UTPT-95-01





New third-family flavor physics:
Vertex corrections




B. HOLDOM1


Department of Physics
University of Toronto
Toronto, Ontario
CANADA M5S 1A7





 10 Feb 1995
ABSTRACT


As described previously, a new massive gauge boson (X) coupling only
to the third family produces a tantalizing pattern of deviations away
from the standard model. These include increasing / and
b h
decreasing the (M ) extracted from / . We review the status of
s Z h
these X-Z mixing effects. We then calculate X boson induced vertex
corrections to Z partial decay widths and to tt production in pp
colliders.





1 holdom@utcc.utoronto.ca


The third family, and its place in the standard model, is not as well tested as the first

and second families. In fact there is a growing realization that the precision electroweak

experiments still a leave a large opening for new flavor physics associated with the third

family. There are also reasons for believing that the lightest new nonuniversal gauge

interactions, should they exist, will couple preferentially to the heaviest family. Some

model independent analyses have appeared [1] and these are useful in elucidating the full

range of possible effects. It is also useful to have definite theoretical targets to aim at

when trying to understand the significance of experimental results. (For other examples

of targets see [2].) This note will explore further a minimal extension of the standard

model to include a nonuniversal flavor interaction [3]. A massive U(1) gauge boson

denoted by X is introduced which couples to the third family but not to lighter families.

We do not consider here the effects of the small fermion mass mixing between families; for

some discussion see [3] and [4].

The X boson we consider couples to the following current.

J X (1)
 = t5t + b5b +  + LL

The existence of an X boson was noted first in a dynamical model for the top mass [5],

before its implications for precision experiments were considered.2 The fact that the X

boson has axial couplings to quarks is related to the manner in which the X receives a

mass. It also leads to X-Z mass mixing generated by a top quark loop, and this is what

causes a shift of the Z couplings to the third family [3]. The fact that the X boson has

vector rather than axial couplings to the is dictated by the observed universality in the

Z partial widths to e, , and . The latter constrains the Z axial coupling much more

than the vector coupling since g g + g g and g 0.07g . The issue of a small
A A V V V A

universality breaking correction to will be a main topic of this paper.


It is often speculated that the third family, and the top quark in particular, is

somehow wrapped up in the physics of electroweak symmetry breaking. In our extension

the new third family flavor physics is tied up with electroweak symmetry breaking; the

source of the mass of the X boson is the same as for the W and Z masses. All these boson

masses arise due to their axial couplings to fermions, and this gives us the relation [3]


gX e
= (2)
MX 4csMZ

2 There are no gauge anomalies in the model of [5]. The point is that the X boson also couples to a fourth
family, and the mass generation mechanism for the fourth family determines the remaining lighter third
family fields and their X charges.


2


where c = cos and s = sin . With this relation the corrections induced by the X
W W

boson depend only weakly on the unknown X mass.

We are motivated in our study in the following ways.
 It seems that the standard model is far from confirmed by present data, since a
new gauge boson with a coupling-to-mass ratio the same as the known weak bosons

can still be added without conflict with the "precision data".
 An extension of the standard model with a new nonuniversal gauge interaction is
completely at odds with the GUT/supersymmetry orthodoxy; this should provide

incentive understand the effects of such an extension, if only to rule it out.
 The present precision data [6] actually favors this extension over the non-extended
standard model.3

The main effects of the X boson arise through the mixing with the Z, which causes a

shift of the Z couplings to the third family. The relative sizes of the various shifts are

determined by the X couplings and only the overall magnitude depends on the estimate of

the X-Z mixing in [3]. All the X boson induced shifts quoted below can be simply scaled

if a different estimate of the X-Z mixing is used. Perhaps the result of most interest is

/ = 0.021 to be compared to the experimental value of 0.0200.009. To obtain the
b b

latter we attribute any experimental deviation from the standard model value of / to
b h

a shift in . The increase in increases the predicted / and thus decreases the
b b h

(M ) extracted from / by an amount (M ) = -0.014. is unaffected due to
s Z h s Z h

the canceling shifts in and (M ), and as noted in [3], the result is that the extracted
b s Z

(M ) is brought into line with low energy measurements. In this way a single new
s Z

physics contribution to the Zbb vertex resolves the two features of the present data

potentially embarrassing to the standard model [3][7]. Whether or not the s
measurement is embarrassing is controversial, but see [8].

For the we find that X boson substantially increases the asymmetry parameter,

A /A = 0.21. Here the jury is still out since the two independent measurements of

A /A , from the forward-backward asymmetries (assuming e- universality) and from
e
the tau polarization studies, give results not in good agreement; A /A = 0.570.26 and

0.060.11 respectively. For the neutrino we obtained / = -0.015, which is not in


conflict with -0.0140.023. The latter is inferred from the measured ratio inv/
assuming three light neutrinos. Note that the experimental values for all these



3 Of course the larger theory in which this extension is embedded may have additional effects. But note
that the larger theory proposed in [5] is not a standard ETC theory.


3


universality breaking shifts are obtained from observables which are quite insensitive to

possible oblique (universal) corrections. The X boson also produces a slight shift in A ,
b
but here there is no well-measured observable insensitive to oblique corrections.

For this and other observables it is necessary to know the oblique corrections, and this

in turn requires a global fit. Without actually performing a global fit it is possible to see

that the impact of an X boson on other observables will be small. For example the shift

in A /A = -0.0054 from the X boson can be compared to the present 4% error in the
b b
observable Ab ( A ). The shift in the leptonic widths (mostly ) implies a -0.09%
FB b
shift in the total width to be compared with its present 0.15% error. We have seen
Z

that (M ) should be allowed to vary in a global fit, which will imply that any shift in
s Z h

is minimal. Thus the total shift in , and the parameter extracted from , will be
Z Z

small. A global analysis has been performed in [9] which accounts for oblique as well as

nonoblique corrections to Zbb couplings. The resulting constraints, not surprisingly, are

completely compatible with the X boson shifts g b = -g b = -0.0038 (here gb is
L R L, R

defined as in [9]). Note that the global fit in [9] did not allow (M ) to vary, which was
s Z

instead fixed at 0.012.4

Lastly we may ask how the values of sin2 extracted from various observables should
W

be corrected to account for the existence of the X boson. For the observable Ab =
FB

(3/4)A A most of the dependence on sin2 comes from A , and thus the presently
b e W e
extracted value of sin2 needs only be reduced by 0.0001 to obtain the true value. The
W

sin2 extracted from A = (3/4)A A must be increased by 0.0019. This brings the
W FB e
sin2 from A (average over three leptons) even closer to the sin2 from Ab . The
W FB W FB

sin2 extracted from the average polarization P = -A must be increased by 0.0038.
W
This would put this sin2 about 1.5 above the average of the other LEP measurements.
W

We already noted that A /A extracted from polarization studies does not support the
e
X boson hypothesis, and we now see that the source of the discrepancy is P . It also

appears that improvements in the measurement of the average polarization are

beginning to be limited by systematic uncertainties [6].

We finally turn to the vertex corrections, diagrams in which the X boson is attached

directly to the third family fermions. These effects are generally smaller than the X-Z

mixing effects. In fact they occur at order q2 in a momentum expansion, where q is the

momentum flowing into the vertex, whereas the X-Z mixing effects occur at order q0. On


4 We also do not understand the claim made in [7] that the anomaly in b
b should be accounted for by a gR
and not a g b b b b
L. Since b is less sensitive to a gR, if gL = 0 then the required gR would imply a 4%
decrease in A .
b



4


the other hand the vertex corrections affect all of the Z couplings to the third family, and

not just those couplings proportional to the X boson couplings. Of particular interest

then is the vertex correction to the axial Z coupling, which is not affected by X-Z

mixing. As we have said, the partial width data is very sensitive to any such correction.

The X induced vertex corrections are also potentially interesting in their effect on tt

production in pp colliders. Here it is not obvious that the corrections to the g tt vertex

are negligible, since the energies involved may be comparable to the X mass, and the X

may be strongly interacting.





Figure 1 : Diagrams contributing to Z or gluon vertex corrections induced by the
X boson. The fermion is a member of the third family.

We will start with the Z decay case in which we can safely ignore the mass of the

fermion (b, , ) compared to the Z and X masses. Here we can obtain analytical

expressions for the corrections both before and after expanding in M /M . We consider
Z X

the diagrams in Fig. (1) and write the massive X propagator in unitary gauge. We
express the integrals in terms of scalar integrals and find that the individual diagrams are
finite. The self-energy graphs contribute a finite field renormalization which is accounted
for in the usual way. In this case of ignoring the fermion mass it makes no difference
whether we consider vector or axial-vector couplings for the X and Z. For Z decay into
any of the third family fermions these corrections increase each partial width by the same
amount, while the asymmetries are unaffected. For the partial widths

2
g 2
= X - 4 1 + 1 ln 1 + r ln r + dilog 1 + r
162 r (3)

+ 2 3 + 2r ln r - 7 - 4r

where r (MZ/M )2
X and the dilog function is defined by


x ln(y)
dilog(x) = dy . (4)
1 -y
1

If we expand this result in powers of r we find



5


2
g
= X r 11 - 6 ln r . (5)
722

Note the term nonleading in ln(r) is significant. If we use (2) to determine the value of gX
then the linear dependence on r is canceled. For example with M /M = 5 we obtain
X Z

/ = 0.0015.

This correction to the partial width must be added to the correction coming from X-Z

mixing. For both and the mixing effect is about 10 times as large. But for the
b

X-Z mixing only affects the vector Z coupling which gives / = 0.0022 [3], and thus


the vertex correction is of comparable importance. This is the case even though the

vertex correction is formally suppressed by (M /M )2. Our total correction to / is
Z X

in the 0.003 to 0.004 range. The precision of the current measurements is starting to

become comparable; the current data gives / 0.004  0.004.


It is amusing to note that the significance of the data used in determining R /
h

has been recently called into question [10]. The distribution of the individual values of

R , R , R from each of the four experiments were found to be inconsistent with gaussian
e 

statistics. The conclusion of [10] was that there must be substantial systematic effects

unaccounted for in the original error estimate. An alternative conclusion, if there is

indeed a problem, is that the interpretation of the data is wrong; if the assumption of

universality is relaxed then the apparent problem with the data largely disappears.

We now turn to the case of top production. For nonzero quark mass the self-energy

graphs are no longer finite, and we renormalize by requiring that the correction to the

gluon vertex vanish at s q2 = 0. Although in our model the X boson couplings to the

quarks are axial, we will also consider vector X couplings for comparison. We have to

resort to evaluating integrals numerically, and we present the results in Fig. (2). We plot

the correction as a function of t s/M 2, and to determine g we use (2) with M /M =
X X X Z

5. The different lines represent different choices of u 4m2/s < 1 where m is the quark

mass. We also display the m = 0 case, which is given by (3) with r replaced with t. Thus

we find that quark mass effects give much more enhancement for vector rather than axial

X couplings. In fact in the axial case for u = 0.8 the result is very similar to m = 0; for

decreasing u the result decreases further (but not much smaller than the u = 0.7 line)

until it again approaches the m = 0 line for small u. Our conclusion is that the vertex

corrections induced by the X boson with axial quark coupling make only a very minor

correction to tt production.

In contrast we have found that the vertex corrections contribute to a potentially



6


observable shift in the Z partial width to . Our results for the vertex corrections could

be easily adapted to cases other than the specific X boson we have chosen to study. For

example if the coupling of a new gauge boson was not constrained by (2), then the vertex

corrections considered here could be much more significant. As for our specific X boson

and the X-Z mixing effects, it will be interesting to keep an eye on certain trends in the

data as the data improves.


0.99 0.9 0.8 0.7 0.99
0.03





0.9


0.02







0.01

0.7


0.8





0.5 1 t 1.5

Figure 2 : t s/M 2 and the curves are labelled by the value of u 4m2/s. The
X
heavy solid line corresponds to m = 0. The X boson with axial coupling to the top
quark (thin solid lines) is to be compared with the vector coupling case (dotted
lines).





7


Acknowledgements
I thank John Terning for describing to me his global fit to the data. This research was
supported in part by the Natural Sciences and Engineering Research Council of Canada.


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