%Paper: 
%From: "Mike Booth, Enrico Fermi Institute" <booth@curie.uchicago.edu>
%Date: Fri, 29 Jan 93 19:32:06 -0600
%Date (revised): Wed, 03 Feb 93 19:56:21 -0600
%Date (revised): Wed, 03 Feb 93 20:00:56 -0600
%Date (revised): Wed, 03 Feb 93 20:18:07 -0600
%Date (revised): Wed, 3 Feb 93 20:09:05 CST

%
% The W-boson EDM in the Standard Model, Mike Booth EFI-93-01.
%
% This document is set up to include two figures into the text and to
% come out in landscape mode.  However, in the unfortunate spirit of
% the least common denominator, the default is to do *neither* of
% those.  However, it's easy to change the switches just below
% these comments:
%
% To produce landscape, uncomment "\TwoUptrue".
%
% To include the figures uncomment "\IncludeFigstrue".
%
% If for some reason you want to use the metafont version of the
% figures, you'll need to create the font (use the makefile include)
% and comment out the line "\input epsf".
%

%W-boson EDM in the Standard Model


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%% These switches control the behaviour of the file
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%%%%%% begin newprep.sty   %%%%%%%%%
\catcode`@=11
% Mike Booth's preprint stuff
% inspired by Markus@T31 (artcustom.sty)
% trying to be compatible with revtex (except for \pubnumber, etc)
%
% New version of \caption with narrower text width. In addition the
% caption text is typeset in \small\sl.

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
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%
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% \@ifundefined{@paperid}{\begin{flushright}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%

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%\baselineskip here controls spacing between lines in address
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% than between lines of the address itself.  How do I do that?
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%
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%


\def\settitleparameters{
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%%%%%% end   newprep.sty   %%%%%%%%%


%%%%%% begin slash.sty   %%%%%%%%%
% a bunch of slash macros, compiled and twiddled by Mike Booth
% Slashing is a bit tricky, since it varies from letter to letter
% and I don't know enought tex details to take advantage of that.

% here's a modified version of the TeXsis macro
% it uses the ``not'' character, moves it back, and raises it
% \slashraise and drops it the depth of the character to
% try to handle p, q and g.  It works pretty well, but
% doesn't quite get q right.
% (might try using smash instead of kerning)

% Things to look at for writing a better macro \mathaccent,
% etc, which first appear around pp152-157 of the TeX book.
% Also look at fontdim

% I think the box around the slashed character is too big, but
% I'm not sure how to fix that.

% a couple of things to twiddle
\newdimen\slashraise \slashraise=1pt
%\mathchardef\fslash="3236
\mathchardef\fslash="0236


% This is it, baby
\def\slash@char#1#2{\setbox0=\hbox{$#2$}           % set a box for #2
   \dimen0=\wd0                                 % and get its size
   \dimen2=-\dp0 \advance\dimen2 by \slashraise
   \setbox1=\hbox{$\m@th#1\mkern-13mu\fslash$}
	 \dimen1=\wd1               % get size of "/"
   \ifdim\dimen0>\dimen1                        % #2 is bigger
						%so center / in box
      \rlap{\hbox to \dimen0{\hfil\raise\dimen2\box1\hfil}}%
      #2                                        % and print #2
   \else                                        % / is bigger
      \rlap{\hbox to \dimen1{\hfil$#2$\hfil}}   % so center #2
      \raise\dimen2\box1                              % and print /
   \fi}                                         %

\def\slashchar#1{\mathrel{\mathpalette\slash@char#1}}
% this seems to look the best, so...
\let\slsh=\slashchar
%%%%%% end   slash.sty   %%%%%%%%%



%\input{mydefs}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% mydefs.tex
\catcode`@=11

% if using overcite, change the citation style to a boxed number
\ifx\@ove@rcfont\undefined
  % may need to define citen
  \ifx\citen\undefined
     % cheater's citen
     \def\citen#1{\begingroup \def\@cite##1##2{{##1}}%
	\@citex[]{#1}\endgroup}
  \fi
\else
  \def\@cite#1{$\@ove@rcfont\m@th^{[\hbox{#1}]}$}
\fi

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  \advance \marginparwidth by -\dimen1  \evensidemargin\oddsidemargin
  \hsize\textwidth}
\def\partder#1#2{{\partial #1\over\partial #2}}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% misc definitions
%
%

%\def\slash#1{\mathord{\mathpalette\c@ncel{#1}}}
%\def\steepslash{\c@ncel}

\def\efi{\address{Enrico Fermi Institute\\
              University of Chicago, Chicago, Illinois 60637}}
\def\efiphys{\address{%
  Enrico Fermi Institute and Department of Physics\\
              University of Chicago, Chicago, Illinois 60637}}

\def\bar{\overline}
%
\def\eqb{\begin{equation}}
\def\eqe{\end{equation}}
\def\unit{{\bf 1}}
\def\hardfill#1{\vrule depth \z@ height\z@ width #1}
\def\mpty{\mbox{}}
\def\Dn{d_n}
\def\gam#1{\gamma_{#1}}
\def\gamO{\gam{o}}
\def\delt#1{\delta^{#1}}

\def\Pslsh{\slsh{P}}
\def\pslsh{\slsh{p}}
\def\lslsh{\slsh{l}}
\def\kslsh{\slsh{k}}

\let\goesto\rightarrow
\def\Dleft{\overleftarrow{D}}
\def\Dright{\overrightarrow{D}}
\def\inv#1{{1 \over #1}}
\def\invA{\inv{A}}

%\def\Or#1{{\cal O}(#1)} % order
%\def\Order{{\cal O}}
%\def\OG#1{\Order_{G^{#1}}}
%\def\Op{O}
%\def\W#1{W^{(#1)}}
%\def\ZO{Z_O}

\def\Or#1{O(#1)} % order
\def\Order{O}
\def\OG#1{\Order({G^{#1}})}
\def\Op{{\cal O}}
\def\W#1{{\cal W}^{(#1)}}
\def\ZO{Z_{\Op}}

\def\BGf{background field }
\def\logdiv{\brax\inv{P^4}\ketx}
\def\CA{C_A}
%
% single index stuff
\def\Amu{A_{\mu}}
\def\amu{a_{\mu}}
\def\anu{a_{\nu}}
\def\Pmu{P_{\mu}}
\def\Dmu{D_{\mu}}
\def\Dnu{D_{\nu}}
\def\Dsigma{D_{\sigma}}
\def\Dalpha{D_{\alpha}}
\def\Palpha{P_{\alpha}}
% end of single index stuff\def\BRA#1{\left\langle #1\right|}
\def\bra#1{\langle #1 |}
\def\KET#1{\left| #1\right\rangle}
\def\ket#1{| #1\rangle}
\def\VEV#1{\left\langle #1\right\rangle}
\def\braket#1#2{\langle#1 | #2\rangle}
\def\vev#1{\langle 0 | #1 | 0\rangle}
\def\brax{\bra{x}}
\def\bray{\bra{y}}
\def\ketx{\ket{x}}
\def\kety{\ket{y}}
% end of vev stuff
% various traces
\def\Tr{{\rm Tr}}
\def\tr{{\rm tr}}
\def\trD{\tr_{\rm D}}
\def\trCD{\tr_{\rm C, D}}
\def\trC{\tr_{\rm C}}
% end of traces
\def\Im{{\rm Im}}
\def\F{{\bf F}}%
\def\ss#1{#1}
\def\Loop{F} % until I can come up with a symbol
\def\SigmaR{\Sigma^{\rm ren}}
\def\lr{\leftrightarrow}
\def\Mu#1{m_{u_{#1}}}
\def\Md#1{m_{d_{#1}}}

\let\bigsum=\sum
\def\Amp{{\cal A}}

\def\GammaIR{\Gamma^{\rm IR}}
\def\Sigmab{\bar\Sigma}
\def\ecm{e\,{\rm cm}}

%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% beginning of paper:
%\wider{60pt}
%\pubdate{\today}
\ifTwoUp
  \twoup
\fi

%% begin timestamp.sty
\def\Month{\ifcase\month\or
 January\or February\or March\or April\or May\or June\or
 July\or August\or September\or October\or November\or December\fi
}
\newcount\hr
\newcount\min
\newcount\xx
\def\TimeStamp{\hr=\time \divide \hr by 60
        \xx=\hr \multiply\xx by 60
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	\hbox{\ifnum \hr < 10 0\fi\number\hr :\number\min}}

%% end timestamp.sty

%\ifdraft
%  \pubdate{{\ }\\ \hbox{Revised \number\day\ \Month\
%	\number\year\ \TimeStamp}}
%\else
% %\pubdate{February, 1992\\ \hbox{Revised October, 1992}}
%  \pubdate{Revised October, 1992}
%\fi

%\font\crest=uofc-shield% scaled 500
%\publabel{\crest C}
%\publabel{C}
\pubnumber{EFI-93-01\\ 
\pubdate{January, 1993}
\begin{document}
\begin{titlepage}
\title{The Electric Dipole Moment of the W and Electron in the
	Standard Model}
\author{Michael J.~Booth%
 \footnote{Electronic address: booth@yukawa.uchicago.edu}
}
\efiphys
\maketitle
\begin{abstract}
I show that the electric
dipole moment of the W-boson $d_W$ vanishes to two loop order in the
standard Kobayashi-Maskawa Model of $CP$ violation.  The argument is a
simple generalization of that used to show the vanishing of the quark
electric dipole moment.  As a consequence, the electron electric
dipole moment vanishes to {\em three} loop order.
Including QCD corrections may give a
non-vanishing result; I estimate $d_W \approx 8\cdot 10^{-31}\ecm$,
which induces an electron EDM $d_e \approx 8 \cdot 10^{-41}\ecm$,
considerably smaller than a previous calculation.
%The result shows that the standard model
%prediction for the electron electric dipole moment is
%even smaller than previously thought.
\end{abstract}
\pacs{11.30.Er, 12.15.Ji, 13.10.+q, 13.40.Fn}
\end{titlepage}
\section{Introduction}

Any model of $CP$ violation will in general induce ---
through loop effects --- $P$ and $T$ violating electric
dipole moments (EDMs) for elementary particles, including
the $W$-boson.
The $P$ and $T$ violating interaction of the $W$ boson with
a photon can be described by the effective Lagrangian\cite{Peccei}
\eqb
\label{lagrangian}
{\cal L}^{CP}_{W\gamma} = i\lambda_1 \tilde F_{\mu\nu}W^\mu W^\nu +
	i{\lambda_2\over M_W^2} \tilde F_{\mu\nu}W^\mu{}_\sigma
	W^{\sigma\nu}.
\eqe
Here $\tilde F_{\mu\nu} = \frac{1}{2}\epsilon_{\mu\nu\alpha\beta}
	F^{\alpha\beta}$,
	$W_{\mu\nu} = \partial_\mu W_\nu - \partial_\nu W_\mu$
and $W_\nu$ is the $W$-boson field%
\footnote{
  As an aside, note that the first term in
  eqn. (\ref{lagrangian}) is not $SU(2)$ gauge invariant, so it
  should be accompanied by operators containing the Higgs field and its
  coefficient should be proportional to $SU(2)$ breaking terms ---
  it would not be present in an unbroken theory.
}.
In terms of the Lagrangian eqn (\ref{lagrangian}),
the coefficient of the W-boson EDM (WEDM) is then given by $d_W =
	(\lambda_1 + \lambda_2/M_W^2)(e/ 2 M_W)$.

Although it is possible in principle to measure the WEDM directly, for
example in scattering experiments\cite{Queijeiro}, it will also
contribute
through loops in lower energy phenomena.
%At the phenomenological level,
It was proposed years ago as an explanation of $CP$ violation
in $K^0\ \bar{K}^0$ mixing\cite{Salzman^2}.  In most models, the WEDM
(and related operators) give the dominant contribution
to the the electron EDM.  This is believed to be the case in
the Standard Kobayashi-Maskawa Model (KM model) of $CP$ violation,
where the contribution of the WEDM to the electron EDM has been
calculated to be $d_e \simeq 10^{-38}\ecm$\cite{Hoogeveen},
although this is in conflict with an estimate of $d_W$%
\cite{ChangWEDM} which is itself on the order of $10^{-38}\ecm$.

In this note I will show that in fact $d_W$ vanishes to
two-loop order in the KM model.  This in turn implies that
$d_e$ vanishes to three-loop order, contradicting the result
of ref. (\citen{Hoogeveen}). The remainder of this paper
is organized as follows:  in section two I
review the mechanics of $CP$ violation in the KM model. In section
three I demonstrate the vanishing of $d_W$ and in the following
section I estimate the three-loop QCD contribution to
$d_W$ and $d_e$.
Finally, in section five I conclude.

%In fact, as is well known, the KM model stands out among models
%because it predictions for $CP$ violation are so small.  In the past
%it has been shown that direct two-loop contributions to the
%quark\cite{Shabalin} and lepton\cite{Donoghue} EDMs vanish.
%Here I will show that the WEDM also vanishes to two-loop order.
%As a consequence, the electron EDM is expected to be signifigantly
%smaller
%than previously estimated.
%It is at least a four loop effect
%and is thus estimated to be ??.

\section{The KM Model}

Unitarity of the CKM matrix and rephasing invariance imply that
all $CP$ violating effects in the standard model are governed by
the quantity $\Phi_{u_1 d_1}^{u_2 d_2} =
 \Im(V_{u_1d_1} V^*_{u_2d_1} V_{u_2d_2} V^*_{u_1d_2})$.
Here $V_{ud}$ is KM matrix element and $u_i, d_j$ are arbitrary
up and down-type quarks.
$\Phi$ has two important properties\cite{Jarlskog,Isi}: it is
antisymmetric in $(u_1, u_2)$ and $(d_1, d_2)$ and the sum over any
index, {\it eg}. $u_1$, vanishes.  For three families there is the
particularly simple relation\cite{Jarlskog}
$\Phi_{u_1 d_1}^{u_2 d_2} = J\, \sum_{\gamma k} \epsilon_{u_1 u_2
\gamma} \epsilon_{d_1 d_2 k}$.
Any $CP$ violating amplitudeis given by summing $\Phi$ together with
the Feynman diagrams
$\Amp_{u_1 u_2}^{d_1 d_2}$
% $\Amp(\Mu1,\Mu2,\Md1,\Md2)$
% (and possibly
% factors of the form $|V_{u d}V_{u' d'}|$) for each
for each configuration of quarks that contribute to the
%for the particular
process in question.  $\Amp$ is a function of quark masses and
possibly KM angles (not phases).  In the case where the weak
interactions
occur along a single quark line --- as is true for the diagrams
which generate $d_W$ --- there is no additional dependence on the
KM angles so that one has
$\Amp_{u_1 u_2}^{d_1 d_2} = \Amp(\Mu1^2,\Mu2^2,\Md1^2,\Md2^2)$.
Since the weak interactions in the KM model are purely left handed,
quark masses enter the diagrams quadratically, through the
denominators of the rationalized propagators%
\footnote{
  Strictly speaking, this is true only in the unitary gauge.  For
  other gauge choices, mass dependence will also enter through the
  unphysical Higgs vertices.  However, it is possible to arrange
  the calculation so that these masses drop out.
}.
Because of the antisymmetry of $\Phi$, only those
parts of Feynman diagrams which are not symmetric under the exchange
of up or down quark masses will contribute to $CP$ violation. This
anti-symmetrization, which is the GIM mechanism,
leads to cancellations between the contributions of
different quarks and is responsible for suppressing most $CP$
violating effects in the KM model\cite{BBS}.

\section{The WEDM in the KM Model}
The two loop contributions to the WEDM in the KM model are
shown in figure~1.  An additional set of five is
generated by interchanging the roles of the up and down quarks in the
loop, for a total of ten diagrams.  I will only consider
the first set, since the treatment of the second set is exactly
parallel.

\ifIncludeFigs
\begin{figure}
     \ifEPSF
	\centerline{\epsfbox{wedm-fig1.eps}}%
     \else
       \begin{center}
         \input wedm-fig1
       \end{center}
     \fi
   \caption{\label{wedm}
   The two-loop contributions to $d_W$.
   }
\end{figure}
\fi

It is easy to eliminate from further consideration
the diagram where the photon attaches to
the bottom of the loop in figure \ref{wedm}:
the photon momentum $k$ appears only in the down quark line, so
the two up-quark propagators have the same momentum dependence.
Consequently, the
diagram is symmetric in the up-quark masses and as discussed
earlier will not contribute to the WEDM; it will be clear from the
renormalized form of the self-energy that this statement remains true
after renormalization.

Instead of studying the four remaining diagrams directly, consider a
simpler set of diagrams obtained by cutting the $d_2$ quark line
in figure \ref{wedm}.
%opening up the quark line at the ``x'' in figure \ref{wedm}.
They are shown in figure \ref{1loop}.

\ifIncludeFigs
\begin{figure}
   \ifEPSF
     \centerline{\epsfbox{wedm-fig2.eps}}
   \else
     \begin{center}
       \input wedm-fig2
     \end{center}
   \fi
    \caption{\label{1loop} Vertex sub-graphs of the diagrams of
	figure 1. }
\end{figure}
\fi
%It is easy to see that the only contributions we must consider are
%where the photon is attached inside the external $W$ lines.  Attaching
%it outside corresponds to the case where the photon attaches to the
%bottom of the diagram in figure 1.  This diagram is symmetric in the
%?up quark masses and thus does not contribute.

%The diagrams of figure \ref{1loop} may be combined as (keeping only
%the chiral projectors from the W-vertices)
The diagrams of figure \ref{1loop}
combine to give (keeping only
the chiral projectors from the W-vertices)
\begin{equation}
\Amp_\mu = L {1\over \pslsh - {\kslsh\over2} - \Mu2}
  \Gamma_\mu(p, k)
  {1\over \pslsh + {\kslsh\over2} - \Mu1} R
\end{equation}
where $\Gamma_\mu$ is the complete vertex function defined by
\begin{eqnarray}
\Gamma_\mu(p,k) &=&
  \gamma_\mu
    {1\over\pslsh-{\slsh k\over2} -\Mu2}\Sigma(p-k/2) +
  \mpty \Sigma(p+k/2){1\over\pslsh+{\slsh k\over2} -\Mu1}\gamma_\mu
  \nonumber \\
&&\mpty +\GammaIR_\mu(p,k),
\end{eqnarray}
$\Sigma$ is the renormalized self-energy function and $\GammaIR$ is the
irreducible vertex function.  These have been computed many times
before%
\cite{Shabalin,ShabalinV,Others} and their detailed forms
are not required.  However,
because the results are not generally well known, I will sketch
the calculation.
%simply show the results.
Since the EDM is a static effect, I will
expand the vertex in powers of the photon momentum $k$, keeping only
the first term\footnote{
  An equivalent analysis of the vertex function using a
  different approach has recently been presented in
  ref. \citen{Me}.
}.


After renormalization, the self-energy $\Sigma$ takes the form
\begin{equation}
\Sigma_{u_2 u_1} =
        (\pslsh - m_{u_2})\bar\Sigma_{u_2u_1}(\pslsh - m_{u_1})
\end{equation}
%
with
\begin{equation}
\bar\Sigma = F_0^{(1)}(p^2) ( \pslsh R + m_{u_2}R + m_{u_1}L )
 + F_0^{(2)}(p^2)\pslsh L.
\end{equation}
Here $F_0^{(1)}$ and $F_0^{(2)}$ are %defined by?
\begin{eqnarray}
F_0^{(1)} & = & {p^2 F_0 + \Mu1\Mu2 X_0\over
	(p^2 - \Mu1^2) (p^2 -\Mu2^2)},\\
F_0^{(2)} & = & {\Mu1\Mu2F_0 + (\Mu1^2 + \Mu2^2 - p^2)X_0\over
	(p^2 - \Mu1^2) (p^2 -\Mu2^2)}
\end{eqnarray}
and
\begin{eqnarray}
F_0 &= &f(p^2) -
	{\Mu1^2f(\Mu1^2) - \Mu2^2 f(\Mu2^2) \over \Mu1^2 - \Mu2^2},\\
X_0 & = & \Mu1\Mu2{f(\Mu1^2) - f(\Mu2^2) \over \Mu1^2 - \Mu2^2}.
\end{eqnarray}
Finally, $f(p^2)$ is the function which occurs in the unrenormalized
self energy, $\Sigma_0(p) = f(p^2)\pslsh L$.
Similarly, the irreducible vertex has the form
\begin{eqnarray}
\GammaIR_\mu(p,k) = -Q_u \partder{}{p_\mu} \Sigma +
 f_2 \{\pslsh, \sigma_{\mu\nu} k^\nu \}L,
\end{eqnarray}
where $f_2(p^2)$ does not depend on the down-quark masses.

In terms of the above functions, $\Amp_\mu$ has the relatively
simple form
(to \Order(k))
\begin{eqnarray}
\Amp_\mu(p,k) &=& \{\pslsh, \sigma_{\mu\nu} k^\nu \} R\,
{p^2 f_2(p^2) + p^2 F_0^{(1)} - \Mu1\Mu2 F_0^{(2)} \over
	(p^2 - \Mu1^2)(p^2 - \Mu2^2)} \nonumber\\ &&\mpty
- Q_u \partder{}{p_\mu} L\Sigmab(p)R.
\end{eqnarray}
Recalling the expression for $\Sigmab$ one sees that
$\Amp_\mu$ is a symmetric function of $\Mu1$ and $\Mu2$.
It is this symmetry which is at the heart of all the vanishing
two loop effects in the KM model.
%Since this holds for any $W$ momentum $p$,
It follows that the diagrams of figure 1 combine to produce
a symmetric function of the up-quark masses
%Thus,
so that
as I intended to show,
the W-boson EDM vanishes to two loops in the Standard Model.

\section{Estimates}
In order to obtain a non-zero WEDM, it is necessary to destroy the
symmetric way in which the quark propagators enter the diagrams.
This can be accomplished by including QCD loop corrections.
The resulting
3-loop diagrams are rather difficult to calculate. One way to
simplify these calculations would be to perform an effective
lagrangian expansion, including gluonic operators, and then
integrate out the gluon fields.
However,
in view of the small expected size of the WEDM in the KM model,
a cruder estimate will suffice.

Because the GIM mechanism effectively cuts off the
the momentum integrals, it is reasonable to estimate them
by their infrared limits.  Gluon loops will typically
introduce logarithmic mass dependence.  This is also true of
the functions $f(p^2)$ and $f_2(p^2)$.  Moreover, the entire
diagram has a superficial logarithmic divergence. Consequently
I expect the mass dependence of the diagrams contributing to
$d_W$ to be logarithmic.
% this is born out by explicit calculations in similar problems.
Because of the insensitivity of the
logarithm --- even for the widely separated scales in the KM model
--- the GIM cancellation will be weak and
%(here one should take $m_u$ and $m_d$ to be constituent masses)
I can obtain a reasonable (over-) estimate of $d_W$ by setting these
logs to 1.  I then obtain
\eqb
d_W \approx J \big( {1\over 16 \pi^2}\big)^2 \big({g_W^2 \over8}\big)^2
	\big({\alpha_s\over 4\pi}\big)
	\big({e\over 2 M_W}\big) \simeq 8 \cdot 10^{-30} \ecm.
\eqe
In ref. (\citen{ChangWEDM}) Chang {\it et al.} estimated the vanishing
two-loop contribution to be
\eqb
d_W = J \big({g_W^2\over 8 \pi^2}\big)^2 \big({e\over 2 M_W}\big)
	{m_b^4 m_s^2 m_c^2\over M_W^8}.
\eqe
However, this estimate is unduly pessimistic because it assumes the
quark mass dependence of the diagrams is polynomial, leading to
the small mass ratios in their estimate.

In order to estimate the electron EDM $d_e$, I use the
analysis of Marciano and Queijeiro\cite{Marciano}, who have updated the
original study by Salzman and Salzman\cite{Salzman^2}.
They calculate the contribution of
$d_W$ to $d_e$ and obtain the relation
\eqb
d_e \approx \big({g_W^2 \over 32 \pi^2} \big) \big({m_e\over M_W})
  \big[\ln{\Lambda^2\over M_W^2} +\Order(1)\big] \, d_W,
\eqe
where $\Lambda$ is a cutoff describing the scale of ``new physics''.
Setting the term in brackets to 1, I find for the KM model
\eqb
d_e \approx 8 \cdot 10^{-41}\ecm,
\eqe
which is considerably smaller than the value obtained by Hoogeveen
\cite{Hoogeveen}, whose calculation of $d_e$ contains the
graphs of figure \ref{wedm} as sub-graphs and should thus vanish.
\section{Conclusions}

%I have shown that the WEDM vanishes to two loop order in the
%KM model of $CP$ violation.
%This result shows that the W-boson
The vanishing of the WEDM to two-loop order in the KM model
shows that the W-boson
is on the same footing as all other fundamental particles in the
KM model: none of them possess electric dipole moments to two-loop
order.  This was shown years ago for the quarks\cite{Shabalin,Donoghue}
and leptons\cite{Donoghue} and more recently it was shown that
the chromo-electric dipole moment of the gluon (the Weinberg operators)
also vanishes\cite{Me}.

In order to generate a non-vanishing EDM for these particles,
it is necessary to destroy the symmetric way in which the quark
propagators enter the diagrams.  This can be done by considering
higher-order operators, or at the cost of another loop
by including QCD corrections.  Either way, this leads
to an extra suppression of $CP$ violating effects in the standard
model.

\noteadded
After completing this work, I became aware of the work of
Khriplovich and Pospelov (ref. \citen{Khriplovich}) who also consider
this problem.  Inspired by the same work
(E.~P.~Shabalin, ref. \citen{Shabalin}) we reach the same conclusion
about the vanishing $d_W$ and $d_e$.  My work can be viewed as
an extension of theirs in so far as I obtain actual estimates
of the QCD loop contribution to $d_W$ and $d_e$.


\acknowledgements
I am grateful to Scott Willenbrock for bringing reference
\citen{Khriplovich} to my attention.
This work was supported in part by DOE grant
AC02 80ER 10587.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%70
%% References:
\def\jvp#1#2#3#4{#1~{\bf #2}, #3 (#4)}
\def\PR#1#2#3{\jvp{Phys.~Rev.}{#1}{#2}{#3}}
\def\PRD#1#2#3{\jvp{Phys.~Rev.~D}{#1}{#2}{#3}}
\def\PRL#1#2#3{\jvp{Phys.~Rev.~Lett.}{#1}{#2}{#3}}
%\def\PLB#1#2{Phys. Lett.~B {\bf #1}, #2}
\def\PLB#1#2#3{\jvp{Phys. Lett.~B}{#1}{#2}{#3}}
\def\NPB#1#2#3{\jvp{Nucl.~Phys.~B}{#1}{#2}{#3}}
\def\SJNP#1#2#3{\jvp{Sov.~J.~Nucl.~Phys.}{#1}{#2}{#3}}
\def\AP#1#2#3{\jvp{Ann.~Phys.}{#1}{#2}{#3}}
\def\PL#1#2#3{\jvp{Phys.~Lett.}{#1}{#2}{#3}}
\def\NuovoC#1#2#3{\jvp{Nuovo.~Cim.}{#1}{#2}{#3}}
\begin{thebibliography}{99}


\bibitem{Peccei} K.~Hagiwara, R.~D.~Peccie and D.~Zeppenfeld,
	\NPB{282}{253}{1987}.

\bibitem{Queijeiro} A.~Queijeiro, \PLB{193}{354}{1987}.

\bibitem{Salzman^2} F.~Salzman and G.~Salzman, \PL{15}{91}{1965};
	\NuovoC{41A}{443}{1966}.

\bibitem{Hoogeveen} F.~Hoogeveen, \NPB{341}{322}{1990}.

\bibitem{ChangWEDM} D.~Chang, W.-Y.~Keung and J.~Liu,
	\NPB{355}{295}{1991}.

\bibitem{Shabalin} E.~P.~Shabalin, \SJNP{28}{75}{1978}.

%\bibitem{Donoghue} J.~Donoghue, \PRD{18}{1632}{1978}.

\bibitem{Jarlskog} C.~Jarlskog, \PRL{55}{1039}{1985};
  \jvp{Z.~Phys.~C}{29}{491}{1985};
  Dan-di Wu, \PRD{33}{860}{1986}.

\bibitem{Isi} I.~Dunietz, \jvp{Ann.~Phys.}{184}{350}{1988};
	I.~Dunietz, O.~W.~Greenberg and Dan-di Wu,
	\PRL{55}{2935}{1985}.

\bibitem{BBS} M.~J.~Booth, R.~A.~Briere and R.~G.~Sachs,
	\PRD{41}{177}{1990}.

% vertex calculations
\bibitem{ShabalinV} E.~P.~Shabalin, \SJNP{32}{129}{1980}.

%\bibitem{Nanop} D.~V.~Nanopoulos, A.~Yildiz and P.~H.~Cox,
%	\AP{127}{126}{1980}.

\bibitem{Others} D.~V.~Nanopoulos, A.~Yildiz and P.~H.~Cox,
	\AP{127}{126}{1980};
	N.~G.~Deshpande and G.~Eilam, \PRD{26}{2463}{1983};
	N.~G.~Deshpande and M.~Nazerimonfared,
                        \NPB{213}{390}{1983};
        S.-P.~Chia, \PLB{130}{315}{1983}.
        %S.-P.~Chia and G.~Rajagopal, \PLB{156}{405}{1985}.

\bibitem{Me} M.~J.~Booth, ``A note on Weinberg Operators in
the Standard Model'', preprint EFI-92-13-Rev.

\bibitem{Marciano} W.~J.~Marciano and A.~Queijeiro,
	\PRD{33}{3449}{1986}.

\bibitem{Donoghue} J.~Donoghue, \PRD{18}{1632}{1978}.

\bibitem{Khriplovich} I.~B.~Khriplovich and M.~Pospelov,
	\SJNP{53}{638}{1991}.

% \bibitem{DPF} M.~J.~Booth, ``Dimension Eight Operators, CP Violation
% 	And The Neutron Electric Dipole Moment'',
% 	preprint EFI-92-67, to appear in the
%         {\it Proceedings of the 1992 Division of Particles
% 	and Fields Meeting}, World Scientific (Singapore).

\end{thebibliography}

\ifIncludeFigs\else
% get the labels right
 \newpage
  \begin{figure}
     \caption{\label{wedm}
     The two-loop contributions $d_W$.
     }
  \end{figure}
  \begin{figure}
      \caption{\label{1loop} Vertex sub-graphs of the diagrams of
	figure 1. }
  \end{figure}
\fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%70

\end{document}

\font \wedm=wedm
\documentstyle{article}
\pagestyle{empty}
\begin{document}
\begin{figure}
\unitlength=1mm
\begin{center}
\input wedm-fig1
\end{center}
\end{figure}
\end{document}
\font \wedm=wedm
\documentstyle{article}
\pagestyle{empty}
\begin{document}
\begin{figure}
\unitlength=1mm
\input wedm-fig2
\end{figure}
\end{document}
\font \wedm=wedm
\unitlength=1mm
% Note: these Feynman diagrams are ~30 x 25mm
\begin{center}
\begin{picture}(45,25)
  \put(0,0){\wedm b}
%  \put(0,0){T} % This shows that diagram extends left past (0,0)
  \put(-1,0){$\scriptstyle W$} % external line
  \put(29,0){$\scriptstyle W$} % external line
  \put(10, 0){$\scriptstyle d_2$}
  \put(13, 9){$\scriptstyle W$}
  \put(11, 16){$\scriptstyle d_1$}
  \put( 5, 8){$\scriptstyle u_2$}
  \put(23, 8){$\scriptstyle u_1$}
%  \put(-1, 13){$\scriptstyle \gamma$} % left photon line
%  \put(29, 13){$\scriptstyle \gamma$} % right photon line
%  \put(13, 20){$\scriptstyle \gamma$} % middle photon
  \put(13, -6){$\scriptstyle \gamma$} % bottom middle photon
\end{picture}
\end{center}

\begin{picture}(45,25)
  \put(0,0){\wedm l}
%  \put(0,0){T} % This shows that diagram extends left past (0,0)
  \put(-1,0){$\scriptstyle W$} % external line
  \put(29,0){$\scriptstyle W$} % external line
  \put(10, 0){$\scriptstyle d_2$}
  \put(13, 9){$\scriptstyle W$}
  \put(11, 16){$\scriptstyle d_1$}
  \put( 5, 8){$\scriptstyle u_2$}
  \put(23, 8){$\scriptstyle u_1$}
  \put(-1, 13){$\scriptstyle \gamma$} % left photon line
%  \put(29, 13){$\scriptstyle \gamma$} % right photon line
%  \put(13, 23){$\scriptstyle \gamma$} % middle photon
%  \put(13, -6){$\scriptstyle \gamma$} % bottom middle photon
\end{picture}
%
\begin{picture}(45,25)
  \put(0,0){\wedm r}
%  \put(0,0){T} % This shows that diagram extends left past (0,0)
  \put(-1,0){$\scriptstyle W$} % external line
  \put(29,0){$\scriptstyle W$} % external line
  \put(10, 0){$\scriptstyle d_2$}
  \put(13, 9){$\scriptstyle W$}
  \put(11, 16){$\scriptstyle d_1$}
  \put( 5, 8){$\scriptstyle u_2$}
  \put(23, 8){$\scriptstyle u_1$}
%  \put(-1, 13){$\scriptstyle \gamma$} % left photon line
  \put(29, 13){$\scriptstyle \gamma$} % right photon line
%  \put(13, 20){$\scriptstyle \gamma$} % middle photon
%  \put(13, -6){$\scriptstyle \gamma$} % bottom middle photon
\end{picture}

\begin{picture}(45,25)
  \put(0,0){\wedm m}
%  \put(0,0){T} % This shows that diagram extends left past (0,0)
  \put(-1,0){$\scriptstyle W$} % external line
  \put(29,0){$\scriptstyle W$} % external line
  \put(10, 0){$\scriptstyle d_2$}
  \put(13, 9){$\scriptstyle W$}
  \put(11, 16){$\scriptstyle d_1$}
  \put( 5, 8){$\scriptstyle u_2$}
  \put(23, 8){$\scriptstyle u_1$}
%  \put(-1, 13){$\scriptstyle \gamma$} % left photon line
%  \put(29, 13){$\scriptstyle \gamma$} % right photon line
  \put(16, 21){$\scriptstyle \gamma$} % middle photon
%  \put(13, -6){$\scriptstyle \gamma$} % bottom middle photon
\end{picture}
%
\begin{picture}(45,25)
  \put(0,0){\wedm M}
%  \put(0,0){T} % This shows that diagram extends left past (0,0)
  \put(-1,0){$\scriptstyle W$} % external line
  \put(29,0){$\scriptstyle W$} % external line
  \put(10, 0){$\scriptstyle d_2$}
  \put(13, 9){$\scriptstyle d_1$}
  \put(11, 16){$\scriptstyle W$}
  \put( 5, 8){$\scriptstyle u_2$}
  \put(23, 8){$\scriptstyle u_1$}
%  \put(-1, 13){$\scriptstyle \gamma$} % left photon line
%  \put(29, 13){$\scriptstyle \gamma$} % right photon line
  \put(16, 21){$\scriptstyle \gamma$} % middle photon
%  \put(13, -6){$\scriptstyle \gamma$} % bottom middle photon
\end{picture}

%
\font \wedm=wedm
\unitlength=1mm
% Note: our Feynman diagrams are 40mm x 25mm
\begin{picture}(50,20)
  \put(0,0){\wedm 2}
%  \put(0,0){T} % This shows that diagram extends left past (0,0)
  \put(-3,6){$\scriptstyle W$}
  \put(41,6){$\scriptstyle W$}
  \put(19,3){$\scriptstyle W$} % for the loop
  \put(-3,-4){$\scriptstyle d_2$}
  \put(41,-4){$\scriptstyle d_2$}
  \put(19,-4){$\scriptstyle d_1$}
  \put( 8,-4){$\scriptstyle u_2$}
  \put(30,-4){$\scriptstyle u_1$}
%  \put( 8,8){$\scriptstyle \gamma$}
  \put(20,18){$\scriptstyle \gamma$}
%  \put(30,8){$\scriptstyle \gamma$}
\end{picture}
%
\begin{picture}(50,20)(0,0)
  \put(0,0){\wedm 4}
%  \put(0,0){T} % This shows that diagram extends left past (0,0)
  \put(-3,6){$\scriptstyle W$}
  \put(41,6){$\scriptstyle W$}
  \put(19,3){$\scriptstyle d_1$} % for the loop
  \put(-3,-4){$\scriptstyle d_2$}
  \put(41,-4){$\scriptstyle d_2$}
  \put(19,-4){$\scriptstyle W$}
  \put( 8,-4){$\scriptstyle u_2$}
  \put(30,-4){$\scriptstyle u_1$}
%  \put( 8,8){$\scriptstyle \gamma$}
  \put(19,18){$\scriptstyle \gamma$}
%  \put(30,8){$\scriptstyle \gamma$}
\end{picture}
\\[2\baselineskip]

\begin{picture}(50,20)
  \put(0,0){\wedm 1}
%  \put(0,0){T} % This shows that diagram extends left past (0,0)
  \put(-3,6){$\scriptstyle W$}
  \put(41,6){$\scriptstyle W$}
  \put(19,3){$\scriptstyle W$} % for the loop
  \put(-3,-4){$\scriptstyle d_2$}
  \put(41,-4){$\scriptstyle d_2$}
  \put(19,-4){$\scriptstyle d_1$}
  \put( 8,-4){$\scriptstyle u_2$}
  \put(30,-4){$\scriptstyle u_1$}
  \put( 8,8){$\scriptstyle \gamma$}
%  \put(19,8){$\scriptstyle \gamma$}
%  \put(30,8){$\scriptstyle \gamma$}
\end{picture}
%
\begin{picture}(50,20)
  \put(0,0){\wedm 3}
%  \put(0,0){T} % This shows that diagram extends left past (0,0)
  \put(-3,6){$\scriptstyle W$}
  \put(41,6){$\scriptstyle W$}
  \put(19,3){$\scriptstyle W$} % for the loop
  \put(-3,-4){$\scriptstyle d_2$}
  \put(41,-4){$\scriptstyle d_2$}
  \put(19,-4){$\scriptstyle d_1$}
  \put( 8,-4){$\scriptstyle u_2$}
  \put(30,-4){$\scriptstyle u_1$}
%  \put( 8,8){$\scriptstyle \gamma$}
%  \put(19,8){$\scriptstyle \gamma$}
  \put(30,8){$\scriptstyle \gamma$}
\end{picture}

