Fermion Masses and Mixings from Dihedral Flavor Symmetries with Preserved Subgroups
aa r X i v : . [ h e p - ph ] S e p Fermion Masses and Mixings from Dihedral Flavor Symmetrieswith Preserved Subgroups
A. Blum ∗ , C. Hagedorn † , and M. Lindner ‡ Max-Planck-Institut f¨ur KernphysikPostfach 10 39 80, 69029 Heidelberg, Germany
We perform a systematic study of dihedral groups used as flavor symmetry. The key feature here isthe fact that we do not allow the dihedral groups to be broken in an arbitrary way, but in all casessome (non-trivial) subgroup has to be preserved. In this way we arrive at only five possible (Dirac)mass matrix structures which can arise, if we require that the matrix has to have a non-vanishingdeterminant and that at least two of the three generations of left-handed (conjugate) fermions areplaced into an irreducible two-dimensional representation of the flavor group. We show that thereis no difference between the mass matrix structures for single- and double-valued dihedral groups.Furthermore, we comment on possible forms of Majorana mass matrices. As a first application wefind a way to express the Cabibbo angle, i.e. the CKM matrix element | V us | , in terms of grouptheory quantities only, the group index n , the representation index j and the index m u,d of thedifferent preserved subgroups in the up and down quark sector: | V us | = ˛˛˛ cos “ π ( m u − m d ) j n ”˛˛˛ whichis | cos( π ) | ≈ . n = 7, j = 1, m u = 3 and m d = 0. We prove that two successful modelswhich lead to maximal atmospheric mixing and vanishing θ in the lepton sector are based onthe fact that the flavor symmetry is broken in the charged lepton, Dirac neutrino and Majorananeutrino sector down to different preserved subgroups whose mismatch results in the prediction ofthese mixing angles. This also demonstrates the power of preserved subgroups in connection withthe prediction of mixing angles in the quark as well as in the lepton sector. I. INTRODUCTION
Experiments have shown that the quark and lepton mix-ing angles are completely different in size and hierarchy.They are small in the quark sector θ q ≈ . ◦ , θ q ≈ . ◦ , θ q ≈ . ◦ with the Cabibbo angle θ q being the largest [1], while inthe lepton sector two of them are large, if not maximal, θ l ≈ ◦ , θ l ≈ . ◦ , together with the third angle θ l being smaller than 9 . ◦ at the 2 σ level [2].From the viewpoint of model building two special struc-tures for the lepton mixing angles are very interesting:a.) the case of tri-bimaximal (TBM) mixing [3] and b.)the case of µτ symmetry (MTS) [4] in the neutrino sec-tor. For tri-bimaximal mixing the sines of the mixingangles are given bysin ( θ l ) = 13 , sin ( θ l ) = 12 and sin ( θ l ) = 0while µτ symmetry only enforcessin ( θ l ) = 12 and sin ( θ l ) = 0 ∗ [email protected] † [email protected] ‡ [email protected] leaving the angle θ l undetermined. Both ansaetze forthe mixing angles are compatible with the best fit valuesat the 2 σ level.The Standard Model (SM) can only accommodate, butnot explain these data. The possible special structure forthe lepton mixing matrix together with the hierarchy ofthe quark mixing angles is quite a strong hint for a flavorsymmetry G F which is broken in a non-trivial way.Requiring that at least two of the three fermion genera-tions can be unified by G F and avoiding extra Goldstoneor gauge bosons from breaking G F spontaneously leadsus to the conclusion that the best choice for G F is a dis-crete , non-abelian symmetry.Recently, in a series of papers [5] it has been shown thatthe discrete symmetry A , which is the symmetry groupof even permutations of four distinct objects, is able toexplain TBM mixing in the lepton sector, if it acts onthe fermion generations in the following way: the threeleft-handed lepton doublets transform as the irreduciblethree-dimensional representation of A , while the left-handed conjugate charged leptons transform as the threesinglets of A , , and (also called 1, 1 ′ and 1 ′′ ).In reference [6] this phenomenologically successful modelhas been extended to the quark sector by using the sym-metry group T ′ , which is the double-valued group of A .In the quark sector it allows for some non-trivial connec-tions among the CKM matrix elements and the quarkmasses.The key point of these studies lies in the fact that theHiggs fields which necessarily also transform non-triviallyunder the flavor group in order to form A ( T ′ ) invariantYukawa couplings do not acquire arbitrary vacuum expec-tation values (VEVs), but VEVs which break the flavor group A ( T ′ ) down to certain non-trivial subgroups , suchas Z , Z and Z . The fields which preserve a Z (in caseof A , Z in case of T ′ ) only couple to neutrinos at theleading order, while the other fields breaking the flavorsymmetry of the model down to a Z subgroup only cou-ple to charged fermions at this level. This exactly leadsto TBM mixing and explains large lepton mixing anglesas the result of two different subgroups, whereas smallquark mixing angles come from mass matrices which areinvariant under the same subgroup.These results trigger the following questions: Are A andits double-valued group T ′ the only groups in which suchresults can be achieved? Or are there other groups amongthe non-abelian discrete symmetries which can lead tothe same or to a similar result by preserving some non-trivial subgroup?In order to answer these questions at least partly, we in-vestigate in this paper the dihedral groups D n and theirdouble-valued groups D ′ n as flavor symmetries [7] . Theyare non-abelian for n > n >
1, respectively. Thegroup D n is the symmetry group of a regular planar n -gon. The dihedral groups and their double-valued groupsare well suited for a general study, since they form a seriesof groups with similar properties, e.g. they all containonly one- and two-dimensional representations. Amongthe discrete groups there are several such series of groups:the permutation groups S n of n distinct objects, the al-ternating groups A n of even permutations of n objects,and the two series of subgroups of SU (3), ∆(3 n ) and∆(6 n ) [8]. The groups S n and A n are only interest-ing for small n ( S , S , S and A , A , A ), since withincreasing n the dimension of their non-trivial represen-tations increases beyond three. So they are only appro-priate for case by case studies and not for a general studylike ours. ∆(3 n ) and ∆(6 n ) are more similar to thedihedral groups and therefore more interesting, but alsoless known in particle theory [9].In our general study we test whether the groups D n and D ′ n can be used to get a similar result as in the men-tioned A and T ′ model, i.e. whether one can inducecertain mixing patterns in the lepton and/or quark sec-tor by breaking to different subgroups. In order to doso, we first need to establish the group theory of thegroups D n and D ′ n , in particular we carefully study allpossible subgroups. We show for each of them all rep-resentations and directions which leave them invariant.In a next step we calculate all possible mass matriceswhich can then arise in such a model. Thereby the gaugegroup is taken to be the one of the SM for simplicity.In order to keep the calculation tractable we restrict our-selves to mass matrices with a non-vanishing determinantand for Dirac mass matrices we assume in a first stepthat all Higgs fields which transform under the flavorsymmetry are copies of the SM Higgs doublet. Further-more we do not discuss cases in which all left-handed andleft-handed conjugate fields transform as singlets underthe flavor group, since such structures can be producedby the use of an abelian group as well. Note that this does not exclude the possibility that the Majorana massmatrix originates from a coupling which involves onlyfields transforming as one-dimensional representation un-der the flavor group, as the Majorana mass term stemssolely from the coupling of two either left-handed or left-handed conjugate fields. We present all mass matricesand discuss differences among Dirac and Majorana massmatrix structures.As one interesting application we show that a predictionof the Cabibbo angle θ q , more precisely of one of theelements of the quark mixing matrix V CKM , in terms ofonly group theoretical quantities, i.e. the index n of thegroup D n ( D ′ n ), the index of the representation and thebreaking direction in the flavor space, becomes possible.Several smaller dihedral groups [10] and their double-valued groups [11] were already used in the literature toconstruct models with flavor symmetries. We commenton some of them by comparing the mass matrices theyuse with ours. We show that the prediction of θ l =45 ◦ and θ l = 0 of models using a flavor symmetry D × Z aux ) [12] and D × Z aux ) [13] results from the factthat in these models non-trivial subgroups of D and D are preserved, respectively.We only briefly touch the question of the VEV alignmentand the choice and stabilization of the desired vacuumstructure in our conclusions, since the study of the Higgspotentials is beyond the scope of this paper. Neverthe-less, we emphasize that of course only the proof that anadvocated VEV structure is realized in a certain poten-tial and the proof of its stability can make the theoryviable. A detailed study of potentials of SM Higgs dou-blets which transform under D n or D ′ n will be presentedelsewhere [14].The paper is structured as follows: in Section II wepresent the basic group theory of the dihedral groups D n and D ′ n ; Section III contains the analysis of the sub-groups of D n and D ′ n ; in Section IV we study the stepwisebreaking of the dihedral symmetries and show all possi-ble breaking chains for the single-valued groups D n . Westudy in Section V all Dirac as well as Majorana mass ma-trices with a non-vanishing determinant which can arisefrom the distinct breakings found in Section III and men-tion some possible applications in Section VI. A compar-ison of our findings to the literature is given in SectionVII. Finally, we conclude in Section VIII and commenton differences among models using flavor-charged Higgsdoublets and gauge singlets (flavons) as well as on somesimple Higgs potentials. The various appendices containfurther group theoretical results which are needed forour calculations such as Kronecker products and Cleb-sch Gordan coefficients as well as the decomposition ofrepresentations of the dihedral groups into representa-tions of their subgroups and the breaking chains for thegroups D ′ n . II. PROPERTIES OF DIHEDRAL GROUPSA. Single-valued Groups D n All D n groups are non-abelian apart from D ( ∼ = Z )and D ( ∼ = Z × Z ). They only contain real one- andtwo-dimensional irreducible representations. If its index n is even, the group D n has four one- and n − n being odd D n hastwo one- and n − two-dimensional representations. Inthe following we denote the one-dimensional representa-tions with and the two-dimensional ones with wherethe indices i and j are i = 1 , , ..., n − for D n with n odd and i = 1 , ... , .., n − D n with n even. The representation is always the trivial one,i.e. the one whose characters are 1 for all classes . Theorder of the group D n is 2 n . The generators A and B ofthe one-dimensional representations are A=B=1 for and A=1, B=-1 for . For even n we also have A=-1,B=1 for and A=B=-1 for . The generators of thetwo-dimensional representations are [15]:A = e( πin ) j
00 e − ( πin ) j ! , B = (cid:18) (cid:19) (1)with j = 1 , . . . , n − n even and j = 1 , . . . , n − for n odd. They fulfill the relations:A n = , B = , ABA = B . (2)Note that we have chosen complex generators for thetwo-dimensional representations. Since the representa-tions themselves are real, there exists a unitary matrix U which links their generators to the complex conjugates: U = (cid:18) (cid:19) . For any (cid:18) a a (cid:19) ∼ the combination U (cid:18) a ⋆ a ⋆ (cid:19) = (cid:18) a ⋆ a ⋆ (cid:19) transforms as instead of (cid:18) a ⋆ a ⋆ (cid:19) ,as would be the case for real generators A and B. B. Double-valued groups D ′ n The groups D ′ n are the double-valued counterparts of thegroups D n . All groups D ′ n with n > D ′ is isomorphic to Z . The simplest non-abelian double-valued dihedral group, D ′ , is also called the quaterniongroup. Hence, one often uses the notation Q n insteadof D ′ n . The group D ′ n is of order 4 n . Similar to the The character of a representation for a certain group elementis just the trace of the corresponding representation matrix in-dependent of the choice of basis. For one-dimensional repre-sentations the representation matrix is only a complex number(unequal zero) which is then also the character of this represen-tation. groups D n they only contain one- and two-dimensionalirreducible representations. The group D ′ n has four one-dimensional and n − n even, the one-dimensional representations are real,i.e. their characters are real. Furthermore the two-dimensional representations with j even are real, i.e.not only their characters are real, but there also existsa set of real representation matrices. In contrast to thisthe representations with j odd are pseudo-real, i.e.their characters are real, but one cannot find a set ofrepresentation matrices which are also real. If n is oddthe one-dimensional representations are real, while and are complex conjugated to each other. Asfor n even, the representations with j even are realand with j odd are pseudo-real. Compared to the groups D n one has to add the pseudo-real and complex repre-sentations to get D ′ n . Therefore the real representationsare usually called even, while the pseudo-real and com-plex ones are named odd representations [16]. The gen-erators for the one-dimensional representations are thesame as for D n , if n is even. If n is odd, the generatorsof the one-dimensional representations are A=B=1 for , A=1, B=-1 for , A=-1, B= − i for and A=-1,B= i for . The generators and their relations for thetwo-dimensional representations also have a similar formas in case of D n [16, 17]:A = e( πin ) j
00 e − ( πin ) j ! , B = (cid:18) (cid:19) (3)for j even, andA = e( πin ) j
00 e − ( πin ) j ! , B = (cid:18) ii (cid:19) (4)for j odd. They fulfill:A n = R , B = R , R = , ABA = B , (5)with R being in case of an even representation and − for an odd one. Comparing the generators of even andodd representations one recognizes that the generator Bcontains an extra factor i for odd representations. Sincealso for D ′ n all two-dimensional representations are realor pseudo-real, i.e. not complex, there has to exist asimilarity transformation U between the representationmatrices and their complex conjugates. If the index j ofthe representation is even, U is the same as for therepresentations of D n . For j being odd, U = (cid:18) −
11 0 (cid:19) such that (cid:18) − a ⋆ a ⋆ (cid:19) = U (cid:18) a ⋆ a ⋆ (cid:19) transforms in the sameway as (cid:18) a a (cid:19) ∼ with j odd. III. NONTRIVIAL SUBGROUPS
In this section we determine the subgroups of a general D n or D ′ n group, using the generators given in SectionII. This can be done systematically by determining foreach representation the eigenvalues and eigenvectors ofthe group elements. All group elements, which have thesame eigenvector corresponding to the eigenvalue of 1,form a subgroup. We then determine the group struc-ture, which is simple, as all subgroups turn out to beeither dihedral or cyclic.For the one-dimensional representations we only need tolook at which elements of the group are represented bya 1 - these elements then form a subgroup. For the two-dimensional representations, we need to determine allrepresentation matrices which have an eigenvalue of 1.We only need to consider two general matrices: A x andBA y and calculate their eigenvalues as a function of x or y and the index j of the representation , and then cal-culate the corresponding eigenvectors. All eigenvectorswith eigenvalue 1 turn out to have the same structure: (cid:18) e − πi j mg (cid:19) (6)where g is the order of G F (2 n for single-valued, 4 n fordouble-valued groups) and m is an integer. We thus havefor a given two-dimensional representation a class of sub-groups, parameterized by m , where one of the generatorsof the subgroup will be BA m . For group elements rep-resented by the unit matrix, which appear in unfaithfulrepresentations , an arbitrary eigenvector correspondsto an eigenvalue of 1. Hence we have for unfaithfultwo-dimensional representations an additional subgroupmade up of all group elements represented by the unitmatrix. To make sure that we have determined all sub-groups, we need to consider possible combinations of 2 ormore representations. Further subgroups will necessarilybe subgroups of the subgroups determined above. Sinceall of the subgroups encountered so far are either dihe-dral or cyclic, we know that all further subgroups willalso be either dihedral or cyclic. As it turns out, we needat most 2 different representations to reach any possiblesubgroup of our original D n or D ′ n .We come to a physical interpretation of our results byusing them to determine how the VEV of a scalar fieldtransforming non-trivially under G F will break that sym-metry. A VEV of a scalar transforming under a givenrepresentation conserves the subgroup of elements whichleave the VEV invariant, i.e. the VEV is an eigenvec-tor to the eigenvalue 1 and these we determined above.For two-dimensional representations there can be severalsubgroups and therefore the structure of the scalar VEV A representation is unfaithful if the number of distinct represen-tation matrices is smaller than the order of the group. is important. We denote an arbitrary VEV by < > ,while a VEV proportional to the eigenvector of Eq.(6)will be denoted by < > ′ . Subgroups corresponding toa combination of two representations will be conservedby a combination of VEVs. We get the following resultsfor D n groups: D n < > −→ D n D n < > −→ Z n = < A >D n < > −→ D n = < A , B >D n < > −→ D n = < A , BA >D n < > −→ Z j = < A n j > (j | n ) D n < > −→ nothing (j ∤ n ) D n < > ′ −→ D j = < A n j , BA m > (j | n ; m = 0 , , ..., n j − D n < > ′ −→ Z = < BA m > (j ∤ n ; m = 0 , , ..., n − D n < > + < > −→ Z n = < A > (One can also use < > + < > or < > + < > .) D n < > + < > ′ −→ Z = < BA m > (j | n, n j odd ; 0 ≤ m ≤ n j − m even for ; for m odd) and for D ′ n groups we get: D ′ n < > −→ D ′ n D ′ n < > −→ Z n = < A >D ′ n < > or < > −→ Z n = < A > ( for odd n ) D ′ n < > −→ D ′ n = < A , B > ( for even n ) D ′ n < > −→ D ′ n = < A , BA > ( for even n ) D ′ n < > −→ Z j = < A n j > (j | n ) D ′ n < > −→ nothing (j ∤ n ) D ′ n < > ′ −→ D ′ j2 = < A n j , BA m > (j even, j | n ; m = 0 , , ..., n j − D ′ n < > ′ −→ Z = < BA m > (j even, j ∤ n ; m = 0 , , ..., n − D ′ n < > + < > −→ Z n = < A > ( for even n )(One can also use < > + < > or < > + < > .) D ′ n < > + < > ′ −→ Z = < BA m > ( n even ; j | n, n j odd ; 0 ≤ m ≤ n j − m even for ; for m odd) D ′ n < > + < > ′ −→ Z = < A n > ( n even ; n j odd ; 0 ≤ m ≤ n j − m odd for ; for m even) Some of these results can also be found in references[18] and [19]. We now know the minimal VEV struc-ture needed to break G F down to a given subgroup. Themaximal VEV structure preserving that subgroup is thenachieved by allowing VEVs for all representations, whichhave at least one component transforming trivially underthe subgroup in question. Therefore we list the transfor-mation properties of the representations of our originaldihedral group under a given subgroup. These can befound by expressing the generators of the subgroup interms of the generators A and B of G F . As a generalfeature we remark that two-dimensional representationsbecome reducible in several cases. The complete list ofdecompositions is given in Appendix B along with themaximal VEV structure. IV. BREAKING CHAINS
We have seen, that a dihedral group will in general haveseveral nontrivial subgroups. We consider all possiblebreaking patterns of a dihedral group, where the symme-try breaking happens in an arbitrary number of steps.The first step in every chain, will be one of the break-ings induced by a single VEV, which we considered inSection III. For the next step we need to consider howa second (different) VEV will further break our symme-try group, by considering the intersection of the groupelements leaving both VEVs invariant, thereby findingthe subgroup which is conserved by both VEVs. We de-termine the group structure of these elements - if thenew subgroup is in fact smaller than the old one, wehave found a viable next step in the breaking sequence.This procedure is then iterated until we reach either a Z (which has no further subgroups, being the smallestnontrivial group), or until we reach a Z j with arbitraryindex j. Z j can always be further broken down to a Z k ,where k is a divisor of j, as we discuss below.We use the notation m j for the phase factor m in theVEV < > ′ . This classification according to differentphase factors becomes important if a breaking sequencecontains VEVs of two distinct two-dimensional represen-tations.Note the breaking patterns marked with a star. Theseare not just breaking sequences in their own right, butalso can be used as building blocks within or at the endof other sequences. In general, we can reduce the orderof a dihedral or cyclic group step by step, until we havereached the subgroup which we want to conserve, as longas the conditions given for the starred breaking sequencesare fulfilled at each step.We find two paths in the breaking sequences, one alongthe dihedral groups and one along the cyclic groups, inthe minimal case eliminating one prime divisor of the or-der in each step. At any point in the sequence we canstep over from the dihedral to the cyclic path (from whichthere is of course no turning back). The cyclic path endsat Z q , q being the smallest prime factor of n , while thedihedral path, will end at D or D ′ as these groups arenontrivial. D ∼ = Z is simple, while D ′ ∼ = Z , is not justnontrivial but also non-simple, so that we can break onestep further down to the Z group generated by A n .The breaking chains for D n are: D n < > −→ Z n < > −→ Z n < > −→ Z j (j | n ) D n < > −→ Z n < > −→ Z j D n < > −→ D n < > −→ Z n < > −→ Z j (j | n ) D n < > −→ D n < > −→ Z n < > −→ Z j (j | n ) D n < > −→ D n < > −→ Z j (j | n ) D n < > −→ D n < > ′ −→ D j < > −→ Z k (j | n ; k | j)( m j even for and for m j is odd) D n < > −→ D n < > ′ −→ D j < > ′ −→ Z (j | n ; k ∤ j; m j = m k even ) D n < > −→ D n < > ′ −→ Z (j ∤ n )( m j even for and for m j is odd) D n < > ′ −→ D j < > −→ Z j ( m j arbitrary ) D n < > ′ −→ D j < > −→ Z (j ∤ n )( m j even for and for m j is odd) D n < > ′ −→ D j < > −→ Z k (k | j; m j arbitrary ) D n < > ′ −→ D j < > ′ −→ Z (k ∤ j; m j = m k ) D n < > ′ −→ Z ∗ D n < > −→ Z j < > −→ Z k (k | j) ∗ D n < > ′ −→ D j < > ′ −→ D k (k | j; m j = m k ) The corresponding results for D ′ n are given in AppendixC. The conditions on the indices can be divided into twotypes: One concerns the divisibility of indices. These ap-pear, when we need to ensure, that we have not brokentoo far, i.e. that the subgroup we want to break to is ac-tually contained in the subgroup we have already brokento. The second type of condition concerns restrictionson the phase factors m j . These appear, if the directionin which we have broken is important. Several break-ing directions occur, if we deal with distinct subgroupsshowing the same structure - for example in D n we havetwo D n subgroups: < A , B > and < A , BA > . The Z subgroup generated by BA m , for example, is only a sub-group of the first in case of m even and only a subgroupof the second, if m is odd. Hence we need to impose re-strictions on the phase factor m .Finally, note that in any of the chains given above we caninterchange and and again receive a viable break-ing sequence. This may cause some of the requirementsto change, as shown along with the breaking chains. V. MASS MATRICES
We can determine the Dirac mass matrices M which aregenerated when G F is broken to one of the subgroups wedetermined in Section III by Higgs bosons transformingnon-trivially under G F . Majorana mass matrices are dis-cussed in Section V F. For simplicity we assume that theHiggs bosons are SU (2) L doublets like the one in the SM.The group theoretical tools we need, i.e. the Kroneckerproducts and Clebsch Gordan coefficients, are given inAppendix A. As mentioned in the Introduction, we re-strict ourselves in a fairly major way: We do not considermass matrices with a zero determinant, that is with atleast one zero eigenvalue.We need to decide how the (SM) fermions will trans-form under G F . Their transformation properties are notlimited by our choice of subgroup. The only limitationwe impose here, is that we do not want all fermions totransform under one-dimensional representations of G F ,since the resulting structures could also be obtained byan abelian G F . Therefore we will not allow for this pos-sibility (for Dirac mass terms) , leaving us with twogeneral options for the transformation properties of the(SM) fermions. The first possibility is L ∼ ( , , ) , L c ∼ ( , )which we call the three singlet structure. The secondpossibility is L ∼ ( , ) , L c ∼ ( , )which we call the two doublet structure. Note, thatwe do not discuss explicitly the case where the left-handed fermions transform under one one- and one two-dimensional representation with the left-handed conju-gate fermions transforming under three one-dimensionalrepresentations, since we only need to transpose the massmatrices of the three singlet structure to switch the trans-formation properties of left-handed and left-handed con-jugate fermions.The mass matrices arise, when the Higgs bosons acquire aVEV. We do not choose their transformation properties,as we did for the fermion fields. In fact, we do not evenlimit the number of Higgs bosons. In this way the massmatrix structure is entirely determined by the propertiesof G F and its subgroup, and not by our choice of scalarfields. When determining the mass matrix generated bybreaking down to a certain subgroup, we reference Ta-ble III to Table VI in Appendix B and determine whichrepresentations are allowed a VEV, while keeping thatsubgroup intact. We then start by assuming that ourmodel contains a Higgs boson for each of these possiblerepresentations and that all of them acquire a VEV,with a structure conserving the relevant subgroup. We For the case of Majorana mass terms: see below. We do not need to consider the case of two Higgs bosons trans-forming under the same representation of the flavor group: Thesetwo Higgs bosons would have identical quantum numbers, andonly the linear combination acquiring a VEV would appear inthe mass matrices. However, two Higgs fields transforming in thesame way can be very important for other aspects of a model,especially when discussing the Higgs potential, as shown in ref-erence [20]. can then easily eliminate Higgs bosons from our model,by setting their VEVs to zero in the mass matrix.We first give our general results in Section V A. In Sec-tion V B we give the conventions and notation we usein Section V C and Section V D, where we discuss thethree singlet and two doublet structures, respectively, forsingle-valued groups D n . The resulting mass matriceswill be discussed subgroup by subgroup. In Section V Ewe discuss why no new Dirac mass matrix structures ap-pear for double-valued groups. Finally, we present thepossible Majorana mass matrices in Section V F. A. General Results
We encounter a very limited number of distinct Diracmass matrix structures, i.e. in total only five distinctstructures are possible. We display them for down-typefermions (down-type quarks and charged leptons). Thechanges for up-type fermions, i.e. up-type quarks andneutrinos, are discussed in Section V B . The first pos-sible structure is a diagonal mass matrix A B
00 0 C (7)The second type are semi-diagonal mass matrices, of theform A B C (8)where the squared mass matrix M M † has eigenvalues | A | , | B | and | C | . We also encounter a block matrixstructure A B C D E (9)for which the squared mass matrix has the characteristicpolynomial ( λ − | A | )( λ − λ ( | B | + | C | + | D | + | E | ) +( | B | + | C | )( | D | + | E | ) − | BD ∗ + CE ∗ | ). These threestructures are so common, that we will not give themexplicitly each time. Instead, we will give the type ofmatrix, followed by a listing of the nonzero entries, wherewe use the notation and nomenclature introduced here.Note, that in some cases the entries in the 2-3-submatrixare further correlated, e.g. there exist equalities among We have to distinguish these two cases, since our Higgs fields arealways assumed to transform as the SM Higgs doublet. In theSM the field H itself couples to the down-type fermions, whileits conjugate ǫ H ⋆ is coupled to up-type ones. This differenceis relevant here, since, for example, we decided to use complexgenerators for the two-dimensional representations. them.Finally, we find two structures, which appear only once,both times for the subgroups Z = < BA m > . One hasone texture zero and the general structure C − Ce − iφ k A D De − iφ k B E Ee − iφ k (10)with the group theoretical phase φ = 2 πmn . (11)In this case the squared mass matrix has the characteris-tic polynomial ( λ − | C | )( λ − λ ( | A | + | B | + 2( | D | + | E | )) + ( | A | + 2 | D | )( | B | + 2 | E | ) − | AB ∗ + 2 DE ∗ | ).This mass matrix only shows up for a three singlet struc-ture. The other mass matrix structure has no texturezeros and is of the form A C Ce − iφ k B D EBe − iφ j Ee i (k − j) φ De − i (j+k) φ (12)where the squared mass matrix has the characteristicpolynomial ( λ −| D − Ee i k φ | )( λ − ( | A | +2 | B | +2 | C | + | D + Ee i k φ | ) λ + ( | A | + 2 | C | )(2 | B | + | D + Ee i k φ | ) − | AB ∗ + C ( D ∗ + E ∗ e − i k φ ) | ). This mass matrix only ap-pears with a two doublet structure.In all cases the characteristic polynomial can be factor-ized into a linear and a quadratic one in λ .We additionally find one case, where all matrix elementsare distinct. We do not consider it, as it corresponds toa smaller symmetry being fully broken: D n breaks downto Z q , where for all two-dimensional representations showing up in the model j is a multiple of q , i.e. j = c j q ,c j an integer. We can then replace the original D n sym-metry by a D nq and all the representations by j , asthey are in fact all unfaithful representations of the origi-nal D n symmetry. Breaking the original symmetry downto Z q then corresponds to fully breaking the smaller sym-metry . As we want to consider conserved subgroups,we dismiss this case, when it shows up.Note, that such a case still allows for some non-trivial cor-relations among the mass matrix elements. Nevertheless,they are then not determined by a preserved subgroup,but only by the fact that we use a non-abelian symmetry. To be more precise: In all cases the representations of the Higgsfields have the property that their index is divisible by q , butthis is not necessarily true for the representations under whichthe fermions transform. However, one can then always find someother representations for the fermions which reproduce exactlythe same matrix structure and which have the property that alsotheir index is divisible by q such that the case can be reduced toa smaller symmetry which is fully broken. B. Conventions and Notation κ ’s and α ’s denote Yukawa couplings, < φ i > denotesthe VEV of the Higgs field transforming as . For theVEVs of Higgs fields transforming as we have two pos-sibilities: Either they are allowed to acquire an arbitraryVEV, in which case we denote the VEV by (cid:18) < ψ >< ψ > (cid:19) or they are allowed only a certain VEV structure. Asdiscussed in Section III there is only one such structure.We write the VEV of Higgs fields transforming as andacquiring this VEV structure as < ψ j > (cid:18) e − πi j mn (cid:19) Note that the VEV structure only determines the relativephase between the two doublet components, so that weare in general free to decide which component we wantto include the phase factor in. We will on several occa-sions want to make use of this freedom, to simplify theappearance of our mass matrix. For example, if we havea left-handed fermion transforming as or and theleft-handed conjugate fermions transforming as , weneed a Higgs boson transforming as n -j) to form aninvariant Yukawa coupling. We then write the VEVs ofthese Higgs fields as < ψ n − j > (cid:18) ( − m e − πi j mn (cid:19) In this way, only the phase factor e − πi j mn shows up in themass matrix and not its complex conjugate. Similarlythe VEV of Higgs fields transforming as n -(j+k)) iswritten as < ψ n − (j+k) > (cid:18) e − πi (j+k) mn (cid:19) so that it contains the same phase as that of the Higgsfields transforming as .To get the corresponding mass matrices for up-typefermions a few changes have to be implemented. Ob-viously, they have different Yukawa couplings, i.e.: κ → κ u and α → α u (13)Furthermore all VEVs need to be complex conjugated.This corresponds to the following substitutions: < φ i > → < φ i > ∗ (14) < ψ > → < ψ > ∗ and < ψ > → < ψ > ∗ or < ψ j > → < ψ j > ∗ e πi j mn These changes are only necessary, if up-type fermionscouple to Higgs bosons which transform in the same wayunder SU (2) L × U (1) Y as the Higgs bosons which cou-ple to the down-type fermions, like it is in the SM. Ifthey couple to Higgs bosons with other transformationproperties under SU (2) L × U (1) Y , as for example in theMinimal Supersymmetric Standard Model (MSSM) withthe Higgs doublets h u and h d , such changes are obsoleteand obviously then also the VEVs entering the up- anddown-type fermion mass matrices can never be the same.We have in general left all minus signs and phases in themass matrices, even if they can be trivially rotated away. C. Three Singlet Structure
To get a mass matrix with a nonzero determinant, atleast one two-dimensional representation has to get aVEV, otherwise the second and third columns of themass matrix will be zero. This means we can ignoreall subgroups, where no two-dimensional representationis allowed a VEV, leaving us with three subgroups toconsider Z = < BA m > , D q and Z q . The two possibletwo-dimensional representations that can show up are and n -k) coming from the product of with a one-dimensional representation.We first consider the subgroup, where all two-dimensional representations acquire a VEV, that is Z = < BA m > . Z = < BA m > Due to the large number of possible combinations of one-dimensional representations, we shall only give rows asbuilding blocks for a mass matrix. We group them ac-cording to the index j, i.e. according to the transfor-mation properties of the first generation of left-handedconjugate fermions. We give the row vectors for the p -throw of the mass matrix, depending on the index i p , i.e.the transformation properties of the p -th generation ofleft-handed fermions. We assume, that m is even. To getto an odd m , we need to switch and and φ and φ , as can be inferred from Table III.j = 1 , i p = 1( κ p < φ > , α p < ψ k > , α p < ψ k > e − πi k mn )j = 2 , p = 1(0, α p < ψ k > , α p < ψ k > e − πi k mn )j = 3 , i p = 1( κ p < φ > , α p < ψ k > , α p < ψ k > e − πi k mn )j = 1 , p = 2(0, α p < ψ k > , − α p < ψ k > e − πi k mn )j = 2 , i p = 2( κ p < φ > , α p < ψ k > , − α p < ψ k > e − πi k mn )j = 4 , i p = 2( κ p < φ > , α p < ψ k > , − α p < ψ k > e − πi k mn ) j = 1 , i p = 3( κ p < φ > , α p < ψ n − k > , α p < ψ n − k > e − πi k mn )j = 2 , p = 3(0, α p < ψ n − k > , α p < ψ n − k > e − πi k mn )j = 3 , i p = 3( κ p < φ > , α p < ψ n − k > , α p < ψ n − k > e − πi k mn )j = 1 , p = 4(0, − α p < ψ n − k > , α p < ψ n − k > e − πi k mn )j = 2 , i p = 4( κ p < φ > , − α p < ψ n − k > , α p < ψ n − k > e − πi k mn )j = 4 , i p = 4( κ p < φ > , − α p < ψ n − k > , α p < ψ n − k > e − πi k mn )These building blocks end up giving us only one generaloption for a mass matrix, that is a mass matrix with onezero entry. Mass matrices with more than one zero entryor no zero entry give a determinant of zero, and so willnot be considered here. This is due to the fact, that thereis a correlation between the entry in the first column andthe relative sign of the entries in the second and thirdcolumn. A nonzero entry in the first column necessarilyimplies either a relative sign between the entries in theother two columns (for j=2 and j=4) or no relative sign(for j=1 and j=3) - we can not however have both casesin the same mass matrix. To see what a typical massmatrix looks like, consider an example. Let j = i = 1,i = 2 and i = 3. The mass matrix then reads: α < ψ k > − α < ψ k > e − πi k mn κ < φ > α < ψ k > α < ψ k > e − πi k mn κ < φ > α < ψ n − k > α < ψ n − k > e − πi k mn (15)The scalar representations and the position of the zero(within the first column) can vary, depending on ourassignment of fermion representations, but the generalstructure will always be the one texture zero structure ofEq.(10). D q = < A nq , BA m > We continue with those cases, where not all two-dimensional representations can acquire a VEV. For thesubgroups D q , we only need to consider the case, where nq is odd. As the only two-dimensional representationfor the fermions is , the only relevant two-dimensionalrepresentations for Higgs fields are and n -k) .Now we can read off Table IV that the Higgs bosonstransforming under these representations can onlyacquire a VEV if q divides k and q divides n − k,respectively. If nq is even, these two conditions areequivalent, i.e. either both Higgs fields can receive aVEV and we are effectively dealing with a smaller G F which is then broken down to Z = < BA m > , or neithercan receive a VEV, and we end up having the secondand third columns equal to zero.If, however nq is odd, these two conditions are mutuallyexclusive (except where q =1, but D is the same as Z = < BA m > ), and we consider them separately below.We do not need to consider the case where neither ofthe two conditions are fulfilled, as then no Higgs fieldtransforming under a two-dimensional representationwill acquire a VEV and we are again left with a massmatrix containing two zero column vectors.As in Section V C 1, we consider the structure of the p -th row, depending on i p , j and k. The simplest entryis the M p element. We have, since only φ can acquirea VEV that M p = κ p < φ > if i p = j or M p = 0otherwise. We give the other entries as building blockrow vectors (cid:0) M p M p (cid:1) . They are, for q dividing ki p = 1 : ( α p < ψ k > , α p < ψ k > e − πi k mn )i p = 2 : ( α p < ψ k > , − α p < ψ k > e − πi k mn )i p = 3 , q dividing n -ki p = 1 , p = 3:(( − m α p < ψ n − k > , α p < ψ n − k > e − πi k mn )i p = 4:(( − m +1 α p < ψ n − k > , α p < ψ n − k > e − πi k mn )As in Section V C 1 the multiple possibilities can bereduced to one general form. This is again due to thefact, that we have a correlation between the elementin the first column and those in the second and thirdcolumns. Only those rows, for which i p = j can havea nonzero element in the first column. So if we choosethe element in the first column to be nonzero, we havealso determined whether the elements in the second andthird columns are zero. We must have at least one zeroentry in the second and third column, otherwise weare only breaking a smaller G F down to its subgroup Z = < BA m > , a case we have already discussed. So,we need to choose j in such a way, that zero elementsin the second and third column are not correlated witha zero element in the first column, otherwise we wouldend up with a zero row vector. This means that j = 3 orj = 4 for q dividing k , while if q divides n − k we mustchoose j = 1 or j = 2. We can then not have anotherrow, where i p = j, because there again the elements inthe second and third column would be zero and two rowvectors would be linearly dependent. So exactly one ofthe i p must be equal to j, say i . And we can say evenmore: To ensure a nonzero determinant, i and i mustbe unequal, since otherwise the second and third rowvector will be linearly dependent. We can then write theparticle content more exactly as L ∼ ( , , ) and L c ∼ ( , ). All mass matrices will then be of theblock form of Eq.(9). We give as an example the entries for i = 3, i = 1, i = 2 and q dividing k: A = κ < φ > (16) B = α < ψ k >C = α < ψ k > e − πi k mn D = α < ψ k >E = − α < ψ k > e − πi k mn Z q = < A nq > The major difference between the subgroups Z q and D q is that in the former all doublets will acquirearbitrary VEVs, and hence the mass matrices willexhibit less symmetry. We have only, as for the D q subgroups, to consider nq odd. The reasons howeverare slightly different: As can be inferred from TableIII, all one-dimensional representations can acquire aVEV if nq is even, so either again none of the relevantHiggs bosons that transform under a two-dimensionalrepresentation of the flavor group can acquire a VEV (if q does not divide k), or we are faced with the case whereall Higgs fields relevant for Yukawa terms (that is boththose transforming under one-dimensional and undertwo-dimensional representations) can acquire a VEV -this corresponds to the case discussed in Section V A ofa smaller flavor symmetry being fully broken.So we can again set nq to be odd and start by giving theelements in the first column. They are M p = κ p < φ > if i p = j, M p = κ p < φ > if p × = or M p = 0otherwise, whereas the elements in the second and thirdcolumns are, for q dividing k:i p = 1: ( α p < ψ > , α p < ψ > )i p = 2: ( α p < ψ > , − α p < ψ > )i p = 3 ,
4: (0, 0)and for q dividing n -ki p = 1 ,
2: (0, 0)i p = 3: ( α p < ψ n − k > , α p < ψ n − k > )i p = 4: ( α p < ψ n − k > , − α p < ψ n − k > )We can reduce the large number of possibilities. Firstwe set j, which must either be in { , } or in { , } . If wewant the M p element to be nonzero, then i p must be inthe same set as j and we also know whether the elementsin the second and third column are zero or not. If wenow choose j in such a way, that a nonzero element inthe first column implies nonzero elements in the secondand third column (and thereby a zero element in thefirst column implies a zero row vector), we are left tochoose between a mass matrix with 9 distinct nonzeroelements and a mass matrix with at least one zero rowvector and thereby a zero determinant. We thereforeneed to choose j in such a way, that a nonzero elementin the first column implies a zero in the second and third0column, i.e. j = 3 or j = 4 if q divides k , j = 1 or j = 2if q divides n − k. If we however choose two elementsin the first column to be nonzero, then those two rowvectors will be linearly dependent, and we will have azero determinant. So, we need to choose one i p in thesame set as j, while the other two must lie outside thatset - and, again, they cannot be equal, to ensure linearindependence of the corresponding row vectors. Thegeneral structure will then always be a block matrix. Wegive as an example the entries for the case where j = 3,i = 4, i = 1, i = 2 and q divides k: A = κ < φ > (17) B = α < ψ >C = α < ψ >D = α < ψ >E = − α < ψ > D. Two Doublet Structure
In our discussion of the two doublet structure we fre-quently use an additional index, p, given by × = .Without loss of generality we assume j ≥ k.For the two doublet structure we will discuss all possiblesubgroups, as they all give viable mass matrices.Anotherdifference compared to the three singlet structure is thatthe mass matrices given in this section are also potentialcandidates for Majorana mass matrices, if we imposethe conditions j = k and i = l. If a mass matrix can alsobe used as a Majorana mass matrix, we will mentionthis and briefly note, which Yukawa couplings need tobe equal in that case and which terms drop out due toanti-symmetry. Z n = < A > We read off Table III, that only < φ > and < φ > canget a VEV when conserving this subgroup. This limitsour freedom in choosing representations for the fermions:The two doublets must couple to form a or a ,otherwise the second and third row vectors of the massmatrix will be zero. This imposes the condition j = k.Also, we need p = 1 or 2, otherwise the first row vectorwill turn out to be zero. These restrictions leave us witha semi-diagonal mass matrix structure with entries: A = κ < φ p > (18) B = κ < φ > + κ < φ >C = κ < φ > − κ < φ > If p = 1 this is also a possible structure for a Majoranamass matrix. In this case the anti-symmetric part, i.e.the terms containing < φ > drop out. Z n = < A > From the Table III we infer that this subgroup only allowsfor one-dimensional representations to acquire a VEV.So, we need the product of the two doublets to contain atleast one one-dimensional representation. We are therebyleft with three possibilities:Case (i): j = k, j + k = n gives a semi-diagonal matrix. A = κ < φ p > (19) B = κ < φ > + κ < φ >C = κ < φ > − κ < φ > Case (ii): j + k = n , j = k = n gives a block structure. A = κ < φ p > (20) B = κ < φ > + κ < φ >C = κ < φ > + κ < φ >D = κ < φ > − κ < φ >E = κ < φ > − κ < φ > Case (iii): j = k, j + k = n gives a diagonal structure. A = κ < φ p > (21) B = κ < φ > + κ < φ >C = κ < φ > − κ < φ > Cases (i) and (ii) are also a possibility for Majorana massmatrices. In this case the anti-symmetric terms contain-ing < φ > drop out. Z q = < A nq > This subgroup requires quite an amount of case differen-tiation, since we want to make the discussion as generalas possible, and allow all possible relations between q andthe other indices of the model. We will first discuss thecase where nq is even, and then, at the end of this subsec-tion, discuss the slight changes induced by nq being odd.As an ordering principle in our discussion, we have takenthe structure of the resulting mass matrix, as only thethree characteristic types discussed in Section V A willshow up.Most of the conditions deal with the question, whichtwo-dimensional representations are allowed a VEV. Thistranslates directly into deciding whether q divides the in-dex of that representation. The two-dimensional repre-sentations which can show up are and , n -j) and n -k) from the coupling of two-dimensional withone-dimensional representations and , n -(j+k)) and from the coupling of the two two-dimensionalrepresentations. We will only give mass matrices for thecase where × contains . Mass matrices for thecase where it contains n -(j+k)) can be obtained byreplacing ψ j+k by ψ n − (j+k) and then switching the com-ponents of the doublet (see Appendix A 2).1 j k j+k j-k Structure q divides Det[M]=0 q divides × Diagonal q divides × Semi-diagonal q divides × Det[M]=0 q divides × Det[M]=0 q divides × × Block q divides × × × × Full
TABLE I: Index relations and corresponding mass matrixstructure As q must divide n for Z q to be a subgroup of D n , q dividing j + k and q dividing n − (j + k) are equivalent.As already noted, q dividing j is equivalent to q dividing n − j, if nq is even, which we assume for this discussion.To ensure a nonzero determinant, q must at least divideeither j − k or j + k: If not, the two by two submatrixin the lower right-hand corner of the mass matrix will bezero. This implies directly that q must divide either bothj and k , or neither of the two. If q divides j and k how-ever, then it also divides j+k and j − k - hence all relevanttwo-dimensional representations can acquire an arbitraryVEV and all one-dimensional representations can acquirea VEV anyway. As this leads to the case where the massmatrix contains 9 distinct entries, we disregard this case.We summarize our findings in Table I and discuss thedifferent cases below in detail.Diagonal MatrixThis structure appears in the following case: q must di-vide j + k but q does not divide (j − k), j or k . Note thatthis case is not possible for q =2, since the sum and thedifference of two numbers are either both odd or botheven, nor is it possible for j = k. For j + k = n this gives A = κ < φ p > (22) B = κ < ψ >C = κ < ψ > If j + k= n the mass matrix entries are A = κ < φ p > (23) B = κ < φ > + κ < φ >C = κ < φ > − κ < φ > Semi-diagonal MatrixThis structure shows up, if q divides (j − k), but notj + k, j or k . This is not possible if q = 2, nor is itpossible for j+k = n , since this contradicts the conditions This includes the case j=k, as all numbers divide zero, corre-sponding to the fact, that and can get a VEV for anarbitrary q . q ∤ (j + k) = n and nq being even. This leaves two cases:For j = k the mass matrix entries are A = κ < φ p > (24) B = κ < ψ − k >C = κ < ψ − k > while for j = k we get A = κ < φ p > (25) B = κ < φ > + κ < φ >C = κ < φ > − κ < φ > which is a candidate for a Majorana mass matrix if weomit the anti-symmetric terms.Block MatrixThis structure shows up, if q divides (j − k) and j + k, butnot j and k . This forces q to be even, as q must divide2j = (j − k) + (j + k) while not dividing j , that is a factorof 2 is relevant for making a number divisible by q . Incase j + k = n , j = k we get A = κ < φ p > (26) B = κ < φ > + κ < φ >C = κ < ψ − k >D = κ < ψ − k >E = κ < φ > − κ < φ > In case j + k = n and j = k, we get A = κ < φ p > (27) B = κ < ψ >C = κ < ψ − k >D = κ < ψ − k >E = κ < ψ > and if j+k = n and j = k, we get A = κ < φ p > (28) B = κ < ψ >C = κ < φ > + κ < φ >D = κ < φ > − κ < φ >E = κ < ψ > Finally, concerning the case j = k, j+k = n : this leads usto the conditions q | n and q ∤ n = j = k, which implieseither q = n (already covered) or q even and n odd. Themass matrices are the same in both cases, with entries: A = κ < φ p > (29) B = κ < φ > + κ < φ >C = κ < φ > + κ < φ >D = κ < φ > − κ < φ >E = κ < φ > − κ < φ > nq even q either divides both j and ( n − j) or neither,and we need no further case differentiation.If however nq is odd, we need to pay closer attention. Incase i and l are both in { , } , the discussion is as above,since n -j) and n -k) will not show up as the rep-resentation of a Higgs field. In case i and l are both in { , } we can also use the above discussion if we sub-stitute j by ( n − j) and k by ( n − k) in the conditions,which changes nothing concerning the conditions regard-ing sums and differences, as q will always divide n .Further changes occur due to the fact that φ , ∼ are not allowed a VEV anymore, see Table III.Finally, we are left with the following: We drop for amoment the condition that j ≥ k and instead impose thecondition i ∈ { , } and l ∈ { , } . This means that M will be zero, since p = 3 or 4 and φ p is then not allowed aVEV due to nq being odd. If we now want to avoid hav-ing a zero column or row vector in our mass matrix, both ψ n − j and ψ k must acquire a VEV, i.e. q must divide kand ( n − j). By assumption however, q does not divide n , which leads us straight to the conclusion that q doesnot divide j + k or | j − k | , thereby leaving the two-by-twomatrix in the lower right-hand corner of the mass matrixzero, which we have excluded. Z = < BA m > As this structure also strongly depends on the one-dimensional representations under which the fermionstransform, we have reduced it entirely to building blocks,to avoid having to deal with too many subcases. Aswe can read off Table III, all Higgs bosons transform-ing under a two-dimensional representation will acquirea structured VEV. The only constraints that arise aretherefore due to the Higgs bosons transforming underone-dimensional representations. This means that the M entry is of special interest. We will first write downthe general structure and then use this to explain, why M has to be nonzero to ensure a nonzero determinant. κ w κ X κ Y e − πi k mn κ X κ u κ vκ Y e − πi j mn κ ve − πi (j − k) mn κ ue − πi (j+k) mn (30)where: u = < φ > if j + k = n , m even u = < φ > if j + k = n , m odd u = < ψ min [j+k ,n − j − k] > if j + k = n v = < φ > if j = k v = < ψ j − k > if j = k w = < φ p > where p=1,3 if m even, p=1,4 if m oddNote that some of the phase factors degenerate to signs, in case of j=k or j+k= n . X i and Y i depend on thetransformation properties of the doublets and singletsinvolved, i.e. X and Y depend on i and k, X and Y depend on j and l. Building blocks for X and Y aregiven in Table II. The same table can be used for X and Y , substituting l for i and j for k.For the discussion of the M element let us assume that m is even; the reasoning for an odd m will be analogous. w = 0 implies that p=2 or 4, that is i=1 or 3 while l=2or 4. Switching i and l is also possible - the reasoning isagain analogous. If we now consult the table for the X and Y entries, we see that this implies a relative minussign between the X and Y entries, while there is norelative minus sign between the X and Y entries, so thesum of the second and third row vector is proportionalto the first row vector. This is true also with nontrivialphases. Therefore w = 0 must hold.If j = k and i = l, the matrix can be made symmetricby imposing κ = κ and can then also show up as aMajorana mass matrix. D n = < A , B > and D n = < A , BA > The discussion for these subgroups is very similar to thatof the subgroup Z n , as only one-dimensional representa-tions can receive a VEV, however in this case not all ofthem (see Table IV). We receive the mass matrices for D n = < A , B > by simply eliminating all terms contain-ing φ and φ from the mass matrices for Z n , makingsure that we do not end up with a zero determinant. Toprevent this must be allowed a VEV, that is p mustbe 1 or 3. We can then distinguish between the followingthree sub-cases as for Z n :Case (i): j = k gives a semi-diagonal structure. A = κ < φ p > (31) B = κ < φ >C = κ < φ > Case (ii):j = k, j + k = n gives a block structure. A = κ < φ p > (32) B = κ < φ >C = κ < φ >D = κ < φ >E = κ < φ > Case (iii): j = k, j + k = n gives a diagonal structure. A = κ < φ p > (33) B = κ < φ >C = κ < φ > Note that in cases (i) and (iii) we get two degenerateeigenvalues. We can get the mass matrices generated bybreaking to D n = < A , BA > by substituting φ by φ in the matrices above. In this case p must either be 1 or3 i = 1 i = 2 i = 3 i = 4 „ X Y « = „ < ψ k >< ψ k > « „ < ψ k > − < ψ k > « „ ( − m < ψ n − k >< ψ n − k > « „ ( − m +1 < ψ n − k >< ψ n − k > « TABLE II: Building blocks for matrix of two doublet structure under the subgroups Z = < B A m > < φ > . Case (i) and (ii) are applicablefor Majorana mass matrices - in this case p=1 since imust be equal to l. D q = < A nq , BA m > This case is very similar to the Z q case. The two dif-ferences are: a) Less one-dimensional representations areallowed a VEV and b) the two-dimensional representa-tions are only allowed a structured VEV (see Table IV).The first difference implies restrictions on the index p - ifthe entry in the upper left-hand corner is zero, the deter-minant is zero too. So, we need to impose the conditionthat is allowed a VEV which depends on the evennessand oddness of m and nq , see Table IV.The second difference means replacing the arbitrary dou-blet VEV components by one with same absolute valueand fixed relative phase. The case where we have 9 inde-pendent entries thereby does not show up here - insteadthe full matrix structure now corresponds to breaking asmaller G F down to Z = < BA m > . We briefly commenton this at the end of this subsection.We will do the entire discussion assuming an even m .This means p has to be either 1 or 3 ( nq even). To getthe mass matrices for an odd m exchange φ and φ ,taking care to also switch indices in the conditions. Therelative signs that occur in this case are encoded in fac-tors of ( − m , which are therefore left in, even thoughthe rest of the discussion concerns only even m .As for the case of Z q there are no major differences be-tween nq being even or odd, except that we need to imposethe condition p = 1, if nq is odd, as then only is al-lowed a VEV. Furthermore, if i and l are in { , } , onemust also replace j and k by n − j and n − k in the condi-tions, respectively. The structuring we use is the same asin Section V D 3, that is the mass matrices are classifiedaccording to their structure.Diagonal MatrixFor this structure we must have q dividing j + k but q not dividing (j − k), j or k .This is not possible for q =2nor for j = k. For j+ k = n this gives A = κ < φ p > (34) B = κ < ψ j+k >C = κ < ψ j+k > e − πim (j+k) n If j+ k = n the mass matrix entries are A = κ < φ p > (35) B = κ < φ >C = ( − m κ < φ > Note, that for both of these matrices the squared massmatrix
M M † has two degenerate eigenvalues.Semi-diagonal MatrixFor this structure q must divide (j − k), but not j + k, jor k . This is not possible if q = 2, nor is it possible forj + k = n . We are left with two cases: If j = k we end upwith A = κ < φ p > (36) B = κ < ψ j − k >C = κ < ψ j − k > e − πim (j − k) n while for j = k we get A = κ < φ p > (37) B = κ < φ >C = κ < φ > which is a candidate for a Majorana mass matrix if i = l.Both these matrices give degenerate eigenvalues in thesquared mass matrix.Block MatrixThis structure shows up, for q dividing (j − k) and j + k,but not j and k . This forces q to be even. For j + k = n ,j = k we get A = κ < φ p > (38) B = κ < φ >C = κ < ψ j − k >D = κ < ψ j − k > e − πim (j − k) n E = ( − m κ < φ > In case j + k = n and j = k, we get A = κ < φ p > (39) B = κ < ψ j+k >C = κ < ψ j − k >D = κ < ψ j − k > e − πim (j − k) n E = κ < ψ j+k > e − πim (j+k) n = n and j = k, we get A = κ < φ p > (40) B = κ < ψ >C = κ < φ >D = κ < φ >E = κ < ψ > e − πim j n and if j + k = n and j = k = n , that is for q even and n odd, we get A = κ < φ p > (41) B = κ < φ >C = κ < φ >D = κ < φ >E = ( − m κ < φ > Full MatrixIf q divides all relevant indices, that is if it divides j andk and thereby automatically divides (j − k) and j + k, weare actually breaking a smaller G F down to its subgroup Z = < BA m > , as discussed in Section V C 2. This isbecause Z = < BA m > is equivalent to D , where theabove conditions are automatically fulfilled, as 1 dividesany integer. We therefore do not need to consider thiscase. E. Mass Matrices in D ′ n One can immediately see, that the Dirac mass ma-trix structures for double-valued dihedral flavor symme-tries are very similar to those generated by single-valuedgroups. A general correspondence between subgroups of D n and D ′ n can be established, by looking at the allowedVEVs in Table III to Table VI: D n Z n Z n Z q D D n D q Z (q=2) Z D ′ n Z n Z n Z q Z D ′ n D ′ q2 Z − As one can see n always has to be replaced by 2 n in thediscussion. The only relevant difference between single-and double-valued groups is the existence of odd repre-sentations. Therefore, at least one of the fermion genera-tions should transform as an odd representation in orderto find possible new mass matrix structures.Odd two-dimensional representations do not get a struc-tured VEV and in case they get a VEV, all VEVs willbe arbitrary. This is only possible for the subgroup Z q .Similarly, for the odd one-dimensional representations,i.e. and of D ′ n with n odd, only two subgroups al-low non-vanishing VEVs, namely Z n and Z q for nq beingeven. Note that in both cases and simultaneouslyget a VEV.The only changes that appear are additional signs dueto differences in the Clebsch Gordan coefficients (see Ap-pendix A 2 b). If the Higgs fields transform as odd rep-resentations, such additional signs are not relevant, since they can be absorbed into the VEVs, as these are arbi-trary anyway.Furthermore one sees, that if at least one of the fermionrepresentations is an odd two-dimensional representa-tion, this results in the 3rd column being multiplied by-1 for the mass matrix structures of Eq.(7), Eq.(8) andEq.(9). These additional signs can be rotated away byredefining the left-handed conjugate fermions. We wouldneed the mass matrix structure of Eq.(10) or Eq.(12) forthis sign change to have phenomenological consequences.These mass matrix structures however only appear forthe subgroup Z = < BA m > , which has no counterpartfor double-valued groups, so that these two structures donot arise for double-valued groups.Strictly speaking, we encounter fermion mass matrixstructures like Eq.(10) and Eq.(12) also here, but all suchcases can be reduced to a single-valued group. In the sim-plest case we have used only even representations for thefermions. Then it is clear that one could also have useda single-valued group right from the beginning. Anothercase occurs, if all the fermions transform as odd represen-tations. Then the Higgs fields have to transform as evenones. One cannot simply reduce this case to a single-valued group, as one does not find odd representations in D n groups. However, one always finds some equivalentassignment for the fermions using only even representa-tions which leads to the same mass matrix structure andwhich can then be reduced to the case of a single-valuedgroup. Furthermore, one finds cases in which the massmatrix is allowed to have arbitrary entries, but there ex-ists no smaller symmetry of the original group which isfully broken. This is the same case as already mentionedfor the D n groups, if the subgroup is Z q . Similarly tothere, we can find equivalent assignments for the fermionswhich result in the same matrix structure and which in-deed correspond to a smaller group being fully broken.We thus conclude that no new Dirac mass matrix struc-tures appear for double-valued groups.For an odd n , we need to take into account that and are complex (conjugated). This implies that × = , so we have to replace < φ > in themass matrices by < φ > , and vice versa, even wherethey show up only implicitly as < φ p > . This also leadsto differences, when switching to up-type mass matrices: φ ∗ transforms as . So if we encounter a < φ > inthe down-type mass matrix, we need a < φ > ∗ for theup-type mass matrix. For odd two-dimensional represen-tations an additional minus sign is introduced along withthe second component of the VEV when switching to theup-type mass matrix, due to the matrix U introducedin Section II B. All these changes do not lead to newstructures. F. Majorana Mass Matrices
Majorana mass matrices correspond to Yukawa couplingsinvolving two identical fermions, either L or L c . The rel-5evant Higgs fields are then SU (2) L triplets or gauge sin-glets, respectively, whereby VEVs of total singlets can bereplaced by direct mass terms. The fact that we coupleidentical fermions, forces Majorana mass matrices to besymmetric.One comment is in order concerning our exclusion cri-terion of demanding a nonzero determinant: This is nolonger phenomenologically motivated in this case, sinceMajorana masses are only allowed for neutrinos and thedata still allow one neutrino to be massless. Neverthe-less, we restrict ourselves in this way in order to keep thediscussion manageable.We have already mentioned when and how Majoranamass terms can show up if the Majorana fermions trans-form under one two- and one one-dimensional represen-tation of G F , as these correspond to mass matrices ofthe two doublet type. For D ′ n , we have to mention thatin case of odd two-dimensional representations the termscontaining < φ > are anti-symmetric. So, wheneverwe remark in Section V D that the anti-symmetric termscontaining < φ > drop out, it is instead the terms con-taining < φ > that will drop out. This does not lead toany new structures compared to the Dirac case.However, other structures can appear, since we allow ei-ther L or L c to transform under three one-dimensionalrepresentations of G F . Thereby we also need to considerthe case where all fermions in the Yukawa term trans-form under one-dimensional representations.For D n with n arbitrary and for D ′ n with n even weknow that, in addition to the Majorana mass matrix be-ing symmetric, all diagonal entries will be nonzero, as twoidentical one-dimensional representations always coupleto form a trivial representation.We first discuss the case of n even for D n and D ′ n . Look-ing at Table III to Table VI we see that depending on thepreserved subgroup the following one-dimensional rep-resentations can get a VEV: only , and withi = 1 or all one-dimensional representations. Especially,the case in which three representations get a VEV is ex-cluded. Concerning the assignment of the fermions wecan distinguish the following three cases: either all threegenerations transform in the same way or two of themtransform as the same representation or all three trans-form as different representations. If the three genera-tions are assigned to ( , , ), all mass matrix entriesare non-zero, i.e. the Majorana mass matrix is a gen-eral symmetric matrix with 6 independent parameters,as × = holds. For the assignment ( i , i , i )there exist two possible structures: either the matrix hasa block structure or it has 6 independent entries. In thefirst case one has to ensure that i × i is not allowed aVEV by the preserved subgroup, while in the second case i × i should also acquire a VEV. In the last case, allfermions transform under different one-dimensional rep-resentations, which allows apart from the block and thearbitrary structure the possibility of having a matrix withnon-vanishing entries on the diagonal only. The case onlyoccurs, if the preserved subgroup only allows to ac- quire a VEV and therefore the flavor symmetry is notbroken in the Majorana mass sector.For the case of D n with n odd, we only have two one-dimensional representations to choose from. Thereforeat least two generations of fermions have to transformunder the same representation, forbidding the structurewith non-vanishing entries only on the diagonal.For D ′ n with n odd, structures not found above couldonly arise from fermion assignments involving the rep-resentations and , as and are complex andhence × and × = . This means if is not allowed a VEV, we can have zero elements on thediagonal. However, we find that if is not alloweda VEV, only the trivial representation is allowed aVEV for n odd, i.e. the dihedral symmetry is unbrokenin the Majorana mass sector. The structure arising from L ( c ) ∼ ( i , i , i ) with i ∈ { , } and i , i ∈ { , } ,i = i is then the same as in case of the two doubletassignment, if the two odd one-dimensional representa-tions and are replaced by the doublet, i.e. it is asemi-diagonal matrix with the obvious restriction that C equals B .As already mentioned, if is the only representationwhich gets a VEV, the flavor symmetry is unbroken inthe Majorana sector. This happens in models in whichthe flavor symmetry is only spontaneously broken at alow energy scale, for example at the electroweak scale.All the possible mass matrix structures have been enu-merated above. VI. APPLICATIONS
In this section we show that we can predict the Cabibboangle, i.e. | V us | , in the quark sector in terms of grouptheoretical quantities, i.e. the index n of the group, theindex of the representations i, j, k and l under whichthe generations of left- and left-handed conjugate fieldstransform, Q ∼ ( , ) , d c , u c ∼ ( , ) , and the breaking direction in flavor space, m u and m d .In Section V D 4 we showed that in case of a preservedsubgroup Z = h BA m i of D n the resulting mass matrix M d for the down quark sector is of the form M d = A d C d C d e − i φ d k B d D d E d B d e − i φ d j E d e − i φ d (j − k) D d e − i φ d (j+k) (42)where we defined: φ d = 2 πn m d , m d = 0 , , , ... . such that we break down to a subgroup Z = h BA m d i of D n . A d , B d , ... are in general independent complex num-6bers which are products of VEVs and Yukawa couplings. M d M † d can be diagonalized by the unitary matrix U d = cos( θ d ) e i β d θ d ) e i β d − sin( θ d ) √ i φd j √ θ d ) √ − sin( θ d ) √ e − i φ d j − √ θ d ) √ e − i φ d j (43)with β d = arg (cid:2) A d B ⋆d + C d ( D d + E d e i φ d k ) ⋆ (cid:3) and θ d de-pending on A d , B d , ... in a non-trivial way. The masseigenvalues for the first, second and third generation arethen: m d = 12 [ | A d | + 2 ( | B d | + | C d | ) + | D d + E d e i φ d k | (44) − ( −| A d | + 2 ( | B d | − | C d | ) + | D d + E d e i φ d k | ) sec(2 θ d )] , m s = | D d − E d e i φ d k | , m b = 12 [ | A d | + 2 ( | B d | + | C d | ) + | D d + E d e i φ d k | +( −| A d | + 2 ( | B d | − | C d | ) + | D d + E d e i φ d k | ) sec(2 θ d )] . The mass matrix M u for the up-type quarks has the form: M u = A u C u e i φ u k C u B u e i φ u j D u e i φ u (j+k) E u e i φ u (j − k) B u E u D u (45)with φ u = 2 πn m u , m u = 0 , , , ... . such that Z = h BA m u i is the subgroup of D n whichis preserved by the Higgs fields coupling to the up-typequarks. A u , B u , .... are in general complex quantities like A d , B d , ... .The unitary transformation of the left-handed fields isgiven by: U u = θ u ) e i β u sin( θ u ) e i β u − e i φu j √ − sin( θ u ) √ θ u ) √ √ − sin( θ u ) √ e − i φ u j cos( θ u ) √ e − i φ u j (46)where β u = arg (cid:2) A u B ⋆u + C u ( D u + E u e − i φ u k ) ⋆ (cid:3) − φ u jand θ u is a function of the parameters A u , B u , ... .The masses for the first, second and third generationread: m u = | D u − E u e − i φ u k | , (47)m c = 12 [ | A u | + 2 ( | B u | + | C u | ) + | D u + E u e − i φ u k | − ( −| A u | + 2 ( | B u | − | C u | ) + | D u + E u e − i φ u k | ) sec(2 θ u )] , m t = 12 [ | A u | + 2 ( | B u | + | C u | ) + | D u + E u e − i φ u k | +( −| A u | + 2 ( | B u | − | C u | ) + | D u + E u e − i φ u k | ) sec(2 θ u )] . Taking Eq.(43) and Eq.(46) we arrive at the followingform for the CKM mixing matrix V CKM : V CKM = U Tu U ⋆d = 12 − e i φ d j x − s d − x + e i φ d j x − c d i α c d c u + y + s d s u − e − i φ d j y − s u i α c u s d − y + c d s u − y + c u s d + 2 e i α c d s u e − i φ d j y − c u y + c d c u + 2 e i α s d s u (48)where we defined: x ± = (1 ± e i δφ j ), y ± = (1 ± e − i δφ j ), δφ = φ u − φ d , α = β u − β d and used the abbreviations: s d,u = sin( θ d,u ) and c d,u = cos( θ d,u ).As one can see the element | V us | solely depends on thegroup theoretical quantities n , j, m u and m d : | V us | = 12 (cid:12)(cid:12) i δφ j (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) cos (cid:18) π ( m u − m d ) j n (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (49)Note further that only the transformation properties ofthe left-handed fields which form a doublet under the fla-vor group D n are relevant, since only their representationindex j appears in the expressions Eq.(48) and Eq.(49) . To be correct, the phases β u,d also depend on the index k of The other two mixing angles θ q and θ q can be tuned bythe use of the two unconstrained angles θ u and θ d . TheJarlskog invariant J CP [21] depends on the phase α . Inthis way the experimental value of J CP of 3 . × − can be reproduced.In case of n = 7 and ( m u − m d ) j = 3, e.g. m d = 0, m u = 1 and j = 3, we arrive at a value of | cos( π ) | ≈ . | V us | which is only 2% smaller than the mea-sured value 0 . +0 . − . [1]. In other words the size ofthe Cabibbo angle can be explained by group theoreti-cal means derived from a flavor symmetry D n which is the doublet representation under which the left-handed conju-gate fields transform. Nevertheless these phases depend on manyother parameters A u,d , B u,d , ... such that they are primarily notdetermined by the group theory of the flavor symmetry. distinct subgroups, Z = h BA m u i and Z = h BA m d i with m u = m d , in the up and downquark sector. This obviously requires that up-type anddown-type quarks do not couple to the same Higgs fields,i.e. a further separation mechanism is needed here. Inthe SM one can simply assume an extra Z symmetry: Q i → Q i , u ci → u ci and d ci → − d ci for the fermions and for the Higgs fields: ϕ u i → ϕ u i and ϕ d i → − ϕ d i where ϕ u i denote all Higgs fields which can couple tothe up-type quarks, while ϕ d i only couple to down-typequarks. In order to obtain two distinct VEV configura-tions for the fields ϕ u i and ϕ d i which either break to Z = h BA m u i or Z = h BA m d i we also need a separationmechanism in the Higgs sector, but this issue will not bediscussed here.Eq.(44) and Eq.(47) for the masses of the up-type anddown-type quarks already show some possible source offine-tuning, since the smallness of the up quark mass m u as well as of the strange quark mass m s must be dueto the smallness of the expression | D u − E u e − i φ u k | and | D d − E d e i φ d k | , respectively. A further study of thisissue is however beyond the scope of this paper and willbe treated in detail elsewhere [22].There is a second way to generate such a mixing matrix:If we change the transformation properties of the left-handed conjugate fields to d c , u c ∼ ( i , i , i ) , we arrive at a mass matrix which is of the form of thetranspose of the one shown in Eq.(10). The preservedsubgroup is then also of the form Z = h BA m i , as ex-plained in Section V C 1. The number of free parametersis again five and therefore the chances to find a numericalsolution which can reproduce all the masses and mixingparameters correctly apart from the predicted value forone of the elements of V CKM are similar as in the caseabove. However, in a complete model one or the othermass matrix may be more easily accommodated with thesuitable number of Higgs fields, VEV alignments, etc..From the theoretical viewpoint the model in which theleft- as well as the left-handed conjugate fields are par-tially unified into a doublet and a singlet representationof the discrete group might be favorable.In all cases a thorough check whether one can accom-modate all the quark masses and the rest of the mix-ing angles is necessary, although we believe that this canbe done due to the number of free parameters. Further-more a Higgs potential which allows for the breaking intothe two different directions needs to be constructed andproven to be stable against corrections. And also an ex-tension of this idea to include leptons is desirable. Allthese issues combined with a numerical study are dele-gated to a future publication [22].
VII. COMPARISON WITH THE LITERATURE
The fact that preserved subgroups of a dihedral groupcan play an important role in obtaining certain mixingpatterns is often not explicitly discussed. Several resultsfrom the literature can however easily be obtained fromthe group theoretical considerations we have performedin this paper. Realizing this can help to understand howand why flavor symmetry models work, as the followingexemplary discussion shows.In reference [12] a D flavor symmetry is used, withan additional Z ( aux )2 for separating the three differentYukawa sectors: the charged lepton sector, the Diracneutrino sector and the Majorana neutrino sector. Thetransformation properties of the fermions are chosen togive a two doublet structure in all three sectors. Thenone breaks down to a different subgroup in each sector: D is conserved in the charged lepton sector. D isconserved in the Dirac neutrino sector, while it is brokendown to Z in the Majorana neutrino sector. As theauthors use a different basis, the actual mass matricesdiffer from ours. The resulting physical mixing angleshowever are unaffected, i.e independent of the basis.Here we just describe the result for the mass matricesin their basis. The mass matrix for the charged leptonsas well as the Dirac neutrino mass matrix are diagonal,i.e. they do not contribute to the lepton mixing. Twotexture zeros in the right-handed neutrino mass matrixare obtained, which only appear because a gauge singlettransforming as a further non-trivial D singlet is absentfrom the model. They are removed when the seesawmechanism is applied. The resulting mass matrix forthe light neutrinos is µτ symmetric, i.e it has a maximalmixing angle θ l , a zero mixing angle θ l and a freemixing angle θ l . The texture zeros in the right-handedneutrino mass matrix effect the relations between theeigenvalues of the light neutrino mass matrix, i.e. theyenforce a normal hierarchy. Nevertheless, this modelis not complete, as quarks are not discussed in thiscontext. [23] goes on to discuss soft breaking of the D in the scalar potential. More precisely, the Z subgroupis broken explicitly. Thus, the resulting doublet VEV nolonger conserves Z , which can lead to sizable deviationsfrom a maximal θ l , while leaving θ l = 0.A similar result is reproduced by a D flavor symmetryin reference [13]. In this case, the authors work in thesame group basis we have used, so a comparison is morestraightforward. Nevertheless, they present their Diracmass matrices in the basis ¯ LR , while our results arealways given in the basis of LL c . The D ( ∼ = S ) flavorsymmetry is again joined by a Z symmetry, whichcreates the same three sectors. D is then broken to Z in the charged lepton sector. In the Dirac neutrinosector the entire flavor symmetry is conserved, while itis broken down to a Z in the Majorana neutrino sector- in this case all three equivalent subgroups < B > , < BA > and < BA > are mentioned. The chargedlepton mass matrix and the Dirac mass matrix for the8neutrinos are again diagonal, i.e. the non-trivial mixingstems from the right-handed neutrino mass matrix. Herethe maximal scalar field content is allowed, resultingin the same light neutrino mass matrix as for the D model. The additional parameter which arises fromhaving the full scalar field content does not affect thestructure of the neutrino mass matrix obtained from theType I seesaw formula, can however affect the neutrinomass hierarchy, as in this case the normal hierarchy isnot necessarily predicted.We finally mention some further examples, wherenontrivial subgroups of dihedral flavor symmetriesare conserved and explicitly mentioned. [24] and [25]mention the possibility of breaking their dihedral flavorsymmetry down to a nontrivial subgroup. [24] uses a D flavor symmetry, which is then broken down to Z to achieve maximal atmospheric mixing and θ l = 0.[25] uses a D symmetry. If this is broken down to Z ,maximal mixing can be achieved. However, withoutseparating the Yukawa sectors or explicitly breaking theresidual symmetry, the resulting maximal mixing anglesin the neutrino and charged lepton sector will cancel inthe leptonic mixing matrix.The use of a dihedral symmetry is not only limited tothe determination of the leptonic mixing angles. Thebreaking chains of Section IV can also be employed toexplain the fermion mass hierarchy, by giving mass todifferent generations at different breaking scales. [26]breaks D ′ down to Z and then further down, while [27]breaks a D flavor symmetry down to Z and then fullybreaks it. VIII. CONCLUSIONS
We have determined the possible Dirac and Majoranamass matrix structures that arise if a dihedral flavorgroup D n is broken down to a non-trivial subgroup byVEVs of some scalar fields. In order to exploit the non-abelian structure of the dihedral groups we only consid-ered assignments where at least two of the left-handed orleft-handed conjugate (SM) fermions are unified into anirreducible two-dimensional representation, i.e. L ∼ ( , , ) , L c ∼ ( , ) (50)or L ∼ ( , ) , L c ∼ ( , ) (51)We constrained ourselves by the requirement that themass matrices have to have a non-vanishing determinantto reduce the number of cases so that a general discussionbecomes possible. The number of different mass matrixstructures we encounter is then limited. We find the fol-lowing mass matrix structures: C − Ce − iφ k A D De − iφ k B E Ee − iφ k (52) and A C Ce − iφ k B D EBe − iφ j Ee i (k − j) φ De − i (j+k) φ (53)Both of these matrix structures are associated with aconserved subgroup Z . The other mass matrices whichcan arise from a conserved subgroup are either diagonal,semi-diagonal or of block form.All these mass matrices can result from a Dirac massterm and several ones also from a Majorana mass.For Majorana mass terms we additionally allow the as-signment that all involved fermions can transform as one-dimensional representations. Furthermore, we also stud-ied the case of an unbroken dihedral group.It turns out that the double-valued groups D ′ n do not leadto additional structures. We observe one slight differencein case of a Majorana mass term for D ′ n with n odd, ifthe fermions transform as three one-dimensional repre-sentations. Due to the odd one-dimensional representa-tions a semi-diagonal mass matrix can arise. Althoughthis structure cannot be deduced with one-dimensionalrepresentations in the groups D n ( n arbitrary) and D ′ n ( n even), this structure is not new, as it can be repro-duced, if the fermions which transform as the two oddone-dimensional representations are unified into a two-dimensional representation.Note that all these matrix structures base on the assump-tion that for each representation µ which is/contains atrivial representation under the residual subgroup of theflavor symmetry there is a Higgs field present in the the-ory which transforms as µ . In this way the structures ofthe mass matrices only depend on the choice of the orig-inal group, the subgroup as well as the transformationproperties of the fermions, but not on the choice of theHiggs content. Therefore our results are less arbitrary.Reducing the Higgs content is also possible by simply set-ting the corresponding VEV to zero in the mass matrix.In some cases this just reduces the number of parametersand therefore increases the predictive power of the model[12], while in other cases this is essential in order to getthe desired mixing pattern [6, 20, 28].In general one observes that for several subgroups themass matrices (can) exhibit 2 − φ is set to zero. This can be used to obtain maximal atmo-spheric mixing in the leptonic sector.Interestingly, it is not only possible to predict such spe-cial mixing angles as 45 ◦ or zero with a dihedral flavorsymmetry, as we showed in the very first application ofour general study. There we were able to describe oneelement of the CKM mixing matrix, namely | V us | , whichcorresponds to the Cabibbo angle, only in terms of grouptheoretical quantities, like the index n of the dihedralgroup D n , the index j of the representation under whichthe (left-handed) quarks transform as well as the indices9of the preserved subgroups of D n , m u and m d : | V us | = ˛˛˛˛ cos „ π ( m u − m d ) j n «˛˛˛˛ = | cos( 3 π | ≈ . Since these dihedral symmetries have already been usedseveral times in the literature, it was interesting to seewhether the successful models, like the ones by Grimusand Lavoura which can predict µτ symmetry with thehelp of the flavor symmetry D × Z ( aux )2 [12] or D × Z ( aux )2 [13], already make use of our described idea. Andindeed these two models are examples in which the threesectors represented by the charged lepton mass matrix,the Dirac neutrino mass matrix and the Majorana massmatrix for the right-handed neutrinos preserve differentnon-trivial subgroups of the employed flavor symmetry.This mismatch of the different preserved subgroups thenleads to the non-trivial mixing pattern.The notation L and L c of the fermions in our discussionalways implied that they transform under the SM gaugegroup. However, all structures remain the same, if weconsider the MSSM as framework. Concerning a GUTlike SU (5) the fifteen fermions of one generation areunified into and ¯ . As a consequence, for example,the (Dirac) mass matrix for the up-type quarks stemsfrom the couplings of two s. Therefore only the twodoublet structure with i = l and j = k applies here.Similar changes occur, if we consider other (unified)gauge groups. As our study includes all the possiblemass matrix structures arising from the two assignmentsEq.(50) and Eq.(51), choosing a gauge group other thanthe one of the SM under which the mass terms areinvariant does not give rise to new cases.In our study we always assumed that the involved Higgsfields (in case of the Dirac mass matrix) are copiesof the SM Higgs SU (2) L doublet. This might causeproblems in a complete model, since it is well-knownthat multi-Higgs doublet models suffer from FCNCswhose bounds usually demand that the Higgs fields havemasses above 10 TeV. However, all structures we havepresented can also be achieved in models in which onlythe SM Higgs doublet (or in the MSSM the Higgs fields h u and h d ) exists and the flavor symmetry is broken bysome gauge singlets, usually called flavons, mostly at ahigh energy scale. Then one has to deal in general withnon-renormalizable operators consisting of two fermions,the usual Higgs field and appropriate combinations ofthe flavon fields. According to their mass dimensionthe operators then have different suppression factorscompared to the multi-Higgs doublet case where alloperators arise at the renormalizable level. In an explicitrealization one therefore has to check whether all resultswhich are produced in a multi-Higgs doublet modelcan be reproduced by the usage of non-renormalizableoperators in a natural way. For example, the topYukawa coupling has to be large and should come froma renormalizable term.We mainly concentrated on the explanation of the mix- ing pattern of the fermions by the flavor symmetry andits breaking to subgroups. However it is also interestingto ask whether the hierarchy among the fermion masses,especially among the up-type quarks, could also beaccommodated in this framework, for example by somestepwise breaking of the symmetry. Obviously, it isan advantage, if this could also be done. If not, theFroggatt-Nielsen mechanism [29] could be used as anexplanation. For this an extra U (1) flavor symmetry isneeded. It is in general non-trivial to combine a certainassignment of the fermions under the discrete flavorgroup with a certain set of U (1) charges being necessaryfor the mass hierarchy. As mentioned above the usageof flavon fields is already a kind of implementationof the Froggatt-Nielsen mechanism, as their existencegenerally leads to non-renormalizable operators, whosesuppression according to the number of insertions offlavon fields might be the origin of the fermion masshierarchy.And finally we want to comment on the VEV alignment.As we saw, there exists a specific VEV structure for scalarfields transforming under a two-dimensional representa-tion which is necessary for conserving certain subgroups.This VEV structure < ψ j > (cid:18) e − πi j mn (cid:19) (55)can in fact arise naturally from the extremization of ascalar potential. For example: • a D -invariant two Higgs potential, with the tworeal scalar fields transforming as SM gauge singletsand as under D , • the equivalent potential for D , with two real scalarfields transforming as SM gauge singlets and as under D , • the most simple phenomenologically viable D in-variant potential for SU (2) L doublet scalars, whichis now a three Higgs potential with the three scalarfields transforming as SU (2) L doublets and as and under D One finds in all cases that the VEV structure given aboveis the only one that can extremize the potential whilebreaking the flavor symmetry, if one assumes real pa-rameters in the potential and possibly also no correla-tions among the parameters of the potential. Without the additional singlet the potential exhibits an acciden-tal U (1) symmetry, apart from the gauge symmetries [14]. Here we preferred to use real representation matrices which thenhave real eigenvectors belonging to the eigenvalue 1 such that wecould solve the extremization conditions under the assumptionof real parameters and also real VEVs. In this way all allowedVEV configurations correspond to the specific VEV structure. SU (2) L doublet potential in D no such explicit statement can bemade. Nevertheless, the above VEV structure is still al-lowed.These statements make us confident that there is someway to get the announced VEV structure, even if morescalar fields exist in the model.Regarding the success of models like [6, 20] the cru-cial issue of the VEV alignment might be solved ina more efficient way in the context of supersymmetricmodels with flavon fields. The main reason is that the(super-)potential itself is significantly simplified by usinggauge singlet scalars which transform non-trivially un-der G F . Furthermore, as these fields are gauge singletswhich break the flavor symmetry at a high energy scaleΛ > m SUSY , the condition that the F-terms of some fla-vored gauge singlet fields vanish determine the equationsfor the VEVs. These can then be often solved analyti-cally.
Note added:
After completing this work the paper [30] byC.S. Lam appeared in which he also deals with the factthat non-trivial subgroups of some discrete flavor sym-metry can help to explain a certain mixing pattern andalso very briefly mentions that the Cabibbo angle mightbe the result of some dihedral group.
Acknowledgments
A.B. acknowledges support from the Studienstiftung desDeutschen Volkes. C.H. was supported by the “Sonder-forschungsbereich” TR27.
APPENDIX A: MORE GROUP THEORY1. Kronecker Products
The products of the one-dimensional representations of D n are: ×
11 12 13 1411 11 12 13 1412 12 11 14 1313 13 14 11 1214 14 13 12 11 where the representations only exist in groups D n with aneven index n . The products × transform as × = and for n even there are also × = with k = n − j.If 4 is a divisor of n , the product of the representation withj = n with any one-dimensional representation of the group alsotransforms as .The products × are of the form + + with j =min(2 i , n − D n has an index n whichis divisible by four one also finds the structure × = P j=1 for i = n . This shows that there is at most one representation ineach group D n with this property. The mixed products × can have two structures: a. ) × = + with k = | i − j | and l = min(i + j , n − (i + j)) and b. ) × = + + withk = | i − j | for i + j = n .For D ′ n with n even the one-dimensional representations have thesame product structure as for D n while for n being odd they are: ×
11 12 13 1411 11 12 13 1412 12 11 14 1313 13 14 12 1114 14 13 11 12 due to the fact that the two one-dimensional representations and are complex conjugated to each other. The rest of theformulae for the different product structures are the same as in thecase of D n ,i.e. in each formula above which is given for D n onehas to replace n by 2 n .The Kronecker products can also be found in reference [31].
2. Clebsch Gordan Coefficients a. for D n For × = the Clebsch Gordan coefficient is trivially one.For × the Clebsch Gordan coefficients are:for i = 1: „ ` ´` ´ « ∼ and for i = 2: „ ` ´` − ´ « ∼ If the index n of D n is even, the group has two further one-dimensional representations whose products with are ofthe form:for i = 3: „ ` ´` ´ « ∼ n -j and for i = 4: „ ` ´` − ´ « ∼ n -j For the products × the covariant combinations are: „ « ∼ , „ − « ∼ and «„ « 1CCA ∼ or «„ « 1CCA ∼ n -2i If the index n of D n is even and i = n (4 has to be a divisor of n ),there is a second possibility: × = P j=1 . The Clebsch Gordancoefficients are „ « ∼ , „ − « ∼ , „ « ∼ , „ − « ∼ For the products × with i = j there are the two structures × = + with k = | i − j | and l = min(i + j , n − (i + j)) or × = + + with k = | i − j | , if i +j = n (obviously n hasto be even). The Clebsch Gordan coefficients for × = + are: «„ « 1CCA ∼ or «„ « 1CCA ∼ and «„ « 1CCA ∼ or «„ « 1CCA ∼ n -(i+j) For the structure × = + + with i+j = n the ClebschGordan coefficients are „ « ∼ , „ − « ∼ and «„ « 1CCA ∼ or «„ « 1CCA ∼ b. for D ′ n For n even the Clebsch Gordan coefficients for the products × = are the same as in the case of D n , i.e. for i = 3 , n instead of n .If n is odd, the same holds for j odd whereas for j even the ClebschGordan coefficients of the products × and × have tobe interchanged.The Clebsch Gordan coefficients for the products × are thesame as for D n , if i is even. Similarly, the ones of × withi = j are the same, if i, j are both even or one is even and one is odd,if n is even. For n being odd the only difference is that in the casethat the product is of the form × = + + the ClebschGordan coefficients for the covariant combination transforming as and are interchanged.Concerning the structure of the products × = + + with j = min(2 i , n − „ − « ∼ „ « ∼ and «„ − « 1CCA ∼ or «„ − « 1CCA ∼ n -2i If i = n ( n even), then one has × = P j=1 . The ClebschGordan coefficients are „ − « ∼ , „ « ∼ , „ − « ∼ , „ « ∼ × for i , j being odd is either + with k = | i − j | andl = min(i + j , n − (i + j)) or + + with k = | i − j | , ifi + j = n . The Clebsch Gordan coefficients in the first case are: «„ − « 1CCA ∼ or «„ −
10 0 « 1CCA ∼ and «„ − « 1CCA ∼ or «„ − « 1CCA ∼ n -(i+j) In the second one the Clebsch Gordan coefficients are: „ − « ∼ , „ « ∼ and «„ − « 1CCA ∼ or «„ −
10 0 « 1CCA ∼ APPENDIX B: DECOMPOSITION UNDERSUBGROUPS
The decomposition of representations of G F under its subgroupsis given in Table III, Table IV, Table V and Table VI. We haveused the following non-standard convention for the representationof Z n : The representation transforms as e πin (k) , so that denotes the trivial representation and n + k) = 1k .We will denote the components of the two-dimensional representa-tion by ∼ „ a k b k « For two-dimensional representations under dihedral subgroups, wefind in the tables the general identification ∼ . However,dihedral subgroups will have less two-dimensional representationsthan the original group, so we need to make the following identifi-cations, if the dihedral subgroup has no representation :In D j , j even:( e πim j n a j2 + b j2 ) ∼ , ( − e πim j n a j2 + b j2 ) ∼ (B1)( e πim j n a j + b j ) ∼ , ( − e πim j n a j + b j ) ∼ (B2) e πim k n a k b k ! ∼ ( if k < j2 ) (B3) b k e πim k n a k ! ∼ ( if j > k > j2 ) (B4)In D j , j odd:( e πim j n a j + b j ) ∼ , ( − e πim j n a j + b j ) ∼ (B5) e πim k n a k b k ! ∼ ( if k ≤ j −
12 ) (B6) b k e πim k n a k ! ∼ ( if j > k > j −
12 ) (B7)If k > j, ∼ . If j divides k, then transforms justas , i.e. as + . For D ′ j one can make the same identificationsas for D j , j even, if one makes the substitutions j →
2j and n → n . For j dividing k, then transforms as , i.e. as + . APPENDIX C: BREAKING CHAINS FOR D ′ n We give the possible breaking sequences for a double-valued di-hedral group D ′ n . The breaking sequences for D n along with adiscussion of the conventions used is given in Section IV. D n → Subgroup V EV allowed ? Z n = < A > → yes → yes → n → n → a k ∼ , b k ∼ n -k) Z j = < A n j > → yes → yes → n if n j even → n if n j even → a k ∼ , b k ∼ if j | k Z = < BA m > → yes → → m if m even → m +1 if m odd → ( e πim k n a k + b k ) ∼ , ( − e πim k n a k + b k ) ∼ „ e − πi k mn « Z n = < A > → yes → yes → yes → yes → a k ∼ , b k ∼ n -k) TABLE III: Transformation properties of the representations of a dihedral group under its abelian subgroups, as determined in SectionIII. The rightmost column shows whether a representation has a component, which transforms trivially under the subgroup, i.e. if ascalar field transforming under this representation can acquire a VEV, while conserving this subgroup. If only a specific VEV structure isallowed, it is given explicitly, otherwise an arbitrary VEV is allowed. D ′ n < > −→ Z n < > −→ Z n < > −→ Z j ( j | n ) D ′ n < > −→ Z n < > −→ Z j D ′ n < > −→ Z n < > −→ Z j (j | n ; n odd ) D ′ n < > −→ D ′ n < > −→ Z n < > −→ Z j (j | n ; n even ) D ′ n < > −→ D ′ n < > −→ Z n < > −→ Z j (j | n ; n even ) D ′ n < > −→ D ′ n < > −→ Z j (j | n ; n even ) D ′ n < > −→ D ′ n < > ′ −→ Z < > −→ Z (j ∤ n )( m j even and for and exchanged m j is odd) D ′ n < > −→ D ′ n < > ′ −→ Z < > ′ −→ Z (j ∤ n ; m j = m k )( m j even for and for m j is odd) D ′ n < > −→ D ′ n < > ′ −→ D ′ j2 < > −→ Z j (j | n )( m j even - for and exchanged m j is odd) D ′ n < > −→ D ′ n < > ′ −→ D ′ j2 < > −→ Z k (j | n ; k | j)( m j even for and for m j is odd) D ′ n < > −→ D ′ n < > ′ −→ D ′ j2 < > ′ −→ Z < > −→ Z (j | n ; m j = m k ; k ∤ j) ( m j even - for and exchanged, m j is odd) D ′ n < > −→ D ′ n < > ′ −→ D ′ j2 < > ′ −→ Z < > ′ −→ Z (j | n ; m j = m k = m l ; k ∤ j)( m j even for and for m j is odd) D ′ n < > ′ −→ Z < > −→ Z ( m j arbitrary ) D ′ n < > ′ −→ Z < > ′ −→ Z ( m j = m k ) D ′ n < > ′ −→ D ′ j2 < > −→ Z j ( m j arbitrary ) D ′ n < > ′ −→ D ′ j2 < > −→ Z j (j | n )( n odd or m j odd - if → , then n is odd or m j is even.) D ′ n < > ′ −→ D ′ j2 < > −→ Z (j ∤ n ; n even )( m j odd for and for m j is even) D ′ n < > ′ −→ D ′ j2 < > −→ Z < > −→ Z ( n even )( m j even - for and exchanged m j is odd) D ′ n < > ′ −→ D ′ j2 < > −→ Z < > ′ −→ Z ( n even ; m j = m k )( m j even for and for m j is odd) D ′ n < > ′ −→ D ′ j2 < > −→ Z k (k | j; m j arbitrary ) D n → Subgroup V EV allowed ? D n = < A , B > → yes → → yes → → D n = < A , BA > → yes → → → yes → D j = < A n j , BA m > → yes → → ( n j even, m even ) yes ( n j even, m odd ) ( n j odd, m even ) ( n j odd, m odd ) → ( n j even, m odd ) yes ( n j even, m even ) ( n j odd, m odd ) ( n j odd, m even ) → e − πi k mn ! ( if j | k)TABLE IV: Transformation properties of the representations of a dihedral group under its non-abelian subgroups. For further details seecaption of Table III. D ′ n < > ′ −→ D ′ j2 < > ′ −→ Z < > −→ Z ( n even ; m j = m k ; k ∤ j)( m j odd for and for m j is even) D ′ n < > ′ −→ D ′ j2 < > ′ −→ Z < > ′ −→ Z ( m j = m k ( = m l ); k ∤ j) ∗ D ′ n < > −→ Z j < > −→ Z k (k | j) ∗ D ′ n < > ′ −→ D ′ j2 < > ′ −→ D ′ k2 (k | j; m j = m k ) [1] W. M. Yao et la. [Particle Data Group], J. Phys. G ,1 (2006).[2] M. Maltoni, T. Schwetz, M. A. Tortola andJ. W. F. Valle, New J. Phys. , 122 (2004)[arXiv:hep-ph/0405172 v5].[3] P. F. Harrison, D. H. Perkins and W. G. Scott, Phys.Lett. B (2002) 167; P. F. Harrison and W. G. Scott,Phys. Lett. B (2002) 163; Z. z. Xing, Phys. Lett. B (2002) 85; P. F. Harrison and W. G. Scott, Phys.Lett. B (2003) 76.[4] T. Fukuyama and H. Nishiura, hep-ph/9702253;R. N. Mohapatra and S. Nussinov, Phys. Rev. D ,013002 (1999); E. Ma and M. Raidal, Phys. Rev. Lett. ,011802 (2001); C. S. Lam, Phys. Lett. B , 214 (2001);P.F. Harrison and W. G. Scott, Phys. Lett. B , 219(2002); T. Kitabayashi and M. Yasue, Phys. Rev. D ,015006 (2003); Phys. Lett. B , 133 (2005); W. Grimusand L. Lavoura, Phys. Lett. B , 189 (2003); J.Phys. G , 73 (2004); A. Ghosal, hep-ph/0304090;W. Grimus, A. S. Joshipura, S. Kaneko, L .Lavoura,H. Sawanaka, M. Tanimoto, Nucl. Phys. B , 151(2005); R. N. Mohapatra, JHEP , 027 (2004); A. de Gouvea, Phys. Rev. D , 093007 (2004); R. N. Mo-hapatra and W. Rodejohann, Phys. Rev. D , 053001(2005); R. N. Mohapatra, S. Nasri and H. B. Yu, Phys.Lett. B , 231 (2005); Phys. Rev. D , 033007 (2005);Y. H. Ahn, S. K. Kang, C. S. Kim and J. Lee, Phys. Rev.D , 093005 (2006); Phys. Rev. D , 013012 (2007);W. Grimus and L. Lavoura, J. Phys. G , 1757 (2007).[5] E. Ma, Phys. Rev. D , 031901 (2004); G. Altarelliand F. Feruglio, Nucl. Phys. B , 64 (2005);Nucl. Phys. B , 215 (2006); K. S. Babu andX. G. He, arXiv:hep-ph/0507217; I. de MedeirosVarzielas, S. F. King and G. G. Ross, Phys. Lett. B ,153 (2007); X. G. He, Y. Y. Keum and R. R. Volkas,JHEP (2006) 039; E. Ma, Phys. Rev. D ,057304 (2006); F. Bazzocchi, S. Kaneko and S. Morisi,arXiv:0707.3032 [hep-ph].[6] F. Feruglio, C. Hagedorn, Y. Lin and L. Merlo, Nucl.Phys. B , 120 (2007).[7] J. S. Lomont, Applications of Finite Groups , Acad.Press (1959) 346 p.; P. E. Desmier and R. T. Sharp, J.Math. Phys. , 74 (1979); J. Patera, R. T. Sharp andP. Winternitz, J. Math. Phys. , 2362 (1978). D ′ n → Subgroup V EV allowed ? Z n = < A > → yes → yes → n → n → a k ∼ , b k ∼ n -k) Z n = < A > → yes → yes → yes → yes → a k ∼ , b k ∼ n -k) Z j = < A n j > → yes → yes → n if n j even → n if n j even → a k ∼ , b k ∼ if j | k (k can be odd ) Z = < BA m > → yes → → ( n even, m even ) yes ( n odd, m odd ) ( n even, m odd ) ( n odd, m even ) → ( n even, m even ) ( n odd, m odd ) ( n even, m odd ) yes ( n odd, m even ) → ( e πim k n a k + b k ) ∼ , ( − e πim k n a k + b k ) ∼ e − πi k mn ! ( if k even ) Z = < A n > → yes → yes → n if n even → n if n even → a k ∼ , b k ∼ if k even TABLE V: Transformation properties of the representations of a double-valued dihedral group under its abelian subgroups. For thedecomposition of the two-dimensional D ′ n representations under its subgroup Z one has to mention that for k even splits up into and under Z , while for k being odd the representations are and . For further details see caption of Table III. [8] W. M. Fairbairn , T. Fulton and W. Klink, J. Math.Phys. , 1038 (1964); A. Hanany and Y. H. He, JHEP , 013 (1999); T. Muto, JHEP , 008 (1999);W. M. Fairbairn and T. Fulton, J. Math. Phys. , 1747(1982); C. Luhn, S. Nasri and P. Ramond, J. Math. Phys. , 073501 (2007).[9] G. C. Branco, J. M. Gerard and W. Grimus, Phys.Lett. B , 383 (1984); K. C. Chou and Y. L. Wu,hep-ph/9708201; D. B. Kaplan and M. Schmaltz, Phys.Rev. D , 3741 (1994); I. de Medeiros Varzielas,S. F. King and G. G. Ross, Phys. Lett. B , 153 (2007);Phys. Lett. B , 201 (2007); E. Ma, Mod. Phys. Lett. A , 1917 (2006); arXiv:0709.0507 [hep-ph]; T. Kobayashi,H. P. Nilles, F. Ploger, S. Raby and M. Ratz, Nucl. Phys.B , 135 (2007).[10] W. Grimus and L. Lavoura, Phys. Lett. B , 189(2003); W. Grimus, A. S. Joshipura, S. Kaneko,L. Lavoura and M. Tanimoto, JHEP , 078 (2004);G. Seidl, hep-ph/0301044; T. Kobayashi, S. Raby andR. J. Zhang, Nucl. Phys. B , 3 (2005); P. H. Framp- ton and T. W. Kephart, Phys. Rev. D , 086007 (2001);C. D. Carone and R. F. Lebed, Phys. Rev. D , 096002(1999); E. Ma, Fizika B , 35 (2005); S. L. Chen andE. Ma, Phys. Lett. B , 151 (2005); C. Hagedorn,M. Lindner and F. Plentinger, Phys. Rev. D , 025007(2006); P. Ko, T. Kobayashi, J. h. Park and S. Raby,Phys. Rev. D , 035005 (2007); J. Kubo, Phys. Lett. B , 303 (2005); Y. Kajiyama, J. Kubo and H. Okada,Phys. Rev. D , 033001 (2007); note that we excludedpapers about D as it is isomorphic to S .[11] M. Frigerio, S. Kaneko, E. Ma and M. Tanimoto, Phys.Rev. D , 011901 (2005); P. H. Frampton and A. Rasin,Phys. Lett. B , 424 (2000); K. S. Babu and J. Kubo,Phys. Rev. D , 056006 (2005); Y. Kajiyama, E. Itouand J. Kubo, Nucl. Phys. B , 74 (2006); P. H. Framp-ton and T. W. Kephart, Int. J. Mod. Phys. A , 4689(1995); Phys. Rev. D , 1 (1995); P. H. Frampton andO. C. W. Kong, Phys. Rev. Lett. , 781 (1995); Phys.Rev. D , 2293 (1996); Phys. Rev. Lett. , 1699 (1996);M. Frigerio and E. Ma, arXiv:0708.0166 [hep-ph]. D ′ n → Subgroup V EV allowed ? D ′ n = < A , B > → yes → → yes → → D ′ n = < A , BA > → yes → → → yes → D ′ j2 = < A n j , BA m > → yes → → ( n j even, m even ) yes ( n j even, m odd ) ( n j odd, m even ) ( n j odd, m odd ) → ( n j even, m odd ) yes ( n j even, m even ) ( n j odd, m odd ) ( n j odd, m even ) → e − πi k mn ! ( if j | k) , k even TABLE VI: Transformation properties of the representations of a double-valued dihedral group under its non-abelian subgroups. Forfurther details see caption of Table III. [12] W. Grimus and L. Lavoura, Phys. Lett. B , 189(2003).[13] W. Grimus and L. Lavoura, JHEP , 013 (2005).[14] C. Hagedorn, in preparation .[15] J. S. Lomont,
Applications of Finite Groups , Acad.Press (1959) 346 p.[16] P. E. Desmier and R. T. Sharp, J. Math. Phys. , 74(1979).[17] J. Patera, R. T. Sharp and P. Winternitz, J. Math.Phys. , 2362 (1978).[18] S. Califano, Vibrational States , Wiley, London (1976)365p.[19] P. H. Frampton and A. Rasin, Phys. Lett. B , 424(2000).[20] G. Altarelli and F. Feruglio, Nucl. Phys. B , 215(2006).[21] C. Jarlskog, Phys. Rev. Lett. , 1039 (1985).[22] A. Blum, C. Hagedorn, A. Hohenegger, in preparation . [23] W. Grimus, A. S. Joshipura, S. Kaneko, L. Lavoura andM. Tanimoto, JHEP , 078 (2004).[24] F. Caravaglios and S. Morisi, arXiv:hep-ph/0503234.[25] C. Hagedorn, M. Lindner and F. Plentinger, Phys. Rev.D , 025007 (2006).[26] P. H. Frampton and T. W. Kephart, Phys. Rev. D , 1(1995).[27] C. D. Carone and R. F. Lebed, Phys. Rev. D , 096002(1999).[28] G. Altarelli and F. Feruglio, Nucl. Phys. B , 64(2005).[29] C. D. Froggatt and H. B. Nielsen, Nucl. Phys. B ,277 (1979).[30] C. S. Lam, arXiv:0708.3665 [hep-ph].[31] P. H. Frampton and T. W. Kephart, Int. J. Mod. Phys.A10