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%From: papavass@MAFALDA.PHYSICS.NYU.EDU (Joannis Papavassiliou)
%Date: Mon, 6 Mar 95 16:46:10 EST









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\centerline{\normalsize\bf GAUGE INVARIANCE AND}
\baselineskip=22pt
\centerline{\normalsize\bf ANOMALOUS GAUGE BOSON COUPLINGS}
\baselineskip=16pt
%\centerline{\normalsize\bf MANUSCRIPT BY COMPUTER}
%\centerline{\footnotesize\sf (For subsequent 20\% photoreduction to 17.8
%$\times$ 11.9 cm text area)\footnote{The \LaTeX\ source file for this document
%%may
%be used as a template for your article, and can be requested by e-mailing
%{\sf worldscp@singnet.com.sg}.}}
%\vfill
\vspace*{0.6cm}
\centerline{\footnotesize JOANNIS PAPAVASSILIOU}
\baselineskip=13pt
\centerline{\footnotesize\it Department of Physics, New York University,
4 Washington Place}
\baselineskip=12pt
\centerline{\footnotesize\it New York, NY 10003, USA}
\centerline{\footnotesize E-mail: papavass@mafalda.physics.nyu.edu}
\vspace*{0.3cm}
\centerline{\footnotesize and}
\vspace*{0.3cm}
\centerline{\footnotesize KOSTAS PHILIPPIDES}
\baselineskip=13pt
\centerline{\footnotesize\it Department of Physics, New York University,
4 Washington Place}
\baselineskip=12pt
\centerline{\footnotesize\it New York, NY 10003, USA}
\centerline{\footnotesize E-mail: kostas@mafalda.physics.nyu.edu}
%\centerline{\footnotesize\it Group, Company, Address, City, State ZIP/Zone,
%Country}

%\vfill
\vspace*{0.9cm}
\abstracts{ Using the S--matrix pinch technique we obtain to one loop order,
gauge independent $\gamma W^-W^+$ and  $Z W^-W^+$ vertices in the context of
the standard model, with all incoming momenta off--shell. We show that the
vertices so constructed satisfy simple QED--like Ward identities.
These gauge invariant vertices give rise to
expressions for the
magnetic dipole and electric quadrupole
form factors of the W gauge boson,
which, unlike previous treatments,
satisfy the crucial properties of infrared finiteness and
perturbative unitarity.}


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\section{Introduction}
A new and largely unexplored frontier on which the ongoing search for new
physics will soon focus is the study of the structure of the three-boson
couplings.
A general parametrization of the trilinear gauge boson vertex for
on--shell $W$s and off--shell  $V=\gamma,~Z$ is
\bea
\Gamma_{\mu\alpha\beta}^{V}= & -ig_{V}~\Big[~
 ~f \left[~ 2g_{\alpha\beta}\Delta_{\mu}+ 4(g_{\alpha\mu}Q_{\beta}-
g_{\beta\mu}Q_{\alpha})~\right]
 +~ 2\Delta\kappa_{V}~(g_{\alpha\mu}Q_{\beta}-g_{\beta\mu}Q_{\alpha})
\nonumber \\
& ~~+~  4\frac{
%\big{\Delta Q_{V}}}{\big{M_{W}^{2}}}
% XXX: bigs superfluous
{\Delta Q_{V}}}{{M_{W}^{2}}}
(\Delta_{\mu}Q_{\alpha}Q_{\beta}-
\frac{1}{2}Q^{2}g_{\alpha\beta}\Delta_{\mu})
{}~~\Big]~ +~ ... ~~~,
\eea
with $g_{\eg}= gs$, $g_{Z}= gc$, where $g$ is the  $SU(2)$
gauge coupling, $s\equiv sin\theta_{W}$ and $c\equiv cos\theta_{W}$,
 and the ellipses denote omission of C, P, or T violating terms.
 The four-momenta $Q$ and $\Delta$
 are related to the incoming momenta $q$, $p_{1}$ and $p_{2}$ of
the gauge bosons $V,~W^-$and $W^+$ respectively, by
$q=2Q$, $p_{1}=\Delta -Q$ and $p_{2}=-\Delta - Q$~
\cite{Bardeen}.
The form factors $\Delta\kappa_{V}$ and $\Delta Q_{V}$,
also defined as
$
\Delta\kappa_{V}= \kappa_{V} + \lambda_{V} - 1
$ ,
and
$
 \Delta Q_{V}= -2\lambda_{V}
$,
are compatible with C, P,
and T invariance, and
are  related to the magnetic dipole moment $\mu_{W}$ and the electric
quadrupole moment $Q_{W}$, by the following expressions:
\be
\mu_{W} = \frac{e}{2M_{W}}(2+ \Delta\kappa_{\gamma})
{}~~\mbox{,}~~
Q_{W}= -\frac{e}{M^{2}_{W}}(1+\Delta\kappa_{\gamma}+\Delta Q_{\gamma})~.
\ee
In the context of the standard model,
 their canonical, tree level values are
 $f=1$ and  $\Delta\kappa_{V}=\Delta Q_{V}=0$.
To determine the radiative corrections to these quantities
one must cast the resulting one--loop expressions in the following form:
\be
\Gamma_{\mu\alpha\beta}^{V}= -ig_{V}[
 a_{1}^{V}g_{\alpha\beta}\Delta_{\mu}+ a_{2}^{V}(g_{\alpha\mu}Q_{\beta}-
g_{\beta\mu}Q_{\alpha})
 + a_{3}^{V}\Delta_{\mu}Q_{\alpha}Q_{\beta}]~,
\label{1loopParametrization}
\ee
where $a_{1}^{V}$, $a_{2}^{V}$, and $a_{3}^{V}$ are
 complicated functions of the
momentum transfer $Q^2$, and the masses of the particles
appearing in the loops.
It then follows that  $\Delta\kappa_{V}$ and $\Delta Q_{V}$
are given by the
following expressions:
\be
\Delta\kappa_{V}=\frac{1}{2}(a_{2}^{V}-2a_{1}^{V}-Q^{2}a_{3}^{V})
{}~\mbox{,}~~
\Delta Q_{V}= \frac{M^{2}_{W}}{4}a_{3}^{V}~.
\label{1loopdeltakappaandQ}
\ee
Calculating the one-loop expressions for $\Delta\kappa_{V}$ and
 $\Delta Q_{V}$ is a non-trivial task, both from
the technical and the conceptual point of view.
If one
calculates
just the Feynman diagrams contributing to the $\gamma W^{+}W^{-}$
vertex and then extracts from them the contributions to
$\Delta\kappa_{\gamma}$ and $\Delta Q_{\gamma}$,
one arrives at
expressions that are
plagued with several pathologies, gauge-dependence being one of them.
Indeed, even if the two W are considered to be on shell, since the incoming
photon is not, there is no {\sl a priori}
 reason why a gauge-independent (g.i.) answer
should emerge. In the context of the renormalizable $R_{\xi}$ gauges
the final answer depends on the
choice of the gauge fixing parameter $\xi$, which enters into the one-loop
calculations through the gauge-boson propagators
( W,Z,$\gamma$, and unphysical Higgs particles).
In addition, as shown by an explicit calculation
performed
in the Feynman gauge ($\xi=1$), the answer for
$\Delta\kappa_{\gamma}$
is {\sl infrared divergent} and
violates perturbative unitarity,
e.g. it grows monotonically for $Q^2 \rightarrow \infty$
{}~\cite{Lahanas}.
All the above pathologies may be circumvented
if one adopts the pinch technique (PT)
\cite{Cornwall}.
The application of this method gives rise to
new expressions,
$\hat{\Delta}\kappa_{\gamma}$ and $\hat{\Delta} Q_{\gamma}$,
which are
gauge fixing parameter ($\xi$) independent,
ultraviolet {\sl and} infrared finite, and
well behaved for large momentum transfers $Q^{2}$.

\section{The pinch technique}
The S-matrix pinch technique is an algorithm that allows the construction
of modified g.i. $n$-point functions, through the
order by order rearrangement of
Feynman graphs, contributing to a certain physical
and therefore ostensibly g.i. amplitude,
(an S-matrix in our case). This rearrangement separates the S--matrix
into kinematically distinct pieces, akin to self--energies, vertices
and boxes,
 from which the new effective $n$-point functions can be
extracted. The resulting expressions are independent of the initial
gauge choice
and of the specific process one employs.
 For the case of vertices, the PT
 amounts into identifying and appending  to the usual vertex--graphs
those parts of the box graphs that exhibit a vertex--like structure.
These parts are called pinch parts;
they emerge every time a gauge--boson propagator, a three gauge
boson vertex or a scalar--scalar--gauge boson vertex contributes an
integration
momentum $k_{\mu}$ to the original graph's numerator.
Such momenta,
when contracted with a $\gamma$ matrix, trigger
 an elementary identity of the form
\be
k_{\mu}\gamma^{\mu} P_{L}\equiv \ksl P_{L} =
S_{i}^{-1}(p+k) P_{L} - P_{R}S_{j}^{-1}(p) + m_{i}P_{L} - m_{j}P_{R}~.
\label{BrokenPinch}
\ee
The first term on the r.h.s. of Eq.(\ref{BrokenPinch}) removes the
internal fermion propagator and generates a pinch term, while
the second vanishes on shell. Collecting all vertex-like
pinch parts and appending them to the usual vertex graphs
gives rise to a g.i. sub--amplitude, with the same kinematical properties
as a vertex.

\section{Gauge--invariant gauge boson vertices
 and their Ward identities}

We consider
the S-matrix element for the process
\begin{equation}
e^-  + \nu  +  e^- \to
e^- + e^-  +  {\overline \nu}     ~~.
\label{process}
\end{equation}
and isolate the part $T(q,p_1,p_2)$ of the S--matrix which depends
only on the momentum transfers $q$, $p_1$, and $p_2$.
Since the final result (with pinch contributions included)
is g.i., we choose to work in the Feynman gauge
($\xi_{i}=1$); this particular gauge simplifies the calculations
because it removes all longitudinal parts from the
tree-level gauge boson propagators. So, pinch contributions can
only originate from appropriate momenta furnished by the
tree--level gauge boson vertices.
Applying the pinch technique algorithm we isolate all
vertex--like parts contained
in the box diagrams and allot them to the usual vertex
graphs.
The final expressions for
one loop g.i. trilinear gauge boson vertices are :
\bea
\frac{1}{g^3s}
\hGa^{\gamma W^{-}W^{+}} _{\mu \alpha \beta} & =&
\Gamma^{\gamma W^-W^+}_{\mu \alpha \beta} |_{\xi _i =1}
 + q^{2}~B_{\mu\alpha \beta}
 +U_W^{-1}(p_1)^{\rho}_{\alpha} B^+_{\mu \rho \beta}
  +U_W^{-1}(p_2)^{\rho}_{\beta} B^-_{\mu \alpha \rho}  \nonumber \\
&& - 2 \Gamma_{\mu\alpha\beta} \left[~
I_{WW}(q)+s^2I_{W\gamma}(p_1)+c^2I_{WZ}(p_1)+s^2I_{W\gamma}(p_2)
+c^2I_{WZ}(p_2) ~\right]
\nonumber \\
&& + p_{2 \beta} g_{\mu \alpha} ~ {\cal M}^-
  +p_{1 \alpha} g_{\mu \beta}~  {\cal M}^+ ~~,
\label{gigWW}
\eea
\bea
\frac{1}{g^3c}
\hGa^{ZW^{-}W^{+}} _{\mu \alpha \beta} & =&
\Gamma^{Z W^-W^+}_{\mu \alpha \beta} |_{\xi _i =1}
+ U_Z^{-1}(q)^{\rho}_{\mu} B_{\rho \alpha \beta}
 +U_W^{-1}(p_1)^{\rho}_{\alpha} B^+_{\mu \rho \beta}
  +U_W^{-1}(p_2)^{\rho}_{\beta} B^-_{\mu \alpha \rho}  \nonumber \\
 && -  2 \Gamma_{\mu\alpha\beta} \left[~
I_{WW}(q)+s^2I_{W\gamma}(p_1)+c^2I_{WZ}(p_1)+s^2I_{W\gamma}(p_2)
+c^2I_{WZ}(p_2) ~\right] \nonumber \\
&& + q_{\mu} g_{\alpha \beta}~M_Z^2~ {\cal M}
+ p_{2 \beta} g_{\mu \alpha} ~M_W^2~ {\cal M}^-
  + p_{1 \alpha} g_{\mu \beta}~M_W^2~  {\cal M}^+
\label{giZWW} ~~.
\eea
The quantities $I_{ij}$ and ${\cal M}^{\pm}$
are integrals with two and three scalar propagators, respectively;
their explicit expressions
have been reported elsewhere
\cite{PaPhi}.

The g.i. vertices satisfy the following simple Ward
identities (WI), relating them to the
g.i. $W$ self energy and $\chi WW$ vertex constructed also via the PT :
\be
q^\mu \hGa ^{ZW^-W^+} _{\mu \alpha \beta}
+ iM_Z \hGa ^{\chi W^-W^+} _{\alpha \beta}
 =  gc ~\left[ \hPi ^W _{\alpha \beta} (1) - \hPi ^W _{\alpha \beta} (2)
\right]~~,
\label{qZWW}
\ee
\be
q^\mu \hGa ^{\gamma W^-W^+} _{\mu \alpha \beta}
 =  gs ~\left[ \hPi ^W _{\alpha \beta} (1) - \hPi ^W _{\alpha \beta} (2)
\right] ~~.
\label{qZWW}
\ee
These WI
 are the one--loop generalizations of the respective {\it tree level} WI;
their validity is
crucial for the gauge independence of the S--matrix.
It is important to emphasize that they make no
reference to ghost terms, unlike the corresponding Slavnov-Taylor
identities satisfied by the conventional, gauge--dependent vertices.

For the case of {\it on--shell}
{}~$W$s one sets $p_1^2=p_2^2=M^2_W$ and neglects all
terms proportional to $p_{1\alpha}$ and $p_{2 \beta}$,
as well as the left over pinch terms of the $W$ legs. Then the
$\gamma WW$ vertex reduces to the form
\be
\frac{1}{g^3s}
\hGa^{\gamma W^{-}W^{+}} _{\mu \alpha \beta}  =
\Gamma^{\gamma W^-W^+}_{\mu \alpha \beta} |_{\xi _i =1}
 + q^{2}~B_{\mu\alpha \beta}(q,p_1,p_2)
 - 2 \Gamma_{\mu\alpha\beta}
I_{WW}(q)~~.
\label{gigWW}
\ee
This is of course the same answer one obtains by applying the PT
{\it directly} to the S--matrix of $e^{+}e^{-} \rightarrow W^{+}W^{-}$.
Thus for the form factors ${\Delta\kappa}_{\gamma},~{\Delta Q}_{\gamma}$
the only function we need is $B_{\mu\alpha \beta}$, given below
\be
g^2 B_{\mu\alpha \beta} = \sum_{V=\gamma Z} g_{V}^2
\int\frac{d^4k}{i(2\pi)^4}
\frac{g_{\alpha \beta} ~(k-\frac{3}{2}(p_1-p_2))_{\mu}
-g_{\alpha \mu}~(3k+2q)_{\beta}
-g_{\beta \mu}~(3k-2q)_{\alpha}}
{\left[(k+p_1)^2-M_W^2\right]\left[(k-p_2)^2-M_W^2\right]
\left[k^2-M_V^2\right]}
{}~.
\ee

\section{Magnetic dipole and electric quadrupole form factors for the $W$}

Having constructed the g.i. $\gamma WW$ vertex we proceed to extract
its contributions to the magnetic dipole and electric quadrupole form
factors of the $W$.
We use carets to denote the g.i. one--loop contributions.
Clearly,
\be
\hat{\Delta}\kappa_{\gamma} = \Delta\kappa_{\gamma}^{(\xi=1)}
+ \Delta\kappa_{\gamma}^{P}~,
\ee
and
\be
\hat{\Delta}Q_{\gamma} = \Delta Q_{\gamma}^{(\xi=1)}+  \Delta Q_{\gamma}^{P} ~.
\ee
where $\Delta Q_{\gamma}^{(\xi=1)}$ and $\Delta Q_{\gamma}^{(\xi=1)}$
are the contributions of the usual vertex diagrams in
the Feynman gauge \cite{Lahanas}, whereas
$\Delta Q_{\gamma}^P$ and $\Delta Q_{\gamma}^P$the analogous
contributions from the pinch parts.
By performing the momentum integration in $B_{\mu\alpha\beta}$, we find for
$p_1^2=p_2^2=M_W^2$
\be
B_{\mu\alpha\beta}= -\frac{Q^2}{8\pi^2 M^2_W}\sum_{V=\gamma, Z} g^2_{V}
\int_{0}^{1}da \int_{0}^{1}(2tdt) \frac{F_{\mu\alpha\beta}}{L_{V}^{2}}~,
\ee
where
\be
F_{\mu\alpha\beta}
= 2(\frac{3}{2} + at)g_{\alpha\beta}{\Delta}_{\mu}+
2(3at+2)[g_{\alpha\mu}Q_{\beta}-g_{\beta\mu}Q_{\alpha}]~,
\label{F}
\ee
and
\be
L^{2}_{V} = t^{2}-t^{2}a(1-a)(\frac{4Q^{2}}{M^{2}_{W}}) +
 (1-t)\frac{M^{2}_{V}}{M^{2}_{W}}~,
\ee
from which follows that
\be
{\Delta\kappa}_{\gamma}^{P}= -\frac{1}{2} \frac{Q^{2}}{M^{2}_{W}}
\sum_{V} \frac{\alpha_{V}}{\pi} \int_{0}^{1}da \int_{0}^{1}(2tdt)
\frac{(at-1)}{L^{2}_{V}}~,
\ee
and
\be
{\Delta Q}_{\gamma}^{P} = 0 ~~~.
\ee
We observe that $\Delta\kappa_{\gamma}^{P}$ contains an
infrared divergent term, stemming from the double
integral shown above, when $V=\gamma$. This term cancels exactly
against a similar infrared divergent piece
contained in $\Delta\kappa_{\gamma}^{(\xi = 1)}$,
{}~\cite{Lahanas} thus rendering
$\hat{\Delta}\kappa_{\gamma}$ infrared finite.
After the infrared pieces have been cancelled, one notices that
the remaining contribution of $\Delta\kappa_{\gamma}^{P}$
decreases monotonically
as $Q^2 \rightarrow \pm \infty$; due to the difference
in relative signs this contribution cancels
asymptotically against the monotonically increasing
contribution from
$\Delta\kappa_{\gamma}^{(\xi = 1)}$.
Thus by including the pinch part the
unitarity of $\hat{\Delta}\kappa_{\gamma}$ is restored and
$\hat{\Delta}\kappa_{\gamma} \rightarrow 0$ for large values of $Q^2$.
It would be interesting to determine how these quantities could be
directly extracted from future $e^{+}e^{-}$ experiments.

\section{Conclusions}
We showed how to use the PT in order to construct
g.i.
gauge--boson vertices,
which satisfy naive QED--like Ward identities.
These vertices give rise to
magnetic dipole and electric quadrupole form factors for the $W$,
which can, at least in principle, be promoted to physical
observables.
%give rise to
%gauge independent one--loop corrections $\hat{\Delta}\kappa_{\gamma}$
%and $\hat{\Delta}Q_{\gamma}$
%to the magnetic dipole and electric quadrupole form factors of the $W$,
%which are infrared finite and respect unitarity.

\section {Acknowledgment}
The authors are happy to acknowledge useful discussions with
J.~M.~Cornwall, A.~Lahanas, C.~Papadopoulos,
A.~Sirlin, and D.~Zeppenfeld.
This work was supported in part by the National Science Foundation under
Grant No..



%\normalsize
\section{References}

\begin{thebibliography}{9}
\bibitem{Bardeen} W.~A.~Bardeen, R.~Gastmans, and B.~Lautrup,
 Nucl. Phys.; B 46 (1972) 319.
\bibitem{Lahanas} E.N.~Argyres, G.~Katsilieris, A.B.~Lahanas,
C.G.~Papadopoulos,  and V.C.Spanos, Nucl. Phys. B 391 (1993) 23.
\bibitem{Cornwall}
J.~M.~Cornwall,
in {\sl Proceedings of the French-American Seminar on Theoretical
Aspects of Quantum Chromodynamics},
Marseille, France, 1981, edited J.~W.~Dash (Centre de Physique
Th\'eorique, Marseille, 1982).
\bibitem{PaPhi} J.~Papavassiliou and K.~Philippides,
Phys. Rev. D 48 (1993) 4255.
\end{thebibliography}


\end{document}




Semileptonic top decays in the context of the Weinberg Model
(WM) \cite{Wein} have been the focal point of extensive
study \cite{Gunion}.
In the WM the new basic ingredient is the possibility
of inducing CP violating effects in the leptonic sector, due to the presence
of the additional charged Higgs sector. The way such CP violating effects
arise can be seen from the relevant Lagrangian term
\begin{equation}
{\cal L}= \frac{gm_{t}}{\sqrt{2} M}{\bar{t}}_{R}b_{L}
\frac{c_{1}c_{2}s_{3}-s_{2}c_{3}e^{i\delta}}{s_{1}c_{2}}H^{+}
-\frac{gm_{\tau}}{\sqrt{2} M}{\bar{\nu}}_{L}\tau_{R}
\frac{c_{1}s_{2}s_{3}+c_{2}c_{3}e^{i\delta}}{s_{1}c_{2}}H^{+}
+h.c.~~,
\label{Lagr}
\end{equation}
involving Yukawa couplings between the extra charged Higgs $H^{+}$ and the
fermions (quarks and leptons).
We note that the constants $s_{i}$, $c_{i}$ and $\delta$
appear in the CKM-like matrix operating in the charged Higgs sector and are
not elements of the usual CKM matrix; $M$ is the mass of the $W$.
The possibility for additional CP violating effects has been studied in the
context of the decay mode $t\rightarrow b\tau\nu$.
The observable considered is the partial decay rate
asymmetry (PRA), namely
\begin{equation}
{\cal A}=\frac{\Gamma (t\rightarrow b \tau^{+}\nu_{\tau})
-\Gamma (\bar{t}\rightarrow \bar{b}\tau^{-}\bar{\nu_{\tau}})}
{\Gamma (t\rightarrow b \tau^{+}\nu_{\tau})
+\Gamma (\bar{t}\rightarrow \bar{b}\tau^{-}\bar{\nu_{\tau}})}~.
\label{DefA}
\end{equation}

At one loop the PRA receives contributions through interference terms
between one-loop Standard Model (SM) graphs for the process
$t\rightarrow W^{+}b\rightarrow b\tau^{+}\nu_{\tau}$,
and the tree-level WM graph for the
process $t\rightarrow H^{+}b\rightarrow b\tau^{+}\nu_{\tau}$.
Consequently, the entire effect is proportional to $m_{t}m_{\tau}$.
Due to helicity mismatches \cite{Soni1} only the
longitudinal parts of the SM graphs contribute to the PRA. In addition,
due to the fact that the Higgs couplings are complex numbers, it is only the
imaginary parts of such longitudinal contributions which is relevant.
So, ${\cal A}$ is proportional to
\begin{equation}
{\cal A}\sim \int dq^{2}f(q^{2})Im(G_{L})~,
\label{Phase}
\end{equation}
where $f(q^{2})$ is a phase space function \cite{Soni2}
and $G_{L}$ is the longitudinal component of any one-loop graph.
In addition, it is important to notice the presence of the phase space
integral, whose range extends from $m_{\tau}^{2}$ all the way up to
${(m_{t}-m_{b})}^{2}$.

In computing the one-loop contribution to ${\cal A}$ the only graphs
considered was the $W$ self-energy graphs, containing fermionic loops (
although, as we will see in the next section, they are not the
only graphs contributing to ${\cal A}$). The original motivation
for singling
out the $W$ propagator with fermionic loops was the
expectation that due to the
general {\sl resonant} nature of such graphs, significant enhancement of the
PRA might take place. As it was soon realized however \cite{Soni1}
this
resonant behavior could not
be exploited, because it is only the longitudinal parts of
the self-energy graphs which contribute to the PRA,
and
it is only the transverse (but not the longitudinal) parts of the $W$
self-energy, which displays resonant behavior. So, when the
$W$ propagator is
decomposed in the form
\begin{equation}
{G}_{\mu\nu}=(g_{\mu\nu}-\frac{q_{\mu}q_{\nu}}{q^{2}})G_{T}+
\frac{q_{\mu}q_{\nu}}{q^{2}}G_{L}~,
\label{Decomp}
\end{equation}
with
\begin{equation}
G_{T}= \frac{1}{q^{2}-M^{2}+i\epsilon_{T}}~,
\label{Pt}
\end{equation}
and
\begin{equation}
G_{L}= \frac{1}{M^{2}+i\epsilon_{L}}~,
\label{Pl}
\end{equation}
where
\begin{equation}
\epsilon_{T}=(\frac{g^{2}}{32\pi})
\frac{m_{c}^{2}{(2q^{2}+m_{c}^{2})(q^{2}-m_{c}^{2})}^{2}}{q^{4}}~,
\label{Et}
\end{equation}
and
\begin{equation}
\epsilon_{L}= (\frac{3g^{2}}{32\pi})
\frac{m_{c}^{2}{(q^{2}-m_{c}^{2})}^{2}}{q^{4}}~.
\label{El}
\end{equation}
{}From Eq(\ref{Pt}) and Eq(\ref{Pl}) follows that
\begin{equation}
Im(G_{T})= -\epsilon_{T}{|G_{T}|}^{2},~~~
Im(G_{L})= -\epsilon_{L}{|G_{L}|}^{2}~.
\label{ImPt}
\end{equation}
We notice that
due to rescattering the
$\tau\nu$ loop should not contribute
for CPT to be an exact symmetry, so that the next
threshold is due to the $cs$ loop. Finally, when $ImG_{L}$ of
Eq(\ref{ImPt}) is
inserted in Eq(\ref{Phase}) (instead of the resonant $ImG_{T}$ which does not
contribute), the result is very small (${\cal A}\sim 10^{-8}$).

In an attempt
to exploit the resonant character of $ImG_{T}$,
one then proceeded
to compute {\sl two loop} contributions \cite{Soni1} to ${\cal A}$.
In the two-loop calculation the helicity mismatch argument operating at
one-loop is not valid any more. Thus, the resonant $ImG_{T}$ starts
contributing. So, in this calculation one hopes to compensate the suppression
from the extra powers of the coupling constant (due to the second loop)
with the resonant contributions now present, in such a way that the two-loop
resonant contributions are effectively comparable
to one-loop contributions.
In estimating ${\cal A}$ the values of
$s_{i}$,$c_{i}$, and $\delta$
have been maximized, subject to all experimental constraints.
In particular, for $M_{H^{+}}=200 GeV$, $s_{1}=0.252$,
$s_{2}= 8.29\times 10^{-3}$, $s_{3}=0.707$, and $\delta=\frac{\pi}{2}$,
we have that ${\cal A} = - 3.9\times 10^{-5}$.


\section{New one-loop contributions}

As already indicated in the previous section, there is an entire class of
graphs which contribute to ${\cal A}$ at one-loop,
which have not been
included in the original
 calculations. Such contributions originate from
imaginary parts of self-energy, vertex, and box diagrams, which contain
gauge boson loops instead of fermionic loops. The reason such graphs
contribute to ${\cal A}$ is due to the fact that ${\cal A}$ receives
contributions through the entire phase space integration range, from
$m_{\tau}^{2}$ to $m_{t}^{2}$. There are two types of such thresholds:

i) bosonic thresholds, opening when $q^{2}>M^{2}$,
($W\rightarrow W\gamma$); clearly, the imaginary
parts of such graphs contribute in the phase space
integration for $q^{2}>M^{2}$.

ii) top thresholds, corresponding to $t\rightarrow Wb$, from vertex and box
(but not $W$ self-energy) graphs.
The imaginary parts of such graphs
are non-vanishing for every value of $q^{2}$, as long as
$m_{t}^{2}>M^{2}+m_{b}^{2}$, which is of course true.

As before, only the longitudinal components contribute to ${\cal A}$
at one-loop. Moreover, such contributions are non-resonant, just as the
longitudinal $W$ self-energy graphs containing fermion loops.
However, since
there is no suppression factor $\frac{m_{c}^{2}}{M^{2}}$ in this case,
such graphs are in general expected to contribute significantly; as we will
see shortly, this is indeed the case.

Having realized the relevance of the new thresholds, their computation is in
principle straightforward. All one needs to do is isolate the longitudinal
contributions and then compute their imaginary parts. It turns out that
the process of isolating the longitudinal parts is significantly
facilitated if one uses a particular type of gauges. So, instead of
using the common choice of the renormalizable $R_{\xi}$ gauges,
we will work in the context of the background field gauges (BFG) \cite{Abb},
using appropriate Feynman rules.
The reason for this choice is the fact
that in the BFG framework, the self-energy and vertices satisfy the
following set of naive, QED-like Ward identities:
\begin{equation}
q^{\mu}q^{\nu}{\hat{\Pi}}_{\mu\nu}
-2Mq^{\mu}{\hat{\Theta}}_{\mu}+M^{2}\hat{\Omega}=0 ~~,
\label{WIa}
\end{equation}
\begin{equation}
q^{\mu}{\hat{\Pi}}^{\mu\nu}-M{\hat{\Theta}}_{\nu}=0 ~~,
\label{WIb}
\end{equation}
\begin{equation}
q^{\mu}{\hat{\Gamma}}_{\mu}-M\hat{\Lambda}=0 ~~.
\label{WIc}
\end{equation}
where
${\hat{\Pi}}_{\mu\nu}$ is the $W^{+}W^{-}$ self-energy,
${\hat{\Theta}}_{\mu}$ is the $\phi^{+}W^{-}$ mixing term,
$\hat{\Omega}$ the $\phi^{+}\phi^{-}$ self-energy,
${\hat{\Gamma}}_{\mu}$ is the $Wtb$ (or $W\tau\nu$) vertex
and $\hat{\Lambda}$ is the $\phi tb$ (or $\phi\tau\nu$) vertex,
all of them computed to one-loop, in the context of the BFG.~
$\phi^{+}$ is the charged would-be Goldstone boson.
All the above quantities depend in general on the
gauge-fixing parameter $\xi_{Q}$, used to gauge-fix the quantum
field inside the loops. However, since the final answer is guaranteed to be
$\xi_{Q}$-independent, provided {\sl all} graphs are included, any choice
for  $\xi_{Q}$ is legitimate; in particular, we choose  $\xi_{Q}=1$.

Returning to the Ward identities,
it is relatively straightforward to
exploit them, in order to decompose the amplitude in transverse and
longitudinal pieces, without detailed knowledge of the explicit closed
expressions of the individual graphs \cite{Pap}.
We define $\Gamma_{0}^{\mu}=
\frac{g}{2\sqrt{2}}\gamma^{\mu}(1-\gamma_{5})$ and
$\Lambda_{0}=
\frac{g}{2M\sqrt{2}}[m_{1}(1-\gamma_{5})-m_{2}(1+\gamma_{5})]$;
when sandwiched between on shell external spinors
$u_{1}(p_{1})$ and $u_{2}(p_{2})$, with $q=p_{1}-p_{2}$ the identity
${\bar{u}}_{1}q_{\mu}\Gamma_{0}^{\mu} u_{2}=
{\bar{u}}_{1}M_{w}\Lambda_{0} u_{2}$ holds.
Furthermore,
we define
\begin{equation}
{\hat{\Gamma}}_{\mu}^{t}
= {\hat{\Gamma}}_{\mu}+ \frac{q_{\mu}}{q^{2}}M\hat{\Lambda}~,
\label{Gt}
\end{equation}
and
\begin{equation}
{\hat{\Pi}}_{\mu\nu}^{t}=
{\hat{\Pi}}_{\mu\nu}-
\frac{q_{\mu}q_{\nu}}{q^{2}}M\hat{\Theta}~.
\label{pit}
\end{equation}
Both ${\hat{\Gamma}}_{\mu}^{t}$ are
${\hat{\Pi}}_{\mu\nu}^{t}$ are transverse, e.g.
\begin{equation}
q^{\mu}{\hat{\Gamma}}_{\mu}^{t}=0, ~~~q^{\mu}{\hat{\Pi}}_{\mu\nu}^{t}=0~~.
\label{qw}
\end{equation}
Using the identity
\begin{equation}
\frac{1}{M^{2}}= \frac{1}{q^{2}}+ \frac{q^{2}-M^{2}}{q^{2}M^{2}}~,
\label{Ident}
\end{equation}
we obtain \cite{Pap} for the propagator-like contribution
${T}_{1}$ of the
$S$-matrix element
\begin{equation}
{T}_{1}=
{\Gamma}^{\mu}_{0}[\frac{1}{q^{2}-M^{2}}]{\hat{\Pi}}_{\mu\nu}^{t}
[\frac{1}{q^{2}-M^{2}}]{\Gamma}^{\nu}_{0}+
\Lambda_{0}[\frac{1}{q^{2}}]\hat{\Omega}[\frac{1}{q^{2}}]\Lambda_{0}~,
\label{T1}
\end{equation}
and for the vertex-like piece ${T}_{2}$
\begin{equation}
{T}_{2}=
\Gamma^{\sigma}_{0}
[\frac{g_{\sigma}^{\mu}}{q^{2}-M^{2}}]
{\hat{\Gamma}}_{\mu}^{t}
-\Lambda_{0}[\frac{1}{q^{2}}]\hat{\Lambda}~.
\label{T2}
\end{equation}
It is important to notice that the longitudinal parts of
Eq(\ref{T1}) and Eq(\ref{T2})
are multiplied by the kinematic factor $\frac{1}{q^{2}}$, instead of
$\frac{1}{q^{2}-M^{2}}$; they are therefore manifestly non-resonant,
in the entire range of
the phase space integration, even at $q^{2}=M^{2}$.


\section {Calculations and results}

By virtue of this decomposition, we only need to calculate self-energy and
vertex graphs with a charged $\phi$ (but not $W$) coming in;
this represents
a significant calculational
simplification. On the other hand, since no such simple
decomposition exists for
box-like parts of the $S$-matrix,
we will compute the imaginary contributions of box diagrams
directly, and then isolate their longitudinal parts.
It turns out that graphs containing a $Z$ or a $\phi_{z}$
inside their loops
are numerically
suppressed. Since all such graphs form a gauge-invariant subset, their
omission does not interfere with the gauge independence of the final
answer.

The effect of these contributions
is additionally enhanced due to the presence of large logarithms
of the form $ln(\frac{m_{t}^{2}}{m_{b}^{2}})$,
$ln(\frac{m_{t}^{2}}{m_{\tau}^{2}})$, and
$ln(\frac{m_{t}^{2}}{M^{2}})$, which originate from vertex and box
diagrams.
After collecting all contributions and integrating over the available
phase space, using the same values for the constants
$s_{i}$, $c_{i}$ and $\delta$ and $M_{H^{+}}$ as before,
we finally find
${\cal A} = - 2.0\times 10^{-5}$.

We notice that:

i) The result of these new threshold is comparable to the outcome of the
two loop resonant calculation, and at least two
orders of magnitude larger then
the one-loop fermionic contributions.

ii) The new result comes with the same relative sign as the two-loop
result; therefore, the entire effect is to further enhance
the value of the PRA.

\section {Conclusions}

In this paper we addressed issues related to the calculation of
the PRA in the WM. We focused on semileptonic top decays,
on the dominant channel $t\rightarrow b\tau\nu$.
We showed that due to the fact that the PRA receives contributions from
the entire kinematically available phase space, new one loop contributions,
not previously considered, arise.
Such contributions are non-resonant and gauge-invariant.
 It turns out that the PRA so obtained is
two orders of magnitude larger than the one calculated form the
non-resonant fermionic contributions to the $W$ self-energy,
and are comparable to the two loop result.
%As it has already been pointed out \cite{Soni1} however,
% the PRA may not be
%the most appropriate observable, and other alternatives have been proposed,
%which seem to give more sizable effects.

\section {Acknowledgment}
The author thanks Professor
Albero Sirlin for suggesting this problem to him.
This work was supported by the National Science Foundation under
Grant No..

%\normalsize

\section{References}

\begin{thebibliography}{9}
\bibitem{Wein} S. Weinberg, {\it Phys. Rev. Lett.} {\bf 37} (1976) 657.
\bibitem{Gunion} R. Cruz, B. Grzadkowski, and J. F. Gunion,
{\it Phys. Lett.} {\bf B289} (1992) 440.
\bibitem{Soni1} D. Atwood, G. Eilam, A. Soni, R. P. Mendel,
and R. Migneron, {\it Phys. Rev. Lett.} {\bf 70} (1993) 1364.
\bibitem{Soni2} D. Atwood, G. Eilam, A. Soni, R. P. Mendel,
and R. Migneron, {\it Phys. Rev.} {\bf D49} (1994) 289.
\bibitem{Abb} L. F. Abbott, {\it Nucl. Phys.} {\bf B185} (1981) 189.
\bibitem{Pap} J. Papavassiliou, {\it Phys. Rev.} {\bf D50} (1994) 5958.
\end{thebibliography}


\end{document}


