Four-Fermion Dynamics and Fermion Masses


Bob Holdom*
Department of Physics, University of Toronto, Toronto, ON M5S1A7



Four-fermion operators with large anomalous dimension may feed down mass to quarks and
leptons from a heavy fourth family. No technicolor sector is required. A model illustrates the
origin of the large top mass along with quark mass hierarchies and mass mixings.



1. Introduction

We would like to explore the possibility that effects of physics at very high scales
can feed down to low scales via 4-fermion operators having large anomalous scaling.
There have been various suggestions in the literature that 4-fermion operators may
have an anomalous dimension of order two in strongly coupled gauge theories with
small or vanishing -function. This would make effective 4-fermion operators
"relevant" at low energies, even though they may be generated on much higher energy
scales. For this to happen there must be a new strong interaction which is perhaps
stuck at some nonperturbative infrared fixed point. Our main object in this paper will
be to show how such operators can have very interesting implications for the origin of
fermion mass.
Operators displaying anomalous scaling were utilized in the walking technicolor
context; in that case a two-technifermion-two-quark operator T T qq displays the
anomalous dimension m of the mass operator TT , which is close to unity.1) If this
operator is generated by extended technicolor (ETC) at a scale ETC the anomalous
scaling results in an enhancement of order ETC/ when the operator is renormalized
at the scale  <
 7 Oct 1995 ETC. The result is an enhanced quark mass. Similarly a four-
technifermion operator T T T T could have an anomalous dimension roughly 2m which
could make it a relevant operator; here the resulting enhancement feeds into
technipion masses. The phenomenon of 4-fermion operators as relevant operators was
also studied in strongly interacting quenched QED in the ladder approximation.2)
Perhaps the main problem facing dynamical theories of fermion mass is the issue
of a large t quark mass in association with a small parameter. In ETC theories the
isospin breaking required in the operator T T qq splitting typically also shows up in the
four-technifermion operators T T T T , thus implying that the technifermion sector will
make a large contribution to . The problem is severe for walking technicolor
because in that case the four-technifermion operators are enhanced more than the
operator which feeds mass to the t.3) What we are looking for is the opposite
situation, in which the operator contributing to the t mass is enhanced by anomalous


* Based on a talk given at Yukawa International Seminar '95: From the Standard Model to Grand
Unified Theories, Kyoto, Japan, 21-25 Aug 1995. E-mail address: holdom@utcc.utoronto.ca


scaling at least as much or more than any other operator which contributes to .
In this work we will not introduce a new unbroken gauge symmetry such as
technicolor. Instead we have in mind a family gauge symmetry which, at high
energies, serves to enhance certain operators involving quarks and leptons. If the t
quark mass is associated with such an operator, that operator must be large enough
on scales as low as a TeV. This implies that at least some remnant of the family
gauge symmetry must survive down to such energies. All fermions involved in the
operator responsible for the t mass must feel this gauge symmetry. But eventually
any such new gauge symmetry must be broken, since the t does not couple to a new
massless gauge interaction. We are thus led to consider a new strong gauge
interaction broken close to a TeV, and this in turn leads us to the question of how
electroweak symmetry breaking itself could be associated with such dynamics.

2. The t mass

We will suppose that the remnant family gauge symmetry is U(1)X, and that this
gauge symmetry breaks such that MX/gX 1 TeV. A possible origin of this breaking
will be described in Section 4. Above some higher scale , say 100 to 1000 TeV,
U(1)X becomes embedded into a larger gauge family symmetry. The fermions which
feel U(1)X will be the third family, along with a new fourth family of quarks and
leptons having standard quantum numbers. We will assume that the 1 TeV dynamics
produces an effective 4-fermion interaction which is above the critical value necessary
for the formation of fermion mass, and that the fermions receiving this large mass are
members of the fourth family. In the absence of any fine-tuning, these masses will be
of order a TeV as well. This dynamical mass in turn implies electroweak symmetry
breaking.
Let us consider two quark doublets, which will end up describing the third family
quarks (t, b) and the fourth family quarks (t , b ). We will label the two doublets by
Q (U, D) and Q (U , D ), where we use capital letters to emphasize that these are
not the mass eigenstates. Q and Q are taken to have equal and opposite vectorial
X charges. The reason for these choices will become clear when we discuss the larger
family symmetry and the resulting mass matrices below. Now let us consider possible
effective 4-fermion operators which may already be present in the theory at the higher
scale . We assume that the U(1)X remains strong (at a fixed point perhaps)
throughout the 1 TeV to range and enhances some 4-fermion operators through
anomalous scaling. In analogy with the technicolor analysis we would expect that the
following operators are enhanced.

QLQRQRQL QLQRQRQL QLQRQRQL

QLQRQLQR QLQRQLQR QLQRQLQR (1)

(Here and below we will keep the "+ h.c." implicit.) Notice that the last three are of
the LRLR form and involve i2 in the contraction of the SU(2)L and SU(2)R
indices. These three operators must have a dynamical origin at some higher scale.
We take all these operators to be SU(2)LU(1)Y invariant, and four of them may or
may not be SU(2)R violating depending on whether there is a 3 inserted in the


contraction of SU(2)R indices; the fourth and fifth operators vanish when the 3 is
inserted.
We will pursue the idea that it is the sixth operator which is responsible for the t
and b masses. We will suppose that the t and b masses correspond to the mass terms
ULUR and DLDR respectively. Then the sixth operator can feed mass down to the t
and b with masses described by ULUR and DLDR . In particular the operator
QLDRQLUR , which contains the term ULDRDLUR , feeds mass from b to t, while the
operator QLURQLDR feeds mass from t to b. The coefficients of these two operators
at 1 TeV must be roughly in the ratio of mt to mb.
Fig. 1 illustrates the form of the diagrams
which produce the enhancement of the t mass.
This now looks promising since the t mass
operator QLDRQLUR , and all the other enhanced
operators, are such that they do not produce a
direct contribution to t b mass-splitting. By a
direct contribution we mean a diagram with only
one insertion of the operator and with only the t Fig. 1. Enhancement of the t mass. The
fermion in the loop is the b ' and the
and b masses involved. Operators which would
gauge boson is the X.
directly contribute to the splitting, for example
QLURURQL , are not expected to be enhanced by the X interaction. (The factors
QLUR and QRQL are not U(1)X singlets and do not have a large positive anomalous
dimension.) In this way the t b mass splitting may be less than the tb mass
splitting.
The basic reason for why this picture can work is that the t mass is still fairly
small compared to the TeV mass expected for the t and b . The operator responsible
for the t mass is thus small in the sense that effects relying on multiple insertions of
the operator are suppressed. In fact to produce a contribution to there must be at
least four insertions of the operator. For example a t b mass splitting would require
two insertions, and goes like the mass-splitting squared. There are other dangerous
operators, such as (Q
R3QR)2, which can cause a direct contribution to , but to
produce this operator requires three loops and four insertions of the t mass operator.
Notice that the U(1)X interactions are isospin conserving, and thus the typically
large contributions to the (Q
R3QR)2 operator found in ETC theories are avoided.
The only remaining question involves the first two operators in (1); two insertions of
the isospin violating versions of these operators can produce a contribution to
(Q
R3QR)2. The isospin violation in the first two operators can in turn be induced
by two insertions of the t mass operator. We will say more about a possible origin of
the first two operators below.
There has been a detailed analysis4) of a contribution to which may be carried
over to our situation. These authors consider the effects of a single isospin-violating
4-fermion operator which feeds mass down to the t quark from a massive
technifermion (the analog of our t mass operator). They then find the new
contribution to when ignoring all other possible isospin-breaking operators (which
would be expected in a conventional ETC theory). Although nonnegligible, this
correction is typically within the experimental bounds.


3. The t and b masses

To understand electroweak symmetry breaking
we must return to the question of how the t and b
receive their mass. Here another ingredient seems G
to be necessary. The first two operators in (1),
QLQRQRQL and QLQRQRQL , must exist in the
theory with a certain sign and with sufficient
strength. The sign must be such as to resist the
formation of the QLQR or QLQR masses. The
situation is then reminiscent of the Schwinger-
Dyson analysis of a gauge theory with a constant
coupling in the presence of a 4-fermion
interaction with coupling G.5)6) In that work the
generation of mass in a channel attractive with Fig. 2. Phase diagram for 4-fermion
respect to the gauge interaction was studied. The coupling G and gauge coupling .
phase diagram in Fig. 2 was obtained such that chiral symmetry breaking takes place
to the right of the line, with the ultraviolet cutoff going to infinity as the line is
approached. Of interest for us is that no dynamical mass formation was found to
occur for G < -Gcritical/4,6) even for a gauge coupling larger than the usual critical
coupling necessary for chiral symmetry breaking. Although this result was derived in
the ladder approximation, it suggests that there is a mechanism for preventing the
QLQR and QLQR masses from forming in our case.
We have mentioned that the X boson receives a mass via a mechanism to be
described in the next section. Now consider the effective theory on energy scales
below the X mass. Integrating out the effects of the X boson will add to and modify
the set of dominant effective 4-fermion operators. In particular the operators
QLQRQRQL and QLQRQRQL will be generated in addition to new contributions to
the two repulsive operators mentioned above. In our situation where the X coupling
is large (e.g. X > critical), effects to all order in X would have to be considered. It is
safe to say that the resulting size and sign of the coefficients of the effective
QLQRQRQL and QLQRQRQL operators is unknown.
In the effective theory below a TeV the Schwinger-Dyson equation will be
dominated by the effective 4-fermion operators; in this sense the strong interaction
effects have been reduced to the values of just a few numbers. Whether or not mass
forms depends on whether these coefficients have the right sign and are above some
critical value. We assume that this is the case, and that the only channel in which the
dynamical mass forms is QLQR . There is no reason for all the effective 4-fermion
interactions in the theory to respect a discrete symmetry under which underlined
fields are interchanged with non-underlined fields. Then the QLQR mass may be
naturally preferred over the QLQR mass. Since we do not expect the 4-fermion
interactions to arise very close to some critical value, we do not expect any significant
hierarchy between the scale of the operators and the resulting mass. In this situation
higher dimension operators may also be playing a role, e.g. QLQRQRQLQLQRQRQL ,
and such operators with the appropriate sign may also explain why the QLQR mass
forms, but not the QLQR mass. Such terms are analogous to the quartic terms in a


potential for two scalar fields, which can cause a nonvanishing vacuum expectation
value to develop for one of the fields but not the other.

4. The family gauge symmetry

We now consider the theory at the higher scale where U(1)X becomes embedded
into a larger family gauge symmetry. A minimal choice for this family symmetry
turns out to be U(1)VSU(2)V , so that the complete gauge symmetry above is

U(1)VSU(2)VSU(3)CSU(2)LU(1)Y (2)

The complete fermion content consists of the two sets of fermions (Q, L) and (Q , L ).
Each set is a standard family of quarks and leptons, but now each such "quark" Q or
Q and "lepton" L or L carries an SU(2)V index. (Q, L) transform as (+, 2) under
U(1)VSU(2)V while (Q, L ) transform as (-, 2 ). U(1)X is a combination of U(1)V and
the 3 piece of SU(2)V such that the X charges are +1 for (Q1, L1) (the subscript is the
SU(2)V index), -1 for (Q 1 , L 1) and zero for the subscript 2 fields. In this way the X
boson couples to the 3rd and 4th families, but not to the 1st or 2nd families. A right-
handed neutrino Majorana mass NRNR is invariant under the above gauge symmetry.
We therefore expect that right-handed neutrinos do not exist in the theory at scale ,
since it is natural for them to have a much larger mass. The dynamical origin for the
breakdown of U(1)VSU(2)V at scale is not something we will need to speculate
upon here.
We can expect that the physics at some even higher scale M will generate a full
set of operators invariant under the gauge symmetry in (2). We will refer to these
operators as M-operators. For scales between and M we assume that the SU(2)V
coupling and perhaps also the U(1)V coupling are strong (perhaps stuck at some
infrared fixed point), so that it is these interactions which cause the bulk of the
anomalous scaling of 4-fermion operators. A set of operators we would expect to be
enhanced and perhaps relevant at the lower scale is the same six operators we listed
in (1), except that now each field carries an SU(2)V index contracted within each
Lorentz-scalar factor. These six operators contain the previous set, i.e. when each
SU(2)V index is equal to 1.
The first two, QLQRQRQL and QLQRQRQL , serve the useful purpose, if they have
the appropriate sign, of resisting the formation of U(1)VSU(2)V singlet masses such
as QLQR and QLQR . We have already seen the role they can play in helping to
generate the fourth family mass. In fact these operators would be generated with the
right sign by an U(1)A interaction under which the Q and Q have equal and opposite
axial charges. The mass of the U(1)A boson could be of order M, or as low as , or
somewhere in between. One attractive feature of the U(1)A is that it is an isospin
symmetric interaction, meaning that this contribution to the QLQRQRQL and
QLQRQRQL operators is isospin symmetric. We will leave the existence of the U(1)X
as an open question for now. The sixth operator, responsible for the t mass, can have
its dynamical origin in the M-scale physics, and it is here that SU(2)R symmetry
breaking must be realized.
The gauge group and fermion content we have described would have a number of
global symmetries leading to unwanted Goldstone bosons, if it were not for other M-


operators. It turns out that the following two-quark-two-lepton M-operators are
sufficient to break all the global symmetries.

QLURLLER QLURLLER

QLURLLER QLURLLER (3)

Their U(1)VSU(2)V structure allows us to assume that these are also enhanced
operators. We will see below that the last two operators need to be smaller than the
first two (and incidently the last two do not respect the possible U(1)A symmetry).
Note that there are no purely leptonic operators of the LRLR form because of the
absence of the right-handed neutrinos.
U(1)VSU(2)V breaks at scale and we may expect new contributions to the set
of effective operators induced by the -scale dynamics. The new operators need not
respect U(1)VSU(2)V , and we will refer to them as -operators. In particular an
important role in the generation of quark masses will be played by the -operators:

QLQRQLQR QLQRQLQR

QLQRQLQR QLQRQLQR (4)

These operators break U(1)V , but they can still respect SU(2)V since abQaQb is an
SU(2)V-singlet. SU(2)V dynamics plays some role in their formation, and thus we
might expect the SU(2)V-singlet operators to dominate (in the sense of most attractive
channel). We will assume this to be true, keeping in mind that SU(2)V-breaking will
become clear on lower scales due to renormalization effects induced by U(1)X.
Our -operators break a continuous symmetry and thus we would like to
determine the analog of the pion decay constant f. The first question is what are the
coefficients of the -operators? The scale may be defined as the scale at which the
underlying degrees of freedom start becoming apparent. For a 4-fermion operator
generated by an explicit gauge exchange, its coefficient is g2/2; here is the mass
of the gauge boson and g2 4 for gauge interactions strong enough to break chiral
symmetries. We will assume a similar result for the dynamically generated -
operators, and take the generic size of a -operator to be 4/2. This is consistent
with a unitarity upper bound of 8/2 on the size of a 4-fermion operator.7)
We use this crude estimate to illustrate the following point. Roughly speaking f is
determined by some 3-loop diagram involving a -operator "vertex" and its hermitian
conjugate, in analogy with the derivation of the Pagels-Stokar formula. By using
4/2 as the size of the -operators, assigning a factor of 1/(4)2 to each loop, and by
cutting off the integrations at we find


f
(4)2 (5)

The point is that the resulting f may be a few orders of magnitude less than the scale
at which the -operator originates.
From this we see that our -operators contribute very little to the gauge boson
masses in SU(2)VU(1)V/U(1)X, which are of order . But now we note that the -
operators also break U(1)X, and thus our estimate of f becomes an estimate of the


ratio MX/gX. This then is our mechanism for the breaking of U(1)X, via physics
occurring at the higher scale . We will assume that MX/gX is of order a TeV.

5. Quark mass matrices

Before we can discuss the remaining quark masses, we must first discuss the
mass. We note that the E LEREREL and ELEREREL M-operators may be smaller
than analogous operators in the quark sector, which were discussed above. If nothing
else, the contribution of the QCD corrections to the scaling of the quark operators can
be significant if acting over a sufficiently large momentum range. Thus the E LER or
E LER masses may be resisted less than the QLQR or QLQR masses. Because of this
different dynamics, we shall feel free to assume that the mass corresponds to a
dynamical E LER mass. In addition the absence of right-handed neutrinos means that
the lepton masses will likely contribute to , and thus the mass should be
sufficiently smaller than the t and b masses to avoid a problem.
We now consider mass matrices, with the following elements.


QL2QR2 QL2QR2 QL2QR1 QL2QR1
QL2QR2 QL2QR2 QL2QR1 QL2QR1 (6)
QL1QR2 QL1QR2 QL1QR1 QL1QR1
QL1QR2 QL1QR2 QL1QR1 QL1QR1

The 1 and 2 subscripts are SU(2)V indices. With the operators discussed so far the
up-type mass matrix takes the following form.

0 F2 0 0
G2 E D 0 (7)
0 C B F1
0 0 G1 A

The entry A comes directly from the dynamical Q L1QR1 mass. For the other entries
we give the corresponding M-operator or -operator responsible.

B : UL1DR1DL1UR1

C : UL1DR1DL1UR2

D : UL2DR1DL1UR1

E : UL2DR1DL1UR2

F1 : EL1ER1UL1UR1 (8)

F2 : EL1ER1UL2UR2

G1 : EL1ER1UL1UR1

G2 : EL1ER1UL2UR2


All these are pieces of SU(2)V-singlet operators. C and D are -operators and are
expected to be similar in size, while B, F, and G are M-operators. The F and G
operators are feeding down mass from while the rest are feeding down mass from b .
We expect B to be enhanced relative to (F, G), both because of QCD renormalization
effects and because of our expectation that the b mass is larger than the mass. And
as we have said, G is small compared to F.
We may assume that the E operator does not arise as a
-operator in the same way as the C and D operators, since
it is more strongly resisted by the U(1)V force. (In fact the Fig. 3. A circle (square) is a
C and D operators are not resisted at all by U(1)V at one -operator (M-operator).
loop.) But the E operator will still arise from a two-loop
diagram involving both the QLDRQLUR and QLDRQLUR -operators and the
QLDRQLUR M-operator, as shown in Fig. 3. Then E is naturally suppressed.
As far as the resulting masses are concerned, we need the size of the operators
when renormalized at scale . Strong U(1)X effects, and to a lesser extent QCD
effects, will cause the various operators to scale differently as they are run down from
to a TeV. Generally we expect that the most enhanced operators will be those with
the most subscript 1 fields appearing in U(1)X-invariant and Lorentz-scalar
combinations. The result is that B and (F1, G1) are enhanced more than the sets (F2,
G2) and (C, D), while E is enhanced the least or not at all. A typical relative
enhancement factor is / (1 TeV). Putting all this together we find a natural
hierarchical pattern of masses emerging.
The down-type matrix will receive contributions from loop diagrams involving the
operators we have already discussed. The right-handed neutrinos are not in the
theory at the scale and thus we may ignore diagrams involving internal right-
handed neutrinos and -operators. The down-type matrix then takes the form


H 0 I 0
0 E D 0 (9)
I C B 0
0 0 0 A

The entries correspond to the following.

B : DL1UR1U L1DR1

C : DL1UR1U L1DR2

D : DL2UR1U L1DR1

E : DL2UR1UL1DR2 (10)

H : DL2UR1UL1DR2

I = E R1EL1(DL1DR2 - DL2DR1)

The B M-operator and the C and D -operators are the SU(2)R-transformed partners
of the B, C, and D operators. The former three must all be small compared to the
latter three, and thus the -operators must also reflect the SU(2)R breaking. C and


D may in fact be generated by the QLDRQLUR and QLDRQLUR -
operators, which contain C and D, along with the third M-operator
in (1). This is shown in Fig. 4. The same figure describes the Fig. 4.
generation of the I operator from the QLDRQLUR -operator and
the L LERQLUR M-operator.
We have used the E label in both the up and down matrices because although the
two operators are different, they are generated the same way and are expected to be of
similar size. The H operator also arises in a similar way, except that the two-loop
diagram involves the other two -operators, QLQRQLQR and QLQRQLQR . (The
SU(2)R structure of the these operators does not matter here.) The interesting thing
about the H operator is that it basically feeds mass from t to d. It has a partner
which would feed mass from b to u, but since mb mt we have neglected this effect in
the up-type matrix. Noting that the E operator is feeding mass from t to s, and the
fact that the H and E operators are expected to be of similar size, leads to the not
completely ridiculous relation
md/mt ms/mt (11)

It is amusing that the d mass is connected with the t mass, while the u mass is
connected to the mass. Note finally that the c quark is only receiving a ms-size
contribution from the E operator, leaving the bulk of the c mass to be produced via
mixing with the t.
Of particular interest for the suppression of flavor changing neutral currents is the
possible suppression of the (12) and (21) entries in the down-type mass matrix. They
do in fact receive contributions from the through a loop involving two M-operators
where one of the M-operators is either a B or a G operator, both of which are
suppressed. A loop involving two -operators where one operator is different from
those discussed above, either L LERQLUR or LLERQLUR , could also contribute. We
will assume for now that all these effects are small. Finally the (14) and (41) entries
can be generated by the same -operators which were involved in generating the
operator H; but the position of these entries makes them quite unimportant and we
set them to zero.

6. Quark mixings

We would now like to see if the model allows for a realistic quark mass spectrum
and Kobayashi-Maskawa mixing matrix. The model has produced mass matrices of a
certain form, and so it is of interest to see how constraining these forms are. We will
choose sample values for the elements of the mass matrices, but we are of course not
claiming to show that the model actually produces these values. We would like the
resulting quark masses to be reasonable when renormalized at a TeV. We choose the
up-type matrix to be the following matrix, which gives the following set of masses:
(0.022, 0.74, 160, 1000) GeV.

0 .16 0 0
-.01 .1 10 0 (12)
0 -10 160 16
0 0 -1 1000


The (33) entry is fixed by the t mass. The (22) entry is the same as in the down-
type matrix and is thus basically fixed by the s mass. Then the (23) entry is fixed in
order to obtain the c mass through mixing with the t. The (12) entry is determined if
we want the bulk of Cabibbo mixing to occur in the up sector, and finally the (21)
entry is determined by the u mass. Here we see the small effect of the G2 operator.
The (34) and (43) entries coming from F1 and G1 are of little consequence, and we
have set them equal to 100 times the F2 and G2 entries. The resulting left and right-
handed mixing matrices Lu and Ru are the following.

.98 .22 0 0 1.0 -.013 0 0
-.22 .97 .062 0 .013 1.0 -.062 -.0002 (13)
.013 -.061 1.0 .016 .0008 .062 1.0 .0016
-.0002 .0010 -.016 1.0 0 0 -.0016 1.0

For the down-type matrix we take


.005 0 -.015 0
0 .1 .07 0 (14)
.015 -.07 3 0
0 0 0 1000

Here the physical masses are basically the diagonal components of this matrix. The
(23) and (32) entries are determined such that we obtain the correct Vcb, while the
(13) and (31) entries are determined by Vub. For the left-handed mixing matrix Ld we
find the following.

1.0 -.0035 -.0050 0
.0036 1.0 .023 0 (15)
.0049 -.023 1.0 0
0 0 0 1.0

Rd is the same except that the off-diagonal entries have opposite sign. The main
interest here is that the mixing between the d and s quarks is small, which provides a
useful suppression of K-K mixing induced by the scale physics. In addition we
have been able to produce a realistic Kobayashi-Maskawa mixing matrix, V LuTLd,
as follows.

.98 -.22 .0030 -.0002
.22 .97 -.040 .0010 (16)
.0051 .039 1.0 -.016
0 -.0004 .016 1.0

Our conclusion is that if the operators we considered are actually the dominant ones,
then the quark mixing matrices are fairly well determined. We also see that the basic
hierarchical patterns which emerged in the last section are consistent with the sample
mass matrices displayed here.

7. Lepton masses

For completeness we briefly consider lepton masses. We have noted that the
E L1ER1mass ( mass) can play an important role in feeding mass into the quark


sector. There are three operators of the RLLR form which can feed the mass down
to the other three charged leptons.

E R1EL1EL1ER1 E R1EL1EL2ER2 E R1EL1EL2ER2 (17)

The existence of the first two operators is implied through one-loop effects involving
the M-operators in (3). The first is enhanced through U(1)X-induced anomalous
scaling relative to the other two. The third operator can actually be generated by an
explicit gauge interaction, namely a broken SU(2)V generator. But effects very similar
to those generating the second operator can also contribute to the third, and so it
appears that the third operator must be larger than the second. Thus the three
operators listed are generating the , e, and  masses respectively. As for the
neutrinos, a Majorana N L1NL1 mass would be a mass. If this was somehow
dynamically generated then we would be left with three light neutrinos (NL1, NL2,
NL2) (, , e) which do not receive mass via any 4-fermion operator.

8. Z-X mixing

The most interesting observable implications of the model are those associated
with the X boson and its mixing with the Z, which occurs via a t loop. As discussed
previously,8) this causes the Z couplings to the third family to be shifted. In light of
recent data, this is interesting if the shift in the Zbb vertex roughly cancels the
standard model correction to this vertex, which produces a -2% correction. The
latter correction also involves a t loop along with an electroweak gauge boson. Thus
for the two effects to roughly cancel we find three conditions. 1) The X should have
axial couplings to the t to produce a mass mixing between the Z and the X. 2) The t
and the b should have the same sign axial X coupling to produce an effect of the right
sign. 3) gX/MX should be similar to the coupling-to-mass ratio of the electroweak
gauge bosons to produce an effect of the right magnitude.
A similar effect has been noted in the Z t X t Z
context of more conventional ETC models.9)
Here a diagonal ETC gauge boson mixes with
the Z via a technifermion loop. Since the t
technifermions must preserve isospin to good Z Z
approximation, the ETC gauge boson must
have isospin breaking couplings to the
technifermions for this mixing to occur. It is t
not surprising then10) that such a model leads W W
to a large contribution to , via diagrams
involving technifermion loops and the ETC b
gauge boson. In our case the X boson has Fig. 5. Some contributions to .
isospin conserving couplings, which implies
that the analogous diagrams involving the heavy, approximately degenerate, 4th-
family quarks cause little problem. Instead the more important diagrams which
contribute to are those involving the t, shown in Fig. 5.
If the contribution to the Zbb vertex is arranged to be the same in both cases, it
may be seen that these latter contributions to are suppressed relative to the ETC


contributions by roughly the ratio of the t loop to the techniquark loop. We may
write this ratio as f 2
t /f2 where ft determines the t contribution to the Z mass,



M 2 2
Z = e2 (f2+f ) (18)
4s2c2 t

An NJL-type estimate of ft has been given elsewhere.11)

(1TeV)2
f 2 2
t 3 m ln (19)
82 t m2t

We may also follow that reference11) to estimate our contribution to , where the
three diagrams in Fig. 5 correspond to the three terms in parenthesis:

g 2f 4
(8 + 4 - 1) X t
MX f 2 (20)
0.003 for MX/gX = 1 TeV

Such a shift is not yet excluded by the data.
Let us return to the Zbb vertex. We have seen in the model that the third family
quarks are to good approximation composed of the fields QL1 and QR1, which have
opposite X charges. Thus the t and the b have equal axial couplings to the X, which
means that the first two conditions for producing a desirable correction in the Zbb
vertex are satisfied. As for the last condition we need the ratio gX/MX. We have seen
that the t and b masses imply the bulk of the W and Z masses; they also imply a
contribution to MX/gX equal to 4csMZ/e.8) But we have also noted that there may
be another contribution to the X mass from -operators, which then implies that

g 2
X < G (21)
M
X 2 2

We will see below that MX/gX 1 TeV produces a correction to the Zbb vertex of the
right magnitude. This value for MX/gX appears reasonable, since we don't expect a
large hierarchy between the X mass and the (t ,b ) masses.
Our discussion of quark masses led us to require that the heavy be composed of
the fields LL and LR. This leaves to be composed of LL and LR, which implies
purely vector X couplings to the . This in fact is necessary to avoid upsetting the
constraints on the Z partial width to the .8) In summary the model implies that the
X couples to the following third family current,

J
X (22)
 = t(L -R )t + b(L -R )b + (L +R ) + L 

where R,L (1  5)/2.
The Z couplings to the third family are shifted by an amount g X
Z ZJ where

m2
g Z
Z = - r gX (23)
m2X


r is the ratio of the ZX mixing diagram involving a t loop, and the Z mass diagram
involving t and b loops. That may be written as

g 2
r X ft
e f2 (24)
4 c s

The final result is

2 f 2
g 2 gX t
Z - 4 c s
e MZ MX f (25)

-0.003 for MX/gX = 1 TeV

We note that our estimate of r here is larger than in our previous work,8) but this may
be compensated by the new contributions to MX/gX. The implications for
electroweak corrections may be extracted from the previous work in which
gZ = -0.0028.

9. Conclusion

We began by showing how anomalous scaling could enhance the 4-fermion
operators responsible for the t mass more than other dangerous isospin violating
operators. There are two questions related to the new strong interaction responsible
for the anomalous scaling. How does this gauge symmetry break close to a TeV, and
how is this breaking associated with electroweak symmetry breaking? We have tried
to present plausible answers to both these questions. At a higher scale this new
strong interaction becomes embedded into a larger family gauge symmetry which also
couples to the two light families. The quark mass matrix can then be related to
various 4-fermion operators enhanced by varying amounts due to anomalous scaling,
and the structure of these operators leads to natural mass hierarchies and mixings.
The 4-family model we have presented illustrates these points well, but it remains to
be seen whether the various dynamical assumptions made are correct.
Most of the operators we have discussed require a dynamical origin, and in
particular the SU(2)R breaking manifest in these operators could have a dynamical
origin in some initially SU(2)R-symmetric high-energy theory. We discuss this issue in
the Appendix. We have noted that our model has no new unbroken gauge symmetry.
But a less minimal version of the model could have more families with a larger family
gauge symmetry. Most of the discussion of this paper would still apply, except that
there would remain in the end another sector of fermions involving a new unbroken
gauge symmetry.
Lastly it is conceivable that some of the 4-fermion operators we have discussed
could remain relevant over a very large energy range, in which case our scale M could
refer to some kind of unification scale or even Planck mass. What is interesting in this
regard is that the SU(3)CSU(2)LU(1) gauge symmetry could remain as is, up to
scale M, since the 4-fermion operators can eliminate what would otherwise have been
Goldstone bosons. Our new strong interactions will of course significantly affect the


running of the standard model gauge couplings. But given that the fermions come in
standard model families, the relative running of the three couplings is not affected at
leading order in these couplings, to all orders in the new strong couplings. Thus the
basic tendency for the three standard model couplings to become more equal at some
large scale remains.


Appendix


We would like to consider the question of a dynamical breakdown of SU(2)R. How
likely is it that some dynamically generated fermion 4-point function takes the
following form in SU(2)LSU(2)R space?


Qa c b d 0 0
LQRQLQR 0 1 (26)
-1 0 ab 1 0 cd

To illustrate the main point we can consider a toy scalar field potential V, effectively
replacing the 4-point function by a local field abcd(x). Note that the first two indices
on the are SU(2)L and the last two are SU(2)R. In constructing the terms in V we
note that the ith index on a field must be contracted with the ith index of the
conjugate field, because the different indices represent different flavors of the
underlying 4-fermion operator.

V = -abcdabcd
+ A(abcdabcd)2 + Babcdabefbhefghcd
+ Cacbdaebfgehfgchd + Dacdbaefbgefhgcdh (27)
+ Eabcdabcefghefghd + Fabdcabecfgehfgdh
+ Gadbcaebcfeghfdgh + Hdabceabcefghdfgh

We may assume G = E and H = F since other than the weak interactions, the strong
interactions at the scale are left-right symmetric. We then seek conditions on the
coefficients which ensure that abcd of the form (26) is a global minimum. We also
note that C D and E F correspond to either of the interchanges QR Q R or QL
Q L . This is not a symmetry of the interactions but for now we set C = D and E =
F for simplicity; we can obtain similar results in the more general case. We also take
the coefficients to be real. Then abcd of the form in (26) will be a global minimum
of the potential if the following conditions are satisfied.

E < 0, -E < C < -2E, A + B > -3E - C, B <
-E/2 (28)

There is still an unwanted degeneracy at the minimum, as abcd ab cd is a global
minimum for any belonging to

1 0
, 1 0 , 0 1 , 0 1 (29)
0 1 0 -1 1 0 -1 0

and for any belonging to

1
0 , 0 0 , 0 1 , 0 0 (30)
0 0 0 1 0 0 1 0


or vice versa. But we have so far ignored the effects of the weak interactions. Their
effects at leading loop order will correspond to additional quadratic terms in our
effective potential.

V 2 i i 2 3 3
weak gLeafbabcdefcd + gY ecfdabcdabef (31)

These terms are sufficient to pick out the following two SU(2)LU(1)Y invariant
possibilities as global minima.


0 0 0 1
abcd 0 1 or 0 1 (32)
-1 0 ab 1 0 cd -1 0 ab 0 0 cd

These two possibilities would result in either a large top or bottom mass respectively.
This last degeneracy is also broken weakly, for example by a quark loop with
electroweak gauge boson exchange.


Acknowledgements


I would like to thank the Yukawa Institute for Theoretical Physics and my host,
T. Kugo, for their hospitality and support during my three month stay. This research
was also supported in part by the Natural Sciences and Engineering Research Council
of Canada.



References

1) B. Holdom, Phys. Rev D24 (1981) 1441; B. Holdom, Phys. Lett. B150 (1985) 301; K. Yamawaki,
M. Bando, and K. Matumoto, Phys. Rev. Lett. 56 (1986) 1335; T. Appelquist, D. Karabali, and
L.C.R. Wijewardhana, Phys. Rev.Lett. 57 (1986) 957; T. Appelquist and L.C.R. Wijewardhana,
Phys. Rev D35 (1987) 774; T. Appelquist and L.C.R. Wijewardhana, Phys. Rev D36 (1987) 568.
2) W. Bardeen, C.N. Leung, S.T. Love, Nucl. Phys. B273 (1986) 647.
3) S. Chivukula, Phys. Rev. Lett. 61 (1988) 2657.
4) Y. Sumino, talk presented at the Yukawa International Seminar '95, August 1995, Kyoto.
5) T. Appelquist, M. Einhorn, T. Takeuchi, and L.C.R. Wijewardhana, Phys. Lett. 220B, 223 (1989);
V.A. Miransky and K. Yamawaki,Mod. Phys. Lett. A4 (1989) 129; K. Matumoto, Prog. Theor.
Phys. Lett. 81 (1989) 277.
6) W. Bardeen, C.N. Leung, S.T. Love, Nucl. Phys. B323 (1989) 493.
7) T.Asaka , N.Maekawa , T.Moroi , Y.Shobuda , and Y.Sumino, talk given at 5th Workshop on
Japan Linear Collider, Tsukuba, Japan, Feb. 1995, .
8) B. Holdom, Phys. Lett. B339 (1994) 114; B351 (1995) 279.
9) Guo-Hong Wu, Phys.Rev.Lett.74 (1995) 4137; K. Hagiwara, N. Kitazawa, KEK-TH-433, Apr 1995,
.
10) T. Yoshikawa, (Hiroshima U.) HUPD-9514, Jun 1995,  .
11) R. Sekhar Chivukula, B.A. Dobrescu, J. Terning, Phys.Lett. B353 (1995) 289.



