JHEP04(2008)077 2 √ ǫ/ + , 4 π April 7, 2008 ≈ April 18, 2008 March 27, 2008 epton mixing 23 rinos, Neutrino February 18, 2008 θ flavor symmetries Accepted: Revised: Published: Received: hanism. The lepton mass ton mixing. In our models, n future neutrino oscillation are rather general in the sense s and/or neutrinos. Moreover, oggatt-Nielsen mechanism and os can mix maximally. We also that is of the order of the Cabibbo ǫ ta ¨rbr,D-97074 W¨urzburg, it¨at [email protected] , 0 and find several thousand explicit such models that ≈ 13 θ ) for the reactor angle and a new sum rule Published by Institute of Physics Publishing for SISSA 3 ǫ ( O . 13 θ 2. We focus on solutions that can give close to tribimaximal l Discrete and Finite Symmetries, Solar and Atmospheric Neut . We present a systematic group space scan of discrete Abelian 0 [email protected] ≃ C θ ¨rbr,Germany W¨urzburg, E-mail: Institut Theoretische f¨ur Physik und Astrophysik Univers [email protected] Physics. Florian Plentinger, Gerhart Seidl and Walter Winter for the atmospheric angle,experiments. allowing the models to be tested i Keywords: for lepton mass modelssmall that neutrino produce masses nearly arematrices tribimaximal emerge generated lep from by higher-dimension the operatorsare via type-I predicted the as seesaw Fr powers mec of aangle single expansion parameter Abstract: with a very small reactor angle Group space scan oftribimaximal flavor lepton symmetries mixing for nearly that large leptonic mixings can comein from the the charged neutrino lepton sector,find both a left- new and relation right-handed neutrin provide an excellent fit to current neutrino data. The models JHEP04(2008)077 4 5 6 1 3 2) 12 13 [12] √ (1.1) / HPS U [11] can be and examples 4 = arctan(1 PMNS A 12 U θ e tribimaximal leptonic e ref. [10]) tells us that eV can be naturally con- 1 . = 0. The actually observed rinos signal physics beyond − ergy scale close to the GUT g matrix epton mixing [15]. s based on w mechanism [7, 8], in which     13 10 ximal mixing matrix 2 θ 1 [3], and accelerator [4] neutrino 2 are given by √ 1 0 eutrinos are massive. Since neu- . . . √ − 2 23 θ − 3 3 3 1 1 1 √ √ √ 10 6 6 ∼ 2 3 1 1 √ √ q − − vanishes, i.e.,     13 – 1 – = θ HPS , see refs. [16] and [17]). These models, however, have U 4 A symmetries ≈ may then be expressed in terms of deviations from tribi- n Z PMNS and the atmospheric angle PMNS U U 12 θ GeV [9]. 16 10 × 2 4, whereas the reactor angle ≈ π/ = , the solar angle GUT M 23 θ Many models have been proposed in the literature to reproduc HPS Current neutrino oscillation data (for a recent global fit se U using the double covering group of 4. Scanning approach 5. Results of the group6. space scan Summary and conclusions A. Mass and mixing parameters oscillation experiments, have very welltrinos established are that massless n inthe the SM. standard In model fact, (SM), the smallness massive of neut neutrino masses 1. Introduction During the past decade, solar [1], atmospheric [2], reactor 2. Lepton masses and mixings 3. Flavor structure from Contents 1. Introduction leptonic mixing angles in mixing using non-Abelian flavor symmetries (for early model maximal mixing [13, 14] as “nearly” or “near” tribimaximal l and In the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) lepton mixin nected with grand unified theoriesthe (GUTs) absolute [5, neutrino 6] mass via scale thescale becomes seesa suppressed by an en approximated by the Harrison-Perkins-Scott (HPS) tribima as JHEP04(2008)077 4, π/ trices. ≈ [19] (see in terms C θ t points of + CKM 12 PMNS V θ U 2. By expressing . 0 o “extended QLC” the notation for the ≃ ǫ flavor symmetries using veral thousand explicit 0. gatt-Nielsen mechanism. generally have the merit n or the atmospheric angle, can of flavor symmetries. eal lepton mass matrices, ing matrix 0. For these textures, the ing renormalization group discrete flavor symmetries ≈ Z necessary flavor symmetry utrinos that lead to nearly enomenological observation hand, is implied by the idea mmary and conclusions. ) in total 1981 qualitatively d leptons can assume any of ≈ 13 archical pattern of the lepton te saw mechanism (for a related te small neutrino masses, we relation for the reactor angle, r symmetries that yield nearly θ ing and the mass hierarchies of ete Abelian flavor symmetries. terizations of es the relation h sum rules [24], with stress on es are in the flavor basis all ex- d models). An interesting con- 13 t are produced by the Froggatt- es in terms of flavor symmetries, θ dict the observed fermion mass hier- – 2 – , we have derived in ref. [31] for the CP-conserving ǫ is of the order of the Cabibbo angle ǫ , which serves as a single small expansion parameter of the ma ǫ 0. Here, , 2 2 is the Cabibbo angle. QLC has been studied from many differen . ,ǫ,ǫ 0 4 π ≃ C θ [26], with respect to statistical arguments [27], by includ In refs. [30, 31], we have suggested a generalization of QLC t In this paper, we describe the systematic construction of se The paper is organized as follows: in section 2, we introduce C θ also the lepton mass ratios as powers of ref. [20] for modelsnection including between quarks the and quark ref. andof [21] the quark-lepton lepton for sector, unifie complementarity on (QLC) thethat [22], other the which measured solar is mixing the angle ph very accurately satisfi neutrino masses become small dueapproach to see the ref. canonical [33]). type-I The see pressed matrix by elements powers of of these textur case (a discussion of nonzero phases can be found in ref. [32] where distinct mass matrix texturestribimaximal for neutrino mixing the with charged a leptons small and reactor ne angle archies and the Cabibbo-Kobayashi-Maskawa (CKM) quark mix generally difficulties (for a discussion see ref. [18]) to pre view: as a correction tophenomenological bimaximal aspects mixing [23], [25], together in wit conjunction with parame of effects [28], and in model building realizations [29]. (EQLC), where the mixing anglesthe of values the left- and right-hande This suggests a model buildinge.g. interpretation of via the the textur Froggatt-Nielsen mechanism [34]. lepton mass models, in which nearlycharged tribimaximal leptons lepton and mix neutrinos emergeOur from products motivation of is discr thatthat models they with need only Abelianbreaking. a flavor We very symmetries perform simple a group scalar space sector scan of to products achieve of discre the the results of EQLCi.e., from we ref. consider [31] theassume and CP-conserving only restrict case. the canonical to type-I In the seesawmass order mechanism. matrices case to results The of genera hier from r higher-dimensionNielsen operators mechanism. tha Moreover, we are interestedtribimaximal only lepton in mixing flavo with a very small reactor angle lepton masses and mixings.and Next, describe in the sectionThen, generation 3, we we of outline specify lepton in our Our section mass general 4 terms results, our a via approach listand to the of our the sum Frog explicit rules group example for space models, theare s PMNS a shown angles new in including a section new 5. sum rule Finally, f we present in section 6 our su JHEP04(2008)077 , = , ton diag D T ν (2.1) (2.4) (2.2) (2.3) (2.6) (2.5) GeV, diag eff U 2 , M M 10 diag eff diag eigenvalues ℓ GeV. After M , i∼ ν M 14 ) and U    H , tion (we follow τ h x 10 plus three right- , b = δ i 13 13 ,m Y ∼ − c c µ eff are diagonalized by e L 23 23 + h.c. (1) 13 c s − c eff j ,m + h.c. s U e B ν GeV the Dirac neutrino c j c i M 3 is the generation index. x , M 2 ν m b , × δ x ν M i c i T b δ R 2 i e ij 10 L ν , e ) U and 13 ij R . [30]) ∼ 13 ) e type-I seesaw mechanism [7]. s s R = 1 R diag M nos, the seesaw mechanism leads R 23 D i ( 23 = diag( c SU(2) Y M M s 13 2 1 M , i ( 12 c as a product of the form R × 12 2 T 1 s D H − 12 diag s U ℓ x c h , s c j − M − U x − ν M 1 = = c i j 23 breaking scale − K R ν R 23 s i D x c ν L Hℓ M b 12 U 2 M 12 ij c x D − ) σ are the Dirac Yukawa coupling matrices of the c = SU(3) i − D D M , M – 3 – ij D B ′ , are the first, second, and third neutrino mass x ) − Y = † 3 b δ is the Majorana mass matrix of the right-handed x SM M D i b δ D ( i e x U G m = R Y e ( U − 13 and 13 M s develops a vacuum expectation value eff c j diag , are unitary mixing matrices, whereas D s − ℓ e ν 23 i and c j M Y 23 s H e M U e , c 13 i ij 2 D 12 c ℓ ) 12 c eV in agreement with observation. The leptonic Dirac mass ∗ U ℓ , are the left-handed leptons, the right-handed charged lep c 12 c i 2 ,m c − H M ν = 1 − − ( ij , and 23 ) m D − 23 R ℓ c s Y 12 = ( , U , and 12 s ′ , and the Majorana mass matrices c i s − e D − , M D is the charged lepton and ′ , † ℓ = mass ℓ T , are diagonal mass matrices with positive entries. The mass , U    M U ) Y L Y ), where i D i 3 = L diag eff , e H diag x , U i ℓ and h ′ ,m b ν ℓ U M ℓ 2 M is a CKM-like matrix that reads in the standard parameteriza = ℓ , U M ℓ = ( is the SM Higgs doublet, U ℓ x ,m i and b 1 U U M ℓ = H , m ℓ diag M R where 2. Lepton masses and mixings We assume the SM with gauge group where doublets, and the right-handed SM singlet neutrinos, and handed neutrinos that generate smallThe neutrino masses lepton via Yukawa couplings th and mass terms are charged leptons and neutrinos, and Here, neutrinos with entries of the order of the where M of the charged leptons and neutrinos are given by electroweak symmetry breaking, with entries of the ordermatrices 10 eigenvalues. We can always write a mixing matrix diag( mass matrix. After integrating outto the right-handed the neutri effective Majorana neutrino mass matrix where here throughout the conventions and definitions given in ref and the mass terms of the leptons become JHEP04(2008)077 3 1) , , 2 but and 2 , (2.7) (3.1) (3.2) (3.3) φ k i ij , non- n θ v , , e . Under = 1 Z i 1 , and the r F i runs over φ → i 23 G e i ≃ x θ x ij k ˆ )= θ n in eq. (2.7) is sume for each ]. The PMNS i m f π h 1 under c j 2 ν , i singlets. − = diag( k PMNS and only determined ,...,r n Hℓ b i U SM 2 k 2 Maj ) may be different. We n G σ ∼ , r i K is a flavor symmetry. We i 1 1 ij r ) ( F − are e form ′ D mod G ting out heavy fermions with ∼ k Y . , k choose a convention, where for n ,...,m c , i } y )( f 2 1 m lie all in the first quadrant, i.e., 1), where the index , k ij Maj + n , b − } ǫ k x 2 Z charge of the particle and K , ν α x i , the atmospheric angle k denote diagonal phase matrices that x i 23 = 1 j , where =1 + h.c. (3.4) q ˆ θ n 13 m k x F , e , c k j 2 is of the order of the Cabibbo angle, θ symmetries, i.e., Z ( x 1 . ν G K are all in the range [0 x 13 PMNS (Π α c 0 i n )= i ˆ k θ b ν U × x e , Z ,...,n i ×···× n m − ≃ ij 2 K 2 x c j ) 12 and = , SM ˆ n ′ C e θ R 1 i ν x θ Z , G ℓ Y – 4 – . The CKM-like matrix U . Moreover, the ( 0 ∗ D that carries a charge † and 2 ,...,q ℓ ∈ { ≃ k × L i 2 = diag( φ H k U 1 x − F ∈ { x n ij 6= ij n ,q x B ˆ θ f ) D i = 1 i k ′ j Z ℓ q K M symmetries and ( Y , r ) v/M = i k k ij )( 1 n ∼ c F φ , the reactor angle ,q = k ij with ǫ i , and PMNS Z factors and the a i k the group order (i.e., the number of elements) of G ǫ 12 j ǫ x ij U p ) and =1 n θ ˆ n k θ x 3 m k charges is equivalent with =1 Z n Z ϕ , ℓ ]. In eq. (2.5), i m k i k (Π π p n =1 , e 2 2 1 m k (Π Z x 2 , = cos ϕ − − , where i [0 )= F . The phases in ij = Π , e i m = acquire nonzero universal vacuum expectation values ∈ c x 1 | M , which we identify in the parameterization of eq. (2.6) as k ϕ x , Y F is a direct product of discrete i of two δ b n δ the charges are non-negative and lie in the range x ij e L ≃ G f k ˆ ,R,ν θ F | ′ k y ,...,p n G i 2 + Z th entry in each row vector denotes the a single scalar flavon field . In order to spontaneously break the flavor symmetries, we as k j ,p ], and k is the number of x k i 1 is a CKM-like matrix parameterized as in eq. (2.6), and = sin 2 π ,D,D n n = diag( p ′ , ( b . m U Z ij x δ [0 s ℓ,ℓ ∼ When the D , we assign to the leptons the charges ∈ c i → = e F ij x ˆ b are θ x with where universal mass are generated by the Froggatt-Nielsen mechanism by integra matrix reads contains the Majorana phases δ described by the solar angle 3. Flavor structure from Dirac CP-phase assume that where Let us next extend the SM gauge group to will denote by G where the is the generation index (seeeach section group 2). In the following, we The sum renormalizable lepton Yukawa couplings and mass terms of th is a singlet under all other modulo factor JHEP04(2008)077 , , : k o c i n D µ ν Z m (3.6) (3.5) for a , M : ℓ ǫ and e M , m i symmetries, , i.e., where ,ℓ fit to nearly c i : 1 in perfect ½ e , to reproduce n ǫ . ′ 3 matrices with R Z : = k ij Y c × ′ 2 ǫ ℓ ǫ U =1 = and m k er products of , , 3 , , operators can be built Π ere ′ } } } D m k k k ≈ : n n n ,Y e three textures defined in ′ n eq. (3.6) with the list of e information on the order pton textures for dels introduced in section 3 ted discussion of anomalies ij ℓ 2 he cancellation of anomalies ) on mass spectrum Y he Yukawa coupling matrices m R we pick some flavor symmetry , as itable extra matter fields which mod : mod mod ij M k 1 j k k j j ) ( it models using flavor symmetries. q r r R m [31], we have, based on assumptions − − − , M . The important point is that the k i k k k ij i i b k p q r ǫ ∗ n confidence level (CL)). − − − f , , , =1 and ( σ k k k m k , n n n Π ij according to eqs. (3.2) and (3.3). Then, we ) ≈ D – 5 – F mod mod mod ij G M ) ( k k k j j j , q D r r ij M + ) + + flavor symmetries are global, but it might be important ( ℓ k i k k i i n p r q M Z . The lepton textures are therefore the 3 { { { , ′ are zero. Although these rotations do not show up in the R k ij a Y c i ǫ e =1 m k = min = min = min and , are dimensionless order unity Yukawa couplings. The modul , Π ′ R k ij k ij k ij ′ D c b Y a ≈ ,Y ′ ij ℓ ) Y in eq. (3.5) is a consequence of the cyclic nature of the ℓ and , k M and the complex conjugated fields ′ ( n D k : 1 and a normal neutrino mass hierarchy n ,Y as in eq. (3.1) and assign to the three generations of leptons 2 ′ f ǫ ℓ : Y F , following eq. (3.6). Next, to find models that can give a good 4 R G ǫ M Note that in our models, the In ref. [31], the textures have been extracted in the basis wh We define a “texture” as the matrix collecting the leading ord = τ matrix elements approximating ( In what follows, weeq. will, (3.6) for a a “texture certain set”. model call the set of th for our symmetries could, e.g.is, be achieved however, by beyond considering su theand scope other of phenomenology, see this ref. paper [36]). (for a recent4. rela Scanning approach In this section, we willthose describe which how give we nearly identify tribimaximalgroup among lepton the mixing. mo First, to gauge them to survive quantum gravity corrections [35]. T determine from the charge assignments the corresponding le all possible charge combinations under certain Yukawa coupling orunity mass coefficients matrix, thereby ignoring th and agreement with current neutrino data (at the 3 tribimaximal lepton mixing, we1981 representative compare texture sets the given in textures ref.of [31]. found EQLC, In i determined ref. fits of the order one Yukawa couplings and the minimum takesfrom both into account that the higher dimension nearly tribimaximal lepton mixing along with a charged lept m charges of the leptonsand determine the a right-handed Majorana hierarchical neutrino pattern mass of matrix. t observables, they are important for formulating our explic and function mod whereas the rotations acting on the JHEP04(2008)077 equal (4.1) 4 (to = 3), n ies the ǫ ≥ m CL [31]) , thereby n ½ σ 6= . 5 ′ with ℓ Z U n × ǫ 4 4. A complete scan Z , = 2), 30 (for × 3 ) we set an entry ǫ , 2 extures in our reference 2 m Z , rge assignments onto the = r symmetries that produce rge assignment that yields les and phases entering the e models introduced in sec- oying the texture reduction F = 1 ample, we will demand that and that the charged lepton ill use this reduced reference ion of the models for the dif- he charged leptons that have G ove reference list to a reduced ce list” of 43278 qualitatively m he charged lepton textures. In ast one entry odel that allows an excellent fit , as predicted by the flavor symmetry. , , whereas we can always choose In total, we find in the scan 6021 n ce to match onto flavor symmetry ǫ }  for ls for 4 π ,π k = 1), 45 (for n 0 le overall power of Z , ǫ, factor m 2 ∈ { =1 ,ǫ m k ′ ℓ 2 0 All these models allow for an excellent fit to  1. Valid models are selected as described in , ϕ 5 (for charged leptons). This is different from ref. [31], CL and most of them actually at 1 ′ = Π 3 ℓ 1 ∈ – 6 – ≥ σ F ′ n , ϕ ℓ 23 G ′ ℓ m > ,θ 3 for both neutrinos and charged leptons. For specific flavor 2 b δ 1 ′ ℓ 13 ≥ n ,θ ′ can be nontrivial, i.e., where we can have 9 for ℓ 12 ′ θ ℓ such that the left- and right-handed lepton sectors are now ≤ U c i k e n is summarized in figure 1. = 0 (for a definition of the notation, see section 2). This appl ′ F = 4), with ℓ 2 3 (for neutrinos only) or when G α ≥ all possible combinations that satisfy m ′ n = ℓ U ′ ℓ 1 α We can now impose extra assumptions on the properties of the t = Note that in our reference list (after factoring out a possib For an application of relatedAdditionally, we flavor groups have also see included ref. the [37]. number of valid mode 1 2 3 ′ ℓ 3 while the phases can take the values been extracted in bases where We therefore include now in our considerations textures of t leading to a multitudecomplete of analogy new with explicit ref. representationsmatrices [31], for we t assume for the mixing ang all treated on thefrom same refs. footing. [30, As 31], adifferent we lepton consequence, texture finally sets. after arrive empl We use atmodels this an list for as enlarged our nearly referen “referen tribimaximaltextures lepton contained mixing. in this Any referenceto flavor list nearly cha provides tribimaximal a lepton valid mixing. m list to search fornone interesting of flavor the symmetry textures models.textures is contain For completely (after ex factoring anarchic out (or common democratic) factors) at le section 4 by matchingreduced the reference list textures of generated 17772such by non-anarchic models texture the sets. that flavor reproduce cha ferent 2093 groups texture sets. The distribut ϕ hypothesis of EQLC to the nearly tribimaximal neutrino mixing (at 3 and 24 (for have sufficient structure in thelist texture). of This 17772 reduces distinct thelist ab texture to sets. perform the In group the space next scan section, of we flavor w symmetries. 5. Results of the group spaceLet scan us now present thenearly results tribimaximal of lepton the mixing. grouption space We 3 assume scan and throughout for choose flavo th as flavor symmetries to zero when where such an entry is set to zero when models, however, we will always show the actual suppression has been performed for groups up to order 40 (for JHEP04(2008)077 re as 24 | ugh ) we (5.1) F n | ≤ ǫ G F of solar | different G 2 ǫ 9 | ≤ | ∼ F to the entries G 10 ℓ | 2 atm M m ∆ , / : 1 2 ⊙ 4 dels generally increases n mixing as a function of ǫ m ading to : , and in 2 } ǫ 1 io ∆ = , ǫ, e neutrinos have a normal mass 3 2 ble common overall factor m ,ǫ or a discussion and references see 0 : { 2 m : 1 (cf. eq. (3.2)). Note that we use F , m G – 7 – different possibilities for arbitrary texture sets. : 1 and the lepton mass ratios 9 2 ◦ ǫ 6 1 : · 4 . , the more possibilities for the charge assignments we ǫ | to the matrix entries 6+9 13 F = θ G R τ | M m : up to some periodic modulation. It is interesting to give a ro has to be in order to more or less reproduce an arbitrary textu µ | and F F m G : D G | e M for increasing group order. In the left (right) panel we have m F , i.e., we have 4 G } 1 45). Number of flavor models leading to nearly tribimaximal lepto , ǫ, 2 |≤ ,ǫ F 3 G ,ǫ In figure 1, we can observe the trend that the number of valid mo 4 We are interested here in an SU(5) compatible fit. ≤ | 4 ,ǫ 0 Figure 1: the flavor group with a very small reactor angle (24 a figure of merit: The larger over atmospheric neutrino mass squared difference.hierarchy, renormalization Since th group effects areref. negligible [31]). (f with the group order possibilities for the charge assignments in i.e., we have have a normal neutrino mass hierarchy with a rat { restrict ourselves in estimate on how large On the other hand, we have nine different charges per group, le set. For this purpose, note that (after factoring out a possi JHEP04(2008)077 5 Z with × from (5.5) (5.4) (5.2) (5.3) 4 9 ◦ . , Z Z 2 2 ǫ ǫ × × 3   5 Z 4 4 5 5 Z × − − and higher. In 2 = 2 2 7 Z 24, while we have 1 1 F Z √ √ ense that they can amples. In table 1, G g angles in the range   |≤ F + + um rules for the leptonic G l (i.e., more than 99%) of 2 2 n. summarized in table 2 in , th complete flavor charge ǫ ǫ trino sector, and that the ◦ , such as √ √ of our models, these matrix ation, but also between the eviation of about +7 ure set previously identified . The rough guideline for k ≤ | nts. , such as 52 ǫ n R 0 + + k ssibilities for the texture sets, ddition, we can have maximal n ≈ M 4 4 π π , (see eq. (3.4)) lie in the interval 23 = = ′ R . The complete information on the . and ′ Y 23 23 R , 60 , θ Y ◦ D CL). The models are thus characterized 1 ≃ and σ 6 , . and , M ′ · , θ , θ ℓ D , 2 2 13 2 ′ ǫ 9 ǫ M D 15 ,Y ′  4 ℓ – 8 – ,Y = 1 2 Y ′ ℓ , θ & Y ◦ 13 | − factors with moderate . For example, we find for model #6 the sum rules F 39 2 ǫ n 1 G texture set, we roughly estimate that the number of √ | Z . , a charged lepton mass spectrum as in eq. (5.1), and  , θ factors with large enough 12 R 2 θ = ǫ any n M  . Z 13 4 9 ◦ 45. In fact, figure 1 seems to suggest that − 34 2 | ≤ 1 , θ √ F 2 4 ǫ G  . With respect to the expansion parameter − + ≤ | /ǫ 2 2 ǫ ǫ √ √ − − and 1 ǫ 4 4 π π = = = 45 is already entering the regime estimated in eq. (5.2). We have checked that for the 6021 valid models practically al Out of our set of 6021 valid models, let us now consider a few ex Table 2 demonstrates that our models are very general in the s All 22 models in tables 1 and 2 lead to the following PMNS mixin Some of the models in table 1 allow to extract very easily new s | 12 12 F θ θ G i.e., possibilities for the charges should exceed the number of po have. In order to reproduce while model #8 exhibits the relations This means that, for instance, four should be sufficient, or two the Yukawa coupling matrix elements of | elements can therefore indeed be viewed as order onewe coefficie show 22 explicitassignment valid and models by the listing resulting the textures flavor for group wi figure 1, we have much less possibilities on the l.h.s. where 1 on the r.h.s. 24 “natural” order one Yukawa couplings choosing these 22 models wasin to give EQLC one [31], example for with each text distinct between by a very small reactor angle close to zero, and a significant d in agreement with neutrino oscillation data (at 3 corresponding mass and mixingthe parameters appendix. of the 22 models is exhibit maximal mixings inleft- the and/or charged right-handed neutrinos lepton can and/ormixings mix in the maximally. all neu In sectors, a not1st only and between the the 2nd 2nd as and well 3rd as gener between the 1st and the 3rd generatio maximal atmospheric mixing. mixing angles using an expansion in JHEP04(2008)077 5 Z 9 9 9 7 7 7 7 8 8 9 × Z Z Z Z Z Z Z Z Z Z 3 F × × × × × × × × × × Z G 5 5 5 6 6 5 5 5 5 5 × Z Z Z Z Z Z Z Z Z Z 2 Z 3) 2) 0) , , , 1 2 1 , , , 0) 0) 0) 6) 3) 5) 0) 0) 0) 0) 8) 5) 0) 5) 6) 2) 2) 2) 2) 6) 1) 4) 1) 2) 1) 5) 5) 7) 5) 3) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (1 (1 (0 , , , (1 (2 (1 (0 (0 (0 (1 (2 (3 (1 (0 (3 (1 (5 (0 (3 (4 (3 (3 (4 (1 (1 (4 (1 (0 (2 (2 (2 (2 (0 3 3 3 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2) 2) 2) ,p , r ,q , , , 8) 1) 1) 1) 0) 6) 4) 4) 4) 4) 0) 4) 1) 4) 5) 1) 2) 6) 6) 7) 1) 8) 6) 5) 3) 0) 2) 0) 1) 3) 2 2 2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 0 0 , , , , r ,q ,p (0 (0 (0 (3 (2 (2 (3 (2 (3 (2 (1 (4 (0 (4 (5 (4 (2 (3 (3 (3 (2 (1 (1 (2 (2 (0 (3 (3 (0 (4 1 1 1 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (1 (0 (1 r q p , , , 8) 0) 6) 0) 3) 4) 6) 4) 0) 8) 6) 4) 6) 3) 0) 3) 1) 5) 2) 4) 6) 8) 1) 2) 1) 0) 2) 7) 0) 8) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0) 1) 1) , , , (1 (2 (3 (2 (4 (1 (2 (1 (4 (0 (4 (1 (1 (2 (1 (2 (0 (1 (4 (0 (3 (2 (2 (0 (2 (2 (2 (3 (3 (3 0 1 0 , , , (0 (1 (1       L               2 4 2   4 2 2 2 3 4 2 5 4 1 1 1 1 − ǫ ǫ ǫ ǫ 1 1 1 1 1 1 1 ǫ ǫ 1 ǫ ǫ ǫ ǫ ǫ B 2 4 2 3 2 ǫ ǫ 1 1 1 1 4 2 3 4 2 2 2 5 4 1 1 1 1 1 1 – 9 – ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ 1 1 1 1 ǫ ǫ ǫ ǫ ǫ ǫ 2 ǫ ǫ ǫ ǫ 1 1 1 1 /M 2 3 2 2 2 ǫ ǫ ǫ ǫǫǫ ǫ 2 2 2 2 5 2 5 1 1 1 1 ǫ ǫ ǫ ǫ ǫ ǫ ǫǫǫ ǫ ǫǫǫ ǫ ǫ ǫ ǫ 1 1 R ǫ ǫ ǫ ǫ ǫ ǫ ǫ       2 2       2           ǫ 2 3 2 M ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ               i     2 3 3 2 ǫ   2 2   ǫ ǫ ǫ ǫ ǫ 3 3 1 1 1 ǫ ǫ ǫ ǫ ǫ ǫ ǫ H 3 1 1 1 1 h 2 ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ 1 ǫ 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 1 1 1 1 / 1 2 2 2 2 ǫǫǫ ǫ ǫ 2 2 2 2 3 2 3 ǫǫǫ ǫ ǫ ǫǫ ǫ 1 ǫǫǫ ǫ ǫ ǫ ǫ ǫ ǫǫ 1 1 1 1 D ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ       2     3         M 2     2 4 ǫ 2 3 2 2 ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ                       i 2 2 2 5 2 2 2 2 2 2 ǫ ǫ ǫ ǫ 1 1 1 1 1 1 1 1 1 2 2 2 5 2 2 1 ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ 1 1 1 ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ H 4 4 2 2 2 2 4 4 3 2 2 4 3 4 4 2 2 2 2 h 4 2 5 ǫ ǫ 4 2 4 2 2 2 3 6 3 ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ / ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ℓ 3 3 3 3 3 4 3 3 4 4 4 4 2 2 5 5 3 2 3 3 2 2 4 ǫ ǫ 2 4 3 2 2 4 3 4 ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ M                     ǫ ǫ ǫ ǫ ǫ ǫ ǫ   8 9 7 5 6 4 2 3 1 11 10 # JHEP04(2008)077 5 5 Z Z 7 6 6 6 9 9 7 9 7 × × Z Z Z Z Z Z Z Z Z 4 4 × × × × × × × × × Z Z 6 5 5 5 3 4 5 5 5 × × Z Z Z Z Z Z Z Z Z 2 2 Z Z 2) 4) 3) 1) 0) 0) , , , , , , 0 3 0 1 1 0 n mixing. Shown are , , , , , , 3) 0) 3) 0) 0) 5) 3) 4) 1) 6) 2) 3) 4) 8) 3) 2) 8) 0) 1) 4) 3) 2) 1) 6) 5) 0) 0) , , , , , , , , , , , , , , , , , , , , , , , , , , , (0 (0 (0 (0 (0 (0 , , , , , , (0 (0 (2 (1 (1 (1 (0 (2 (0 (0 (1 (4 (4 (0 (3 (0 (3 (0 (0 (4 (1 (1 (0 (0 (4 (1 (4 , , , , , , , , , , , , , , , , , , , , , , , , , , , 4) 0) 2) 1) 1) 2) and the resulting textures. , , , , , , 1) 3) 4) 3) 5) 8) 3) 0) 1) 5) 4) 2) 2) 1) 3) 2) 8) 0) 2) 3) 5) 4) 0) 3) 2) 5) 3) , , , , , , , , , , , , , , , , , , , , , , , , , , , 1 1 0 3 0 0 F , , , , , , (3 (0 (0 (1 (2 (3 (2 (4 (2 (1 (1 (3 (4 (1 (1 (2 (0 (2 (1 (0 (3 (4 (0 (3 (0 (0 (0 G ored out of the textures. , , , , , , , , , , , , , , , , , , , , , , , , , , , (1 (1 (1 (0 (1 (0 , , , , , , 3) 3) 1) 5) 0) 5) 3) 8) 2) 0) 1) 4) 5) 6) 1) 1) 7) 1) 4) 1) 0) 4) 6) 3) 2) 6) 3) , , , , , , , , , , , , , , , , , , , , , , , , , , , 4) 1) 3) 2) 4) 1) , , , , , , (2 (2 (2 (2 (1 (1 (1 (0 (2 (1 (2 (0 (3 (0 (3 (3 (2 (3 (4 (0 (2 (3 (2 (0 (4 (1 (4 2 1 1 2 2 0 , , , , , , (1 (1 (0 (0 (0 (1               3 3 3 2 3       ǫ   5 2 3 2 2 2 1 1 1 ǫ ǫ ǫ ǫ ǫ ǫ 1 1 1 1 1 1 1 2 3 2 5 1 1 ǫ ǫ 1 1 ǫ ǫ ǫ 3 2 2 2 3 ǫ ǫ 1 ǫ ǫ ǫ ǫ 5 2 3 3 2 2 3 2 1 ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ 2 3 3 2 ǫ ǫ ǫ ǫ ǫ ǫ ǫ – 10 – ǫ ǫ 1 ǫ ǫ ǫ ǫ ǫǫǫ 1 1 1 1 3 2 2 3 2 ǫ ǫ ǫ ǫ 2 2 3 2 3 2 2 2 2 1 1 1 1 ǫ ǫ ǫ ǫ ǫ ǫ ǫ 1 1 2 5 3 2 2   ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ   ǫ ǫ ǫ ǫ ǫ ǫ ǫ 3 2       ǫ         ǫ 2 2 2     ǫ ǫ ǫ ǫ ǫ ǫ ǫ                 2 3 2       3 2 2 2 2 2 2 3 2 1 1 1 ǫ ǫ ǫ ǫ ǫ ǫ ǫ 2 1 1 1 1 1 1 ǫ ǫ 1 1 2 2 3 2 ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ 2 3 1 1 1 1 1 1 ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ 2 2 2 2 4 3 3 2 2 2 ǫ ǫ ǫ ǫ 2 2 2 3 2 2 2 ǫ ǫǫ 1 1 ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ 3 2 2 2 2 ǫ ǫ ǫ ǫ 1 1 ǫ ǫ ǫ ǫ ǫ ǫ ǫ 1 ǫ ǫ ǫ ǫ ǫ           2         2 2 2 2     ǫ ǫ ǫ 2 2 ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ         2               2 2 2 1 1 ǫ 1 1 1 1 1 1 2 2 3 2 2 2 2 2 2 2 2 2 2 ǫ ǫ ǫ ǫ 1 1 1 1 1 1 1 ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ 4 2 3 2 2 4 4 4 2 4 4 5 ǫ ǫ ǫ 4 4 2 2 4 2 2 2 2 2 2 5 5 3 4 4 5 4 4 2 ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ A list of 22 valid flavor models for nearly tribimaximal lepto 3 3 4 3 3 3 3 3 3 4 4 4 ǫ ǫ ǫ 4 4 5 5 4 5 2 2 2 3 2 3 4 4 4 4 4 4 4 5 3 ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ         2               ǫ ǫ ǫ ǫ 21 22 20 18 19 17 15 16 14 12 13 Table 1: the explicit flavor charges under the flavor symmetry group Possible common overall suppression factors have been fact JHEP04(2008)077 n Ε ions 0.4 (5.6) is very models 13 θ 0.3 natural, it 13 . θ . Nevertheless, 2 ◦ ǫ 0.2 1 3 4 exhibit variations . 0 s − 23 θ ) under 1% variations ≪ 0.1 2 l-known QLC relation ǫ 3 A experiments [39]. In gs. This means that by ǫ √ t cancellations between ν ( and xing angles (in degrees), for + ian discrete symmetries as 23 Θ e atmospheric angle follows 12 relations for 4 54 52 50 48 46 π ≃ O 2, which predicts a deviation θ mal mixing (the octant) with remains 2. This prediction makes these √ = 13 Ε / θ √ 13 C . The points correspond to a random 23 / θ 0.4 s. For example, the deviation from ǫ , by about 1%. The plot in the middle θ C ′ . Note that a similar suppression of R θ + Y 2 C θ 4 π 0.3 or at 90% CL otherwise [40]. , θ ∼ and ≈ ) 5 , 3 . ′ ǫ 2 D 23 ( − θ . Unlike for models #6 and #8, the relation 0.2 , Y O picks up only a small relative correction that is ′ ℓ 3 C 10 CL by the T2K or NO Y θ – 11 – = 13 & θ σ ∼ 13 0.1 13 0, the variation of θ 2 → 2 ǫ 13 Θ , θ ) in eq. (5.6) is not due to an accidental cancellation betwee 1.0 0.5 0.0 1.5 3 2 are to leading order just as in the two previous examples. 4 ǫ ǫ ( Ε 23 − . The stability of the relation θ 3 C 0.4 2 ≃ O in eqs. (5.4) and (5.5), however, are not stable under variat θ ǫ CL for sin √ 13 13 ∼ and σ θ θ in the limit − 0.3 ◦ 12 4 π , however, is different from that in eqs. (5.4) and (5.5): Now, θ 1) 13 = 2. The reactor angle, on the other hand, becomes for these two θ − 5 √ 0.2 12 . / θ C Solar (left), reactor (middle), and atmospheric (right) mi θ 0.1 − in eq. (5.6) is stable under variations of the Yukawa couplin 4 π 13 The relations for Let us therefore have a look at model #5, which has the sum rule ≈ 12 θ Θ 30 45 40 35 12 varying the order one Yukawa couplings, variation of the order one Yukawa couplings in for model #5 in table 1 as a function of the expansion parameter small due to a suppression by a factor Figure 2: shows that the relation The relation for the reactor angle has beenin found both for models the #6 model and #8 in the ref. new [38]. sum Th rule The sum rules for addition, one can measure thea sign of neutrino the factory deviation at from 3 maxi from maximal mixing by anmodels amount testable of in approximately future neutrino oscillationmaximal experiment mixing can be established at 3 small due to an apparent suppression by a factor suppressed by a factor large mixing angles. For both the modelsθ #6 and #8, the solar angle satisfies the wel of the order one Yukawa couplings is shown in figure 2. While of the order (0 of the Yukawa couplingslarge and contributions must from therefore different be sectors. a result To of render exac these might, thus, be necessarydemonstrated to in extend ref. these [41]. models by non-Abel JHEP04(2008)077 , ℓ ), 6 M Z × up to 3 1 testable n Z Z 24 models = (1) models ) by letting = 2 U F 4. n F G Z π/ (1) symmetries. G atisfy a new sum × U nnection between 1 of the flavor groups n dicts a new relation excellent fit to nearly senger fermions. Z A experiments or at a 0. Moreover, they lead ν hough the groups = ≈ table 1 (with varying nd the phenomenological eptonic mixings can come n F ed Majorana mass matrix. ations in any lepton sector. roups is a relevant feature texture sets. chanism. Upon flavor sym- atic scan of the group space 13 Z ector, maximal mixings can arameterized by powers of a G ierarchical Yukawa coupling most general CP-conserving θ nd NO o normal hierarchical neutrino e t flavor models for nearly tribi- e expansions is = 18 with corresponding Abelian stic flavor models. | F 2 that is of the order of the Cabibbo (1) (for . G (1) charges vary in the whole range 0 | U U ≃ × C θ up to (1) get in these examples highly suppressed by flavor symmetries. In our models, small neu- 2 ≃ U n R n ǫ Z – 12 – Z M × 1 (1) models correspond to the group n ) and U k Z and n Z = D F = M G factors and a maximum group order of 45. As a consequence, F are in all three above examples in eqs. (5.4), (5.5), and (5.6 ), groups, we have also performed a scan of G n D n 23 Z θ Z M and 2 for the atmospheric mixing angle, which makes these models (1) (for 12 √ U θ / C θ 1. Thereby, we have found 129 models for the cyclic groups and + . This analysis already indicates that the cyclic character − ), for the reactor angle. Moreover, we found models that all s 8 4 π 3 ǫ C k and arises from integrating out heavy Froggatt-Nielsen mes θ ≈ ( (1) symmetries. All of the 24 . C O θ U 23 = 40 and product groups θ We wish to point out that in this paper we have established a co In analogy with the All flavor models realize the assumptions of EQLC and yield an A characteristic property of our flavor models is that large l Among several explicit models, we have found a model that pre | = ,...,n F 1 , 13 G in this sense stable, since the leading term in the respectiv a model building top-down approach using flavor symmetries a neutrino factory. the sum rules for In order to seein whether the construction the of cyclic valid character models, of we the have compared discrete single g in future neutrino oscillation experiments such as the T2K a | 0 groups, i.e., with (the absolute value of) the respective individual for which we have obtained in total 21 models. However, even t using rule factors may be essential for facilitating the construction of reali 6. Summary and conclusions In this paper, we havemaximal constructed lepton several mixing thousand from explici productstrino of masses emerge onlymetry from the breaking, canonical the type-I Froggatt-Nielsenand seesaw mechanism mass me produces matrix textures h ofsingle the small leptons. symmetry breaking These parameter textures are p produce textures that are very similar to those of model #6 in angle and varying first row in tribimaximal neutrino mixing with a very small reactor angl from the charged leptonsarise and/or in neutrinos. both In the theGenerally, Dirac we neutrino mass can matrix s have maximal or mixings in between the any two heavy gener right-hand θ we have found 6021 valid models that reproduce 2093 distinct to the hierarchical charged lepton massmasses. spectrum as In well our as t case analysis, of we real have lepton mass restrictedfor matrices. ourselves groups We with to have up performed the to a four system JHEP04(2008)077 ) R 3 ) ) ) ) ) ) ) ) ) R 23 ) , ϕ ) ) ) 0) ) 0) 0) 0) 0) ) ) 0) 2 2 0) 0) 2 4 π 2 2 0) R 2 0) , 0) 4 , , π 4 4 ,π ,π ,θ ,π π π , ,ǫ , , ,ǫ ,ǫ 0 ,ǫ 0 0 0 0 0 ,ǫ , 2 2 2 R 13 , 2 4 4 4 π π , , π, , , π, , π, π , π, , π, , , ,π,π , , ϕ ,π,π 4 0 π , ǫ, 0 0 0 0 0 0 0 , ,ǫ , 0 0 0 0 0 , ,ǫ,ǫ 0 R 1 ,ǫ ,θ 4 π , , ǫ,ǫ, ǫ,ǫ, 2 2 2 , , , , ǫ, ǫ,ǫ , ǫ,ǫ ǫ, ( ( ( ǫ ǫ ( ǫ ( (0 (0 ( π, π, π, π, π, ( π, R 12 (0 ( ( ( , ϕ (0 (0 (0 (0 ( (0 (0 (0 ( ( ( ( ( θ R ( δ ( ) ) ′ ′ D 2 D ) 23 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 2 2 2 4 4 π π 2 0) 0) 0) 0) 0) 0) , α ,θ 0) 0) 0) 4 4 π π ,ǫ ′ ,ǫ ′ ,ǫ , , ,ǫ , , , , , , ,π ,ǫ , ,π ,π flavor models with re- 4 2 π 2 2 2 2 4 4 π 0 0 D π D 1 0 0 0 13 0 0 0 0 , , , , , ,ǫ,ǫ , , π, , , , π, , ,π,π ,π,π ,π,π ,ǫ ,ǫ 2 2 ,θ ǫ,ǫ, ǫ,ǫ, 2 π, π, 4 ǫ,ǫ ǫ,ǫ,ǫ π π, ǫ,ǫ ǫ,ǫ , α ǫ ′ ǫ (0 (0 ( ( (0 ′ ( ( (0 (0 (0 ǫ (0 (0 ( ( (0 (0 ( (0 ( ( ( ( (0 (0 ( ( , “#” labels in both tables D 12 extures are predicted from e current work successfully D s and mixing parameters of he extraction of viable lep- t the exact form of the mass θ δ ( ( enesis. (for further detailed examples ) ion, the inclusion of CP-violating D 3 ) ) ) ) ) ) ) ) ) ) ) D 23 ) ) ) , ϕ ) ) ) ) ) ) ) ) ) 0) 2 0) 0) 0) 4 4 4 π π π 4 π 4 4 4 4 4 4 4 D 2 π π π π π π π , ,π ,π ,π ,π ,π ,π ,π , ,θ ,π ,π ,ǫ , , , ,ǫ , 0 , , , , , , , 0 0 0 0 0 0 0 0 0 0 2 2 2 2 2 D 13 4 , , , , , , , , , , π, π , , π, , , ϕ 4 4 4 4 4 4 4 π π π π π π π ,ǫ 0 0 0 0 0 0 0 0 0 0 0 0 0 , ,ǫ ,ǫ ,ǫ ,ǫ D 1 ,θ , , , , , 2 4 π ǫ, ǫ, ǫ, ǫ, ǫ, ǫ, ǫ, 4 4 4 π π π ǫ ( ( ( ( ( ( ( π, π, π, π, π, π, π, π, ( (0 D 12 ( ( ( ( (0 (0 (0 , ϕ (0 (0 ( ( ( ( ( ( ( ( θ ( D δ ( – 13 – ) ) ′ ′ ℓ 2 ℓ 23 ) ) ) ) ) ) ) 2 0) 0) 0) 2 2 2 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) ,θ 0) 0) 0) , α , , , , ,ǫ ,ǫ ′ , , , , , , , , , , , , , , ,π ′ ,ǫ 2 2 2 ,ǫ ,ǫ 2 ℓ 1 ℓ 13 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , , , , , , , ,ǫ , ,ǫ ,ǫ ,θ ǫ, ǫ, ǫ,ǫ, ǫ, ǫ, , α ǫ,ǫ,ǫ 4 4 4 4 ǫ, ǫ,ǫ ǫ, π π π π ′ ( ( ( ( ( ′ (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (0 ( ( ( ( ℓ ( ( ( ( ℓ 12 δ θ ( ( ) ) ) ) ) ) ℓ 23 ℓ 2 ) ) ) ) ) ) ) ) ) ) ) ) ) 2 2 2 ) 4 2 2 π 4 π 2 2 2 2 2 2 0) 0) 0) 0) 0) 0) 0) 0) ,θ , ,ǫ ,ǫ ,ǫ , α , , , ,π , , ,ǫ 2 ,ǫ ,ǫ 2 2 2 ℓ 1 2 ℓ 13 0 2 0 0 0 0 0 0 ,ǫ,ǫ , , , , π, , π, , π, , π, ,ǫ ,π,π ,π,π ,ǫ,ǫ ,ǫ,ǫ ,ǫ,ǫ ,ǫ ,ǫ ,ǫ ,ǫ 2 ,θ , α 2 π, ǫ,ǫ,ǫ 2 2 2 π, ǫ, ǫ, π,π,π π,π,π ℓ ǫ,ǫ ǫ (0 (0 (0 ( (0 (0 (0 (0 ǫ ( (0 ( (0 ǫ (0 ǫ (0 (0 ( ǫ ( ( ( (0 ( ( ℓ 12 δ ( ( ( ( ( θ ( L ) ) − D ) ) ) ) ) ) ) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) B ,ǫ ,ǫ ,ǫ ,ǫ ,ǫ ,ǫ ,ǫ ,ǫ ,ǫ , , , , , , , , 1 1 1 1 1 1 1 1 1 /m 1 1 1 1 1 1 1 1 , ǫ, , ǫ, , , ǫ, , , ǫ, , ǫ, , ǫ, , ǫ, , ǫ, /M 2 2 2 2 2 2 2 2 2 2 D i ǫ, ǫ, ǫ, ǫ,ǫ, ǫ, ǫ, ǫ, ǫ, ǫ, ǫ, ǫ, ǫ, ǫ, ǫ, ǫ, ǫ, ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ( ( ( ( ( ( ( ( R i ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( m We believe that it would be interesting to study our sample of m 9 6 7 8 3 4 5 1 2 # 11 12 13 10 bottom-up approach of ref. [31]. While ref. [31] deals with t ton mass textures thatmatches are the in textures agreement ontoflavor with explicit symmetries observation, and flavor th their models, breaking. where the t spect to further model building aspects,phases, anomaly as cancellat well as in view of lepton flavorA. violation or Mass leptog and mixing parameters In this appendix, wethe list matrices the that complete are information generatedthe on same by the model. the mas models The inmatrices data of in table the table 1. 22 2 Here models allowson to following such the fully reconstructions, notation reconstruc of see section also 2 ref. [31]). JHEP04(2008)077 ) ) ) ) ) ) ) ) ) ) 0) 2 0) 0) 0) 0) 0) 4 π 2 2 0) 2 0) 2 , , ,π ,π , ,π ,ǫ , , , ,ǫ 0 0 0 0 ,ǫ 0 0 2 4 ,ǫ π 4 2 π 2 , , , , π, , 2 , , , π, , π, 0 , , ,ǫ,ǫ he Fed- 0 0 0 0 0 0 ,ǫ 0 0 0 ,ǫ ,ǫ 2 (2002) 2 2 , , , 2 , , , ǫ,ǫ,ǫ ǫ, ǫ ǫ 4 ǫ,ǫ ǫ π ǫ ( ( (0 π, π, π, ( ( ( ( ( ( (0 (0 (0 (0 (0 (0 ( ( ( 89 B 539 ss matrices and ) ) ) ) ) ) 2 ) ) ) ) 2 ) 2 2 0) 0) 0) 0) 0) 0) 0) ,ǫ , program of Deutsche ,ǫ ,ǫ , , ,π , , ,ǫ , ,ǫ ,π ,π ,π 4 2 4 π π Phys. Lett. 0 4 2 π 0 0 0 0 0 0 0 0 0 de use of the freedom to , , ,ǫ,ǫ , ,ǫ,ǫ , , , , , , , , , π, , Theoretical Astrophysics ,ǫ 2 2 2 2 4 π ǫ, π, 2 ǫ ǫ ǫ ǫ,ǫ (0 ǫ (0 (0 (0 ( (0 (0 (0 (0 (0 ct number 05HT1WWA2. ( ( ǫ ( ( ( ( ( Phys. Rev. Lett. ( , s 1147 ]. e-I data ) ) ) ) 2. ) ) ) ) ) ) ) ) Evidence for oscillation of atmospheric 0) Determination of solar neutrino 0) 0) 0) 0) , 4 4 4 4 2 2 2 π π π π 0) 4 π , , , ,π ,π , ,π ,π , , , , , , 0 0 0 0 0 0 ,ǫ ,ǫ 0 0 ,ǫ 2 2 2 2 4 2 π , , , , , π, , , , , 0 0 4 = 1 π 0 0 , 0 0 0 0 0 ,ǫ , , 0 0 ,ǫ ,ǫ ,ǫ ,ǫ 2 , , j , , 4 4 4 π π π 4 4 4 ǫ, π π π ǫ π, π, π, π, π, ( ( ( ( (0 ( ( ( ( (0 (0 ( ( (0 (0 ( ( ( hep-ex/9807003 3 and – 14 – Measurement of day and night neutrino energy , 2 , ) ) ) ) ) ) ) ) 2 2 2 2 2 2 0) 0) 0) 0) 0) 0) 0) 0) 0) 0) ,ǫ , , , , , , , , , , ,π ,ǫ ,ǫ ,ǫ ,ǫ ,ǫ ,ǫ = 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , , , i , , , , , , , , , , 4 π 4 4 π π (1998) 1562 [ 4 ǫ, ǫ, ǫ, ǫ, ǫ, π (0 (0 (0 (0 (0 (0 (0 (0 ( (0 ( ( ( ( ( ( ( ( ]; 81 ]. = 0 for ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 2 2 2 2 collaboration, S. Fukuda et al., collaboration, Y. Fukuda et al., 4 4 4 π π π 0) 4 D j π 0) 0) , , , , ,ǫ ,ǫ ,ǫ ,ǫ , , , ,π α 2 2 2 2 2 2 2 2 2 of Deutsche Forschungsgemeinschaft. G.S. is supported by t 0 0 0 , ,ǫ ,ǫ ,ǫ ,ǫ ,π,π = ,ǫ ,ǫ ,ǫ ,ǫ 2 2 2 2 π, 2 2 2 2 π, π,π,π π,π,π π,π,π π,π,π π,π,π ǫ,ǫ ′ ǫ (0 ( ǫ ǫ ǫ ( (0 ǫ ǫ ǫ ǫ ( ( ( ( ( ( ( ( ( ( D i ( ( ( ( ϕ hep-ex/0205075 Phys. Rev. Lett. = , Supplementary information for the reconstruction of the ma ′ nucl-ex/0204009 ) ) ) ℓ i 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) ,ǫ ,ǫ ,ǫ , , , ϕ collaboration, Q.R. Ahmad et al., 1 1 1 1 1 1 , ǫ, , ǫ, , ǫ, , ǫ, , ǫ, , ǫ, , ǫ, , ǫ, , ǫ, , ǫ, , ǫ, , ǫ, = 2 2 2 2 2 2 2 2 2 2 2 2 ǫ, ǫ, ǫ, ǫ, ǫ, ǫ, ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ( ( ( ℓ i ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ϕ SNO spectra at SNO and constraints on neutrino mixing parameter Super-Kamiokande (2002) 179 [ oscillation parameters using 1496 days of Super-Kamiokand Super-Kamiokande neutrinos 011302 [ [1] [2] Yukawa couplings of the modelsset in table 1. 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