June 1995



NEW THIRD-FAMILY FLAVOR PHYSICS
AND THE TOP MASS


B. HOLDOM*

Department of Physics, University of Toronto

Toronto , Ontario, CANADA M5S 1 A7

ABSTRACT

A new massive gauge boson (X) coupling to the third family produces a tantalizing
pattern of deviations away from the standard model. These include increasing
b/h and decreasing the s(MZ) extracted from h/ . We indicate how the X
boson may be related to a dynamical origin of the top mass.


When you ask those working on supersymmetric theories why they work on
supersymmetric theories, they often refer to the predicted (M ) using sin2 as
s Z W
input. The minimal supersymmetric model gives (M ) between 0.125 and 0.140
s Z
for superpartner masses between 0.1 and 1 TeV, when possible GUT threshold
effects are ignored.1 This is to be compared to the world average: (M ) = 0.117
s Z
 .005. The cleanest low-energy determinations of (M ) are on the low side of
s Z
the world average. Deep inelastic scattering and lattice calculations of quarkonia
spectra yield 0.112  .005 and 0.115  .002 respectively. When compared to these
numbers, the supersymmetry prediction is not completely compelling. The
theoretical errors associated with extracted from the hadronic width and from
s
jet studies are less clear. (There is also a recent extraction of from production
s
using sum rules, 0.109  .001, 2 but the true theoretical error is open to question
 26 Jun 1995 here as well.)
There is also the LEP determination, and in particular the clean determination
provided by a measurement of R / . This gives (M ) = 0.125  .005.
h s Z
When averaged with the other determinations, it helps to move the world average
higher. Instead of doing this, it is of interest to consider R along with another
quantity measured at LEP, R / . Here there is a well known 2%
b b h
discrepancy with the standard model prediction, and I attribute this to the
following positive shift in .
b

/ = 0.028  0.012 (1)
b b

The point is that if this signals new physics in the Zbb vertex, then this same
physics will increase , and thus R . To compensate, the extracted from R
h s


* Invited talk given at the Top Quark Workshop, Iowa State University, May 2526, 1995.


must decrease. A 2% increase in corresponds to the following shift in .3
b s

(M ) -0.013 (2)
s Z

This would bring the determined from R in line with the other values
s
mentioned above. Another way to see this is shown in the following figure in
which R is plotted versus R . (I use data reported in winter 1995 conferences.)
b
The standard model prediction with (M ) = 0.112 is located between 3 and 4
s Z
away from the central value. Increasing (M ) moves the standard model
s Z
prediction horizontally. That may be compared with holding (M ) = 0.112 and
s Z
instead increasing by 2% above the standard model value. This brings about
b
good agreement with the data, thus showing how the R and R measurements
b
both favor a new contribution to .
b


0.230


2
= 1


2
= 4


2
= 9



Rb
0.220
mH
0.112 up 2%
b
1000 300 60
140
mt
200

s = 0.112 0.125


0.210
20.60 20.70 20.80 20.90
R

The question is whether this simple picture is supported by a global fit to all
electroweak data. Until recently many global analyses in the literature did not
allow (M ) to vary along with the Zbb coupling g b. One of the first which did
s Z Z
was provided by Matsumoto.4 When S, T, g b and were all allowed to vary he
Z s
found that central value for floated downward, (M ) = 0.112  .009, in
s s Z


agreement with the above discussion. For a more recent global fit with various
assumptions about new physics see reference 5.
Although none of this constitutes a truly serious problem for the standard
model, we are led to at least consider the possibility of new flavor physics
associated with the third family. Would the latter be all that surprising? Perhaps
not, at least in a dynamical symmetry breaking context. Consider the natural
generation of quark and lepton masses due to strongly interacting gauge theories.
The usual scenario has new fermions receiving mass of order 1 TeV, associated
with the dynamical breakdown of electroweak symmetry. This mass is fed down to
some quark q via new physics characterized by some scale . The point is that the
larger the m , the smaller the ; the mass scale of new interactions is inversely
q
related to the mass of particles it couples to. Thus we should not be surprised if
new flavor physics shows up first with the third family.
What does this new flavor physics look like? It is likely to involve a broken
family gauge symmetry of some sort, and when gauge symmetries break one often
has broken diagonal generators which correspond to massive U(1) gauge fields.
Thus a likely remnant of a broken family symmetry would be a massive gauge
boson coupling to the third family but not lighter families. We will refer to this as
the X boson. 3
We can thus expect the following effect. Due to mixing between the Z and the
X, for example through a top loop, the Z couplings to the b will be shifted.


b
Z t X
(3)



We may deduce some required properties of the X by comparing this contribution
to the standard model correction to the Zbb vertex, which also involves a t inside a
loop. This latter contribution drives down by about 2%, which is precisely the
b
effect not seen in the data. This must then be more or less canceled by the new
physics contribution. We see that a) the ratio g /M must be similar to that of
X X
the electroweak gauge bosons, b) the X should have an axial coupling to the t in
order to produce mass mixing with the Z, and c) t and b can have the same sign
axial coupling to the X. The latter is of interest if X is to originate from a family
symmetry. Since we also expect that the X couples to the and , the Z couplings

to these leptons will also be shifted.
Is it likely that the X couples to the third-family fermions and to no other
fermions? (We are ignoring small mass mixing effects.) It does not seem likely if
one considers gauge anomalies and the fact that the X is emerging from a gauged
family symmetry. This leads us to consider a fourth family and a X coupling to
"third-family number minus fourth-family number". This generator easily emerges


from a family symmetry; it also clearly avoids gauge anomalies.
This fourth family can play a useful role; in fact we will assume that members
of the fourth family develop the required 1 TeV dynamical masses. And we will
require that this dynamics breaks not only the electroweak symmetry, but also the
X boson gauge symmetry. For this to happen the fermion mass eigenstates are not
the same as the states which have vector couplings to the X. In particular, let q
and q be the two quark doublets with equal and opposite vector X charges.
Suppose the condensate which forms is q q
L R + h.c. = 0 . This defines the mass
eigenstates for the fourth family quarks with m m 1 TeV, whose Dirac fields
t b
(t , b ) are each composed of [q , q ].
L R
The result is that the X has axial couplings to (t , b ). This in turn implies that
vacuum polarization graphs involving the t and b will produce mass for both the
Z and the X. And this determines the coupling to mass ratio for the X.3
gX e
= (4)
MX 4csMZ

The third-family quarks (t, b) are then composed of [ q , q ], which implies that
L R
(t, b) also have axial X couplings. For the and on the other hand, we take
them to be composed of [ , ] and [ , ] respectively, which implies that the
L R L R
has vector couplings to the X. The reason for this choice is made clear below. The
result is that the X couples to the following third-family current.

J X (5)
 = t5t + b5b +  + LL

By comparing the Z-X mixing diagram involving the t loop to the Z mass diagram
involving the t and b we find that the Z couplings to the third family are shifted
by an amount g ZJ X where3
Z 

m 2
g t
Z - e (6)
8 c s mq

We may express the various shifts in the Z couplings in terms of the asymmetry
parameters A = 2gfg f/( gf2 + gf2) and the partial decay widths f2 + gf2 .
f v a v a f g v a
For the charged leptons the axial coupling is much larger than the vector coupling.
Thus the observed similarity between e, , and places a strong constraint on

g . This is the reason we have chosen the to have vector couplings to the X
a
boson.
Besides the shifts in the quantities and as we have described, we also have
b s
the shifts in the following table along with the relevant observable. The latter are
chosen 3 specifically to look for universality breaking corrections involving the third
family, and are quite insensitive to possible oblique corrections. In the case of A
there are two such independent observables. Note also that certain systematic
errors cancel in the ratios.


0
A
A 3 P A
FB
and
A 0 e, FB
A A e
FB 4 P


+ +
inv e 
=






e,


The following table shows the results. Our estimate of Z-X mixing produces
shifts in and of the desired magnitude, as discussed above. For A the X
b s
boson produces a large shift; this is not inconsistent with the average of the two
experimental determinations, which in turn are not in good agreement with each
other. For and the experimental shifts are consistent with zero, but they are

also not inconsistent with the X boson.


Measurement X boson

/ +0.028 0.012 +0.021
b b

(M ) -0.014
s Z

+0.32 0.19
A /A
+0.02 0.09 +0.21

/ -0.0140.023 -0.015


/ +0.003 0.004 0.0022 + 0.0015




We also have to consider other corrections to the Z vertex where an X is
exchanged between the two third-family fermions.6 We may write the shifts in the
Z couplings in the following way.

q 2
gZ = constant + O( ) (7)
M 2X

The Z-X mixing contributes to the constant term, while the additional vertex
corrections contribute to the q2 term, where q is the 4-momentum entering the
vertex. These latter corrections are then suppressed, and they are only important
in the case of where they produce the term 0.0015 in the / entry of the

table.
We motivated the existence of new flavor physics from the large size of the top
mass. In fact the existence of an X boson with the properties we have described
originated in a model7 in which the top mass arose in a dynamical context different


from extended technicolor. Such a model must not only explain the top mass, but
also explain why the electroweak breaking condensates, in this case the four family
quarks, preserve isospin symmetry to good approximation. The trick is to keep the
SU(2)U(1) breaking physics of the condensate distinct from the isospin breaking
physics responsible for the t mass. This is where ETC theories run into problems. 8
If some ETC interaction is able to produce an operator of the form UUtt with a
coefficient much larger than the one for DDbb , to produce a large top mass, then
the same interactions are very likely to produce a UUUU operator with a
coefficient much larger than the one for DDDD . These latter operators affect the
size of condensates through the gap equations, and in particular they make it very
difficult to understand why UU DD .
Our model introduces some new ingredients in an attempt to overcome this
problem. One is to replace technicolor with "hypercolor", with the main
distinction being that hypercolor breaks at a TeV via the same condensate which
breaks SU(2) U(1). The X boson is a broken diagonal generator of hypercolor.
The third and fourth families, originally part of hypercolor multiplets, now emerge
as singlets under the unbroken subgroup of hypercolor. The condensate involves
the fourth family as we have already described. If hypercolor is a walking theory
then some four-hyperfermion operators originating at higher scales can be expected
to be strongly enhanced. Among such operators are those which break isospin (but
not of course SU(2)U(1)). And among these operators is one, which we do not
give here, which contains tq q t but not bq q b or q q q q . Because it is strongly
enhanced by hypercolor, when combined with the fourth family condensate it can
generate a large top mass. It does not produce a b mass nor does it contribute to
the Zbb vertex. And as well, the presence of this operator by itself is consistent
with t t b b . Other more dangerous operators may be present, but because
of their different structure they are not enhanced by hypercolor scaling effects
nearly so strongly. One of the main features of this picture is that isospin breaking
originates dynamically, via SU(2) breaking, at a scale of order 1001000 TeV. For
R
more details see references 3, 6, and 7.

Acknowledgements

This research was supported in part by NSERC of Canada.

References
1. J. Bagger, K. Matchev, D. Pierce, Phys.Lett . B348 (1995) 443.
2. M.B. Voloshin, TPI-MINN-95/1-T, .
3. B. Holdom, Phys. Lett. B339 (1994) 114.
4. S. Matsumoto, KEK-TH-418, .
5. P. Bamert, C.P. Burgess and I. Maksymyk, McGill-95/18, .
6. B. Holdom, Phys. Lett. B351 (1995) 279.
7. B. Holdom, Phys. Lett. B336 (1994) 85.
8. S. Chivukula, Phys. Rev. Lett. 61 (1988) 137.



