Generation Symmetry and E_{6} Unification

# Generation Symmetry and E6 Unification

## Abstract

The group for grand unification is combined with the generation symmetry group . The coupling matrices in the Yukawa interaction are identified with the vacuum expectation values of scalar fields which are representations of the generation symmetry. These values determine the hierarchy of the fermions as well as their mixings and CP-violation. This generation mixing appears in conjunction with the mixing of the standard model fermions with the heavy fermions present in the lowest representation of . A close connection between charged and neutral fermions is observed relating for instance the CKM mixings with the mass splittings of the light neutrinos. Numerical fits with only few parameters reproduce quantitatively all known fermion properties. The model predicts an inverted neutrino hierarchy and gives rather strict values for the light and heavy neutrino masses as well as for the decay parameter. It also predicts that the masses of the two lightest of six ‘right handed’ neutrinos lie in the low TeV region.

###### pacs:
11.30.Hv, 12.10.Dm, 12.15.Ff, 14.60.Pq
12

## I Introduction

The origin of the properties of quarks and leptons, the masses and the mixings of the three generations, forms still an open problem in particle physics. Grand unified theories Pati:1974yy () provide a general understanding of the structure and the quantum numbers of the standard model states and suggest new ideas. The hope is to find a consistent scheme which provides for intimate relations between the many observables. Lagrangians considered for a single generation have the fundamental property of chiral symmetry, i.e. they are invariant under chiral transformations of the fermion fields taken together with appropriate transformations of the Higgs fields. For more generations chiral symmetry requires an extended symmetry which relates the different generations: the fermions should be members of a non-Abelian generation symmetry.. There are two possibilities to enforce generation symmetry: one can either enlarge the number of Higgs fields by giving them generation indices or one can identify the coupling matrices in front of the Higgs fields as vacuum expectation values of new scalar fields carrying generation quantum numbers.

We choose here the second alternative in connection with the grand unified symmetry group E6 ()-recentE6 () . For three generations the coupling matrices are matrices. The corresponding 9 scalar fields are taken to be hermitian and are expressed in terms of the hermitian matrix

 Φ(x)=χ(x)+iξ(x) . (I.1)

Here denotes a symmetric and an antisymmetric matrix. Clearly, this choice implies that conventional Yukawa interactions containing these fields are effective ones with dimension 5 and thus have to be understood on a deeper level. This can be done by the introduction of additional heavy spinor fields. We will see, that by integrating out these heavy fields one finds interesting consequences for the relation between quark and neutrino mass matrices.

The introduction of the hermitian matrix field coupled to the fermion fields suggests to use the group to describe the generation symmetry. In addition we will make use of a discrete parity like symmetry, generation parity . From the point of view of chiral symmetry the use of instead of would be more consequent. We will not treat this extension because the subgroup of , in combination with , is sufficient to reach our aims. Together with the unification group it can be dealt with in a very economical way. In the literature non-Abelian continuous nonAbelGenSym () and discrete A4 () flavor symmetries have been discussed and applied in various models. In our approach fermion fields are taken to transform as 3-vectors under the generation group . Consequently, the symmetric part of has to transform as and the antisymmetric part as . Spontaneous symmetry breaking of this generation symmetry leads to vacuum expectation values of . By an orthogonal transformation the symmetric matrix can be taken diagonal. As we will see its elements describe the up quark hierarchy:

 (I.2)

Here denotes the scale at which the appropriate effective Yukawa interaction of dimension 5 is formed. We take the mass ratios to be valid at a high scale. At the weak scale these ratios are somewhat modified. Taking gives good agreement with the experimental mass determinations. In the following we will use this parameter for expressing small quantities.

There remains only the vacuum expectation value of the antisymmetric matrix to produce all mixing and CP violating properties of quarks and leptons. (The use of a purely antisymmetric and hermitian mixing matrix was suggested in antE6 ()). We take for the particular form which was abstracted from the analysis of the fermion spectrum and the CKM matrix in Stech:2003sb () using symmetry for grand unification. It has a particular symmetry: it changes sign by exchanging the second with the third generation:

 1M′⟨iξ⟩=A=i λA ⎛⎜⎝0σ−σ−σ01/2σ−1/20⎞⎟⎠ . (I.3)

The relative factor between the (1,2) and (2,3) elements determines the particle mixings. Its value gives a good description of the mixings of quarks and neutrinos as will be shown in this article. The coupling constant can be incorporated into the mass scale (except for renormalization group considerations).

As already mentioned above, besides the symmetry we also introduce a parity symmetry , ‘generation parity’. The (extended) standard model fermions are taken to be even under this symmetry while the Higgs fields directly coupled to these fermion fields as well as the fields and have negative generation parity. To obtain the spontaneous symmetry breaking of we use in the Lagrangian for the field invariant potentials up to 4th order in the fields and . Adding also the invariant Coleman-Weinberg potential a complete breaking of the generation symmetry can be achieved. Moreover, by selecting properly the coefficients of these potentials the numerical values of (I.2) and (I.3) give the absolute minimum of the total potential.

It is clearly a challenge to obtain the masses, mixings and CP properties of all fermions with only the two generation matrices and together with a few vacuum expectation values of the corresponding Higgs fields. Suited for this task is the grand unification symmetry Stech:2003sb (). Here we can postulate that particle mixings are caused by the mixing of the standard model particles with the heavy particles occurring in the lowest representation of . The up quarks cannot mix since they have no heavy partners in this representation. Thus, it follows immediately that - apart from a constant factor - their mass spectrum is simply given by [as presented in (I.2)].

As we will see the breaking of generation symmetry and symmetry gives a hold on the complete fermion spectrum which includes new heavy fermions. In particular, we find a very strong hierarchy for the heavy right handed neutrinos: their mass matrix is proportional to the square of the up quark mass matrix. By integrating out these heavy neutrinos, their masses appear in the denominator compensating the square of the Dirac mass matrix in the nominator. This mechanism gives the light neutrino spectrum a less pronounced hierarchy than the one of the up quarks. A possible consequence is an inverted neutrino spectrum. Such an interesting situation is not obtained in more conventional treatments of grand unified theories. It will be seen that very few parameters are sufficient to describe all the known properties of charged and neutral fermions in a quantitative way.

The use of as a grand unified theory has many virtues E6 ()-recentE6 (). The fermions are in the lowest representation of this group and an elegant cyclic symmetry connects quark fields, lepton fields and anti-quark fields. In particular, as shown in Stech:2003sb (), combines the mixings among fermion generations with the mixing of the standard model particles with heavy charged and neutral states.

Thus, our starting symmetry at the GUT scale is

 E6×SO(3)g×Pg . (I.4)

In order to introduce notations and conventions we add at this place a short description of the representation of . In the grand unification model the fermions are contained in the representation of the group, i e. they are described by two component (left handed) Weyl fields for each generation:

 ψαr ,    α=1,2,3  ,    r=1,⋯,27 . (I.5)

denotes the flavor index and labels the generations. These fields - even under - describe the fermions of the standard model plus additional quark and anti-quark fields with the same charge as the down quarks and new heavy charged and neutral leptons. All fermions are in singlet and triplet representations of the maximal subgroup of

 SU(3)L×SU(3)R×SU(3)C≡G333 , (I.6)

which plays an important role in our approach. In terms of we have

 27=QL(x)+L(x)+QR(x) , (I.7)

where the quantum number assignments are:

 Quarks:   QL(x)=(3,1,¯3) , Leptons:   L(x)=(¯3,3,1) , Anti−quarks:   QR(x)=(1,¯3,3) . (I.8)

For each generation one has

where . In this description acts vertically (index ) and horizontally (index ) and is a color index. It is seen, that in each generation there exist 12 new fields extending the standard model: colored quarks , charged leptons and 4 new neutral leptons , which all must correspond to heavy, but not necessarily very heavy, particles.

The group can serve as an intermediate gauge symmetry below the breaking scale (the unification scale). It can be unbroken only at and above the scale where the two electro-weak gauge couplings combine. In the non-supersymmetric model, which we adopt here, this point occurs at  GeV according to the extrapolation of the standard model couplings. Interestingly, this is just the scale relevant for the neutrino masses using the see-saw mechanism. Above this scale the united electroweak couplings and the QCD coupling run at first separately until the electroweak coupling bends to meet the QCD coupling at the unification point. This happens at the scale  GeV (see Fig. 1 and more details in Appendix A.1).

Before symmetry breaking equivalent forms of (I.9) can be obtained by applying left and right -spin rotations. We fix the basis by using vacuum expectation values (VEV) of the lowest Higgs representation, the of . In this representation possible vacuum expectation values are restricted to the 5 neutral members sitting in positions corresponding to the ones of the neutral leptons in (I.9). Below the scale only two light doublets are assumed to be active. We incorporate them in two different scalar -plet fields. Thus we introduce two scalar fields and with the following transformation properties under

 H27∼(27,1,−) ,     ~H27∼(27,1,+) . (I.10)

Only can couple to the fields in the form . Thus, it is convenient to choose a basis in which the VEV forms a diagonal matrix. In the scalar potential of the Lagrangian the bilinear terms with respect to both scalar - plets can contain the singlet part of the fields :

 μ2H†27H27+~μ2~H†27~H27+μ′χ(1)(H†27~H27+H27~H†27) . (I.11)

After develops a VEV, and for sufficiently small (i.e. ), the mass eigenstates and following from (I.11) are linearly related to and

 H27=Hu+zHd ,     and     ~H27=−zHu+Hd , with    z=−μ′⟨χ(1)⟩(μ2−~μ2) ,   |z|≪1. (I.12)

We take the light up type doublet to be in : and the light down type doublet to occur in . These Higgs doublets are important for generating the correct fermion mass pattern. determines the scale of the up quarks and the scale of the down quarks and charged leptons of the standard model. is expected to be of the same order of magnitude as . The factor , which vanishes before -symmetry breaking, is responsible for the small values of bottom and tau masses compared to the top mass. Components of , which are standard model singlets have large VEVs. For instance provide high (Dirac type) masses for all new quarks and leptons with the exception of the ‘right handed’ neutrinos and . Their masses arise from a different mechanism involving both and . is not a new scale parameter but is determined by the onset of , the meeting point of the electroweak gauge couplings: with the result (see Appendix A.1):

 MI=1.27⋅1013 GeV ,      e33=MIg2(MI)=2.27⋅1013 GeV . (I.13)

We will see that this value of provides for the correct mass scale of the light neutrinos.

Due to the linear Yukawa coupling all fermions - except the ’right handed’ neutrinos and - are Dirac particles and (so far) have the same hierarchy as the up quarks. The hierarchy of the ’right handed’ neutrinos, on the other hand, becomes super strong due to the combined action of and . Particle mixings will occur by taking into account the antisymmetric generation matrix combined with an antisymmetric (in indices) Higgs field. These mixings modify the hierarchy of down quarks and charged and neutral leptons in agreement with the experimentally observed particle spectra.

## Ii Generation Symmetry

For the construction of the Yukawa interaction symmetric under the existence of a set of new fields is necessary. First of all, as mentioned in the introduction, we introduce scalar fields represented by the matrix . The members of the symmetric part transforms as [under ] and the ones of the antisymmetric part according to . To have a renormalizable interaction we need new heavy vector like fermionic fields and which transform as

 F∼(27,3,−) ,     ¯F∼(¯¯¯¯¯¯27,3,−) , F′∼(27,3,−) ,    ¯F′∼(¯¯¯¯¯¯27,3,−) . (II.14)

The Yukawa interaction involving the Higgs fields arises from the vertices

 (ψT(g(1)χ(1)+g(5)χ(5))¯F) ,  M(FT¯F) ,  (FTH27ψ) . (II.15)

In (II.15) we used matrix notation with regard to generation indices, but suppressed indices and Clebsch Gordan coefficients. and denote the singlet and -plet parts of , respectively. By integrating out the massive fields one gets the wanted effective interaction

 (II.16)

The corresponding diagram is shown in Fig.  2a.  Clearly, the antisymmetric matrix does not contribute to the symmetric (in indices) field . It couples, however to the antisymmetric Higgs representation of with negative generation parity. The vertices are

 ((ψTg(3)iξ¯F′) , M′(F′T¯F′) ,  (F′THA351)ψ) ⇒  g(3)M′(ψTiξHAψ) . (II.17)

The corresponding diagram containing the heavy fields is shown in Fig.  2b.

The symmetry forbids couplings such as , , etc. In the following we will not separate the singlet and 5-plet parts of but simply use the combination as occurring in (i.e ) and absorb and the coupling in the values of and , respectively. Now we can introduce number valued generation matrices by taking the vacuum expectation values of and as described in the introduction:

 1M⟨χ⟩=G ,       1M′⟨iξ⟩=A . (II.18)

As mentioned before, the generation matrix in Eq. (II.18) combined with vacuum expectation values of the Higgs field gives Dirac masses to all fermions except the two heavy leptons and . The latter require Higgs fields transforming as with respect to , components, which are not contained in and . Instead of introducing another high dimensional Higgs field it is plausible to use the Higgs field together with . This avoids the appearance of a new unknown generation matrix. With the help of the generation symmetry it is possible to derive the relevant generation matrix of the ’composite’ plet in terms of the matrix. This increases the predictive power of the model. To obtain the corresponding effective interaction from a renormalizable interaction, another massive Dirac field is required. It is a vector in generation space and an singlet: , with even under and odd under . In addition we need a total singlet field which is odd under . It can be identified with the singlet part of with vacuum expectation value . The vertices are

 (FTH†27N) ,   ⟨χ(1)⟩(~NN)=MN(~NN) ,   (~N~H†27F) . (II.19)

Integrating out the fields the vertex emerges. Below the masses of the states the effective Yukawa interaction

 1M21MN(ψTχH†27)(~H†27χψ) (II.20)

is generated. The corresponding diagram is shown in Fig.  3. Because in (II.20) the indices are not shown, one has to keep in mind that the combinations and are singlets. It is easily seen that by this interaction only neutral leptons can get masses, notably the right handed neutrinos and . They are coupled to the large elements of and , which are standard model singlet fields and thus will get large vacuum expectation values. The most interesting feature is however the appearance of the square of . It implies that after generation symmetry breaking the mass hierarchy of these heavy neutrinos is dramatic, namely equal to , the square of the mass hierarchy of the up quarks which is already a very strong one.

and , which both appear now in the Yukawa interaction, can be expressed in term of the mass eigenstates and . From (I.12) we have and , with the mixing parameter . We choose the VEVs of and as follows

 ⟨Hu⟩=⎛⎜ ⎜⎝e11  0 00  0 00−zϵ32 e33⎞⎟ ⎟⎠,  ⟨Hd⟩=⎛⎜ ⎜⎝0000ϵ2200ϵ320⎞⎟ ⎟⎠. (II.21)

Thus, the VEV of the Higgs field has the diagonal form taken by convention

 ⟨H27⟩=⟨Hu+zHd⟩≃⎛⎜ ⎜⎝e11 0 00zϵ22 00 0 e33⎞⎟ ⎟⎠ , (II.22)

while ’s VEV has the structure

 ⟨~H27⟩=⟨Hd−zHu⟩≃⎛⎜ ⎜⎝−ze11  0     0   0  ϵ22     0   0  ϵ32 −ze33⎞⎟ ⎟⎠ . (II.23)

Since has no off diagonal element one expects a large element for . We take , where is identified with . In principle, however, together with and could be of a lower scale (but still ). The dominant VEVs of and fix now the mass terms of the ‘right handed’ neutrinos and

 F{2,3}G2((L32)TL33)+F{3,3}G2((L33)TL33) ,
 with  F{2,3}≃e33ϵ32MN≃e33 ,  F{3,3}≈−z(e33)2MN≈−ze33 . (II.24)

is the only Majorana mass term occurring so far.

The effective Yukawa interaction below now reads

 LeffY = Gαβ(ψαTH27ψβ)+Aαβ(ψαTHA351ψβ)+ 1MN(G2)αβ(ψαTH†27~H†27ψβ) . (II.25)

This effective Yukawa interaction together with the VEV configurations (II.22) and (II.23) contains all the necessary information about the generation structure: Generation hierarchy and generation mixing are now completely fixed. At this stage we do not need to specify the scales of . However, these scales become important when we take renormalization effects into account.

Let us now discuss the breaking of the generation symmetry . The Lagrangian for the field is

 LΦ=12Tr{(∂μΦ−e[Bμ,Φ])(∂μΦ−e[Bμ,Φ])}−V(χ,ξ) ,

with , where denote the vector potential and the 3 antisymmetric generators of the generation symmetry . stands for an invariant potential. Gauge invariance allows to choose , the symmetric part of , to be a diagonal matrix with 3 real elements. This defines a direction in symmetry space for a possible spontaneous symmetry breaking. Normalizing for this basis one can write

 Φ=⎛⎜⎝χ1000χ2000χ3⎞⎟⎠+i√2⎛⎜⎝0ξ3−ξ2−ξ30ξ1ξ2−ξ10⎞⎟⎠ . (II.26)

In this form the vertices we have shown above simplify drastically since the orthogonal transformation diagonalizing can be absorbed by the fields , and .

The scalar potential in (II) can easily be made to have a minimum for specific values of the three invariants: the trace of , the trace of the square of the traceless part of and the trace of . At this minimum we have then two relations for the 3 fields forming and one relation for the other 3 fields forming . There still remains the freedom of such transformations which do respect this minimum and keep diagonal. This remaining symmetry is necessarily a discrete subgroup of . It is the discrete symmetry group . As a generation symmetry has been suggested in many publications A4 (). In our approach this symmetry occurs naturally in connection with the starting symmetry . It is the remaining symmetry after using appropriate potentials invariant under and the choice of a symmetry breaking direction.

Radiative corrections add to the potential new invariant parts which are of logarithmic type. The Coleman-Weinberg potential Coleman:1973jx () is of this form. By including it the total potential can lead to a complete spontaneous symmetry breaking of which results in vacuum expectation values for and of the form (I.2), (I.3). We demonstrate this here by using the potential

 V(χ,ξ)=−M22Tr[χ2]−M′24Tr[(iξ)2]+c1(Tr[χ])4+ cT(Tr[χ2T])2+c3Tr[χ3] Tr[χ]+cξ (Tr[(iξ)2])2+ cχξ0Tr[χ.(iξ)2] Tr[χ]+cχξ1Tr[χ.iξ.χ.iξ]+ 3e264π2[M4B1ln(d1M2B1M2)+M4B2ln(d2M2B2M′2) +M4B3ln(d3M2B3M′2)] . (II.27)

Here denotes the traceless part of and stands for the vector boson masses as field dependent functions (we take in (II.27 for simplicity). The coefficients in (II.27) can be tuned such that the symmetry breaks spontaneously and produces and with . To achieve this one has to use the six relations following from the first derivatives of the potential at the proposed minimum: and . Because of the large hierarchy a high accuracy of this calculations is necessary. We use here i.e., the mass scale for forming the antisymmetric matrix is large compared to the one for forming the symmetric matrix (this is required for the gauge coupling unification, see the renormalization group treatment in the appendix). Setting then e.g. , and all other coefficients are determined by putting the first derivatives of the potential to zero. These coefficients are sufficiently small ( for instance) to allow perturbative treatments. The log terms are small near the minimum. With this choice of the potential the six eigenvalues of the matrix for the second derivatives of the potential are all positive at the wanted values of . The minimum obtained is an absolute one (but degenerate with respect to different signs of the three elements in ).

The above formulae for the vacuum expectation values and the corresponding potential would have a different form if we would have used other values for the couplings and . However, also for this more general situation potentials can be constructed to produce the required spontaneous symmetry breaking.

For the three vector boson masses at the minimum of the potential one gets

 MB∼⎧⎪⎨⎪⎩1.40 eM0.505 eM′0.505 eM′⎫⎪⎬⎪⎭ . (II.28)

## Iii Charged Fermion Masses and Mixings

As we have already mentioned, the diagonal generation matrix (i.e. ) is taken such that the up quark masses have their observed hierarchy. The top quark mass at is determined by the vacuum expectation value and the coupling at this scale

 mt=e11λt=e11 Gu^u3,3=170.9 GeV . (III.29)

The notation (the superscript of ) indicates that at low energy one has to distinguish between different channels. The renormalization group effects are treated in the Appendix. We take for a value such that the top coupling constant near is close to one.

 λt(MZ)=1.30  ,   e11=131.5 GeV . (III.30)

Having for all diagonal -couplings the same spacing at the scale , we are able to calculate the spacing for each channel at the scale . As obtained in the Appendix we find

 Gu^u(MZ)=Diag(r−1tσ4 ,r−1tσ2 ,1)⋅λt(MZ) , with    rt=0.717 . (III.31)

Comparing with the measured values of up and charmed quark masses suggest

 σ=0.050 . (III.32)

Eq. (III.31) gives for this value of   MeV and  GeV in good agreement with experiment. We will use (III.32) in all further calculations.

The case for the down quark and charged lepton masses is not as straightforward. Here the light fermions will mix with the heavy states and . We take this as the source of the particle mixings. It involves the indices and (-spin mixing) of and of the Higgs field . According to Eq. (II.25) this mixing of light and heavy fermions occurs together with the generation mixings by the coupling matrix (i.e. the matrix ) .

i) The down quark masses and mixings

The matrix elements for down quarks are part of a mass matrix. It has the form

 (III.33)

In this equation the constants stand for the vacuum expectation values of :

 fij=⟨(HA)ij⟩ ,    i,j=2,3 . (III.34)

and have values of the order of the weak scale, while and will be very large because they arise from standard model singlets. Further below we will find that the vacuum expectation values of the standard model singlet components of , like , have values which are small compared to (smaller than ). This allows to make use of the see-saw formula and also justifies the neglection of in (III.33). Integration of the heavy -states leads to the down quark mass matrix at the scale

We distinguished different and matrices when coupled to different channels etc. Different renormalization factors arise when starting from the original and matrices at high scales. The calculations are deferred to the appendices. The matrices appearing in (III.35) are defined in (A.12) and (A.14).

For the matrix element in (III.35) with the generation index we write

 zϵ22Gd^d33=ϵ22λb=m0b . (III.36)

The down quark mass matrix leads to 7 observables : 3 mass eigenvalues, 3 mixing angles and the CP violating phase. According to (III.35) we have 3 parameters for a fit of the experimental results. We use the notation of the various coupling constants provided in (A.12) and (A.14). Taking then

 m0b=2.95 GeV ,      λd^dAf22≃−0.23 GeV , f32f23e33λd^DAλD^dAλD^D=1.62⋅10−4 GeV , (III.37)

one gets

 md(MZ)≃2.6 MeV ,   ms(MZ)≃50 MeV , mb(MZ)≃2.89 GeV , |Vus|≃0.228 , |Vcb|≃0.042 , |Vub|≃0.0039 . (III.38)

The angles in the quark unitarity triangle have the values

 αq≃97o ,     βq≃23o ,     γq≃60o . (III.39)

Our results are very satisfying. Quark masses, the CKM mixing angles and the three angles of the unitarity triangle have values within experimental errors Yao:2006px (); Xing:2007fb ().

ii) The charged lepton masses and mixings

The charged lepton sector is constructed in a similar way. The light leptons mix with the heavy ’s through the vacuum expectation values of the multiplet (the vacuum expectation values of are considered to be negligible because of its representation for the symmetry):

 ⟨Hi{j,k}A⟩=fi{j,k} . (III.40)

With the abbreviations

 f2{1,3}=g22 ,   f2{1,2}=g23 ,   f3{1,3}=g32 (III.41)

and applying again the effective Lagrangian (II.25) the mass matrix for charged leptons has the form

 (III.42)

Integrating out the heavy states leads at the scale to the matrix

 (III.43)

Here appear again renormalization group coefficients defined in (A.12), (A.14). We choose

 zGe−e+33ϵ22≃ϵ22λτ=m0τ=1.593 GeV , λe−e+Ag22≃−0.1898 GeV , g23g32e33λe−E+AλE−e+AλL¯L≃3.16⋅10−4 GeV . (III.44)

These three numbers determine the charged lepton masses and, as in the quark case, determine also the 3 mixing angles and the CP violating phase. The charged lepton mixings are not directly observable, but will play an important role in the discussion of the neutrino properties. The masses come out precisely

 me(MZ)=0.487 MeV ,    mμ(MZ)=102.8 MeV , mτ(MZ)=1.747 GeV . (III.45)

For the mixing angles and the CP violating phase we obtain

 θe12≃4.8o ,     θe23≃5.1o ,     θe13≃0.35o ,     γe≃−23o . (III.46)

We have defined these four parameters in complete analogy to the quark CKM mixing angles and the unitarity angle gamma.

iii) Estimates of , , , and

So far only the combinations , and are known numerically. However, there is one relation due to the known mass of the vector boson of the standard model. It connects all vacuum expectation values which belong to doublets:

 (e11)2+(ϵ22)2+(f22)2+(g22)2+(f23)2+(g23)2=(174.1 GeV)2 . (III.47)

Because of the nearly identical interaction of down quarks and charged leptons we can estimate the ratio from the fit values given in (III.37) and (III.44) by taking the large standard model singlet VEV’s and to be equal. One gets which is close to as one could have expected. Using Eq. (III.47) the only parameter needed then is the ratio . This is a ratio of two weak scale quantities with the same weak isospin quantum numbers. We therefore take it to be equal to one. This choice gives a value for small enough to justify the application of the see-saw mechanism we used above. It also provides for small values for the couplings and necessary for renormalization group stability of the neutrino sector to be discussed in the appendices. All couplings and VEV’s introduced are now fixed:

 f23≃39 GeV ,     g23=ϵ22≃76 GeV , zGe−e+3,3≃λτ≃1.6 GeVϵ22≃2⋅10−2 ,  λb≃4⋅10−2 , f32e33=g3