Origin of Symmetric PMNS and CKM Matrices Werner Rodejohanna and Xun-Jie Xua,b aMax-Planck-Institut fu¨r Kernphysik, Postfach 103980, D-69029 Heidelberg, Germany bInstitute of Modern Physics and Center for High Energy Physics, Tsinghua University, Beijing 100084, China ([email protected] and [email protected]) ThePMNSandCKMmatricesarephenomenologicallyclosetosymmetric,andasymmetricform could be used as zeroth-order approximation for both matrices. We study the possible theoretical origin of this feature in flavor symmetry models. We identify necessary geometric properties of discrete flavor symmetry groups that can lead to symmetric mixing matrices. Those properties are actuallyverycommonindiscretegroupssuchasA ,S or∆(96). Asanapplicationofourtheorem, 4 4 we generate a symmetric lepton mixing scheme with θ = θ = 36.21◦;θ = 12.20◦ and δ = 0, 12 23 13 realized with the group ∆(96). 5 1 I. INTRODUCTION (cid:18)0.845(cid:19) (cid:18)0.592(cid:19) (cid:18)0.172(cid:19) 0  0.791 0.512 0.133  2 (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) b The properties of the fermion mixing matrices are ex- |UPMNS|= 00..522514 00..649585 00..768024 . (3) e pected to give important hints on the underlying flavor (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) F physics. Flavor symmetries [1] are an attractive and  0.521 0.698 0.782  0.254 0.455 0.604 most often studied approach to explain the rather differ- 2 2 ent structure of the Pontecorvo-Maki-Nakagawa-Sakata Here the upper (lower) values in each entry are upper (PMNS) and Cabibbo-Kobayashi-Maskawa (CKM) mix- (lower)boundsofthematrixelements. TheCKMmatrix ] ing matrices. Literally hundreds of models have been h hasbeenmeasuredtoahighprecision(hereweshowthe proposed in the literature, applying many possible dis- p 1σ range) and the relations |U | = |U |, |U | = |U | crete groups in order to explain lepton and quark mix- 12 21 23 32 - are still well compatible with data. The relation |U |= p ing. Instead of adding simply another model to that list, 13 e we study in this paper an interesting possible property |U31| is, however, not fulfilled by data. As a symmetric h mixing matrix requires that [2, 11] of both the CKM and PMNS matrices. Namely, despite [ thefactthattheCKMmixingisasmallwhilethePMNS |U |2−|U |2 =|U |2−|U |2 =|U |2−|U |2 =0, 3 mixing is large, both can to reasonable precision be es- 31 13 12 21 23 32 (4) v timated to be symmetric. The symmetric form of the 1 we have an interesting option, namely, that some flavor CKMmatrixhasearlybeennoticedandstudiedinmany 9 symmetry or other mechanism generates |U12| = |U21|, references [2–10]. After neutrino oscillation was well es- 9 |U | = |U | but U = U = 0. Higher order correc- 23 32 13 31 2 tablished the possible symmetric PMNS matrix also at- tions, which are frequently responsible for the smallest 0 tractedsomeattention[11–17]. Thesymmetricformdis- mixing angles, are then the source of non-zero |U | (cid:54)= . cussed in these references includes the manifestly sym- 13 1 |U |,aswellasforCPviolation. Rathertrivially,matri- 0 metric case (U =UT) and the Hermitian case (U =U†). ces31with only one mixing angle are symmetric, the same 5 It is easy to get the relation holds for the unit matrix. 1 The symmetry conjecture for the PMNS mixing is less : v (U =UT)⇒(|U|=|U|T)⇐(U =U†) (1) compatible with data, as shown by the 3σ bounds in Eq. i (3) [20]. Similar to the quark sector, the 13- and 31- X elementsareincompatiblewithsymmetry(theothertwo bytakingabsolutevalues,whichimpliesanyphysicalpre- r a dictionfrom|U|=|U|T canalsobeusedintheothertwo relations between the elements are also not favored by casesU =UT orU =U†. Bothofthemarespecialcases data),andasimilarsituationasmentionedaboveforthe of |U| = |U|T, which is what we mean by symmetric CKMmatrixmightberealized. Ofcourse,onecouldalso imagine that an originally symmetric mixing matrix is mixing matrix from now on. modifiedbyhigherordercorrections,VEV-misalignment, Using the global fits of the CKM [18] and PMNS [19] RG-effects or other mechanisms that have been studied matrices, one finds: in the literature. For completeness, we give the phenomenological pre- (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) 0.97441 0.22597 0.00370 dictionofasymmetricmixingmatrix,usingthestandard  0.97413 0.22475 0.00340  parametrizationoftheCKMandPMNSmixingmatrices (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)   0.22583 0.97358 0.0426  [11]: |U |=  CKM  0.22461 0.97328 0.0402  (cid:18)0.00919(cid:19) (cid:18)0.0416(cid:19) (cid:18)0.99919(cid:19) |U |= sinθ12 sinθ23 13 (cid:112) 0.00854 0.0393 0.99909 1−sin2δ cos2θ cos2θ +cosδ cosθ cosθ 12 23 12 23 (2) (5) 2 Thisistheuniquephysicalpredictionofboth|U|=|U|T 23] and U =UT. Note that |U|=|U|T has only one predic- U†SU =D ; (9) tion as the unitarity requires that the relation in Eq. (4) ν ν S isfulfilled,soonceweset|U13|2 =|U31|2 weimmediately U(cid:96)†TU(cid:96) =DT , (10) get|U|=|U|T. ItisalsotheuniquepredictionofU =UT because any 3-by-3 unitary U with |U| = |U|T can be whereallDarediagonalmatrices. NotethatUν obtained transformed to a new U(cid:48) satisfying U(cid:48) = U(cid:48)T simply by from Eq. (9) does not include Majorana phases which rephasing [2, 11]. Note that this does not hold for more rephase each column of Uν. Eq. (9) is independent of than 3 generations. The Hermitian case U =U† has not such rephasing which means the Majorana phases are only the prediction of Eq. (5) but also CP conservation not determined by flavor symmetries in this approach. [13D].esTphitues,thitewaopupladrepnrteddicetvisaitnioθn13f=romtanthθ1e2irtsaynmθ2m3.etric FofortwMoaZjo2r,ani.ae.nZeu2t⊗rinZos2 Gtoν fhuallsytdoebteermaidnieretchtepmroidxuincgt forms, one can still use it as an attractive zeroth-order in neutrino sector[24]. For quarks, we have the same Ansatz and attempt to study its theoretical origin. One framework but since they are Dirac fermions, we do not option,putforwardin[15],isthatasingleunitarymatrix have to be limited to Z2⊗Z2. In this case note that ST V diagonalizes all mass matrices of quarks and leptons andUνT inEqs.(6,7)shouldbereplacedwithS† andUν†. at leading order, in addition to the SU(5) relation m = d mT between the down quark and charged lepton mass (cid:96) II. A THEOREM FOR |U|=|U|T matrices. While it is difficult to embed this in realistic massspectrainGUTs,thepredictionsofthisscenarioare thatU =V†V =1whileU =VTV issymmetric. Inthispaper,asmentionedabove,wedefinesymmetric CKM PMNS In this paper, we study the the origin of symmetric mixing matrix as mixingmatricesfromanunderlyingflavorsymmetry. We |U|=|U|T (11) prove a theorem which links geometric properties of dis- cretesymmetrygroupstothesymmetricformofthemix- rather than the original definition of U = UT or the ingmatrices. Thistheorem,explainedindetailinSection HermitiancaseU =U†. Thephenomenologyisthesame II, holds only for subgroups of SO(3) with real represen- for |U| = |U|T and U = UT, but more general than the tations, and can interestingly be realized in the most of- Hermitian case. ten studied groups A and S . We find a modification 4 4 Beforeweformulateourtheoremthatlinksgeometrical that holds in the complex case in Section III, that could properties of the flavor symmetry group to a symmetric be applied to subgroups of SU(3), for instance to ∆(96). mixingmatrix,wewillfirstdefinethegeometricconcepts We use this to reproduce a previously studied, and actu- which will be used. ally symmetric, mixing scenario for the PMNS matrix in TheZ symmetriesusedinneutrinosectorareactually Section IV. 2 just reflections or 180◦ rotations (the difference between Sinceouranalysislinksthepropertiesofthesymmetry them is trivial, any 180◦ rotation in 3-dimensional space group with the mixing matrix, we end our introduction can be changed toa reflectionif weadd anoverallminus with a summary on how the generators of the group can sign and vice versa). In 3-dimensional flavor space, be related to the matrices diagonalizing the mass ma- going w.l.o.g. in the diagonal neutrino basis, the Z trices, following the strategy developed in [21–23]. In 2 transformations correspond to putting minus signs to general, if a flavor symmetry group G is applied to, for neutrino mass eigenstates: ν → −ν . The combined instance,theleptonsector,thenitmustbebrokentotwo i i Z ⊗ Z in 3-dimensional flavor space corresponds to residualsymmetriesG andG actingonthechargedlep- 2 2 (cid:96) ν two reflections with respect to the direction of neutrino ton sector and neutrino sector: mass eigenstates. Or, in a picture we are more familiar (cid:40) G : STM S =M with, if there are planes whose normal vectors are G→ ν ν ν (6) neutrino mass eigenstates, the Z symmetries are just G : T†M T =M . 2 (cid:96) (cid:96) (cid:96) the mirror symmetries of those planes. Since Z ⊗ Z 2 2 contains commutative mirror transformations or since Heretheleft-handedneutrino(assumedtobeMajorana) the mass eigenstates are orthogonal, the mirrors should mass matrix M is invariant under the transformation ν STM S forS ∈G andM (definedbym m†,wherem be perpendicular to each other, as shown in Fig. 1 by ν ν (cid:96) (cid:96) (cid:96) (cid:96) translucent squares. If the transformation from G is is the charged lepton mass matrix) is the effective mass (cid:96) real (we will discuss the complex case later) in flavor matrix of left-handed charged leptons, invariant under space, it is an SO(3) transformation which can always T†M T. Then the diagonalizing matrices U and U de- (cid:96) ν (cid:96) be represented by a rotation. If the rotation axis is on fined by the bisecting plane, which is defined as the plane that M =U D UT; (7) bisects the two mirror squares, or as the boundaries of ν ν ν ν M =U D U†; (8) octants in the diagonal neutrino basis, then we define (cid:96) (cid:96) (cid:96) (cid:96) that G bisects the two Z ⊗Z . We show two bisecting (cid:96) 2 2 canbedirectlydeterminedbyS andT accordingto[21– planes in Fig. 1, while all bisecting planes are shown in 3 Figure2. Thecompletecollectionofallsixpossiblebisecting planes and their geometrical relation with the mirror planes. Figure 1. The geometrical relation of the mirror planes and Arotationalsymmetrywithitsaxisononeofthesebisecting the bisecting planes. The mirrors are placed on y−z, z−x planescangive,accordingtoTheoremA,asymmetricmixing or x−y planes. The two round disks are called bisecting |U|=|U|T. planes because they bisect all the square mirror planes. The bisecting planes are boundaries of octants. hedral group T and octahedral group O which are just the widely used A and S flavor symmetry groups, re- 4 4 Fig. 2. This gives now all definitions necessary for our spectively (for geometrical interpretations on A and S 4 4 theorem. see, for e.g. [25]). We can see in Fig. 3 that if we choose the three 180◦ rotational axes as x, y and z axes, which Theorem A: penetratethetetrahedronthroughthetwocentralpoints IfanSO(3)subgroupGcontainstwonon-commutative of two edges, then the bisecting planes are determined, Abelian subgroups G and G , and if G is isomorphic as shown by dark blues circles. The tetrahedron is also ν (cid:96) ν to Z ⊗Z while G bisects the Z ⊗Z , then G as a invariantunderthe120◦ rotationmarkedinFig.3,which 2 2 (cid:96) 2 2 flavorsymmetrycanproduceasymmetricmixingmatrix. is a bisecting rotation since the axis is on three bisecting planes. In explicit formulae, we say the tetrahedron is The definition of bisection and symmetric mixing are invariant under the following rotations, given previously. The subgroups G and G are required ν (cid:96)   to be Abelian because the residual flavor symmetries are 0 −1 0 alwaysAbelian[21–23],andnon-commutativesothatthe Rbs =0 0 −1; (12) mixing is non-trivial. 1 0 0 The proof of this theorem will be obvious after we in- troduce the general SO(3) rotation and the diagonaliza- S =diag(1,−1,−1); S =diag(−1,1,−1). (13) tion below (see Eqs. (20) and (24)). The Z symmetries 1 2 2 areappliedtoMajorananeutrinosandthebisectingrota- Here R is the 120◦ bisecting rotation and S (i = 1,2) bs i tiontochargedleptons. However,onecanalsoapplythe are the 180◦ rotations around the x and y axes. Eq. theoremtoquarksandobtainasymmetricCKMmixing. (12) can be obtained by requiring that R (1,0,0)T = InthecaseofDiracfermions,Z2 isnotnecessarybutsuf- (0,0,1)T, which means R rotates the x basxis to the z bs ficient. We will comment further on CKM mixing later. axis, as well as the other two relations R (0,1,0)T = bs Becausetheaxisofabisectingrotationcanberotatedon (−1,0,0)T and R (0,0,1)T =(0,−1,0)T. bs itsbisectingplanes, thereareinfinitebisectingrotations. If R and S (i = 1,2) are the residual symmetries bs i Hence, Theorem A can produce infinite symmetric mix- of the charged lepton sector and neutrino sector respec- ing matrices with one degree of freedom. Note that the tively, i.e. unitary matrix with the constraint |U| = |U|T has only one prediction, see Eq. (5). R† M R =M ; STM S =M , (14) bs (cid:96) bs (cid:96) i ν i ν Actually a lot of discrete flavor symmetries satisfy the conditionsrequiredbyTheoremA,forexamplethetetra- thenaccordingtoEqs.(9,10)wecancomputeU andU (cid:96) ν 4 Figure3. Tetrahedronsymmetry. Thedarkbluecirclesshow thebisectingplanes. Theaxesofthe120◦ rotationalsymme- Figure 4. Octahedron symmetry. Similar to Fig. 3, accord- triesofthetetrahedronareonthoseplanes,thereforeaccord- ing to Theorem A the octahedral group can also be used to ingtoTheoremAthetetrahedralgroupasaflavorsymmetry produce a symmetric mixing matrix |U|=|U|T. can produce a symmetric mixing matrix |U|=|U|T. we will show below. The neutrino and charged lepton from Rbs and Si. The result is mass matrices are now invariant under transformations   of (S , S ) and R respectively: 1 ω ω2 1 2 bs 1 U(cid:96) = √ −1 −ω2 −ω ; Uν =1. (15) STM S =M ; (i=1,2) (17) 3 1 1 1 i ν i ν R† M R =M . (18) bs (cid:96) bs (cid:96) We see that U is the Wolfenstein matrix, up to trivial (cid:96) signs. ThereforeinthiscasethePMNSmatrixU =U†U Diagonalizing these matrices with the transformation (cid:96) ν is symmetric. i.e. |U|=|U|T. M =U D UT , M =U D U† (19) As another example we show in Fig. 4 that the ν ν ν ν (cid:96) (cid:96) (cid:96) (cid:96) octahedral symmetry, which is isomorphic to the widely used S4 symmetry, also has the required properties for givesthePMNSmixingmatrixU =U(cid:96)†Uν. Accordingto Theorem A. Figs. 3 and 4 show that the properties TheoremA,U willbesymmetricwithproperorderingof required by Theorem A are quite common in discrete the eigenvectors. groups with 3-dimensional irreducible real representa- Since we choose the basis in which S1 and S2 are di- tions. agonal, Mν is constrained to be diagonal by Eq. (17), hence U is diagonal. As discussed at the end of the In- ν Now we present the theorem in explicit formulae. For troduction, diagonalization of (S1,S2) and Rbs will give simplicity, we choose a basis under which the mirror has Uν and U(cid:96). So actually the key point of Theorem A can a normal vector (1,0,0)T, (0,1,0)T or (0,0,1)T so the be stated as follows: mirror symmetry is just a reflection with respect to the For any SO(3) matrix R, if R = Rbs there must be y−z, z−x or x−y planes. Then the mirror transfor- an unitary matrix U which is symmetric (|U| = |U|T) mations through y−z, and z−x planes are and can diagonalize R. The converse is also true which means R=R is the necessary and sufficient condition bs S1 =diag(−1,1,1) and S2 =diag(1,−1,1) for |U|=|U|T. Thus, in the basis we choose, we have a bisecting ro- respectively. Under this basis, the normal vectors n = tation to generate |U | = |U |T and the mirror symme- (n ,n ,n )T of the six bisecting planessatisfyone of the (cid:96) (cid:96) 1 2 3 tries to make U diagonal, therefore we get a symmetric six conditions: ν PMNSmatrix. Intheabovediscussionwehaveexplained the theorem in a specific basis, however, the physical re- |n |=|n |, (i,j =1,2,3; i(cid:54)=j), (16) i j sultisindependentofanybasis. Onecanchooseanother i.e. n = ±n , n = ±n or n = ±n . The bisecting basis where the mirrors are not on the x−y, y−z and 1 2 2 3 1 3 rotationsR withsuchanaxishavespecialformswhich z − x planes, in which case the neutrino sector is not bs 5 diagonal and in general |U | (cid:54)= |U |T. However, the geo- As for the explicit form of the bisecting rotation R , (cid:96) (cid:96) bs metrical relation of the bisecting planes and the mirror we should first introduce the general rotation. The most planes makes sure that the product U†U is symmetric. general rotation in Euclidean space which rotates the (cid:96) ν whole space around an axis n=(n ,n ,n )T (n.n=1) 1 2 3 by an angle φ is  n2+c(cid:0)n2+n2(cid:1) (1−c)n n +sn −sn +(1−c)n n  1 2 3 1 2 3 2 1 3 R(n,φ)=(1−c)n1n2−sn3 c+n22−cn22 sn1+(1−c)n2n3 , (20) sn +(1−c)n n −sn +(1−c)n n c+n2−cn2 2 1 3 1 2 3 3 3 where c=cosφ and s=sinφ. One can check that Eq. (20) does rotate the whole Here|U |2isnot|U ||U |buteachelementx of|U |2is R R R ij R space around n by an angle φ while keeping n invariant. theabsolutevaluesquaredoftheij-elementofU . Note R For example, when n=n ≡(0,0,1)T we have thattheorderofthecolumnsin(24)canbechangedsince z reordering of the columns of a diagonalization matrix is   c s 0 just a matter of permutation of eigenvectors. For n = 1 R(nz,φ)=−s c 0, (21) n3, we recommend to write it in this order so that once 0 0 1 one takes n =n one immediately obtains a symmetric 1 3 matrix. For the other cases such as n =n etc., we can 1 2 whichisthefamiliarformofarotationinthex−y plane always reorder the columns to get a symmetric matrix. around the z axis. From Eq. (24), the proof of Theorem A is easy. One For each of the six conditions in Eq. (16) we can get justsetsEq.(24)equaltoitstransposeandfindsn2 =n2. a bisecting rotation matrix from Eq. (20). We use the 1 3 Thereareothertwopossiblepermutationofthecolumns symbol R(±ij) to denote these bisecting rotations: where (n2,n2,n2)T is the first or the last column of U , 1 2 3 R R(±ij) ≡R(n| ,φ). from which we can get n22 =n23 or n21 =n22. ni=±nj This completes our proof of Theorem A. As an example, for n =n , we have 1 3   d a p III. GENERALIZATION TO THE COMPLEX R(13) = b h a, (22) CASE q b d The previous theorem only applies for flavor sym- where a = sn +(1−c)n n , b = (1−c)n n −sn , 1 1 2 1 2 1 metries with real representations, while some groups d=c+n2(1−c). The remaining parameters p, q and h 1 used in flavor symmetry model building enjoy complex aredeterminedbyRRT =1ifa,b,darefixed. Ingeneral representations. For the complex case, we cannot find a they are not equal to each other, but their precise forms clear geometrical picture as was possible for real repre- are not important here. The point we should notice here sentations in 3-dimensional Euclidean space. However is, if R has n = n , then the 12-element equals the 1 3 we can somewhat generalize the previous theorem to the 23-element, the 21- the 32- and the 11- the 33-element. complex case by finding some connections between the Conversely,ifanSO(3)matrixhastheformof(22),then real and complex case [26]. In the following discussion, it must be a bisecting rotation with its axis on the x=z all unitary matrices are elements of SU(3) since the plane. Thiscanbeseenbysolving(22)asanequationfor difference between U(3) and SU(3) is a trivial phase. (n,φ) (the solution always exists since (20) contains all possible SO(3) matrices) and finding that the solutions Theorem B: always have n =n . 1 3 If an SU(3) matrix T can be rephased to a real matrix R can be diagonalized by R as follows, UR†RUR =diag(eiφ,1,e−iφ), (23) T =diag(eiα1,eiα2,eiα3)Rdiag(eiβ1,eiβ2,eiβ3), (25) where the eigenvalues only depend on φ while U only R andifthe R isoneofthebisectingrotationswith n =n i j dependsonn. Asonecanchecknumericallyorbydirect [27] from Theorem A, and if further analytic calculation, |U | has the following form, R α +β =0(k (cid:54)=i,j), (26)   k k 1−n2 2n2 1−n2 1 1 1 1 |UR|2 = 21−n22 2n22 1−n22 . (24) then T gives a symmetric mixing matrix [28]. 1−n2 2n2 1−n2 3 3 3 6 Note on Theorem B: k in Eq. (26) is the remain- could serve as a starting point or zeroth-order approxi- ing number among {1,2,3} when |n | = |n | picks out mation. i j two numbers for i and j. Since the rephasing matrices The mixing scheme can be produced in ∆(96) group diag(eiα1,eiα2,eiα3)anddiag(eiβ1,eiβ2,eiβ3)shouldbein which can be defined by three generators a, b and c with SU(3), it must hold α +α +α =β +β +β =0. So the following properties [32]: 1 2 3 1 2 3 actually Eq. (26) is equivalent to α +α =−β −β . i j i j As an example, consider that the bisecting rotation is a3 =b2 =(ab)2 =c4 =1, R(13) in Eq. (22), then we have α2+β2 =0. In this case caca−1 =a−1c−1a=bcb−1 =b−1cb, T is cbc−1b−1 =bc−1b−1c. (34)   f +ig aη pη 1 5 T(13) = bη3 h aη2 , (27) In a 3-dimensional faithful representation, a, b and c can qη bη f −ig be represented by [32] 6 4     where η are some phases, i.e. |η | = 1. The 22-element 0 1 0 0 0 −1 i i isstillhthe(real)and11-elementistheconjugateofthe a(3) =0 0 1, b(3) = 0 −1 0 , 33-element, as a result of α +β = 0. We also have 1 0 0 −1 0 0 2 2 η η η η = 1 because α +α +β +β = 0. T(13) can   1 2 3 4 1 3 1 3 i 0 0 be diagonalized by a unitary matrix which we call UT13 c(3) =0 −i 0. (35) and one can check that 0 0 1  t + g 1−t t − g  2 2sϕ 2 2sϕ The mixing scheme is produced if a Z subgroup gener- |U |2 = 1−t h−cϕ 1−t , (28) 3 T13  2t − 2sgϕ 11−−cϕt 2t + 2sgϕ  abtyedS1by=Tb=,Sa22(=cb)a22cc2aancd2 aisZa2p⊗plZie2dstuobgcrhoaurpgegdenleepratotends and neutrinos respectively. To be precise, where c =cosϕ, s =sinϕ and ϕ ϕ   c =(−1+2f +h)/2, (29) 0 0 i ϕ 1−h Z3 :T =−1 0 0 (36) t=1− . (30) 0 i 0 2(1−c ) ϕ We can see that indeed |UT13|=|UT13|T. and the two Z2 are generated by     0 0 1 −1 0 0 IV. APPLICATION Z2⊗Z2 :S1 =−0 1 0, S2 = 0 1 0 . 1 0 0 0 0 −1 In this section we will apply our theorems to an ac- (37) tual mixing scheme. After the T2K neutrino experiment Fromourtheorem,itiseasytoseethatthiscanproduce measured a large non-zero θ in 2011 [29], many models symmetric mixing. In the diagonal neutrino basis we 13 have been proposed to explain the result. Refs. [30, 31] transform T to Td, scannedaseriesofdiscretegroups(∆(6n2)andΓ ),and one of the found schemes was quite close to the TN2K re-  i √i i  2 2 2 sult at the time. In the standard parametrization, the T =−√1 0 √1  (38) d  2 2  angles are −i √i −i 2 2 2 2 θ =θ =tan−1 √ =36.21◦, (31) 23 12 and (S ,S ) to (S ,S ): 3+1 1 2 1d 2d and S =diag(−1,−1,1); S =diag(−1,1,−1). (39) 1d 2d (cid:18) (cid:19) 1 1 θ13 =sin−1 2 − 2√3 =12.20◦. (32) Then, according to our theorem, we see that Td can be rephased via a transformation defined in Eq. In total the PMNS matrix is (25) to a bisecting rotation R with α2 = β2 = 0, √ √ (α ,α ) = (π/2,−π/2) and (β ,β ) = (0,0). Or, in a  16(cid:0)3+ 3(cid:1) √13 16(cid:0)3− 3(cid:1) sim1ple3r way, Td has the form of1Eq3. (27). So the mixing U = −√1 √1 √1 . (33) matrix should be symmetric.  3√ 3 3√  1(cid:0)3− 3(cid:1) −√1 1(cid:0)3+ 3(cid:1) 6 3 6 Adynamicalrealizationofthemixingschemein∆(96) Whilethismixingschemeisruledoutbycurrentdata,it has been studied in Ref. [33]. That model is rather com- fulfills our criterion of a symmetric mixing matrix, and plicatedusingboth3-and6-dimensionalrepresentations. 7   Herewepresentasimplermodelwhichonlyusestwoad- ω2 1 ω 1 ditional sets of scalar fields φν,φ(cid:96), and features all par- U(cid:96) = √ −ω −1 −ω2 . (54) 3 ticles in the same 3-dimensional representation of ∆(96) −i −i −i of (36) and (37). Thus, the PMNS matrix is (cid:96),(cid:96)c,ν,φν,φ(cid:96) ∼3. (40)  i√+ω −√ω2 −−√i+ω  6 3 6 We use the representation in (36) and (37) rather than U = 1√+i −√1 −1√−i . (55) (38) and (39) because the Clebsch–Gordan (CG) coeffi- PMNS  6 3 6  i+√ω2 −√ω −−i√+ω2 cients are simpler. The result does not depend on the 6 3 6 basis. The CG coefficients we will use in this representa- It is related to the matrix in Eq. (33) via tion are 3⊗3→1:δ (41) UPMNS =diag(eiβ1,eiβ2,eiβ3)Udiag(1,eiα1,eiα2), (56) ij 3⊗3⊗3→1:(cid:15)ijk (42) where β1 = 105◦,β2 = 225◦, β3 = 165◦, α1 = 45◦, 3⊗3⊗3⊗3→1:δijmn (43) α2 =90◦. Here α1, α2 would be the Majorana phases if the couplings yν and yν in Eq. (46) were real. Therefore 3⊗3⊗3⊗3→1:δ δ δ δ . (44) 1 2 im jn in jm even though the Dirac-type CP is conserved in this Here (cid:15) is the Levi-Civita tensor (or order 3 antisym- model, generally there is still CP violation due to the ijk metric tensor) and δ is defined as non-zero Majorana phases, unless the phases of yν and ijmn 1 yν are tuned to exactly cancel α , α . (cid:40) 2 1 2 1 (i=j =m=n), δ = (45) ijmn 0 otherwise. Our theorem can also be applied to the quark sector. Onejustassignsthebisectingrotationalsymmetrytothe The invariant Lagrangian in the lepton sector is residualsymmetryofup-typequarks(ordown-type)and the mirror symmetries to that of down-type quarks (or L=y(cid:96)(cid:15) φ(cid:96)(cid:96) (cid:96)c +y(cid:96)δ φ(cid:96)φ(cid:96)(cid:96) (cid:96)c +y(cid:96)δ δ φ(cid:96)φ(cid:96)(cid:96) (cid:96)c up-type),thentheCKMmixingwillbesymmetric. How- 1 ijk i j k 2 ijmn i j m n 3 im jn i j m n +yνδ φνφνν ν +yνδ δ φνφνν ν . (46) ever building a realistic model for the CKM mixing is a 1 ijmn i j m n 2 im jn i j m n somewhat more difficult task. Compared to the lepton Aftersymmetrybreaking,φν andφ(cid:96) obtainthefollowing sector where hundreds of flavor symmetry models have VEVs: been proposed, for the quark sector much fewer models exist. ThisisduetothefactthatthesmallCKMmixing (cid:10)φ(cid:96)(cid:11)=v(cid:96)(1,−1,i), (cid:104)φν(cid:105)=vν(1,0,1). (47) angles do not have straightforward geometric interpreta- tion,whichisthebasisofdiscreteflavorsymmetrybuild- In the charged lepton sector M =m m† is (cid:96) (cid:96) (cid:96) ing. AmongtheexistingmodelsfortheCKMmixing,we   cannot find one that fulfills our criteria (exceptions are u x+iy −ix−y of course the trivial cases in which one interprets the M(cid:96) =|v(cid:96)|2x−iy u ix−y , (48) CKM matrix as the unit matrix or as a matrix which ix−y −ix−y u onlyconsistsoftheCabibboangle),andscanningalldis- crete groups for the flavor symmetry of quarks is out of while the neutrino mass matrix is the main purpose of this paper. Anyway, when looking   A 0 B for flavor groups to build models for the quark sector, Mν =|vν|2 0 0 0 , (49) our theorem could be a guidance because when a mix- B 0 A ing scheme generated from a flavor symmetry is close to realistic CKM mixing, then it must be also close to a where A=y1ν +y2ν;B =y2ν with symmetric form. u=2|y(cid:96)|2+2|y(cid:96)|2+|y(cid:96)+y(cid:96)|2 (50) 1 3 2 3 V. CONCLUSION x=|y(cid:96)|2−3|y(cid:96)|2−2Re[y(cid:96)∗y(cid:96)], (51) 1 3 2 3 A possible zeroth-order, but surely aesthetically at- y =2Re[y(cid:96)∗y(cid:96)]. (52) tractive, mixing Ansatz for the CKM and PMNS matri- 1 2 ces is that they are symmetric. The origin of symmetric M and M can be diagonalized by the following unitary PMNS and CKM matrices from the viewpoint of flavor (cid:96) ν matrices symmetry models has been the focus of our paper. We have proposed a theorem on the relation between  √1 0 −√1  symmetric mixing matrices and geometric properties of 2 2 Uν = 0 1 0 , (53) discrete flavor symmetry groups. An illustrative connec- √1 0 √1 tion between the rotation axes of the geometric body 2 2 8 associatedtothesymmetrygroupexists, andshowsthat groups and possible features of the mixing matrices may popular subgroups of SO(3) such as A and S can lead have further applications. 4 4 to symmetric mixing matrices. Groups with complex irreducible representations do not easily allow for a geometrical interpretation, but a ACKNOWLEDGMENTS partial generalization of our theorem is possible, which can then apply to SU(3) subgroups such as ∆(96). A previously studied mixing scheme that turns out to cor- We thank Yusuke Shimizu for useful discussions. WR respond to a symmetric PMNS matrix was used as an is supported by the Max Planck Society in the project explicit example. MANITOP. 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