Algebraic Structure of Lepton and Quark Flavor Invariants and CP Violation
aa r X i v : . [ h e p - ph ] J u l Algebraic Structure of Lepton and Quark Flavor Invariants and CP Violation
Elizabeth E. Jenkins and Aneesh V. Manohar Department of Physics, University of California at San Diego, La Jolla, CA 92093 (Dated: June 19, 2018 12:16)Lepton and quark flavor invariants are studied, both in the Standard Model with a dimension fiveMajorana neutrino mass operator, and in the seesaw model. The ring of invariants in the leptonsector is highly non-trivial, with non-linear relations among the basic invariants. The invariantsare classified for the Standard Model with two and three generations, and for the seesaw modelwith two generations, and the Hilbert series is computed. The seesaw model with three generationsproved computationally too difficult for a complete solution. We give an invariant definition of the CP -violating angle ¯ ϑ in the electroweak sector.timestamp: 6/19/2018 12:16 I. INTRODUCTION
The observation of neutrino oscillations requires thatthe Standard Model [1–3] be modified to account for neu-trino masses. The leading theory of neutrino mass isthe seesaw model [4], which contains additional fermionswhich are singlets under the SU (3) × SU (2) × U (1) gaugegroup. An attractive feature of the seesaw theory isthat it naturally gives rise to leptogenesis [5] in CP -violating heavy neutrino decay. The generated leptonasymmetry then produces a baryon asymmetry via Stan-dard Model sphaleron processes. Interestingly, the lightneutrino masses favored by experiment are precisely inthe range needed to produce a baryon asymmetry of theright magnitude [6, 7]. The baryon asymmetry is knownto ∼
10% accuracy from the precision cosmic microwavebackground data of WMAP [8].At energies well below the mass scale M of the heavyfermion singlets in the seesaw model, one constructs alow-energy effective theory obtained by integrating outthe heavy Majorana neutrino singlets. The Lagrangian ofthe low-energy effective theory is given by the renormal-izable Lagrangian of the Standard Model plus additionalhigher-dimensional terms obtained from integrating outthe heavy neutrinos. The leading term is a dimension-five operator [9] which produces a Majorana mass termfor the neutrinos of the Standard Model when the Higgsdoublet acquires a vacuum expectation value v . This op-erator is the unique dimension-five operator which canbe constructed from Standard Model fields. Thus, it isnatural for this dimension-five operator to be the first ob-served effect of new physics beyond the Standard Model.The low-energy effective theory contains additional op-erators at dimension six [10, 11]. The leading effectof these operators is a flavor-nondiagonal correction tothe weakly-interacting neutrino kinetic energy term afterelectroweak symmetry breakdown. This contribution re-sults in a small O ( v/M ) nonunitary contribution to thelepton mixing matrix U PMNS . Unfortunately, for GUT-scale values of the seesaw scale M , this nonunitarity of U PMNS is far too small to be observed experimentally.For the purposes of this paper, the Standard Modellow-energy effective theory is the SU (3) × SU (2) × U (1) gauge theory with only left-handed doublet neutrinos,plus an additional dimension-five gauge invariant oper-ator which gives a Majorana mass to the neutrinos af-ter spontaneous symmetry breaking, and the high-energy(renormalizable) theory is the seesaw model.Flavor violation of quarks and leptons by StandardModel weak interactions is parameterized by unitary 3 × V → e − i Φ U V e i Φ D (1)in the quark sector, where Φ U = diag( φ u , φ c , φ t ) andΦ D = diag( φ d , φ s , φ b ). Physical quantities are basisindependent, and must be invariant under the rephas-ing Eq. (1). CKM rephasing invariants have beenstudied extensively in the literature [12–15], the best-known example being the CP -odd Jarlskog invariant J = Im V V V ∗ V ∗ . Rephasing invariance also existsfor the lepton mixing matrix. In a previous paper [15],we extended the analysis of rephasing invariants to givea complete classification of these invariants for the Stan-dard Model, and for the seesaw model.The parameterization of the flavor structure in termsof masses and mixing angles is convenient for computingdecay rates and scattering amplitudes. However, if onewants to understand the origin of flavor structure, themore fundamental quantities are the flavor matrices inthe Lagrangian from which the masses and mixing anglesare derived by diagonalization. A well-known difficultyis that the flavor matrices are basis-dependent, since onecan make unitary transformations on the quark and lep-ton fields in the Lagrangian. For example, the Yukawamatrix for charge 2 / Y U → U U c T Y U U Q (2)where U Q and U U c are unitary transformations on thequark doublet and singlet fields. One cannot directlycompare a mass-matrix prediction with experiment, sincethe mass matrices are basis-dependent. Observable quan-tities must be independent of this change of basis, i.e. in-variant under Eq. (2), and such quantities are sometimesreferred to as weak basis invariants [16, 17]. One cancheck the predictions of a flavor model by comparing in-variant quantities with their corresponding experimentalvalues.Classifying invariants also is important in analyzingtheories which explain flavor by a dynamical mechanism.The idea can be illustrated by a simple example — con-sider a low-energy theory which has a 3 × X which transforms as an SU (3)adjoint, X → U XU † . Imagine that X is a dynamicalvariable in some high-energy theory, and that the low-energy value of X is given by minimizing an effective po-tential V ( X ) generated by the high-energy theory. It iswell-known (see Sec. V) that the only independent invari-ants are I = h X i and I = h X i , where h∗i denotes amatrix trace, so the potential can be written as V ( I , I )and minimizing it leads to the equation0 = ∂V∂I X a + 34 ∂V∂I d abc X b X c , (3)where X = X a T a . Eq. (3) implies that X a = k d abc X b X c where the constant of proportionality is k = − (3 / ∂V /∂I ) / ( ∂V /∂I ) evaluated at the minimum.The solutions of this equation are either (i) the trivialsolution X = 0, or (ii) X can be brought to the diagonalform X = − k − = − √ k T (4)with symmetry breaking in the T direction. Thus the SU (3) symmetry is either (i) unbroken or (ii) broken tothe SU (2) × U (1) subgroup. Symmetry breaking to U (1) is not allowed. Examples of this type were studied in theearly literature on unified theories [18, 19] in the contextof understanding gauge symmetry breaking patterns byminimizing Higgs potentials. A recent example from fla-vor physics needing the classification of invariants can befound in Ref. [20].There is an extensive literature on quark and lepton in-variants (see, e.g. [16, 17, 21–24]). The main emphasis inprevious work has been the study of CP violation. CP -violating invariants analogous to the Jarlskog invariantwere written down. The vanishing of the CP -violatinginvariants was sufficient to guarantee the vanishing of CP violation in the CKM and PMNS mixing matrices.In this paper, we take a different approach, studyingall the invariants, and treating the problem using themethods of invariant theory [25–27], which considers thering of polynomials that are invariant under the actionof a group. Polynomial invariants also are the relevant objects for physics applications, since an effective La-grangian is written as a polynomial in the basic variableswhich describe the theory. A basic result of invarianttheory is that the ring of invariants has a finite numberof generators. There can be non-trivial relations amongthe invariants, known as syzygies [28], so that the invari-ant ring need not be a free ring. The number of invariantsof a given degree is encoded in the Hilbert series. Thecomplete classification of the invariant ring is, in general,a very difficult computational problem.In this paper, we study the invariants of the StandardModel low-energy theory and the seesaw theory in boththe quark and lepton sectors. In the quark sector, thecomplete structure of the invariant ring is given, and therelation between the polynomial invariants and rephasinginvariants also is given. The structure of the invariantring in the lepton sector is considerably more involvedthan in the quark sector. The classification of leptoninvariants is given for the low-energy effective StandardModel theory for two and three generations. For thehigh-energy seesaw theory, the classification is given fortwo generations. For three generations, we have beenunable to completely classify all the relations or to de-termine the Hilbert series because the problem is com-putationally too difficult. The simpler invariants (i.e. ofsmall degree) are given for this case.The paper is organized as follows. Section II definesthe high-energy seesaw theory and its low-energy effec-tive theory. The flavor-symmetry breaking matrices and ϑ -angles of each theory are given, together with theirtransformation properties under flavor symmetry and CP . Section III defines the mass and mixing matricesof the high-energy and low-energy theories. The high-energy theory contains three mixing matrices, the quarkCKM mixing matrix V CKM , its analogous lepton mixingmatrix V and a mixing matrix for the heavy neutrinosinglets W . The low-energy theory contains two mixingmatrices, the quark CKM mixing matrix V CKM and thelepton PMNS matrix U PMNS . Section III explains thecounting of mixing angles and phases for the mixing ma-trices for arbitrary numbers of Standard Model fermionsand neutrino singlets in both the high- and low-energytheories. Finally, rephasing invariance of the mixing ma-trices is discussed. Section IV provides a brief introduc-tion to the mathematics of invariant theory that we needfor our analysis. Several model theories are consideredto elucidate the mathematical results. The next two sec-tions consider the classification of flavor invariants forthe high-energy seesaw theory and its low-energy effec-tive theory. Section V reviews the classification of thequark mass matrix invariants, which are identical to thequark invariants of the Standard Model for both theo-ries. Sections VI and VII consider the classification of For example, the chiral Lagrangian is a polynomial in the quarkmass matrix M . lepton mass matrix invariants for two and three genera-tions of fermions, respectively, in both the low-energy ef-fective theory and the seesaw theory. The complete clas-sification is given for the low-energy effective theory fortwo and three generations. The lepton invariant analysisof the full seesaw theory is significantly more complex.The complete classification is given for two generationsof fermions, and partial results for three generations aregiven. II. FLAVOR SYMMETRIES
We consider the SU (3) × SU (2) × U (1) gauge the-ory with n g generations of Standard Model fermionsand n ′ g generations of gauge singlet fermions (neutrinosinglets). The fermion multiplets are Q i = ( , ) / , U ci = ( ¯3 , ) − / , D ci = ( ¯3 , ) / , L i = ( , ) − / and E ci = ( , ) , i = 1 , . . . , n g , and N cI = ( , ) , I =1 , . . . , n ′ g . All fermion multiplets are left-handed Weylfields. The fermion multiplets with n ′ g = n g have a natu-ral embedding in the spinor representation of SO (10),so the usual choice is n ′ g = n g . Theories with n ′ g = n g also are possible, however. Experimentally, we know that n g = 3, but there is no experimental limit on n ′ g . Big-bang nucleosynthesis constrains the number of neutrinoflavors to be less than four; however, this only constrainsneutrinos which are light enough to be present at tem-peratures of order an MeV.The flavor symmetry of the fermion sector of the high-energy theory is SU ( n g ) × U ( n ′ g ) × U (1) , since there is aseparate SU ( n g ) flavor symmetry for each of the five mul-tiplets Q , U c , D c , L and E c , a U ( n ′ g ) flavor symmetry forthe singlets N c , and two additional non-anomalous U (1)flavor symmetries. Out of the six possible U (1) symme-tries, only three linear combinations are non-anomalousunder SU (3) × SU (2) × U (1): N c number which is in-cluded in U ( n ′ g ), ( B − L ), and ( E c + D c − U c ) num-ber. The three additional anomalous U (1) groups can betreated as symmetries if the three ϑ -angles ϑ , , of the SU (3), SU (2) and U (1) gauge groups transform underarbitrary chiral phase transformations ψ → e iα ψ ψ on thefields ψ = Q, U c , D c , L and E c as ϑ → ϑ − n g (2 α Q + α U c + α D c ) ,ϑ → ϑ − n g (3 α Q + α L ) , (5) ϑ → ϑ − n g (cid:18) α Q + 43 α U c + 13 α D c + 12 α L + α E c (cid:19) . Eq. (5) does not depend on n ′ g or α N c , since N c are gaugesinglets. With the transformation Eq. (5), the chiral fla- The ϑ angles multiplying F ˜ F terms are not to be confused withangles θ of the quark and lepton mixing matrices. There are noinstantons in the U (1) sector, but the ϑ angle can have physicalconsquences in the presence of topological defects. vor symmetry becomes U ( n g ) × U ( n ′ g ), with a separateflavor factor for each of the six fermion multiplets.The U ( n g ) × U ( n ′ g ) flavor symmetry of the fermionand gauge kinetic energy terms is explicitly broken bygauge-invariant renormalizable terms — Yukawa cou-plings between fermion multiplets and the Higgs doubletand Majorana mass terms of the fermion singlets. Theflavor symmetry-breaking Lagrangian is given by L = − U ci ( Y U ) ij Q j H − D ci ( Y D ) ij Q j H † − E ci ( Y E ) ij L j H † − N cI ( Y ν ) Ij L j H − N cI M IJ N cJ + h.c. , (6)where H = (1 , / is the Higgs doublet, and gauge andLorentz indices have been suppressed. The Yukawa cou-plings Y U,D,E are n g × n g matrices, whereas the neutrinoYukawa coupling Y ν is an n ′ g × n g matrix. The singletneutrino Majorana mass matrix M is a symmetric n ′ g × n ′ g matrix. In the Standard Model without neutrino singlets,renormalizable terms proportional to Y ν and M are ab-sent.Under the chiral flavor symmetry transformations ψ →U ψ ψ , where U ψ are unitary matrices in flavor space forthe fermion fields ψ = Q , U c , D c , L , E c and N c , theYukawa coupling matrices, the Majorana mass matrixand the ϑ angles transform as Y U → U U c T Y U U Q ,Y D → U D c T Y D U Q ,Y E → U E c T Y E U L ,Y ν → U N c T Y ν U L ,M → U N c T M U N c ,ϑ → ϑ − U Q − arg det U U c − arg det U D c ,ϑ → ϑ − U Q − arg det U L ,ϑ → ϑ −
16 arg det U Q −
43 arg det U U c −
13 arg det U D c −
12 arg det U L − arg det U E c . (7)Under CP , each matrix is transformed to its complexconjugate, and each ϑ angle changes sign, Y U,D,E,ν → Y ∗ U,D,E,ν ,M → M ∗ ,ϑ , , → − ϑ , , . (8)Under the chiral flavor symmetry transformation, the ϑ angles are shifted by Eq. (7). The invariant angle ¯ ϑ QCD is defined by¯ ϑ QCD = ϑ + arg det Y U + arg det Y D . (9)The analogous angles ¯ ϑ , can not be separately defined,but one can define an invariant ϑ -parameter in the elec-troweak sector¯ ϑ EW = ϑ + 2 ϑ + 83 arg det Y U + 23 arg det Y D +2 arg det Y E . (10)After electroweak symmetry breaking, the QED ϑ -angleis 2 ¯ ϑ QED = ¯ ϑ EW . The factor of two arises because thegenerators for a non-abelian gauge theory are normalizedto Tr T a T b = δ ab / n ′ g massive Majorana neutrino singlets withmasses of O ( M ), the heavy Majorana neutrino massscale, and all other fermions are strictly massless. It isnatural that M be of order the GUT scale, the scaleat which the GUT gauge symmetry breaks to the Stan-dard Model gauge group, under which the N c fields areuncharged. When the Higgs field gets a vacuum expec-tation value v/ √
2, the Yukawa matrices generate Diracmass matrices for the quarks and leptons, m U,D,E,ν = Y U,D,E,ν v √ . (11)with the same flavor transformation properties as theYukawa couplings. The Dirac and Majorana mass matri-ces of the ( n g + n ′ g ) left-handed neutrino fields combineto form a neutrino mass term − N I ( M N ) IJ N J , ≤ I , J ≤ n g + n ′ g (12)where the ( n g + n ′ g ) × ( n g + n ′ g ) neutrino mass matrix M N is equal to the symmetric matrix M N ≡ (cid:18) m νT m ν M (cid:19) . (13)The ( n g + n ′ g ) neutrino fields N I are ( ν i , N cI ). The( n g + n ′ g ) mass eigenstates of Eq. (13) give the Majoranamass-eigenstate neutrino fields, which are linear combi-nations of ν i and N cI . The heavy neutrinos with masses O ( M ) are predominantly N c with an O ( v/M ) admixtureof ν , and the light neutrinos with masses O ( v /M ) arepredominantly ν with an O ( v/M ) admixture of N c .A low-energy effective field theory can be obtainedfrom the seesaw theory by integrating out the n ′ g heavyMajorana neutrino mass eigenstates. In this low-energytheory, the leading flavor symmetry-breaking Lagrangianis given by L EFT = − U ci ( Y U ) ij Q j H − D ci ( Y D ) ij Q j H † − E ci ( Y E ) ij L j H † + 12 ( L i H ) ( C ) ij ( L j H ) + h.c. , (14)where the coefficient of the dimension-five operator [9] isgiven by C = Y Tν M − Y ν (15)to lowest order in the 1 /M expansion. When the elec-troweak gauge symmetry breaks, the dimension-five op-erator yields an effective n g × n g Majorana mass matrix m = − C v / CP , the flavor matrices Y U,D,E and ϑ angles ϑ , , of the low-energy effective theorytransform under chiral flavor symmetry and CP as inEq. (7) and Eq. (8), respectively, whereas C transformsas C → U LT C U L ,C → C ∗ , (17)respectively.We will analyze the flavor structure of both the seesawtheory and its low-energy effective theory. The analysisdepends only on the flavor transformation properties ofthe Yukawa coupling and Majorana mass matrices (i.e.the fermion mass matrices). Thus, it applies to any the-ory which has Dirac and Majorana mass matrices withthe same transformation properties as given here, regard-less of whether the Dirac mass terms are proportionalto Yukawa couplings in the theory, or are generated bysome mechanism from more fundamental parameters ofthe theory. III. MASSES, MIXING ANGLES AND PHASES
In this section, we define the mass and mixing param-eters of the high-energy seesaw theory and its low-energyeffective theory. Most of the section is a review of well-known results, and serves to define the parameters andnotation which are needed later. The mass matrices ofthe high and low energy theories in the weak eigenstatebasis are transformed to the mass eigenstate basis by fla-vor rotations to obtain the fermion masses and mixingmatrices. The counting of mixing angles and phases forthe case n ′ g = n g follows the analysis of Ref. [15]. Thecounting of physical parameters is given here for the cases n ′ g > n g and n ′ g < n g , for completeness.Any complex matrix M can be written in the form M = U Λ U ′ where U and U ′ are unitary matrices, and Λis a diagonal matrix with real, non-negative entries. If M is also a symmetric matrix, then it can be written in theform M = M T = U T Λ U , where U is a unitary matrix. A. High-Energy Theory
The flavor matrices of the high-energy seesaw theoryare written in Eq. (6) in the weak eigenstate basis. Theseflavor matrices are related to the mass eigenstate basisby Y U = U U c Λ U U U ,Y D = U D c Λ D U D ,Y E = U E c Λ E U E ,Y ν = U N c Λ ν U ν ,M = U ′ N c T Λ N U ′ N c , (18)where Λ U,D,E , Λ ν and Λ N are n g × n g , n ′ g × n g and n ′ g × n ′ g diagonal matrices respectively, with real, non-negative entries; U U c ,D c ,E c and U U,D,E,ν are n g × n g unitary matrices, and U N c and U ′ N c are n ′ g × n ′ g uni-tary matrices, which transform the mass eigenstate ba-sis to the weak eigenstate basis. Performing the chi-ral flavor transformation Eq. (7) with U U c T = U U − , U D c T = U D − , U E c T = U E − , U Q = U U − , U L = U E − ,and U N c = U ′ N c − brings the flavor matrices to the form Y U = Λ U ,Y D = Λ D V − ,Y E = Λ E ,Y ν = W − Λ ν V,M = Λ N , (19)where V CKM ≡ U U U D − , V ≡ U ν U E − and W ≡ U N c − ( U ′ N c ) T are the three unitary matrices which de-scribe flavor mixing in the seesaw theory. V CKM is theCabibbo-Kobayashi-Maskawa mixing matrix in the quarksector. As is well-known, this n g × n g matrix correspondsto the mismatch between the unitary field redefinitionson U and D in the quark doublets Q required to diagonal-ize Y U and Y D . V is the analogue of the CKM matrix inthe lepton sector; it is the n g × n g matrix correspondingto the mismatch between the unitary field redefinitionson ν and E in the lepton doublets L required to diago-nalize Y ν and Y E . W is an n ′ g × n ′ g mixing matrix in thelepton sector corresponding to the mismatch between theunitary field redefinitions on N c required to diagonalize M and Y ν .To proceed further, it is necessary to consider the threecases n ′ g = n g , n ′ g < n g and n ′ g > n g individually. Wefirst specialize to the case n ′ g = n g considered previouslyin Ref. [15] and review the analysis given there. The anal-ysis is then generalized to the cases n ′ g = n g . The quarksector only depends on the number of quark generations n g , but the lepton sector analysis depends on whether n ′ g = n g , n ′ g < n g or n ′ g > n g . n ′ g = n g The real diagonal matrices Λ
U,D,E,ν,N are invariant un-der the rephasings,Λ ψ → e − i Φ ψ Λ ψ e i Φ ψ , ψ = U, D, E, Λ ν → e − i Φ ν Λ ν e i Φ ν , Λ N → η N Λ N η N , (20)where Φ U,D,E,ν are real diagonal matrices, and η N is adiagonal matrix with allowed eigenvalues ±
1. Only ± N c . Underthese rephasings, the mixing matrices V CKM , V and W transform as V CKM → e − i Φ U V CKM e i Φ D , Matrices Masses Angles PhasesΛ U n g D n g V CKM n g ( n g − ( n g − n g − n g n g ( n g − ( n g − n g − n g generations.The Λ U and Λ D rows give the parameters if Y U or Y D areconsidered separately, and the third row gives the additional parameters if both Y U and Y D are considered together. Thereare ( n g − mixing parameters (angles plus phases), and atotal of ( n g + 1) parameters. V → e − i Φ ν V e i Φ E ,W → e − i Φ ν W η N . (21) Quark Sector:
The parameter counting in the quark sec-tor is well-known, and is summarized here for complete-ness. The matrices Λ U and Λ D each contain n g eigen-values, which correspond to the U -quark and D -quarkmasses, respectively, and are CP even. The quark mix-ing matrix V CKM is an n g × n g unitary matrix with n g parameters. It is conventional to divide these parame-ters into angles and phases — angles are even under CP ,whereas phases are odd under CP . If the V CKM matrixis CP invariant, it is an n g × n g real orthogonal matrixwith n g ( n g − / V CKM has n g ( n g − / n g ( n g + 1) / e iχ e i Φ V ( θ i , δ i ) e i Ψ , (22)where χ is an overall phase, Φ = diag(0 , φ , · · · , φ n g ),and Ψ = diag(0 , ψ , · · · , ψ n g ). The phase redefinitionsΦ U and Φ D of V CKM in Eq. (21) can be chosen to removethe 2 n g − χ , φ i , ψ i , i = 2 , · · · , n g . Thus, V CKM has n g ( n g + 1) / − (2 n g −
1) = ( n g − n g − / V CKM = V ( θ i , δ i ) interms of a standard functional form V , where the n g ( n g − / θ i ∈ [0 , π/
2] and the ( n g − n g − / δ i ∈ [0 , π ). The CKM matrix for n g = 3 is given by [29] V ( θ , θ , θ , δ ) ≡ c s − s c × c s e − iδ − s e iδ c c s − s c
00 0 1 (23) There are n g phases each in Φ U and Φ D , but the transformationΦ U = Φ D ∝ leaves V CKM invariant. where s i ≡ sin θ i and c i ≡ cos θ i . It is now conventionalto call the angles θ , θ , θ rather than θ , θ , θ . Thestandard form Eq. (23) has det V = 1. Lepton Sector:
The matrices Λ N and Λ E each have n g eigenvalues which are CP even. The lepton mixing ma-trices V and W are n g × n g unitary matrices, which canbe parametrized by V = e iχ e i Φ V ( θ i , δ i ) e i Ψ / ,W = e iχ ′ e i Φ ′ V ( θ ′ i , δ ′ i ) e i Ψ ′ / . (24)We use the same standard functional form V as for thequark sector, but with different numerical values for thearguments θ i and δ i . The factor of two in Ψ and Ψ ′ willbe explained below.The rephasing transformations Φ ν , Φ E and η N ofEq. (21) can be used to (i) eliminate χ , χ ′ and ψ i , (ii) re-strict ψ ′ i to the range [0 , π ) rather than [0 , π ), and (iii)eliminate either Φ or Φ ′ , but not both . It is convenient touse the same domain [0 , π ) for all phases, which is whyΨ ′ was scaled by a factor of 2.First consider amplitudes which depend only on Y ν and Y E , but not on M . In this case, the mixing matrix W is no longer observable and can be set to unity. Themixing matrix V has (2 n g −
1) allowed phase redefini-tions: n from Φ ν , n from Φ E , and minus one, becauseΦ ν = Φ E ∝ V . Thus, the parame-ter counting for the mixing matrix V is identical to thatfor V CKM in the quark sector, with n g ( n g − / n g − n g − / M and Y ν and not on Y E , the mix-ing matrix V is no longer observable and can be set tounity. The mixing matrix W has n g allowed phase re-definitions Φ ν . Thus, there are n g ( n g − / n g ( n g + 1) / − n g = n g ( n g − / M , Y ν and Y E are considered together, thenthe mixing matrices V and W together can have 2 n g allowed phase redefinitions due to Φ ν and Φ E . As com-pared with the case of only V or only W , where therewere 2 n g − n g phase redefinitions possible, we have( n g −
1) fewer phase redefinitions, and hence ( n g − n g −
1) additionalphases occur because the same phase redefinition Φ ν waspresent for both V and W , and so cannot be chosen toremove phases from both V and W . Thus, there are anadditional ( n g −
1) phases if all three mass matrices areconsidered together. These phases can be included in ei-ther V or W . The standard form of the mixing matriceswhich uses the Φ ν phases to eliminate the Φ phases from V is given by V = V ( θ i , δ i ) ,W = e − i ¯Φ V ( θ ′ i , δ ′ i ) e i Ψ ′ / , (25) The use of the same symbols θ i for the quark and lepton sectorsshould cause no confusion, since we do not need to deal withmixing in both sectors simultaneously. Matrices Masses Angles PhasesΛ N n g ν n g E n g V : Y ν , Y E n g ( n g − ( n g − n g − W : M, Y ν n g ( n g − n g ( n g − n g − n g n g ( n g − n g ( n g − n ′ g = n g gen-erations. The Λ N , Λ ν and Λ E rows give the parameters if M or Y ν or Y E are considered separately. The V and W rowsgive the additional parameters if both Y ν and Y E , or both M and Y ν are considered together, respectively. The lastrow gives the additional parameters to those in the previousrows when all three matrices M , Y ν and Y E are consideredtogether. There are 2 n g ( n g −
1) mixing parameters (anglesand phases), and a total of n g (2 n g + 1) parameters. whereas the standard form of the mixing matrices whichuses the Φ ν phases to eliminate the Φ ′ phases from W isgiven by V = e i ¯Φ V ( θ i , δ i ) ,W = V ( θ ′ i , δ ′ i ) e i Ψ ′ / . (26)In Eq. (25), V has the canonical CKM form with n g ( n g − / θ i and ( n g − n g − / δ i , whereas inEq. (26), W has the canonical PMNS form with n g ( n g − / θ ′ i and n g ( n g − / n g − n g − / δ i and the ( n g −
1) phases ψ ′ i . In either basis, there are ( n g −
1) additional phases¯Φ ≡ Φ − Φ ′ which cannot be removed, and are observable.This parameter counting for n ′ g = n g is summarized inTable II. ϑ Angles:
Once the mixing matrices have been put instandard form, one can perform additional phase rota-tions which leave the mixing matrices invariant to elim-inate ϑ angles. The only allowed transformation is anoverall phase rotation with Φ U = Φ D = φ Q
1, i.e. baryonnumber. Under this phase transformation, ϑ → ϑ ,ϑ → ϑ − n g φ Q ,ϑ → ϑ + 32 n g φ Q . (27)The transformation leaves ϑ and ϑ + 2 ϑ unchanged,so there are two physical ϑ angles remaining: ϑ QCD , thestrong interaction CP -angle in the basis where the quarkmass matrices are real and diagonal, and ϑ EW = ϑ +2 ϑ ,the electroweak CP -angle in the basis where the quarkand charged lepton mass matrices are real and diagonal. Matrices Masses Angles PhasesΛ N n ′ g ν n ′ g E n g V : Y ν , Y E n g ( n g − n g ( n g − − n ′ g + 1 W : M, Y ν n ′ g ( n ′ g − n ′ g ( n ′ g − n ′ g − U n g − n ′ g ( n g − n ′ g )( n g − n ′ g − ( n g − n ′ g )( n g − n ′ g + 1)Total n g + 2 n ′ g n g n ′ g − n ′ g n g n ′ g − n g TABLE III: Parameters in the lepton sector for n g fermion generations and n ′ g < n g neutrino singlets. The total number ofparameters is equal to the sum of the first six rows minus the last row. The parameters in U n g − n ′ g are removed from V .Matrices Masses Angles PhasesΛ N n ′ g ν n g E n g V : Y ν , Y E n g ( n g − ( n g − n g − W : M, Y ν n ′ g ( n ′ g − n ′ g ( n ′ g + 1) − n g ¯Φ n g − U n ′ g − n g ( n ′ g − n g )( n ′ g − n g − ( n ′ g − n g )( n ′ g − n g + 1)Total 2 n g + n ′ g n g n ′ g − n g n g n ′ g − n g TABLE IV: Parameters in the lepton sector for n g fermion generations and n ′ g > n g neutrino singlets. The total number ofparameters is equal to the sum of the first six rows minus the last row. The parameters in U n ′ g − n g are removed from W . n ′ g < n g For n ′ g < n g , the n ′ g × n g diagonal matrix Λ ν can bewritten as Λ ν ≡ h ¯Λ ν i , (28)where 0 denotes the n ′ g × ( n g − n ′ g ) zero matrix, and ¯Λ ν is a diagonal n ′ g × n ′ g matrix with n ′ g real non-negativeeigenvalues. This matrix is invariant under h ¯Λ ν i → e − i Φ ν h ¯Λ ν i " e i Φ ν U n g − n ′ g , (29)where U n g − n ′ g denotes an arbitrary ( n g − n ′ g ) × ( n g − n ′ g )unitary matrix. The rephasing transformations of thelepton mixing matrices are V → " e − i Φ ν U − n g − n ′ g V e i Φ E ,W → e − i Φ ν W η N . (30)instead of Eq. (21). The additional unitary transformation matrix inEq. (30) can be used to eliminate parameters in V .The parameter counting for n ′ g < n g is summarizedin Table III. The number of CP -even parameters is( n g n ′ g + n g + n ′ g ) and the number of CP -odd parametersis ( n g n ′ g − n g ), consistent with the results of Ref. [10]. n ′ g > n g For n ′ g > n g , the n ′ g × n g diagonal matrix Λ ν can bewritten as Λ ν ≡ " ¯Λ ν , (31)where 0 denotes the ( n ′ g − n g ) × n g zero matrix, and ¯Λ ν is adiagonal n g × n g matrix with n g real positive eigenvalues.This matrix is invariant under " ¯Λ ν → " e − i Φ ν U n ′ g − n g ¯Λ ν e i Φ ν , (32)where U n ′ g − n g denotes an arbitrary ( n ′ g − n g ) × ( n ′ g − n g ) unitary matrix. The rephasing transformation of thelepton mixing matrices is V → e − i Φ ν V e i Φ E ,W → " e − i Φ ν U n ′ g − n g W η N . (33)instead of Eq. (21).The additional unitary transformation matrix inEq. (33) can be used to eliminate parameters in W .The parameter counting for n ′ g > n g is summarizedin Table IV. The number of CP -even parameters is( n g n ′ g + n g + n ′ g ) and the number of CP -odd parametersis ( n g n ′ g − n g ), consistent with the results of Ref. [10]. B. Low-Energy Effective Theory
The flavor matrices in the low-energy effective the-ory are written in Eq. (14) in the weak eigenstate basis.These matrices are related to the mass eigenstate basisby Y U = U U c Λ U U U ,Y D = U D c Λ D U D ,Y E = U E c Λ E U E ,C = U ′ Tν Λ U ′ ν . (34)Performing chiral flavor transformations in the low-energy theory with U U c T = U U c − , U D c T = U D c − , U E c T = U E c − , U Q = U U − , U L = U E − brings theflavor matrices to the form Y U = Λ U ,Y D = Λ D V − ,Y E = Λ E ,C = (cid:0) U − (cid:1) T Λ U − , (35)where V CKM ≡ U U U D − and U − ≡ U ′ ν U E − are thetwo unitary matrices which describe flavor mixing in thelow-energy effective theory. V CKM is the CKM mixingmatrix in the quark sector. U PMNS is the PMNS mixingmatrix in the lepton sector, which is the lepton mixingmatrix which is physically measurable at low energies.The real diagonal matrices Λ
U,D,E, are invariant un-der the rephasingsΛ ψ → e − i Φ ψ Λ ψ e i Φ ψ , ψ = U, D, E, Λ → η ν Λ η ν , (36)which correspond to arbitrary phase redefinitions of thefermion mass eigenstate fields U c , D c , E c , U , D and E , and the discrete rephasings ν → η ν ν , where η ν is adiagonal matrix with allowed eigenvalues ± V CKM → e − i Φ U V CKM e i Φ D ,U PMNS → e − i Φ E U PMNS η ν . (37) Matrices Masses Angles PhasesΛ E n g n g U PMNS n g ( n g − n g ( n g − n g n g ( n g − n g ( n g − n g generations. The Λ E and Λ rows givethe parameters if m E or m are considered separately. The U PMNS row gives the mixing angles and phases of the PMNSmixing matrix.
The quark mixing matrix V CKM has the angles andphases given in Table I as before. The counting of pa-rameters in the lepton sector is summarized in Table V,and is well-known. U PMNS contains n g ( n g − / θ i . The number of phases of U PMNS is n g ( n g +1) / n g phase redefinitions Φ E , for a total of n g ( n g − / n g − n g − / δ i and( n g −
1) phases ψ i . The canonical parametrization of U PMNS is U PMNS = V ( θ i , δ i ) e i Ψ / , (38)Ψ = diag(0 , ψ , . . . , ψ n ).For n g = 3, the low-energy lepton mixing matrix isgiven by U PMNS = V (cid:16) θ ( U )1 , θ ( U )2 , θ ( U )3 , δ ( U ) (cid:17) × e iψ ( U )2 /
00 0 e iψ ( U )3 / , (39)where the superscript ( U ) denotes quantities in thePMNS matrix. IV. INVARIANT THEORY
In the previous sections, we have discussed the param-eters (masses, angles and phases) for the low- and high-energy theories. We would like to analyze the theoriesusing invariant quantities written directly in terms of theoriginal parameters of the theory, the matrices Y U,D,E,ν and M . The structure of the invariants is highly non-trivial, and depends in an interesting way on the numberof generations.To study the invariants, it is useful to introduce severalmathematical results from invariant theory [25–27]. Thegeneral problem is the following: one has a set of variables x , . . . x n which transform (reducibly or irreducibly) un-der the action of a group G . The set of polynomials in { x i } with complex coefficients form a ring C [ x , . . . , x n ].The polynomial ring C [ x , . . . , x n ] is a free ring on thegenerators x , . . . , x n , i.e. it is given by taking linear com-binations of all possible products of powers of the gener-ators with coefficients in C , and there are no non-trivialrelations among the generators.The ring C [ x , . . . , x n ] G ⊆ C [ x , . . . , x n ] is the set of G -invariant polynomials, i.e. those polynomials which areunchanged by the action of G . This is clearly a ring, sincesums and products of invariant polynomials are also in-variant polynomials. A highly non-trivial result, if G isa reductive group, is that C [ x , . . . , x n ] G is finite gener-ated. Let the generators be I , . . . I r , each of which is a G -invariant polynomial in the original variables x , . . . , x n .Then, any G -invariant polynomial can be written as apolynomial P ∈ C [ I , . . . , I r ]. However, C [ x , . . . , x n ] G need not be a free ring in the generators I , . . . I r ; therecan be non-trivial relations among them.In the following sections, we analyze the invariant ringfor the quark and lepton sectors of the Standard Modeleffective theory and the seesaw model. It is useful to firstlook at some simple examples before discussing the caseof interest. We start with a famous result on symmetricpolynomials, and then discuss three examples involvingcontinuous groups which are closer in structure to thequark and lepton invariant problem. The first model isa theory which has a freely generated ring, with no rela-tions. The second theory has one non-trivial relation, andis similar in structure to the ring for quark invariants forthree generations studied in Sec. V B and for lepton in-variants in the Standard Model for two generations stud-ied in Sec. VI A. The third example is only slightly morecomplicated, but leads to an intricate structure of in-variants, with many relations, and a complicated Hilbertseries. This is similar to what we find for lepton invari-ants in the Standard Model for three generations, and inthe seesaw model for two and three generations. A. Symmetric Polynomials
The classic example from invariant theory is the studyof symmetric polynomials. The permutation group S n acts on a polynomial f ( x , . . . , x n ) in C [ x , . . . , x n ] by p : f ( x , . . . , x n ) → f ( x p (1) , . . . , x p ( n ) ) (40)where ( p (1) , . . . , p ( n )) is a permutation of (1 , . . . , n ). Apolynomial in C [ x , . . . , x n ] S n is invariant under the ac-tion of any permutation. A standard result [30] is thatthe invariant ring is generated by the elementary sym-metric polynomials I = x + x + . . . x n = X i x i , A reductive group is defined by the property that every repre-sentation is completely reducible. A Lie group which is a directproduct of simple compact Lie groups and U (1) factors is reduc-tive, as is any finite group. I = x x + x x + . . . + x n − x n = X i Consider a theory with two couplings m and m whichtransform under a G = U (1) × U (1) symmetry as m → e iφ m , m → e iφ m . (43)We look at the ring C [ m , m ∗ , m , m ∗ ] U (1) × U (1) of allpolynomials which are U (1) × U (1) invariant. It is clearthat they can be written as linear combinations of mono-mials of the form( m m ∗ ) r ( m m ∗ ) r (44)where r and r are integers. Thus, the ring of invariantpolynomials is generated by the invariants I = m m ∗ and I = m m ∗ , and there are no relations between thesegenerators.The Hilbert series H ( q ) is defined as H ( q ) = ∞ X r =0 c r q r (45)where c r is the number of invariants of degree r , and c = 1. In our example, c = 0; c = 2 since m m ∗ and m m ∗ are the two degree-two invariants; c = 0; c = 3 since ( m m ∗ ) , ( m m ∗ )( m m ∗ ) and ( m m ∗ ) arethe three degree-four invariants; and so on. It is easy tosee that the Hilbert series is H ( q ) = 1 + 2 q + 3 q + 4 q + 5 q + . . . = ∞ X n =0 ( n + 1) q n = 1(1 − q ) . (46)Another derivation of the Hilbert series is the follow-ing. The generators I = m m ∗ and I = m m ∗ are0both of degree two, and the invariants of higher orderare given by multiplying together arbitrary powers of I and I . The product (cid:0) I + I + . . . (cid:1) (cid:0) I + I + . . . (cid:1) (47)gives each invariant once, which leads to the Hilbert series H ( q ) = (cid:0) q + q + . . . (cid:1) (cid:0) q + q + . . . (cid:1) = 1(1 − q ) , (48)in agreement with Eq. (46).In the general case of a semisimple Lie group, it isknown that H ( q ) has the rational form H ( q ) = N ( q ) D ( q ) , (49)where the numerator N ( q ) and denominator D ( q ) arepolynomials. Furthermore, the numerator is of degree d N and is of the form N ( q ) = 1 + c q + . . . c d N − q d N − + q d N (50)where the coefficients are non-negative, c r ≥ 0, and N ( q )is palindromic, i.e. N ( q ) = q d N N (1 /q ) . (51)The denominator takes the form D ( q ) = p Y r =1 (1 − q d r ) , (52)and is of degree d D = P r d r . The number of denomina-tor factors p is equal to the number of parameters. Thenumber of parameters is defined as the minimal codi-mension of an orbit, and agrees with the usual physicsusage of the term. Model I has p = 2 parameters, be-cause we start with four objects m , m , m ∗ and m ∗ (or equivalently, the real and imaginary parts of m and m ), and have two phase redefinitions Eq. (43), whicheliminates two variables. In other words, one can alwaysmake a phase redefinition to make m and m real andnon-negative, and these are the two independent param-eters. In our example, N ( q ) = 1, d = d = 2 and thenumber of denominator factors is two. The number ofdenominator factors p is equal to the number of param-eters.There is a theorem due to Knop [31] which says thatdim V ≥ d D − d N ≥ p (53)where dim V is the dimension of the vector space onwhich the group transformations act; d D and d N are thedegrees of the denominator and numerator; and p is thenumber of parameters. In most cases, the upper boundis an equality, but not always. (We will see an examplefor the quark invariants involving only the U -quark mass matrix.) In Model I, the vector space basis is m , m ∗ , m , m ∗ , so dim V = 4, p = 2, d N = 0 and d D = P d r = 4,and we see that Knop’s theorem gives 4 ≥ − ≥ 2, withan equality for the upper bound.One also can construct a multi-graded Hilbert series.Let c r r r r be the number of invariants of order r in m , order r in m ∗ , order r in m , and order r in m ∗ .Then h ( q , q , q , q ) = X c r r r r q r q r q r q r = 1(1 − q q )(1 − q q ) , (54)and the usual Hilbert series is H ( q ) = h ( q, q, q, q ). Themulti-graded series gives more information about thestructure of the invariants. However, it is importantto remember that the results quoted above for H ( q ),Eqs. (49)–(53), do not hold in general for the multi-graded case. C. Model II Consider a theory with couplings m and m withcharges one and two, respectively, under a G = U (1)symmetry, m → e iφ m , m → e iφ m . (55)The ring of invariant polynomials C [ m , m ∗ , m , m ∗ ] U (1) is generated by the four basic invariants I = m m ∗ , I = m m ∗ , I = m m ∗ and I = m ∗ m . These generators,however, are not all independent, since I I = I I , sothat C [ m , m ∗ , m , m ∗ ] U (1) is not a free ring generatedby I − .It is straightforward to show that the multi-gradedHilbert series is h ( q , q , q , q ) = 1 − q q q q (1 − q q )(1 − q q )(1 − q q )(1 − q q ) , (56)where q , q , q and q count powers of m , m ∗ , m and m ∗ , respectively.The denominator of the multi-graded Hilbert series isgenerated by the invariants I − , whereas the numeratorcompensates for the fact that I I and I I count asonly one invariant at order q q q q , rather than two,because I I = I I . The numerator of the multi-gradedHilbert series does not have the special properties of thenumerator of the Hilbert series H ( q ) discussed in theprevious example.In this example, dim V = 4, dim G = 1, and there arethree parameters since the phase transformation Eq. (55)eliminates one of the original four real variables in m and m . The Hilbert series H ( q ) = h ( q, q, q, q ) is H ( q ) = 1 + q (1 − q ) (1 − q ) , (57)1which has a palindromic numerator with d N = 3, anda denominator with d D = 7, and p = 3 is equal to thenumber of denominator factors and to the number of pa-rameters. Knop’s theorem gives 4 ≥ − ≥ 3, with anequality for the upper bound.Expanding Eq. (57) in a series in q gives the invariantsof each degree. We see that there are two generators ofdegree two, I and I , and one generator of degree three,which can be chosen to be I + I , corresponding to thedenominator factors (1 − q ) and (1 − q ), respectively.Expanding out the denominator would give a coefficientof q of +1. There are two invariants of degree three, I ± I . The missing degree-three invariant I − I iscounted by the + q term in the numerator, so that thecoefficient of q in the expansion of H ( q ) is 2. When thedenominator factors are expanded in a series, they canoccur to any power, so one can have arbitrary powers of I , I and I + I . However, the q factor in the numeratoroccurs only once. This means that powers of I − I higher than the first can all be eliminated in terms ofpolynomials P ( I , I , I + I ) which have already beenincluded. This statement follows from the identity( I − I ) = ( I + I ) − I I = ( I + I ) − I I . (58)There exists a similar identity for the Jarlskog invariantwhich will be derived in Sec. V.The generator I + I of the denominator is not ho-mogeneous in the multi-grading; I is of degree q q and I is of degree q q , which is why Eq. (56) can not bewritten in a form similar to Eq. (57) with positive coe-ficients in the numerator and one less generator in thedenominator. D. Model III Consider yet another model with three couplings m , m and m with charges 1, 2 and 3, respectively, undera U (1) symmetry, m → e iφ m , m → e iφ m , m → e iφ m . (59)The structure of the invariants is considerably more com-plicated than in the previous examples, even though thetheory is only slightly more complicated. All the invari-ant polynomials are generated by thirteen invariant gen-erators I = m m ∗ ,I = m m ∗ ,I = m m ∗ ,I = m m ∗ ,I = m ∗ m ,I = m m ∗ ,I = m ∗ m , I = m m ∗ ,I = m ∗ m ,I = m m m ∗ ,I = m ∗ m ∗ m ,I = m m m ∗ ,I = m ∗ m ∗ m . (60)There are 35 relations between products of invariants I i I j given by: I I = I I , I I = I I , I I = I I , I I = I , I I = I I , I I = I I , I I = I I I , I I = I I , I I = I , I I = I I , I I = I I , I I = I I , I I = I I I , I I = I I , I I = I , I I = I I I , I I = I I I , I I = I I , I I = I I , I I = I I I , I I = I , I I = I I I , I I = I I , I I = I I , I I = I I , I I = I I I , I I = I I I , I I = I I I , I I = I I I , I I = I I I , I I = I I I , I I = I I , I I = I I I , I I = I I I and I I = I I I . The newfeature here is that these relations are not independent—there are relations among the relations (known as syzy-gies in the mathematics literature), e.g. multiplying bothsides of I I = I I and I I = I I gives I I I I = I I I , (61)which is also obtained by multiplying the relations I I = I I and I I = I I , and using I I = I I I . TheHilbert series is H ( q ) = 1 + q + 3 q + 4 q + 4 q + 4 q + 3 q + q + q (1 − q ) (1 − q )(1 − q )(1 − q ) . (62)Here dim V = 6, dim G = 1, and the number of param-eters is 5. From the Hilbert series, d N = 10, d D = 16,and p = 5. The number of parameters is equal to p , andKnop’s theorem gives 6 ≥ − ≥ 5, with an equalityfor the upper bound.There are thirteen invariants in Eq. (60). However,there are only five denominator factors in Eq. (62), soonly five basic invariants, two of degree two, and one eachof degrees three, four and five, generate a free ring. Theother invariants must satisfy non-trivial relations (thosegiven below Eq. (60)), and this is reflected by the com-plicated numerator in Eq. (62), which implies that theinvariant ring has a non-trivial structure, with many re-lations. The different terms in the numerator show thatthere are many invariants which can be eliminated whenraised to higher powers, or multiplied by lower order in-variants, by relations analogous to Eq. (58). There is oneinvariant of degree two (the + q term), three in degreethree (the +3 q term), etc. This model shows that evena relatively simple theory given by Eq. (59) can lead toa set of invariants with an interesting syzygy structure.Furthermore, the number of invariants and relations ofeach degree is encoded in the Hilbert series.2 V. QUARK INVARIANTS We can now address the first problem of interest —flavor invariants in the quark sector. We are interestedin polynomials in m U , m U † , m D and m D † where m U → U U c T m U U Q ,m D → U D c T m D U Q , (63)under the chiral flavor transformations. To cancel U U c and U D c , one must consider the combinations X U ≡ m U † m U ,X D ≡ m D † m D , (64)which both transform as adjoints X U,D → U † Q X U,D U Q . (65)Thus, the invariants are traces of products of X U and X D . The structure of the invariants depends non-trivially on the number of generations, so we considerthe cases n g = 2 and n g = 3 separately. A. n g = 2 First, consider invariants involving only X U . The basicinvariants are h X U i , h X U i , h X U i , . . . (66)where h∗i denotes a matrix trace. This series of tracesterminates after n g terms for an n g × n g matrix, by theCayley-Hamilton theorem which states that every matrixsatisfies its characteristic equation. For an arbitrary 2 × A , the Cayley-Hamilton theorem gives A = h A i A + h(cid:10) A (cid:11) − h A i i . (67)Taking the trace of both sides gives the trivial result (cid:10) A (cid:11) = (cid:10) A (cid:11) . Multiplying by A and taking the traceimplies (cid:10) A (cid:11) = h A i (cid:10) A (cid:11) − h A i , (68)so that h A n i , n ≥ h A i and h A i . Thus, there are two independent invariants, I , = h X U i and I , = h X U i , which can be constructedfrom X U alone. Both of these invariants are CP even.The two invariants contain the same information as theeigenvalues of X U , i.e. the two U -type quark masses. Forinvariants constructed only from m U , the number of pa-rameters is p = 2, the two eigenvalues of X U . The vector One could equally well work with the Yukawa matrices, whichdiffer by factor v/ √ space has dim V = 8, because m U and m U † are both2 × I , and I , are of degree two andfour, respectively, in m U , so the Hilbert series is H ( q ) = 1(1 − q )(1 − q ) . (69)Here d N = 0, d D = 6 are the degrees of the numeratorand denominator, respectively, and the number of de-nominator factors is p = 2, which is equal to the numberof parameters. Knop’s theorem gives 8 ≥ − ≥ 2, whichholds, but this time the upper bound is not an equality.Similarly, there are two independent CP -even invari-ants I , = h X D i and I , = h X D i which involve only X D . These two invariants contain the same informationas the eigenvalues of X D , namely the two D -type quarkmasses.Invariants containing both X U and X D can be writtenas traces of the form h X U r X Ds X U r X Ds . . . i , (70)for integers r i and s i . The Cayley-Hamilton theorem fora 2 × r i and s i greater than one in Eq. (70) can be reduced, so we areleft with traces of the form h X U X D . . . X U X D i = h ( X U X D ) r i . (71)Again, invariants with r > I , = h X U X D i , which is CP even.In summary, the basic quark invariants for n g = 2quark generations, which generate all the invariants, are: I , = h X U i = h m U † m U i ,I , = h X D i = h m D † m D i ,I , = h X U i = h (cid:0) m U † m U (cid:1) i ,I , = h X U X D i = h m U † m U m D † m D i ,I , = h X D i = h (cid:0) m D † m D (cid:1) i . (72)Writing the invariants in terms of the usual quark massesand the Cabibbo angle gives I , = m u + m c ,I , = m d + m s ,I , = m u + m c ,I , = m u m s + m c m d + ( m s − m d )( m c − m u ) cos θ,I , = m d + m s . (73)Knowing the five invariants allows one to determine thefour masses and θ , because m i ≥ 0, and θ lies in the firstquadrant.Using u and d to count powers of m U and m D givesthe multi-graded Hilbert series h ( u, d ) = 1(1 − u )(1 − u )(1 − d )(1 − d )(1 − u d ) . (74)3The Hilbert series H ( q ) = h ( q, q ) is H ( q ) = 1(1 − q ) (1 − q ) . (75)In this example, p = 5 (four masses and one mixing an-gle, see Table I), dim V = 16, since there are four 2 × d N = 0, and d D = 16. The number of denom-inator factors is the number of parameters, and Knop’stheorem gives 16 ≥ − ≥ 5, with the upper bound anequality.The denominator factors in Eq. (75) show that thereare two generators of degree two, and three of degreefour, which agrees with Eq. (72).If one started with X U and X D as the basic objects,then dim V = 8. In this case, the Hilbert series is given byreplacing q → q in Eq. (75), since we now count powersof X U , X D rather than m U , m D , so d N = 0, d D = 8 andKnop’s inequality becomes 8 ≥ − ≥ B. n g = 3 For an arbitrary 3 × A , the Cayley-Hamiltontheorem states that A = A h A i − A h h A i − (cid:10) A (cid:11)i + 16 h h A i − (cid:10) A (cid:11) h A i + 2 (cid:10) A (cid:11)i . (76)Taking the trace of both sides gives the trivial result (cid:10) A (cid:11) = (cid:10) A (cid:11) . Multiplying by A and taking the tracegives (cid:10) A (cid:11) = 16 h A i − h A i (cid:10) A (cid:11) + 43 (cid:10) A (cid:11) h A i + 12 (cid:10) A (cid:11) , (77)so that h A n i , n ≥ h A i , h A i , and h A i .Thus, the invariants involving X U alone are I , = h X U i , I , = h X U i and I , = h X U i , and invariantsinvolving X D alone are I , = h X D i , I , = h X D i and I , = h X D i , all of which are CP even.Invariants containing both X U and X D are of the formEq. (70), but now with r i = 1 , s i = 1 , 2, so thatone has traces of products of X U , X U , X D , X D . This re-striction still leads to an infinite number of invariants.However, many of these invariants are not independent. For arbitrary 3 × A , B and C , one has theidentity0 = h A i h B i h C i − h BC i h A i − h AB i h A i h C i− h AC i h A i h B i + 2 h ABC i h A i + 2 h ACB i h A i− (cid:10) A (cid:11) h B i h C i + 2 h AB i h AC i + (cid:10) A (cid:11) h BC i +2 h C i (cid:10) A B (cid:11) + 2 h B i (cid:10) A C (cid:11) − (cid:10) A BC (cid:11) − (cid:10) A CB (cid:11) − h ABAC i (78)which can be derived by substituting A → A + B + C intoEq. (77), and picking out the order A BC terms. Thisidentity eliminates h ABAC i , i.e. traces where the samematrix is repeated, so that in invariants Eq. (70), X U , X U , X D and X D can each occur at most once. Forexample, h X U . . . X U . . . i can be replaced by (cid:10) X U . . . (cid:11) ,and (cid:10) X U . . . X U . . . (cid:11) can be replaced by (cid:10) X U . . . (cid:11) , whichcan then be eliminated using Eq. (76).Writing out all of the possibilities gives the basic quarkinvariants for n g = 3 quark generations. There are 11 CP -even invariants, ten of which are I , = h X U i ,I , = h X D i ,I , = h X U i ,I , = h X U X D i ,I , = h X D i ,I , = h X U i ,I , = h X U X D i ,I , = h X U X D i ,I , = h X D i ,I , = h X U X D i , (79)and one CP -odd invariant I ( − )6 , = h X U X D X U X D i − h X D X U X D X U i . (80)The eleventh CP -even invariant is I (+)6 , = h X U X D X U X D i + h X D X U X D X U i . (81)All the invariants in the quark sector can be written aspolynomials in these basic invariants.The multi-graded and one-variable Hilbert series are h ( u, d ) = 1 + u d (1 − u )(1 − u )(1 − u )(1 − d )(1 − d )(1 − d )(1 − u d )(1 − u d )(1 − u d )(1 − u d ) ,H ( q ) = h ( q, q ) = 1 + q (1 − q ) (1 − q ) (1 − q ) (1 − q ) , (82)4respectively. This case has p = 10 parameters, consistingof 6 masses, three angles and one phase, which agreeswith the number of denominator factors. The originalvariable space has dim V = 36, from the two 3 × d N = 12 and d D =48, respectively, and Knop’s inequality is 36 ≥ − ≥ 10, which is satisfied, with the upper bound being anequality. If one started with X U and X D as the basicobjects, then dim V = 18, and the Hilbert series is givenby replacing q → q in Eq. (82), so d N = 6, d D = 24,and Knop’s inequality becomes 18 ≥ − ≥ CP -odd invari-ant Eq. (80). The Hilbert series implies that the otherdegree-twelve invariant, Eq. (81), cannot be an indepen-dent invariant. Indeed, it can be written as a polynomialin the other CP -even invariants,3 I (+)6 , = I , I , − I , I , I , − I , I , I , +3 I , I , I , − I , I , I , + 3 I , I , I , − I , I , I , + I , I , I , + 3 I , I , + 3 I , I , + I , I , I , − I , I , , (83)and so can be eliminated.The Hilbert series numerator only has an entry q , butthere is no q term. This means that I ( − )6 , is an indepen-dent invariant, but the square and all higher powers of I ( − )6 , are not. The square of the CP -odd invariant I ( − )6 , is CP -even, and can be written as a polynomial (with 241terms out of a possible 305 terms) in the CP -even invari-ants of Eq. (79). The most general polynomial invariantin the quark sector can be written as P + I ( − )6 , P (84)where P and P are polynomials in the CP -even invari-ants Eq. (79).This example illustrates how the structure of the in-variants is encoded in the Hilbert series. For many pur-poses, the details of the relations, such as Eq. (83), or theformula for (cid:16) I ( − )6 , (cid:17) are not important; all one needs toknow is that I ( − )6 , occurs linearly, and I (+)6 , can be elimi-nated.The quark sector parameters are determined by theten CP -even parameters I , , I , , I , , I , , I , , I , , I , , I , , I , , I , , and the single CP -odd parameter I ( − )6 , . From the CP -even invariants, one can determinethe U -type quark masses m u,c,t and D -type quark masses m d,s,b , which are real and non-negative, and four combi-nations of the CKM parameters, cos θ , cos θ , cos θ and cos δ , all of which are CP even. Since the CKM an-gles θ , θ , θ lie in the first quadrant, these anglesare determined uniquely by their cosines. However, cos δ does not determine the phase δ uniquely, because it can-not distinguish between δ and − δ . Under CP , δ ↔ − δ .Thus, one Z piece of information, the sign of δ , is miss-ing. This sign is provided by the invariant I ( − )6 , . Theonly information needed is the sign of I ( − )6 , , which is why (cid:16) I ( − )6 , (cid:17) can be written in terms of the other CP -even in-variants. This discussion corresponds to the well-knownresult that the unitarity triangle can be obtained by mea-suring the lengths of its sides, which are CP -conserving,rather than the angles, which are CP -violating. Know-ing the sides determines the triangle up to a two-foldreflection ambiguity, which is fixed by the sign of I ( − )6 , ,or, equivalently, the sign of the Jarlskog invariant, so thatthe only additional information contained in the Jarlskoginvariant is the sign. The relations between the invari-ants are similar to those obtained by studying rephasinginvariants [15].The invariant I ( − )6 , also can be written as I ( − )6 , = 13 D [ X U , X D ] E , (85)and is proportional to the Jarlskog invariant J [12], I ( − )6 , = 2 iJ ( m c − m u )( m t − m c )( m t − m u ) × ( m s − m d )( m b − m s )( m b − m d ) , (86)where J = Im ( V CKM ) ( V CKM ) ∗ ( V CKM ) ( V CKM ) ∗ . (87) I ( − )6 , vanishes if two U -type quarks or two D -type quarksare degenerate. It is well-known that quark CP viola-tion vanishes for degenerate U -type or D -type quarks. I ( − )6 , is odd under the exchange of two U -type or two D -type masses, e.g under m u ↔ m c , whereas the invariantsin Eq. (79) are even under exchange, so I ( − )6 , cannot bewritten in terms of the other invariants. (cid:16) I ( − )6 , (cid:17) is evenunder exchange, and can be written in terms of the otherinvariants.It is, of course, well-known that CP conservation inthe quark sector requires J = 0, or equivalently, I ( − )6 , =0. What is new is the structure of the ring of all in-variant polynomials, and the relation between the CP -conserving and CP -violating invariants.5 VI. LEPTON INVARIANTS FOR TWOGENERATIONS The structure of the lepton invariants, like the quarkinvariants, depends on the number of generations, so wefirst consider the case of n g = 2 generations in this sec-tion. The case of n g = 3 generations is considered inSection VII. We will outline the derivation of the results,but not give all the details. A. The Standard Model Effective Theory We now study the lepton invariants in the StandardModel low-energy effective theory with a neutrino Ma-jorana mass term. The structure of the lepton invari-ants is considerably more complicated than the quarkinvariants. The lepton sector of the low-energy theorycontains the flavor symmetry breaking matrices Y E and C , so we are interested in polynomials in m E , m E † , m and m ∗ = m † , since m is a symmetric matrix. Thesematrices transform as m E → U E c T m E U L ,m E † → U E c † m E † U L ∗ ,m → U LT m U L ,m ∗ → U L † m ∗ U L ∗ , (88)under chiral flavor transformations. To cancel U E c , onemust consider the combinations X E ≡ m E † m E ,X ∗ E = X ET ≡ m ET m E ∗ , (89)which transform as X E → U † L X E U L ,X ET → U LT X ET U L ∗ . (90)It also is convenient to define X ≡ m ∗ m , (91)which transforms as X → U L † X U L , (92)as well as (cid:16) m ∗ ( X En ) T m (cid:17) , which transforms as (cid:16) m ∗ ( X En ) T m (cid:17) → U L † (cid:16) m ∗ ( X En ) T m (cid:17) U L . (93) The invariants involving only X E are I , = h X E i and I , = h X E i , whereas the invariants involving only m and m ∗ are I , = h X i and I , = h X i .The invariants involving X E , m and m ∗ are of theform h m ∗ ( X Er ) T m X Es . . . m ∗ ( X Er n ) T m X Es n i (94)for integers r i and s i . The Cayley-Hamilton theorem im-plies that all powers r i and s i greater than one in Eq. (94)can be rewritten in terms of lower order invariants. Thus,one needs to consider traces of matrix products contain-ing the matrices X E , X , and (cid:0) m ∗ X ET m (cid:1) at mostonce.In summary, the generators of the invariants are: I , = h X E i = h m E † m E i ,I , = h X i = h m ∗ m i ,I , = h X E i = h (cid:0) m E † m E (cid:1) i ,I , = h m ∗ X ET m i = h m X E m ∗ i = h m ET m E ∗ m m ∗ i = h m E † m E m ∗ m i ,I , = h X i = h ( m ∗ m ) i ,I , = h m ∗ X ET m X E i = h m ∗ m ET m E ∗ m m E † m E i , (95) I ( − )4 , = h m ∗ X ET m X E m ∗ m i− h m ∗ X ET m m ∗ m X E i = h m ∗ m ET m E ∗ m m E † m E m ∗ m i− h m ∗ m ET m E ∗ m m ∗ m m E † m E i , where I ( − )4 , is CP odd, and the rest are CP even. Thesquare of the CP -odd invariant, (cid:16) I ( − )4 , (cid:17) , is not inde-pendent; it can be expressed in terms of polynomials inthe other CP -even invariants. In addition, the CP -eveninvariant I (+)4 , , obtained by the substitution − → + in I ( − )4 , , is not independent, and thus is not included in theabove list.There are six parameters: four masses, one angle andone phase, see Table V. The four masses, one mixingangle, and one phase, can be determined from I , , I , , I , , I , , I , and I , up to a sign ambiguity in the phase,just as for the case of three generations of quarks alreadydiscussed. The sign of the phase is fixed by the sign of I ( − )4 , .The multi-graded Hilbert series is h ( y, z ) = 1 + y z (1 − y )(1 − y )(1 − z )(1 − z )(1 − y z )(1 − y z ) , (96)6where y counts powers of m E and z counts powers of m .The single variable Hilbert series is H ( q ) = h ( q, q ) = 1 + q (1 − q ) (1 − q ) (1 − q ) . (97)The q term in the numerator shows that there is onedegree-eight invariant I ( − )4 , which occurs, but that thesquare of this invariant is not independent and can beeliminated.The number of denominator factors p = 6 is equal tothe number of parameters, and d N = 8, d D = 22. Thenumber of variables is dim V = 14, since we have one 2 × × ≥ − ≥ CP -even invariants inEq. (96). The numerator shows that there is an invariantof degree eight, whose square can be eliminated, which is I ( − )4 , . The structure of the invariants for n g = 2 is similarto that for quarks for n g = 3.Weak-basis invariants for two generations in thelow-energy effective theory were studied previously byBranco, Lavoura and Rebelo [32]. They defined an in-variant Q , related to I ( − )4 , by2 i Im Tr Q = I ( − )4 , , (98)and showed that Q = 0 is a necessary and sufficient con-dition for CP conservation. This is consistent with ourresults, since the only CP -odd generating invariant is I ( − )4 , . B. The Seesaw Model In this section, we analyze the lepton invariants in theseesaw theory for n g = n ′ g = 2 generations of fermions.There are three matrices in the lepton sector, m ν , m E and M , and their complex conjugates m ν † , m E † and M † = M ∗ . From Eq. (7), we see that only m E trans-forms under U E c , so it must always occur in the combi-nation X E = m E † m E , (99)which transforms as X E → U † L X E U L (100) It is worth emphasizing that in our notation m ν refers to theDirac mass matrix m ν = Y ν v/ √ 2, not the Majorana mass matrix m of the effective theory. under the chiral flavor symmetry transformations. Themass matrices m ν , m † ν , M and M ∗ transform as m ν → U N c T m ν U L ,m † ν → U L † m † ν U N c ∗ ,M → U N c T M U N c ,M ∗ → U N c † M ∗ U N c ∗ . (101)It is useful to define X ν ≡ m † ν m ν ,Z ν = m ν m † ν ,Z νT = Z ν ∗ = m ∗ ν m νT , (102)which transform as X ν → U † L X ν U L ,Z ν → U N c T Z ν U N c ∗ ,Z νT → U N c † Z νT U N c , (103)as well as X N ≡ M ∗ M,Z N = M M ∗ ,Z X = m ν X E m ν † (104)which transform as X N → U † N c X N U N c ,Z N → U N c T Z N U N c ∗ ,Z X → U N c T Z X U N c ∗ . (105)Note that Z N T = Z N ∗ = X N .The invariants involve three mass matrices, m E , m ν and M . One first can consider the simpler problem ofstudying invariants which only depend on two out of thethree matrices. The first case, invariants involving only m E and m ν , consists of invariants formed from tracesof X E and X ν only, with no insertions of M or M ∗ .These invariants are the same as the invariants in thequark sector with the substitutions X U → X ν and X D → X E . The second case, invariants involving only m ν and M , are invariants which do not contain X E . These havethe same structure as invariants constructed in the low-energy theory, with the replacements m → M , m E → m Tν , i.e. X E → Z Tν .The most general invariant involving all three matriceshas the structure h M ∗ A M A T . . . M ∗ A n − M A T n i , (106)where A i = A i = m ν P ( X E , X ν ) m ν † , where P isa polynomial in X E and X ν . This result can be ob-tained by representing the chiral transformations of thematrices graphically, as shown in Fig. 1. Products ofmatrices such as Eq. (106) also occurred when studyingrephasing invariants [15]. For rephasing invariants, one7 m † ν X E m ν M ∗ M FIG. 1: Graphical representation of the chiral transformationproperties of the lepton mass matrices X E , m ν , m † ν , M and M ∗ . A solid line represents U N c , and a dashed line U L . Theinvariants are obtained by forming graphs with no externallines. can factor long products into smaller ones, each involvingat most four mixing matrices, using reconnection identi-ties. This factorization is no longer possible for the caseof mass-matrix invariants, which leads to an interestingand highly non-trivial structure for the invariants.The basic invariants can be constructed using Eq. (106)and eliminating higher powers of matrices by the Cayley-Hamilton identity Eq. (67). The generators are: I , , = h X E i = h m E † m E i ,I , , = h X ν i = h m ν † m ν i ,I , , = h X N i = h M ∗ M i ,I , , = h X E i = h m E † m E m E † m E i ,I , , = h X ν X E i = h m ν † m ν m E † m E i ,I , , = h X ν i = h m ν † m ν m ν † m ν i ,I , , = h Z ν Z N i = h m ν m ν † M M ∗ i ,I , , = h X N i = h M ∗ M M ∗ M i ,I , , = h Z X Z N i = h m ν m E † m E m ν † M M ∗ i ,I , , = h M ∗ Z ν M Z νT i = h M ∗ m ν m ν † M m ν ∗ m νT i ,I , , = h M ∗ Z ν M Z X T i = h M ∗ m ν m ν † M m ν ∗ m ET m E ∗ m νT i ,I ( − )2 , , = h M ∗ Z ν Z X M i − h M ∗ Z X Z ν M i = h M ∗ m ν m ν † m ν m E † m E m ν † M i− h M ∗ m ν m E † m E m ν † m ν m ν † M i ,I ( − )0 , , = h Z N Z ν M Z νT M ∗ i − h M ∗ Z ν Z N M Z νT i = h M M ∗ m ν m ν † M m ν ∗ m νT M ∗ i− h M ∗ m ν m ν † M M ∗ M m ν ∗ m νT i ,I , , = h M ∗ Z X M Z XT i = h M ∗ m ν m E † m E m ν † M m ν ∗ m ET m E ∗ m νT i ,I ( − )2 , , = h Z N Z X M Z νT M ∗ i − h M ∗ Z X Z N M Z νT i = h M M ∗ m ν m E † m E m ν † M m ν ∗ m νT M ∗ i− h M ∗ m ν m E † m E m ν † M M † M m ν ∗ m νT i ,I ( − )2 , , = h M ∗ Z ν Z X M Z νT i − h M ∗ Z X Z ν M Z νT i = h M ∗ m ν m ν † m ν m E † m E m ν † M m ν ∗ m νT i− h M ∗ m ν m E † m E m ν † m ν m ν † M m ν ∗ m νT i ,I ( − )4 , , = (cid:10) M ∗ Z N Z X M Z TX (cid:11) − (cid:10) M ∗ Z TX Z N M Z X (cid:11) ,I ( − )4 , , = (cid:10) M ∗ Z ν Z X M Z TX (cid:11) − (cid:10) M ∗ Z X Z ν M Z TX (cid:11) . (107)There are several invariants which can be immediatelyeliminated because they are polynomials in lower orderinvariants and which have not been listed above. Theseinvariants include I (+)2 , , , I (+)0 , , , I (+)2 , , , I (+)2 , , , I (+)4 , , and I (+)4 , , , which are related in an obvious way to the invari-ants in Eq. (107) with superscripts ( − ). The degree-eightinvariants I (+)2 , , and I (+)0 , , are eliminated by the identities0 = I , , I , , I , , − I , , I , , I , , − I , , I , , − I , , I , , + 2 I (+)2 , , , I , , I , , − I , , I , , − I , , I , , − I , , + 2 I (+)0 , , , (108)and the degree-ten invariants I (+)2 , , and I (+)2 , , are elimi-nated by the identities0 = I , , I , , I , , − I , , I , , − I , , I , , I , , − I , , I , , + 2 I (+)2 , , , I , , I , , I , , − I , , I , , − I , , I , , I , , − I , , I , , + 2 I (+)2 , , . (109)The degree-twelve invariants I (+)4 , , and I (+)4 , , are alsopolynomials in lower order invariants, but we do not in-clude the explicit identities here.In Eq. (107), there are three CP -even invariants ofdegree two, five of degree four, two of degree six, one ofdegree eight, and one of degree ten, for a grand total of 12basic CP -even invariants. In addition, there are two CP -odd invariants of degree eight, two of degree ten and twoof degree twelve, for a total of 6 basic CP -odd invariants.All of the invariants can be written as polynomials inthese 18 basic invariants.The multi-graded Hilbert series is h ( x, y, z ) = ND ,N = 1 + 2 x y z + y z + x y z + x y z + x y z + x y z − x y z − x y z − x y z − x y z − x y z − x y z − x y z ,D = (cid:0) − x (cid:1) (cid:0) − x (cid:1) (cid:0) − y (cid:1) (cid:0) − y (cid:1) (cid:0) − z (cid:1) (cid:0) − z (cid:1) (cid:0) − x y (cid:1) (cid:0) − y z (cid:1) (cid:0) − x y z (cid:1) (cid:0) − y z (cid:1) (cid:0) − x y z (cid:1) , (110)where x , y , z count powers of m E , m ν and M , respec-tively. The Hilbert series H ( q ) = h ( q, q, q ) is H ( q ) = 1 + q + 3 q + 2 q + 3 q + q + q (1 − q ) (1 − q ) (1 − q )(1 − q ) , (111)which has a palindromic numerator. The number of de-nominator factors p = 10 is equal to the number of pa-rameters, and d N = 20 and d D = 42. The number ofvariables is dim V = 22, because we have two 2 × × ≥ − ≥ 10, andthe upper bound is an equality. The 10 parameters inthe lepton sector of the seesaw model for n g = n ′ g = 2generations correspond to 2 charged lepton masses, 4 Ma-jorana neutrino masses of the two light and the two heavyneutrinos, 2 angles and 2 phases.One can see from the Hilbert series that the struc-ture of invariants is far more complicated than in thequark case. The denominator factors (1 − q ) (1 − q ) of Eq. (111) corresponds to the generators I , , , I , , , I , , , I , , , I , , , I , , , I , , , I , , . At degree six, inaddition to products of lower order invariants, there aretwo new invariants, I , , and I , , . These two invari-ants correspond to the (1 − q ) factor in the denomina-tor, and the + q term in the numerator. Since there isonly one power of (1 − q ) factor in the denominator, weknow that there will be non-trivial relations involving thedegree-six invariants. At degree eight, there are 3 new in-variants from the +3 q term in the numerator in additionto products of lower degree invariants which make up thedenominator. These are the three degree-eight invariantsin Eq. (107). There are three new invariants of degreetwelve (from the +3 q ), but only two degree-twelve in-variants in Eq. (107). The third degree-twelve invariantis the square of the degree-six invariant correspondingto the + q term in the numerator, so the square of this CP -even invariant cannot be removed. We have notedearlier that there must be non-trivial relations involvingthe degree-six invariants. These relations first occur atdegree 14,0 = I , , I , , I ( − )2 , , + I , , I ( − )0 , , I , , − I , , I ( − )2 , , − I , , I , , I ( − )2 , , − I , , I ( − )0 , , I , , − I , , I ( − )2 , , − I , , I ( − )0 , , I , , + 2 I , , I ( − )2 , , + 2 I , , I ( − )2 , , I , , I , , I ( − )2 , , − I , , I ( − )2 , , + I , , I , , I ( − )2 , , − I , , I , , I ( − )2 , , − I , , I , , I ( − )0 , , − I , , I , , I ( − )2 , , +2 I , , I ( − )2 , , − I , , I ( − )2 , , + 2 I , , I ( − )0 , , , (112) and are non-linear relations involving the two degree-sixinvariants. One can proceed to higher degrees — thereare six relations of degree 16, etc., and verify the num-ber of independent invariants at each degree agrees withEq. (111). The details of the relations are not important.The main purpose of giving Eq. (112) is to show thatthere can be non-linear relations among the generatinginvariants. To completely unravel all of the non-linearrelations requires going beyond degree 20, the highestpower of q in the numerator of Eq. (111). VII. LEPTON INVARIANTS FOR THREEGENERATIONS In this section, we consider the lepton invariants inthe low-energy and high-energy theories for three gen-erations of fermions. The number of invariants is fargreater than for two generations, and there are many re-lations between them. For the low-energy theory, we givethe Hilbert series, and the invariants which correspond tothe denominator factors. For three generations, even theHilbert series proved too difficult to compute. For thiscase, we make some general remarks, and discuss someinvariants considered previously by Branco et al. [32, 33],and by Davidson and Kitano [24]. A. The Standard Model Effective Theory The invariants involving only X E are I , = h X E i , I , = h X E i and I , = h X E i , whereas the invariantsinvolving only m and m ∗ are I , = h X i , I , = h X i and I , = h X i .The invariants involving X E , m and m ∗ are of theform h m ∗ ( X Er ) T m X Es . . . m ∗ ( X Er n ) T m X Es n i (113)for integers r i and s i . The Cayley-Hamilton theoremimplies that all powers r i and s i greater than two inEq. (113) can be rewritten in terms of lower order invari-ants. Thus, one needs to consider traces of matrix prod-ucts containing the matrices X E , X , (cid:0) m ∗ X ET m (cid:1) ,and (cid:16) m ∗ (cid:0) X E (cid:1) T m (cid:17) at most twice. Identity Eq. (78)cannot be used to eliminate traces with multiple powersof m , because h m Am B i gets converted to traces of theform (cid:10) m AB (cid:11) which are no longer invariant. There aremany basic invariants, which involve a single trace, upto degree m m E , and we do not list them all here. The9ones up to degree twelve, which are sufficient for the de-nominator of the Hilbert series (and hence to determinethe parameters) are: I , = h X E i = h m E † m E i ,I , = h X i = h m ∗ m i ,I , = h X E i = h (cid:0) m E † m E (cid:1) i ,I , = h X E X i = h m E † m E m ∗ m i ,I , = h X i = h ( m ∗ m ) i ,I , = h X E i = h (cid:0) m E † m E (cid:1) i ,I ′ , = h X E X i = h (cid:0) m E † m E (cid:1) m ∗ m i ,I , = h m ∗ X ET m X E i = h m ∗ m ET m E ∗ m m E † m E i ,I , = h X E X i = h m E † m E ( m ∗ m ) i ,I , = h X i = h ( m ∗ m ) i ,I , = h m ∗ X ET m X E i = h m ∗ m ET m E ∗ m (cid:0) m E † m E (cid:1) i ,I ( ± )4 , = h m ∗ X ET m m ∗ m X E i ± h m ∗ m m ∗ X ET m X E i = h m ∗ m ET m E ∗ m m ∗ m m E † m E i± h m ∗ m m ∗ m ET m E ∗ m m E † m E i ,I , = h m ∗ ( X ET ) m X E i = h m ∗ (cid:0) m ET m E ∗ (cid:1) m (cid:0) m E † m E (cid:1) i ,I ( ± )6 , = h m ∗ X ET m m ∗ m X E i± h m ∗ m m ∗ X ET m X E i = h m ∗ m ET m E ∗ m m ∗ m (cid:0) m E † m E (cid:1) i± h m ∗ m m ∗ m ET m E ∗ m (cid:0) m E † m E (cid:1) i ,I ( ± )8 , = h m ∗ ( X ET ) m m ∗ m X E i± h m ∗ m m ∗ ( X ET ) m X E i = h m ∗ (cid:0) m ET m E ∗ (cid:1) m m ∗ m (cid:0) m E † m E (cid:1) i± h m ∗ m m ∗ (cid:0) m ET m E ∗ (cid:1) m (cid:0) m E † m E (cid:1) i . (114)The multi-graded Hilbert series is h ( y, z ) = ND ,N = − y z − y z − y z − y z − y z − y z − y z − y z − y z − y z − y z − y z − y z − y z − y z − y z − y z − y z + y z + y z + 2 y z + y z + y z + y z + y z + y z + 3 y z + 3 y z + 3 y z + y z + 3 y z + 2 y z + y z + 2 y z + 2 y z + 1 ,D = (cid:0) − y (cid:1) (cid:0) − y (cid:1) (cid:0) − y (cid:1) (cid:0) − z (cid:1) (cid:0) − z (cid:1) (cid:0) − z (cid:1) (cid:0) − y z (cid:1) (cid:0) − y z (cid:1) (cid:0) − y z (cid:1) (cid:0) − y z (cid:1) × (cid:0) − y z (cid:1) (cid:0) − y z (cid:1) , (115)where y counts powers of m E and z counts powers of m . The single-variable series H ( q ) = h ( q, q ) is H ( q ) = 1 + q + 2 q + 4 q + 8 q + 7 q + 9 q + 10 q + 9 q + 7 q + 8 q + 4 q + 2 q + q + q (1 − q ) (1 − q ) (1 − q ) (1 − q ) (1 − q ) . (116)The number of denominator factors p = 12 is equal tothe number of parameters, and d N = 36 and d D = 66.The number of variables is dim V = 30, because we haveone 3 × × ≥ − ≥ 12, and the upper bound is an equality. Note that thenumerator is palindromic. The 12 parameters consist of 3charged lepton masses, 3 Majorana light neutrino masses,3 angles and 3 phases.The Hilbert series Eq. (116) has a complicated numer-ator, which shows that the structure of the invariant ringis highly non-trivial. From the denominator of Eq. (116),we see that there are two generators of degree two, three of degree four, four of degree six, two of degree eight, andone of degree 10, which can be multiplied freely, with norelations. These account for most of the invariants inEq. (114), but there remains one CP -even invariant eachof degrees 6, 10, 12, and one CP -odd invariant each ofdegrees 8, 10, 12. These contribute q + q + 2 q + 2 q to the numerator in Eq. (116). The coefficient of q inthe numerator of Eq. (116) is 2. Where does the otherdegree-eight invariant not in Eq. (114) come from? Thedegree-six invariant that corresponds to the numeratorfactor q can be multiplied by either of the two degreeinvariants, I , or I , , to give two additional degree-8 in-variants. One of these can be written as a polynomial inlower order invariants; the other survives. One can con-0tinue this analysis to arbitrarily high order — the entireinvariant structure is encoded in a very compact way inthe Hilbert series Eq. (116). An explicit example of theconstruction just discussed is given in Sec. VI B for thehigh-energy theory with n g = 2, which provides a simplerexample of an invariant ring with non-trivial relations.For three generations, Branco, Lavoura and Rebelo [32]defined four invariants:2 iI = I ( − )4 , iI = (cid:10) X E m ∗ m m ∗ m m ∗ X TE m (cid:11) − c.c.2 iI = (cid:10) X E m ∗ m m ∗ m m ∗ X TE m m ∗ m (cid:11) − c.c.2 iI = det (cid:2) m X E m ∗ + m ∗ X TE m (cid:3) − c.c. (117)of degrees (4 , , , 8) and (6 , CP conserva-tion. The CP -violating invariants of Eq. (114) corre-spond to the denominator factors of the Hilbert series.There are additional CP -violating invariants not listedwhich correspond to terms in the numerator. B. The Seesaw Model The invariants involve three mass matrices, m E , m ν and M . One first can consider the simpler problem ofstudying invariants which only depend on two out of thethree matrices. The first case, invariants involving only m E and m ν , consists of invariants formed from tracesof X E and X ν only, with no insertions of M or M ∗ .These invariants are in direct analogy to the invariantsof the quark sector with the substitutions X U → X ν and X D → X E . The second case, invariants involvingonly m ν and M , are invariants which do not contain X E .These have the same structure as invariants constructedin the low-energy theory, with the replacements m → M , m E → m Tν , i.e. X E → Z Tν .The most general invariant involving all three matriceshas the structure h M ∗ A M A T . . . M ∗ A n − M A T n i , (118)where A i = A i = m ν P ( X E , X ν ) m ν † , where P isa polynomial in X E and X ν . The generating invariantsare given by using Eq. (118). In this case, there are avery large number of generating invariants. They includeall those discussed earlier in the seesaw theory for twogenerations, as well as many other.For n g = n ′ g = 3 generations, there are 21 parameterswhich consist of 9 masses, 6 angles and 6 phases. The 9masses are the 3 charged lepton masses, 3 light Majorananeutrino masses and 3 heavy Majorana neutrino masses.There are 3 angles in the mixing matrix V and 3 anglesin the mixing matrix W . There is one δ -type phase in V and in W , two Majorana phases Ψ ′ in W , and 2 phases ¯Φwhich are not removeable when V and W are consideredtogether. We have been unable to construct the multi-gradedand one-variable Hilbert series in this case. However, itis clear that the structure of the invariant relations is ex-tremely complicated. There are a number of constraintson the form of the one-variable Hilbert series. The de-nominator must be a product of p = 21 factors. Thenumerator must be palindromic, and d N and d D mustsatisfy the Knop inequality 48 ≥ d D − d N ≥ 21 sincedim V = 48. The number of variables dim V = 48 resultsbecause there are two 3 × m E and m ν with 9independent entries each, one 3 × M with 6 independent entries, and the complex conjugatesof the three matrices.Ref. [33] defined six invariants in the seesaw theory,2 iI = (cid:10) Y ν Y † ν M ∗ M M ∗ ( Y ν Y † ν ) T M (cid:11) − c.c.2 iI = (cid:10) Y ν Y † ν M ∗ M M ∗ M M ∗ ( Y ν Y † ν ) T M (cid:11) − c.c.2 iI = (cid:10) Y ν Y † ν M ∗ M M ∗ M M ∗ ( Y ν Y † ν ) T M M ∗ M (cid:11) − c.c.(119)which involve CP -violating phases which are relevant forleptogenesis, as well as2 i ˜ I = (cid:10) Y ν X E Y † ν M ∗ M M ∗ ( Y ν X E Y † ν ) T M (cid:11) − c.c.2 i ˜ I = (cid:10) Y ν X E Y † ν M ∗ M M ∗ M M ∗ ( Y ν X E Y † ν ) T M (cid:11) − c.c.2 i ˜ I = (cid:10) Y ν X E Y † ν M ∗ M M ∗ M M ∗ ( Y ν X E Y † ν ) T M M ∗ M (cid:11) − c.c.(120)which involve the other phases.Ref. [24] defines an invariant2 iI = (cid:10) κ † κκ † ( Y Tν Y ∗ ν ) − κ ( Y † ν Y ν ) − (cid:11) (121)for leptogenesis, where κ is m with factors of the Higgsvacuum expectation value removed. This is not a poly-nomial in the basic variables of the seesaw model. It canbe related to the invariants considered here using the for-mulæ given below.Invariants in the seesaw model can be related to thoseof the low-energy effective theory. The basic relation isEq. (15), which relates the neutrino mass matrices inthe seesaw model to the Majorana mass matrix m inthe low-energy effective theory. Clearly, the relations be-tween the invariants cannot be polynomial, since inversepowers of M are involved, but one can write the low-energy invariants in terms of a rational function of thehigh-energy invariants. The basic identities are:det A A − = h A i − A det A = 12 h A i − (cid:10) A (cid:11) (122)for 2 × A A − = A − A h A i − (cid:10) A (cid:11) + 12 h A i det A = 13 (cid:10) A (cid:11) − (cid:10) A (cid:11) h A i + 16 h A i (123)1for 3 × C = Y Tν M − Y ν = Y Tν ( M ∗ M ) − M ∗ Y ν (124)to obtain the desired relations using A = M ∗ M , andsubstituting for C (i.e. m ) in the expressions for thelow-energy invariants. The expressions are valid as longas det M ∗ M = 0, i.e. as long as the singlet neutrinos areheavy and the transition to a low-energy effective theoryis valid. VIII. CONCLUSIONS We have used the mathematics of invariant theory toclassify the independent invariants of the Standard Modeleffective theory and its high-energy seesaw model andto study the non-trivial structure of relations (syzygies)among the invariant generators. The complete classi-fication of invariants and the Hilbert series have beenobtained for the Standard Model effective theory witha dimension-five Majorana neutrino mass operator. A complete solution also has been obtained for the renor-malizable seesaw model with n g = n ′ g = 2 fermion gen-erations. The lepton sector of the seesaw model involvesthree different mass matrices, the charged lepton massmatrix, the Dirac Mass matrix of the weakly-interactingdoublet neutrinos and the Majorana mass matrix of thegauge-singlet neutrinos. The invariant structure is verycomplicated. In the case of n g = n ′ g = 3 generationsof fermions, we have been unable to find the Hilbert se-ries for the invariant generators, and thus the structureof the syzygy relations for three generations remains anopen problem. Acknowledgments AM would like to thank Professor Nolan Wallach forextensive discussions on invariant theory. 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