Radiative generation of quark and lepton mass hierarchies from a top-quark mass seed

PHYSICAL REVIEW D VOLUME 41, NUMBER 5 1 MARCH 1990

Radiative generation of quark and lepton mass hierarchies from a top-quark mass seed

Xiao-Gang He and Raymond R. Volkas Research Centre for High Energy Physics, School of Physics, University of Melbourne, Parkville 3052 Australia

Dan-Di Wu Institute of High Energy Physics, Academia Sinica, Beijing 100 039, People's Republic of China

(Received 24 July 1989)

The possibility that of all the fermions in the standard model only the top quark gains mass at the tree level is investigated. Quark and lepton mass hierarchies are introduced by inducing the other fermion masses at various orders of perturbation theory. Lepton- and baryon-number-conserving scalar diquarks and leptoquarks are invoked to carry out these radiative corrections. We show that the simplest model one can construct has the following cascade: tree level-+top; one loop-bottom; two loop+charm, tau; three loop-+strange, mu, up, down; four loop-+electron. Neutrinos are strictly massless because we do not introduce right-handed chirality components. Phenomenological constraints on the family-lepton-number-violating scalar diquarks and leptoquarks are considered. We elucidate the symmetry structure of the model we arrive at, thereby demonstrating the importance of a certain Z , discrete symmetry in preventing the appearance of unwanted Higgs-fermion-Yukawa couplings.

The search for an explanation of the mass pattern of quarks and leptons is a long-standing puzzle which con- tinues to present important challenges for particle theor- ists. ' A prominent feature of this pattern is the presence of mass hierarchies. One such hierarchy is that between different generations. For example, the mass of the tau is about four orders of magnitude greater than the mass of the electron. Within each generation there are other hierarchies such as that which exists between the lepton and quark members. The standard model (SM) has no predictive power for fermion masses. It is thus worthwhile searching for physics beyond the SM which could shed some light on the fermion mass problem.

There are many ideas in the literature regarding such new physics. For example, compositeness may be an explanation, or perhaps one should look to the properties of six-dimensional Calabi-Yau manifolds (see Ref. 1 for a discussion of the flavor problem). In this paper we will examine the possibility that all quark and lepton masses arise radiatively, with the top-quark mass as the (tree- level) seed.

Such a scenario is motivated by the fact that the top quark may be the only fermion whose mass is of the same order as M w and Mz. Indeed it is tempting to look for models in which the top quark naturally acquires a larger mass than other f e r m i ~ n s . ~ In this paper the top quark will be the only fermion to gain mass via the standard tree-level mechanism involving the vacuum expectation value (VEV) of one standard Higgs doublet. The fermion-Higgs-Yukawa couplings, which generate the other fermion masses in the SM, will be forbidden by a Z 3 discrete symmetry (see later). We will introduce baryon- and lepton-number-conserving scalar diquarks and leptoquarks in order to generate nonzero masses for the other fermi0ns.j Another method of radiativelv inducing fermion masses is through several copies of the

SM doublet, some of which do not develop VEV's .~ Al- though Higgs-induced tree-level flavor-changing neutral processes are forbidden in the model of Ref. 4, despite the need for several Higgs doublets,' by a discrete symmetry, we prefer to use only one doublet.

We envisage a cascade mechanism, whereby quarks and leptons gain mass at various orders of perturbation theory from masses induced at the preceding order of ap- proximation. In this way we hope to explain at least some of the qualitative features of the observed mass spectrum.

Let us remind ourselves of the quark and lepton mass spectrum (see Table I ) (Ref. 6). We will not introduce right-handed neutrinos, so they will be exactly massless for the same reason as they are in the SM. It is con- venient to consider particular cascade patterns for the four heaviest particles t , b, T , and c first, and to worry about the other particles later. Now, T and c have rough- ly the same mass, and they are about three or four times lighter than the b. We will thus assume that T and c obtain mass at the same order of perturbation theory. We will consider two cases, represented by the cascades

number of loops: 0,1,2,.

In Case (1 ) the mass difference between c, T, and b is as- cribed to the hypothesis that c and r acquire mass at the two-loop level, while b acquires mass at the one-loop level. Case (2) envisages b, c, and T all obtaining mass at the one-loop level, with the slight hierarchy between them put down to small "accidental" differences in certain coupling constants.

We now construct radiative corrections which lead to

1630 @ 1990 The American Physical Society

41 RADIATIVE GENERATION OF QUARK AND LEPTON MASS . . . 1631

TABLE I. The mass spectrum of quarks and leptons. All the running quark masses are evaluated at 1 GeV except for the bound quoted for the top-quark mass, where we have used its physical mass.

Particle Mass or mass limit (GeV)

these patterns. The generic diagram is given by Fig. 1. The cross on the internal fermion line represents the top- quark-mass insertion which comes from the standard Yu- kawa coupling

where 4 - (1 ,2 , -1 ) under SU(3),8SU(2),@U(l) , , and ( & ) - U. The cross on the scalar line indicates mixing between two diquark or leptoquark scalars (depending on whether the external fermions are quarks or leptons). The scalar 7 couples to left-handed fermions, while 7' couples to right-handed fermions. In this way the left- right transition, which indicates the generation of a mass, is induced.

The up-quark mass matrix arising from Eq. (3) is given by (M,U,,, I i j = r i 6 3 j . This matrix has only one nonzero eigenvalue (i.e., it has rank I), which we identify with the tree-level top-quark mass. The radiative effects we will now consider will lead to corrections to the fermion mass matrices of the form

Each contribution arb," will raise the rank of the matrix by 1, provided a: is linearly independent of {am J and 6,"

FIG. 1. The generic diagram used for radiatively inducing one-loop fermion masses in this paper. The external fermion lines correspond to either quarks or leptons. If they are quarks then the 7 particles are baryon-number-conserving scalar diquarks; if they are leptons then the 7 particles are baryon- and lepton-number-conserving scalar leptoquarks. The cross on the internal scalar line indicates mixing between 7 (which couples to left-handed fermions) and 7' (which couples to right-handed fermions). The internal fermion line contains the tree-level top-quark-mass insertion.

is linearly independent of b y ) , where (aTb;") is the set of previously computed corrections. The nature of these contributions will be discussed more fully later in the paper.

Consider the generation of b and c masses. The first issue is the nature of the diquark q . Its coupling to left- handed quarks is generically given by ?jFLqjL7,, where i, j= 1,2,3 are generation indices, which means that there are two choices:

There is an important difference between 7: and 7; in that the former incorporates three different vertices (up- up-7, up-down-7 and down-down-71, whereas the latter only has one (up-down-7). Thus the singlet only con- nects uplike quarks with downlike quarks. T o generate the b mass one has to introduce 7, such that

~7~ - (3,1, - $1, 7; djR q 2 invariant . (5)

y2 plays the role of 7' in this case. T o obtain a nonzero b mass at the one-loop level, one can therefore introduce either 7; and q 2 or 7; and 77,. The nature of the fields Fi and Ff which mix these particles is then given by

For simplicity one would want to make the identifications

For Case (2) one also needs to generate the c quark mass at the one-loop level. This necessitates the introduction of

7,-(3,1,-$), i ~ u , , r ] , + H . c . invariant , (10)

where a = 1,2 is an index for the first two generations. A left-handed up-up-r] vertex is also needed, which means that one has to choose 7; for Case (2).

Let us summarize the situation thus far. Case ( 1 ):

Case (2):

1632 XIAO-GANG HE, RAYMOND R. VOLKAS, AND DAN-DI WU 41

At this point we comment that since Hif is antisym- metric in flavor space in Case (2), there is no d 3 L u 3L COU-

pling. After one-loop corrections, the massive eigenstate is a combination of d l , and dz, If we still want to identify this state as the physical b L , the mixing matrix in the charged current will be phenomenologically unacceptable and therefore Case (2) is not interesting. There are also other reasons that this case is not interesting, as we will see later.

Simplicity implies the identifications

We now come to a crucial issue. Is there a symmetry which is compatible with the couplings in Eqs. i3), (1 I) , and (12), which forbids the SM couplings qiL uUR +,qiLdJR $,fiLljR 4, and which is consistent with the identifications of Eq. (13)?

To study this question we consider the following phase transformations:

where a,. . . ,a are, for the moment, unrelated, and where the phase transformation of tR is required to be strictly different from that of u,, .

The compatibility of Eq. (3) with the symmetry implies that

The nonexistence of the other SM Higgs- fermion-Yukawa couplings means that

The transformation laws for the 77 particles are

i i - 2 a ) t 77:,17;-+e ~ 1 7 1 7 s 9

1 i - 8 - 7 1 772-e 172 ,

1 i - e - y 1 772-e 772 9

This then implies that

Now if one wanted to impose Eq. (13), one would further require

- 2 a + y + 6 = 2 n 5 r for some n 5 E § . (19)

However, this is in contradiction with Eq. (16b). The next simplest possibilities for F 1 are

( A ) F i = ( P : T ~ ~ or F: =new triplet Higgs field ,

( B ) Fs =d:d2 or F ; = S =new singlet Higgs field ,

where $J,,& are different Higgs doublets. There is a problem with "Ff=new triplet Higgs field" in that phenomenological constraints from the p parameter of weak interactions would require its VEV to be quite small. Since this would require the coupling constant h , to be made artificially large we consider this possibility no further. The introduction of two Higgs doublets may be vi- able, but for reasons of simplicity and because several Higgs doublets may lead to unwanted flavor-changing neutral processes, we will not pursue this line of development either. Also, as was noted earlier, the Kobayashi- Maskawa matrix has an unacceptable form in this model. Clearly the simplest allowable scenario is the introduction of a new Higgs field Fs =S-(1 ,1 ,0) , which trans- forms like

There are no impediments to taking the VEV of S to be fairly high.

The criterion of simplicity thus favors the cascade of Eq. ( I ) , with the Lagrangian

How do the leptons and the other quarks gain mass? We require that c and T pick up a mass from the now nonzero b-quark mass (see Fig. 2 ) . T o do so we need to introduce the particles 77,, q4 , and 175 with the following couplings:

To complete the spectrum we need just one more leptoquark 176 so that

So, the final Lagrangians are

This Lagrangian generates all of the Feynman diagrams depicted in Fig. 2. The cascade we obtain is as follows: tree level--+top; one loop-bottom; two loop-charm, tau; three loop-strange, mu, up, down; and four loop-+electron. We will now describe, by reference to Fig. 2, how the rank of the up-quark mass matrix is in- creased as one goes to higher loops; the construction of down-quark and lepton mass matrices can be obtained in a similar fashion. At the tree level (putting M U P r M ) we

RADIATIVE GENERATION OF QUARK AND LEPTON MASS . . . 1633

FIG. 2. This figure shows how and at what level of perturbation theory the various quarks and leptons acquire nonzero masses. The mechanism depicted resembles a cascade, in that nonzero masses acquired at one order of perturbation theory are used to generate masses at the next order. To simplify the diagrams we have defined effective mass insertions in the middle column by writing the number of loops above the cross representing the left-right mass transition. The effective mass matrices are defined also. Thus the one-loop correction to the down-quark mass matrix a,'b] is proportional to ( 2, H , ) , r k )H:u. The various other effective mass matrices can be easily read off the Feynman diagrams.

2 .

have that Mi, = r,6,, which has rank 1, as can be easily checked. At the one-loop level the only correction to M is displayed in Fig. 3. This graph, however, does not lead to an increase in the rank of the matrix; it only provides a small correction to the parameters r,. At the two-loop level the correction G2Mlj=c)d,'G,, arises. This is proportional to

T ,-*-.,5 /' " , L + s " " : i "a

a ; < n:" G' < charm

where sib,!, is the rank-1, one-loop down-quark mass matrix. Thus the entries in M which were zero up to the one-loop level receive nonzero contributions at the two- loop level. These corrections, however, only increase the rank of the matrix by 1, so only the charm quark gains a mass. The up quark obtains a nonzero mass at the three- loop level from the two-loop-level lepton mass matrix:

where e i f,!, is the two-loop lepton mass matrix. The way the various fermion masses seed the generation of nonzero masses at the next order of perturbation theory is summarized by Fig. 4. (One can check that higher- order diagrams with different topology to those depicted in Fig. 2 do not change the conclusions obtained above.) We should also make the comment that unlike fermion masses, fermion mixing angles do not fall into a hierarchical pattern in this approach. Nevertheless, the observed values for these angles can be reproduced by taking ap- propriate values for some of the a priori unknown parameters in our model.

We must now check that there is a symmetry which is compatible with the couplings introduced in Eq. (24) and with the inequalities i 16a)-( 16c). Some straightforward algebra yields the relations

~ = 4 a - 2 y - 6 - t 2 n , r r for some n , E § , (25a)

p=3a+B- y --26+2n,.rr for some n , € 9 , (25b)

- 2 a + y + 6 = f N ~ for some N E S . ( 2 5 ~ )

Equation ( 2 5 ~ ) is crucial, as can be seen by comparing it with Eq. (16b). We must impose the constraint

We can now summarize the transformation laws im- plied by all of the above:

FIG. 3. The Feynman diagram depicting the one-loop correction to the up-quark mass matrix. This correction does not change the rank of the matrix; it just renormalizes the T,.

1634 XIAO-GANG HE, RAYMOND R. VOLKAS, AND DAN-DI WU - 4 1

NO. of Looos 0 1 2 3 4 spin-0 particles 77 are not, in general. For example, v4

FIG. 4. A summary of the cascade produced by our model. A nonzero mass for the particle at the base of the arrow induces a nonzero mass for the particle at the tip.

Actually the transformations in Eq. (27) consist of four independent symmetries: one discrete Z 3 symmetry and three U(1) symmetries (there are three because of the three undetermined parameters a, P, and y ). Indeed these three U(1) invariances turn out to be just U ( l ) y @ U ( l jL@U(l 1,. It is in fact the Z3 discrete symmetry [with group elements ( 1, w , w2) where w = ei2"N/3" 1 which is responsible for preventing the SM couplings ?jlL uaR 4, qjLdiR 6, and fTL I,R 4 (see Table 11). It is the only symmetry in the model which distinguishes between tR and u a R , thus giving the top quark its special status.

At this point a few phenomenological comments are in order. Although the Yukawa interactions of the neutral standard Higgs particle is flavor diagonal, those of the

and 775 can mediate the process KL -+pe via a tree-level diagram. These effects are an interesting experimental signature and their nonobservation provides stringent bounds on the parameters involving 77 particles. Clearly these processes can be suppressed either by making the relevant fermion-77 Yukawa coupling constants small, or by making the 77 particles heavy. We choose to do the latter, because the underlying philosophy of our model precludes the appearance of unnaturally small Yukawa coupling constants. A rough calculation for the process KL +pe provides a lower bound for the mass of 77, or q 5 of 1 to 40 TeV, corresponding to choices of -0.2 to -5 for the Yukawa coupling constants. This bound is indi- cative of the mass range allowed for all of the 77's.

Why do we choose a range of 0.2-5 for the Yukawa couplings? This is because we preclude small Yukawa coupling constants in the theory, and we demand that all such coupling constants be of order 1. What does "of order 1" mean exactly? I t is really quite a vague concept. We think it is reasonable to call numbers between 0.2 and 5 "of order 1." There is an element of subjectivity about this choice; a larger range may also be reasonable. All one can really say is that, for example, a Yukawa coupling constant of 0.001 would definitely be anomalously small. But what about 0.08, for instance? The border between "natural" and "unnatural' is not well defined. The choice of 0.2 made above was dictated by the ratio m, / ( 4 ) > 0.2. From the beginning we were allowing the top-quark mass to be as low as 50 GeV (the current rough experimental limit). Therefore, for consistency we should allow all other Yukawa coupling constants to possibly be as small as 0.2. Upper bounds may be motivated by demanding that perturbation theory still be valid (our analysis explicitly uses perturbation theory, of course). This would restrict Yukawa coupling constants to be less than about 5, and quartic-Higgs self-interaction coupling constants to be less than about 15.

This discussion also explains why we were not too strict about what mass ratios we would associate with a radiative origin. I t is clear that, for instance, the strange-quark mass should be considered of radiative origin, when compared to the top-quark mass. I t is not clear, a priori, whether b, c, and T ought to be separated by an order of perturbation theory. Many cascade patterns are, in principle, compatible with both naturalness and radiative mass generation. In this approach one can only hope to gain a partial understanding of the hierarchical pattern of fermion masses. At any rate, with Yu- kawa coupling constants in the range alluded to above, we are able to reproduce all the fermion masses.

In order to counteract the effect of having the 7 parti-

TABLE 11. The Z , transformation properties of all the particles in our model. Note that this symmetry discriminates between tR and u , ~ .

Group element Fermion Diquark Leptoquark Higgs boson

!!! RADIATIVE GENERATION O F QUARK A N D LEPTON MASS. . . 1635

cles in the TeV range, one must postulate fairly large mixing between them, so that the radiatively induced masses turn out to be sufficiently large. From Eqs. (211, (22) , and (23) one sees that ( S ), h h,, h, , and h, have to be made large enough to accomplish this. Because of the fact that we are dealing with a many-parameter theory, it is difficult to be more precise about what impli- cations this may have.

In summary then, we have succeeded in constructing an extension of the SM, which has every fermion mass other than the top-quark mass generated radiatively. Our model differs from many of the others in the literature in that no exotic massive fermions (such as a sequen- tial fourth generation) need to be postulated. This model naturally suppresses scalar-induced flavor-changing neutral processes involving the standard Higgs doublet. However, other scalar particles in the model do mediate such processes, which results in a strong constraint on the theory. The model affords us some understanding of the hierarchical structure of the quark and lepton mass spectrum. In common with other models of radiative mass generation, the understanding is qualitative rather than quantitative because the diquark and leptoquark Yukawa and Higgs couplings are a priori arbitrary. Our model does produce a hierarchy in masses for the heavy fermions, which is in qualitative agreement with the observed values. However, it does not lead to an understanding of 'why the up- and down-quark masses are significantly smaller than the strange-quark and muon masses, since u, d, s, and p gain mass at the same order of perturbation theory. The electron, however, is naturally

the lightest charged fermion in the theory. Also, we have to incorporate a somewhat wide range of mass ratios be- ing generated at the same order of perturbation theory. This requires there to be a spread about a central value of 1 for the coupling constants and scalar mass ratios in the theory. This may require, for instance, that some of the coupling constants approach the limit at which perturbation theory breaks down. T o what extent this is "unnatural" is partly subjective; a definitive answer is difficult to obtain. The issues raised above are serious, and warrant further investigation of models utilizing our basic idea, with a view to ameliorating the difficulties of our theory. For example, another line of development would be to consider generalizations of the transformation in Eq. (14) by introducing more flavor dependence. In other words, perhaps it is not only the top quark that has distinctive transformation properties under some family-dependent discrete, or even continuous, symmetry.

As a concluding comment, we feel it is significant and interesting that one can indeed generate all fermion masses in a hierarchical manner from just the top-quark mass.

This work was supported by the Australian Research Council. D.D.W. would like to thank Professor B. H. J. McKellar and the Research Centre for High Energy Physics at Melbourne University for financial support. R.R.V. would like to thank G . C . Joshi for useful discus- sions.

'R. D. Peccei, invited talk given at the Zuoz Spring School, Balakrishna, A. L. Kagan, and R. N. Mohapatra, Phys. Lett. Zuoz, Switzerland, 1988 [Report No. DESY 88-078 (unpub- B 205, 345 (1988); K. S. Babu and X.-G. He, ibid. 219, 342 lished)]. (1989).

lV. A. Miransky, M. Tanabashi, and K. Yarnawaki, Mod. Phys. 4E. Ma, Phys. Rev. Lett. 62, 1228 (1989). Lett. A 4, 1043 (1989); R. Barbieri and L. J. Hall, Nucl. Phys. 5S. Weinberg and S. L. Glashow, Phys. Rev. D 15, 1958 (1977); B319, 1 (19891. E. A. Paschos, ibid. 15, 1966 (1977).

,B. S. Balakrishna, Phys. Rev. Lett. 60, 1602 (1988); B. S. 6J. Gasser and H. Leutwyler, Phys. Rep. 87, 77 (1982).

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Radiative generation of quark and lepton mass hierarchies from a top-quark mass seed