January 19, 2017 7 Twenty years after the discovery of µ τ symmetry. 1 − 0 2 n a J 8 Takeshi Fukuyama 1 1 ] h p Research Center for Nuclear Physics (RCNP), - Osaka University, Ibaraki, Osaka, 567-0047, Japan p e h [ 1 v 5 Abstract 8 9 4 It has passed 20 years after we proposed µ τ symmetry in light neutrino mass matrix. − 0 Thismodelissimplebutreproducedthecharacteresticpropertiesofleptonsector. Afterthat, . 1 duringthe experimentaldevelopments,there haveappearedsomany extensions butmost of 0 those phenomenological models are lacking systematic outlooks towards more fundamental 7 1 theories. Inthispaper,wetrytoconsiderrathersystematicmodelextensionsandapplication : v to GUT model. i X r a 1E-mail:[email protected] 1 Introduction Twenty years ago we proposed first in the world µ τ symmetry in the neutrino mass matrix − model [1], 0 A A ± M = A B C . (1) ν   A C B  ±    in the charged lepton diagonal base. Here A,B,C are real and its components are invariant under µ τ exchange. (1,2) and (1,3) components are equal up to phase convention. This − matrix, therefore, has been called µ τ symmetric mass matrix. This leads immediately to − θ = π/4 and θ = 0 (double sign in the same order as (1)). This matrix represents the 23 13 ∓ characteresticpatternofthemixingangleswhichisquitedifferentfromthatofquarksector. The vanishing (1,1) component leads to the small mixing angle (SMA) solution on θ (See Eq.(11)), 12 which had survived with large mixing angle (LMA) solution at that time. However, KamLAND [2] selected the larger part of solar neutrino angles, and we may set nonzero parameter D in place of the vanishing (1,1) component without breaking µ τ symmetry. In 2013, Daya-Bay[3] − made surprise the unexpectedly large θ . It is impressing that our minimal SO(10) model [4] 13 discussed in section 4 has suffered from large θ before Daya-Bay. 13 The observed data of leptonic mixing matrix nowaday are summarized as [5] sin2θ = 0.304 0.014 12 ± ∆m2 = (7.53 0.18) 10 5 eV2 21 ± × − sin2θ = 0.51 0.05 (normal mass hierarchy) (2) 23 ± sin2θ = 0.50 0.05 (inverted mass hierarchy) 23 ± ∆m2 = (2.44 0.06) 10 3 eV2 (normal mass hierarchy) 32 ± × − ∆m2 = (2.51 0.06) 10 3eV2 (inverted mass hierarchy) 32 ± × − sin2θ = (2.19 0.12) 10 2 13 − ± × Even in these refined data, our µ τ symmetric model does not lose its significance since such − simpleand realsymmetric modelis basically an idealized modelandremains valid as thezero’th order approximation of more sophiscated models. Indeedtherehave appearedavastvariety ofpapersduringnewexperimentaldevelopments. Unfortunately, most of those phenomenological models are lacking systematic analyses valid for theoretical developments from phenomenological model to more fundamental one. In this paper we reconsider µ τ symmetry in these experimental backgrounds and try to − fill this deficit. This paper is organized as follows. In section 2 we review the original µ τ symmetric − model. This model is extended in section 3. In section 4 we argue the correlaton with GUTs. Section 5 is devoted to discussions. 2 µ τ symmetric model − SinceneutrinooscillationexperimentsarewhollyinsensitivetotheMajoraphases,thePontecorvo- Maki-Nakagawa-Sakata (PMNS) mixing matrix is in general written in the form c c , c s , s e iδ c c , c s , s e iδ 13 12 13 12 13 − 13 12 13 12 13 − U = c s s c s eiδ, c c s s s eiδ, s c U , U , s c  23 12 23 12 13 23 12 23 12 13 23 13   µ1 µ2 23 13  − − − ≡ s s c c s eiδ, s c c s s eiδ, c c U , U , c c  23 12− 23 12 13 − 23 12− 23 12 13 23 13   τ1 τ2 23 13      (3) Here c = cosθ , s = sinθ as usual. Neutrino mass matrix is written as ij ij ij ij m , 0, 0 1 − M = U 0, m , UT (4) ν  2  0, 0, m  3    in thecharged lepton diagonalbase, wherewecan set 1form usingtherephasing. Itsexplicit 1 − components are (M ) = m c2 c2 +m s2 c2 +m s2 e 2iδ, (5) ν 11 − 1 12 13 2 12 13 3 13 − (M ) = m c c U +m s c U +m s c s e iδ, (6) ν 12 − 1 12 13 µ1 2 12 13 µ2 3 13 13 23 − (M ) = m c c U +m s c U +m s c c e iδ, (7) ν 13 − 1 12 13 τ1 2 12 13 τ2 3 13 13 23 − (M ) = m U2 +m U2 +m c2 s2 , (8) ν 22 − 1 µ1 2 µ2 3 13 23 (M ) = m U2 +m U2 +m c2 c2 . (9) ν 33 − 1 τ1 2 τ2 3 13 23 As is easily checked, θ = π/4 and θ = 0 is the uniquesolutions for real (M ) = (M ) , 23 ± 13 ν 12 ∓ ν 13 (M ) = (M ) relations (double sign corresponds). ν 22 ν 33 If we adopted θ = π/4 and θ = 0, then neutrino mass matrix becomes 23 13 m 0 0 1 − M = U 0 m 0 UT ν  2  0 0 m  3    c2m +s2m 1 c s (m +m ) 1 c s (m +m ) − 1 1 1 2 √2 1 1 1 2 −√2 1 1 1 2 =  √12c1s1(m1+m2) 21(−s21m1+c21m2+m3) 12(s21m1−c21m2+m3) ,  −√12c1s1(m1+m2) 21(s21m1−c21m2+m3) 21(−s21m1+c21m2+m3)    (10) where c (s ) cosθ (sinθ ) for brevity. In (10) we assumed further 1 1 12 12 ≡ c2 m +s2 m = 0, (11) − 12 1 12 2 and we obtain the mass matrix of (1). Neutrino masses are expressed in terms of A,B,C, 1 m = B C 8A2+(B C)2 , 1 − 2 ± − ± h p i 1 m = B C + 8A2+(B C)2 , (12) 2 2 ± ± m = Bh C. p i 3 ∓ The double sign corresponds to (1). Eq.(11) indicates that neutrinoless double beta decay does not happen in this limit, since m U m U = 0. (13) ee ei i ei h i ≡ i X (11) favored the small mixing angle solution for θ which still had survived at that time. Also 12 the vanishing (1,1) component is interested in connection with seesaw invariant mass matrix [1]. 0 A A 0 A A 0 A A 0 A A − (a) = A B C (b) = A B B (c) = A B C (d) = A B C         A C C A B C A C B A C B        −          (14) (c) is transformed to (d) by the interchange of C to C and these are physically equivalent as − follows. If we leave θ as a free parameter and keep the assumtion (11), then M is reduced to 23 ν 0 c √m m s √m m 2 1 2 2 1 2 − c √m m ( m +m )c2 +m s2 (m m +m )c s , (15)  2 1 2 − 1 2 2 3 2 1− 2 3 2 2  s √m m (m m +m )c s ( m +m )s2+m c2  − 2 1 2 1− 2 3 2 2 − 1 2 2 3 2    where c (s ) cosθ (sinθ ) for brevity. Therefore, (c) and (d) are corresponding to s = 2 2 23 23 23 ≡ π/4 and π/4, repecetively. θ has been determined from the mixing factor sin22θ and they 23 23 − are equivalent. (a) and (b) are also substantially same and from Eq.(10) they are enforced to m 0. This is the case of inverted hierarchy. This symmetric and seesaw invariant concepts is 3 ≈ extended to two-zero texture [6]. Eq.(11) was imposed as it enables us to fix all three masses by the same three parameters as (12). You can easily generalize this simple model as D A A ± Mν′ =  A B C . (16) A C B  ±    Eqs.(11) and (12) in this case are generalized to 2√2A tan2θ = , (17) 12 B C D ± − 1 m = B C +D 8A2+(B C D)2 , 1 − 2 ± − ± − h p i 1 m = B C +D+ 8A2+(B C D)2 , (18) 2 2 ± ± − m = Bh C. p i 3 ∓ If B C D = A, U goes to tri-bi-maximal case [7], ± − 2 1 0 √6 √3 UTB =  −√16 √13 −√12 . (19) 2 1 1    −√6 √3 √2    3 Extension of µ τ symmetric model − Theoriginalµ τ symmetricmodelwasrealandcannotinvolveCPphases. Soanaiveextension − istoextendittoacomplexandsymmetricmassmatrixretainingµ τ symmetry. Thereasoning − why we adhere to a symmetric matrix will be explained in section 4. Eq.(1) is a real symmetric matrix. Its naive extension is D A A ∗ ± M = A B C (double sign corresponds), (20) ν   A C B  ± ∗ ∗    where D,C , A,B . This matrix is diagonalized by the unitary matrix [8], ∈ R ∈ C u u u 1 2 3 U = v v v (double sign correspond to that of (20)) (21)  1 2 3  v v v  ± 1∗ ± 2∗ ± 3∗    as M M = UD2U . (22) ν ν† ν † Then z w w ∗ ± MνMν† = UDν2U† =  w∗ x y . (23) w y x ∗  ±    In Eqs. (20), (21), and (23), double sign corresponds. Here x,z , y,w . In this complex ∈ R ∈ C form we have, in addition to θ = π, θ = 0, another solution, 23 4 13 π π θ = , δ = for minus signature (24) 23 4 ±2 π π θ = , δ = for plus signature (25) 23 −4 ±2 δ = π is interesting since it is the global minimum (though 1 σ) [9]. −2 One strategy for extending µ τ symmetry is the following: The extensions not only − explain the leptonic CP phase but also must include quark sector. This is because we are considering GUT as its more fundamental final correspondent. One of such examples preserves µ τ symmetry up to phase but breaks the symmetric property of mass matrices [10] like, − D A A f f f Mf = Pf†MˆfPf ≡ Pf† A′f Bf Cf Pf (26) A C B  ′f f f    where P = diag(eiαf1,eiβf2,eiγf3) (27) and f includes up-type and down-type quark. First we diagonalize Mˆ by two orthogonal f matrices O and O as f1 f2 OT Mˆ O = diag(m ,m ,m ) (28) f1 f f2 f1 f2 f3 cosϕ sinϕ 0 fi fi − O = U sinϕ cosϕ 0 (i = 1,2) (29) fi TB fi fi  0 0 1     where U is a tri-bi-maximal mixing matrix (19). We found that these matrices are consistent TB with the experimental data of CKM mixing matrix. This is the extension to quark sector but is left on the same phenomenological level as the original work of lepton sector. Hereafter we restrict ourselves in symmetric mass matrices again. If we involve quark sector, it must reveal somehighersymmetric(morefundamental)newcharacter. Inthissence,thoughthereareavast variety of these extensions, most of these phenomenological extensions have no systematic idea leading to more fundamental theoretical models. For the route from (low energy) phenomeno- logical model to (high energy) more fundamental one, some symmetry must play an important role. Let us explain it in the well known example: QED lagrangian has U(1) and Lorentz invari- ances, 1 θ LQED = F Fµν + ǫµνρσF F +L . (30) −4 µν 16π2 µν ρσ matter Under T transformation, E E and B B, and therefore θ θ, if we preserve T → − → → − invariance. In another word, T invariance requires θ = 0. Lorentz invariance breaks to spatial rotation invariance for dielctics, ǫ 1 θ Ldielectric = E2 B2+ E B. (31) 2 − 2µ 4π2 · Thus,permeability (µ)andpermittivity (ǫ)characterize this symmetrybreaking. Eq.(31)((30)) maybeconsideredasaphenomenological(afundamental)model. Accordingtothisgeneralidea, how the phenomenological µ τ symmetry is incorporated to higher symmetry group or more − comprehensive model ? It is natural to incorporate quark sector in this higher symmetric world. We consider here A group as a candidate for it [11]. A is the four degreed symmetry group 4 4 with even permutation whose elements we denote as (a ,a ,a ,a )[12]. A is generated by the 1 2 3 4 4 S and T and their products, which satisfy S2 = T3 = (ST)3 = 1. (32) The three-dimensional unitary representation, in a basis where the element S is diagonal, is built up from: 1 0 0 0 1 0 S = 0 1 0 , T = 0 0 1 . (33)     − 0 0 1 1 0 0  −        Let us practice the transformation, for instance, ST to V (a ,a ,a )T 1 2 3 ≡ a a 2 2 TV = a , STV = a . (34)  3   3  − a a  1   − 1      The rule of the game for reading the permutation group of four degree from three dimensional vector is to make plus element change to a and do minus signs interchange, and ST corre- 4 sponds to (a ,a ,a ,a ). Thus S means the 2 3 symmetry and T does cyclic permutation 4 1 3 2 ↔ or equivalently Z . Mathematically this is the elementary example of Sylow’s theorem [13]. 3 The order of A is 12 = 22 3, and it is the product of normal subgroup V , composed of 4 4 × (1,2)(3,4), (1,3)(2,4), (1,4)(2,3), 1 and Z . Thus A (2 3) symmetry Z . Namely, 3 4 3 ⊃ ↔ × µ τ symmetry may be considered the residual symmetry broken from A . This fact is very 4 − important for the model buiding of more fundamental theories. So far we have not considered Z , and let us consider how Z symmetry appears. Correspondingto this extension, we general- 3 3 ize from the lepton sector to the quark-lepton sector, denoting their fields as ψ (i: generation), i and call 2-3 symmetry instead of µ τ summetry in that case. − WeassignZ chargeofeachgenerationoffermionssoastobecompatiblewith2-3symmetry 3 [14], ψ ψ , ψ ωψ , ψ ωψ , (35) 1L 1L 2L 2L 3L 3L → → → where ω3 = +1. Then, the bilinear terms qLiuRj, qLidRj, lLiνRj, lLieRj and νRicνRj are transformed as follows: 1 ω2 ω2 ω2 ω ω , (36)   ω2 ω ω     where u ν L L q = . l = . (37) L L dL ! eL ! Therefore, if we assume two SU(2) doublet Higgs scalars H and H , which are transformed as 1 2 H ωH , H ω2H , (38) 1 1 2 2 → → the Yukawa interactions are given as follows H = Yu q H˜ u +Yd q H d Yukawa (A)ij Li A Rj (A)ij Li A Rj AX=1,2(cid:16) (cid:17) + Yν l H˜ ν +Ye l H e (39) (A)ij Li A Rj (A)ij Li A Rj AX=1,2(cid:16) (cid:17) + YR νc Φ˜0ν +Ye νc Φ0ν +h.c., (1)ij Rj Rj 2ij Rj Rj (cid:16) (cid:17) where H+ H+ H = A , H˜ = A . (40) A A HA0 ! −HA− ! Therefore, 0 0 0 0 ∗ ∗ Yu , Yd , Yν , Ye , YR = 0 , Yu , Yd , Yν , Ye , YR = 0 0 (41) (2) (1) (2) (1) (1)  ∗ ∗  (1) (2) (1) (2) (2)  ∗  0 0 0  ∗ ∗   ∗      In (41), thesymbol denotes non-zeroquantities. Here, in orderto give heavy Majorana masses ∗ of the right-handed neutrinos ν , we have assumed an SU(2) singlet Higgs scalar Φ0, which is R transformed as H . Mass matrices are sums of Y and Y and their (1,1) element must be 1 (1) (2) vanished: 0 ∗ ∗ . (42)   ∗ ∗ ∗  ∗ ∗ ∗    Thus zero-texture model becomes another useful character as well as symmetric property. Then how far can we go along this line of thought ? In this case, neutrino mass matrix may have the special property of seesaw mechanism [15], and the concept of the seesaw invariance plays an important role [1, 6]. Two-zero texture is interesting from the parameter counting. Mass matrix M is determined by 9out-putparameters, 3masses, 3angles and 3CP phases(one Dirac δ and ν 2 Majorana phases) in the charged lepton flavour diagonal base. Two-zero texture gives four constraints and 9-4=5 in-put free parameters [16, 17, 18]. Among others, the following textures are very important, 0 0 0 0 ∗ ∗ M(1) = and M(2) = 0 . (43) ν   ν   ∗ ∗ ∗ ∗ ∗ 0  ∗ ∗   ∗ ∗ ∗      (2) (1) They are related by µ τ excahange, M = P M P . Here ν 23 ν 23 − 1 0 0 P = 0 0 1 . (44) 23   0 1 0     As mentioned above, five parameters remain free in two-zero texture model. Therefore, if θ , ∆m2 , ∆m2 are determined, δ, ρ, σ are predicted. ij sol atm m s t t m eiδ 1 13 23 12 3 ≈ m s t /t m eiδ (45) 2 12 23 12 3 ≈ ∆m2 /∆m2 s2 t2 t2 1/t2 (46) solar atm ≈ 13 23| 12− 12| (2) (1) for M case [18], where t tanθ . For M case is obtained by replacing t by 1/t . ν ij ij ν 23 23 ≡ − (1) (2) Thus M and M give similar results. This fact is used in GUT formulation as as will be ν ν discussed in the next section. 4 Mass matrix model and GUT GUT models basically search the vertical structure of quark-lepton of one generation. Inter- family (horizontal) relations like Yukawa structure are not predicted. µ τ symmetry may be − clue to this extension. In the previous section, we considered that 2 3 symmetry and Z suggest zero tex- 3 ↔ ture solution. So we consider GUT implimented witbh texture. For that purpose we set two requirements GUT model itself must have few ambiguities and be predictive enough. • The reliable mass texture model should be adopted. • The SO(10) grand unified theory (GUT) can provide the most promising framework to unify quarks and leptons, because the entire SM matter contents of each generation (including a right-handed neutrino) can be unified in a single irreducible representation, 16. A particular attentionhasbeenpaidtotherenormalizableminimalSO(10)model,wheretwoHiggsmultiplets 10 126 are utilized for the Yukawa couplings with the matter representation [19]. The { ⊕ } couplings to the 10 and 126 Higgs fields can reproduce realistic charged fermion and neutrino mass matrices using their phases thoroughly [20, 21]. 126 Higgs is selected since it includes (10, 1, 3) and (10, 1, 3) under the Pati-Salam subgroup which induce type I and type II seesaw mechanism, respectively. Yukawa coupling is given by W = Yij16 H 16 +Yij 16 H 16 , (47) Y 10 i 10 j 126 i 126 j where 16 is the matter multiplet of the i-th generation, H and H are the Higgs multiplet i 10 126 of 10 and 126 representations. The Yukawa coupling, after SO(10) symmetry is broken down to the standard model, is given as follows: M = c M +c M u 10 10 126 126 M = M +M d 10 126 M = c M 3c M (48) D 10 10 126 126 − M = M 3M e 10 126 − M = c M T T 126 M = c M . R R 126 Here c , c , c , c are comlex constants. It should be remarked that mass matrices are 10 126 T R complex andsymmetricmatrices becauseofthegrouppropertyof10,126 representations. Here we proceed to incorporate two-zero texture i n this model. We adopted the two-zero texture (2) mass M of (43). ν Unfortunately, the data fittings have been performed by inputting quark sector spectrum and outputted the lepton sector. In this approach, neutrino mass texture is contaminated by the special base adopted in quark sector and shows no clear texture in M . The reason why ν we adopted the quark sector as input data is that the leptoin sector had been more ambiguous than the quark sector. Nowaday, however, the situation changed. The lepton parameters are more accurate rather than the quark ones, and a large threshold correction is expected in the quark sector in SUSY models. In that sence, it is better to perform the fitting by inputting the parameters in the lepton sector. The formulation is presented in not only such a practical purpose, but also to make clear the property of the solution with v 1016 GeV. Here v is R R ≈ the typical intermediate enrygy scale, and usualy adopted 1013 GeV spoils the gauge coupling unifications [22]. Real data fitting revealed that in v 1016 GeV solution, the down quark R ≈ mass is smaller than the observation, and (1,1) and (1,2) elements of (M ) are smaller than ν the other elements in the fit result. The deficit of down quark mass can be considered as the threshold correction. Under the assumtion of (M ) =(M ) lead us to [23] ν 11 ν 12 ∆m2sol cos2θ sin22θ 4sin2θ ∆m2sol cos4θ +cos2θ tan2θ cosδ = ∆m2atm 13 12− 13 ∆m2atm 12 12 23. (49) PMNS 4sin3θ 1+ ∆m2sol c(cid:16)os2θ sin2θ tanθ (cid:17) 13 ∆m2atm 12 12 23 (cid:16) (cid:17) In Fig.1 we plot the relation between δ and θ in the assumption. Of course those two PMNS 23 mass matrix elements are not exactly zero in the fits, and provides a guide to understand the