A translational flavor symmetry in the mass terms of Dirac and Majorana fermions
aa r X i v : . [ h e p - ph ] F e b A translational flavor symmetry in the mass terms of Dirac and Majorana fermions
Zhi-zhong Xing
1, 2, 3, ∗ Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China Center of High Energy Physics, Peking University, Beijing 100871, China
Requiring the effective mass term for a category of fundamental Dirac or Majorana fermions ofthe same electric charge to be invariant under the transformation ψ α L(R) → ψ α L(R) + n α z ψ in theflavor space, where n α and z ψ stand respectively for the flavor-dependent complex numbers andthe spacetime- and flavor-independent element of the Grassmann algebra, we show that n α can beidentified as the elements U αi in the i -th column of the unitary matrix U used to diagonalize thecorresponding Hermitian or symmetric fermion mass matrix M ψ , and m i = 0 holds accordingly. Wefind that the reverse is also true. Given the very facts that the charged leptons, up- and down-typequarks all have a strong mass hierarchy and current experimental data allow the lightest neutrinoto be (almost) massless, the zero mass limit for the first-family fermions and the translational flavorsymmetry behind it are expected to be a natural starting point for model building. The standard model (SM) of particle physics hasproved to be a huge success, but its flavor sector remainsunsatisfactory in the sense that all the flavor parametersare theoretically undetermined. Going beyond the SM,one has to introduce some more free parameters in con-nection with the massive neutrinos. One way out of thissituation is to determine or constrain the flavor structuresof leptons and quarks with the help of certain proper fam-ily symmetries, such that some testable predictions forfermion masses and flavor mixing parameters (or theircorrelations) can be achieved. So far a lot of efforts havebeen made along this line of thought toward building phe-nomenologically viable models [1, 2], but the true flavordynamics is still unclear. At the present stage a logicalcandidate for the underlying flavor symmetry should atleast help interpret some salient features of the observedfermion mass spectra and flavor mixing patterns.In 2006 Friedberg and Lee put forward a novel idea toconstrain the flavor texture of three Dirac neutrinos [3].In the basis of a diagonal charged-lepton mass matrix,they required the Dirac neutrino mass term L ν to keepunchanged under the transformation ν α → ν α + z (for α = e, µ, τ ), where z is a spacetime-independent elementof the Grassmann algebra and anticommutates with theneutrino fields ν α . The constraint of this translational symmetry of L ν on the neutrino mass matrix M ν is sostrong that det ( M ν ) = 0 holds, implying that one ofthe neutrino masses m i (for i = 1 , ,
3) must vanish. Avery instructive neutrino mixing pattern U = U TBM O in correspondence to m = 0 can then be obtained [3–5],where U TBM stands for the well-known “tribimaximal”flavor mixing pattern [6–8] and O is a unitary rota-tion matrix in the complex (1 ,
3) plane. In this scenario,however, a proper symmetry breaking term has to beintroduced to assure m > m as indicated by currentneutrino oscillation data [9].In fact, as early as in 1979, ’t Hooft had imposed aquite similar translational displacement φ ( x ) → φ ( x ) + Λon the Lagrangian L φ for a renormalizable scalar field theory, where Λ is spacetime-independent and commu-tates with φ [10]. It turns out that both the mass andthe self-coupling parameter of φ have to vanish in order toguarantee the invariance of L φ , which can be referred toas a Goldstone-type symmetry [11, 12], under the abovetransformation. So one generally expects that the trans-lational symmetry of an effective Lagrangian may providea simple and natural way to understand why the mass ofa fermion or boson in this physical system is vanishing orvanishingly small , although its deep meaning remainsa puzzle at present.In this paper we are going to show that the effectivemass terms of Dirac or Majorana fermions may all have akind of translational symmetry under the discrete shifts ψ α L(R) → ψ α L(R) + n α z ψ in the flavor space, if and onlyif m i = 0 holds and n α = U αi are the elements in the i -column of the unitary matrix U used to diagonalizethe corresponding Hermitian or symmetric fermion massmatrix M ψ . We find that the reverse is also true. Giventhe very facts that all the fundamental fermions of thesame nonzero electric charge have a rather strong masshierarchy and current neutrino oscillation data allow thelightest neutrino to be (almost) massless, the zero masslimit for the first-family fermions and the translationalsymmetry behind it can serve as a very natural startingpoint for building viable models toward understandingthe observed patterns of both the Cabibbo-Kobayashi-Maskawa (CKM) quark flavor mixing matrix [13, 14] andthe Pontecorvo-Maki-Nakagawa-Sakata (PMNS) leptonflavor mixing matrix [15–17]. First of all, let us consider the effective mass termof the three known neutrinos (i.e., ν e , ν µ and ν τ ) by If there were no flavor mixing in the lepton or quark sector, theflavor and mass eigenstates of a fermion would be identical witheach other. In this case the kinetic energy and mass terms of afermion field, which does not involve any self-interactions, wouldbe closely analogous to those of a scalar field. assuming that they have the Majorana nature −L M = 12 X α X β (cid:2) ν α L h m i αβ (cid:0) ν β L (cid:1) c (cid:3) + h . c . , (1)where α and β run over the flavor indices e , µ and τ ,“ c ” denotes the charge conjugation, and h m i αβ = h m i βα are the elements of the 3 × M ν . Since M ν can be diagonalized via the unitarytransformation U † M ν U ∗ = diag { m , m , m } , we havethe expressions h m i αβ = X i (cid:0) m i U αi U βi (cid:1) , (2)for i = 1 , U allows us to prove X α (cid:2) U ∗ αj h m i αβ (cid:3) = m j U βj , X β (cid:2) h m i αβ U ∗ βj (cid:3) = m j U αj , X α X β (cid:2) U ∗ αj h m i αβ U ∗ βj (cid:3) = m j , (3)which are essentially equivalent to one another and allproportional to the neutrino mass m j (for j = 1 , ν α L → ν α L + U αj z ν , (4)where z ν is a spacetime- and flavor-independent elementof the Grassmann algebra. Then L M becomes −L ′ M = −L M + 12 m j " z ν z cν + X α (cid:2) U αj ν α L (cid:3) z cν + z ν X β (cid:2) U βj (cid:0) ν β L (cid:1) c (cid:3) . (5)It becomes clear that L ′ M = L M will hold under the abovetransformation if and only if m j = 0 holds. Namely, oneof the three neutrinos must be massless if the effectiveMajorana neutrino mass term L M keeps invariant underthe discrete shifts of ν α L made in Eq. (4), which helpsprovide a novel link between the two sides of one coin(i.e., the mass and flavor mixing issues of the Majorananeutrinos). The point is that the three flavor-dependentcoefficients U αj of z ν constitute the j -th column of theunitary matrix U used to diagonalize M ν , which corre-sponds to m j = 0. In comparison, most of the popularglobal discrete flavor symmetries can help predict very Of course, an effective Majorana mass term for the right-handedneutrino fields in the canonical seesaw mechanism [18–22] can bediscussed in an analogous way. specific neutrino mixing patterns but leave the neutrinomass spectrum unconstrained [25–27].Given the fact of m > m , one is left with either m = 0 (normal ordering) or m = 0 (inverted ordering)for the neutrino mass spectrum. In either case the othertwo neutrino masses can be determined by inputting theexperimental values of two independent neutrino mass-squared differences, and only a single nontrivial Majo-rana CP phase of U survives [28].In the basis of a diagonal charged-lepton mass matrix,the unitary matrix U appearing in Eqs. (2)—(5) is justthe PMNS matrix U PMNS which link the neutrino masseigenstates ν i (for i = 1 , ,
3) to the neutrino flavor eigen-states ν α (for α = e, µ, τ ). Focusing on the possibility of m = 0, one may adopt the original Kobayashi-Maskawa(KM) parametrization [14] for U PMNS so as to make theexpressions of U α in the first column of U PMNS as simpleas possible. Explicitly, U PMNS = c s c s ˆ s ∗ − s c c c c + s ˆ s c c ˆ s ∗ − s c − s s c s c − c ˆ s c s ˆ s ∗ + c c P ν (6)with the definitions c i ≡ cos θ i , s i ≡ sin θ i , ˆ s ≡ s e i φ and P ν ≡ diag { , e i σ , } , where φ and σ denote theDirac and Majorana CP-violating phases, respectively.If U e = 2 / √ U µ = U τ = − / √ θ = arcsin(1 / √ ≃ . ◦ and θ = 45 ◦ are taken),for example, one will obtain U PMNS = 1 √ √ − √ −√ − √ √ c ˆ s ∗ − ˆ s c P ν ; (7)namely, U PMNS = U TBM O P ν with O being a unitaryrotation matrix in the complex (2 ,
3) plane. This sim-ple flavor mixing pattern, which was first proposed in2006 [29, 30], remains favored in today’s neutrino phe-nomenology. A combination of Eqs. (2) and (7) allowsus to reconstruct the Majorana neutrino mass matrix inthe chosen basis: M ν = a a aa a aa a a + c − cc b + 2 c − b − c − b b − c , (8)in which a ≡ (cid:0) m c + m ˆ s ∗ (cid:1) / b ≡ (cid:0) m ˆ s + m c (cid:1) / c ≡ ( m ˆ s − m ˆ s ∗ ) c / √ m ≡ m e σ are de-fined. The simple structure of M ν depends on the simplechoice of U α and is suggestive of certain simple flavorsymmetries which can be used for model building.Provided M l is neither diagonal nor Hermitian, onecan diagonalize M l M † l with the help of a unitary trans-formation U † l M l M † l U l = diag { m e , m µ , m τ } . In this casethe PMNS lepton flavor mixing matrix is expressed as U PMNS = U † l U ν , where U ν is equivalent to the unitarymatrix U used to diagonalize M ν in the above discus-sions. Of course, the mass term of the charged leptons orDirac neutrinos may also possess a possible translationalflavor symmetry of this kind, so may the mass term ofthe up- or down-type quarks. We proceed to discuss the Dirac fermion mass termsin an opposite way. If the massive neutrinos are of theDirac nature, it is possible to treat them on the samefooting as the charged leptons and quarks. Without lossof any generality, a Dirac fermion mass matrix M ψ canalways be taken to be Hermitian after a proper choice ofthe flavor basis in the SM or its extensions which have noflavor-changing right-handed currents [31]. In this caseone may diagonalize M ψ via the unitary transformation V † M ψ V = diag { λ , λ , λ } with λ i being the eigenval-ues of M ψ (i.e., m i = | λ i | are physical masses of thecharged or neutral fermions under consideration, eitherfor leptons or for quarks). The effective mass term fora given category of the fundamental fermions with thesame electric charge can be written in the chosen basisas follows: −L D = X α X β (cid:2) ψ α L h m i αβ ψ β R (cid:3) + h . c . , (9)in which the subscripts α and β run over the flavor indicesof the Dirac neutrinos, charged leptons, up-type quarksor dow-type quarks, and h m i αβ = h m i ∗ βα = X i (cid:0) λ i V αi V ∗ βi (cid:1) (10)holds thanks to the Hermiticity of M ψ . Now let us requirethat L D keep unchanged under a translational transfor-mation of the left- and right-handed fermion fields in theflavor space, ψ α L(R) → ψ α L(R) + n α z ψ , (11)where n α and z ψ are the flavor-dependent complex num-bers and the spacetime- and flavor-independent elementof the Grassmann algebra, respectively. We find that L D will be invariant with respect to the transformation madein Eq. (11) if and only if the conditions X α (cid:2) n ∗ α h m i αβ (cid:3) = 0 , X β (cid:2) h m i αβ n β (cid:3) = 0 , X α X β (cid:2) n ∗ α h m i αβ n β (cid:3) = 0 (12)are satisfied. Given these constraints, it is easy to showthat the determinant of M ψ vanishes, indicating that oneof the three fermion masses m i (for i = 1 , ,
3) must beexactly vanishing. Substituting Eq. (10) into Eq. (12), we immediately obtain X i " λ i X α ( n ∗ α V αi ) V ∗ βi = 0 , X i λ i V αi X β (cid:0) V ∗ βi n β (cid:1) = 0 , X i λ i X α ( n ∗ α V αi ) X β (cid:0) V ∗ βi n β (cid:1) = 0 . (13)If the sum of n ∗ α V αi over α (or equivalently, the sum of V ∗ βi n β over β ) in Eq. (13) were nonzero, one would be leftwith an interesting but phenomenologically-disfavoredrelationship V βi /V βj = constant for β taking differentflavors in connection with m k = 0, where i , j and k runcyclically over 1, 2 and 3. So this nontrivial solution hasto be discarded.It is therefore straightforward to obtain the other non-trivial solution to Eq. (13): X α ( n ∗ α V αi ) = X β (cid:0) V ∗ βi n β (cid:1) = 0 , (14)in correspondence to λ i = 0. One can see that n α ∝ V αj and n β ∝ V βj with j = i satisfy Eq. (14). Imposing thenormalization condition on n α , we simply take n α = V αj and n β = V βj associated with the j -th mass eigenvalue λ j , and substitute them into Eq. (13). We are then leftwith the consistent results λ j V ∗ βj = 0 , λ j V ∗ αj = 0 , λ j = 0 . (15)In other words, the invariance of a Dirac fermion massterm L D under the translational transformation made inEq. (11) implies that the three flavor-dependent complexnumbers n α can be fully determined by the elements of V in its j -th column corresponding to λ j = 0.The reasoning made above for the Dirac fermions canalso be extended to the Majorana neutrinos, or vice versa.While the possibility of m = 0 (or m = 0) is still al-lowed by current experimental data, none of m e = 0, m u = 0 and m d = 0 are true in nature. But the observedstriking hierarchies m e ≪ m µ ≪ m τ , m u ≪ m c ≪ m t and m d ≪ m s ≪ m b [9] indicate that the zero mass limitfor the first-family charged fermions is actually a reason-able starting point for model building, and the nonzerobut small values of m e , m u and m d can be naturally at-tributed to either the tree-level perturbations [32–34] orthe loop-level corrections [35, 36].Note that m u = 0 used to be the most economical so-lution to the strong CP problem in quantum chromody-namics (QCD) [37, 38], but it has been discarded today.In any case m u = 0 can be regarded as a straightfor-ward consequence of the translational symmetry of L D for the up-type quarks as discussed above, and it is wellin tune with the naturalness principle advocated by ’tHooft [10]. The finite values of m u , m d , m e and m (or m ) can therefore be generated from some slight break-ing of such an unconventional flavor symmetry. To keepthe flavor mixing pattern obtained in the zero mass limitunspoiled, the simplest phenomenological way to breakthe translational symmetry of L D (or L M ) is just to adda diagonal and flavor-universal mass term [3–5].To illustrate, let us consider the up- and down-quarksectors in the m u = m d = 0 limits by simply taking V q = 1 √ √ − √ −√ − √ √ c q s q − s q c q , (16)where c q ≡ cos ϑ q and s q ≡ sin ϑ q (for q = u or d). Inthis case the CKM flavor mixing matrix turns out to be V CKM = V † u V d = ϑ sin ϑ − sin ϑ cos ϑ , (17)where ϑ = ϑ d − ϑ u is a nontrivial but small quark mix-ing angle. Note that the structural parallelism between V u and V d (or equivalently, between M u and M d ) assuresthat the resulting V CKM should not deviate far from theidentity matrix . Combining Eqs. (10) and (17), we findthat the reconstructed up-type quark mass matrix M u has the same form as M ν in Eq. (8) with the replace-ments a ≡ (cid:0) m c c + m t s (cid:1) / b ≡ (cid:0) m c s + m t c (cid:1) / c ≡ ( m c − m t ) c u s u / √
6, so does the down-type quarkmass matrix M d with the corresponding replacements.To generate the masses for the first-family quarks to-gether with the other two flavor mixing angles and CPviolation, one needs to properly break the translationalsymmetry in the up- and down-quark sectors.If the translational symmetry of a Dirac or Majoranamass term is obtained at a superhigh energy scale, itmay also be broken at the electroweak scale due to thequantum corrections described by the renormalization-group equations (RGEs) [39]. But it has been shown thata nonzero value of m (or m ) of O (10 − ) eV cannot begenerated from m = 0 (or m = 0) unless the two-loop RGE running effects are taken into account for theMajorana neutrinos [40, 41], and such a tiny mass and theassociated Majorana CP phase do not play any seeablerole in neutrino physics. The masses of the first-familyDirac fermions are in general insensitive to the two-loopRGE-induced quantum corrections either [42–44]. It has been a common belief in particle physics thatbehind different families of the fundamental fermions issome kind of flavor symmetry which can help understand In comparison, the charged-lepton and neutrino mass matricesmust be quite different in their textures so as to give rise tosignificant effects of lepton flavor mixing. the salient features of fermion mass spectra and flavormixing patterns. Although the origin of the massive Ma-jorana neutrinos may be quite different from that of thecharged leptons and quarks, their flavor structures arelikely to share the same symmetry. In this connectionmany flavor symmetries (either Abelian or non-Abelian,either continuous or discrete, either local or global) havebeen tried in the past decades, but new ideas are alwayscalled for in order to pin down the true flavor dynamics.That is why we highlight a possible translational flavorsymmetry associated with the mass terms of fundamentalDirac or Majorana fermions in this work.If the effective mass term for a category of Dirac orMajorana fermions of the same electric charge keeps un-changed under the transformation ψ α L(R) → ψ α L(R) + n α z ψ in the flavor space, where n α and z ψ stand respec-tively for the flavor-dependent complex numbers and thespacetime- and flavor-independent element of the Grass-mann algebra, we have shown that n α can be simplyidentified as the elements U αi in the i -th column of theunitary matrix U used to diagonalize the correspond-ing Hermitian or symmetric fermion mass matrix M ψ ,and m i = 0 holds accordingly. 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