The Double Cover of the Icosahedral Symmetry Group and Quark Mass

Home , Binary icosahedral group, Binary tetrahedral group, Icosahedral symmetry, Perfect group, Tetrahedral symmetry

arXiv:1011.4928v3 [hep-ph] 8 Feb 2011 cshda group icosahedral yohssta h oa nl sgvnb tan g by point h given crystallographic solids, is a Platonic not angle five is solar the it the on because that based part hypothesis in groups least discrete at the applications of one investigated is we which which in [24] work ∆(3 fHrio-ekn-ct HS tibmxml iig[3,a shown as [33], mixing “tri-bimaximal” [HPS] Harrison-Perkins-Scott of urst iges h successful the singlets, to quarks arx a hfe h aaimfradesn h tnadMod Standard angle the by mixing addressing above for quark from paradigm the bounded the from is shifted differences angle has th striking third of matrix, its the angles with and mixing pattern, large the This are of two matrix that mixing revealed [8] have data oscillation trino o-bla aiysmere.Lpo ao oeshv encons been have models flavor T Lepton [9]) also symmetries. (see family symmetries non-Abelian family continuous with [10] mechanism Nielsen h iayttaerlgroup trip one tetrahedral and binary representations one-dimensional the three has which [11], oeiggop hc losfrsioilrpeettos h mo The representations. t spinorial models for also allows which are Se group, There covering theory. theory. unified flavor grand complete fully a c a building obtain within model and cases flavor quarks some and the theory incorporate group to discrete of reviews Recent n h cshda ymtygroup symmetry icosahedral the and rdc of product a rtsgetdi 2]adltrepoe ihna within explored later and [26] in suggested first was 1] nftr ok ewl rvd oecmrhnieoeve o overview comprehensive more a provide will we work, future In [14]. o h aib nl ae ntegle ai hti orc oth to correct is that ratio golden the on based angle Cabibbo the for bytebasic the obey I hswr epoietegoptertcltosnee o building for needed tools theoretical group the provide we work this h etn n urstgte ihnthe within together quarks and leptons the si the in As eaigtegle ai n h oa iigageicuea include g angle the mixing in solar appears the ratio and golden ratio the golden since the building relating model flavor lepton for ′ hc sioopi to isomorphic is which , nti ae,w netgt h xeso fteioaerlsym icosahedral the of extension the investigate we paper, this In oepeiey hl h ur etrpeiul niae ao m flavor a indicated previously sector quark the while precisely, More ncrancss oee,teqak r otesl accommoda easily most are quarks the however, cases, certain In ihtemaueeto etiooclain 16,a nrgigpatt intriguing an [1–6], oscillations neutrino of measurement the With ewl e htt noprt h urs ti ihyadvantageous highly is it quarks, the incorporate to that see will We Z ⊗ h obeCvro h cshda ymtyGopadQuark and Group Symmetry Icosahedral the of Cover Double The n 2 n ∆(6 and ) 9 ur ao oe ae nta eainta ssmlrt h quark the to similar is that relation that on based model flavor quark T A ′ oetn h ru oisdul oe,a ntecs ftetra of case the in as cover, double its incorpora To to group leptons. the the extend for to group symmetry family viable a aiysmer ru nteqaksco.Teioaerl( icosahedral The sector. quark the in group symmetry family ASnmes 12.15Ff,12.60.Jv yields numbers: PACS that mixings and masses quark well-known the the of of model a construct ai ru hoeia ol o ao oe-uligbas model-building flavor for tools theoretical group basic ae the case, 4 eivsiaeteie httedul oe fterotation the of cover double the that idea the investigate We and U n 2 relation (2) I 2 1] h eiietpoutof product semidirect the [15], ) ′ S n consider and , 3 eateto hsc,Uiest fWsosn aio,W Madison, Wisconsin, of University Physics, of Department U [20], 2 etrsaeepce ob osbewithin possible be to expected are textures (2) A PSL 4 2 U h lentn ru ffu lmns[1,tebnr tetrahedr binary the [11], elements four of group alternating the , T 2 ur etr models. texture quark (2) ⊗ n sinn h ihe eeainqak odult n h th the and doublets to quarks generation lighter the assigning and , U (2 2 I 2 ur etrs[2 a eotie oehrwt h etns lepton the with together obtained be can [32] textures quark (2) I , = ′ )[1,teqaenoi ymtis[2,tedhda symmetries dihedral the [22], symmetries quaternionic the [21], 7) iaL vrt n lxne .Stuart J. Alexander and Everett L. Lisa I safml ymtygopfrteqak.Ti td sbsdo ou on based is study This quarks. the for group symmetry family a as safml ymtygopfrtelpos h cshda symme icosahedral The leptons. the for group symmetry family a as hc sioopi to isomorphic is which , 1 ⊕ 3 I hrfr,a okn xml,w locntutasupersymm a construct also we example, working a as Therefore, . ′ Dtd coe 5 2018) 25, October (Dated: .INTRODUCTION I. rmwr [34]. framework θ sol 1 = Z /φ 3 Z where , and 2 Z × Z D A 7 2 10 5 φ [16], do h iayioaerlgroup icosahedral binary the on ed rmwr n[7,woas rvddaprediction a provided also who [17], in framework tpoieteapei h erhda ( tetrahedral the is example prominent st e ersnain ycnieigisdul cover, double its considering By representation. let 2]( four-family (a [24] er group metry oe htpeit cos predicts that model 1+ (1 = l(M ao puzzle. flavor (SM) el ea oesicuebt ursadlpos in leptons, and quarks both include models veral e fteoiia ru setne oisdouble its to extended is group original the if ted nb on n[83] h aua etse is step next natural The [28–30]. in found be an rce ae ntettaerlsmer group symmetry tetrahedral the on based tructed eteqak,i shgl advantageous highly is it quarks, the te erl( hedral A U f e iisadsuggested and digits ree dlbidn rmwr ae nteFroggatt- the on based framework odel-building h ag etnmxn nlssgetdiscrete suggest angles mixing lepton large the , lioaerlsmer ru sthe is group symmetry icosahedral al Z aiNkgw-aaaPneov (MNSP) Maki-Nakagawa-Sakata-Pontecorvo e op norwr,w xlrdteinteresting the explored we work, our In roup. dbe oprtvl nxlrdfrphysics for unexplored comparatively been ad a ou nyo ur iig(e ..[31]). e.g. (see mixing quark on only focus hat fteCbboKbysiMsaa(CKM) Cabibbo-Kobayashi-Maskawa the of s I oer fteioaern lentv ideas Alternative icosahedron. the of eometry 5 2-nprdqakflvrmdl ae on based models flavor quark (2)-inspired n[12–14]. in oextend to ru a rvosypooe as proposed previously was group ) 2 ayo h ucsflpredictions successful the of many I ′ r flpo iighseegd h neu- The emerged. has mixing lepton of ern ′ n osdrflvrmdlbidn o both for building model flavor consider and Z × √ since , h aib nl frgoa t,se[7]). see fits, global (for angle Cabibbo the A 5) 4 2 / ymty epoiethe provide We symmetry. ) stegle ai.Ti hypothesis This ratio. golden the is 2 I [17], etro the of sector ,576 USA 53706, I, I ( I A ′ a obe ersnain that representations doublet has oisdul cover double its to 5 A S oisdul oe,tebinary the cover, double its to ) 4 5 [18], oe a efudi [27]). in found be can model θ sol T S asTextures Mass = A 3 ′ lgroup al Z ⊗ 5 φ/ 1] h semidirect the [19], MADPHYS-10-1566 I saueu setting useful a as ′ [25]. 2 3 and co prediction ector Z ⊗ r generation ird I D ′ ec,in Hence, . T 9 n r group, try A previous r ′ oe of model 4 2,25], [23, [12–14], group ) etric I ′ . 2

The outline for this paper is as follows. We begin by describing the basic features of the icosahedral symmetry group and its double cover ′. Next, we will outline the group presentation and the construction of group invariants ′ I I ′ for . We then present our quark ﬂavor model based on 9 and address the issue of vacuum alignment for the ﬂavonI sector. Finally, we present our conclusions and outlook.I ⊗ Z

II. THEORETICAL BACKGROUND

We begin with a brief overview of the icosahedral symmetry group and its double cover, the binary icosahedral group ′. The basic theory of the icosahedral symmetry group is knownI in the mathematical literature, and can be found inI the papers [24, 35–39]. Finite groups are also covered extensively in several texts [40–43], and reviews [30]. The icosahedral symmetry group is the group of all rotations that preserve the orientation of the icosahedron, which is the Platonic solid consistingI of twenty equilateral triangles. (Here and in what follows we will only consider proper rotations, and therefore ignore inversions.) is the finite subgroup of SO(3) that is isomorphic to 5, the alternating group of five elements. contains 60 elements:I the identity, rotations by 2π/5 and 4π/5 aboutA an axis through each of the twelve vertices,I rotations by 2π/3 about an axis through the center of each of the twenty faces, and rotations by π about the midpoint of each of the thirty edges (resulting in fifteen distinct rotations). These five types of rotations form five conjugacy classes, which are denoted as follows: 2 1, 12C5, 12C5 , 20C3, 15C2. (1)

k Here we follow the standard procedure and use Schoenflies notation: Cn is a rotation by 2kπ/n, and the number in k front gives the number of group elements in the conjugacy class. The elements of Cn are known as order n elements, i.e., they result in the identity after n operations. Given the basic results in discrete group theory that the number of elements in the group is equal to the sum of the squares of the irreducible representations, and that the number of conjugacy classes is equal to the number of irreducible representations, the 60 elements of result in the condition: I 1+12+12+15+20=60=12 +32 +32 +42 +52. (2) Thus, has five irreducible representations: 1, 3, 3′, 4, and 5. The presence of the two distinct triplet representations suggestsI that may be a good candidate for a family symmetry group, as explored in our previous investigation [24]. We note hereI that in contrast to ′, which has three one-dimensional representations, the only one-dimensional representation of is the identity. ThisT fact, which will be of significance when we turn to the construction of flavor models, results becauseI is a perfect group; i.e., it is equal to its commutator subgroup [41, 43]. I ′ The double cover of the icosahedral group, which is denoted as the binary icosahedral group , has double the number of group elements as , for a total of 120. The group elements are grouped into the five conjugacyI classes of and four additional conjugacyI classes: I 2 R, 12C5R, 12C5 R, 20C3R, (3) where R is I, and I denotes the identity. The absence of another copy of 15C2 occurs because the additional group − ′ elements in double the order of the existing C2 conjugacy class from 15 to 30, instead of forming a new conjugacy class. This informationI can be used to calculate the additional irreducible representations of ′: I 1+12+12+20+15=60=22 +22 +42 +62. (4) Hence, ′ has the spinorial representations 2, 2′, 4′, and 6 in addition to the 1, 3, 3′, 4, and 5 representations. I ′ The character table of , which can also be found in [36, 37, 39], is given in Table I. We see that the golden ratio, I φ =(1+ √5)/2 (where φ 1=1/φ), appears for the order five elements for the doublet and triplet representations. − From the character table, it is straightforward to deduce the form of the tensor products of the irreducible representations, which are known in the literature [36, 37, 39]. The tensor products involving the doublet and triplet representations are as follows (for the complete list, see [36, 37, 39]): ′ ′ ′ ′ 3 3 = 1 3 5, 3 3 = 1 3 5, 3 3 = 4 5, ⊗ ⊕ ⊕ ′ ⊗ ′ ⊕ ′ ⊕ ⊗ ′ ⊕ 2 2 = 1 3, 2 2 = 1 3 , 2 2 = 4, ⊗ ⊕ ′ ′ ⊗ ′ ′⊕ ′ ⊗ ′ ′ 2 3 = 2 4 , 2 3 = 2 4 2 3 = 6, 2 3 = 6. (5) ⊗ ⊕ ⊗ ⊕ ⊗ ⊗ Observe that the basic U(2) relation, 2 2 = 1 3 (and an analogous relation involving the primed representations), holds within the ′ group. In what follows,⊗ we⊕ will discuss the construction of quark flavor models based on this relation in analogyI with the well-known U(2) flavor models [32]. First, however, we will need the ′ group presentation and the construction of group invariants, which is the subject of the next section. I 3

′ 1 3 3′ 4 5 2 2′ 4′ 6 I 1 1 3 3 4 5 2 2 4 6 12C5 1 φ 1 φ 1 0 φ 1 φ 1 1 2 − − − − 12C5 1 1 φ φ 1 0 φ 1 φ 1 1 − − − − − 20C3 1 0 0 1 1 1 1 1 0 − − 30C2 1 1 1 0 1 0 0 0 0 − − R 1 3 3 4 5 2 2 4 6 − − − − 12C5R 1 φ 1 φ 1 0 φ φ 1 1 1 2 − − − − − 12C5 R 1 1 φ φ 1 0 1 φ φ 1 1 − − − − 20C3R 1 0 0 1 1 1 1 1 0 − − − TABLE I: The character table of the binary icosahedral group ′, in which φ is the golden ratio: φ = (1+ √5)/2. I

III. GROUP PRESENTATION AND GROUP INVARIANTS

The elements of discrete groups such as the binary icosahedral group can be generated by a set of basic elements that satisfy certain relations; the elements and the rules together are known as the “presentation” of the group. For the case of and ′, there are several equivalent group presentations (see e.g. [36–39]). Following our previous work I I on ( 5) [24], we prefer to use the presentation of Shirai [37], in which all group elements can be expressed in terms of theI A order two generator S and the order 5 generator T , such that

2 5 2 3 −1 −1 3 S, T : S = T = R, (T ST ST STST ) = I. (6)

Here R is the identity I in the case of the single-valued representations 1, 3, 3′, and 5, and R is I for the double- ′ ′ − valued representations 2, 2 , 4 , and 6. We note that while S and T are contained in the conjugacy classes C2 and C5, respectively, the order three combination of S and T given above is contained in the C3R conjugacy class (not C3), which is why it cubes to the identity rather than R. The Shirai presentation can be related to other presentations that are standard in the literature, such as the presentation of Threlfall involving order three and order ﬁve generators found for example in [40]:

a, b : a3 = b5 = (ab)2 = R, (7) and the presentation described in the work of Cummins and Patera [36], which involves two generators of order two and one order three generator:

3 2 2 3 2 3 A1, A2, A3 : A1 = A2 = A3 = R, (A1A2) = (A2A3) = (A1A3) = R. (8)

The generators of the Shirai presentation can be related (up to similarity transformations) to the Threlfall generators 4 by S = ab and T = (ab)b (ab), and to the Cummins and Patera generators by S = A2A1A3A2 and T = A1A2A3. The generators S and T of the Shirai presentation have been given for the single-valued representations in [24, 37]. Rather than present the complete list for the double-valued representations, we provide the Shirai presentation generators for the doublets 2 and 2′, which can also be found in [37]:

1 i(φ 1) φ + i 1 φ + i i(φ 1) S2 = − ,T2 = − 2 i φ i(1 φ) 2 i(φ 1) φ i − − − −

1 iφ φ (1 i) 1 (1 i) φ iφ S2′ = − − ,T2′ = − − − . 2 (1 + i) φ iφ 2 iφ (1 + i) φ − − − − The Shirai presentation generators for the 4′ and the 6 can also be found in [37]. Since we will not need them for the ﬂavor model-building at leading order that we will consider in this paper, we do not state them here (we will consider more general scenarios in future work [34]). For completeness of presentation, however, we also state here the Shirai presentation generators for the triplets 3 and 3′, which are given by

1 1 1 φ φ 1 φ φ 1 − 1 1 1 S3 =  φ φ 1  ,T3 =  φ φ 1  , (9) 2 1 2 −1 1 φ 1 φ  φ −   φ −  4

1 1 φ φ 1 φ φ 1 1 −1 1 −1 − ′ ′ S3 =  φ 1 φ  ,T3 =  φ 1 φ  . (10) 2 − 1 2 1 1 φ 1 φ  φ   − − φ  While the group presentations of and ′ are well known in the literature, the further group theoretical tools that are needed for flavor model buildingI are theI rules for constructing group invariants in a form that is suitable for physics applications. For , we did this exercise for the Shirai presentation [24]. Here we will present the basic relations for ′ that we have calculatedI in the Shirai presentation that are necessary for flavor model building at leading order, andI defer the complete list of ′ tensor products for future work [34]. I We begin with the tensor product 2 2 = 1 3. Defining the two distinct doublets as 2 = (a1,a2)T and 2 = (b1,b2)T , it is straightforward to show that up⊗ to normalization⊕ factors, the singlet is given by the antisymmetric combination

1 = a2b1 a1b2, (11) − and the triplet 3 is given by the symmetric combination

T 3 = ( ia1b1 + ia2b2,a1b1 + a2b2,ia2b1 + ia1b2) . (12) − For 2′ 2′ = 1 3′, the singlet is again given by Eq. (11), and the triplet 3′ takes the (symmetric) form ⊗ ⊕ ′ T 3 = ( ia1b1 ia2b2, a1b1 + a2b2,a2b1 + a1b2) . (13) − − − The tensor product of 2 2′ = 4 results in the following form for the quartic: ⊗ T 4 = ( a2b1 a1b2,ia2b1 ia1b2, ia1b1 ia2b2, a1b1 + a2b2) . (14) − − − − − − ′ For 2 3 = 2 4 , in which the initial doublet is (a1,a2)T and the initial triplet is (b1,b2,b3)T , the doublet is ⊗ ⊕ T 2 = ( a2b1 + ia2b2 a1b3, a1b1 ia1b2 + a2b3) , (15) − − − − and the 4′ (which we will not need in this paper) takes the form

′ 1 T 4 = (√3(ia1b1 + a1b2),ia2b1 + a2b2 2ia1b3, ia1b1 + a1b2 2ia2b3, √3( ia2b1 + a2b2)) . (16) √3 − − − − Similarly, for 2′ 3′ = 2′ 4′, the 2′ is ⊗ ⊕ ′ T 2 = (ia2b1 a2b2 a1b3, ia1b1 a1b2 + a2b3) , (17) − − − − and the 4′ is given by

i 2 2 2 1 2 2 √ 1 3 φ a b φ a b + 5a b − − 1 √ 1 1 1 2 2 3 ′ 1  3( iφa b + φ a b a b )  4 = − 1 − , (18) √ 2 1 2 2 1 3 √5  3( iφa b φ a b a b )   i − −2 −   2 a1b1 + φ aab2 + √5a2b3   − φ  in which φ = (1+ √5)/2 is the golden ratio. Once again, for completeness, let us also recall that the singlet in the 3 3 = 1 3 5 takes the form (as does the singlet in the analogous relation for 3′ 3′): ⊗ ⊕ ⊕ ⊗ 1 = a1b1 + a2b2 + a3b3, (19) in which we have written the initial triplets as (a1,a2,a3)T and (b1,b2,b3)T . (For the explicit form of the 3 and the 5 and their primed counterparts, see [24].)

IV. QUARK FLAVOR MODEL BUILDING AND U(2) TEXTURES

In this section, we turn to a discussion of quark ﬂavor model building based on ′ and present a viable model of ′ I the quark masses and mixings based on 9. Let us begin by recalling the results of our previous study [24] I ⊗ Z of the icosahedral group ( 5), in which we constructed a simple toy lepton ﬂavor model that resulted in a solar I A 5

c c c c 0 0 0 0 Field Q Q3 u d t b Hu,d ρ σ χ η η2 ψ σ χ η ψ ′ 2 1 2 2 1 1 1 1 1 1 2 2 3 1 1 2 3 I 8 5 8 6 8 5 2 8 5 2 8 9 1 α α α α α 1 α α α 1 α α α α 1 α Z

′ TABLE II: The charge assignments for the quark, Higgs, flavon, and driving field supermultiplets in our 9 model, with 2πi/9 0 0 0 0 I ⊗ Z α = e . The flavon fields are the ρ, σ, χ, η, η2, and ψ, while the driving fields are σ , χ , η , and ψ .

mixing angle given by tan θsol = 1/φ. In this analysis, we embedded the lepton doublets Li as a 3 and the lepton c 3′ 3 3 1 3 5 singlets ei as a , and constructed an effective neutrino seesaw matrix based on the relation = and a charged lepton mass matrix based on 3 3′ = 4 5. To break the symmetry, we introduced⊗ a flavon⊕ field⊕ξ that transforms as a 5 for the neutrino sector, and⊗ two charged⊕ lepton sector flavon fields χ and ψ, which transform as a 4 and 5, respectively. The breaking of by these flavon fields resulted in a toy model in which the solar mixing angle is obtained from the neutrino sector, theI maximal atmospheric mixing angle is obtained from the charged leptons, and the reactor angle is zero at leading order. Issues that were not addressed in [24] included the generation of the flavon field vacuum expectation values as well as higher order corrections; in forthcoming work, we will discuss these issues and construct other examples of lepton flavor models [34]). One result from our previous study is that it is not easy to generate the strongly hierarchical charged fermion masses within the group. More precisely, in our example the two flavon fields of the charged lepton sector had a delicate balance of vacuumI expectation values in order to generate a structure in which only the τ lepton has a nonvanishing mass at leading order. The challenge ultimately results in part because only has one irreducible representation that I is one-dimensional, which is the singlet representation 1. This is in sharp contrast to the case of ( 4), which has three one-dimensional representations: 1, 1′, and 1′′. The use of these three representations allowsT forA the generation of mass hierarchies in the lepton sector through an additional U(1)FN or other symmetries that result in specific higher-dimensional operators [11, 14]. Even with this freedom in 4 models, it is known that the quark sector is more easily accommodated by extending the group to ′, and embeddingA the quarks in doublet and one-dimensional representations [12–14]. Hence, we expect a similar extensionT is necessary for the icosahedral symmetry group, for which the generation of mass hierarchies is more difficult than that of the tetrahedral symmetry case. For this reason, we now turn to the binary icosahedral group ′, and construct a supersymmetric model in which the quark superfields are embedded in doublet and singlet represenI tations of ′. The standard approach is to assign the quark superfields of the third generation to singlet representations, I

T c c Q3 = (t,b) 1, t 1, b 1, (20) → → → and assign the lighter generations to doublet representations. Within ′, there are two distinct doublets, 2 and 2′, which allows for some flexibility in model-building. In this paper, we willI assign the first and second generation quark superfields to the 2 representation, as follows:

T T c c c T c c c Q = (Q1,Q2) = ((u, d), (c,s)) 2, u = (u ,c ) 2, d = (d ,s ) 2. (21) → → → This assignment will allow us to construct models with the U(2) relation, 2 2 = 1 3. We note that a replacement of 2 by 2′ in Eq. (21) would also result in an analogous relation, 2′ 2′ = 1 ⊗3′, which⊕ would result in a similar quark flavor models (at least at leading order). An alternative assignment⊗of Q to⊕ the 2 and uc, dc to the 2′ representations (or permutations) would yield models based on the relation 2 2′ = 4, which would necessitate a different pattern of flavor symmetry breaking. We defer the investigation of this possib⊗ ility to forthcoming work [34]. To break the family symmetry, we first recall that in U(2) flavor models with this assignment of the Standard Model quarks, the family symmetry is broken in two stages: first U(2) is broken to U(1) by nonvanishing vacuum expectation values of triplet and doublet fields, and next the U(1) is broken by fields that are singlets under the original family symmetry. This basic flavon sector is retained in U(2)-inspired models based on discrete non-Abelian family symmetries such as ′ (see for example [14]), and we will also use it here. Hence, we now introduce a flavon T ′ chiral superfield ψ that transforms as a 3 and two flavon chiral superfields η1,2 that each transform as a 2 under , as well as a set of ′ singlet superfields ρ, σ, and χ. I I To obtain realistic quark mass matrices, it is necessary to augment with an additional symmetry, which we take ′ I to be a 9 group. The 9 charge assignments for the SM matter superfields and flavon sector superfields of our Z I ⊗ Z model in presented in Table II. We have assumed that the MSSM Higgs doublets Hu,d are inert with respect to the ′ 9 symmetry. We note that the choice of 9 as an additional symmetry of the quark sector has also been made I ⊗ Z ′ Z in the 3 9 model of [14], though our flavon field content and charge assignments are different. With these chargeT assignments,⊗ Z ⊗ Z the superpotential terms that result in effective Yukawa terms involving the up-type quarks take 6 the following form:

′ ′′ c yu2 c yu2 c yu2 c yu3 c W = y 1Q3t H + Q3u η1H + Q3u η2ρH + Q3u η2ψH + Qt η2H u u u M u M 2 u M 2 u M u ′ ′′ ′′′ yu4 c yu4 c yu4 c yu4 c + Qu σσH + Qu ρχH + Qu ηη2H + Qu χψH , (22) M 2 u M 2 u M 2 u M 2 u in which M represents the (presumably high) cutoff scale of the effective theory, and the yui are dimensionless (order one) couplings. Similarly, the effective Yukawa couplings for the down-type quark superfields are given by

′ ′′ yd1 c yd2 c yd3 c yd4 c yd4 c yd4 c W = Q3b χH + Q3d η2χH + Qb η2χH + Qd ρσH + Qd χχH + Qd σψH , (23) d M d M 2 d M 2 d M 2 d M 2 d M 2 d in which the ydi again represent dimensionless couplings. To break the flavor symmetry, we assume that the flavon fields develop vacuum expectation values as follows (this form will be justified later in the paper):

1 2 3 T v3 T 1 2 T T ψ = (ψ , ψ , ψ ) = ( i, 1, 0) , η1 = (η1 , η1 ) = (v21, 0) , h i h i 2 − h i h i 1 2 T T η2 = (η2 , η2 ) = (v22, 0) , ρ = v , χ = v , σ = v , (24) h i h i h i ρ h i χ h i σ in which the vacuum expectation values are of the order

v3 v21 v22 v v v λ2, ρ χ σ λ3, (25) M ∼ M ∼ M ∼ M ∼ M ∼ M ∼

where λ sin θc =0.22 is the Cabibbo angle. Upon ﬂavor and electroweak symmetry breaking (with Hu,d = vu,d), the quark≡ mass matrices that result from Eq. (22) and Eq. (23) take the form h i

2 vσ ′ vρvχ 6 0 y 4 2 y 4 2 0 0 y˜u4λ 0 − u M − u M −  2  6 ′′ 4 ′′′ 5 2 vσ ′ vρvχ ′′ v21v22 ′′′ v3vχ v22  4 3  u = 4 2 2 2 2 3 vu y˜u λ y˜u4λ +˜yu4λ y˜u λ vu, (26) M yu M + yu4 M yu4 M + yu4 M yu M ≡   2 v21 ′ v22vρ  0y ˜ 2λ y˜ 1   0 y 2 + y 2 y 1  u u  u M u2 M u    and

2 vρvσ ′ vχ 3 0 y 4 2 y 4 2 0 0 y˜ 4λ 0 − d M − d M − d  2  ′′ 3 v 3 2 2 = vρvσ ′ χ ′′ v3vσ v22vχ v  y˜d4λ y˜ 4λ y˜d3λ  λ v . (27) d 4 2 2 2 3 2 d d d M yd M + yd4 M yd4 M yd M ≡   2  v22vχ vχ   0y ˜d2λ y˜d1   0 y 2 2 y 1     d M d M  It is straightforward to diagonalize these mass matrices using perturbation theory, which yields the following results for the quark masses at leading order:

2 2 2 y˜u1 y˜u4 8 y˜u2y˜u3 ′′ 4 y˜u2 + y˜u3 4 mu | || | λ vu, mc y˜u4 λ vu, mt y˜u1 + | | | | λ vu, (28) ≃ y˜ 2y˜ 3 y˜ 1y˜ 4 ≃ y˜ 1 − ≃ | | 2 y˜ 1 u u u u u u | 2 − | 2 | | 2 y˜d4 7 ′′ 5 y˜d2 + y˜d3 4 3 md | ′′ | λ vd, ms y˜d4 λ vd, mb y˜d1 + | | | | λ λ vd. ≃ y˜ ≃ | | ≃ | | 2 y˜ 1 | d4| | d | Hence, the quark mass ratios are predicted have the appropriate powers of the Cabibbo angle λ, which are given as 8 4 4 2 3 follows: : mu : mc : mt λ : λ : 1 and md : ms : mb λ : λ : 1. We also have mb : mt λ : 1, which is consistent ∼ ∼ ′ ∼ with low to moderate values of tan β = vu/vd, as in the model of [14]. This range of tan β has advantages in terms of model-building compared with larger values of tan β,T such as the avoidance of large radiative corrections. Similarly, the leading order elements of the CKM mixing matrix also have the appropriate powers of λ:

y˜d4 y˜u1y˜u4 2 ∗ y˜d3 y˜u3 2 ∗ Vud Vcs Vtb 1, Vus ′′ λ ′′ λ Vcd, Vcb λ Vts, ∼ ∼ ∼ ∼−y˜d4 − y˜u2y˜u3 y˜u1y˜u4 ∼− ∼ y˜d1 − y˜u1 ∼− − ∗ ∗ ∗ y˜u4y˜u1 y˜u3 y˜d3 4 y˜d4 y˜d3 y˜u3 3 Vub ′′ λ , Vtd ′′ ∗ ∗ + ∗ λ . (29) ∼ y˜ 2y˜ 3 y˜ 1y˜ 4 y˜ 1 − y˜ 1 ∼ y˜ −y˜ y˜ 1 u u − u u u d d4 d1 u 7

From Eqs. (28)–(29), we see that the well-known U(2) relations Vtd/Vts = md/ms and Vub/Vcb = mu/mc are ′ | | | | reproduced at leading order, as is the case in the model of [14]. p p T ′ Therefore, we see that the embedding of the quarks as given in Eq. (20) and Eq. (21) within can result in a viable quark flavor model at leading order, provided the assumption of the specific flavon sector asI given in Table II. In principle, we can go further and compute higher order corrections to the results of Eq. (28) and Eq. (29), which will modify the leading order relations. However, we do not do so in this paper because we are not including the leptons, and the flavons needed to break the family symmetry might contribute in the quark sector at higher order. We comment that in the lepton model we presented in [24], the lepton sector flavons are fields that transform either as a 4 or a 5 of , and so these fields will not contribute at leading order to the quark mass matrices of the U(2)-inspired model presentedI here simply due to icosahedral symmetry. Clearly, for alternate embeddings in which the quarks of the lighter generations are embedded in both the 2 and 2′, any lepton sector flavon field that transforms as a 4 of is allowed by icosahedral symmetry to couple to the quarks at leading order, though additional discrete symmetriesI can also be imposed to forbid such couplings. We now turn to the important question of justifying the family symmetry breaking pattern of Eq. (24). To address the dynamics of the flavon sector, we follow the standard approach and recall that a global U(1)R charge is present in the supersymmetric sector of the theory (it is broken to a discrete R-parity when supersymmetry breaking terms are included), such that the superpotential terms satisfy the constraint that the total R-charge of any allowed term is +2. Hence, it is necessary to introduce additional “driving” fields which have a U(1)R charge of +2 and thus couple linearly to the flavon fields in the superpotential. Our choice of driving fields is presented in Table II. These fields include two ′ singlets σ0 and χ0, one ′ doublet η0, and one ′ triplet ψ0. With these charge assignments, the leading orderI superpotential couplings involvingI the flavon fieldsI and the driving fields take the form

0 0 0 0 0 0 0 0 0 0 Wﬂ = Mηη1η + g1η η2ρ + g2σ σρ + g3σ χχ + g4χ ρρ + g5χ σχ + g6χ ψψ + g7η η2ψ + g8η1η2ψ + g9χψψ . (30)

With the assumption that the soft supersymmetry breaking scalar mass-squared parameters of the driving fields are all positive at the symmetry breaking scale, the driving fields do not develop vacuum expectation values, and hence it is only necessary to enforce that the F terms of the driving fields vanish to obtain a local minimum of the flavon field potential. These F terms are given as follows:

∂Wfl ∂Wfl 1 1 2 2 3 3 = g2σρ + g3χχ, = g4ρρ + g5σχ + g6(ψ ψ + ψ ψ + ψ ψ ), (31) ∂σ0 ∂χ0 ∂Wfl 2 1 1 2 2 3 2 ∂Wfl 1 2 1 2 1 3 1 = M η1 + g7η2( iψ + ψ )+ g7η2 iψ + g1ρη2, = M η1 + g7η2(iψ + ψ )+ g7η2iψ g1ρη2 , ∂η0 1 η − ∂η0 2 − η − ∂Wfl 1 1 2 2 1 ∂Wfl 1 1 2 2 2 ∂Wfl 2 1 1 2 3 = g8( iη1η2 + iη1η2)+ g9χψ , = g8(η1 η2 + η1 η2)+ g9χψ , = g8(iη1η2 + iη1η2 )+ g9χψ . ∂ψ0 1 − ∂ψ0 2 ∂ψ0 3

The requirement that the F terms of Eq. (31) vanish at the minimum is consistent with the form of the ﬂavon vacuum expectation values of Eq. (24), in which the vevs satisfy the following relations:

1/3 1/3 2 1/3 g4 g3g5 g2g4 g1v22vρ 2g1v22 g8 g3g5 vσ = vρ, vχ = vρ , v21 = , v3 = , (32) −g5 g4g2 g3g5 − Mη Mη g9 g4g2 in which the flat directions vρ and v22 are presumably lifted by supersymmetry breaking terms. For order one values of the coupling ratios g4/g5 and g2/g3, the singlet vacuum expectation values are naturally vρ vσ vχ. For the 3 ∼ ∼ doublet and triplet flavons, a tuning of M /g1 λ M and g8/g9 λ is required to obtain v21 v22 v3 with order η ∼ ∼ ∼ ∼ one values of g4/g5 and g2/g3. Hence, the vacuum structure specified by Eq. (24) and Eq. (25) can be obtained within a specific subset of the multidimensional parameter space of the flavon sector potential.

V. CONCLUSIONS

With the advent of the large angles of the lepton mixing data, the paradigm of discrete non-Abelian family symmetries has become standard in efforts to address the fermion mass puzzle of the Standard Model. Of the possible discrete non-Abelian symmetry groups to consider as a basis for flavor model building, we have argued in this paper that the double cover of the rotational icosahedral symmetry group, the binary icosahedral group ′, is well-suited for this purpose due to its nontrivial triplet and doublet representations. In comparison to other discreteI groups based on the Platonic solids, there has been a relative paucity of appearances of ′ within the physics literature, which is due at least in part to the fact that the icosahedron does not obey translationalI symmetry. Therefore, we have presented 8 the basic group theory of ′, which is available primarily in the mathematics literature, in a form that is suitable for flavor model building. AsI an example, we applied ′ as a family symmetry of the quark sector and constructed a ′ I viable 9 quark flavor model that reproduces many of the desirable features of the well-known U(2) flavor models and otherI ⊗Z models based on the fundamental relation 2 2 = 1 3. By analyzing the dynamics of the flavon fields that break the family symmetry, we have demonstrated⊗ that the⊕ successful leading order predictions of this model can be accommodated within a subset of the full parameter space of the theory. With respect to other discrete non-Abelian family symmetry groups that have been used for flavor model building, the icosahedral group and its double cover are relatively large groups (with group orders of 60 and 120, respectively) which have larger representations. Therefore, icosahedral symmetry provides a very rich setting in which to address the fermion mass puzzle of the Standard Model. Additional studies are needed to determine the full power of ′ as a viable family symmetry group for the quark and lepton sectors. We defer further investigations of this intriguingI framework to future work.

Acknowledgments

We thank D. J. H. Chung for helpful conversations and comments. This work was supported by the University of Wisconsin Alumni Research Foundation and the U. S. Department of Energy contract DE-FG-02-95ER40896.

[1] Y. Fukuda et al., [Super-Kamiokande Collaboration], Phys. Rev. Lett. 81 (1998) 1562; Phys. Rev. Lett. 82 (1999) 2430; T. Kajita for the collaboration, Nucl. Phys. Proc. Suppl. 77 (1999) 123. See also T. Kajita and Y. Totsuka, Rev. Mod. Phys. 73 (2001) 85. [2] S. H. Ahn, et al., [K2K Collaboration], Phys. Lett. B 511 (2001) 178; Phys. Rev. Lett. 90 (2003) 041801. [3] J.N. Bahcall, W.A. Fowler, I. Iben and R.L. Sears, Astrophys. J. 137 (1963) 344; J. Bahcall, Phys. Rev. Lett. 12 (1964) 300; R. Davis, Jr., Phys. Rev. Lett. 12 (1964) 303; R. Davis, Jr., D.S. Harmer and K.C. Hoﬀman, Phys. Rev. Lett. 20 (1968) 1205; J.N. Bahcall, N.A. Bahcall and G. Shaviv, Phys. Rev. Lett. 20 (1968) 1209; J.N. Bahcall and R. Davis, Jr., Science 191 (1976) 264. [4] Y. Fukuda et al., [Super-Kamiokande Collaboration], Phys. Rev. Lett. 81 (1998) 1158; [Erratum-ibid 81 (1998) 4279]; B.T. Clevel et al., Astrophys. J. 496 (1998) 505; W. Hampel et al., [GNO Collaboration], Phys. Lett. B 447 (1999) 127; Q.A. Ahmed. et al., [SNO Collaboration], Phys. Rev. Lett. 87 (2001) 071301; Phys. Rev. Lett. 89 (2002) 011301. [5] K. Eguchi, et al., [KamLAND collaboration], Phys. Rev. Lett. 90 (2003) 021802. K. Inoue, [KamLAND collaboration], New J. Phys. 6 (2004) 147; S. Abe et al., [KamLAND collaboration], Phys. Rev. Lett. 100 (2008) 221803. [6] M. Apollonio, et al., [CHOOZ Collaboration], Euro. Phys. J. C 27 (2003) 331. [7] T. Schwetz, M. Tortola and J.W.F. Valle, New J. Phys. 10 (2008) 113011. [8] B. Pontecorvo, Sov. Phys. JETP 7 (1958) 172 [Zh. Eksp. Teor. Fiz. 34 (1958) 247]; Z. Maki, M. Nakagawa and S. Sakata, Prog. Theor. Phys. 28 (1962) 870. [9] Z.G.Berezhiani and M.Yu.Khlopov, Yadernaya Fizika (1990) V. 51, 1157-1170 [English translation: Sov.J.Nucl.Phys. (1990) V. 51, 739-746]; Z.G.Berezhiani and M.Yu.Khlopov, Yadernaya Fizika (1990) V. 51, 1479-1491 [English translation: Sov.J.Nucl.Phys. (1990) V. 51, PP. 935-942]; Z.G.Berezhiani and M.Yu.Khlopov, Yadernaya Fizika (1990) V. 52, 96- 103 [English translation: Sov.J.Nucl.Phys. (1990) V. 52, 60-64]; Z.G.Berezhiani, M.Yu.Khlopov, and R.R.Khomeriki, Yadernaya Fizika (1990) V. 52, 538-543 [English translation: Sov.J.Nucl.Phys. (1990) V.52, 344-347]; Z.G.Berezhiani and M.Yu.Khlopov, Z.Phys.C- Particles and Fields (1991), V. 49, 73-78; A.S.Sakharov and M.Yu.Khlopov, Yadernaya Fizika (1994) V. 57, 690-697. [English translation: Phys.Atom.Nucl. (1994) V. 57, 651-658]. [10] C.D. Froggatt and H.B. Nielsen, Nucl. Phys. B 147 (1979) 277. [11] E. Ma and G. Rajasekaran, Phys. Rev. D 64, 113012 (2001) [arXiv:hep-ph/0106291]; K. S. Babu, E. Ma and J. W. F. Valle, Phys. Lett. B 552, 207 (2003) [arXiv:hep-ph/0206292]; E. Ma, Phys. Rev. D 70, 031901 (2004) [arXiv:hep-ph/0404199]; G. Altarelli and F. Feruglio, Nucl. Phys. B 720, 64 (2005) [arXiv:hep-ph/0504165]; E. Ma, Phys. Rev. D 72, 037301 (2005) [arXiv:hep-ph/0505209]; A. Zee, Phys. Lett. B 630, 58 (2005) [arXiv:hep-ph/0508278]; G. Altarelli and F. Feruglio, Nucl. Phys. B 741, 215 (2006) [arXiv:hep-ph/0512103]; E. Ma, Mod. Phys. Lett.A 21, 2931 (2006) [arXiv:hep-ph/0607190]; S. F. King and M. Malinsky, Phys. Lett. B 645, 351 (2007) [arXiv:hep-ph/0610250]; S. Morisi, M. Picariello and E. Torrente- Lujan, Phys. Rev. D 75, 075015 (2007) [arXiv:hep-ph/0702034]; M. Honda and M. Tanimoto, Prog. Theor. Phys. 119, 583 (2008) [arXiv:0801.0181 [hep-ph]]; G. Altarelli, F. Feruglio and C. Hagedorn, JHEP 0803, 052 (2008) [arXiv:0802.0090 [hep-ph]]; P. H. Frampton and S. Matsuzaki, arXiv:0806.4592 [hep-ph]; F. Feruglio, C. Hagedorn, Y. Lin and L. Merlo, Nucl. Phys. B 809, 218 (2009) [arXiv:0807.3160 [hep-ph]]; P. Ciafaloni, M. Picariello, E. Torrente-Lujan and A. Urbano, Phys. Rev. D 79, 116010 (2009) [arXiv:0901.2236 [hep-ph]]; C. Hagedorn, E. Molinaro and S. T. Petcov, JHEP 0909, 115 (2009) [arXiv:0908.0240 [hep-ph]]; F. Feruglio, C. Hagedorn and L. Merlo, JHEP 1003, 084 (2010) [arXiv:0910.4058 [hep-ph]]; J. Berger and Y. Grossman, JHEP 1002, 071 (2010) [arXiv:0910.4392 [hep-ph]]; F. Feruglio, C. Hagedorn, 9

Y. Lin and L. Merlo, arXiv:0911.3874 [hep-ph]; I. K. Cooper, S. F. King and C. Luhn, Phys. Lett. B 690, 396 (2010) [arXiv:1004.3243 [hep-ph]]; A. Kadosh and E. Pallante, JHEP 1008, 115 (2010) [arXiv:1004.0321 [hep-ph]]. [12] A. Aranda, C. D. Carone and R. F. Lebed, Phys. Lett. B 474, 170 (2000) [arXiv:hep-ph/9910392]; A. Aranda, C. D. Carone and R. F. Lebed, Int. J. Mod. Phys. A 16S1C, 896 (2001) [arXiv:hep-ph/0010144]; A. Aranda, C. D. Carone and R. F. Lebed, Phys. Rev. D 62, 016009 (2000) [arXiv:hep-ph/0002044]. [13] F. Feruglio, C. Hagedorn, Y. Lin and L. Merlo, Nucl. Phys. B 775, 120 (2007) [Erratum-ibid. 836, 127 (2010)] [arXiv:hep-ph/0702194]; M. C. Chen and K. T. Mahanthappa, Phys. Lett. B 652, 34 (2007) [arXiv:0705.0714 [hep-ph]]; A. Aranda, Phys. Rev. D 76, 111301 (2007) [arXiv:0707.3661 [hep-ph]]; P. H. Frampton and T. W. Kephart, JHEP 0709, 110 (2007) [arXiv:0706.1186 [hep-ph]]; M. C. Chen and K. T. Mahanthappa, arXiv:0710.2118 [hep-ph]; S. Sen, Phys. Rev. D 76, 115020 (2007) [arXiv:0710.2734 [hep-ph]]; P. H. Frampton, T. W. Kephart and S. Matsuzaki, Phys. Rev. D 78, 073004 (2008) [arXiv:0807.4713 [hep-ph]]; M. C. Chen and K. T. Mahanthappa, Phys. Lett. B 681, 444 (2009) [arXiv:0904.1721 [hep-ph]]; M. C. Chen, K. T. Mahanthappa and F. Yu, Phys. Rev. D 81, 036004 (2010) [arXiv:0907.3963 [hep-ph]]; L. Merlo, arXiv:1004.2211 [hep-ph]. [14] G. J. Ding, Phys. Rev. D 78, 036011 (2008) [arXiv:0803.2278 [hep-ph]]. [15] E. Ma, Mod. Phys. Lett. A 21, 1917 (2006) [arXiv:hep-ph/0607056]; I. de Medeiros Varzielas, S. F. King and G. G. Ross, Phys. Lett. B 648, 201 (2007) [arXiv:hep-ph/0607045]; C. Luhn, S. Nasri and P. Ramond, J. Math. Phys. 48, 073501 (2007) [arXiv:hep-th/0701188]; E. Ma, Phys. Lett. B 660, 505 (2008) [arXiv:0709.0507 [hep-ph]]; J. A. Escobar and C. Luhn, J. Math. Phys. 50, 013524 (2009) [arXiv:0809.0639 [hep-th]]; [16] C. Luhn, S. Nasri and P. Ramond, Phys. Lett. B 652, 27 (2007) [arXiv:0706.2341 [hep-ph]]; C. Hagedorn, M. A. Schmidt and A. Y. Smirnov, Phys. Rev. D 79, 036002 (2009) [arXiv:0811.2955 [hep-ph]]. [17] Y. Kajiyama, M. Raidal and A. Strumia, Phys. Rev. D 76, 117301 (2007) [arXiv:0705.4559 [hep-ph]]. [18] E. Ma, Phys. Lett. B 632, 352 (2006) [arXiv:hep-ph/0508231]; C. Hagedorn, M. Lindner and R. N. Mohapatra, JHEP 0606, 042 (2006) [arXiv:hep-ph/0602244]; Y. Cai and H. B. Yu, Phys. Rev. D 74, 115005 (2006) [arXiv:hep-ph/0608022]; H. Zhang, Phys. Lett. B 655, 132 (2007) [arXiv:hep-ph/0612214]; Y. Koide, JHEP 0708, 086 (2007) [arXiv:0705.2275 [hep-ph]]; F. Bazzocchi, L. Merlo and S. Morisi, Nucl. Phys. B 816, 204 (2009) [arXiv:0901.2086 [hep-ph]]. Y. Daikoku and H. Okada, Phys. Rev. D 82, 033007 (2010) [arXiv:0910.3370 [hep-ph]]. C. Hagedorn, S. F. King and C. Luhn, JHEP 1006, 048 (2010) [arXiv:1003.4249 [hep-ph]]. R. de Adelhart Toorop, F. Bazzocchi and L. Merlo, JHEP 1008, 001 (2010) [arXiv:1003.4502 [hep-ph]]; G. J. Ding, arXiv:1006.4800 [hep-ph]; Y. Daikoku and H. Okada, arXiv:1008.0914 [hep-ph]. [19] P. F. Harrison and W. G. Scott, Phys. Lett. B 557, 76 (2003) [arXiv:hep-ph/0302025]; J. Kubo, A. Mondragon, M. Mondragon and E. Rodriguez-Jauregui, Prog. Theor. Phys. 109, 795 (2003) [Erratum-ibid. 114, 287 (2005)] [arXiv:hep-ph/0302196]. T. Kobayashi, J. Kubo and H. Terao, Phys. Lett. B 568, 83 (2003) [arXiv:hep-ph/0303084]; S. L. Chen, M. Frigerio and E. Ma, Phys. Rev. D 70, 073008 (2004) [Erratum-ibid. D 70, 079905 (2004)] [arXiv:hep-ph/0404084]; F. Caravaglios and S. Morisi, arXiv:hep-ph/0503234; S. Morisi and M. Picariello, Int. J. Theor. Phys. 45, 1267 (2006) [arXiv:hep-ph/0505113]; Y. Koide, Phys. Rev. D 73, 057901 (2006) [arXiv:hep-ph/0509214]. N. Haba and K. Yoshioka, Nucl. Phys. B 739, 254 (2006) [arXiv:hep-ph/0511108]; N. Haba, A. Watanabe and K. Yoshioka, Phys. Rev. Lett. 97, 041601 (2006) [arXiv:hep-ph/0603116]; R. N. Mohapatra, S. Nasri and H. B. Yu, Phys. Lett. B 639, 318 (2006) [arXiv:hep-ph/0605020]; Y. Koide, Eur. Phys. J. C 50, 809 (2007) [arXiv:hep-ph/0612058]; A. Mondragon, M. Mondragon and E. Peinado, Phys. Rev. D 76, 076003 (2007) [arXiv:0706.0354 [hep-ph]]; C. Y. Chen and L. Wolfen- stein, Phys. Rev. D 77, 093009 (2008) [arXiv:0709.3767 [hep-ph]]; M. Mitra and S. Choubey, Phys. Rev. D 78, 115014 (2008) [arXiv:0806.3254 [hep-ph]]; D. A. Dicus, S. F. Ge and W. W. Repko, Phys. Rev. D 82, 033005 (2010) [arXiv:1004.3266 [hep-ph]]. [20] K. S. Babu and S. Gabriel, arXiv:1006.0203 [hep-ph]. [21] S. F. King and C. Luhn, Nucl. Phys. B 820, 269 (2009) [arXiv:0905.1686 [hep-ph]]. S. F. King and C. Luhn, Nucl. Phys. B 832, 414 (2010) [arXiv:0912.1344 [hep-ph]]; R. Zwicky and T. Fischbacher, Phys. Rev. D 80, 076009 (2009) [arXiv:0908.4182 [hep-ph]]. [22] P. H. Frampton and O. C. W. Kong, Phys. Rev. D 53, 2293 (1996) [arXiv:hep-ph/9511343]; P. H. Frampton and A. Rasin, Phys. Lett. B 478, 424 (2000) [arXiv:hep-ph/9910522]; M. Frigerio, S. Kaneko, E. Ma and M. Tanimoto, Phys. Rev. D 71, 011901 (2005) [arXiv:hep-ph/0409187]; M. Frigerio, arXiv:hep-ph/0505144; K. S. Babu and J. Kubo, Phys. Rev. D 71, 056006 (2005) [arXiv:hep-ph/0411226]; M. Frigerio and E. Ma, Phys. Rev. D 76, 096007 (2007) [arXiv:0708.0166 [hep-ph]]. [23] C. D. Carone and R. F. Lebed, Phys. Rev. D 60, 096002 (1999) [arXiv:hep-ph/9905275]; W. Grimus and L. Lavoura, Phys. Lett. B 572, 189 (2003) [arXiv:hep-ph/0305046]; W. Grimus, A. S. Joshipura, S. Kaneko, L. Lavoura and M. Tanimoto, JHEP 0407, 078 (2004) [arXiv:hep-ph/0407112]; E. Ma, Fizika B 14, 35 (2005) [arXiv:hep-ph/0409288]; S. L. Chen and E. Ma, Phys. Lett. B 620, 151 (2005) [arXiv:hep-ph/0505064]; C. Hagedorn, M. Lindner and F. Plentinger, Phys. Rev. D 74, 025007 (2006) [arXiv:hep-ph/0604265]; Y. Kajiyama, J. Kubo and H. Okada, Phys. Rev. D 75, 033001 (2007) [arXiv:hep-ph/0610072]; F. Feruglio, C. Hagedorn, Y. Lin and L. Merlo, Nucl. Phys. B 775, 120 (2007) [Erratum-ibid. 836, 127 (2010)] [arXiv:hep-ph/0702194]; P. Ko, T. Kobayashi, J. h. Park and S. Raby, Phys. Rev. D 76, 035005 (2007) [Erratum-ibid. D 76, 059901 (2007)] [arXiv:0704.2807 [hep-ph]]; A. Blum, C. Hagedorn and M. Lindner, Phys. Rev. D 77, 076004 (2008) [arXiv:0709.3450 [hep-ph]]; H. Ishimori, T. Kobayashi, H. Ohki, Y. Omura, R. Takahashi and M. Tanimoto, Phys. Lett. B 662 (2008) 178 [arXiv:0802.2310 [hep-ph]]; H. Ishimori, T. Kobayashi, H. Ohki, Y. Omura, R. Takahashi and M. Tanimoto, Phys. Rev. D 77 (2008) 115005 [arXiv:0803.0796 [hep-ph]]; A. Adulpravitchai, A. Blum and C. Hagedorn, JHEP 0903, 046 (2009) [arXiv:0812.3799 [hep-ph]]; C. Hagedorn and R. Ziegler, arXiv:1007.1888 [hep-ph]. [24] L. L. Everett and A. J. Stuart, Phys. Rev. D 79, 085005 (2009) [arXiv:0812.1057 [hep-ph]]. [25] A. Adulpravitchai, A. Blum and W. Rodejohann, New J. Phys. 11, 063026 (2009) [arXiv:0903.0531 [hep-ph]]. [26] A. Datta, F. S. Ling and P. Ramond, Nucl. Phys. B 671, 383 (2003) [arXiv:hep-ph/0306002]. 10

[27] C. S. Chen, T. W. Kephart and T. C. Yuan, arXiv:1011.3199 [hep-ph]. [28] G. Altarelli and F. Feruglio, New J. Phys. 6, 106 (2004) [arXiv:hep-ph/0405048]; R. N. Mohapatra et al., Rept. Prog. Phys. 70, 1757 (2007) [arXiv:hep-ph/0510213]; A. Strumia and F. Vissani, arXiv:hep-ph/0606054. E. Ma, arXiv:0705.0327 [hep-ph]. G. Altarelli, arXiv:0705.0860 [hep-ph]. G. Altarelli, arXiv:0711.0161 [hep-ph]. F. Feruglio, C. Hagedorn, Y. Lin and L. Merlo, arXiv:0808.0812 [hep-ph]; G. Altarelli, arXiv:0905.2350 [hep-ph]. G. Altarelli, Nuovo Cim. C 32N5-6, 91 (2009) [arXiv:0905.3265 [hep-ph]]; E. Ma, arXiv:0908.1770 [hep-ph]; S. F. King, AIP Conf. Proc. 1200, 103 (2010) [arXiv:0909.2969 [hep-ph]]; G. Altarelli and F. Feruglio, arXiv:1002.0211 [hep-ph]; H. Ishimori, T. Kobayashi, H. Ohki, H. Okada, Y. Shimizu and M. Tanimoto, Prog. Theor. Phys. Suppl. 183, 1 (2010) [arXiv:1003.3552 [hep-th]]; M. Zralek, Acta Phys. Polon. B 41, 1477 (2010). [29] C. H. Albright, A. Dueck and W. Rodejohann, arXiv:1004.2798 [hep-ph]. [30] P. O. Ludl, arXiv:0907.5587 [hep-ph]; W. Grimus and P. O. Ludl, J. Phys. A 43, 445209 (2010) [arXiv:1006.0098 [hep-ph]].; P. O. Ludl, J. Phys. A 43, 395204 (2010) [arXiv:1006.1479 [math-ph]]. [31] P. H. Frampton and O. C. W. Kong, Phys. Rev. Lett. 75, 781 (1995) [arXiv:hep-ph/9502395]; A. Blum, C. Hagedorn and A. Hohenegger, JHEP 0803, 070 (2008) [arXiv:0710.5061 [hep-ph]]; A. Blum and C. Hagedorn, Nucl. Phys. B 821, 327 (2009) [arXiv:0902.4885 [hep-ph]]. [32] R. Barbieri, G. R. Dvali and L. J. Hall, Phys. Lett. B 377, 76 (1996) [arXiv:hep-ph/9512388]; R. Barbieri, L. J. Hall, S. Raby and A. Romanino, Nucl. Phys. B 493, 3 (1997) [arXiv:hep-ph/9610449]; A. Romanino, arXiv:hep-ph/9701349. R. Barbieri, L. J. Hall and A. Romanino, Phys. Lett. B 401, 47 (1997) [arXiv:hep-ph/9702315]; C. D. Carone and L. J. Hall, Phys. Rev. D 56, 4198 (1997) [arXiv:hep-ph/9702430]; M. Tanimoto, Phys. Rev. D 57, 1983 (1998) [arXiv:hep-ph/9706497]; G. Eyal, Phys. Lett. B 441, 191 (1998) [arXiv:hep-ph/9807308]; L. J. Hall and N. Weiner, Phys. Rev. D 60, 033005 (1999) [arXiv:hep-ph/9811299]; R. Barbieri, L. Giusti, L. J. Hall and A. Romanino, Nucl. Phys. B 550, 32 (1999) [arXiv:hep-ph/9812239]; R. Barbieri, P. Creminelli and A. Romanino, Nucl. Phys. B 559, 17 (1999) [arXiv:hep-ph/9903460]; T. Blazek, S. Raby and K. Tobe, Phys. Rev. D 62, 055001 (2000) [arXiv:hep-ph/9912482]; A. Aranda, C. D. Carone and P. Meade, Phys. Rev. D 65, 013011 (2002) [arXiv:hep-ph/0109120]; S. Gopalakrishna and C. P. Yuan, arXiv:hep-ph/0402096. Y. Kobashi and A. Sugamoto, arXiv:hep-th/0407118. S. Gopalakrishna and C. P. Yuan, Phys. Rev. D 71, 035012 (2005) [arXiv:hep-ph/0410181]; Z. Han, Phys. Rev. D 73, 015005 (2006) [arXiv:hep-ph/0510125]; T. Goto, Y. Okada, T. Shindou and M. Tanaka, Phys. Rev. D 77, 095010 (2008) [arXiv:0711.2935 [hep-ph]]. [33] P. F. Harrison, D. H. Perkins and W. G. Scott, Phys. Lett. B 530, 167 (2002) [arXiv:hep-ph/0202074]. [34] L. L. Everett and A. J. Stuart, in preparation. [35] N. B. Backhouse and P. Gard, J. Phys. A 7 2101 (1974). [36] C. J. Cummins and J. Patera, J. Math. Phys. 29, 1736 (1988). [37] K. Shirai, J. Phys. Soc. Jpn. 61 2735 (1992). [38] R. B. Hoyle, Physica D 191, 261(2004). [39] C. Luhn, S. Nasri and P. Ramond, J. Math. Phys. 48, 123519 (2007) [arXiv:0709.1447 [hep-th]]; C. Luhn and P. Ramond, J. Math. Phys. 49, 053525 (2008) [arXiv:0803.0526 [hep-th]]. [40] H. Coxeter and W. Moser, Generators and Relations for Discrete Groups, Springer-Verlag, Berlin (1957). [41] J. S. Lomont, Applications of Finite Groups, Academic Press, New York (1959). [42] M. Hamermesh, Group Theory and its Applications to Physical Problems, Addison-Wesley, Reading (1964). [43] P. Ramond, Group Theory: A Physicist’s Survey, Cambridge University Press (2010).