PHYSICAL REVIEW D 101, 035020 (2020)
Models of lepton and quark masses
Steven Weinberg* Theory Group, Department of Physics, University of Texas, Austin, Texas 78712, USA
(Received 15 December 2019; accepted 27 January 2020; published 19 February 2020)
A class of models is considered in which the masses only of the third generation of quarks and leptons arise in the tree approximation, while masses for the second and first generations are produced respectively by one-loop and two-loop radiative corrections. So far, for various reasons, these models are not realistic.
DOI: 10.1103/PhysRevD.101.035020
I. INTRODUCTION coupling constants to be relatively small, more or less like the electroweak couplings. If these new gauge couplings In the Standard Model the masses of quarks and leptons together with those of the Standard Model all descended take values proportional to the coupling constants in the from some theory such as a string theory or a unified gauge interaction of these fermions with scalar fields, constants that theory in which they were all equal at some very high energy, in the context of this model are entirely arbitrary. But the then in order to have small couplings at accessible energies peculiar hierarchical pattern of lepton and quark masses the new gauge group would have to be a direct product of seems to call for a larger theory, in which in some leading simple subgroups with smaller beta functions than for the approximation the only quarks and leptons with nonzero mass SUð3Þ of QCD—that is, most likely only SOð3Þ and/or are those of the third generation, the tau, top, and bottom, with SOð2Þ. After some attempts, what seems to work best is the other lepton and quark masses arising from some sort of SOLð3Þ ⊗ SORð3Þ, with the three generations of left- radiative correction. Such theories were actively considered handed quark and lepton SUð2Þ ⊗ Uð1Þ doublets forming [1] soon after the completion of the Standard Model, but separate representations (3, 1) of SOLð3Þ ⊗ SORð3Þ, and interest in this program seems to have lapsed subsequently [2]. the three generations of right-handed quarks and charged This paper will explore in detail a class of models of this leptons furnishing separate representations (1, 3). [We label sort, based on a different symmetry group. These models representations of SOð3Þ by their dimensionality.] Though are not realistic, for reasons that will be spelled out later, but we shall concentrate on this gauge group, our analysis will it is hoped that they may help to revive interest in this deal with problems that would have to be encountered in any program, and to lay out some of the methods and problems attempt to interpret the hierarchy of quark and lepton masses that it confronts. as radiative corrections. In order for scalar fields to have renormalizable cou- II. GAUGE AND SCALAR FIELDS plings to these quarks and leptons, they would have to form 9 electroweak doublets If the spontaneous breakdown of the electroweak sym- Φþ metry gave masses only to the quarks and leptons of the third ia ; ð1Þ generation in the tree approximation, then nothing in the Φ0 ia Standard Model would generate masses for the first and transforming as (3, 3) representations of SO ð3Þ ⊗ SO ð3Þ. second generations in higher orders of perturbation theory. L R [Here superscripts indicate charges; subscripts i, j, etc. are To get masses for the second and first generations by SO ð3Þ vector indices running over the values 1, 2, 3; emission and absorption of some sort of gauge bosons, L subscripts a, b, etc. are SO ð3Þ vector indices. also running we would need to expand the gauge symmetry group. In R over the values 1, 2, 3.] Emission and absorption of the order for these masses to be much less than the zeroth order corresponding spinless particles also produces radiative masses of the third generation, we would need the gauge corrections to the quark and lepton masses. As we shall see in the next section, while keeping the mass of the *[email protected] Standard Model Higgs boson and the weak coupling con- stant at their known values, we can take all the other scalar Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. particles and the new vector bosons to be heavy enough to Further distribution of this work must maintain attribution to have escaped detection. But the calculation in Sec. IV shows the author(s) and the published article’s title, journal citation, that the radiative corrections to masses do not disappear and DOI. Funded by SCOAP3. when the new scalar and vector bosons become very heavy.
2470-0010=2020=101(3)=035020(12) 035020-1 Published by the American Physical Society STEVEN WEINBERG PHYS. REV. D 101, 035020 (2020)
The only possible renormalizable coupling of these scalars to leptons and quarks is then ν Φþ 2=3 Φþ i ia − qi ia −1=3 L &l Φ ¼ −GL · l − G · q q ; l− Φ0 Ra D −1=3 Φ0 Ra i L ia qi L ia 2=3 0 q Φ þ2 3 − G i · ia q = þ c:c:: ð2Þ U −1=3 Φ− Ra qi L ia
Here and below GL, GD and GU are constants, and again i and a run over the values 1, 2, 3, repeated indices are summed, and superscripts indicate charges.
III. STATIONARY POINTS: A FIRST LOOK
The most general renormalizable potential for the scalars Φ that is invariant under the new SOLð3Þ ⊗ SORð3Þ as well as the electroweak SUð2Þ ⊗ Uð1Þ takes the form þ þ þ Φþ þ þ Φ Φ Φ jb Φ Φ V ðΦÞ¼−μ2 ia · ia þ ia · kc · ld I Φ0 Φ0 Φ0 Φ0 Φ0 Φ0 ia ia ia jb kc ld
× ½b1δijδklδabδcd þ c1δijδklδacδbd þ c2δijδklδadδbc
þ c3δikδjlδabδcd þ b2δikδjlδacδbd þ c4δikδjlδadδbc
þ c5δilδjkδabδcd þ c6δilδjkδacδbd þ b3δilδjkδadδbc ; ð3Þ
2 T T 2 T T where the bn and cn are various real dimensionless VIðϕÞ¼−μ Trðϕ ϕÞþb½Trðϕ ϕÞ þcTrðϕ ϕϕ ϕÞ: ð5Þ constants. The Lagrangian terms (2) and (3) along with the rest of the Lagrangian happen to be invariant under a The dimensionless constants b and c are linear combi- reflection: nations of the coefficients of the quartic terms in the general potential (3): R∶ Φ → −Φ → − l → −l ð Þ qL qL L L; 4 b ¼ b1 þ b2 þ b3; with right-handed fermions and all gauge fields left c ¼ c1 þ c2 þ c3 þ c4 þ c5 þ c6: ð6Þ invariant. We are concerned here only with stationary points of [A trilinear SOLð3Þ ⊗ SORð3Þ-invariant term Detϕ can- the potential for which charge is conserved, so in seeking not arise from (3). This can also be seen as a conse- Φþ ¼ 0 such stationary points we set ia . Inspection of quence of invariance under the reflection (4).] Eq. (3) then shows that every term is symmetric between 0 0 If this were the end of the story, and there were no other Φ and its Hermitian conjugate. It follows that if VðΦ Þ scalar fields with which the fields Φ could interact, then 0 ia is stationary at a real value of Φ under variations that Eq. (5) would be the potential that governs the possible 0 keep Φ real, then at this point it is stationary under all expectation values of these scalars. We will have to variations of Φ0. [In general, if VðzÞ¼Vðz Þ then for λ introduce other scalar fields that do interact with the ðλþϵÞ¼ ðλþϵ Þ real V V can have no terms of first order Φia, but it will be instructive first to consider the impli- in Imϵ.] We can therefore seek stationary points of the cations of the potential (5), returning later to consider the potential (not necessarily all stationary points) by taking effect of interaction with other scalars. ϕ Φ0 ð3Þ ⊗ ð3Þ the possible expectation values ia of ia to be real. To ensure that the SOL SOR gauge symmetry is (Here and below, we use lower case letters to distinguish spontaneously broken at the stationary points of (5),we the possible spacetime-independent c-number expectation would need to take μ2 > 0. In order for this potential to go values of various scalar fields from the fields themselves.) to þ∞ rather than −∞ when ϕ goes to infinity in any Φþ ¼ 0 ϕ ≡ Φ0 For ia and ia ia real, the potential (3) must direction, the other constants in (5) would have to be in take the form of a general renormalizable potential that is either range b>0 and c>−b or c>0 and b>−c=3, invariant under SOLð3Þ ⊗ SORð3Þ and the reflection R, or both. Any ϕ can be diagonalized by an SOLð3Þ ⊗ and so SORð3Þ transformation, so we can characterize the various
035020-2 MODELS OF LEPTON AND QUARK MASSES PHYS. REV. D 101, 035020 (2020) stationary points of the potential according to their and first generations of quarks and leptons to acquire elements when diagonalized. If we assume that b>0 masses from one-loop and two-loop radiative corrections. and −b
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Z i X d4q Σ ðpÞ¼ ϵ 3ϵ 3 ij 4ð2πÞ4 ki lj ½ð − Þ2 þ 2 − ϵ ½ 2 þ μ2 − ϵ kln p q m3 i q n i ½ 2 ðnÞ ðnÞγμð1 þ γ Þ½− ð − Þνγ þ γ ð1 þ γ Þ × gLuLk uLl 5 i p q ν m3 μ 5 þ 2 ðnÞ ðnÞγμð1 − γ Þ½− ð − Þνγ þ γ ð1 − γ Þ gRuRk uRl 5 i p q ν m3 μ 5 þ ðnÞ ðnÞγμð1 þ γ Þ½− ð − Þνγ þ γ ð1 − γ Þ gLgRuLk uRl 5 i p q ν m3 μ 5 þ ðnÞ ðnÞγμð1 − γ Þ½− ð − Þνγ þ γ ð1 þ γ Þ ð Þ gLgRuRk uLl 5 i p q ν m3 μ 5 : 10
X 2 2 2 2 Here g and g are the gauge couplings of SO ð3Þ ⊗ ð Þ ð Þ μ μ − L R L ¼ gLgRm3 ϵ ϵ n n n ln n m3 ln m3 SO ð3Þ, and the sums over n run over the six vector boson mai 2 ab3 ij3uRb uLj 2 2 : R 4π μn − m3 mass eigenvalues defined by Eq. (8). The generators of bjn ð Þ SOLð3Þ and SORð3Þ are denoted tLi and tRa, and in the 14 ½ ¼ ϵ (3, 3) representation have the components tLk ij i kij ½ ¼ ϵ It makes no difference what units for mass we use in and tRc ab i cab. Hence, for instance, the generators tL1 and tR1 produce transitions between the second and third calculating the logarithms, since a change in units only generations. gives a term proportional to the sum (13). We can diago- γ We are interested in the case in which in the tree nalize the matrix mai (which also gets rid of the 5s) by approximation only m3 is nonzero, so to one-loop order multiplying the left- and right-handed quark or lepton fields we can go on the mass shell for the first and second of the first and second generation with independent 2 × 2 ν generations by simply setting p ¼ 0, in which case (after unitary matrices UL and UR; the physical masses of the first- discarding terms in the integrand odd in q) the two-point and second-generation quarks and leptons are then the † function (10) takes the form elements of the diagonal matrix URmUL. The couplings at zero momentum transfer of the field 0 ReΦ33 to the first and second generation of quarks and Σaið0Þ¼maið1 þ γ5Þ=2 þ miað1 − γ5Þ=2; ð11Þ leptons would be generated by the same one-loop diagram, and would be the same as in the Standard Model. At nonzero where momentum transfer the coupling is modified by a form X factor, but this form factor is nearly constant up to momen- 4igLgRm3 ðnÞ ðnÞ μ m ¼ ϵ 3ϵ 3u u tum transfers of order m3 or the smallest n, whichever is ai ð2πÞ4 ba ji Rb Lj kln greater. Z 4 This general class of models provides a plausible d q : ð Þ possible explanation of why the quarks and leptons of × ½ 2 þ 2 − ϵ ½ 2 þ μ2 − ϵ 12 q m3 i q n i the first and second generations should be much less massive than their third-generation counterparts, but so Each term in the sum over vector boson mass eigenval- far we have seen no reason why the first generation should ues is logarithmically divergent, but the sum is convergent, be so much lighter than the second. But we can now easily ð Þ because the completeness of the set of eigenvectors u n describe the sort of vector boson mass matrices that will togetherP with the orthonormality conditions (9) tell us that give masses for the second but not the first generation in ðnÞ ðnÞT n u u is the unit matrix, and in particular one-loop order. If X ðnÞ ðnÞ ¼ 0 ð Þ uLi uRa : 13 2 2 μ 2 ¼ 0&μ 2 ¼ 0 ð15Þ n L ;Ra Li;R for all a and i, and if The logarithmic divergences are independent of μn, and so their sum is proportional to (13), and hence they cancel. μ2 ¼ 0&μ2 ¼ 0 ð16Þ Indeed, as remarked by Barr and Zee [3], renormalizabilty L2;Li R2;Ra makes this sort of cancellation inevitable, as there is no for all i ≠ 2 and all a ≠ 2, then obviously the only counterterm that could cancel an infinity. 2 eigenvectors of μ with L2 or R2 components respectively After combining denominators, Wick rotating, integrat- ð Þ μ 2 2 n ¼ 0 ing over q , and integrating over Feynman parameters, the have only L or R components, so uR2 for all ðnÞ ≠ 0 ðnÞ ¼ 0 mass matrix of the second and first generations is eigenvectors n for which uLi , and uL2 for all
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ðnÞ ≠ 0 negative power of some large mass, just as lepton con- eigenvectors n for which uRa . Inspection of Eq. (14) servation is not respected by dimension five operators shows then that to one-loop order, m1i ¼ 0 and ma1 ¼ 0 for all i and a. The 2 × 2 mass matrix m of the first and added to the renormalizable Lagrangian of the Standard ai R second generations in this order would then be already Model. If symmetry is violated in this way, the first diagonal, with only one nonzero element, the second generation of quarks and leptons could get masses in the generation mass m2 ¼ m22: tree approximation from interactions of the fermion fields with two or more powers of scalar fields, masses that are X μ2 μ2 − 2 2 small only because of the suppression of these nonrenor- ¼ gLgRm3 ðnÞ ðnÞ n ln n m3 ln m3 ð Þ m2 2 u 1 u 1 2 2 : 17 malizable interactions. But because the first generation 4π R L μ − m n n 3 quarks and leptons are much heavier than neutrinos while much lighter than the third generation quarks and leptons, Repeating the same steps that led to Eq. (17), we see that the mass scale whose reciprocal appears in these higher the first-generation quarks and leptons get a two-loop mass dimensional operators would have to be much lighter than X 2 2 2 2 the mass scale in the interactions that give neutrinos their gLgRm2 ðnÞ ðnÞ μn ln μn − m3 ln m3 m1 ¼ u u : ð Þ mass and much heavier than the third-generation fermions. 4π2 R3 L3 μ2 − 2 18 n n m3 In the next section we take up a possibility that is more in the spirit of this paper, that for a suitable choice of a hidden ð3Þ ⊗ It is easy to think of a finite subgroup of SOL sector of scalar fields, the potential accidently has a ð3Þ ⊗ R SOR that if unbroken would ensure the validity of symmetry that unlike (19) is not a subgroup of the gauge conditions (15) and (16) and thereby give vanishing first- group and R, and which has a subgroup that when generation masses in one-loop order. (This unbroken unbroken naturally gives the vector boson mass matrix subgroup must be finite to avoid the appearance of new the form required for radiative corrections to give masses to massless gauge bosons.) We could take this unbroken the second generation of quarks and leptons in one-loop symmetry as invariance under the operators order but to the first generation only in two-loop order. Masses are also generated by radiative corrections due to R expðiπtL1Þ & R expðiπtR1Þð19Þ emission and absorption of scalar bosons. Here again we will keep to the general case in this section, leaving it for where R is the reflection (4). [This reflection has no effect the next sections to consider specific forms for the mass on vector boson masses, but must be included in order for matrix. Φ0 φþ the appearance of the vacuum expectation value of Re 33 The charged scalar fields N that correspond to charged not to break invariance under the transformations (19).] spinless particles of definite mass MNþ are in general linear Φþ Unfortunately, invariance under (19) would imply not only combinations of the previously introduced fields ia: that conditions (15) and (16) are satisfied, so that the first X ð þÞ generation quarks and leptons get no mass in one-loop φþ ¼ u N Φþ ; ð20Þ μ2 ¼ 0 μ2 ¼ 0 N ia ia order, but would also imply that L3;Ra and Li;R3 ia for all i and a, which according to Eq. (18) would imply ðNþÞ also that the first generation quarks and leptons also get no with uia some constant coefficients found by diagonal- mass in two-loop order. Indeed, we could have seen this izing the charged scalar mass matrix. Assuming again that without looking into the details of radiative corrections. only the third generation quarks get masses in the tree Because R is defined to change the sign of all left-handed approximation, the one-loop two-point function for quarks quark or lepton fields, the first symmetry transformation of the first and second generation is here of the same form (19) changes the sign of the left-handed first generation as (11), except that here mai is complex: quark or lepton fields, so if this is an unbroken symmetry Σ ð0Þ¼ ð1 þ γ Þ 2 þ ð1 − γ Þ 2 ð Þ then the first generation does not get a mass from any ai mai 5 = mia 5 = : 21 source, including scalar boson as well as vector boson interactions. Again, we can diagonalize the matrix mai by multiplying So where does the first generation get its masses? It is the left- and right-handed quark or lepton fields of the first possible that the first generation masses have nothing to do and second generation with independent 2 × 2 unitary with vector boson emission and absorption. It should be matrices UL and UR, and the physical masses of the first- noted that the symmetry of the Lagrangian under the and second-generation quarks and leptons are then the R † reflection is an accidental symmetry, in the sense that elements of the diagonal matrix URmUL. it is a consequence of the gauge symmetries of the theory Following the same methods that led to Eq. (12),we and the condition of renormalizability. It therefore need not find the one-loop contribution of charged scalar bosons be respected by operators in the Lagrangian of higher to the masses of first- and second-generation quarks of dimensionality, whose coefficients are suppressed by a charge þ2=3
035020-5 STEVEN WEINBERG PHYS. REV. D 101, 035020 (2020) X þ2 3 iG G m ðNþÞ ðNþÞ The case of neutral scalars is more complicated, because m = ¼ U D b u u ai ð2πÞ4 3a i3 the neutral fields of definite mass are in general linear N Z combinations of the neutral scalar fields of the hidden d4q × ; ð22Þ sector to be introduced in the next section, as well as of the ½q2 þ m2 − iϵ ½q2 þ M2 − iϵ Φ0 b Nþ fields ia introduced in Sec. II and their adjoints. Separating the real and imaginary parts of any complex and for first- and second-generation quarks of charge −1=3: fields of definite mass, we can take all the neutral scalars of X definite mass to be real, and write them as −1=3 iGUGDmt ðNþÞ ðNþÞ m ¼ u3 u 3 X ai ð2πÞ4 a i 0 ðN0Þ 0 ðN0Þ 0 N φ ¼ ½u Φ þ u Φ þ… ð27Þ Z N ia ia ia ia d4q ia : ð Þ × ½ 2 þ 2 − ϵ ½ 2 þ 2 − ϵ 23 q mt i q MNþ i ðN0Þ where the coefficients uia are various complex constants, and the dots indicate linear combinations of scalar fields of Each term in these sums is logarithmically divergent, but the hidden sector. The mass matrix m appearing in the the ultraviolet divergences again cancel in the sum. To see ia two-point function (21) for the first and second generation this, it is easiest to derive the necessary completeness of leptons and quarks of each charge are then relation by requiring that the fields (20) of definite mass have a kinematic Lagrangian term: 2 X L ¼ iGLmτ ðN0Þ ðN0Þ X mai 4 u3a ui3 þ μ þ ð2πÞ − ∂ φ ∂ φ N μ N N; Z N d4q ; ð Þ × ½ 2 þ 2 − ϵ ½ 2 þ 2 − ϵ 28 so that the propagators of these fields have the conventional q mτ i q MN0 i normalization that we assumed in deriving Eqs. (22) and 2 X −1 3 iG m ð 0Þ ð 0Þ (23). In order that this kinematic Lagrangian should agree m = ¼ D b u N u N þ μ þ ai ð2πÞ4 3a i3 with the correct kinematic Lagrangian −∂μΦ ∂ Φ ,itis ia ia Z N necessary that d4q × ; ð29Þ X ½q2 þ m2 − iϵ ½q2 þ M2 − iϵ ðNþÞ ðNþÞ ¼ δ δ b N0 uia ujb ij ab N 2 X þ2 3 iG m ð 0Þ ð 0Þ m = ¼ U t u N u N ai ð2πÞ4 3a i3 Equations (22) and (23) were derived only for the first- and N Z second-generation one-loop masses, where both i and a d4q × ; ð30Þ equal 1 and/or 2, in which case this relation gives ½q2 þ m2 − iϵ ½q2 þ M2 − iϵ X t N0 ðNþÞ ðNþÞ ¼ 0 ð Þ u3a ui3 : 24 Again, to deal with logarithmic divergences, we need a N completeness relation. We define these real neutral scalars The divergences in each term of Eqs. (22) and (23) are so that the kinematic term in the Lagrangian is independent of N, so the total divergence in the sums is 1 X proportional to (24), and hence vanishes. Equations (22) − ∂ φ0 ∂μφ0 2 μ N N: and (23) then give for the mass matrices of first- and N second-generation quarks of charge þ2=3 and −1=3: In order that this should contain the correct kinematic term X 0 μ 0 þ2 3 G G m ð þÞ ð þÞ −∂μΦ ∂ Φ for the neutral scalars introduced earlier, it is m = ¼ U D b u N u N ia ia ai 16π2 3a i3 necessary that N 2 2 2 2 X X M ln M − m ln m ð þÞ ð þÞ ð 0Þ ð 0Þ Nþ Nþ b b ð Þ u N u N ¼ δ δ ; u N u N ¼ 0: ð31Þ × 2 − 2 ; 25 ia jb ij ab ia jb MNþ mb N N X −1=3 GUGDmt ðNþÞ ðNþÞ In the case that concerns us here, in which both i and a are m ¼ u3 u 3 ai 16π2 a i unequal to 3, the second relation tells us that N 2 2 2 2 X M þ ln M þ − mt ln mt ð 0Þ ð 0Þ × N N : ð26Þ u N u N ¼ 0 ð32Þ M2 − m2 3a i3 Nþ t N
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– Φ0 so the logarithmic divergences cancel in Eqs. (28) (30), scalar fields Re ia have nonvanishing expectation values which give only for i ¼ a ¼ 3. These break SOLð3Þ ⊗ SORð3Þ to the ð2Þ ⊗ ð2Þ SOL SOR subgroup with generators tL3 and tR3. 2 X 2 2 2 2 GLmτ ð 0Þ ð 0Þ M 0 ln M 0 − mτ ln mτ This symmetry breaking by itself gives nonvanishing mL ¼ u N u N N N ; ai 16π2 3a i3 2 − 2 values only for the following components of the SO ð3Þ ⊗ N MN0 mτ L SORð3Þ vector boson mass-squared matrix: ð33Þ 2 X 2 2 2 2 2 2 −1 3 G m ð 0Þ ð 0Þ M ln M − m ln m μ 1 1 ¼ μ 2 2 ¼ g λ ; m = ¼ D b u N u N N0 N0 b b ; L ;L L ;L L ai 2 3a i3 2 2 2 2 16π M 0 − m μ ¼ μ ¼ λ N N b R1;R1 R2;R2 gR : ð34Þ λ ¼h Φ0 i 2 X 2 2 2 2 where Re 33 . þ2 3 G m ð 0Þ ð 0Þ M ln M − m ln m m = ¼ U t u N u N N0 N0 t t : To produce additional components of the vector ai 16π2 3a i3 2 − 2 N MN0 mt boson mass matrix, we introduce a number of additional ð35Þ scalar field multiplets that, like the electroweak doublet Φia, transform according to the (3, 3) representation of ð3Þ ⊗ ð3Þ Φ Inspection of Eqs. (25), (26), and (33)–(35) shows that in SOL SOR , but unlike ia are neutral under the ð2Þ ⊗ ð1Þ order for one-loop scalar boson emission and absorption to electroweak SU U , and therefore cannot have give masses to the second generation of quarks and leptons renormalizable interactions with the quarks and leptons. but not the first generation, there would have to be a scalar We will denote these new electroweak-neutral multiplets Φ Φ ΨðNÞ ≥ 1 field of definite mass that includes both 32 and 23 terms as ia with N . For simplicity, we assume that the but none that contain both Φ31 and Φ13 terms. In this case Lagrangian is invariant under independent sign changes for these radiative corrections to give masses to the first ΨðNÞ → −ΨðNÞ for each of the new scalar multiplets, as generation in two-loop order there would have to be scalar well as the reflection R and SOLð3Þ ⊗ SORð3Þ.Then fields of definite mass that contain both Φ12 and Φ21 terms. the most general renormalizable potential for all the We will have to wait until we return to the primary sector scalars is scalar fields in Sec. VI to see whether these conditions are satisfied. 0 V ¼ VI þ VII þ V ; ð36Þ V. HIDDEN SECTOR SCALARS
To give masses only to the third generation of quarks and where VI is given by Eq. (3), VII is the most general leptons in the tree approximation, we have assumed that the renormalizable potential for the hidden sector scalars
X ¼ ½−μ2 ðΨðNÞTΨðNÞÞþ ½ ðΨðNÞTΨðNÞÞ 2 þ ðΨðNÞTΨðNÞΨðNÞTΨðNÞÞÞ VII NTr bN Tr cNTr N X 1 ð Þ ð Þ ð 0Þ ð 0Þ ð Þ ð Þ ð 0Þ ð 0Þ þ ½ξ 0 ðΨ N TΨ N Ψ N TΨ N Þþκ 0 ðΨ N Ψ N TΨ N Ψ N TÞ 2 NN Tr NN Tr N≠N0 ðNÞT ðN0Þ ðNÞT ðN0Þ ðNÞT ðNÞ ðN0ÞT ðN0Þ þ ζNN0 TrðΨ Ψ Ψ Ψ ÞþρNN0 TrðΨ Ψ ÞTrðΨ Ψ Þ ðNÞT ðN0Þ 2 þ σNN0 ½TrðΨ Ψ Þ ; ð37Þ and V0 is the general interaction between the primary and hidden sectors: X þ þ X þ Φþ X þ Φþ Φ Φ ð Þ ð Þ Φ ja ð Þ ð Þ Φ jb ð Þ ð Þ V0 ¼ ξ ia · ib Ψ N Ψ N þ κ ia · Ψ N Ψ N þ ζ ia · Ψ N Ψ N N Φ0 Φ0 jb ja N Φ0 Φ0 ib jb N Φ0 Φ0 ib ja N ia ib N ia ja N ia jb X þ þ X þ Φþ Φ Φ ð Þ ð Þ Φ jb ð Þ ð Þ þ ρ ia · ia Ψ N Ψ N þ σ ia · Ψ N Ψ N : ð38Þ N Φ0 Φ0 jb jb N Φ0 Φ0 ia jb N ia ia N ia jb μ2 Here the coefficients N, bN, etc. are real but otherwise arbitrary, and summation of repeated subscripts is understood.
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Φþ ¼ 0 Φ0 ¼ 0 We again assume that the stationary point of the potential has ia and Im ia , so to find this stationary point we Φ0 ΨðNÞ Φ0 Ψð0Þ can lump Re ia together with the ia . We write Re ia as ia , and express the total potential as a function of the possible c-number expectation values ψ ðNÞ with N ≥ 0 of all these scalar fields (again denoted with lower case letters): X ¼ ½−μ2 ðψ ðNÞTψ ðNÞÞþ ½ ðψ ðNÞTψ ðNÞÞ 2 þ ðψ ðNÞTψ ðNÞψ ðNÞTψ ðNÞÞÞ V NTr bN Tr cNTr NX ðNÞT ðNÞ ðN0ÞT ðN0Þ ðNÞ ðNÞT ðN0Þ ðN0ÞT þ ½ξNN0 Trðψ ψ ψ ψ ÞþκNN0 Trðψ ψ ψ ψ Þ N≠N0 ðNÞT ðN0Þ ðNÞT ðN0Þ ðNÞT ðNÞ ðN0ÞT ðN0Þ þ ζNN0 Trðψ ψ ψ ψ ÞþρNN0 Trðψ ψ ÞTrðψ ψ Þ ðNÞT ðN0Þ 2 þ σNN0 ½Trðψ ψ Þ : ð39Þ
The sums now extend to N ¼ 0 and/or N0 ¼ 0, with where the L are constants, depending on the coefficients 2 2 ξ 0 μ0 ¼ μ , b0 ¼ b, c0 ¼ c, ξ0N ¼ ξN0 ¼ ξN, etc. bN, cN, NN , etc. in the potential (39), but independent of ψ We note that each of the summed SOLð3Þ and SORð3Þ the components of the s. (The summation convention is vector indices occurs just twice in each term, so this potential suspended here.) The condition that V be stationary with has an accidental symmetry under the sign changes ψ ðNÞ ≠ respect to variations in ia is trivially satisfied if i a or if ðNÞ ðNÞ ðNÞ ðNÞ i ¼ a and ψ ¼ 0, while if i ¼ a and ψ ≠ 0 then this ψ → η η ψ ; ð40Þ ii ii ia Li Ra ia condition takes the form η η where Li and Ra are any N-independent sign factors. It is X 2 ðN0Þ 2 natural for the expectation values of the scalar fields to be 2μ ¼ 0 ½ψ N LNi;N j jj : invariant under any subgroup of this group of sign changes, jN0 in the technical sense that the restriction of the ψs to such invariant values lowers the number of equations that need to With a total of D nonvanishing diagonal components, these be satisfied at a stationary point of the potential by the are D linear inhomogeneous equations for the squares of same amount as it lowers the number of free components of the D nonvanishing components. The determinant of L ψ the s. In particular, it is natural to find stationary points does not vanish for generic values of the coefficients b , that are invariant under the subgroup of the group of N cN, ξNN0 , etc., so these equations have a solution, one that is reflections (40) that consists of all the reflections with unique. (We saw a simple example of this in Sec. II.) It is a η ¼−η ¼−η ¼η ¼−η ¼−η η ¼ −η ¼ L1 L2 L3 R1 R2 R3,or L2 L1 more complicated business to find if this solution has −η ¼ η ¼ −η ¼ −η η ¼ −η ¼ −η ¼ 0 L3 R2 R1 R3,or L3 L1 L2 ½ψ ðN Þ 2 positive values for all jj , and if this stationary point is ηR3 ¼ −ηR1 ¼ −ηR2. Invariance under this subgroup just ðNÞ ð0Þ an absolute minimum or even a local minimum of the implies that all ψ (including ϕ ≡ ψ ) are diagonal. ia ia ia potential. In the absence of a specific candidate for a At this point we will greatly simplify our discussion by realistic theory, it does not seem worthwhile to go into this. taking the coefficient σ 0 of the final term in Eq. (37) and NN With diagonal scalar expectation values, the 6 × 6 vector the corresponding coefficient σ in Eq. (38) to vanish. I N boson mass-square matrix takes the block-diagonal form have not been able to think of any symmetry assumption that would have this as a consequence, but setting all σ 0 0 1 NN μ2 00 and σ equal to zero has the very convenient implication 1 N B C ψ ðNÞ μ2 ¼ @ 0 μ2 0 A ð41Þ that with all ia diagonal, the potential (39) is a function 2 squares 2 only of the of the diagonal components. It is then 00μ3 natural to find a stationary point for which in the basis in ð Þ ψ N 2 which all ia are diagonal, there is any desired assortment where the 2 × 2 submatrices μ are ð Þ i of zeroes on the diagonal of any ψ N . ia Not only are such stationary points natural—they all μ2 μ2 Li;Li Li;Ri actually occur. When the ψ ðNÞ are all diagonal, the μ2 ≡ : ð42Þ i μ2 μ2 derivative of the potential (39) with respect to any one Ri;Li Ri;Ri ψ ðNÞ component ia takes the form To find a vector boson mass-squared matrix of the sort X 0 that gives a hierarchy of quark and lepton masses, we can ∂V 2 ðNÞ ðNÞ ðN Þ 2 ¼ δ −2μ ψ þ ψ L 0 ½ψ ; ðNÞ ia N ii ii Ni;N j jj include just three (3, 3) real neutral scalar multiplets, with ∂ψ 0 ia jN nonzero expectation values:
035020-8 MODELS OF LEPTON AND QUARK MASSES PHYS. REV. D 101, 035020 (2020) 0 1 0 1 000 000 eliminated by the Higgs mechanism, but since there is B C B C ð3Þ ⊗ ð3Þ ψ ð0Þ ¼ @ 000A; ψ ð1Þ ¼ @ 0 α 0 A; only one SOL SOR gauge group, another linear combination would be left as real massless spinless 00λ 00β 0 1 particles. If we allow an interaction between the scalar γ 00 fields of the primary and hidden sector, but assume that it is B C ψ ð2Þ ¼ @ 0 δ 0 A: ð43Þ very weak, then the broken symmetry would entail a very light pseudo-Goldstone boson, which is almost as bad. To 000 avoid this, we must not only include the interactions in Eq (36) between the scalar fields of the primary and hidden The vector boson mass-squared matrix here takes the sector, but also take these interactions strong enough to block-diagonal form (41), (42), and now the nonvanishing keep the pseudo-Goldstone boson too heavy to have been elements of the submatrices are observed. With this interaction present, it is necessary to 2 2 2 2 2 2 2 2 reconsider the results in Sec. III for the stationary points μ ¼ g ðλ þ α þ β þ γ þ δ Þ; μ ¼ −2g g αβ; L1;L1 L L1;R1 L R and masses of the scalar fields of the primary sector. μ2 ¼ 2 ðλ2 þ α2 þ β2 þ γ2 þ δ2Þ R1;R1 gR ; In carrying out this analysis, it is both necessary and μ2 ¼ g2 ðλ2 þ β2 þ γ2Þ; μ2 ¼ g2 ðλ2 þ β2 þ γ2Þ; convenient to assume that the expectation values of the L2;L2 L R2;R2 R scalar fields of the hidden sector are much larger than those μ2 ¼ 2 ðα2 þ γ2 þ δ2Þ μ2 ¼ 2 ðα2 þ γ2 þ δ2Þ L3;L3 gL ; R3;R3 gR ; of the primary sector. This will ensure that the masses of the ð3Þ ⊗ ð3Þ μ2 ¼ −2g g γδ: ð44Þ SOL SOR gauge bosons are much larger than the L3;R3 L R W and Z masses. The Goldstone boson that is eliminated by 2 2 the Higgs mechanism is then close to the Goldstone boson But note that μ 2 ¼ μ 2 ¼ 0 for all i and a. All the L ;Ra Li;R associated with the symmetry breaking in the hidden sector, SOLð3Þ ⊗ SORð3Þ gauge bosons are massive, and can be made heavier than the W and Z by arranging that some leaving a pseudo-Goldstone boson in the primary sector. combinations of α, β, γ and δ (as for example just γ) are This assumption also gives the scalar fields of the hidden ψ ðNÞ sufficiently larger than λ. sector large masses, locking in their expectation values ia In one-loop order the emission of L1 and/or R1 gauge with N ≥ 1, independent of the fields of the primary sector. bosons in 2 ↔ 3 transitions produces second-generation With this assumption, the potential for the scalar fields of masses given by Eq. (33), but there still is no mixing of the the primary sector is effectively L2 and R2 gauge boson masses, so one-loop radiative corrections still do not give any mass to the first generation X Φþ Φþ of quarks and leptons. But because there now is a non- V ¼ V − ia · ia μ2 I;eff I Φ0 Φ0 ia vanishing mixing of the L3 and R3 gauge boson masses, ia ia ia the emission of the L3 component of a gauge boson in a X Φþ Φþ 1 → 2 transition followed by the absorption of the R3 − ia · ai μ02 ; ð45Þ Φ0 Φ0 ia component of the same gauge boson gives a mass for the ia ia ai first generation of quarks and leptons, which as shown in Eq. (18) is proportional to the corresponding second- where generation mass, and hence is of two-loop order. X X X μ2 ¼ − ξ ½ψ ðNÞ 2 − κ ½ψ ðNÞ 2 − ρ ½ψ ðNÞ 2 VI. THE PRIMARY SECTOR REVISITED ia N ja N ib N jb j;N≥1 b;N≥1 b;j;N≥1 It is easy to preserve the results of Sec. II when we add ð46Þ the hidden sector of scalar fields, by setting equal to zero all interactions of the scalar fields Φ of the primary sector X ia ð Þ ð Þ ðNÞ 02 02 N N Ψ ≥ 1 — μ ¼ μ ¼ − ζNψ ψ aa ; ð47Þ with the fields ia with N of the hidden sector that ia ai ii N≥1 is, by setting the coefficients ξN, κN, ζN, and ρN in Eq. (38) equal to zero (as well as taking all σN to vanish). But not only would this be an unnatural act, it would also have an and VI is given by Eq. (3). unacceptable consequence. With no interaction between the To find the stationary point of VI;eff, we again take the Φþ Φ0 scalar fields of the primary and hidden sectors, the potential expectation values of ia and Im ia equal to zero, and let ϕ Φ0 would be invariant under separate SOLð3Þ ⊗ SORð3Þ the expectation value ia of Re ia have only the single transformations of the scalar fields of each sector. When nonzero element ϕ33 ¼ λ. (The existence of such a sta- these two symmetries are spontaneously broken, there tionary point was shown in Sec. V.) Setting the term in two Φ0 − hΦ0 i would be sets of massless Goldstone bosons. One Eq. (45) of first order in ia ia equal to zero linear combination of these massless fields would be then gives
035020-9 STEVEN WEINBERG PHYS. REV. D 101, 035020 (2020)
μ2 þ μ2 þ μ02 λ2 ¼ 33 33 ; ð48Þ 2ðb þ cÞ where b and c are given by Eq. (6). Φ0 ≡ Φ0 − hΦ0 i The scalar mass matrix can be read off by considering the terms in Eq. (45) that are quadratic in ia ia ia and Φþ ia and their adjoints: X X ¼ ½2λ2 − μ2 − μ2 jΦþ j2 þ ½2λ2ð þ þ Þ − μ2 − μ2 jΦþj2 VI;quad b1 ia ia b1 c1 c2 i3 i3 ≠3 ≠3 ≠3 i X;a i X þ ½2λ2ð þ þ Þ − μ2 − μ2 jΦþ j2 − μ02 Φþ†Φþ b1 c3 c5 3a 3a ia ia ai ≠3 ≠3 ≠3 aX i ;a X − 2 μ02 ðΦþ†ΦþÞþ½2λ2ð þ Þ − μ2 − μ2 − μ02 jΦþ j2 þ ½2λ2 − μ2 − μ2 jΦ0 j2 i3Re i3 3i b c 33 33 33 b1 ia ia ≠3 ≠3 ≠3 Xi i ;a þ ½2λ2ð þ þ þ þ þ Þ − μ2 − μ2 jΦ0 j2 b1 c1 c2 c5 c6 b3 i3 i3 ≠3 Xi þ ½2λ2ð þ þ þ þ þ Þ − μ2 − μ2 jΦ0 j2 b1 c3 c5 c2 c4 b3 3a 3a ≠3 a X þ½4λ2ð þ Þ − μ2 − μ2 − μ02 jΦ00 j2 þ 2λ2 ðΦ0 Þ2 b c 33 33 33 b3Re ia ≠3 ≠3 X X i ;a þ 2λ2ð þ þ Þ ðΦ0 Þ2 þ 2λ2ð þ þ Þ ðΦ0 Þ2 þ 2λ2ð þ Þ ðΦ00 Þ2 b3 c5 c6 Re i3 b3 c2 c4 Re 3a b c Re 33 ≠3 ≠3 iX X a − μ02 Φ0†Φ0 − 2 μ02 ðΦ0†Φ0 Þ ð Þ ia ia ai i3Re i3 3i : 49 i≠3;a≠3 i≠3
First, using Eq. (48), we see that there are two fields here the scalar fields of the hidden sector. The terms in Eq. (49) þ 0 of zero mass: Φ33 and ImΦ33. These are the Goldstone involving these fields are bosons of broken SUð2Þ ⊗ Uð1Þ and appear physically as X the helicity zero states of the W and Z bosons, just as in the ½ð Φ0 Þ2ð2λ2ð þ Þ − μ2 − μ2 Þ Re i3 b c i3 Standard Model. i≠3 00 Next, note that another field of definite mass is ReΦ , 0 2 2 2 2 0 0 02 33 þðReΦ Þ ð2λ ðb þ cÞ − μ − μ Þ − 2Φ Φ μ ; which plays the same role here as the Higgs boson of the 3i 3i 3i i3 i3 Standard Model. Its squared mass is the coefficient of 00 2 or, using Eq. (48) again, ðReΦ33Þ =2 in Eq. (49). Using Eq. (48), this is X ½ðμ2 − μ2 ÞðReΦ0 Þ2 þðμ2 − μ2 ÞðReΦ0 Þ2 2 ¼ 2½6λ2ð þ Þ − μ2 − μ2 − μ02 33 i3 i3 33 3i 3i mH b c 33 33 i≠3 ¼ 4ðμ2 þ μ2 þ μ02 Þ¼8λ2ð þ Þ ð Þ − 2μ02 Φ0 Φ0 ð Þ 33 33 b c : 50 i3Re 3iRe i3 : 51
The final result is the same as in Sec. II, the only As anticipated, these fields would evidently be massless in difference being that in the derivation μ2 is replaced with the absence of the interaction between primary and hidden μ2 þ μ2 þ μ02 . As noted in Sec. II, our knowledge of the sectors. In the approximation assumed above in this 33 33 ð3Þ ⊗ ð3Þ Higgs boson mass and the weak interaction strength lets us section, that the breaking of SOL SOR is mostly conclude from this result that b þ c has the value 0.032. No due to the expectation values of scalar fields of the hidden other scalar boson mass is given by the same combination of sector, the massless Goldstone boson fields associated with ΨðNÞ ΨðNÞ parameters, so it is plausible that if the constants bn and cn in this breaking are dominated by terms i3 and 3i with Eq. (3) are of order unity then all other scalar bosons are i ≠ 3 and N ≥ 1, and it is these fields and not the primary Φ0 Φ0 much heavier than the Higgs boson. sector fields Re i3 and Re 3i that provide the helicity zero Φ0 Φ0 ð3Þ ⊗ ð3Þ Of particular interest are the fields Re i3 and Re 3i with part of the massive SOL SOR gauge bosons. ≠ 3 Φ0 i , which would be the Goldstone boson fields of Continuing, the terms in Eq. (49) involving Im i3 and ð3Þ ⊗ ð3Þ Φ0 Φ0 ≠ 3 broken SOL SOR if the ia did not interact with Im 3i with i are
035020-10 MODELS OF LEPTON AND QUARK MASSES PHYS. REV. D 101, 035020 (2020) X 2 2 2 0 2 2 X 2 2 2 2 ½½2λ ð þ þ Þ − μ − μ ð Φ Þ τ ð 0Þ ð 0Þ − b1 c1 c2 i3 Im i3 L GLm N N MN0 lnMN0 mτ lnmτ mμ ¼ m22 ¼ u32 u23 ; i≠3 16π2 2 − 2 N MN0 mτ þ½2λ2ð þ þ Þ − μ2 − μ2 ð Φ0 Þ2 b1 c3 c5 3i Im 3i ð57Þ − 2μ02 Φ0 Φ0 ð Þ i3Im i3Im 3i ; 52 −1=3 ms ¼m22 0 0 and the terms in Eq. (49) involving ReΦ or ImΦ with 2 X 2 2 2 2 ia ia G m ð 0Þ ð 0Þ M lnM −m lnm i ≠ 3 and a ≠ 3 are ¼ D b u N u N N0 N0 b b ; ð Þ 16π2 32 23 2 − 2 58 X N MN0 mb ½½2λ2ð þ Þ − μ2 − μ2 ð Φ0 Þ2 − μ02 Φ0 Φ0 b1 b3 ia Re ia iaRe iaRe ai i≠3;a≠3 and
ð53Þ þ2 3 m ¼ m = c 22 2 X 2 2 2 2 and G m ð 0Þ ð 0Þ M lnM − m lnm ¼ U t u N u N N0 N0 t t : ð Þ X 16π2 32 23 2 − 2 59 ½½2λ2ð − Þ − μ2 − μ2 ð Φ0 Þ2 −μ02 Φ0 Φ0 N MN0 mt b1 b3 ia Im ia iaIm iaIm ai : i≠3;a≠3 The first generation of quarks and leptons get masses from ð Þ 54 emission and absorption of scalar bosons in two-loop order. It is striking that the same choice of scalar fields in the Corresponding results for the charged scalars can be found hidden sector leads to both the radiative corrections from the first three lines of Eq. (49). involving vector bosons and those involving scalar bosons Inspection of these results shows that the charged and generating quark and lepton masses for the second and first neutral scalar fields of definite mass contain both the terms Φ Φ ≠ μ02 generation to one-loop and two-loop order, respectively. ia and ai with i a if and only if ia is nonzero. Equation (47) shows that for generic coefficients ζN this ð Þ VII. PROBLEMS will be the case if there are one or more scalar fields Ψ N of the hidden sector whose expectation values ψ ðNÞ have both The results obtained here for radiatively generated ii and aa components nonzero. Equation (43) shows that masses involve many unknown parameters. But we have for the choice we have made of the scalar fields of the noted that the large number of new scalar and vector hidden sector and for their expectation values, the coef- particles in these models can (and must) be supposed to be 02 02 02 ficients μ23 and μ12 are nonzero, but μ13 ¼ 0. It follows that heavy enough to have escaped observation, and where they there are charged and neutral scalar fields of definite mass are sufficiently heavy we can easily find the ratios of many that contain both the terms Φ23 and Φ32, but none that quark and lepton masses. Unfortunately, these predicted contain both the terms Φ13 and Φ31. As we have seen these ratios turn out to be wrong. are just the conditions under which the quarks and leptons First, if the masses of the second-generation quarks and of the second generation but not the first generation get leptons arose from radiative corrections involving the masses from one-loop emission and absorption of scalar SOLð3Þ ⊗ SORð3Þ gauge bosons, and if we did somehow bosons. The one-loop quark masses produced by charged arrange that these gauge bosons were all much heavier scalars is given by Eq. (25) and (26): than the quarks and leptons of the third generations, then according to Eq. (17) the ratios of the masses of the second þ2=3 m ¼ m22 and third generations quarks and leptons would all be c X 2 2 2 2 independent of the various masses of the third generation: ð þÞ ð þÞ M M −m m ¼ GUGDmb N N Nþ ln Nþ b ln b 2 u32 u23 2 2 ; X 16π M þ − m gLgR ðnÞ ðnÞ 2 N N b m2=m3 ≃ u u μ ; ð Þ 4π2 R1 L1 ln n 60 ð55Þ n
−1=3 and so would be the same for leptons and quarks of each m ¼ m22 s charge, X 2 2 2 2 G G m ð þÞ ð þÞ M þ ln M þ − m ln m ¼ U D t u N u N N N t t ; ¼ ¼ ð Þ 16π2 32 23 2 − 2 mc=mt ms=mb mμ=mτ 61 N MNþ mt ð56Þ which is not even approximately true of observed masses. The strong interactions can account for some differences in while the one-loop quark and lepton masses produced by these mass ratios, but these interactions are not very strong neutral scalars are given by Eqs. (33)–(35) as at energies of the order of mc and mt, and of course are
035020-11 STEVEN WEINBERG PHYS. REV. D 101, 035020 (2020) entirely absent for the leptons, while mc=mt is an order of up the somewhat unnatural assumption that the coefficients magnitude smaller than mμ=mτ. σN and σNN0 in Eqs. (38) and (39) all vanish. We might Similarly, if the radiatively generated masses of second- instead assume that for some reason these coefficients generation quarks were dominated by the emission and are relatively small, expecting that this will yield rather absorption of charged scalar bosons, then according to small mixing angles. But in this case it is not clear that it Eqs. (55) and (56) we would have would be possible in a natural way to maintain the starting assumption that only the third generation of quarks and ¼ ð Þ mc=mb ms=mt 62 leptons get masses in the tree approximation. Also, in this case we would need to worry about the possibility of flavor- which is even further from the truth. Finally, if the changing neutral currents. radiatively generated masses of second generation quarks The best that can be hoped for the models discussed in and leptons were dominated by the emission and absorption this paper is that they may perhaps provoke new ideas for a of neutral scalar bosons, then according to Eqs. (57)–(59) realistic theory in which radiative corrections account for and Eq. (7) we would have the masses of the first and second generations of quarks and 3 ¼ 3 ¼ 3 ð Þ leptons, together with guidance in dealing with the prob- mc=mt ms=mb mμ=mτ ; 63 lems that will arise in such a theory. which is worse yet. The above wrong predictions of mass ratios involving ACKNOWLEDGMENTS quarks of charge þ2=3 would be invalidated if the mass of the relevant SOLð3Þ ⊗ SORð3Þ gauge bosons or scalar I am grateful for conversations with Can Kilic and bosons were of the same order of magnitude of the other members of the Theory Group at Austin, and for a top quark mass. In that case, these new bosons might be correspondence with Anthony Zee. This article is based on accessible to observation. work supported by the National Science Foundation under Another unrealistic feature of these results is that they Grants No. PHY-1620610 and No. PHY-1914679, and do not exhibit any Cabibbo-Kobayashi-Maskawa mixing with support from the Robert A. Welch Foundation, Grant angles. Quark mass mixing could be included by giving No. F-0014.
[1] S. M. Barr and A. Zee, Phys. Rev. D 17, 1854 (1978);F. [2] For an interesting exception, see B. A. Dobrescu and P. J. Wilczek and A. Zee, Phys. Rev. Lett. 42, 421 (1979);T. Fox, J. High Energy Phys. 08 (2008) 100. Yanagida, Phys. Rev. D 20, 2986 (1979) and other references [3] S. M. Barr and A. Zee, Ref. [1]. In this connection, also see S. cited therein. Weinberg, Phys. Rev. Lett. 29, 388 (1972).
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