Bottom-Up Discrete Symmetries for Cabibbo Mixing
DDESY 16-125
Bottom-Up Discrete Symmetries for Cabibbo Mixing
Ivo de Medeiros Varzielas, ∗ Rasmus W. Rasmussen, † and Jim Talbert ‡ School of Physics and Astronomy, University of Southampton,Southampton, SO17 1BJ, U.K. Deutsches Elektronen-Synchrotron (DESY), Platanenallee 6, 15738 Zeuthen, Germany Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road,Oxford, OX1 3NP, U.K.
We perform a bottom-up search for discrete non-Abelian symmetries capable of quantizing theCabibbo angle that parameterizes CKM mixing. Given a particular Abelian symmetry structurein the up and down sectors, we construct representations of the associated residual generatorswhich explicitly depend on the degrees of freedom present in our effective mixing matrix. We thendiscretize those degrees of freedom and utilize the
Groups, Algorithms, Programming (GAP) packageto close the associated finite groups. This short study is performed in the context of recent resultsindicating that, without resorting to special model-dependent corrections, no small-order finite groupcan simultaneously predict all four parameters of the three-generation CKM matrix and that onlygroups of O (10 ) can predict the analogous parameters of the leptonic PMNS matrix, regardlessof whether neutrinos are Dirac or Majorana particles. Therefore a natural model of flavour mightinstead incorporate small(er) finite groups whose predictions for fermionic mixing are corrected viaother mechanisms. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected], [email protected] a r X i v : . [ h e p - ph ] M a r I. INTRODUCTION
Non-Abelian (NA) discrete flavour symmetries are powerful tools in the effort to explain the observed structureof fermionic masses and mixings. In particular, they allow for precise predictions of mixing matrices and, whencoupled with other auxiliary symmetries, can also help organize mass patterns. Flavour models employing discretesymmetries are generically classified as ‘direct,’ ‘semi-direct,’ and ‘indirect’ (see [1] for a review). In the context ofdirect or semi-direct models, one might assume that, at very high energies normally at or above the GUT scale, aparent flavour symmetry G F breaks to subgroups in the quark G Q and lepton G L sectors, which then subsequentlybreak to subgroups in the charged lepton G e , neutrino G ν , up G u and down G d sectors: G F → G L → (cid:40) G ν G e G Q → (cid:40) G u G d (1)This schematic simplifies if G F = G L = G Q , in which case the first arrow disappears and one only considers a singlereduction to the final residual symmetries. If G L and G Q have separate origins, G F can be constructed from the directproduct of the groups that give rise to G L and G Q . Regardless of the breaking patterns, the parent symmetries mustbe NA in order for generations to be arranged in irreducible multiplets and , similarly, the final pattern of residualsymmetries present in the Standard Model (SM) Lagrangian must be Abelian and of order N ≥ number of generations(this requirement is due to the generations having distinct masses and non-trivial mixing).Recently, the bulk of theoretical studies have focused on the leptonic sector, perhaps due to the flux of new experi-mental data indicating that the reactor angle θ is nonzero [4] (see [5–7] for global fits to neutrino mixing observables)and hence simple models based on, e.g., the flavour symmetry A [8–13] must be abandoned or substantially modified[14–27]. Unfortunately, all model-independent scans of the lepton sector indicate that only large groups of O (10 ) canquantize θ within 3 σ , and even larger groups are needed to quantize the full PMNS matrix to a similar accuracy[28–35]. This statement is true for both Majorana and Dirac-type neutrinos, and regardless of whether the discreteflavour symmetry G F is a subgroup of SU (3) or U (3), but only applies to direct models that completely predict themixing angles. This result was confirmed additionally from general group theoretical arguments [36] and also from abottom-up approach [37] which we will discuss in detail below.Furthermore, studies addressing the quark sector are generally performed in light of the leptons. That is, peoplehave searched for flavour symmetries in the quark sector [38–41] that have irreducible triplet representations or thatcan originate from the same groups that work for leptons (e.g. subgroups of ∆(6 N )). Inevitably, as one mightpredict given the extremely hierarchical structure of the CKM matrix, no finite group has been found that can predictall angles and phases of the CKM to any accuracy. Small groups such as D and other variants of the Dihedralfamily can predict the Cabibbo angle [42, 43], but still not within 3 σ . Within this context, it is prudent to considerthe possibility that, if a NA discrete flavour symmetry does exist in nature, it is described by a small group whosepredictions for fermionic mixing are modified, perhaps via Renormalization Group running [46–48] or additionalsymmetry breaking effects as have been studied for leptonic mixing [49–51]. We adopt this philosophy in the presentnote, and focus on finite groups that can predict the Cabibbo angle at leading order.We study Cabibbo mixing in the quark sector by utilizing the approach introduced in [37], which effectively invertsthe arrows in Eq. (1). This method of ‘[re]constructing’ finite flavour groups begins by identifying residual Abeliansymmetries present in the Standard Model Yukawa sector and then building explicit representations of the generatorsof said symmetries. By construction, those matrices depend on the same degrees of freedom present in the mixingmatrices. One can then utilize the GAP system for computational finite algebra to close the groups generated bythe representations. This approach essentially realizes an automation of the studies performed in [52–55], and waspreviously applied to a special case of µ − τ perturbed leptonic mixing. It is particularly useful as a model-buildingtool. The authors of [56] reiterated the bottom-up perspective in a non-automated fashion and also conceptuallyextended it to treat general CP symmetries. By considering all possible charge assignments in models with an Abelian G F ∼ Z n × Z n × ... in a Froggatt-Nielsen [2] scenario, aswas done in [3], one can also achieve realistic mass and mixing relations. However, these predictions are in terms of an unquantizedtexture parameter (cid:15) , unlike the NA models we discuss here which fully predict the mixing matrix. While smaller groups can produce viable leading order CKM matrices e.g. S in [44, 45], the Cabibbo angle is not predicted in suchmodels. This paper begins with a generic discussion of the residual discrete symmetries that are present in the quark masssector of the SM in section II. Section III reviews the bottom-up method of [37], and elaborates the specific details ofthis implementation. Results are presented in section IV before we discuss the trends and limitations of our searchmethod in section V. We give closing remarks in section VI.
II. THE SYMMETRIES OF THE QUARK YUKAWA SECTOR
The philosophy of building NA flavour symmetries via the identification of residual Abelian symmetries present inthe SM Lagrangian has been approached both analytically and numerically over the last couple of years (cf. referencesabove). We adopt this philosophy below to identify the relevant residual symmetries G u and G d of the quark masssector, closely following the discussion and notation of [53].The SM Lagrangian for quark masses is given by: − L = ¯ U R ˆ M U U L + ¯ D R M D D L + h.c. (2)where U L,R ≡ ( u, c, t ) TL,R , D L,R ≡ ( d, s, b ) TL,R and ˆ M U ≡ diag { m u , m c , m t } . Hence we are in the basis where the upquarks are diagonal. It is clear from Eq. (2) that the Lagrangian is invariant under the action of a U(1) symmetry foreach active generation and, noting that U L and D L belong to the same SU (2) L doublet, the natural residual symmetryof both up and down quark mass terms is U (1) . We are currently only interested in discrete flavour symmetries, sowe focus on discrete cyclic subgroups and their direct products: G Q → (cid:40) G u ∼ Z un , Z un × Z un G d ∼ Z dm , Z dm × Z dm (3)We assign G u / d to a single cyclic Z n/m (with ( n, m ) the order of the associated generator) in analogy to the usualchoice made for the charged leptons, or to a direct product group in analogy to the maximal Z × Z symmetry thatexists for Majorana neutrino mass matrices (see e.g. [1]). However, in this case our cyclic generators are of course notbound to be of order two in either the up or down sectors. Denoting the generator(s) of Z u as T l and the generator(s)of Z d as S Di , the actions of the above residual symmetries on the left-handed fields that are relevant for mixing arerepresented by : U L → T l U L (4) D L → S Di D L (5)where for three generations T l = diag (cid:0) e i Φ , e i Φ , e i Φ (cid:1) l where Φ j = 2 π φ j n (6)Both φ j and n are integers, with n representing the order of the generator. In the down sector, S Di are given as therotated generators that depend on the explicit degrees of freedom present in the unitary mixing matrix: S Di ( { Θ k , α j } ) = U CKM (Θ k ) S ( α j ) i U † CKM (Θ k ) (7)where S i are diagonal matrices analogous to Eq. (6) with phases α j and { Θ k } are whatever mixing angles and CPviolating phases are present in U CKM .If we wish to assume that G u , G d ⊂ SU (3) (or SU(2) for the limiting case of LO Cabibbo mixing) we can of courseimpose charge constraints on T and S D such that: (cid:88) j Φ j , α j ≡ mod π (8)However, in this study we make no such constraint and thus look at the relevant U(3)/U(2) groups to be less restrictive. The transformation properties on the right-handed fields are not important for this puprose, as there is no physical right-handed mixingin the SM.
III. A BOTTOM-UP TECHNIQUE FOR CLOSING GROUPS
In this section we first briefly review our procedure for finding phenomenologically viable NA discrete symmetries(though we refer the reader to [37] for a more detailed discussion) before outlining some of the specific details of itsapplication in the quark sector.
A. [Re]constructing finite flavour groups
Having assigned residual Abelian discrete symmetries to the up and down sectors, one is in a position to search forNA symmetries in a bottom-up manner by examining the possible groups closed by the combination of their associated‘residual’ generators. To this end, the basic maneuvers executed by our scripts can be summarized as follows:1.
Discretization:
By construction our generators depend on the same degrees of freedom { Θ k } present in themixing matrices under consideration, which are of course continuous. In order to find finite flavour groups onemust impose a discretization on { Θ k } . Two examples to this end are given by:tan(Θ k ) = (cid:114) c − c (9a)Θ k = cπ (9b)where c ≡ ab and ( a, b ) ∈ Integers . In the first scheme there is only the single parameter c ( c ∈ [0 , π ( a ≤ b ) to avoid any degeneracy . Our ‘scans’ are then implementedover the variables ( a, b ) — as the range of the examined ( a, b ) parameter space grows, so does the number ofgenerators we construct and by proxy the number of potential finite groups we can close.2. Experimental Constraints:
We can look to experimental constraints to limit/tune the (a,b) search space inorder to find groups that quantize phenomenologically relevant mixing parameters. Experimental data is oftenpresented with respect to the PDG parameterization of U CKM , and hence the most general constraint we canmake on any matrix element is given by: (cid:107) U P DGmin (cid:107) (cid:53) (cid:107) U ij ( c k ) (cid:107) (cid:53) (cid:107) U P DGmax (cid:107) (10)In words, we can insist that no matrix element be any greater/smaller than the largest/smallest (experimentallydetermined) elements of U P DG . We have inequalities in Eq. (10) as opposed to equalities because the finiterepresentations of the residual symmetries know nothing of the ordering of rows and columns (this constraint isrelatively generic because while the discrete group can predict entries in the mixing matrix, it can not predicttheir placement). Thus while we can constrain matrix elements within xσ (where x is an arbitrary integer),we may not immediately predict the mixing angles to within xσ , and require an additional cut at the end ofthe search. One may of course impose Eq. (10) on multiple elements of a given class. A common practice inthe leptonic sector is to assign the smallest predicted entry of the mixing matrix to the (13) entry, which is thesmallest according to observation. However, in cases where one attempts to predict a specific 2 × GAP Implementation:
We wish to build explicit representations of T l and S Di in GAP, and hence we musttranslate our parameterizations of { Θ k } given by Eq. (9) into GAP objects. For the second discretizationscheme, Eq. (9b), this amounts to creating lists of the following:cos ( c ) = E (2 b ) a + E (2 b ) − a c ) = E (2 b ) a − E (2 b ) − a E (4) (11b) As it turns out, systematic studies of finite subgroups of SU (3) [41] show that Eq. (9b) is a rather comprehensive scheme for discretizingthe possible mixing angles for both quarks and Dirac neutrinos. Therefore will we only consider Eq. (9b) in this study. Note howeverat least one relevant counter-example is the canonical tri-bimaximal mixing form, which would require a discretization along the linesof Eq. (9a), with c = [37]. where E returns the primitive N-th root of unity, E ( N ) ≡ e πiN . Additional parameterizations would obviouslylead to more variants of Eq. (11) .4. Generator Formation:
Form the explicit representations of viable S Di Eq. (7) and T l Eq. (6) via Eq. (11).5.
Close the Groups:
Now that we have the relevant GAP representations of S Di and T l in a specified interval of( a, b, φ j , α j , n, m ) and a user-determined experimental σ -range, we wish to close the groups G F / Q / L generatedby them. GAP is capable of constructing groups directly from matrix representations of generators using the GroupW ithGenerators command. As we will discuss below, for instance, the idea of this paper is to form allgroups closed by G Q = { S D , S D , T , T } (12a) G Q = { S D , S D , T } (12b) G Q = { S D , T , T } (12c) G Q = { S D , T } (12d)Eq. (12b) treats the case where G d ∼ Z dm × Z dm and G u ∼ Z un whereas Eq. (12c) treats the case where G d ∼ Z dm and G u ∼ Z un × Z un , and so on.6. Analyze:
Not all groups closed will be finite, of small-order, NA, etc. GAP contains a host of internal commandsthat, given a group structure, can be used to filter results based on user-defined preferences. For our purposeswe are only concerned with small(ish), finite, NA groups. We impose cuts to that end (details below), and thenidentify the remaining flavour symmetry candidates with the
GroupID and
StructureDescription commands .Our scripts carefully keep track of the parameters { a, b, α j ... } associated to the final group structure we identify,so that we might have explicit information on the representations of the residual generators, which is of courserelevant at the model-building stage. B. Specific details of this study
The CKM mixing matrix is given in the Wolfenstein parameterization [57] by: U CKM = − λ / λ Aλ ( ρ − iη ) − λ − λ / Aλ Aλ (1 − ρ − iη ) − Aλ + O ( λ ) (13)Since λ = . + . − . , A = . + . − . , ¯ ρ = . + . − . and ¯ η = . + . − . [57] (where ¯ ρ = ρ (1 − λ / ... ) and¯ η = η (1 − λ / ... )) [58], we find | U CKM | (cid:39) (cid:0) . . (cid:1) (cid:0) . . (cid:1) (cid:0) . . (cid:1)(cid:0) . . (cid:1) (cid:0) . . (cid:1) (cid:0) . . (cid:1)(cid:0) . . (cid:1) (cid:0) . . (cid:1) (cid:0) . . (cid:1) (14)The hierarchical nature of the quark mixing matrix is now obvious; exterior off-diagonal elements are suppressed byone to two orders of magnitude and the upper 2 × SO (2) rotation aboutthe Cabibbo angle: U LOCKM (cid:39) (cid:18) cos θ C sin θ C − sin θ C cos θ C (cid:19) (15) For example, were we to also consider the first scheme Eq. (9a), the corresponding GAP objects would look like:cos (Θ ( c )) = ER (cid:16) − ab (cid:17) sin (Θ ( c )) = ER (cid:16) ab (cid:17) where ER is a square root operation for a rational number N , √ N . Note that
StructureDescription is not an isomorphism invariant command — two non-isomorphic groups can return the same groupstructure string while isomorphic groups in different representations can return different strings. It is designed to primarily study smallgroups of O ( G ) (cid:46) GroupID is unique.
Such a matrix does not exhibit CP violation. Considering the numerical values of Eq. (14) and the fact that nodiscrete group has been found that quantizes them, it makes sense to study only Eq. (15) with the bottom-uptechnique described above. Given the symmetry assignments of Eq. (3) and Eq. (6), we find explicit forms for theeffective 2-generation S Di . S Di = e iα i cos θ C + e iα i sin θ C (cid:0) e iα i − e iα i (cid:1) cos θ C sin θ C (cid:0) e iα i − e iα i (cid:1) cos θ C sin θ C e iα i cos θ C + e iα i sin θ C (16)In the event that G d ∼ Z dm and not a direct product, the index i is meaningless. We clearly then have 3-5 degreesof freedom that need to be discretized in the down sector via Eq. (9b), { α i , α i , θ C } , and of course 2-4 degrees offreedom in the up sector, { Φ l , Φ l } . Then, for each physical degree of freedom discretized using Eq. (9b), there aretwo corresponding integers c = ab which must be scanned over in the bottom-up approach. For all phases α and Φ werestrict a ∈ {− , , } and b ∈ { ... Max( O ( T l , S i )) } where b, for diagonal matrices, also represents the order of thegenerator. It must be at least two so that generations can be distinguished, and its maximum value is user-definedand specified below for various scans. We vary both the discretization parameter ranges and allowed quantizationrange associated to the physical mixing angle θ C in each scan. IV. RESULTS
In this section we present our results and some discussion given the four assignments for the residual symmetries G d / u . In each subsection we reference tabled results of the groups found when searching within the parameter rangesdiscussed above and/or below. The first column of each table gives the parameter c , which is a direct proxy for theCabibbo angle. The following 2-3 columns give the diagonal entries of the 2 × T l and S i , as discussed in Section II. There are three columns when either G u or G d is a direct product. The fourth (fifth)column gives the unique ID of the given group closed as labeled in the GAP system, and the following column theassociated group structure as given by the StructureDescription command. D N corresponds to the Dihedral groupof order N, Q N to quarternions of order N, and QD N to Quasi-Dihedrals of order N. We also remind the reader ofthe isomorphism structure of Σ(2 N ) groups (see Appendix A),Σ(2 N ) ≡ ( Z N × Z N (cid:48) ) (cid:111) Z (17)and for simplicity we have also arbitrarily named the following groups:Ψ( N, M ) ≡ ( Z N × Z M ) (cid:111) Z Finally, the last column gives the value for sin θ C quantized by the group. In all tables we only present results withnon-trivial charge reassignments in the residual symmetry generators and non-trivial permutations of the parameter c .That is, we do not show results where the same group quantizes the same mixing matrix, but with different diagonalmatrix elements in T l or S i , or results with explicitly different c but equivalent sin( cπ ). A. G d ∼ Z dm , G u ∼ Z un We begin by assigning a single cyclic symmetry to both the up and down sectors, which reflects the simplestpossible discrete symmetry scenario, and scanning over the possible NA finite groups closed with the associatedgenerator representations. We present the scan results in Tables I and II, which are also discussed in more detail herethan in following sections.In Table I we allow for a rather large window for the Cabibbo angle, . ≤ sin θ C ≤ . a, b ∈ { , ... } , choices which whencombined yield 52 values of the parameter c . The order of the residual generators is restricted to O ( T, S ) ≤
4, which(given the choices for the phase parameters described above) yields 19 unique diagonal generators to be distributedto both the up and down sectors. This means there are 19 ·
52 = 988 unique non-diagonal generators S D in thedown-sector and 19 ·
52 = 18772 different combinations of generators that could potentially close NA finite groups.To quicken the scans, we first confirm that O ( S D · T ) < ∞ , as will be the case for any finite group generated by S D We test all such combinations for other symmetry assignments where more generators are considered. and T . Then, as our stated goal is to primarily search for small flavour groups, we restrict the order of the parentgroup to O ( G Q ) ≤
75. Table I gives our results given these ‘bottom-up’ inputs. In this and other Tables containingresults one can clearly identify some cases that are subgroups of other groups listed. One sees that a host of groupstructures are obtained with D , D and Z (cid:111) Z providing the best prediction of sin θ C (cid:39) . c = 1 / D n and D n predict the same Cabibbo angle because the order is linked to an integer multiple of thedenominator of the input parameter c for the groups. We find that other semi-direct products, Ψ, and Q groups allpredict less interesting values for sin θ C .One may also observe that D is generated for different Cabibbo angles (sin θ C (cid:39) . θ C (cid:39) . D is also generated for different Cabibbo angles (sin θ C (cid:39) . θ C (cid:39) . n is capable of predicting n angles depending onwhich subgroups are left as residual symmetries. For the particular cases of 2 n = 46 and 2 n = 62 there would be23 and 31 predictions available, and it just happened that two of those were within the . ≤ sin θ C ≤ . Z residuals and 2-d irreps., we consistently reconstruct Dihedral groups of an order related to thedenominator b of the discretization parameter c , and see that the 2-d irrep. has a familiar geometrical interpretation.In detail, with θ C = ab π , we reconstruct a 2 × g rot of the group with determinant det ( g rot ) = 1 (usuallythis can be T S D , if both T and S D have determinant − g rot can be seen, through the use of trigonometric identities, to be a geometrical rotation byan angle that is an integer multiple of b π of order n : g nrot is the identity. The n elements that are rotations (withpositive determinant) can be obtained by taking the powers of g rot . The other n elements of the Dihedral group arereflections (with negative determinant) and can be obtained by multiplying each of the distinct rotations by one ofthe reflections.These results can be compared to Table II, where we tighten the Cabibbo window to . ≤ sin θ C ≤ . a, b ∈ { , ... } . We now only find 8 allowable values for c ranging from to . We further restrict O ( T, S ) ≤ O ( G Q ) ≤ c will intuitively generate much larger groups than before. Indeed, we nowfind only larger Dihedral groups with the smallest ones being D and D . However, these groups obviously yieldbetter predictions for sin θ C — all groups except D and D showing up in Table II predict angles that fall withinthe PDG allowed ranges in Eq. (14). Were we to allow for an even finer gridding of a / b , we should expect to be ableto find Dihedral groups predicting ever more precise mixing angles. D and other Dihedral groups have been known in the literature for some time [42, 43]. Our approach revealshow generating them is nearly a trivial matter. Consider the original mixing matrix Eq. (15), which represents an SO (2) rotation in the Cabibbo plane. This can obviously be thought of as a circle, and quantizing θ C to a rationalmultiple of π corresponds to carving regular polygons out of said circle. Dihedral groups encode the symmetries ofpolygons ( D is the symmetry of a square, e.g.), so it is no surprise that they show up throughout our scans. It isalso no surprise that a finer gridding in the discretization parameters generates larger groups; the number of sidesof the associated polygons increases. Given the order of the Dihedral groups D and D found in Table II, werestrict O ( G F ) (cid:46)
75, since a Dihedral group will always be able to trivially predict a mixing angle in a given range,if the order is high enough. Also note that Dihedral groups are not found in more universal, top-down scans like thatin [41] because most such studies insist that G Q contain 3-D irreducible representations.Given the final results in Tables I and II, one can then directly reconstruct the explicit generator representations(in an appropriate basis) that work for realistic direct and semi-direct discrete models of flavour. As an example,consider line 13 of Table I, where we immediately read off that that the numerical mixing matrix U LOCKM (cid:39) (cid:18) . . − . . (cid:19) (18)is predicted from the NA finite group Z (cid:111) Z ( SmallGroup (28 , T (28 , = − i i S (28 , D = − i cos π i sin π i sin π i cos π (19)where we made use of some trigonometric identities. In direct and semi-direct flavour models, the vacuum expectationvalues of various ‘flavons’ must be invariant under the operation of the group elements corresponding to these matrices(so that the broken family symmetry reproduces the data at the level of the Standard Model Lagrangian Eq. (2)).Hence the bottom-up method can be particularly useful for model-builders. c T diag S i GAP-ID Group Structure sin θ C [-1, 1] [-1, 1] [22, 1] D . [1, -1] [-1, 1] [44, 3] D . [-i, i] [-i, i] [44, 1] Z (cid:111) Z . [-1, 1] [-1, 1] [24, 6] D . [-i, i] [-1, 1] [24, 8] Ψ(6 , . [-i, i] [-i, i] [24, 4] Z (cid:111) Q . [-1, 1] [-1, 1] [26, 1] D . [1, -1] [-1, 1] [52, 4] D . [-i, i] [-i, i] [52, 1] Z (cid:111) Z . [-1, 1] [-1, 1] [28, 3] D . [-i, i] [-1, 1] [56, 4] Z × D . [1, -1] [-1, 1] [14, 1] D . [-i, i] [-i, i] [28, 1] Z (cid:111) Z . [-1, 1] [-1, 1] [30, 3] D . [1, -1] [-1, 1] [60, 12] D . [-i, i] [-i, i] [60, 3] Z (cid:111) Z . [-1, 1] [-1, 1] [42, 5] D . [-1, 1] [-1, 1] [46, 1] D . [-1, 1] [-1, 1] [50, 1] D . [-1, 1] [-1, 1] [54, 1] D . [-1, 1] [-1, 1] [58, 1] D . [-1, 1] [-1, 1] [62, 1] D . [-1, 1] [-1, 1] [62, 1] D . [-1, 1] [-1, 1] [64, 52] D . [-i, i] [-1, 1] [64, 53] QD . [-i, i] [-i, i] [64, 54] Q . [-1, 1] [-1, 1] [68, 4] D . [1, -1] [-1, 1] [34, 1] D . [-i, i] [-i, i] [68, 1] Z (cid:111) Z . [-1, 1] [-1, 1] [70, 3] D . [-1, 1] [-1, 1] [74, 1] D . [1, -1] [-1, 1] [38, 1] D . [1, -1] [-1, 1] [46, 1] D . U LOCKM , where G d ∼ Z m , G u ∼ Z n with m, n < O ( G Q ) ≤
75. We display outcomeswith distinct groups and sin θ C (for each case there were duplicates where different T and S generators from the ones shownresult in the same group and same physical angle). As a final note, the familiar reader may question why Tables I and II do not contain a greater diversity of groupstructures . For example, it is well-known that A , the alternating symmetry of the tetrahedron, has been used topredict unit (i.e. trivial) mixing in the quark sector [8–12] , which may be a reasonable first-order approximationto U CKM . Yet it is clear that we will never obtain this prediction with our approach. Unit mixing translates to adiagonal down-sector generator Eq. (7), which when combined with the diagonal up-sector generator Eq. (6) willnever close a NA finite group, regardless of the associated charges — diagonal matrices commute. As another example,consider S , the symmetry group of the triangle. We do not find it in Tables I or II, but it has a single two-dimensional In both Tables I and II we have restricted the O ( T, S ) to the same maximum value. One may wonder whether more interesting structurescan be found by allowing one subgroup to have a larger maximum order. We have performed a scan along these lines where O ( T ) ≤ O ( S ) ≤
4. We again put a, b ∈ { , ... } , . ≤ sin θ C ≤ .
24, and restrict ( G Q ) ≤
75. With these inputs we find no new groupstructures and no new predictions for the Cabibbo angle. This is expected from other groups, when both sectors are broken to the same subgroup. c T diag S i GAP-ID Group Structure sin θ C [-1, 1] [-1, 1] [110, 5] D . [1, -1] [-1, 1] [220, 14] D . [-1, 1] [-1, 1] [138, 3] D . [1, -1] [-1, 1] [276, 9] D . [-1, 1] [-1, 1] [166, 1] D . [1, -1] [-1, 1] [332, 3] D . [-1, 1] [-1, 1] [194, 1] D . [1, -1] [-1, 1] [388, 4] D . U LOCKM , where G d ∼ Z m , G u ∼ Z n with m, n ≤ O ( G Q ) ≤ θ C (for each case there were duplicates where different T and S generators from the onesshown result in the same group and same physical angle). irreducible representation, that can be generated by two matrices that fit into the forms of Eq. (6) and Eq. (16) (thetwo-dimensional form of Eq. (7)). This absence is due to limits we put on the Cabibbo quantization window — S predicts a much larger value for sin θ C than .3 (.7071). In Appendix A we look at non-physical values of the Cabibboangle and show that, indeed, many other group structures can be found using our method. In Section V we brieflydiscuss the sensitivity of the method to user-defined parameter choices. B. G d ∼ Z dm × Z dm , G u ∼ Z un We now enlarge the symmetry assignment in the down sector by allowing a direct product of cyclic groups, in analogyto the Z × Z symmetry of the Majorana neutrino mass matrix. We again allow a, b ∈ { , ... } , O ( T, S , S ) ≤ O ( G Q ) ≤
75, but restrict the Cabibbo window to . ≤ sin θ C ≤ .
24 to obtain predictions closer to the experimentalvalue. The results are presented in Table III, where we see that the only new group found in comparison to TableI is Z × D , which also predicts sin θ C (cid:39) . S i are clearly degenerate, and therefore not capable of distinguishing the two generations. In any event, from themodel-building perspective, this group is also not an interesting result as it does no more work than D .It is of course not surprising that we do not find any new quantizations of sin θ C , as this is totally controlled bythe range in a / b scanned and the Cabibbo window, which were chosen to be the same as (or contained within) thoseused for Table I. It is also not concerning that, for example, D is ‘generated’ by three matrices when it is wellknown that Dihedrals can be closed with only two. After all, a finite group G F can be ‘generated’ by as many as O ( G F ) elements! So, when we say that Dihedrals have two generators, we mean that the smallest set of generatingelements for Dihedral groups is O (2). Indeed, due to the internal ordering of group elements, if one asks GAP for thegenerators f i of SmallGroup (28 ,
3) corresponding to D , a three element set is returned : GeneratorsOf Group ( SmallGroup (28 , f , f , f ] (20)However, GAP also knows that there is a smaller subset of these three ‘generators’ that will also do the job: M inimalGeneratingSet ( SmallGroup (28 , f , f · f ] (21)The very same reasoning can also be applied in reverse to Table I, where the group Ψ(6 ,
2) would normally be assignedthree generators to better reveal its structure in terms of three cyclic symmetries (( Z × Z ) (cid:111) Z ), yet can in factbe generated by two. ∆(27) ∈ ∆(3 N ) (( Z × Z ) (cid:111) Z ), a popular group for model-building in the leptonic sector[59–61], is a well known example of this. In creating Table III we have otherwise filtered results where generators carry degenerate eigenvalues. Every group present in the tablecould have also been generated by these physically uninteresting matrices. Even Abelian groups like Z will sometimes return multi-element generator sets with the GeneratorsOfGroup command. c T diag S i S i GAP-ID Group Structure sin θ C [-1, 1] [-1, 1] [1, -1] [52, 4] D . [-i, i] [-i, i] [i, -i] [52, 1] Z (cid:111) Z . [-1, 1] [-1, 1] [1, -1] [28, 3] D . [-1, 1] [-1, 1] [-i, i] [56, 4] Z × D . [-i, i] [-i, i] [i, -i] [28, 1] Z (cid:111) Z . [1, -1] [E(3) , E(3) ] [-1, 1] [42, 4] Z × D . [-1, 1] [-1, 1] [1, -1] [60, 12] D . [-i, i] [-i, i] [i, -i] [60, 3] Z (cid:111) Z . U LOCKM , where G d ∼ Z m × Z m , G u ∼ Z n with m, n < O ( T, S ) < O ( G Q ) ≤ θ C (for each case there were duplicates where different T and S generatorsfrom the ones shown result in the same group and same physical angle). C. G d ∼ Z dm , G u ∼ Z un × Z un We also naively scan the symmetry assignment corresponding to two up-sector residual generators, as opposed totwo (non-diagonal) down-sector generators. Utilizing the same parameter ranges as in Section IV B, we find the exactsame results as those presented in Table III, with T ↔ S . This result is unsurprising, as any physical symmetry mustbe basis independent, and moving between the two symmetry assignments in Sections IV B and IV C requires nothingmore than a basis transformation. To see this, simultaneously rotate the 3 generators of Section IV B with the inverseof the operation in Eq. (7) (where we implicitly chose a basis to work in): { S D , T , T } −→ U † CKM { S D , T , T } U CKM ≡ {
S, T D , T D } (22)where T D and T D are non-diagonal generators analogous to S Di given in Eq. (7). However, we are of course entirelyfree to relabel our generators; T l and S i are both diagonal matrices sourced from equivalent lists of all possible chargepermutations in Eq. (6): { S, T D , T D } −→ T ↔ S { T, S D , S D } (23)We have now arrived at the generator set for the symmetry assignment in Section IV B. D. G d ∼ Z dm × Z dm , G u ∼ Z un × Z un As a final check, we also scan the symmetry assignment where two generators are assigned to the up and downsector. We keep the same input parameters as in Section IV B. Although more groups are closed (given the largernumber of generators), after excluding the redundant cases (with the same angle and same G Q ) the results are againthe same as in the previous two sections — the additional generator in either the up or down sector does no work forus, at least within the parameter ranges we choose. E. Looking for broken symmetries — a consistency check
The groups we find are sourced from the explicit representation of the residual generators, Eq. (6) and Eq. (7).The method is ignorant of what these matrices actually represent, i.e. the symmetry assignments of the physicalLagrangian. Hence, from a completely agnostic perspective, we might also use the bottom-up method to analyze thegenerator associated with the upper 2 × S λDi = e iα i λ + e iα i ( λ − (cid:0) e iα i − e iα i (cid:1) ( λ − λ ) (cid:0) e iα i − e iα i (cid:1) ( λ − λ ) e iα i λ + e iα i ( λ − (24)While this generator reflects a trivial rewriting of the original mixing matrix and only changes the numerical values1of its elements by small amounts (for substantially small λ ), it is a priori entirely plausible that the (exact) structuresof Eq. (7) and Eq. (24) for a given quantized value of θ C /λ generate different parent groups G Q when closed with T .That is, minor numerical shifts of | V LOij | might be sourced by entirely different group structures. However , Eq. (24) reflects quark mixing that is only unitary up to O ( λ ): V λ V † λ = (cid:32) O ( λ ) 00 1 + O ( λ ) (cid:33) (25)and hence does not generate a symmetry of the Lagrangian. One might then be tempted to interpret it as a ‘broken-symmetry’ generator. Regardless, we would not expect such a matrix to actually close a finite mathematical group,as the generator itself should not be of finite order, O ( S λDi ) = ∞ . Indeed, upon running our scripts with Eq. (6) andEq. (24) as the potential group generators, we find that no NA finite flavour groups are closed. V. GENERAL TRENDS AND LIMITATIONS OF THE BOTTOM-UP TECHNIQUE
While the bottom-up technique described above is a powerful tool that can be used to rapidly identify viable NAdiscrete symmetries useful for model-building, we here discuss some of its limitations. Regarding physics, the methodonly applies to direct and semi-direct models, respectively those that either predict all angles in the mixing matrix(in this case leaves no freedom in the 2x2 submatrix) or to those models that predict a column of the mixing matrix(like tri-maximal mixing matrices in the case of leptons [37]). The method does not apply to cases where the specificresidual symmetries are not subgroups of the actual flavour symmetry of the model (referred to as indirect models[1]).The method is also sensitive to the user-defined input parameters, including the scan ranges for the various a / b (related to the discretization of the phases and θ C ), the allowed quantization range for sin θ C , the maximum allowedorder for G u/d , and the maximum allowed order for G F . Widening or increasing any of these parameters quicklyproduces many more group closures, and hence also slows operations. Figure 1 plots an independent variation ofeach of these four ‘tunes’ (given the symmetry assignment in Section IV A) against the number of finite, NA groupsclosed (all closed groups are displayed, whether they are duplications or not). These plots are meant as a qualitativeillustration of the growth of group closures. We see that increasing the scan ranges of a / b ( a max - allowing for a finergridding of θ C ) and widening the allowed range of sin θ C produces a roughly linear increase in group closures, whereasincreasing the allowed order of the parent symmetry eventually plateaus (Figure 1D). This plateau is sensible; therewill only be a limited number of finite groups closed when all constraints are also finite. Had we increased the valueof a max to 35 in Figure 1D, for example, the plateau would occur at 120 groups for M axOrder ( G F ) ≥ O ( G F ). Thisbehavior is less intuitive, though clearly an artifact of our constraint on sin θ C . This can be seen from Table IV wheregenerators of order 3 appear even though the prediction of the Cabibbo angle is un-physical. Therefore, relaxing theCabibbo window would increase the number of groups closed from O ( T, S ) ≤ O ( T, S ) ≤ . ≤ sin θ C ≤ .
3, we also ran two other scans where the effective number ofvalues for the Cabibbo angle, θ C , encoded by rational parameter c , are reduced to four and one (there are 10 activeCabibbo angles in Figure 1c). In both instances we see plateaus beginning at O ( T, S ) ≤ O ( T, S ) ≤
4, and inthe single- c scan the final plateau remains up to O ( T, S ) ≤ O ( T, S ) ≤ c scan).Intriguingly, there are plateaus at 4 × ( of c (cid:48) s ) and 4 × ( of c (cid:48) s ) in all three scans. So, there are plateaus at 4and 16 group closures for one active c , 16 and 64 group closures for four active c ’s, and 40 and 160 group closures for10 active c ’s. We have checked that there are (as must be the case) more closures of Abelian finite groups as O ( T, S )increases, but not the NA groups that we are interested in.Another difficulty (not necessarily associated to the bottom-up technique) arises when the angles considered arevery small, as the order of the predictive group then increases significantly as do the computational weights of theassociated GAP objects. This correlation between group order and associated mixing angle is illustrated clearlyin Table II, where the increase in precision of the predicted angles came with an associated increase in the orderof the groups. This is easy to understand for Dihedral groups due to the associated geometric interpretation interms of polygons. It follows then that to obtain a small angle, one naturally needs to have a respectively small c parameter. For example, to quantize the smallest quark mixing angle - θ q ≈ π/
900 - we obtain Dihedral groups with
Order ( G F ) (cid:38) O (1000). This can then be treated as a lower bound on the order of groups necessary to quantize thefull CKM matrix, and all associated degrees of freedom. Indeed, we performed a short, dedicated “bottom-up” scanto hunt for discrete symmetries capable of quantizing the full CKM matrix, yet do not find any groups identifiableby GAP and its Small Groups
Library, given our parameter ranges and computational expense. These results are2
10 20 30 40 50 60050100150200250300350 � ��� � � � � � � � � � � ��� � � ��� ≤ ��� θ � ≤ �������� ( � � ) < ∞ ����� ( �� � ) ≤ � ��� ( ��� θ � ) � � � � � � � � � � ��� � � ����� ( �� � ) ≤ ��� = � ��� ����� ( � � ) < ∞ �������� ( �� � ) � � � � � � � � � � ��� � � ��� ≤ ��� θ � ≤ ����� = � ��� ����� ( � � ) < ∞
50 100 150 200 25001020304050 �������� ( � � ) � � � � � � � � � � ��� � � ��� ≤ ��� θ � ≤ ����� = � ��� ����� ( �� � ) ≤ � FIG. 1: Tables showing the number of parent groups G F found when varying four inputs to the ‘bottom-up’ approach, namelythe discretization parameters a and b (A), the allowed quantization range of sin θ C (B), the maximum allowed order of theresidual Abelian symmetry groups O ( G u , G d ) (C), and the maximum allowed order of the parent NA symmetry group O ( G F )(D). In each case the other 3 inputs are left fixed to the values shown in the tables. The number of groups given representsthe number of raw groups closed by the method, and does not include any trimming of charge degeneracies, etc. The curvesrepresent first-order interpolations of the data, and are present as a visual aid only — they do not represent any theory. consistent with previous studies and with the naive estimate of the necessary order of the predictive group given above– if a relevant finite group had an order smaller than O (1000), our approach should also be able to find it. VI. CONCLUSIONS
We have applied the bottom-up [re]construction procedure of [37] to scan over possible NA finite groups G Q capableof quantizing the Cabibbo angle of CKM mixing. This study complements other ‘top-down’ scans which, by virtue ofthe restrictions put on the irreducible representations of the parent symmetry or other theory biases (e.g. searchingfor groups that also work for the leptons), do not find or otherwise obscure interesting small groups that can do thesame job. After all, no group has been found that can fully quantize Eq. (14), and theorists interested in using NAfinite groups in the quark sector should therefore consider the possibility that such symmetries, if natural, may makepredictions that are substantially corrected via other mechanisms.Our scans find multiple candidate groups for G Q in Tables I-III , including small semi-direct product structures like Z (cid:111) Z and Z (cid:111) Q , Ψ(6 , S and Σ(32) found in Table IV. Our results seem consistent with former studiesof quark mixing, modulo our starting point of 2 dimensional representations for residual generators in the up anddown sector. For larger groups of O (10 ) we can reproduce the PDG values for the (12) and (21) matrix elements of U P DGCKM . We thus also validate the [re]construction procedure, which may be of further use model-building both withinStandard Model and BSM mixing scenarios.3 c T diag S i GAP-ID Group Structure sin θ C [-1, 1] [-1, 1] [8, 3] D . [E(3) , 1] [-1, 1] [18, 3] Z × S . [-i, 1] [-1, 1] [32, 11] Σ(2 · ) . [E(3) , E(3)] [-1, 1] [6, 1] S . [-i, i] [E(3) , 1] [36, 6] Z × ( Z (cid:111) Z ) . [-i, i] [E(3) , E(3)] [12, 1] Z (cid:111) Z . [-i, i] [-i, i] [8, 4] Q . [-1, 1] [-1, 1] [14, 1] D . [-i, i] [-1, 1] [56, 4] Z × D . [1, -1] [-1, 1] [28, 3] D . [-i, i] [-i, i] [28, 1] Z (cid:111) Z . U LOCKM where G u/d ∼ Z n/m with O ( T, S ) ≤ O ( G F ) ≤
75. We have searched the(non-physical) range . ≤ sin θ C ≤ . VII. ACKNOWLEDGMENTS
This project is supported by the European Union’s Seventh Framework Programme for research, technological devel-opment and demonstration under grant agreement no PIEF-GA-2012-327195 SIFT. RWR appreciates the discussionand helpful comments from Walter Winter. JT is grateful to the University of Southampton, where collaboration onthis project began, and to Prof. G.G. Ross for ongoing discussions and encouragement. JT acknowledges supportfrom the Senior Scholarship Trust of Hertford College, University of Oxford.
Appendix A: Symmetries for other angles
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4, and O ( G Q ) ≤
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