Universal Democracy Instead of Anarchy
aa r X i v : . [ h e p - ph ] A p r CERN-PH-TH/2011-294
Universal Democracy Instead of Anarchy
G.C. Branco a , b , , N. Raimundo Ribeiro b , J.I. Silva-Marcos b , a CERN Theory Division,CH-1211 Geneva 23, Switzerland b Centro de F´ısica Te´orica de Part´ıculas, CFTP, Departamento de F´ısica,Instituto Superior T´ecnico, Universidade T´ecnica de Lisboa,Avenida Rovisco Pais nr. 1, 1049-001 Lisboa, Portugal
PACS numbers : 12.10.Kt, 12.15.Ff, 14.65.Jk
Abstract
We propose for the flavour structure of both the quark and lepton sectors, the principleof Universal Democracy (UD), which reflects the presence of a Z symmetry. In thequark sector, we emphasize the importance of UD for obtaining small mixing and flavouralignment, while in the lepton sector large mixing, including the recently measured valueof U e , is obtained in the UD framework through the seesaw mechanism. An interestingcorrelation between the values of U e and sin ( q ) is pointed out, with the prediction of sin ( q ) ≈ . in the region where U e is in agreement with the DAYA-BAY experiment. E-mail: [email protected] E-mail: [email protected] E-mail: [email protected]
Introduction
The recent discovery [1] of a relatively large U e , together with an indication that atmosphericneutrino mixing is not maximal, has had a significant impact on attempts at understandingthe principle, if any, behind the observed pattern of fermion masses and mixing. Until recently,leptonic mixing was in agreement with the Ansatz of tri-bi-maximal mixing (TBM) [2] in theleptonic sector, which in turn triggered the suggestion of various family symmetries like A [3]which can lead to TBM. Since in this ansatz U e vanishes, the recent discovery of a relativelylarge U e , rules out the TBM scheme. This has led to an intense activity in the construction ofmodels which can accommodate a non-vanishing U e , within a variety of frameworks [4]. It hasalso been suggested that the recent data on leptonic mixing enforces the idea that in fact thereis no symmetry principle behind the observed leptonic mixing and instead anarchy [5] prevailswith the mixing arising from a random distribution of unitary × matrices..In this paper instead of anarchy, we advocate the principle of universal democracy (UD) forfermion masses and mixing, with UD prevailing both in the quark and lepton sectors. TheUD principle reflects the presence of a Z family symmetry, which leads in leading order tofermion mass matrices proportional to the so-called democratic flavour structure, where allmatrix elements have equal value. Previously, the democratic flavour structure has been appliedto the quark sector [6]; here we point out that starting with the same democratic flavourstructure for all leptonic mass matrices, namely charged lepton, Dirac neutrino and right-handed Majorana mass matrices, one can reproduce the observed pattern of leptonic massesand mixing, accommodating in particular the recently measured value of U e .This paper is organized as follows. In the next section we present the UD framework. In section3, we show how to obtain natural democracy through a Z family symmetry and discuss boththe quark and lepton mixing. Our numerical results are combined in section 4 and in section 5we present our conclusions. We suggest that all fermion mass matrices are in leading order proportional to the so-calleddemocratic matrix, denoted by D , with all elements equal to the unit: D = (1)For simplicity, we assume that the neutrino masses are generated in the framework of anextension of the Standard Model (SM), consisting of the addition of three right-handed neutrinos1o the spectrum of the SM: − L = Y i jl L i f l jR + Y i jD L i ˜ f n jR + n TiR C ( M R ) i j n jR + h . c . , (2)where L i , f denote the left handed lepton and Higgs doublets, and l jR , n jR the right handedcharged lepton and neutrino singlets. This automatically leads, through the seesaw mechanism[7] to an effective neutrino mass matrix at low energies, with naturally small neutrino masses.In the UD framework, the fermion mass matrices have the form: Quark Sector: M d = c d [ D + e d P d ] M u = c u [ D + e u P u ] (3) Lepton Sector: M l = c l [ D + e l P l ] M D = c D [ D + e D P D ] M R = c R [ D + e R P R ] (4)where the notation is self-explanatory. In particular, c l , c D , c R are over-all constants, M l , M D , M R denote the charged lepton mass matrix, and neutrino Dirac and right-handed neutrino massmatrices, respectively. Finally, the e i P i denote small perturbations to universal democracy.In order for universal democracy to satisfy ’tHooft naturalness principle [8], there should be asymmetry of the Lagrangian which leads to exact UD. In the quark sector, it is straightforwardto find a symmetry which leads to UD. In particular, since all quarks enter on equal footing inthe gauge sector, it is natural to assume that in leading order the Yukawa couplings obeya S Q L × S u R × S d R family permutation symmetry, acting on the left-handed quark doublets,the right-handed up quarks and right-handed down quarks, respectively. It is clear that thissymmetry leads to UD in the quark sector.However, in the lepton sector, and taking into account the observed large leptonic mixing, theUD extension is not straightforward. For definiteness, let us consider that there is a leptonnumber violation mechanism at high energies leading at low energies to the effective Majoranamass term for the light neutrinos. If one trivially extends to the lepton sector the above symme-try by considering a S L L × S l R acting on the lepton doublets and charged leptons, one obtains acharged lepton mass matrix proportional to D . However, the effective neutrino Majorana massmatrix is then proportional to ( c D + c ′ ) , where one expects c , c ′ to be of the same order,since both terms are allowed by the family symmetry. It can be shown [9] that no large leptonicmixing can be obtained in this case.In the next section, we explore the possibility of obtaining the observed quark and leptonicmixing, including the results of the DAYA-BAY experiment, through a small perturbation ofa Z symmetry imposed on the quark and lepton sectors. This Z symmetry leads to UD inleading order in all fermion sectors. 2 The Z symmetry We impose a Z family symmetry on the Lagrangean, realized in the following way. Quark Sector: Q L i → P † i j Q L i u R i → P i j u R i d R i → P i j d R i ; P = i w ∗ W ; W = √ w w
11 1 w (5)where w = e i p .It can be readily verified that this is indeed a Z symmetry since P = P † , P = . Then, theLagrangean, and in particular the quark mass terms Q L i M ui j u R j and Q L i M ui j u R j , are invariant,if the quark mass matrices M u , d obey the following relation P · M · P = M (6)Notice that we do not have P † · M · P = M . It is crucial for our results that Eq. (6) holds andit immediately follows that det ( M ) = , since det ( P ) is not real. Thus, M must have one ormore zero eigenvalues, and in [9] it was indeed shown that M is proportional to the democraticmatrix D . Lepton Sector:
In the lepton sector, we impose the Z symmetry in exactly the same way: L i → P † i j L j l iR → P i j l jR n iR → P i j n jR (7)It is clear that for a Z symmetry realized in the way indicated in Eqs. (5, 6, 7), all fermionmass matrices, M d , M u , M l , M D and M R , are proportional to D . In particular, in the exact Z limit, M R will not contain a term a , since this term is not allowed by the Z symmetry. Significant features of the observed pattern of quark masses and mixing include hierarchicalquark masses, small mixing and the observed alignment between the spectrum of the masses inthe up and down quark sectors. This observed alignment is rarely mentioned in the literatureand yet it is an important feature observed in the quark sector. So it is worth defining in aprecise manner what alignment means. Let us consider a set of quark mass matrices M d , M u M d and M u are close to a diagonal matrix. Even for this set ofmatrices, and taking into account that in the SM, the Yukawa couplings for the up and downquark sectors are entirely independent, it is as likely that in the basis where M u is close to diag ( m u , m c , m t ) , M d is close to diag ( m d , m s , m b ) , meaning alignment, as in contrast having M d close to diag ( m b , m d , m s ) meaning misalignment. Note that the ordering in one of the sectorsis arbitrary, but the relative ordering is physically meaningful. For a set of random matrices,even if one assumes hierarchical masses and small mixing, the probability of having alignmentis only 1/6. For a set of arbitrary masses, one can verify whether one has small mixing andalignment through the use of weak-basis invariants [10].In order to see this, it is convenient to define the dimensionless matrices with unit trace: h u , d = H u , d Tr [ H u , d ] (8)where H u ≡ M u M † u and similarly for H d . As we have previously mentioned, alignment means thatin the weak-basis where H u = diag ( m u , m c , m t ) , H d is close to diag ( m d , m s , m b ) . Then defining A = h d − h u and taking into account that Tr ( A ) = by construction, in turn implying that | c ( A ) | = Tr [ A ] , one can show that the condition for alignment is | c ( A ) | ≪ (9)where c ( A ) is the second invariant of A , namely c ( A ) ≡ a a + a a + a a , with the a i denotingthe eigenvalues of A . In order to see that the condition Eq. (9) leads to alignment, considerthe limit where m t , m b go to infinity. Assuming alignment, one has H u , d = diag ( , , ) so that c ( A ) = . If one considers instead that there is small mixing but no alignment, then in the weak-basis where H u = diag ( m u , m c , m t ) , one may have H d close to diag ( m b , m s , m d ) . One can checkthat in this case c ( A ) ≈ , indicating total misalignment. A very interesting feature of universaldemocracy is the fact that a small perturbation of UD as indicated in Eq. (3), automaticallyleads to alignment of the heaviest generation. Alignment of the two light generations depends,of course, on the specific breaking of UD through e u P d , e u P d in Eq. (3). It is indeed a salientfeature of the universal democracy hypothesis for the quark sector [6] that it guarantees exactlythese two important phenomenological properties: small mixing and alignment. Z A breaking of Z generates leptonic mass matrices of the form of Eq. (4), where the e i ≪ ( i = l , D , R ) and the P i are of order . We assume that the perturbation P R of the right-handedheavy Majorana neutrinos is such that the inverse of D + e R P R exists, and we find that generically [ D + e R P R ] − is of the form: [ D + e R P R ] − = c P e R [ L + e R Q ] ; c P = q + e R p (10)4here p and q = (cid:229) Q i j are cubic and quadratic polynomials in the elements ( P R ) i j and L , Q are matrices with respectively linear and quadratic elements in the ( P R ) i j . Obviously p , q , L and Q are in general of order . It is possible to have special cases where either p or q vanish,but not both, since we require that the inverse of D + e R P R exists. Furthermore, it is a generalcharacteristic of this inverse that the linear matrix L and the quadratic matrix Q satisfy therelations: D L = D Q D = q D (11)Applying these algebraic relations to the effective neutrino mass matrix formula one obtains atransparent formula for M e f f : M e f f == c (cid:16) q D + e D (cid:2) D Q P T D + P D Q D + e D P D Q P T D (cid:3) + e D e R P D L P T D (cid:17) (12)with c = − c D / c R ( q + e R p ) .This expression obtained for the effective neutrino mass matrix tells us when to expect largemixing for the lepton sector in the case of an aligned hierachical spectrum for the chargedleptons, as well as for the Dirac and heavy Majorana neutrinos. In general, i.e., for a genericperturbation P R in the right-handed Majorana sector, there will be more than one element ( P R ) i j of order one, thus implying that the quadratic polynomial q = (cid:229) Q i j is also of order one. So,if the term in M e f f proportional to e D / e R is small, it is clear that the effective neutrino massmatrix will be, just like the charged lepton mass matrix, to leading order, proportional to D ,and, thus, there will be no large mixing. Therefore, if one wants to avoid small mixing, onemust have the term proportional to e D / e R , in Eq. (12), to be of order one or larger. Next, we give a numerical analysis of our model. In particular, we give a specific point inparameter space which exhibits all features described here and is in excellent agreement withthe experimental evidence and in particular with the recent DAYA-BAY result. In accordancewith the universal democracy hypothesis, given in Eq. (4), we write the leptonic mass matricesas proportional to D plus some matrix with elements of order of a power in l ≡ . .For the charged lepton mass matrix, we have M l = c l [ D + l P l ] , where, M l = c l e i d l
11 1 e i d l ; c l = . MeV d = − . d = − . (13)For the neutrino Dirac and right-handed mass matrices, we have: M D = c D [ D + l P D ] ; M R = c R (cid:2) D + l P R (cid:3) (14)5here l P D = . · l . · l . · l − . · l − . · l . · l − . e i p · l . i · l . i · l and l P R = − . i · l
00 0 − . · l with c D ≡ m top ; c R = . × GeV (15)With these, we obtain the following physical observables: charged lepton, light neutrino massesand heavy Majorana neutrinos: m e = . MeV m n = . eV M = . × GeVm µ = . MeV m n = . eV M = . × GeVm t = . MeV m n = . eV M = . × GeV D m = . × − eV D m = . × − eV (16)and mixing (cid:12)(cid:12) U PMNS (cid:12)(cid:12) = . . . . . . . . . | U e | = . q = . q = . J = . | m ee | = . eV (17)Note that all observables are within the experimental bounds, in particular we have a sufficientlarge | U e | , in agreement with the recent DAYA-BAY experimental results. We have also alarge hierarchy for the heavy Majorana neutrinos due to the perturbation term in M R , which isproportional to higher power in l in Eq. (14).In addition to this point, we have explored a stable region in parameter space around thispoint which is in agreement with experiment. This was done using a numerical analysis with aMonte Carlo type algorithm. We generated random perturbations of at most away fromthe initial parameters, by means of an uniform distribution. Each of the plots in Figs. 1, 2, 3represents the normalized density of points belonging to the data set, as a function of the valuesof a pair of observables. Our parameter space point, given in Eqs. (13, 14), is also depicted inthe figures as the intersection of the dashed horizontal line with the dashed vertical line.6 salient feature illustrated by this numerical analysis is an interesting correlation between thevalues of U e and sin ( q ) . In the region where values of U e are obtained in agreement withthe DAYA-BAY experiment, values of q = ◦ are favoured around sin ( q ) ≈ . . We have presented a unified view of the flavour structure of the quark and lepton sectors, basedon the principle of Universal Democracy coming from a Z symmetry, where in leading orderall fermion mass matrices are proportional to the democratic matrix. In the quark sector, smallmixing arises from the breaking of UD which also generates the masses of the light generations.In the lepton sector, the breaking of UD is also small, but in the presence of the seesawmechanism, this small breaking of UD is able to generate large leptonic mixing, including avalue of U e in agreement with the DAYA-BAY result. The smallness of the breaking of Z isnatural in the ’tHooft sense, since in the limit of exact UD, the Lagrangean acquires the Z symmetry.In conclusion, our analysis shows that the observed pattern of fermion masses and mixing, bothin the quark and lepton sectors, may reflect the principle of Universal Democracy, renderednatural by a Z symmetry. AknowledgmentsReferences [1] DAYA-BAY Collaboration, F. An et al., Phys. Rev. Lett. 108, 171803 (2012), [1203.1669];DOUBLE-CHOOZ Collaboration, Y. Abe et al., Phys. Rev. Lett. 108, 131801 (2012),[1112.6353]; RENO collaboration, J. Ahn et al., [1204.0626].[2] L. Wolfenstein, Phys. Rev. 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Naturalness, Chiral Symmetry and Spontaneous Chiral Symmetry Breaking ,lecture given in Cargese Summer Institute 1979, 135.[9] G.C. Branco, J.I. Silva-Marcos, Phys. Lett. B 526 (2002) 104; Joaquim I. Silva-Marcos,JHEP 0212 (2002) 036 [hep-ph/0204051]; Joaquim I. Silva-Marcos, JHEP 0307 (2003) 012[hep-ph/0204051].[10] G.C. Branco and J.I. Silva-Marcos, Phys. Lett. B 715 (2012) 315 [arXiv:1112.1631].8igure 1: Density of data points as a function of sin q and | U e | Figure 2: Density of data points as a function of sin ( q ) and | U e | m n and || U e ||