q + · · · ) . Similarly, we can construct the modular functions, which behave as an S doublet. Under T , we find the following transformation behaviors, η (2 τ ) → e iπ/ η (2 τ ) ,η ( τ / → η (( τ + 1) / , (80) η (( τ + 1) / → e iπ/ η ( τ / . (81)Also, S transformation is represented by η (2 τ ) → r − iτ η ( τ / ,η ( τ / → √− i τ η (2 τ ) , (82) η (( τ + 1) / → e − iπ/ √− iτ η (( τ + 1) / . (83)Then, we consider Y ( α, β, γ | τ ) = ddτ ( α log η ( τ /
2) + β log η (( τ + 1) /
2) + γ log η (2 τ )) . (84)These functions are the modular functions with the weight 2 if α + β + γ = 0. They transformunder S and T as S : Y ( α, β, γ | τ ) → τ Y ( γ, β, α | τ ) ,T : Y ( α, β, γ | τ ) → Y ( γ, α, β | τ ) . (85)Using Y ( α, β, γ | τ ), we construct the S doublet. For example, we use the (2 ×
2) matrixrepresentations of S and T as ρ ( S ) = 12 (cid:18) − −√ −√ (cid:19) , ρ ( T ) = (cid:18) − (cid:19) . (86)They satisfy ( ρ ( S )) = I , ( ρ ( S ) ρ ( T )) = I , ( ρ ( T )) = I , (87)that is, Γ(3) ≃ S . Using these matrices and Y ( α, β, γ | τ ), we search an S doublet, whichsatisfy, (cid:18) Y ( − /τ ) Y ( − /τ ) (cid:19) = τ ρ ( S ) (cid:18) Y ( τ ) Y ( τ ) (cid:19) , (cid:18) Y ( τ + 1) Y ( τ + 1) (cid:19) = ρ ( T ) (cid:18) Y ( τ ) Y ( τ ) (cid:19) . (88)17heir solutions are written by Y ( τ ) = cY (1 , , − | τ ) , Y ( τ ) = √ cY (1 , − , | τ ) , (89)up to the constant c . They are explicitly written by use of eta-function as Y ( τ ) = i π (cid:18) η ′ ( τ / η ( τ /