Anticorrelation of Mass and Mixing Angle Hierarchies
aa r X i v : . [ h e p - ph ] A p r Anticorrelation of Mass and Mixing Angle Hierarchies
Paul H. Frampton (1) and Jihn E. Kim (2 , (1) Dipartimento di Matematica e Fisica “Ennio De Giorgi”,Universita del Salento and INFN-Lecce, Via Arnesano, 73100 Lecce, Italy (2)
Department of Physics, Kyung Hee University, 26 Gyungheedaero,Dongdaemun-Gu, Seoul 02447, Republic of Korea (3)
Department of Physics and Astronomy, Seoul National University,1 Gwanakro, Gwanak-Gu, Seoul 08826, Republic of Korea
We obtain a relationship between the hierarchies of mixing angles and of masses pertinent to theCabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix and the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) lepton mixing matrix. Using this relationship, we argue that the more severehierarchy of the charge- quark masses requires that the CKM matrix be close to a unit matrixwhereas the milder hierarchy of the neutrino masses allows the PMNS matrix to depart markedlyfrom the CKM matrix and contain large mixing angles of the type that are observed. PACS numbers: 12.15.Ff, 11.30.Ly, 14.60.Pq, 12.60.iKeywords: CKM and PMNS matrices, Quark mass, Neutrino mass, A symmetry I. INTRODUCTION
The Standard Model (SM) of particle physics has many free parameters which are fitted to experimental data ratherthan calculated. Most of these parameters are masses and mixing angles, and in the present article we demonstratea general model-independent relationship. Understanding these from symmetric principles is a theoretical solutionof the flavor problem. On the other hand, a generally accepted idea in the astrophysical community is most of theenergies in the universe are carried by dark energy (DE) and dark matter (DM). The solution of the flavor problemis applicable to the cosmic evolution because the symmetry required from the flavor solution necessarily dictates theproperties of the beyond-the-SM (BSM) particles, in particular DM. Not only DM but also the baryon number in theuniverse (BAU) is related to the theoretical solution because one of three Sakharov conditions [1] for generating theBAU needs C and CP violation. The CP phase in the SM may be directly related to the BAU [2] or may not berelated [3, 4]. Even though the CP phase needed for the BAU may not be that of the SM, the phases of the SM andthe BSM can be related if the symmetry in the full theory is known.Among the SM parameters, the best measured parameters are the gauge coupling constants and some masses m e , m µ , m t and m h . Other masses have rather large error bars. On the other hand, three real angles of the Cabibbo-Kobayashi-Maskawa (CKM) matrix and three real angles of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrixare known rather accurately. The CP phases are not known accurately. Therefore, the flavor symmetry can be graspedby looking at the CKM and the PMNS matrices with the phases as free parameters [5]. The CKM matrix is close toidentity, but the PMNS matrix is of the form V (0) ∼ × × ×× × × × × , (1)where × are non-vanishing entries. The form of Eq. (1) has led to the bimaximal PMNS form [6]. It is almosttri-bimaximal [7], which created a frenzy with regard to the tetrahedral symmetry [8]. II. FORM OF CHARGED CURRENTS
At present all the SM particles of three quark-lepton families are known, and one of the most important theoreticalproblems is the flavor problem. The first SM family consists of fifteen chiral fields, with three colors ( α = 1 , ,
3) forthe quarks, Q αL , ( u α ) cL , ( d α ) cL , ℓ L , e cL , (2)where Q αL = (cid:18) u α d α (cid:19) L , ℓ L = (cid:18) ν e e (cid:19) L , (3)and the 2nd and the 3rd families have an identical structure except for the masses and mixings, which are our presentresearch topic. Except for the three neutrinos, all the SM chiral fields obtain masses by using renormalizable couplingsto the vacuum expectation value (VEV) of a Higgs doublet. Neutrinos remain massless unless additional assumptionsbeyond the Standard Model (BSM) are invoked.We have the irreducible representations of the gauge group SU (3) C × SU (2) L × U (1) Y that we put into the definitionof Yukawa couplings. Let us choose the bases where the fermion doublets with the 3rd component of the weak isospin T = − are the mass eigenstates: d (mass) L = d (0) s (0) b (0) L = dsb L , e (mass) L = e (0) µ (0) τ (0) L = eµτ L . (4)Then, the charged currents (CCs) are given by g √ (cid:16) ¯ u (0) L γ µ d (mass) L + ¯ ν (0) L γ µ e (mass) L (cid:17) W + µ , (5)where u (0) L = u (0) c (0) t (0) L , ν (0) L = ν (0) e ν (0) µ ν (0) τ L . (6)In terms of mass eigenstates, Q em = + quarks ( u, c, t ) and neutrinos ( ν , ν , ν ), the defining weak states are relatedto the mass eigenstates by u (0) c (0) t (0) L = U ( u ) † uct L , ν (0) e ν (0) µ ν (0) τ L = U ( ν ) † ν ν ν L , (7)where the unitary matrices diagonalizing L-handed fields are denoted by U and the unitary matrices diagonalizingR-handed fields are denoted by U . Then, the CKM and the PMNS matrices are given by V (CKM) = U ( u ) U ( d ) † = U ( u ) , V (PMNS) = U ( ν ) U ( e ) † = U ( ν ) . (8)The definitions of U ( u ) and U ( ν ) in Eq. (8) have the required number of parameters. In U ( u ) , two phases of L-handed u (0) quarks for constraints exist because the baryon number phase cannot be used as a constraint. Also, three u (0) masses provide three constraints. Thus, out of 9 parameters in a 3 × U ( ν ) , we do not have any phase constraint because Majorana neutrinosare real, we have nine minus three parameters: 3 real angles, 1 Dirac phase, and 2 Majorana phases.Note that our definition of the PMNS matrix is given in accordance with the CKM matrix by using the W + µ couplingwhile the Particle Data Book (PDG) defines the PMNS matrix with the W − µ coupling [9, 10]. This is because the quarkmixing angles are represented with respect to the mass eigenstates while the leptonic mixing angles are representedwith respect to the weak eigenstates. Thus, what the PDG book represents is our U ( ν ) † . Therefore, in discussing theCKM and the PMNS matrices in unison, considering the charge raising currents due to U ( u ) and U ( ν ) to be the sameis better. We could have chosen the mass eigenstates of Q em = + quarks and neutrinos as the defining fermionswith (0) , but then we would have to specify how neutrinos obtain masses, which necessarily includes BSM physics.On the other hand, our choice of Q em = − quarks and Q em = − H d with Y = − : q αL m d , , , m s , , , m b d α (mass) R + ℓ m e , , , m µ , , , m τ e (mass) R H d h H d i + h . c . (9)where q iL = ( u (0) , d (mass) ) TiL . This is equivalent to defining the Yukawa couplings as Y ( d ) ij = m ( d ) i h H d i δ ij and Y ( e ) ij = m ( e ) i h H d i δ ij .The renormalizable couplings for the Q em = + quarks are given by H † d that carries Y = + .For neutrinos, renormalizable couplings such as Eqs. (9) cannot be written. If non-renormalizable couplings areallowed, however, lepton (L) number can be violated and L-violating Majorana-type neutrino mass terms are possible[11]: k ij M ( ℓ Ti ) m C − ( ℓ j ) n ( H u ) m ′ ( H u ) n ′ ǫ mm ′ ǫ nn ′ , (10)where m, n, m ′ and n ′ are the SU(2) indices and i and j are flavor indices. For a non-vanishing VEV of H u , h H u i = v u / √
2, which has T = − ; neutrinos having T = + lead to the mass m ij = v u M k ij . (11) III. SMALL QUARK MIXING AND LARGE LEPTON MIXING ANGLES
1. The CKM Matrix
The CKM matrix elements areCKM (component ij ) : g √ u iL V (CKM) ij γ µ d jL W + µ , (12)where u ( i ) and d ( i ) are mass eigenstates. The diagonal masses of Q em = + quarks are, for the ( ll ) component,Diagonal ( ll ) : m u ¯ u R u L + m c ¯ c R c L + m t ¯ t R t L + h . c . = m u ¯ u (0) iR ( U † i U j ) u (0) jL + m c ¯ u (0) iR ( U † i U j ) u (0) jL + m t ¯ u (0) iR ( U † i U j ) u (0) jL + h . c . = ¯ u (0) αR ( m i U iα U iβ ) u (0) βL + h . c . (13)Let us pay attention to the largest term m t (¯ u (0) αR U tα U tβ u (0) βL + O ( m c m t )) = m t (¯ u (0) αR U † αt )( U tβ u (0) βL ) + O ( m c ) . (14)Thus, we define t L ≃ U tα u (0) αL ,t R ≃ U tβ u (0) βR . (15)Whatever the diagonalizing matrices U and U may be, we can approximately find the top components by using Eq.(15). With the choice of t L and t R , the mass matrix is of a form with a determinant O ( ǫ ): m t O ( ε ) O ( ε )) O ( ε ) O ( ε )) O ( ε ) O ( ε ) O ( ε ) O ( ε ) 1 , (16)where ε is O( m c m t ) ≈ . × m t and m c ,the matrix must take a form so that the determinant turns out to be O( ε ): m t (cid:18) O ( ε ) O ( ε / ) O ( ε / ) 1 (cid:19) . (17)In this case, the CKM matrix is close to a diagonal matrix [12], which basically results from the huge mass hierarchybetween M t and M c . Extending this form to a 3 ×
2. The PMNS Matrix
The PMNS matrix elements arePMNS (component ij ) : g √ ν (0) iL V (PMNS) ij γ µ e jL W + µ , (18)where ν (0) i and e j are the weak eigenstates. In addition, e j is also the mass eigenstate. For neutrino masses, nocounterpart corresponding to Eq. (13) exists instead, we obtain them from Weinberg’s dimension 5 L-violatingoperator: m i ˜ ν iL C − ν iL . (19)The diagonal masses of neutrinos are, for the ( ll ) component,Diagonal ( ll ) : m ν ¯ ν R ν L + m ν ¯ ν R ν L + m ¯ ν R ν L + h . c . → (cid:0) m ν ν T L C − ν L + m ν ν T L C − ν L + m ν T L C − ν L (cid:1) = 12 h m ν ˜ ν (0) iL ( U ( ν ) ) i C − ( U ( ν ) † ) j ν (0) jL + m ν ˜ ν (0) iL ( U ( ν ) ) i C − ( U ( ν ) † ) j ν (0) jL + m ν ˜ ν (0) iL ( U ( ν ) ) i C − ( U ( ν ) † ) j ν (0) jL i , (20)where in the second line we wrote the low-energy effective lagrangian in the SM in terms of L-handed fields as ν i = ( U ( ν ) † ) iα ν (0) α . (21)If a hierarchy of masses exists, we can follow the case of the CKM hierarchy and obtain m (cid:18) O ( ε ′ ) O ( √ ε ′ ) O ( √ ε ′ ) 1 (cid:19) . (22)In the inverted hierarchy, the masses are almost degenerate, so the PMNS matrix need not be near the identitymatrix. From the data for the normal hierarchy [7], we obtain ∆ m ≃ . × − eV and 0 . × − eV , i.e. , m ≃ . × − , and m ≃ . × − , where ε ′ is O(0 . . ). Therefore, the PMNS matrix being close tothe identity matrix is not possible. This is because no strong hierarchy of neutrino masses exists. With a normalhierarchy of neutrino masses, for example, putting M ( ν ) ≃
0, the squared mass differences △ = 7 . × − eV and △ = 2 . × − eV , which are measured [7], suggest that M ( ν ) = 0 .
048 eV and M ( ν ) = 8 . × − eV,hence, the lepton hierarchy is M ( ν ) /M ( ν ) = 5 . M ( t ) = 173 GeV and M ( c ) = 1 . M ( t ) /M ( c ) = 144 .
2. Thus, the quark hierarchy is much stronger than the lepton hierarchy.
IV. FLAVOR SYMMETRY
If the PMNS matrix is not close to identity, the next question to ask is, “How close are the PMNS-matrix elements?”One may consider the mass matrix first, but experimentally presented ones are on the CC interactions. If a symmetryis the basis for large mixing angles in the PMNS matrix, the first one to consider is the permutation symmetry becausepermutation requires somethings to be identical. The simplest case S has representations of singlets only; hence, itis not suitable for relating mixing angles. The permutation of three objects S has a doublet representation, whichmay equate two entries among the matrix elements.Let us consider a matrix with one zero, V (0) ∼ × × ×× × × × × (23)such that the CC in the leptonic sector is, viz. Eq. (18), ν (0) L V (0) γ µ e L = ν (0) L U ( ν ) † U ( ν ) V (0) γ µ e L , (24)where e L is a column vector of mass eigenstates, and ν (0) L = U ( ν ) ν. (25)Noting Eq. (8), ν (0)3 = U ( ν )3 j ν j , ν (0)3 is composed of a representation of the permutation symmetry S . Negligiblecouplings occur between and in ν (0)3 . The PMNS matrix of the form in Eq. (18) is S symmetric, and it isbimaximal; thus, V (PMNS) ∼ × × ×× × × ± √ √ , (26)which may be obtained from the permutation symmetry S [6].By the same token, let us look for a permutation symmetry with a triplet representation . It is present in thepermutation symmetry S . In the PMNS matrix, can one associate 4 entries such that it contains a triplet, forexample, V (try) ∼ × × × p q r s × × , (27)where we have declared that a permutation symmetry of p, q, r, and s exists. We use s in anticipation of settingthis entry to zero at the end. A triplet is assigned in the second row of V iI , i.e. , at i = 2. Twenty-four elements ofpermutations of p, q, r, s are 123 s , s , s , s , s , s , s , s , s , s , s , s , (28)and 213 s , s , s , s , s , s , s , s , s , s , s , s . (29)Equation (28) represents cyclic permutations while Eq. (29) represents anti-cyclic permutations, which is identicalto the scheme that neglects Eq. (29) and allows plus and minus values of s in Eq. (28). Depending on a nonzerovalue of s , all 24 elements in Eq. (28) and Eq. (29) form the symmetric elements S . The four elements p, q, r, and s must be the same. When only Eq. (28) is considered, the same conclusion is drawn. For p = q = r = ∆ while s takes two values ± ∆. Now, suppose p = q = r = √ and s = 0; then, two values of s collapse to one, and Eq. (28)has only twelve elements. Thus, we argue that the discrete group of twelve elements, A [8], will lead to a PMNSmatrix of the form V ∼ × × × √ √ √ α cos α . (30)For a tri-bimaximal form, we require α = ± o , for which V iI has an additional permutation symmetry: V i ↔ V i for i = 3. Note that Eq. (30) uses the weak eigenstates of neutrinos with superscripts (0) .We can present our arguments in terms of the A representations. A has three singlets , ′ , and ′′ and one triplet . How then three singlets are given is obtained from the decomposition of S representations into those of A , asshown in Table 1. The singlet representations are = p + q + r + s , ′ = p + q − r − s and ′′ = p − q + r − s . Thethree numbers in the first row of Eq. (30) correspond to three singlets. The two numbers in the 3rd row correspondto two singlets ′ and ′′ ; hence, α need not be 45 o . Hence, Eq. (30) can be written in terms of A representations as V ∼ ′ ′′ T ′ ′′ . (31) TABLE I: Branching of S representations , ′ , , and ′ into the A and the S representations [14]. S A S ′ ′ ′ ⊕ ′′
23 3 1 ⊕ ′ ′ ⊕ To realize A symmetry, we assign the Yukawa couplings such that the flavor indices of i respect the requirementsdiscussed in Section IV. The L-violating BSM singlets N i can be three, but here we show an example with just onesinglet N : L ∆ L = m N N . The effects of other singlets is effectively adjusted by the coefficients of the Weinbergoperator, v u m N ˜ ν (0) iL C − ( h ij ) ν (0) jL , (32)where ( h ij ) denotes the Yukawa matrix. In Eq. (32), the hierarchy of h ij is determined by physics above theelectroweak scale, in particular by the VEVs of the SM singlet Higgs fields rendering the heavy neutrinos mass. Forexample, if the heavy neutrinos are completely democratic, the heavy neutrinos have a hierarchy M ≫ M = M ≈ h = h ≫ h . With the A symmetry, the heavy neutrino hierarchy is determined bythe representation property of the singlet Higgs. This feature is discussed in [16]. Because we introduce just oneHiggs doublet H u giving mass to neutrinos, the A symmetry is counted only by the neutrinos. The weak eigenstateneutrinos of Eq. (32) are related to the mass eigenstate neutrinos by, neglecting phases of a tri-bimaximal U , ν (0) θφ = U ( θ, φ ) ν ∼ cos θ, sin θ cos φ, sin θ sin φ T , ± , ν e ν µ ν τ = cos θ ν e + sin θ cos φ ν µ + sin θ sin φ ν τ √ ν e + √ ν µ + √ ν τ ± √ ν µ + √ ν τ , (33)and Eq. (32) expressed in terms of the mass eigenstates is, for cos θ = ±√ √ and φ = 45 o , h v u m N (cid:16) " ±√ √ ν e + 1 √ ν µ + 1 √ ν τ C − " ±√ √ ν e + 1 √ ν µ + 1 √ ν τ + (cid:20) √ ν e + 1 √ ν µ + 1 √ ν τ (cid:21) C − (cid:20) √ ν e + 1 √ ν µ + 1 √ ν τ (cid:21) + (cid:20) ± √ ν µ + 1 √ ν τ (cid:21) C − (cid:20) ± √ ν µ + 1 √ ν τ (cid:21) (cid:17) . (34)Certainly, Eq. (34) looks a bit more complicated than Eq. (32).Generalizing the forms h ij and U is straightforward if the the three heavy neutrino masses are different and thethree CP phases are included. Nevertheless, if the (31) element of U remains zero as in Eq. (34), then the Dirac CPphase does not appear [15]. For the Dirac CP phase to have an effect, a term(s) violating the tri-bimaximal form,notably in the (31) element of U ( θ, φ ), must exist.A model construction is to declare the weak eigenstates of neutrinos with superscripts (0) to be the triplet repre-sentation under the dihedral group A and V (0) in Eq. (24) to be Eq. (31). Of course, we work in bases where the T = − components in the quark and lepton sectors are mass eigenstates. At this stage, all quarks states can betaken as singlets under A . In GUTs, the tensor product × = 2 · ⊕ ⊕ ′ ⊕ ′′ should be considered to declarethe A properties of the quark representations, which will be discussed in the future [16]. VI. CONCLUSION
Our letter suggests a general approach to study the connection between the lepton and the quark parametersprecisely by exploiting the difference between the two sectors. We have compared the mixing angles and the massespertinent to the quark and the lepton mixing matrices. We have shown that the two matrices possess intriguinghierarchies anticorrelated between angles and masses and hope this observation will lead to further progress.ACKNOWLEDGMENTThis work is supported in part by a National Research Foundation (NRF) grant from Korea (NRF-2018R1A2A3074631). [1] A. D. Sakharov, Pisma Zh. Eksp. Teor. Fiz. , 32 (1967) [JETP Lett. , 24 (1967)].[2] L. Covi, J. E. Kim, B. Kyae, and S. Nam, Phys. Rev. D , 065004 (2016).[3] G. Segr`e and H. A. Weldon, Phys. Rev. Lett. , 1737 (1980).[4] M. Fukugita and T. Yanagida, Phys. Lett. B , 45 (1986).[5] J. E. Kim, S-J. Kim, S. Nam, and M. Shim, Jarlskog determinant and data on flavor physics , eprint:1907.12247 [hep-ph].[6] P. F. Harrison, D. H. Perkins, and W. G. Scott, Phys. Lett. B , 137 (1995).[7] I. Esteban, M. C. Gonzalez-Garcia, A. Hernandez-Cabezudo, M. Maltoni, and T. Schwetz, JHEP , 106 (2016). See also,P. F. de Salas, D. V. Forero, C. A. Ternes, M. Tortola, and J. W. F. Valle, Phys. Lett. B , 633 (2018).[8] K. S. Babu, E. Ma, and J. W. F. Valle, Phys. Lett. B , 207 (2003).[9] K. Nakamura and S. Petcov (Particle Data Group), Sec. 14 of the PDG book [10].[10] M. Tanabashi et al. (Particle Data Group), Phys. Rev. D , 030001 (2018).[11] S. Weinberg, Phys. Rev. Lett. , 1566 (1979).[12] S. Weinberg, Trans. New York Acad. Sci. , 185 (1977).[13] H. Fritzsch, Nucl. Phys. B , 189 (1979).[14] H. Ishimori et al. , An Introduction to Non-Abelian Discrete Symmetries for Particle Physicists , Lecture Notes in PhysicsVol. 858 (Springer-Verlag, Berlin, 2012).[15] J. E. Kim and M-S. Seo,
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