1. Introduction


The search for the top quark and the Higgs boson, the only as yet unobserved ingre-

dients of the Standard Model (SM), is expected to dominate the area of particle physics

in the coming years. While the Higgs boson has always been naturally associated to the

breaking of the electro-weak symmetry, the recent experimental discovery that the top

quark is heavier than the W and Z has led physicists to speculate that it also may be sen-

sitive to physics on the electro-weak scale. For this reason, the theoretical determination

of its physical properties in the context of the SM and their systematic comparison with

experiment may reveal an underlying, more fundamental theory.

One of the most efficient ways to study top quarks will be to pair-produce them in

future very energetic e+e- colliders [1], through the reaction e+e- tt. In general,
the leptonic nature of the target allows for clean signals. In addition, due to their large

masses, the produced top quarks are expected to decay weakly (t
t bW+bW-, with
subsequent leptonic decays of the W ), before hadronization takes place; therefore electro-

weak properties of the top can be studied in detail and QCD corrections can be reliably

evaluated in the context of perturbation theory, when the energy of the collider is well

above the threshold for t
t production.

Motivated by this type of experiments, Atwood and Soni presented in a recent paper
arXiv: v1 12 Nov 93 [2] a phenomenological analysis for determining the magnetic and electric dipole moment

form factors of the top quark. Such form factors are defined through the following Lorentz

decomposition of the V t
t vertex, where V represents a boson (a or Z in our case) coupled

to the conserved leptonic current:

i i
V  q  q
 (q2) = F V
1 (q2) + F V
2m 2 (q2) + 5F V
3 (q2) + 5F V
4 (q2) (1.1)
t 2mt

where  i [
2 , ] and q2 = s, i.e. the Mandelstam variable associated to the squared

energy of the center of mass. In the above decomposition, F V
2 is the magnetic dipole

moment (MDM) and F V
4 is the electric dipole moment (EDM) form factor. In particular,

1


F
2 defined at q2 = 0 is the usual definition of the anomalous magnetic moment. In the case

of the top quark production, clearly q2 4m2. Within the SM the tree-level value for both
t

F V
2 and F V
4 is zero. F V
2 becomes non-zero through one-loop quantum corrections, whereas

F V
4 , which violates CP , receives its first non-vanishing contributions at three loops [3].

The upshot of the analysis of [2] was that the dependence of the differential cross section

for the reaction e+e- tt on the real and imaginary (absorptive) parts of the MDM and
EDM form factors, for an incoming photon or Z, can be singled out individually, through

a set of optimally chosen physical observables. Clearly, the possibility of such a detailed

experimental study of the top dynamics is attractive, and could provide valuable probes

for new physics.

The theoretical aspects of the situation are however not entirely clear. As it was

pointed out already in the classic paper by Fujikawa, Lee, and Sanda [4], off-shell form

factors of fermions are in general gauge dependent quantities. In the context of the R

gauges, for example, a residual dependence on the gauge-fixing parameter survives in

the final expressions of form factors, when q2 = 0 [5]. Obviously, in the case of e+e-

annihilation into heavy fermions, the value of q2 must be above the heavy fermion threshold

(q2 4m2, in our case). Consequently, the intermediate photon and Z are far off-shell,
t

and therefore, MDM and EDM form factors may in general be gauge-dependent and not

suitable for comparison with experiment.

A way out of this problem is to define gauge independent off-shell vertices using the

pinch technique (PT). The PT was invented by Cornwall over a decade ago [6] and has

since been applied to a variety of physical problems. The main idea of this method is to

resum via a well-defined algorithm the Feynman diagrams contributing to an ostensibly

gauge-invariant process (like an S-matrix element), in such a way as to form new gauge-

independent proper vertices, and new propagators with gauge-independent self-energies

and only a trivial gauge dependence - that of their tree level parts. In the context of

QCD a gauge invariant gluon self-energy was derived, and its Schwinger-Dyson equation

2


constructed and solved for T = 0 [7], as well as finite T [8]. The plasmon decay rate was

also calculated at finite T using the same method [9]. Later the QCD gauge invariant

three-gluon vertex was calculated at one loop level and was shown to satisfy a simple QED-

like Ward identity [10]. The subleading corrections to the self-energy were calculated by

Lavelle [11]. Finally, the gauge-invariant four-gluon QCD vertex was constructed and

its Ward identity derived [12]. The PT was first extended to the case of non-Abelian

gauge theories with spontaneously broken gauge symmetry (with elementary Higgs) in the

context of a toy field theory based on SU (2), and a gauge independent electro-magnetic

form factor for the neutrino was constructed [13]. The complicated task of applying

the P.T. in the electro-weak sector of the Standard Model was accomplished by Degrassi

and Sirlin [14]. These last authors, in addition to deriving explicit expressions for the

one loop gauge-invariant W W and ZZ self-energies, introduced an alternative description

of the PT in terms of equal time commutators of currents. More recently [15], gauge

independent W +W - and ZW +W - vertices were constructed in the context of the SM;

they are related to the gauge-independent W-propagators introduced in [14] by a very

simple Ward identity, and give rise to expressions for the W magnetic dipole and electric

quadrupole moments, which, unlike previous treatments [16], are independent of the

gauge-fixing parameter and infrared finite.

In this paper we focus on the MDM form factor of the top quark. In particular:

a. In the context of the SM we show that the conventionally defined form factor for the

top quark is gauge dependent in the class of R gauges. The gauge dependence is formally

expressed as a double integral over the usual Feynman parameters, and is proportional to

q2m2
t .
M 4
w


b. We show how one can define a gauge independent MDM form factor using the PT.

c. The gauge dependence found in (a) is computed and turns out to be numerically

very strong; its presence distorts not only the quantitative but also the qualitative behavior

of the answer obtained in (b). More specifically, unphysical thresholds are introduced,

3


the asymptotic behavior of the form factors as q2 is altered, and the numerical
dominance of perturbative QCD, which is present in the gauge independent expressions

of (b), is totally washed out. Of particular interest is the fact that the popular "unitary"

gauge ( the limit of the R gauges as ) gives a completely wrong answer. This
analysis indicates in retrospect that the gauge dependence established in (a) is a serious

pathology and may lead to erroneous conclusions.

d. We extract the real and imaginary parts of the gauge independent MDM form

factors for incoming photon and Z (F
2 and F Z
2 , respectively), and evaluate them numeri-

cally. We perform our computations for values of q2 comfortably above the threshold for

top production, and we ignore therefore non-perturbative effects due to threshold dynamics

[17]. It turns out that even though perturbative QCD provides in general the dominant

contribution, the MDM form factor displays a high sensitivity to the Higgs mass. It is

important to emphasize that the Higgs dependence of the MDM is actually stronger than

that displayed by the e+e- tt cross section as a whole [18].
The paper is organized as follows. In section 2 we review the S-matrix PT and discuss

some of the more important results for our purposes. In addition, we briefly present

Degrassi's and Sirlin's alternative formulation of the PT. In section 3 we perform an explicit

calculation and determine the naive, gauge dependent expression for the MDM form factor.

In section 4 we show how to use the pinch technique to define gauge independent t
t and

Zt
t vertices, which give rise to gauge independent expressions for the MDM form factors.

In section 5 we present a simple QED Ward identity relating the gauge independent t
t

vertex with the inverse top quark propagators, calculated in the Feynman gauge ( = 1).

Section 6 contains the results of our gauge invariant MDM computations, and a discussion

of their significance. We present our conclusions in section 7. Finally, in an Appendix, we

list in detail the contributions of individual Feynman diagrams to the MDM form factors.





4


2. The Pinch Technique


In this section we briefly review the S-matrix pinch technique. In particular, we

outline the method of derivation of the gauge-independent proper self-energy of a gauge

boson and comment on the technical differences that arise when PT is applied to a theory

with symmetry breaking, like the electro-weak theory, as opposed to a theory like QCD.

In addition, we present the main idea of Degrassi's and Sirlin's formulation of the pinch,

and establish some of the notation we will use in the sequel.

The PT is an algorithm that allows the construction of modified gauge independent n-

point functions, through the order by order resummation of Feynman graphs contributing

to a certain physical and therefore ostensibly gauge independent process (an S-matrix

in our case). The simplest example that demonstrates how the P.T. works is the gauge

boson two point function (propagator). Consider the S-matrix element T for the elastic

scattering of two fermions of masses M1 and M2. To any order in perturbation theory T

is independent of the gauge fixing parameter , defined by the free gluon propagator

-g + (1 - )qq
q2
 (q) = (2.1)
q2


On the other hand, as an explicit calculation shows, the conventionally defined proper

self-energy (collectively depicted in graph 1a) depends on . At the one loop level this

dependence is cancelled by contributions from other graphs, like 1b and 1c, which do not

seem to be of propagator type at first glance. That this must be so is evident from the

form of T :

T (s, t, M1, M2) = T1(t) + T2(t, M1, M2) + T3(s, t, M1, M2) (2.2)


where the function T1(t) depends only on the Mandelstam variable t = -(^p1 -p1)2 = -q2,
and not on s = (p1 + p2)2 or on the external masses. The propagator-like parts of graphs

like 1e and 1f, which enforce the gauge independence of T1(t), are called "pinch parts".

The pinch parts emerge every time a gluon propagator or an elementary three-gluon vertex

5


contribute a longitudinal k to the graph's numerator. The action of such a term is to

trigger an elementary Ward identity of the form

k /k = (/p + /k - m) - (/p - m) (2.3)
=S-1(p + k) - S-1(p)
once it gets contracted with a matrix. The first term on the right-hand side of Eq. (2.3)

will remove the internal fermion propagator - that is a "pinch" - whereas S-1(p) vanishes

on shell. This last property characterizes the S-matrix PT we will use throughout this

paper. Returning to the decomposition of Eq. (2.2), the function T1(t) is gauge invariant

and unique, and represents the contribution of the new propagator. We can construct the

new propagator, or equivalently T1(t), directly from the Feynman rules. In doing so it is

evident that any value for the gauge parameter may be chosen, since T1, T2, and T3 are

all independent of . The simplest of all covariant gauges is certainly the Feynman gauge

( = 1), which gets rid of the longitudinal part of the gluon propagator. Therefore, the only

possibility for pinching in this gauge arises from the four-momentum of the three-gluon

vertices, and the only propagator-like contributions come from graph 1b.

To explicitly calculate the pinching contribution of a graph such as 1b it is convenient

to decompose the vertex in the following way, first proposed by 't Hooft. Group theory

factors aside,

 = P + F (2.4)


with

P
 (q + k)g + kg (2.5)

and

F
 2qg - 2qg - (2k + q)g (2.6)

F satisfies a Feynman-gauge Ward identity:




qF = [k2
 - (k + q)2]g (2.7)

6


where the RHS is the difference of two inverse propagators in the Feynman gauge. As for

P (P for "pinch"), it gives rise to pinch parts when contracted with matrices



g(q/ + /
k) =g[(/p + q/ - m) - (/p - /k - m)] (2.8)
=g[S-1(p + q) - S-1(p - k)]
and
g/
k =g[(/p - m) - (/p - /k - m)] (2.9)
=g[S-1(p) - S-1(p - k)]
Now both S-1(p + q) and S-1(p) vanish on shell, whereas the two terms proportional to

S-1(p - k) pinch out the internal fermion propagator in graph 1b, and so we are left with
two "pinch" (propagator-like) parts and one "regular" (purely vertex-like) part:

1 ig2 d4k
pinch part = 2  (- c ] (g
2 A)[ (2)4 k2(k + q)2 ) (2.10)

1 ig2 d4kS(p
regular part = ( c ] - k)F (2.11)
2 A)[ (2)4 k2(k + q)2

with cA the Casimir operator for the adjoint representation [ cA = N in SU (N ) ] and one

factor of 2 for the two pinching terms from Eq. (2.8) and Eq. (2.9). When we add to the

usual propagator graphs (in the Feynman gauge) the pinch contributions of Eq. (2.10),

together with an equal contribution coming from the mirror-image graph of 1.b, we find

the new gauge invariant self-energy ^
(q), given by


^
(q) = (q2g - qq)^(q) (2.12)

where

^
(q) = -bg2 ln(-q2) (2.13)
2

and b = 11N is the coefficient in front of
482 -g3 in the usual one loop -function. We see
that the modified propagator has a gauge independent self-energy and only a trivial gauge

dependence originating from the tree part given by Eq. (2.1). After all pinch contributions

from graph 1.b have been allotted to the new gluon self-energy, the rest of graph 1.b,

7


namely the expression in Eq. (2.11), is genuinely vertex-like and must be added to the

usual QED-like graphs (Fig.2a). The final gauge-invariant vertex ^
a is given by


d4kS(p
^ i c c N d d4kS(p + q
a = g2 [( A ) - k)F+( A+ f ) - k)S(p - k)]
 (2)4 2 k2(k + q)2 2 c f k2
(2.14)

where is the fermion representation matrix, df its dimension, and Cf its Dynkin index.

If we now act on the ^
a given in Eq. (2.14) with q and use Eq. (2.7) it immediately

follows that ^
a satisfies the following QED-like Ward identity:

q ^
a = a[(p) - (^p)] (2.15)

with the usual quark self-energy in the Feynman gauge and ^
p = p + q. [19] The

application of the P.T. in a theory with spontaneous symmetry breaking like the Standard

Model is significantly more involved [13,14]. The main reason is that the massive charged

vector mesons couple to fermions with different masses, and therefore the elementary Ward

identity of Eq. (2.3) gets modified to:

/
k = (/
p + /
k - m1) - (/p - m2) + (m1 - m2) (2.16)
=S-1(p + k) - S-1(p) + (m1 - m2)
The first two terms of Eq. (2.16) will pinch and vanish on shell, respectively, as they did

before. But in addition a term proportional to m1 - m2 is left over. However, as it was
shown in [13], any additional gauge-dependent contributions coming from Eq. (2.16) cancel

exactly against contributions coming from graphs involving charged Goldstone bosons,

whose couplings to fermions are also proportional to m1 - m2.
Finally, we conclude this section with a brief presentation of an alternative formula-

tion of the PT introduced in [14] in the context of the Standard Model. In this approach

the interaction of gauge bosons with external fermions is described in terms of current

correlation functions, i.e. matrix elements of Fourier transforms of time-ordered products

of current operators [20]. This is particularly economical because these amplitudes au-

tomatically include several closely related Feynman diagrams. When one of the current

8


operators is contracted with the appropriate four-momentum, a Ward identity is triggered.

The pinch part is then identified with the contributions involving the equal-time commu-

tators in the Ward identities, and therefore involve amplitudes in which the number of

current operators has been decreased by one or more. A basic ingredient in this formula-

tion are the following equal-time commutators, some of which we will also employ later in

section 3:



(x0 - y0)[J0 (x), J(y)] = c2J (x)4(x
W Z W - y) (2.17)



(x0 - y0)[J0 (x), J(y)] =
W W -J3(x)4(x - y) (2.18)



(x0 - y0)[J0W(x), J(y)] = J (x)4(x
W - y) (2.19)

with J
3 2(J + s2J
Z ). On the other hand



(x0 - y0)[J0 (x), J (y)] = 0 (2.20)
Vi Vj



where Vi, Vj {, Z} To demonstrate the method with an example, consider the vertex
 shown in Fig.1b, where now the gauge particles in the loop are W s instead of gluons

and the incoming and outgoing fermions are massless. It can be written as follows (with

= 1):

d4k
 = (q, k, (x)J (0)]
24 -k - q) d4xeikx < f|T[J
W W |i > (2.21)

When an appropriate momentum, say k, from the vertex is pushed into the integral

over dx, it gets transformed into a covariant derivative d acting on the time ordered
dx

product < f |T[J(x)J (0)]
W W |i >. After using current conservation and differentiating
with respect to the -function terms, implicit in the definition of the T product, we

end up with the left-hand side of Eq. (2.18). So, the contribution of each such term is

9


proportional to the matrix element of a single current operator, namely < f |J3|i >; this is
precisely the pinch part. Calling P the total pinch contribution from the
  of Eq. (2.21),

we find that

P =
 -g3cIWW(Q2) < f|J3|i > (2.22)

where
d4k 1
Iij(q) = i ( ) (2.23)
24 (k2 - M2)[(k + q)2 ]
i - M2j
Obviously, the integral in Eq. (2.23) is the generalization of the QCD expression Eq. (2.8)

to the case of massive gauge bosons.





3. Gauge dependence of the conventional MDM form factor.


In this section we explicitly show that the MDM form factor extracted from the

conventional vertex graphs, calculated in a general renormalizable (R) gauge, depends on

the gauge-fixing parameter , and is therefore not suitable for comparison with experiment.

We then isolate the gauge dependent contribution to the MDM originating from the vertex-

like pinch parts of box graphs, calculated in the same class of gauges. It turns out that

the latter gauge-dependent contributions to the MDM cancel exactly against the gauge-

dependent parts of the conventional vertex graphs, thus giving rise to a gauge independent

expression for the top MDM. It is important to emphasize that even though it has been

proved [14] that the inclusion of the vertex-like pinch parts is necessary in order to define

a gauge independent t
t vertex, it has not been shown explicitly that the conventionally

defined expression for the MDM is indeed gauge dependent. The obvious alternative (which

we will show not to be true) is that, even though the usually defined vertex has an overall

gauge dependence, its part proportional to q could be fortuitously gauge independent.

In addition, the explicit computation of the gauge dependence of the conventionally defined

10


MDM enables one to appreciate quantitatively the significance of the issue at hand. The

tree-level vector-meson propagator (k)i in a general covariant gauge is given by

k 1
i k
 (k) = [g - (1 - i) ] (3.1)
k2 - iM2 k2
i - M2i
with i = w, z, , and M = 0. The propagator Dc(k) of the two charged Goldstone bosons

is given by
1
Dc(k) = (3.2)
k2 - wM2w
and the propagator Dn(k) of the neutral Goldstone boson is

1
Dn(k) = (3.3)
k2 - zM2z
The propagator DH (k) of the physical Higgs particle is gauge-independent at tree-level

1
DH (k) = (3.4)
k2 - m2H
and therefore the graph of Fig.3.16 has no gauge-dependent contribution. Finally, we

denote the quark propagators by


1
Sj(k) = (3.5)
/
k - mj
with j = t, b, for top and bottom quarks. We further define c = cosW and s = sinW ,

where W is the Weinberg mixing angle, = |Utb|2 for the modulus of the C.K.M. mixing
between top and bottom [21], and CZ = 1 - 8s2
3 - 5. Finally, the left-right projectors
are given by PL,R = 1 5

We denote by V (i)
 the -dependent parts of the graphs (i)
 , shown in Fig.3, with


i = 1, 2, .., 16

We use the identity


( - 1) 1 1 1
= [ ] (3.6)
(k2 - M2)(k2 - M2) M2 k2 - M2 - k2 - M2
11


in order to isolate the -dependent parts of the graphs. Gauge-dependent terms propor-

tional to q are set to zero, when considered "sandwiched" between the on-shell target

electrons. In what follows we omit the loop momentum integration symbol ( e )2 d4k ,
2s2 24

and we use dimensional regularization whenever necessary. We define the scalar quantities

D1(i), D2(i), and D3(i) as follows

1
D1(i) = k2 -iM2i 1
D2(i) = (3.7)
[(k + q)2 - M2](k2 )
i - iM2i
1
D3(i) = [(k + q)2 -iM2](k2 )
i - iM2i
Clearly, D2(i = 1) = D3(i = 1).

The most involved graph is (1)
 of Fig.3, given by


(1) =
  (k + q) (k)PLSb(k + p2) PL (3.8)


Its w-dependent part V P (
 w) is given by



V P
 (w) = q2[V 1P
 + V 2P
 + V 3P
 ] (3.9)


with


V 1P PLSb(k + p2)[mtPR - mbPL]
 = (1 - w)[(k + q)2 -M2][(k +q)2 ](k2 )
w - wM2w - M2w
[m
V 2P = (1 bPR - mtPL]Sb(k + p2)PL (3.10)
 - w)[(k + q)2 -M2](k2 )(k2 )
w - M2w - wM2w
V 3P = (1
 - w)2 k[mbPR - mtPL]Sb(k + p2)[mtPR - mbPL]
[(k + q)2 - M2w][(k + q)2 - wM2w](k2 - M2w)(k2 - wM2w)
The superscript P in the formulas above stands for "pinch"; it hints to the fact that these

contributions will eventually cancel against pinch parts coming from box graphs (see next

section). It is important to notice the presence of the factor q2 in the r.h.s. of Eq. (3.9).

Clearly, V P (
 w = 1) = 0.


12


The -dependent contributions V (i)
 from each individual graph (i)
 are:


V (1) = V P ( .
  w) - V (2)
 - V (3)


1
V (2) = V (4)
 M PLSb(k + p2)[mtPR - mbPL]D2(w) - 
w 2
1
V (3)
 = [m V (4)
M bPR - mtPL]Sb(k + p2)PLD2(w) - 
w 2
2
V (4) = k
 M 2 [mbPR - mtPL]Sb(k + p2)[mtPR - mbPL]D3(w)
w




V (5)
 =
3M 2 PLSb(k + p1)[mtPR - mbPL] - [mbPR - mtPL]Sb(k + p2)PL
w (3.11)

+ 2PLD1(w) - V (8)




1
V (6) = [
 3M 2 C 2
Z - 2mtCZSt(k + p1)5 + 2mt5St(k + p2)CZ]D1(z) - V (9)

w

V (7) =
 -U(3)


1
V (8)
 = ( )[m
3 M 2 bPR - mtPL]St(k + p2)St(k + p1)[mtPR - mbPL]D1(w)
w
4 m2
V (9) = ( t )
 3 M 2 5St(k + p2)St(k + p1)5D1(z)
w

V (10)
 = 0

The analogous -dependent contributions from the self-energy graphs (i)
t of the external

top quarks (Fig.(3)) are:





V (11) =
 - 3M2 PLSb(k + p1)[mtPR - mbPL] - [mbPR - mtPL]Sb(k + p2)PL
w


+ 2PLD1(w) - V (14)





1
V (12)
 = - [
3M 2 C 2
Z - 2mtCZSt(k + p1)5 + 2mt5St(k + p2)CZ]D1(z) - V (15)

w

4 3 1
V (13)
 = -(1 - )( )
3  k4

13


1
V (14) = ( )
 3 M 2 St(p1)[mbPR - mtPL]St(k + p1)[mtPR - mbPL]+
w
(3.12)

+[mbPR - mtPL]St(k + p2)[mtPR - mbPL]St(p2) D1(w)

4 m2
V (15) = ( t )[
 3 M 2 St(p1)5St(k + p1)5 + 5St(k + p2)5St(p2)]D1(z )
w

V (16)
 = 0

Adding the above equations together we see that 16 V (i) , so that
i=1  = V P




(w)
 = R
 |(w=1) + V P
 (w) (3.13)


In the equation above R
 |(w=1) represents the "regular"(hence the superscript R), purely
vertex-like contributions in the Feynman gauge (w = 1). The term "regular" or "purely

vertex-like" refers to what is left from the vertex graphs, after the propagator-like pieces

have been removed through pinching. (see Eq. (2.11) and Eq. (2.14))

The calculation of the gauge dependence of the Zt
t vertex  proceeds in an analogous

way. The final answer is exactly the same as in Eq. (3.13), except that one must replace

V P
 by V PZ
 defined as


V PZ = (q2 )[V 1P + V 2P + V 3P ], (3.14)
 - M2z   


e.g.

(w) = R (
  |(w=1) + V PZ
 w) (3.15)


It is important to notice the particular form of the gauge-dependent terms V P (
 w) and

V PZ
 , namely that they are explicitly multiplied by q2 and q2 -M2z, respectively. As we will
see in the next section, this is precisely the characteristic structure of vertex-like contri-

butions originating from box graphs, after pinching. Moreover, the gauge-dependent parts

vanish, when the incoming gauge bosons are on-shell (q2 = 0 and q2 = M 2, respectively),
z

as they should.

14


We next proceed to extract from Eq. (3.13) the form factor F (q2) proportional to

q. We use the Gordon decomposition

1 i

u(p  q
2)u(p1) = 
u(p + ]u(p
2m 2)[(p1 + p2) 1) (3.16)
t 2mt

to convert terms proportional to (p1 + p2) into linear combinations proportional to
 

and q. The final answer is:


F2(q2) = F2(q2)|(w=1) + f2(w, q2) (3.17)


where F2(q2)|(w=1) originates from the first term in the r.h.s. of Eq. (3.13), whereas
f2(w, q2) originates from the second one, and has the following closed form:

2q2m2 1 1-x
f t
2(w, q2) = dx dy xR(1,
M 2 w) + yR(w, 1) - (x + y)R(1, 1)+
w 0 0

m2 m2
[ b (x + y) t + m2
b (x + y)2][R(1, 1) + R(
M 2 - w, w) - R(1, w) - R(w, 1)]
w 2M 2
w
(3.18)

with


R(1, 2) = [(x + y)2m2t - q2xy - (x + y)(m2t + m2b) + x1M2w + y2M2w + m2b]-1 (3.19)

Clearly, F2(q2) of Eq. (3.17) is w-dependent, through the explicit dependence of f2(w, q2)

on w. This concludes the proof that the conventionally defined MDM form factor is

gauge-dependent, for both an incoming photon or Z, and therefore unsuitable in general

for comparison with experiment.

It is interesting to mention that there is no contribution from either parts of Eq. (3.13)

proportional to the CP violating term q5. Therefore, the usual statement that there

is no one-loop contribution to the EDM is indeed a gauge-independent one.





15


4. Gauge-independent MDM form factors via the PT



In this section we show how the use of the PT can give rise to gauge-independent

expressions for the MDM form factors, for both an incoming photon and Z. The important

point is to recognize that the box-like graphs, like the one shown in Fig.2b, contain gauge-

dependent contributions which are kinematically equivalent to a t
t or a Zt
t vertex. When

all these contributions are added to the conventional vertex graphs, the gauge dependence

cancels exactly.

The only kinematically relevant box graph is the one with an intermediate W +W -

pair, shown in Fig.2b . [22]. Its w-dependent contributions B are given by [23]



B(w) = g2e-PLe+g[V 1P
 + V 2P
 + V 3P
 ] + ... (4.1)



The dots in Eq. (4.1) indicate terms that belong to one of the following categories:

a. Propagator-like, to be allotted to the , Z Z, and ZZ self-energies.

b. Vertex-like of the form g2[V 1P
+ V 2P
+ V 3P
]g
tPLt; the V 1P
, V 2P
and V 3P
are

given in Eq. (3.10) after replacing t e and b . These contribution can be combined
with the conventional e+e- and Ze+e- vertex graphs to construct gauge-independent

e+e- and Ze+e- vertices.

c. Purely box-like, to be combined with the rest of the box graphs.

Clearly, none of the categories listed above contributes kinematically to the MDM

form factors.

We now write the expression given in Eq. (4.1) as a linear combination of a e+e-

and a Ze+e- bare vertex, using the simple identity


g2 g2
e+ e+
4 PLe- = g2s2e+e- + 4 [PL - 4s2]e- (4.2)

16


so that B becomes (we omit the external e+ and e- spinors)


g2s2 ( g2 )[PL
B = - [q2(V 1P 4 - 4s2][(q2
q2  + V 2P
 + V 3P
 )] - q2 - M2 - M2z)(V 1P
 + V 2P
 + V 3P
 )]
z

g2s2 ( g2 )[PL
= - V P 4 - 4s2]V PZ
q2  - q2 - M2 
z
(4.3)

If we now add the first term on the r.h.s. of Eq. (4.3) to Eq. (3.13), and the second term

to Eq. (3.15), we find the gauge-independent t
t and Zt
t vertices, ^
 and ^
, respectively.

They are simply

^
 = R
 |(w=1) (4.4)

and

^
 = R
 |(w=1) (4.5)

thus, the gauge-independent MDM form factor ^
F (q2) is given by:


^
F (q2) = F (q2)|w=1 (4.6)


for both incoming photon and Z.

The gauge-independent t
t and Zt
t vertices given in Eq. (4.4) and Eq. (4.5) coincide

with the the conventional vertex graphs of Fig.3, calculated in the w = 1 gauge, but with

the propagator-like pieces of graph 3.1 removed through pinching. In particular, the gauge-

independent MDM form factors can be obtained if one just calculates the conventional

MDM form factors in that same gauge. This results may come as no surprise to the

reader familiar with the PT. Indeed, as we explained in section 2, in calculating gauge-

independent Green's functions (a vertex in our case) any gauge choice is as good as any

other, as long as one identifies and judiciously allots the pinch contributions. If one decides

to calculate the gauge-independent vertex in the w = 1 gauge, the only possibility for

pinching comes from the three-boson vertex of graph 3.1, which gives rise to a propagator-

like piece proportional to PL. In particular, there are no pinch contributions (propagator

17


or vertex-like) coming from box graphs, since there are no longitudinal terms in the bare

gauge boson propagators appearing in the boxes. Since the only term removed from the

vertex has no part proportional to q, and hence no contribution to the MDM, the

answer for the gauge-independent MDM is obviously the one recorded in Eq. (4.6). The

fact that the gauge-independent answer in the MDM case coincides with the result in the

Feynman gauge is in a sense fortuitous, and is certainly not true in general. Indeed, several

other gauge-independent modified Green's functions, constructed via the PT, turned out

to be different from the corresponding result in the Feynman gauge, or any other popular

gauge for that matter.

Finally, we conclude this section with a comparison of the gauge-independent (w = 1)

answer for the MDM, to the answers computed for different values of w, as given in

Eq. (3.17) and Eq. (3.18). We chose mt = 130 GeV and MH = 150 GeV . The results

shown in Fig.4 are rather impressive and convincingly demonstrate the importance of the

issue at hand. Indeed, the residual gauge dependence is not just a theoretical nuisance,

but has a huge impact on the predictions of the theory, affecting them both quantitatively

and qualitatively. In Fig.3 the real part of the MDM form factor for an incoming photon

is plotted as a function of s. The curve marked (a) is the gauge-independent answer,

(b) corresponds to the gauge choice w = 10, (c) to w = 100, (d) to w = 1000, and

(e) to the unitary gauge (w ) [24]. We see how the curves start approaching the
unitary limit (e), as w becomes large. The gauge-dependent answers are typically at least

one order of magnitude larger than the gauge-independent answer of (a). The reason for

this artificial enhancement is the presence of the factor q2 in front of the expressions in
M 2
w

Eq. (3.18), which is huge due to the massiveness of the top quark. Indeed, already at the

threshold of top pair production, for mt = 130 GeV , we have that q2 = 4 m2t
M 2 M 2 10.5. If
w w

instead of t
t we considered bb production the corresponding factor would be of the order of

10-2. The phenomenological implications of such an enhancement are clear. For example,

any dependence of the final answer on the mass of the Higgs boson gets totally altered.

18


Because of the same reason, the asymptotic behavior of the MDM form factor in a generic

R gauge is totally different than that of the gauge independent case; as we see from Fig.4,

the MDM curves increase monotonically as s , whereas, on the contrary, the gauge-
independent curve approaches zero. It is interesting to notice that within the entire class of

R gauges, only the gauge-independent answer decreases asymptotically [25]. In addition,

as we can see in curves (b) and (c), extra -dependent thresholds are introduced, which are

artifacts and do not correspond to anything physical. Even though it is very unlikely that

anyone would calculate in any of the finite gauges ( = 10, 100, 1000) considered above, it

is important to realize that the popular unitary gauge (e) yields a totally wrong answer.

The same is true for another popular gauge, the Landau gauge, with w = 0 (not shown).

There, in addition to the usual artifacts, one has extra infrared divergences, due to the

presence of massless poles.




5. The Ward identity for the gauge-independent t
t vertex


In this section we prove a QED-like Ward identity, which relates the gauge-independent

t
t vertex ^
 to the top quark self-energy t in the Feynman gauge. The Ward identity is

2
q ^
(p1, p2) = [
3 t(p2) - t(p1)] (5.1)


which is the one-loop generalization of the trivial tree-level result Qtq = Qt[( /
p2 -mt)-
( /
p1 - mt)], with Qt = 2 the electric charge of the top quark. It is important to emphasize
3

that it is only after the propagator-like contributions of graph (3.1) have been removed

that the validity of Eq. (5.1) becomes possible. To see that, let us recall the decomposition

of the tree-level W +W - vertex, given by Eq. (2.5) and Eq. (2.6). The graph (1)
 , in the

Feynman gauge, is (we again omit the ( e )2 d4k factor in front):
2s2 24


1 1
(1)
 = (P
 + F
) PL P ] (5.2)
/
k - m L[
b (k2 - M2)[(k + q)2 ]
w - M2w
19


The part of Eq. (5.2) proportional to P
 , after pinching out the internal quark propa-

gator, gives rise to propagator-like contributions, which, as explained in section 2, will be

allotted to the self-energies of the incoming neutral gauge bosons (, ZZ, Z, Z). How-

ever, unlike the QCD case, a residual contribution R survives, because the internal (b)


and external (t) quarks have different masses. This extra contribution is

R
 = {[mtPR - mbPL]Sb(k)PL + PLSb(k)[mbPR - mtPL]}
(k2 - M2)[(k + q)2 ]
w - M2w (5.3)
= -((2) + (3))
 

and cancels entirely against the contributions of graphs (2)
 and (3)
 . The above obser-


vation is crucial for deriving Eq. (5.1). So, the purely vertex-like contribution (V L)
 from

(1)
 is given by

F
(V L)  PLSb(k)PL
 = (k2 -M2)[(k +q)2 ] - ((2)
 + (3)
 ) (5.4)
w - M2w
After these considerations, proving the Ward identity is rather straightforward. In

that vein, it is far more convenient to act with q on each graph individually, instead of

first calculating them, and then acting with q on the final answer.Using Eq. (2.7) we have:

q[(V L) + (2) + (3)] = (1)
   t (p2) - (1)
t (p1)

q(4)
 = (4)
t (p2) - (4)
t (p1)
1
q(5) = [(1)(p (p
 -3 t 2) - (1)
t 1)]

2
q(6) = [(2)(p (p
 3 t 2) - (2)
t 1)]
2 (5.5)
q(7)
 = [(3)
3 t (p2) - (3)
t (p1)]

1
q(8)
 = - [(4)
3 t (p2) - (4)
t (p1)]

2
q(9) = [(5)(p (p
 3 t 2) - (5)
t 1)]

2
q(10) = [(6)(p (p
 3 t 2) - (6)
t 1)]

Adding the left and right hand sides of Eq. (5.5) we arrive at the advertised result. Clearly,

pinching is instrumental for the validity of Eq. (5.1).




20


6. Calculations and results



In this section we report the results of the numerical evaluation of the integrals listed

in the Appendix. These integrals correspond to the vertex graphs of Fig.3, calculated in

the = 1 gauge, which as we explained in section 4, coincides with the gauge-independent

answer in the PT framework. In arriving at these expressions we used the on-shell con-

ditions /
p1 = /
p2 = mt (or equivallently p21 = p22 = m2t), and the Gordon decomposition of

Eq. (3.16).

We have used very accurate integration routines contained in the Mathematica pack-

age. Such routines can handle the singularities on the integration path which appear in the

formulas. The stability of the numerical results has been tested against different choices

of the parameters of the algorithm.

In an attempt to stay close to the notation of [2], we denote by Re(C ) and Im(C)

the real and imaginary parts of the MDM form factor for an incoming photon. Similarly,

Re(CZ ) and Im(CZ ) denote the respective quantities for an incoming Z. The results are

shown in Fig.5, Fig.6, and Fig.7. In what follows we always assume that the aforementioned

quantities can be individually extracted from e+e- experiments, through the methods

discussed in detail in [2]. In all our calculations mt 130 GeV , so the top is expected
to decay before it can form a bound state [26], and therefore non-perturbative QCD

effects should be numerically not very important. Even so, the one-loop QCD vertex

corrections (graph 7 in Fig.3 with gluon) dominate in general the contribution of all
other graphs. This dominance is due to the fact that the strong coupling s, although it

is small enough for perturbation theory to be trusted, it is still significantly larger than

both the electro-magnetic and the weak couplings, which multiply all other graphs. We

used s = 0.1, = 1 , cos
128 w = 0.885, Mw = 80.6 GeV , Mz = 91.1 GeV and

mb = 4.5 GeV . In addition, in the computations we report, we always place ourselves

sufficiently above the respective t
t threshold, so that non-perturbative threshold effects

21


may also be omitted [27]. Of course, since the mass of the top is a free parameter which

we vary in our computations, the location of the threshold (sthr = 4m2) is also variable.
t


In Fig.5, Re(C ), Im(C), Re(CZ ), and Im(CZ) are plotted as function of s for

mt = 130 GeV and two different values for the mass of the Higgs: a light one (mH =

150GeV ) and a heavy one (mH = 500GeV ). In Fig.6, Re(C), Im(C), Re(CZ ), and

Im(CZ) are plotted as function of s with a heavier choice for the top mass, namely mt =

200 GeV , and the same values for the Higgs masses as in Fig.5. Finally, in Fig.7 Re(C ),

Im(C), Re(CZ ), and Im(CZ) are plotted as a function of mH , for two different values of

the top mass, mt = 130 GeV and mt = 200 GeV . We fixed the value of s at 0.5 T eV ,

exactly as in the analysis of [2]. It is important to notice that in all cases shown in Fig.5

and Fig.6 the form factors approach zero, as s increases, at least for the range of the

parameters that we have explored.

As we see in Fig.5, both Re(C) and Im(C) are smooth functions of s slightly

above s = 300 GeV for both values of mH . On the other hand both Re(CZ ) and Im(CZ)

display a spike (peak) at around s = 600 GeV when mH = 500 GeV , whereas they are

smooth and have no spikes when mH = 150 GeV . Exactly the same qualitative behavior

is also displayed by the graphs of Fig.6; the only difference is an overall enhancement in

the values of all quantities shown, since the top is now heavier. Finally, in Fig.7 both

Re(C ) and Im(C) are smooth functions of mH ; on the contrary, Re(CZ ) and Im(CZ)

are rather sensitive to the values of mH , especially if the mass of the top is large.

The Feynman graph responsible for the behavior described above is the non-Abelian

graph denoted by "special" in Fig.3 (and its mirror graph, not shown), which exists only

if the incoming gauge boson is a Z. On the other hand, the Abelian graph in Fig 3.10,

common to both incoming and Z, is of no particular importance. The importance of the

"special" graph is due to the appearance of a threshold effect, as soon as the momentum
s, carried by the incoming Z, reaches the critical value sc, given by sc = Mz + mH,

namely the threshold for Z + H production (e+e- Z Z + H) [28]. It is now easy
22


to understand the behavior of Re(CZ ) and Im(CZ) in Fig.5(c,d) and Fig.6(c,d): in the

first case (mt = 130 GeV ), for mH = 150 GeV we have sc = 241 GeV , which is below

the t
t production threshold sthr = 260 GeV . So, the threshold effect of the "special"

graph does not show up in the plots. On the contrary, for mH = 500 GeV , we have

that sc = 591 GeV , which gives the effect we observe in our plots, for both Re(CZ ) and

Im(CZ). Exactly the same happens in the second case (mt = 200 GeV ), since the value

of sc = 591 GeV is still larger than sthr = 400 GeV . Of course, the value of sc does

not depend on the choices of mt; what depends on the value of mt is whether or not the

HZ-threshold effect at sc appears in the plots.

The threshold effect seen in both Fig.7c and Fig.7d is also due to the "special" graph.

For a given value of s we expect a threshold effect as soon as mH = mH |thr = s - Mz,
thus, for incoming s = 500 GeV one expects an effect at mH |thr = 500 GeV - Mz
409 GeV ; this is indeed what we observe in Fig.7c and Fig.7d. In particular, the position

of mthr remains unchanged as it should, for both values of m
H t we examined. We emphasize

that due to such threshold effect, some of our gauge-invariant form factors are much more

sensitive to the value of the Higgs mass than the cross section as a whole [18]. For example,

Re(CZ) of Fig.7c, for mt = 200 GeV (dark curve), changes from Re(CZ ) = -1.4  10-3
at mH = 200 GeV to Re(CZ ) = -3.9  10-3 at mH = 400 GeV , a change of 280%.

Based on what we said above about the ZH-threshold effect of the "special" graph,

the following possibilities may come up, if one measured Re(CZ ) and Im(CZ) in the

future (smax is the maximum energy of the collider):

a) If 2mt - Mz < mH < smax, a threshold effect should appear in both Re(CZ)
and Im(CZ) when plotted as a function of s, at a position sc = mH + Mz.

b) If mH < 2mt - Mz or mH > smax, then no threshold effect should appear in
either Re(CZ ) or Im(CZ , in the context of the SM. The second case is particularly interest-

ing, especially in view of the current theoretical bias toward a relatively "light" Higgs. For

23


example if mH < 150 GeV , as favored by the Minimal Supersymmetric Standard Model,

one should not see any HZ threshold effects, unless it turns out that mt < 120 GeV [29].

c) If a threshold effect appears in one of Re(CZ ) or Im(CZ ) but not in the other, or if

more than one threshold effect appears in either or both of them, this should be interpreted

as a signal of physics outside the SM.

It is important to keep in mind that if one calculates the form factors in a gauge-

dependent way, i.e. without pinching, in some cases one gets wrong predictions of threshold

effects, related to unphysical, gauge-dependent thresholds. In these cases, the difference

between the gauge-independent and the gauge-dependent prediction becomes, of course,

dramatic.





7. Conclusions


In this paper we addressed some of the theoretical issues involved in the computation

of off-shell form factors in the context of the Standard Model. Such form factors can

be extracted, at least in principle, from cross-sections in future e+e- experiments. In

particular, we explicitly demonstrated that the conventionally defined MDM form factors

of the top quark are gauge-dependent in the class of R gauges. Due to the massiveness of

the top quark, this gauge dependence turned out to be numerically very strong. We used

the S-matrix pinch technique to eliminate the gauge dependence, and defined a gauge-

independent MDM form factor, suitable for comparison with experiment. We proved a

simple QED-like Ward identity, satisfied by the gauge-invariant t
t vertex costructed via

the PT. Finally, we computed the gauge-independent MDM form factors and presented

the results for both real and imaginary (absorptive) parts. A very interesting feature of

such form factors is their high sensitivity to the value of the Higgs mass, which enters the

formulas as a parameter. For this reason, assuming that the top quark will be discovered

24


and detected before the Higgs particle, the form factor we have analyzed in the present

paper contains interesting information about the Higgs boson.

Of particular interest for the study of CP violation will be the experimental measure-

ment of EDM form factors in the future. As we mentioned in the introduction, there are no

EDM contributions within the SM up to three loops, but several models predict non-zero

EDM already at one-loop. We are currently investigating issues of gauge invariance in the

context of some of those models, with focus on the EDM form factors, and will present

our results elsewhere.




8. Acknowledgements


The authors thank Professor A. Sirlin for useful discussions. One of us (C.P.) thanks

the Physics Department of the New York University for its hospitality during the early

stages of this work. This work was supported in part by the National Science Foundation

under Grant No..




9. Appendix


In this Appendix we list all the contributions to the magnetic form factor F (q2) of

the top quark. Using the standard Feynman parametrization formula

1 1 1-x 1
= dx dy (9.1)
ABC 0 0 [Ax + By + C(1 - x - y)]3
and subsequently the convenient change of variables t = x + y, tz = x - y, we can cast
the MDM contributions of the graphs in Fig.3 (except the one denoted "special") in the

following form

1 1
F (q2) = m2 N V dt t AV dz L
2 t (i) (i) + BV
(i)t + CV
(i)t2 (i)(t, z) (9.2)
i 0 0


25


where L(i)(t, z) = L(t, z; m1, m2, q2).

In what follows we will use the short-hand notation L(m1, m2). In order to identify

the contributions to the magnetic form factor F (q2) and F Z(q2), which arise from the t
t

and the Zt
t vertices respectively, the coefficients in Eq. (9.2) carry a vertex label V = , Z.

Obviously the integrand functions L(i)(z) do not have such a label, as they do not depend

on which gauge boson enters the vertex. In the following list, the contributions are listed

in the same order as in Fig.3 and we use again the abbreviations s = sinW , c = cosW

and = |Utb|2. In addition, we define L = (1 - 8s2).
3





3
N = A = 0 B = 1 C = 1
(1+2+3) 4s2 (1+2+3) (1+2+3) (1+2+3)

L(1+2+3) = L(M 2 , m2) (9.3)
w b





3
N = A = 2m2 = = m2
(4) 8 M 2 (4) b B(4) -3m2b - m2t C(4) b + m2
t
ws2

3 c 1
N Z
(4) = AZ BZ CZ + m2
8 M 2 (4) = 2m2
b (4) = -3m2b - m2t (4) = m2
b t
ws2 s - 2s c

L(4) = L(M 2 , m2) (9.4)
w b






N = A = 2 B = = 1
(5) 4 s2 (5) (5) -3 C(5)
3 4 2
N Z
(5) = - AZ s2 BZ s2
8 s3c (5) = -2 + 3 (5) = 3 - 2s2 CZ
(5) = -1 + 3

L(5) = L(m2, M 2 ) (9.5)
b w





1 32 8 32
N = A = 2 B = s4 + s2 C = s4
(6) -4 s2c2 (6) (6) -3 - 9 3 (6) L + 9

26


3
N Z
(6) = - AZ
64 s3c3 (6) = 8 L BZ
(6) = -L(L2 + 11) CZ
(6) = L(L2 + 3)

L(6) = L(m2, M 2) (9.6)
t z





8
N = A = 0 B = = 1
(7) -9 (7) (7) -1 C(7)
1
N Z
(7) = -3 sc L AZ
(7) = 0 BZ
(7) = -1 CZ
(7) = 1


L(7) = L(m2, 0) (9.7)
t






N = A = 0 B = m2 C = m2 + m2
(8) 8 M 2 s2 (8) (8) b - m2t (8) b t
w

3 2 2 8
N Z = AZ = 0BZ = 4(1 s2)(m2 ) CZ = (1 s2)+ m2s2
(8) -64 M2 (8) (8) - t -m2b (8) -4m2t - b
ws3c 3 3 3

L(8) = L(m2, M 2 ) (9.8)
b w





m2
N = t A) = 0 B = 0 C = 1
(9) -4M2 (9) (9) (9)
ws2

3 m2
N Z t
(9) = -32 M2 s3cL AZ
(9) = 0 BZ
(9) = 0 CZ
(9) = 1
w

L(9) = L(m2, M 2) (9.9)
t z





m2
N = t A = 0 B = 2 C =
(10) 4 M 2 s2 (10) (10) (10) -1
W

3 m2
N Z t
(10) = 32 M2 s3cL AZ
(10) = 0 BZ
(10) = 2 CZ
(10) = -1
W

3 m2
N Z t
(10) = 32 M2 s3cL AZ
(10) = 0 BZ
(10) = 2 CZ
(10) = -1
W

L(10) = L(m2, M 2 ) (9.10)
t H


27


For the "special" graph (last one in Fig.3) and its mirror graph (not shown) the original

Feynman parametrization given in Eq 9.1 was maintained, since the change of variables

(x, y) (t, z), used for all other graphs, is not convenient.
Finally, the one-loop QCD contribution is obtained from Eq. 9.6, by multiplying by

the factor s , where
2 s is the QCD coupling at  = Mz , namely


N = [ s ]N
QCD 2 (7)
(9.11)
N Z = [ s ]N Z
QCD 2 (7)





10. References.


1. See for example, Burton Richter, in Proceedings of the 24th International Conference

on High Energy Physics, Munich, Germany, 1988, edited by R Kotthaus and J. Kuhn

(Springer, Berlin 1988). For a more recent review, see also P.M. Zerwas, preprint

DESY 93-112, and references therein..

2. D. Atwood and A. Soni Phys. Rev. D 45, 2405 (1992).

3. For a review on extensions of the SM where EDM couplings appear already at the

one-loop level see W. J. Marciano, in Proceedings of the Summer Study on CP Vio-

lation, Upton, New York, 1990, edited by S. Dawson and A. Soni (World Scientific,

Singapore, 1990), and references therein.

4. K. Fujikawa, B. W. Lee, and A. I. Sanda Phys. Rev. D 6, 2923 (1972).

5. A well known manifestation of this problem is the gauge dependence of the neutrino

electromagnetic form factor and the mean square radius derived from it; see for

example G. Degrassi, A. Sirlin, and W. J Marciano Phys. Rev. D 39, 287 (1989).

6. J. M. Cornwall, in Deeper Pathways in High Energy Physics, edited by B. Kursunoglu,

A. Perlmutter, and L. Scott (Plenum, NewYork, 1977), p.683.

28


7. J. M. Cornwall, Phys. Rev. D 26, 1453 (1982).

8. J. M. Cornwall, W. S. Hou, and J. E. King, Phys. Lett. B 153, 173 (1988).

9. S. Nadkarni, Phys. Rev. Lett 61, 396 (1988).

10. J. M. Cornwall and J. Papavassiliou, Phys. Rev. D 40, 3474 (1989).

11. M. Lavelle, Phys. Rev. D 44, 26 (1991).

12. J. Papavassiliou, Phys. Rev. D 47, 4728 (1993).

13. J. Papavassiliou, Phys. Rev. D 41, 3179 (1990).

14. G. Degrassi and A. Sirlin, Phys. Rev. D 46, 3104 (1992).

15. J. Papavassiliou and K. Philippides, to appear in Phys.Rev.D, November 1993.

16. E. N. Argyres et al, Nucl. Phys. B 391, 23 (1993).

17. M. J. Strassler and M. E. Peskin, Phys. Rev. D 43, 1500 (1991).

18. W. Beenakker, S .C. van der Marck, and W. Hollik Nucl. Phys. B 365, 24 (1992).

19. It is important to emphasize at this point that the gauge-invariant self-energies and

vertices obtained by the application of the S-matrix pinch technique do not depend

on the particular process employed (fermion + fermion fermion + fermion, fermion
+ fermion gluon + gluon, gluon + gluon gluon + gluon, etc.) and are in that
sense universal. This fact can be seen in an explicit calculation, where one can be

convinced that the only quantities entering in the definition of the gauge-independent

self-energies and vertices are just the gauge group structure constants so that the only

difference from process to process is in the external group matrices associated with

external leg wave functions -due to the different particle assignments- which are, of

course, immaterial to the definition of the things inside. A very instructive example

of an explicit calculation, where two different processes give rise to exactly the same

self-energy for the W gauge boson, is given in [14].

20. A. Sirlin, Rev. Mod. Phys. 50, 573 (1978).

29


21. In what follows we set Uts and Utd equal to zero.

22. All other box graphs either give zero pinch contributions (boxes with intermediate

or Z [15]), or are proportional to the electron mass me (boxes with intermediate

W +-, +W -, and +-); the latter should not be included in the definition of the

MDM form factor, since they explicitly depend on the kinematic details of the target,

namely the mass of the target fermions.


23. See also [14].


24. We have computed the result in the unitary gauge in two ways. First, we take

the limit of the expression given in Eq. (3.18). Second, we perform the
calculation for the MDM starting from the beginning with propagators of the form

i(k) = [g - kk ] 1 and no unphysical Goldstone bosons. The two results
M 2 k2
i -M2i
turn out to be identical.


25. The fact that all gauge-dependent MDM form factors are increasing functions of s

does not mean of course that the unitarity of the S-matrix is violated. Indeed, when

forming S-matrix elements and cross sections, the gauge-dependent MDM expressions

of Eq. (3.17) and Eq. (3.18) mix with the MDM-like terms concealed in the box

graphs, and conspire in such a way as to preserve unitarity. For this reason, the

result obtained by the PT respects unitarity by itself, since all contributions that are

kinematically akin to MDM have been extracted from the box graphs and already

included in the calculation.


26. I. Bigi, Y. Dokshitzer, V. Khoze, J. K
uhn and P. M. Zerwas Phys. Lett. 181B, 157

(1986).


27. B. A. Kniehl and A. Sirlin Phys. Rev. D 47, 883 (1993);

S. Fanchiotti, B. A. Kniehl and A. Sirlin Phys. Rev. D 48, 307 (1993).


28. All other thresholds (W +W -, bb, etc) are well below the tt production threshold.

30


29. Of course in such a case the usual arguments about the non-hadronization of the

produced tops will be invalidated, and non-perturbative QCD effects may become

important.





11. Figure Captions


1) Graphs (a)-(c) are some of the contributions to the S-matrix T . Graphs (e) and

(f) are pinch parts, which, when added to the usual self-energy graphs (d), give rise to a

gauge-independent effective self-energy.

2a) The two graphs defining the gauge-invariant vertex . The circle in the first


graph indicates the F part of the three-gluon vertex.


2b) Vertex-like pinch contributions from a box graph, in a general R gauge, with

= 1.

3) All the graphs contributing to the t
t and Zt
t vertices.

4) The top quark MDM as a function of s, computed for several different values

of the gauge-fixing parameter w (b,c,d,e), to be contrasted to the gauge-invariant answer

shown in (a). The correspondance is as follows: w = 10 (b), w = 100 (c), w = 1000 (d),

w (e).
5) The case of a "light" top (mt = 130 GeV ), for two different choices of the Higgs

boson mass: mH = 150 GeV , and mH = 500 GeV (darker curve). In particular:

5a) Real part of MDM for an incoming ,

5b) Imaginary part of MDM for an incoming ,

5c) Real part of MDM for an incoming Z,

5d) Imaginary part of MDM for an incoming Z.


6) The case of a "heavy" top (mt = 200 GeV ), for the same two choices of the Higgs

boson mass as in Fig.5 :

31


6a) Real part of MDM for an incoming ,

6b) Imaginary part of MDM for an incoming ,

6c) Real part of MDM for an incoming Z,

6d) Imaginary part of MDM for an incoming Z.


7) The dependence of MDM on the mass mH of the Higgs boson, for a "light" and a

"heavy" top, at s = 500 GeV :

7a) Real part of MDM for an incoming ,

7b) Imaginary part of MDM for an incoming ,

7c) Real part of MDM for an incoming Z,

7d) Imaginary part of MDM for an incoming Z.





32


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Edinburgh U. 93/530
NYUTH93/05/02

November 1993





                    





 +-   



J. Papavassiliou1 and C. Parrinello2.




1 Department of Physics, New York University, 4 Washington Place,
New York, NY 10003, USA.

2 Department of Physics, University of Edinburgh, Mayfield Road
Edinburgh EH9 3JZ, Scotland, UK




arXiv: v1 12 Nov 93


ABSTRACT


Motivated by the possibility of experimental determination of top quark form
factors in upcoming e+e- experiments, as recently discussed by Atwood and
Soni, we show at the one loop level that the conventionally defined magnetic
dipole moment (MDM) form factor of the top quark is gauge dependent, in
the class of renormalizable (R) gauges. We explicitly calculate its gauge
dependence, which, due to the massiveness of the top quark, turns out to be
numerically very sizable. We show how to use the S-matrix pinch technique
in order to define gauge independent form factors. The real and imaginary
parts of the gauge independent MDM form factor are calculated and their
dependence on the top quark and Higgs boson masses is discussed. Most
noticeably, the dependence on the Higgs mass turns out to be stronger than
that of the cross section as a whole.



