{{2005 International Linear Collider Workshop - Stanford, U.S.A.}}\\ %% Please keep this conference title here \vspace{12pt} \boldmath{ Precision calculations for $\gamma\gamma\to\PW\PW\to4\,\mathrm{fermions}\,(+\gamma)$ }
A.\ Bredenstein
S.\ Dittmaier
M.\ Roth
The ${\cal O}(\al)$ electroweak radiative corrections to $\ggwwffff$ within the electroweak Standard Model are calculated in double-pole approximation (DPA). Virtual corrections are treated in DPA, and real-photonic corrections are based on complete lowest-order matrix elements for $\ggffff{+}\ga$. The radiative corrections are implemented in a Monte Carlo generator called {\sc Coffer$\gamma\gamma$}% \footnote{The computer code can be obtained from the authors upon request.}% , which optionally includes anomalous triple and quartic gauge-boson couplings in addition and performs a convolution over realistic spectra of the photon beams. A brief survey of numerical results comprises ${\cal O}(\al)$ corrections to integrated cross sections as well as to angular and invariant-mass distributions.
aa4f-lcws05.tex
%% ****** Start of file slactemplate.tex ****** %
%%
%%
%%   This file is part of the APS files in the REVTeX 4 distribution.
%%   Version 4.0 of REVTeX, August 2001
%%
%%
%%   Copyright (c) 2001 The American Physical Society.
%%
%%   See the REVTeX 4 README file for restrictions and more information.
%%
%
% This is a template for producing manuscripts for use with REVTEX 4.0
% Copy this file to another name and then work on that file.
% That way, you always have this original template file to use.
%
\documentclass[slac_one]{revtex4}
\usepackage{graphicx}
\usepackage{fancyhdr}
\pagestyle{fancy}
\fancyhead{} % clear all fields
\fancyfoot{} % clear all fields
%\fancyhead[C]{\it {2005 International Linear Collider Workshop -
%Stanford U.S.A.}}
\fancyfoot[LE,LO]{PSN 0501}
\renewcommand{\headrulewidth}{0pt}
\renewcommand{\footrulewidth}{0pt}
\renewcommand{\sfdefault}{phv}
\renewcommand{\baselinestretch}{1.2}
\setlength{\textheight}{235mm}
\setlength{\textwidth}{178mm}%{170mm}
\setlength{\topmargin}{-20mm}

\def\co{\relax}
\def\co{,}
\def\nl{\nonumber\\}
\def\nlc{\co\nonumber\\}
\def\nln{\\*[-1ex]}
\def\eqskipcorr{\vspace{-\abovedisplayskip}
                \vspace{\abovedisplayshortskip}}
\newcommand{\lsim}
{\mathrel{\raisebox{-.3em}{$\stackrel{\displaystyle <}{\sim}$}}}
\newcommand{\gsim}
{\mathrel{\raisebox{-.3em}{$\stackrel{\displaystyle >}{\sim}$}}}
\def\asymp#1%
{\mathrel{\raisebox{-.4em}{$\widetilde{\scriptstyle #1}$}}}
\def\Nlim#1{\mathrel{\raisebox{-.4em}
{$\stackrel{\disp\longrightarrow}{\scriptstyle#1}$}}}
\def\Nequal#1%
{\mathrel{\raisebox{-.5em}{$\stackrel{=}{\scriptstyle\rm#1}$}}}
\newcommand{\dsl}[1]{\not \hspace{-0.7mm}#1}
\def\dsl{\mathpalette\make@slash}
\def\make@slash#1#2{\setbox\z@\hbox{$#1#2$}%
  \hbox to 0pt{\hss$#1/$\hss\kern-\wd0}\box0}

% Abbreviations for environments
\def\beq{\begin{equation}}
\def\eeq{\end{equation}}
\def\beqar{\begin{eqnarray}}
\def\eeqar{\end{eqnarray}}
\def\barr#1{\begin{array}{#1}}
\def\earr{\end{array}}
\def\bfi{\begin{figure}}
\def\efi{\end{figure}}
\def\btab{\begin{table}}
\def\etab{\end{table}}
\def\bce{\begin{center}}
\def\ece{\end{center}}
\def\nn{\nonumber}
\def\disp{\displaystyle}
\def\text{\textstyle}
\def\fs{\footnotesize}
\def\arraystretch{1.3}

% shorthands for greek letters
\def\al{\alpha}
\def\be{\beta}
\def\Ga{\Gamma}
\def\ga{\gamma}
\def\de{\delta}
\def\De{\Delta}
\def\eps{\epsilon}
\def\veps{\varepsilon}
\def\la{\lambda}
\def\om{\omega}
\def\Om{\Omega}
\def\si{\sigma}
\def\Si{\Sigma}
\def\vth{\vartheta}
\def\ieps{\ri\epsilon}

% new commands for cross referencing
\def\refeq#1{\mbox{(\ref{#1})}}
\def\refeqs#1{\mbox{(\ref{#1})}}
\def\refeqf#1{\mbox{(\ref{#1})}}
\def\reffi#1{\mbox{Figure~\ref{#1}}}
\def\reffis#1{\mbox{Figures~\ref{#1}}}
\def\refta#1{\mbox{Table~\ref{#1}}}
\def\reftas#1{\mbox{Tables~\ref{#1}}}
\def\refse#1{\mbox{Section~\ref{#1}}}
\def\refses#1{\mbox{Sections~\ref{#1}}}
\def\refapp#1{\mbox{App.~\ref{#1}}}
\def\citere#1{\mbox{Ref.~\cite{#1}}}
\def\citeres#1{\mbox{Refs.~\cite{#1}}}

%physical units
\newcommand{\TeV}{\unskip\,\mathrm{TeV}}
\newcommand{\GeV}{\unskip\,\mathrm{GeV}}
\newcommand{\MeV}{\unskip\,\mathrm{MeV}}
\newcommand{\pba}{\unskip\,\mathrm{pb}}
\newcommand{\fb}{\unskip\,\mathrm{fb}}

% roman symbols
\newcommand{\ri}{{\mathrm{i}}}
\newcommand{\rd}{{\mathrm{d}}}
\newcommand{\rE}{{\mathrm{E}}}
\newcommand{\rU}{{\mathrm{U}}}
\newcommand{\rL}{{\mathrm{L}}}
\newcommand{\rT}{{\mathrm{T}}}

% calligraphic symbols
%\renewcommand{\O}{\mathswitch{{\cal{O}}}}
\newcommand{\Oa}{\mathswitch{{\cal{O}}(\alpha)}}
\newcommand{\Oaa}{\mathswitch{{\cal{O}}(\alpha^2)}}
\newcommand{\Oaaa}{\mathswitch{{\cal{O}}(\alpha^3)}}
\renewcommand{\L}{{\cal{L}}}
\newcommand{\M}{{\cal{M}}}
%\newcommand{\V}{{\cal{V}}}
%\newcommand{\B}{{\cal{B}}}

%physical particles
\def\mathswitchr#1{\relax\ifmmode{\mathrm{#1}}\else$\mathrm{#1}$\fi}
\newcommand{\Pf}{\mathswitch  f}
\newcommand{\Pfbar}{\mathswitch{\bar f}}
\newcommand{\PV}{\mathswitchr V}
\newcommand{\PW}{\mathswitchr W}
\newcommand{\Pw}{\mathswitchr w}
\newcommand{\PZ}{\mathswitchr Z}
\newcommand{\PA}{\mathswitchr A}
\newcommand{\Pg}{\mathswitchr g}
\newcommand{\PH}{\mathswitchr H}
\newcommand{\Pe}{\mathswitchr e}
\newcommand{\Pp}{\mathswitchr p}
\newcommand{\Pne}{\mathswitch \nu_{\mathrm{e}}}
\newcommand{\Pnebar}{\mathswitch \bar\nu_{\mathrm{e}}}
\newcommand{\Pd}{\mathswitchr d}
\newcommand{\Pdbar}{\bar{\mathswitchr d}}
\newcommand{\Pu}{\mathswitchr u}
\newcommand{\Pubar}{\bar{\mathswitchr u}}
\newcommand{\Ps}{\mathswitchr s}
\newcommand{\Psbar}{\bar{\mathswitchr s}}
\newcommand{\Pc}{\mathswitchr c}
\newcommand{\Pcbar}{\bar{\mathswitchr c}}
\newcommand{\Pb}{\mathswitchr b}
\newcommand{\Pbbar}{\mathswitchr{\bar b}}
\newcommand{\Pt}{\mathswitchr t}
\newcommand{\Ptbar}{\mathswitchr{\bar t}}
\newcommand{\Pep}{\mathswitchr {e^+}}
\newcommand{\Pem}{\mathswitchr {e^-}}
\newcommand{\PWp}{\mathswitchr {W^+}}
\newcommand{\PWm}{\mathswitchr {W^-}}
\newcommand{\PWpm}{\mathswitchr {W^\pm}}

% particle masses
\def\mathswitch#1{\relax\ifmmode#1\else$#1$\fi}
\newcommand{\Mf}{\mathswitch {m_\Pf}}
\newcommand{\Mff}{\mathswitch {m_{\Pf'}}}
\newcommand{\Ml}{\mathswitch {m_\Pl}}
\newcommand{\Mq}{\mathswitch {m_\Pq}}
\newcommand{\MV}{\mathswitch {M_\PV}}
\newcommand{\MW}{\mathswitch {M_\PW}}
\newcommand{\MA}{\mathswitch {\lambda}}
\newcommand{\MZ}{\mathswitch {M_\PZ}}
\newcommand{\MH}{\mathswitch {M_\PH}}
\newcommand{\Me}{\mathswitch {m_\Pe}}
\newcommand{\Mmy}{\mathswitch {m_\mu}}
\newcommand{\Mta}{\mathswitch {m_\tau}}
\newcommand{\Md}{\mathswitch {m_\Pd}}
\newcommand{\Mu}{\mathswitch {m_\Pu}}
\newcommand{\Ms}{\mathswitch {m_\Ps}}
\newcommand{\Mc}{\mathswitch {m_\Pc}}
\newcommand{\Mb}{\mathswitch {m_\Pb}}
\newcommand{\Mt}{\mathswitch {m_\Pt}}
\newcommand{\GW}{\Gamma_{\PW}}
\newcommand{\GZ}{\Gamma_{\PZ}}
\newcommand{\GV}{\Gamma_{\PV}}

% shorthands for SM parameters
\newcommand{\sw}{\mathswitch {s_\Pw}}
\newcommand{\cw}{\mathswitch {c_\Pw}}
\newcommand{\swbar}{\mathswitch {\bar s_\Pw}}
\newcommand{\cwbar}{\mathswitch {\bar c_\Pw}}
\newcommand{\Qf}{\mathswitch {Q_\Pf}}
\newcommand{\Ql}{\mathswitch {Q_\Pl}}
\newcommand{\Qq}{\mathswitch {Q_\Pq}}
\newcommand{\vf}{\mathswitch {v_\Pf}}
\newcommand{\af}{\mathswitch {a_\Pf}}
\newcommand{\gesi}{\mathswitch {g_\Pe}^{\sigma}}
\newcommand{\gem}{\mathswitch {g_\Pe}^-}
\newcommand{\gep}{\mathswitch {g_\Pe}^+}
\newcommand{\GF}{\mathswitch {G_\mu}}
\newcommand{\VA}{\mathrm{\{V,A\}}}
\newcommand{\V}{\mathrm{V}}

% special definitions for this paper
\newcommand{\RacoonWW}{{\sc RacoonWW}}
\newcommand{\Whizard}{{\sc Whizard}}
\newcommand{\Madgraph}{{\sc Madgraph}}
\newcommand{\HDECAY}{{\sc HDECAY}}
\newcommand{\CompAZ}{{\sc CompAZ}}
\newcommand{\ggz}{\gamma\gamma\PZ}
\newcommand{\gww}{\gamma\PW\PW}
\newcommand{\gzz}{\gamma\PZ\PZ}
\newcommand{\zzz}{\PZ\PZ\PZ}
\newcommand{\gphiw}{\gamma\phi\PW}
\newcommand{\ggww}{\gamma\gamma\PW^+\PW^-}
\newcommand{\gzww}{\gamma\PZ\PW\PW}
\newcommand{\ggzz}{\gamma\gamma\PZ\PZ}
\newcommand{\ggvv}{\gamma\gamma\PV\PV}
\newcommand{\AQGC}{{\mathrm{AQGC}}}
\newcommand{\ggffff}{\gamma\gamma\to 4f}
\newcommand{\eeffff}{\Pep\Pem\to 4f}
\newcommand{\eewwffff}{\Pep\Pem\to\PW\PW\to 4f}
\newcommand{\eeffffff}{\Pep\Pem\to 6f}
\newcommand{\ggffffg}{\ggffff\ga}
\newcommand{\eeffffg}{\eeffff\ga}
\newcommand{\ggffffkg}{\ggffff(\ga)}
\newcommand{\ggtoww}{\gamma\gamma\to \PW\PW}
\newcommand{\eetoww}{\Pep\Pem\to \PW\PW}
\newcommand{\ggwwffff}{\gamma\gamma\to \PW\PW\to 4f}
\newcommand{\ggvvffff}{\gamma\gamma\to \PV\PV\to 4f}

% line characterizations for figure captions
\def\solid{\raise.9mm\hbox{\protect\rule{1.1cm}{.2mm}}}
\def\dash{\raise.9mm\hbox{\protect\rule{2mm}{.2mm}}\hspace*{1mm}}
\def\dot{\rlap{$\cdot$}\hspace*{2mm}}
\def\dashed{\dash\dash\dash\dash}
\def\ddashed{\dash\hspace*{-.5mm}\dash\hspace*{1mm}\dash\hspace*{-.5mm}\dash}
\def\dotted{\dot\dot\dot\dot\dot\dot}
\def\dashdotted{\dot\dash\dot\dash\dot}

% various abbreviations
\def\ie{i.e.\ }
\def\eg{e.g.\ }
\def\cf{cf.\ }
\def\sub{{\mathrm{sub}}}
\def\gsub{g^{(\sub)}}
\def\Gsub{G^{(\sub)}}
\def\cGsub{{\cal G}^{(\sub)}}
\def\cGcoll{{\cal G}^{(\coll)}}
\def\ssub{\sigma^{(\sub)}}

\newcommand{\se}{self-energy}
\newcommand{\ses}{self-energies}
\newcommand{\ct}{counterterm}
\newcommand{\cts}{counterterms}
\newcommand{\cs}{cross section}
\newcommand{\css}{cross sections}

\newcommand{\br}{{\mathrm{br}}}
\newcommand{\IBA}{{\mathrm{IBA}}}
\newcommand{\DPA}{{\mathrm{DPA}}}
\newcommand{\QCD}{{\mathrm{QCD}}}
\newcommand{\QED}{{\mathrm{QED}}}
\newcommand{\LEP}{{\mathrm{LEP}}}
\newcommand{\SLD}{{\mathrm{SLD}}}
\newcommand{\SM}{{\mathrm{SM}}}
\newcommand{\Born}{{\mathrm{Born}}}
\newcommand{\born}{{\mathrm{Born}}}
\newcommand{\corr}{{\mathrm{corr}}}
\newcommand{\full}{{\mathrm{full}}}
\newcommand{\onel}{{\mathrm{1-loop}}}
\newcommand{\weak}{{\mathrm{weak}}}
\newcommand{\eefourf}{{\mathswitchr{ee4f}}}
\newcommand{\cut}{{\mathrm{cut}}}
\newcommand{\SB}{{\mathrm{SB}}}
\newcommand{\unpol}{\mathrm{unpol}}
\newcommand{\CMS}{{\mathrm{CMS}}}
\newcommand{\FW}{{\mathrm{FW}}}
\newcommand{\CC}{{\mathrm{CC}}}
\newcommand{\NC}{{\mathrm{NC}}}
\newcommand{\virt}{{\mathrm{virt}}}
\newcommand{\soft}{{\mathrm{soft}}}
\newcommand{\coll}{{\mathrm{coll}}}
\newcommand{\finite}{{\mathrm{finite}}}
\renewcommand{\min}{{\mathrm{min}}}
\renewcommand{\max}{{\mathrm{max}}}
\newcommand{\Ncf}{N_f^{\mathrm{c}}}

\newcommand{\U}{\mathrm{U}}
\newcommand{\SU}{\mathrm{SU}}

% mathematical functions
\def\atn{\mathop{\mathrm{arctan}}\nolimits}
\def\Li{\mathop{\mathrm{Li}_2}\nolimits}
\def\bLi{\mathop{\overline{\mathrm{Li}_2}}\nolimits}
\def\cLi{\mathop{{\cal L}i_2}\nolimits}
\def\Re{\mathop{\mathrm{Re}}\nolimits}
\def\Im{\mathop{\mathrm{Im}}\nolimits}
\def\sgn{\mathop{\mathrm{sgn}}\nolimits}

\hyphenation{brems-strah-lung}

\begin{document}

%Title of paper
\title{{\small{2005 International Linear Collider Workshop - Stanford,
U.S.A.}}\\ %% Please keep this conference title here
\vspace{12pt}
\boldmath{
Precision calculations for
$\gamma\gamma\to\PW\PW\to4\,\mathrm{fermions}\,(+\gamma)$
}} %% Paper title goes here

% Repeat the \author .. \affiliation  etc. as needed
%
% \affiliation command applies to all authors since the last
% \affiliation command. The \affiliation command should follow the
% other information

\author{A.\ Bredenstein, S.\ Dittmaier, M.\ Roth}
\affiliation{Max-Planck-Institut f\"ur Physik,
D-80805 M\"unchen, Germany}

\begin{abstract}
The ${\cal O}(\al)$ electroweak radiative corrections to $\ggwwffff$ within
the electroweak Standard Model are calculated
in double-pole approximation (DPA).
Virtual corrections are treated in DPA, 
and real-photonic corrections are based on complete lowest-order
matrix elements for $\ggffff{+}\ga$. 
The radiative corrections are implemented in a Monte Carlo generator
called {\sc Coffer$\gamma\gamma$}%
\footnote{The computer code can be obtained from the authors upon request.}%
, which optionally includes anomalous
triple and quartic gauge-boson couplings in addition and performs a
convolution over realistic spectra of the photon beams.
A brief survey of numerical results comprises
${\cal O}(\al)$ corrections to
integrated cross sections as well as to
angular and invariant-mass distributions.
\end{abstract}

%\maketitle must follow title, authors, abstract
\maketitle

\thispagestyle{fancy}
\addtocounter{footnote}{1}

% body of paper here - Use proper section commands
% References should be done using the \cite, \ref, and \label commands
% Put \label in argument of \section for cross-referencing
%\section{\label{}}

\section{INTRODUCTION}

As an option at a future $\Pep\Pem$ linear collider, 
a photon (or $\ga\ga$) collider \cite{Ginzburg:1981vm}
found considerable interest in recent years.
It could provide us with information about new physics phenomena,
such as properties of Higgs bosons or of new particles,
which is in many respects complementary in the
$\Pep\Pem$ and $\ga\ga$ modes
(see, e.g., \citeres{Ginzburg:1981vm,DeRoeck:2003gv} and references therein).
Moreover, a $\ga\ga$ collider is a true W-boson-pair factory,
owing to the extremely high W-pair cross section, which tends to a
constant of about $80\pba$ in the high-energy limit (in the absence
of phase-space cuts), opening the possibility of precision studies in the
sector of electroweak gauge bosons. 
For instance, an analysis of anomalous gauge-boson couplings
in $\ggwwffff$ provides direct information on the $\gamma\PW\PW$ and
$\gamma\gamma\PW\PW$ interactions without interference from the
$\PZ$-boson sector. 
Either way, whether one is
interested in W-boson precision physics or in the search for new phenomena,
precise predictions for W-pair production are indispensable for signal
and background studies.
\looseness -1

As described in \citeres{Bredenstein:2004ef,Bredenstein:2005zk} in detail,
we have constructed a Monte Carlo generator
called {\sc Coffer$\gamma\gamma$} for $\ggffff$ and $\ggffffg$, which
particularly focusses on precise predictions for W-pair-mediated final 
states. Anomalous 
$\gamma\PW\PW$, $\gamma\gamma\PW\PW$, and $\gamma\gamma\PZ\PZ$
gauge-boson couplings as well as a loop-induced $\gamma\gamma\PH$ coupling
are optionally included in lowest-order predictions for $\ggffff$.
Moreover, electroweak radiative corrections of ${\cal O}(\alpha)$
are applied to processes $\ggwwffff$ in the so-called 
``double-pole approximation'' (DPA).
In the following we briefly describe the salient features of 
{\sc Coffer$\gamma\gamma$} and show some sample results.
More details and results (also for effects of anomalous couplings)
can be found in \citeres{Bredenstein:2004ef,Bredenstein:2005zk}.
\looseness -1

\section{LOWEST-ORDER PREDICTIONS}

In \citere{Bredenstein:2004ef} we have 
constructed a Monte Carlo event generator for lowest-order predictions
based on complete matrix elements for $\ggffff$~%
\footnote{Results for cross sections based on the full set of
$\ggffff$ diagrams were also presented in \citere{Moretti:1996nv}. 
However, no convolution over a realistic photon beam spectrum was
performed there.}
and $\ggffffg$. 
The final-state fermions are treated as massless, but
all $4f$ final states are supported.
For $4f$ production all helicity amplitudes (with and without
anomalous couplings) are explicitly given
as compact expressions in terms of spinor products.
Moreover, the introduction of finite gauge-boson decay widths and the
issue of gauge invariance are discussed carefully.
The possibility to convolute the cross sections with
realistic photon beam spectra is offered upon using the
parametrization of {\CompAZ} \cite{Zarnecki:2002qr}.
The Standard Model predictions were successfully compared
to results obtained with the multi-purpose packages
{\Whizard} \cite{Kilian:2001qz} and {\Madgraph} \cite{Stelzer:1994ta}.
\looseness -1

Among the various numerical results shown in \citere{Bredenstein:2004ef},
we only highlight the comparison between results that are based on the full
set of tree diagrams on the one hand and on the so-called ``signal'' diagrams
that involve two resonant W~bosons in $\ggwwffff$ on the other hand. 
Of course, the amplitude
for this ``naive W-pair signal'' is not a gauge-invariant quantity.
Nevertheless, its investigation is interesting, 
because such an amplitude is much simpler than the full
amplitudes for $4f$ production and is universal (up to colour factors)
for all relevant final states.
The DPA for the Born amplitude
is obtained from the naive W-pair signal upon deforming the
momenta of the four outgoing fermions in such a way that the
intermediate W-boson states become on shell, while keeping the W~propagators 
off shell.
Since the residues of the W~resonances are gauge independent,
the Born amplitude in DPA is a gauge-invariant quantity.
In \reffi{fig:locs}
the cross sections of the W-pair signal diagrams and the
DPA for $\ga\ga\to\PW\PW\to 4\,$leptons
are compared with the complete lowest-order cross sections.
\begin{figure}
\setlength{\unitlength}{1cm}
\centerline{
\begin{picture}(7,6.9)
\put(-1.2,-13.1){\special{psfile=dpa.lep.eps hscale=90 vscale=90}}
\end{picture}\hspace*{1em}
\begin{picture}(7,6.9)
\put(-1.2,-13.1){\special{psfile=dparel.lep.eps hscale=90 vscale=90}}
\end{picture}}
\vspace*{-.5em}
\caption{Lowest-order cross sections for
$\ga\ga\to\Pem\bar\nu_\Pe\nu_\mu\mu^+/\Pem\bar\nu_\Pe\nu_\Pe\Pep$
including all diagrams, only W-pair
signal diagrams, and in DPA as a function of the centre-of-mass (CM) energy 
$\sqrt{s_{\ga\ga}}$ of the monochromatic photon beams (l.h.s.),
and the corresponding relative deviations from the DPA (r.h.s.).
(Taken from \citere{Bredenstein:2004ef}.)}
\label{fig:locs}
\end{figure}
The plot on the l.h.s.\ shows the cross sections, 
the plot on the
r.h.s.\ the relative deviation from the corresponding DPA.
We do not include the convolution over the
photon spectrum in this analysis
so that effects of the approximations are clearly visible.
For energies not too close to the W-pair threshold,
the DPA agrees with the full lowest-order cross section within
1--3\%, which is of the expected order of $\GW/\MW$.
Near threshold, i.e.\ for $\sqrt{s_{\ga\ga}}-2\MW={\cal O}(\GW)$,
the reliability of the DPA breaks down, since background
diagrams become more and more important and small scales $\ga$,
such as $\sqrt{s_{\ga\ga}-4\MW^2}$, can increase the naive error
estimate from $\GW/\MW$ to $\GW/\ga$.
The cross section of the W-pair signal diagrams, however, shows large
deviations from the full $\ggffff$ cross sections for the whole
energy range, in particular, at high energies. 
The results of \reffi{fig:locs} show that a naive
signal definition is a bad concept for $\ggwwffff$, since deviations
from the full process $\ggffff$
even reach 5--10\% in the TeV range. This is in contrast to the
situation at $\Pep\Pem$ colliders where the naive W-pair signal
(defined in `t~Hooft--Feynman gauge) was a
reasonable approximation
(see, e.g., \citere{Grunewald:2000ju}).

\section{RADIATIVE CORRECTIONS}

\subsection{Strategy of the calculation}

In \citere{Bredenstein:2005zk} we have extended 
our lowest-order calculation \cite{Bredenstein:2004ef}
for $\ggffff$ by including the electroweak radiative corrections of
${\cal O}(\alpha)$ to the W-pair channels $\ggwwffff$ in DPA.
The DPA extracts those contributions of the ${\cal O}(\alpha)$
corrections that are enhanced by two resonant W-boson
propagators, i.e.\ it represents the leading term in an expansion of
the cross section about the two W-propagator poles.
Note that tree-level diagrams for $\ggffff$ with at most one
resonant W~boson are suppressed w.r.t.\ the doubly-resonant
$\ggtoww$ signal by a factor of ${\cal O}(\GW/\MW)\sim {\cal O}(\alpha)$.
Consequently, predictions based on full lowest-order matrix elements for
$\ggffff$ and ${\cal O}(\alpha)$ corrections for $\ggwwffff$ in DPA
should be precise up to terms of ${\cal O}(\alpha/\pi\times \GW/\MW)$,
since corrections typically involve the factor $\alpha/\pi$.
Including a quite conservative numerical safety factor, the relative
uncertainty should thus be $\lsim0.5\%$ for such predictions,
as long as neglected effects are not additionally enhanced.
The naive error estimate can, in particular, be spoiled by the occurrence of
large scale ratios, which exist, e.g., near production thresholds or
at very high energies. The estimate has recently been confirmed
for $\eewwffff$ with CM energies
$170\GeV\lsim\sqrt{s}\lsim300\GeV$ by comparing a full
${\cal O}(\alpha)$ calculation \cite{Denner:2005es}
with the corresponding DPA predictions provided by {\sc RacoonWW}
\cite{Denner:1999gp}.

For energies in the W-pair threshold region ($\sqrt{s_{\ga\ga}}<170\GeV$)
and below, the DPA is unreliable, because diagrams with at most one
resonant W~boson become equally important. Therefore, in this region
we employ an ``improved Born approximation'' (IBA) which is based on
leading universal corrections but does not involve any resonance expansion.
As also confirmed by the full ${\cal O}(\alpha)$ calculation 
\cite{Denner:2005es} for the related process $\eewwffff$, such an IBA
reproduces the ${\cal O}(\alpha)$-corrected cross section typically
within $\sim\pm 2\%$. 
The convolution of the hard $\ga\ga$ cross section
involves both the IBA (in the low-energy tail) and the DPA
(for $\sqrt{s_{\ga\ga}}>170\GeV$).
As shown in \citere{Bredenstein:2005zk},
for CM energies (of the electrons before the Compton backscattering)
$\sqrt{s_{\Pe\Pe}}\lsim 230\GeV$ our prediction possesses an uncertainty
of $\sim 2\%$, because it is mainly based on the IBA, but already
for $\sqrt{s_{\Pe\Pe}}\gsim 300\GeV$ ($500\GeV$)
the IBA contribution is widely suppressed
so that the DPA uncertainty 
sets the precision of $\lsim 0.7\%$ ($0.5\%$) in our calculation.

In detail, we apply the DPA only to the virtual corrections to
$\ggwwffff$, while we base the real-photonic corrections on complete
lowest-order matrix elements for $\ggffffg$. Apart from the treatment
of IR (soft and collinear) singularities, we can use
the calculation of the bremsstrahlung
processes $\ggffffg$ for massless fermions
described in \citere{Bredenstein:2004ef}.
The concept of the DPA was already described in \citere{Aeppli:1993rs}
for the corrections to $\eewwffff$ and later successfully
applied to these processes in different versions
\cite{Denner:1999gp,Jadach:1998tz,Beenakker:1998gr,%
Kurihara:2001um}.
We follow the strategy of {\sc RacoonWW} \cite{Denner:1999gp}
and adapt it to $\gamma\gamma$ collisions where necessary.
The virtual corrections in DPA can be naturally split into
factorizable and non-factorizable contributions.
The former comprise the corrections to on-shell W-pair
production \cite{Denner:1994ca,Jikia:1996uu}
and the decay \cite{Bardin:1986fi} of on-shell W~bosons.
The latter account for soft-photon exchange between the production
and decay subprocesses; the known results for the non-factorizable
corrections \cite{Melnikov:1995fx}
for $\eewwffff$ can be taken over to $\gamma\gamma$ collisions with
minor modifications.

The combination of virtual and real-photonic corrections is non-trivial
for two reasons. First, the finite-fermion-mass effects have to be
restored in the phase-space regions of collinear photon radiation off
charged fermions, and the IR regularization for soft-photon emission
has to be implemented. To this end, we employ the dipole subtraction
formalism for photon radiation \cite{Dittmaier:1999mb}
as well as the
more conventional phase-space slicing approach.
The second subtlety concerns the fact that we apply the DPA only to
the virtual corrections, but not to the real-photonic parts.
Therefore, the cancellation of soft and collinear singularities
has to be done carefully, in order to avoid mismatch.

Finally, the Higgs-boson resonance in the $s$-channel
has to be treated with care. Firstly, the Higgs decay width has to
be introduced in the amplitude without violating gauge invariance.
We separate the gauge-invariant resonance pole and
include the width only in the resonant part which is proportional
to its gauge-invariant residue. Secondly, although the Higgs
resonance is loop-induced, it is not sufficient to include the
interference between its contribution $\M_{\mathrm{Higgs}}$ to the 
amplitude with the Born amplitude, but it is necessary to include the 
square $|\M_{\mathrm{Higgs}}|^2$ in the squared amplitude, in order
to get a proper description of the resonance.

\subsection{Numerical results}

The cross section for $\ga\ga\to\Pne\Pep\Pd\Pubar$,
including the convolution over the photon spectrum,
is shown in \reffi{fig:sqrts.spec}
as a function of CM energy $\sqrt{s_{\Pe\Pe}}$
for a Higgs mass of $\MH=130\GeV$ and in the lower left plot also
for $\MH=170\GeV$.
\begin{figure}
\setlength{\unitlength}{1cm}
\centerline{
\begin{picture}(7.7,8)
\put(-1.7,-14.5){\special{psfile=sqrts.spec.born.300.eps hscale=100 vscale=100}}
\end{picture}
\begin{picture}(7.5,8)
\put(-1.7,-14.5){\special{psfile=sqrts.spec.born.1000.eps hscale=100 vscale=100}}
\end{picture} }
\vspace*{-.5em}
\centerline{
\begin{picture}(7.7,8)
\put(-1.7,-14.5){\special{psfile=sqrts.spec.rel.300.eps hscale=100 vscale=100}}
\end{picture}
\begin{picture}(7.5,8)
\put(-1.7,-14.5){\special{psfile=sqrts.spec.rel.1000.eps hscale=100 vscale=100}}
\end{picture} }
\vspace*{-1em}
\caption{Integrated cross section for $\ga\ga\to\Pne\Pep\Pd\Pubar$
(upper plots) and relative radiative
corrections (lower plots) including the convolution over the photon spectrum
for $\MH=130\GeV$ and $170\GeV$ (l.h.s.).
For $\sqrt{s_{\Pe\Pe}}>300\GeV$ (r.h.s.) the ``best'' curve
for $\MH=170\GeV$ practically coincides with the shown curve for
$\MH=130\GeV$. (Taken from \citere{Bredenstein:2005zk}.)}
\label{fig:sqrts.spec}
\end{figure}
The remaining input parameters as well as the precise setup of the calculation are given in \citere{Bredenstein:2005zk}.
In the upper plots the absolute
prediction is shown and in the lower plots the corrections relative
to the Born cross section. 
The interesting structure in the lower left plot
reflects the shape of the photon spectrum convoluted with the Higgs resonance.
Since the Higgs resonance
is very narrow, a sizable contribution is only possible if
$x_1x_2s_{\Pe\Pe}\approx\MH^2$ where $x_1$ and $x_2$ are the energy fractions
carried by the photons. The correction is
very small at low $\sqrt{s_{\Pe\Pe}}$ where $x_1$ and $x_2$ both
have to be so large in order to match this condition
that the corresponding spectrum is extremely small. Increasing
$\sqrt{s_{\Pe\Pe}}$ allows for lower values of $x_1$ and $x_2$.
For instance, for $\MH=130\GeV$,
the rise at $\sqrt{s_{\Pe\Pe}}\sim 180\GeV$ results from a region
where both $x_1$ and $x_2$ are in the high-energy tail
of the spectrum which is produced by multiple photon scattering.
The peak at  $\sqrt{s_{\Pe\Pe}}\sim 200\GeV$ is caused by events where
one photon comes from the high-energy tail and one from the dominant
peak in the photon spectrum. Finally, at $\sqrt{s_{\Pe\Pe}}\gsim 210\GeV$
both $x_1$ and $x_2$ originate from
the dominant photon-spectrum peak which causes the steep rise
until $\sqrt{s_{\Pe\Pe}}\sim 220\GeV$.
For energies above the Higgs resonance [see \reffi{fig:sqrts.spec}(d)]
the corrections decrease and
stay in the range of a few per cent up to TeV energies.

In \reffi{fig:wdist} we show the invariant-mass and production-angle
distributions of the $\PW^-$ boson as recontructed from the
$\Pd\Pubar$ pair in the process $\ga\ga\to\Pne\Pep\Pd\Pubar$,
both with and without convolution over the photon spectrum.
\begin{figure}
\setlength{\unitlength}{1cm}
\centerline{
\begin{picture}(7.7,8)
\put(-1.7,-14.5){\special{psfile=wmin.eps hscale=100 vscale=100}}
\end{picture}
\begin{picture}(7.5,8)
\put(-1.7,-14.5){\special{psfile=wmin.ang.eps hscale=100 vscale=100}}
\end{picture}}
\vspace*{-1.2em}
\centerline{
\begin{picture}(7.7,8)
\put(-1.7,-14.5){\special{psfile=wmin.rel.eps hscale=100 vscale=100}}
\end{picture}
\begin{picture}(7.5,8)
\put(-1.7,-14.5){\special{psfile=wmin.ang.rel.eps hscale=100 vscale=100}}
\end{picture}}
\vspace*{-1.2em}
\caption{Absolute predictions (upper plots) and relative corrections
(lower plots) for the invariant-mass (l.h.s.) and production-angle 
(r.h.s.) distributions of the 
$\PW^-$ bosons reconstructed from the $\Pd\Pubar$ pairs
in the process $\ga\ga\to\Pne\Pep\Pd\Pubar$ at $\sqrt{s}=500\GeV$
(Taken from \citere{Bredenstein:2005zk}).}
\label{fig:wdist}
\end{figure}
The upper plots show the absolute predictions and the lower plots the
corrections normalized to the Born predictions.
Since we use $\sqrt{s_{\ga\ga}}=500\GeV$ and $\sqrt{s_{\Pe\Pe}}=500\GeV$,
the corrections are shifted upwards when including the photon
spectrum, because the effective energy of the photons is lower.
For the W-invariant-mass distribution,
the shape of the correction, however,
is hardly changed by the convolution over the photon spectrum.
As the shape of the corrections determines a possible shift of the peak of the
invariant-mass distribution, it is of particular importance
in the determination of the W-boson mass. The measurement of the
W-boson mass can, e.g., be used for understanding and calibrating
the detector of a $\ga\ga$ collider.
The distribution in the W-boson production angle
is sensitive to anomalous couplings.
In order to set bounds on these couplings it is mandatory to know
radiative corrections,
because both anomalous couplings and radiative corrections typically
distort angular distributions.
While the correction without the photon spectrum is about $-9\%$
for W bosons emitted perpendicular to the beam, the corrections are
rather small when including the photon spectrum. 
In fact, the relative correction $\de$
is accidentally small at $\sqrt{s_{\Pe\Pe}} \sim 500\GeV$
[cf.\ \reffi{fig:sqrts.spec}(d)]
and might also become larger if other cuts or event selection procedures
are applied.



% figures should be put into the text as floats.
% Use the graphics or graphicx packages (distributed with LaTeX2e)
% and the \includegraphics macro defined in those packages.
% See the LaTeX Graphics Companion by Michel Goosens, Sebastian Rahtz,
% and Frank Mittelbach for instance.
%
% Here is an example of the general form of a figure:
% Fill in the caption in the braces of the \caption{} command. Put the label
% that you will use with \ref{} command in the braces of the \label{} command.
% Use the figure* environment if the figure should span across the
% entire page. There is no need to do explicit centering.

% \begin{figure}
% \includegraphics{}%
% \caption{\label{}}
% \end{figure}

% Surround figure environment with turnpage environment for landscape
% figure
% \begin{turnpage}
% \begin{figure}
% \includegraphics{}%
% \caption{\label{}}
% \end{figure}
% \end{turnpage}


%\begin{figure*}[t]
%\centering
%\includegraphics[width=135mm]{JACpic2.eps}
%\caption{Example of full width figure.} \label{JACpic2-f1}
%\end{figure*}


%\begin{acknowledgments}
%The authors wish to thank JACoW for their guidance in preparing
%this template.
%
%Work supported by Department of Energy contract DE-AC02-76SF00515.
%\end{acknowledgments}

%\begin{thebibliography}{9}   % Use for  1-9  references
\begin{thebibliography}{99} % Use for 10-99 references

\vspace*{-.1em}

%\cite{Ginzburg:1981vm}
\bibitem{Ginzburg:1981vm}
%I.~F.~Ginzburg, G.~L.~Kotkin, V.~G.~Serbo and V.~I.~Telnov,
I.~F.~Ginzburg {\it et al.},
%``Colliding $\gamma$ e And $\gamma \gamma$ Beams Based On The Single Pass Accelerators (Of Vlepp Type),''
Nucl.\ Instrum.\ Meth.\  {\bf 205} (1983) 47 and
%%CITATION = NUIMA,205,47;%%
%
%\cite{Ginzburg:1982yr}
%\bibitem{Ginzburg:1982yr}
%I.~F.~Ginzburg, G.~L.~Kotkin, S.~L.~Panfil, V.~G.~Serbo and V.~I.~Telnov,
%``Colliding Gamma E And Gamma Gamma Beams Based On The Single Pass E+ E-
%Accelerators. 2. Polarization Effects. Monochromatization Improvement,''
Nucl.\ Instrum.\ Meth.\ A {\bf 219} (1984) 5;\\
%%CITATION = NUIMA,A219,5;%%
%
%\cite{Telnov:sd}
%\bibitem{Telnov:sd}
V.~I.~Telnov,
%``Problems Of Obtaining Gamma Gamma And Gamma Epsilon Colliding Beams At
%Linear Colliders,''
Nucl.\ Instrum.\ Meth.\ A {\bf 294} (1990) 72;\\
%%CITATION = NUIMA,A294,72;%%
%
%\cite{Brinkmann:1997sj}
%\bibitem{Brinkmann:1997sj}
R.~Brinkmann {\it et al.},
%``An interaction region for gamma gamma and gamma electron collisions at
%TESLA/SBLC,''
Nucl.\ Instrum.\ Meth.\ A {\bf 406} (1998) 13
[hep-ex/9707017];\\
%%CITATION = HEP-EX 9707017;%%
%
%\cite{Watanabe:1998mn}
%\bibitem{Watanabe:1998mn}
I.~Watanabe {\it et al.},
``$\gamma \gamma$ collider as an option of JLC,''
KEK-REPORT-97-17; \\
%\href{http://www.slac.stanford.edu/spires/find/hep/www?r=kek-report-97-17}{SPIRES entry}
%\cite{Badelek:2001xb}
%\bibitem{Badelek:2001xb}
B.~Badelek {\it et al.}
%[ECFA/DESY Photon Collider Working Group Collaboration],
``TESLA TDR, Part VI, Chapter 1: Photon collider  at
TESLA,''
hep-ex/0108012;\\
%%CITATION = HEP-EX 0108012;%%
%
%\cite{Abe:2001wn}
%\bibitem{Abe:2001wn}
T.~Abe {\it et al.}  [American Linear Collider Working Group Collaboration],
%``Linear collider physics resource book for Snowmass 2001,''
%in {\it Proc. of the APS/DPF/DPB Summer Study on the Future of
%  Particle Physics (Snowmass 2001) } ed. R.~Davidson and C.~Quigg,
SLAC-R-570, {\it Resource book for Snowmass 2001}
[hep-ex/0106055, hep-ex/0106056, hep-ex/0106057, hep-ex/0106058].

%\cite{DeRoeck:2003gv}
\bibitem{DeRoeck:2003gv}
A.~De Roeck,
%``Physics at a gamma gamma, e gamma and e- e- option for a linear collider,''
hep-ph/0311138;\\
%%CITATION = HEP-PH 0311138;%%
%
%\cite{Krawczyk:2003yz}
%\bibitem{Krawczyk:2003yz}
M.~Krawczyk,
%``ECFA summary: Higgs, gamma gamma and e gamma physics,''
hep-ph/0312341;\\
%%CITATION = HEP-PH 0312341;%%
%
%\cite{Brodsky:2004wx}
%\bibitem{Brodsky:2004wx}
S.~J.~Brodsky,
%``High energy photon photon collisions at a linear collider,''
hep-ph/0404186.
%%CITATION = HEP-PH 0404186;%%

%\cite{Bredenstein:2004ef}
\bibitem{Bredenstein:2004ef}
A.~Bredenstein, S.~Dittmaier and M.~Roth,
%``Four-fermion production at gamma gamma colliders: I. Lowest-order predictions
%and anomalous couplings,''
Eur.\ Phys.\ J.\ C {\bf 36} (2004) 341
[hep-ph/0405169].
%%CITATION = HEP-PH 0405169;%%

%\cite{Bredenstein:2005zk}
\bibitem{Bredenstein:2005zk}
  A.~Bredenstein, S.~Dittmaier and M.~Roth,
  %``Four-fermion production at gamma gamma colliders: 2. Radiative corrections
  %in double-pole approximation,''
  hep-ph/0506005.
  %%CITATION = HEP-PH 0506005;%%

%\cite{Moretti:1996nv}
\bibitem{Moretti:1996nv}
M.~Moretti,
%``Four-fermion productions at a gamma gamma collider,''
Nucl.\ Phys.\ B {\bf 484} (1997) 3
[hep-ph/9604303] and
%%CITATION = HEP-PH 9604303;%%
%
%\cite{Moretti:1996cn}
%\bibitem{Moretti:1996cn}
%M.~Moretti,
%``Four leptons final states from gamma gamma fusion,''
hep-ph/9606225;\\
%%CITATION = HEP-PH 9606225;%%
%
%\cite{Baillargeon:1997se}
%\bibitem{Baillargeon:1997se}
M.~Baillargeon, G.~B\'elanger and F.~Boudjema,
%``Semi-leptonic four-fermion final states in polarised gamma gamma  reactions:
%Exact results vs an improved narrow width approximation,''
Phys.\ Lett.\ B {\bf 404} (1997) 124
[hep-ph/9701368] and
%%CITATION = HEP-PH 9701368;%%
%
%\cite{Baillargeon:1997rz}
%\bibitem{Baillargeon:1997rz}
%M.~Baillargeon, G.~B\'elanger and F.~Boudjema,
%``Effects of non-standard tri-linear couplings in photon photon  collisions. I:
%gamma gamma $\to$ W+ W-,''
Nucl.\ Phys.\ B {\bf 500} (1997) 224
[hep-ph/9701372];\\
%%CITATION = HEP-PH 9701372;%%
%
%\cite{Boos:1997jt}
%\bibitem{Boos:1997jt}
E.~Boos and T.~Ohl,
%``Towards a complete calculation of gamma gamma $\to$ 4f,''
Phys.\ Lett.\ B {\bf 407} (1997) 161
[hep-ph/9705374].
%%CITATION = HEP-PH 9705374;%%

%\cite{Zarnecki:2002qr}
\bibitem{Zarnecki:2002qr}
A.~F.~Zarnecki,
%``CompAZ: Parametrization of the photon collider luminosity spectra,''
Acta Phys.\ Polon.\ B {\bf 34} (2003) 2741
[hep-ex/0207021].
%%CITATION = HEP-EX 0207021;%%

%\cite{Kilian:2001qz}
\bibitem{Kilian:2001qz}
W.~Kilian,
``WHIZARD 1.0: A generic Monte-Carlo integration and event generation  package for multi-particle processes. Manual,''
LC-TOOL-2001-039.
%\href{http://www.slac.stanford.edu/spires/find/hep/www?r=lc-tool-2001-039}{SPIRES entry}

%\cite{Stelzer:1994ta}
\bibitem{Stelzer:1994ta}
T.~Stelzer and W.~F.~Long,
%``Automatic generation of tree level helicity amplitudes,''
Comput.\ Phys.\ Commun.\  {\bf 81} (1994) 357
[hep-ph/9401258].\\
%%CITATION = HEP-PH 9401258;%%
%\cite{Murayama:1992gi}
%\bibitem{Murayama:1992gi}
H.~Murayama, I.~Watanabe and K.~Hagiwara,
``HELAS: HELicity amplitude subroutines for Feynman diagram evaluations,''
KEK-91-11.
%\href{http://www.slac.stanford.edu/spires/find/hep/www?r=kek-91-11}{SPIRES entry}

%\cite{Grunewald:2000ju}
\bibitem{Grunewald:2000ju}
M.~W.~Gr\"unewald {\it et al.},
in {\it Reports of the Working Groups on Precision Calculations
for LEP2 Physics}, eds.\ S.~Jadach, G.~Passarino and R.~Pittau
(CERN 2000-009, Geneva, 2000), p.~1
[hep-ph/0005309].

%\cite{Denner:2005es}
\bibitem{Denner:2005es}
  A.~Denner, S.~Dittmaier, M.~Roth and L.~H.~Wieders,
  %``Complete electroweak O(alpha) corrections to charged-current e+ e- $\to$
  %4fermion processes,''
  Phys.\ Lett.\ B {\bf 612} (2005) 223
  [hep-ph/0502063] and
  %%CITATION = HEP-PH 0502063;%%
%
%\cite{Denner:2005fg}
%\bibitem{Denner:2005fg}
%  A.~Denner, S.~Dittmaier, M.~Roth and L.~H.~Wieders,
  %``Electroweak corrections to charged-current e+ e- $\to$ 4 fermion processes:
  %Technical details and further results,''
  hep-ph/0505042.
  %%CITATION = HEP-PH 0505042;%%

%\cite{Denner:1999gp}
\bibitem{Denner:1999gp}
A.~Denner, S.~Dittmaier, M.~Roth and D.~Wackeroth,
%``Predictions for all processes e+ e- $\to$ 4fermions + gamma,''
Nucl.\ Phys.\ B {\bf 560} (1999) 33
[hep-ph/9904472];
%%CITATION = HEP-PH 9904472;%%
%
%\cite{Denner:2000kn}
%\bibitem{Denner:2000kn}
%A.~Denner, S.~Dittmaier, M.~Roth and D.~Wackeroth,
%``O(alpha) corrections to e+ e- $\to$ W W $\to$ 4fermions(+gamma): First  numerical results from RACOONWW,''
Phys.\ Lett.\ B {\bf 475} (2000) 127
[hep-ph/9912261];
%%CITATION = HEP-PH 9912261;%%
%
%\cite{Denner:1999dt}
%\bibitem{Denner:1999dt}
%A.~Denner, S.~Dittmaier, M.~Roth and D.~Wackeroth,
%``W pair production at future e+ e- colliders: Precise predictions from
%RACOONWW,''
Eur.\ Phys.\ J.\ direct C {\bf 2} (2000) 4
[hep-ph/9912447];
%%CITATION = HEP-PH 9912447;%%
%
%\cite{Denner:2000bj}
%\bibitem{Denner:2000bj}
%A.~Denner, S.~Dittmaier, M.~Roth and D.~Wackeroth,
%``Electroweak radiative corrections to e+ e- --> W W --> 4fermions in  double-pole approximation: The RACOONWW approach,''
Nucl.\ Phys.\ B {\bf 587} (2000) 67
[hep-ph/0006307];
%%CITATION = HEP-PH 0006307;%%
%
%\cite{Denner:2001zp}
%\bibitem{Denner:2001zp}
%A.~Denner, S.~Dittmaier, M.~Roth and D.~Wackeroth,
%``Off-shell W pair production: Universal versus non-universal  corrections,''
%in {\it Proc. of the 5th International Symposium on Radiative Corrections (RADCOR 2000)} ed. H.~E. Haber, [hep-ph/0101257].
hep-ph/0101257 and
%%CITATION = HEP-PH 0101257;%%
%
%\cite{Denner:2002cg}
%\bibitem{Denner:2002cg}
%A.~Denner, S.~Dittmaier, M.~Roth and D.~Wackeroth,
%``RacoonWW1.3: A Monte Carlo program for four-fermion production at e^+ e^- colliders,''
Comput.\ Phys.\ Commun.\  {\bf 153} (2003) 462
[hep-ph/0209330].
%%CITATION = HEP-PH 0209330;%%

%\cite{Aeppli:1993rs}
\bibitem{Aeppli:1993rs}
A.~Aeppli, G.~J.~van Oldenborgh and D.~Wyler,
%``Unstable particles in one loop calculations,''
Nucl.\ Phys.\ B {\bf 428} (1994) 126
[hep-ph/9312212].
%%CITATION = HEP-PH 9312212;%%

%\cite{Jadach:1998tz}
\bibitem{Jadach:1998tz}
%S.~Jadach, W.~P\l{}aczek, M.~Skrzypek, B.~F.~Ward and Z.~W\c{a}s,
S.~Jadach {\it et al.},
%``Final state radiative effects for the exact O(alpha) YFS exponentiated  (un)stable W+ W- production at and beyond LEP2 energies,''
Phys.\ Rev.\ D {\bf 61} (2000) 113010
[hep-ph/9907436];
%%CITATION = HEP-PH 9907436;%%
%
%\cite{Jadach:2000kw}
%\bibitem{Jadach:2000kw}
%S.~Jadach, W.~P\l{}aczek, M.~Skrzypek, B.~F.~Ward and Z.~W\c{a}s,
%``Precision predictions for (un)stable W+ W- pair production at and  beyond LEP2 energies,''
Phys.\ Rev.\ D {\bf 65} (2002) 093010
[hep-ph/0007012];
%%CITATION = HEP-PH 0007012;%%
%
%\cite{Jadach:2001uu}
%\bibitem{Jadach:2001uu}
%S.~Jadach, W.~P\l{}aczek, M.~Skrzypek, B.~F.~Ward and Z.~W\c{a}s,
%S.~Jadach {\it et al.},
%``The Monte Carlo event generator YFSWW3 version 1.16 for W pair  production and decay at LEP2/LC energies,''
Comput.\ Phys.\ Commun.\  {\bf 140} (2001) 432
[hep-ph/0103163] and
%%CITATION = HEP-PH 0103163;%%
%
%\cite{Jadach:2001mp}
%\bibitem{Jadach:2001mp}
%S.~Jadach, W.~P\l{}aczek, M.~Skrzypek, B.~F.~Ward and Z.~W\c{a}s,
%``The Monte Carlo program KoralW version 1.51 and the concurrent Monte  Carlo KoralW&YFSWW3 with all background graphs and first order
%corrections to W pair production,''
Comput.\ Phys.\ Commun.\  {\bf 140} (2001) 475
[hep-ph/0104049].
%%CITATION = HEP-PH 0104049;%%

%\cite{Beenakker:1998gr}
\bibitem{Beenakker:1998gr}
W.~Beenakker, F.~A.~Berends and A.~P.~Chapovsky,
%``Radiative corrections to pair production of unstable particles: Results  for
%e+ e- $\to$ 4fermions,''
Nucl.\ Phys.\ B {\bf 548} (1999) 3
[hep-ph/9811481].
%%CITATION = HEP-PH 9811481;%%

%\cite{Kurihara:2001um}
\bibitem{Kurihara:2001um}
Y.~Kurihara, M.~Kuroda and D.~Schildknecht,
%``On e+ e- $\to$ W+ W- $\to$ 4f (+ gamma) at LEP2,''
Phys.\ Lett.\ B {\bf 509} (2001) 87
[hep-ph/0104201].
%%CITATION = HEP-PH 0104201;%%

%\cite{Denner:1994ca}
\bibitem{Denner:1994ca}
A.~Denner, S.~Dittmaier and R.~Schuster,
%``Probing the Higgs mechanism via gamma gamma $\to$ W+ W-,''
Phys.\ Rev.\ D {\bf 51} (1995) 4738
[hep-ph/9411268] and
%%CITATION = HEP-PH 9411268;%%
%
%\cite{Denner:1995jv}
%\bibitem{Denner:1995jv}
%A.~Denner, S.~Dittmaier and R.~Schuster,
%``Radiative corrections to gamma gamma $\to$ W+ W- in the electroweak standard model,''
Nucl.\ Phys.\ B {\bf 452} (1995) 80
[hep-ph/9503442].
%%CITATION = HEP-PH 9503442;%%

%\cite{Jikia:1996uu}
\bibitem{Jikia:1996uu}
G.~Jikia,
%``Electroweak O(alpha) corrections to W+ W- pair production in polarized
%gamma gamma collisions,''
Nucl.\ Phys.\ B {\bf 494} (1997) 19
[hep-ph/9612380].
%%CITATION = HEP-PH 9612380;%%

%\cite{Bardin:1986fi}
\bibitem{Bardin:1986fi}
D.~Y.~Bardin, S.~Riemann and T.~Riemann,
%``Electroweak One Loop Corrections To The Decay Of The Charged Vector Boson,''
Z.\ Phys.\ C {\bf 32} (1986) 121;\\
%%CITATION = ZEPYA,C32,121;%%
%
%\cite{Jegerlehner:1986vs}
%\bibitem{Jegerlehner:1986vs}
F.~Jegerlehner,
%``Vector Boson Parameters: Scheme Dependence And Theoretical Uncertainties,''
Z.\ Phys.\ C {\bf 32} (1986) 425
[Erratum-ibid.\ C {\bf 38} (1988) 519];\\
%%CITATION = ZEPYA,C32,425;%%
%
%\cite{Denner:tx}
%\bibitem{Denner:tx}
A.~Denner and T.~Sack,
%``The W Boson Width,''
Z.\ Phys.\ C {\bf 46} (1990) 653.
%%CITATION = ZEPYA,C46,653;%%

%\cite{Melnikov:1995fx}
\bibitem{Melnikov:1995fx}
K.~Melnikov and O.~I.~Yakovlev,
%``Final state interaction in the production of heavy unstable particles,''
Nucl.\ Phys.\ B {\bf 471} (1996) 90
[hep-ph/9501358];\\
%%CITATION = HEP-PH 9501358;%%
%
%\cite{Beenakker:1997bp}
%\bibitem{Beenakker:1997bp}
W.~Beenakker, A.~P.~Chapovsky and F.~A.~Berends,
%``Non-factorizable corrections to W pair production,''
Phys.\ Lett.\ B {\bf 411} (1997) 203
[hep-ph/9706339] and
%%CITATION = HEP-PH 9706339;%%
%
%\cite{Beenakker:1997ir}
%\bibitem{Beenakker:1997ir}
%W.~Beenakker, A.~P.~Chapovsky and F.~A.~Berends,
 %``Non-factorizable corrections to W-pair production: Methods and analytic
%results,''
Nucl.\ Phys.\ B {\bf 508} (1997) 17
[hep-ph/9707326];\\
%%CITATION = HEP-PH 9707326;%%
%
%\cite{Denner:1997ia}
%\bibitem{Denner:1997ia}
A.~Denner, S.~Dittmaier and M.~Roth,
%``Non-factorizable photonic corrections to e+ e- $\to$ W W $\to$ 4fermions,''
Nucl.\ Phys.\ B {\bf 519} (1998) 39
[hep-ph/9710521].
%%CITATION = HEP-PH 9710521;%%

%\cite{Dittmaier:1999mb}
\bibitem{Dittmaier:1999mb}
S.~Dittmaier,
%``A general approach to photon radiation off fermions,''
Nucl.\ Phys.\ B {\bf 565} (2000) 69
[hep-ph/9904440];\\
%%CITATION = HEP-PH 9904440;%%
%
%\cite{Roth:1999kk}
%\bibitem{Roth:1999kk}
M.~Roth,
%``Precise predictions for four-fermion production in electron positron
%annihilation,''
PhD thesis, ETH Z\"urich No. 13363 (1999),
hep-ph/0008033.
%%CITATION = HEP-PH 0008033;%%



\end{thebibliography}


\end{document}
%\fancyhead[C]{\it {XXII Texas Symposium on Relativistic Astrophysics at Stanford University,
 %December 13-19, 2004}}


%

hep-ph0506024.extracted_bib
\begin{thebibliography}{9}   % Use for  1-9  references
\begin{thebibliography}{99} % Use for 10-99 references

\vspace*{-.1em}

%\cite{Ginzburg:1981vm}
\bibitem{Ginzburg:1981vm}
%I.~F.~Ginzburg, G.~L.~Kotkin, V.~G.~Serbo and V.~I.~Telnov,
I.~F.~Ginzburg {\it et al.},
%``Colliding $\gamma$ e And $\gamma \gamma$ Beams Based On The Single Pass Accelerators (Of Vlepp Type),''
Nucl.\ Instrum.\ Meth.\  {\bf 205} (1983) 47 and
%%CITATION = NUIMA,205,47;%%
%
%\cite{Ginzburg:1982yr}
%\bibitem{Ginzburg:1982yr}
%I.~F.~Ginzburg, G.~L.~Kotkin, S.~L.~Panfil, V.~G.~Serbo and V.~I.~Telnov,
%``Colliding Gamma E And Gamma Gamma Beams Based On The Single Pass E+ E-
%Accelerators. 2. Polarization Effects. Monochromatization Improvement,''
Nucl.\ Instrum.\ Meth.\ A {\bf 219} (1984) 5;\\
%%CITATION = NUIMA,A219,5;%%
%
%\cite{Telnov:sd}
%\bibitem{Telnov:sd}
V.~I.~Telnov,
%``Problems Of Obtaining Gamma Gamma And Gamma Epsilon Colliding Beams At
%Linear Colliders,''
Nucl.\ Instrum.\ Meth.\ A {\bf 294} (1990) 72;\\
%%CITATION = NUIMA,A294,72;%%
%
%\cite{Brinkmann:1997sj}
%\bibitem{Brinkmann:1997sj}
R.~Brinkmann {\it et al.},
%``An interaction region for gamma gamma and gamma electron collisions at
%TESLA/SBLC,''
Nucl.\ Instrum.\ Meth.\ A {\bf 406} (1998) 13
[hep-ex/9707017];\\
%%CITATION = HEP-EX 9707017;%%
%
%\cite{Watanabe:1998mn}
%\bibitem{Watanabe:1998mn}
I.~Watanabe {\it et al.},
``$\gamma \gamma$ collider as an option of JLC,''
KEK-REPORT-97-17; \\
%\href{http://www.slac.stanford.edu/spires/find/hep/www?r=kek-report-97-17}{SPIRES entry}
%\cite{Badelek:2001xb}
%\bibitem{Badelek:2001xb}
B.~Badelek {\it et al.}
%[ECFA/DESY Photon Collider Working Group Collaboration],
``TESLA TDR, Part VI, Chapter 1: Photon collider  at
TESLA,''
hep-ex/0108012;\\
%%CITATION = HEP-EX 0108012;%%
%
%\cite{Abe:2001wn}
%\bibitem{Abe:2001wn}
T.~Abe {\it et al.}  [American Linear Collider Working Group Collaboration],
%``Linear collider physics resource book for Snowmass 2001,''
%in {\it Proc. of the APS/DPF/DPB Summer Study on the Future of
%  Particle Physics (Snowmass 2001) } ed. R.~Davidson and C.~Quigg,
SLAC-R-570, {\it Resource book for Snowmass 2001}
[hep-ex/0106055, hep-ex/0106056, hep-ex/0106057, hep-ex/0106058].

%\cite{DeRoeck:2003gv}
\bibitem{DeRoeck:2003gv}
A.~De Roeck,
%``Physics at a gamma gamma, e gamma and e- e- option for a linear collider,''
hep-ph/0311138;\\
%%CITATION = HEP-PH 0311138;%%
%
%\cite{Krawczyk:2003yz}
%\bibitem{Krawczyk:2003yz}
M.~Krawczyk,
%``ECFA summary: Higgs, gamma gamma and e gamma physics,''
hep-ph/0312341;\\
%%CITATION = HEP-PH 0312341;%%
%
%\cite{Brodsky:2004wx}
%\bibitem{Brodsky:2004wx}
S.~J.~Brodsky,
%``High energy photon photon collisions at a linear collider,''
hep-ph/0404186.
%%CITATION = HEP-PH 0404186;%%

%\cite{Bredenstein:2004ef}
\bibitem{Bredenstein:2004ef}
A.~Bredenstein, S.~Dittmaier and M.~Roth,
%``Four-fermion production at gamma gamma colliders: I. Lowest-order predictions
%and anomalous couplings,''
Eur.\ Phys.\ J.\ C {\bf 36} (2004) 341
[hep-ph/0405169].
%%CITATION = HEP-PH 0405169;%%

%\cite{Bredenstein:2005zk}
\bibitem{Bredenstein:2005zk}
  A.~Bredenstein, S.~Dittmaier and M.~Roth,
  %``Four-fermion production at gamma gamma colliders: 2. Radiative corrections
  %in double-pole approximation,''
  hep-ph/0506005.
  %%CITATION = HEP-PH 0506005;%%

%\cite{Moretti:1996nv}
\bibitem{Moretti:1996nv}
M.~Moretti,
%``Four-fermion productions at a gamma gamma collider,''
Nucl.\ Phys.\ B {\bf 484} (1997) 3
[hep-ph/9604303] and
%%CITATION = HEP-PH 9604303;%%
%
%\cite{Moretti:1996cn}
%\bibitem{Moretti:1996cn}
%M.~Moretti,
%``Four leptons final states from gamma gamma fusion,''
hep-ph/9606225;\\
%%CITATION = HEP-PH 9606225;%%
%
%\cite{Baillargeon:1997se}
%\bibitem{Baillargeon:1997se}
M.~Baillargeon, G.~B\'elanger and F.~Boudjema,
%``Semi-leptonic four-fermion final states in polarised gamma gamma  reactions:
%Exact results vs an improved narrow width approximation,''
Phys.\ Lett.\ B {\bf 404} (1997) 124
[hep-ph/9701368] and
%%CITATION = HEP-PH 9701368;%%
%
%\cite{Baillargeon:1997rz}
%\bibitem{Baillargeon:1997rz}
%M.~Baillargeon, G.~B\'elanger and F.~Boudjema,
%``Effects of non-standard tri-linear couplings in photon photon  collisions. I:
%gamma gamma $\to$ W+ W-,''
Nucl.\ Phys.\ B {\bf 500} (1997) 224
[hep-ph/9701372];\\
%%CITATION = HEP-PH 9701372;%%
%
%\cite{Boos:1997jt}
%\bibitem{Boos:1997jt}
E.~Boos and T.~Ohl,
%``Towards a complete calculation of gamma gamma $\to$ 4f,''
Phys.\ Lett.\ B {\bf 407} (1997) 161
[hep-ph/9705374].
%%CITATION = HEP-PH 9705374;%%

%\cite{Zarnecki:2002qr}
\bibitem{Zarnecki:2002qr}
A.~F.~Zarnecki,
%``CompAZ: Parametrization of the photon collider luminosity spectra,''
Acta Phys.\ Polon.\ B {\bf 34} (2003) 2741
[hep-ex/0207021].
%%CITATION = HEP-EX 0207021;%%

%\cite{Kilian:2001qz}
\bibitem{Kilian:2001qz}
W.~Kilian,
``WHIZARD 1.0: A generic Monte-Carlo integration and event generation  package for multi-particle processes. Manual,''
LC-TOOL-2001-039.
%\href{http://www.slac.stanford.edu/spires/find/hep/www?r=lc-tool-2001-039}{SPIRES entry}

%\cite{Stelzer:1994ta}
\bibitem{Stelzer:1994ta}
T.~Stelzer and W.~F.~Long,
%``Automatic generation of tree level helicity amplitudes,''
Comput.\ Phys.\ Commun.\  {\bf 81} (1994) 357
[hep-ph/9401258].\\
%%CITATION = HEP-PH 9401258;%%
%\cite{Murayama:1992gi}
%\bibitem{Murayama:1992gi}
H.~Murayama, I.~Watanabe and K.~Hagiwara,
``HELAS: HELicity amplitude subroutines for Feynman diagram evaluations,''
KEK-91-11.
%\href{http://www.slac.stanford.edu/spires/find/hep/www?r=kek-91-11}{SPIRES entry}

%\cite{Grunewald:2000ju}
\bibitem{Grunewald:2000ju}
M.~W.~Gr\"unewald {\it et al.},
in {\it Reports of the Working Groups on Precision Calculations
for LEP2 Physics}, eds.\ S.~Jadach, G.~Passarino and R.~Pittau
(CERN 2000-009, Geneva, 2000), p.~1
[hep-ph/0005309].

%\cite{Denner:2005es}
\bibitem{Denner:2005es}
  A.~Denner, S.~Dittmaier, M.~Roth and L.~H.~Wieders,
  %``Complete electroweak O(alpha) corrections to charged-current e+ e- $\to$
  %4fermion processes,''
  Phys.\ Lett.\ B {\bf 612} (2005) 223
  [hep-ph/0502063] and
  %%CITATION = HEP-PH 0502063;%%
%
%\cite{Denner:2005fg}
%\bibitem{Denner:2005fg}
%  A.~Denner, S.~Dittmaier, M.~Roth and L.~H.~Wieders,
  %``Electroweak corrections to charged-current e+ e- $\to$ 4 fermion processes:
  %Technical details and further results,''
  hep-ph/0505042.
  %%CITATION = HEP-PH 0505042;%%

%\cite{Denner:1999gp}
\bibitem{Denner:1999gp}
A.~Denner, S.~Dittmaier, M.~Roth and D.~Wackeroth,
%``Predictions for all processes e+ e- $\to$ 4fermions + gamma,''
Nucl.\ Phys.\ B {\bf 560} (1999) 33
[hep-ph/9904472];
%%CITATION = HEP-PH 9904472;%%
%
%\cite{Denner:2000kn}
%\bibitem{Denner:2000kn}
%A.~Denner, S.~Dittmaier, M.~Roth and D.~Wackeroth,
%``O(alpha) corrections to e+ e- $\to$ W W $\to$ 4fermions(+gamma): First  numerical results from RACOONWW,''
Phys.\ Lett.\ B {\bf 475} (2000) 127
[hep-ph/9912261];
%%CITATION = HEP-PH 9912261;%%
%
%\cite{Denner:1999dt}
%\bibitem{Denner:1999dt}
%A.~Denner, S.~Dittmaier, M.~Roth and D.~Wackeroth,
%``W pair production at future e+ e- colliders: Precise predictions from
%RACOONWW,''
Eur.\ Phys.\ J.\ direct C {\bf 2} (2000) 4
[hep-ph/9912447];
%%CITATION = HEP-PH 9912447;%%
%
%\cite{Denner:2000bj}
%\bibitem{Denner:2000bj}
%A.~Denner, S.~Dittmaier, M.~Roth and D.~Wackeroth,
%``Electroweak radiative corrections to e+ e- --> W W --> 4fermions in  double-pole approximation: The RACOONWW approach,''
Nucl.\ Phys.\ B {\bf 587} (2000) 67
[hep-ph/0006307];
%%CITATION = HEP-PH 0006307;%%
%
%\cite{Denner:2001zp}
%\bibitem{Denner:2001zp}
%A.~Denner, S.~Dittmaier, M.~Roth and D.~Wackeroth,
%``Off-shell W pair production: Universal versus non-universal  corrections,''
%in {\it Proc. of the 5th International Symposium on Radiative Corrections (RADCOR 2000)} ed. H.~E. Haber, [hep-ph/0101257].
hep-ph/0101257 and
%%CITATION = HEP-PH 0101257;%%
%
%\cite{Denner:2002cg}
%\bibitem{Denner:2002cg}
%A.~Denner, S.~Dittmaier, M.~Roth and D.~Wackeroth,
%``RacoonWW1.3: A Monte Carlo program for four-fermion production at e^+ e^- colliders,''
Comput.\ Phys.\ Commun.\  {\bf 153} (2003) 462
[hep-ph/0209330].
%%CITATION = HEP-PH 0209330;%%

%\cite{Aeppli:1993rs}
\bibitem{Aeppli:1993rs}
A.~Aeppli, G.~J.~van Oldenborgh and D.~Wyler,
%``Unstable particles in one loop calculations,''
Nucl.\ Phys.\ B {\bf 428} (1994) 126
[hep-ph/9312212].
%%CITATION = HEP-PH 9312212;%%

%\cite{Jadach:1998tz}
\bibitem{Jadach:1998tz}
%S.~Jadach, W.~P\l{}aczek, M.~Skrzypek, B.~F.~Ward and Z.~W\c{a}s,
S.~Jadach {\it et al.},
%``Final state radiative effects for the exact O(alpha) YFS exponentiated  (un)stable W+ W- production at and beyond LEP2 energies,''
Phys.\ Rev.\ D {\bf 61} (2000) 113010
[hep-ph/9907436];
%%CITATION = HEP-PH 9907436;%%
%
%\cite{Jadach:2000kw}
%\bibitem{Jadach:2000kw}
%S.~Jadach, W.~P\l{}aczek, M.~Skrzypek, B.~F.~Ward and Z.~W\c{a}s,
%``Precision predictions for (un)stable W+ W- pair production at and  beyond LEP2 energies,''
Phys.\ Rev.\ D {\bf 65} (2002) 093010
[hep-ph/0007012];
%%CITATION = HEP-PH 0007012;%%
%
%\cite{Jadach:2001uu}
%\bibitem{Jadach:2001uu}
%S.~Jadach, W.~P\l{}aczek, M.~Skrzypek, B.~F.~Ward and Z.~W\c{a}s,
%S.~Jadach {\it et al.},
%``The Monte Carlo event generator YFSWW3 version 1.16 for W pair  production and decay at LEP2/LC energies,''
Comput.\ Phys.\ Commun.\  {\bf 140} (2001) 432
[hep-ph/0103163] and
%%CITATION = HEP-PH 0103163;%%
%
%\cite{Jadach:2001mp}
%\bibitem{Jadach:2001mp}
%S.~Jadach, W.~P\l{}aczek, M.~Skrzypek, B.~F.~Ward and Z.~W\c{a}s,
%``The Monte Carlo program KoralW version 1.51 and the concurrent Monte  Carlo KoralW&YFSWW3 with all background graphs and first order
%corrections to W pair production,''
Comput.\ Phys.\ Commun.\  {\bf 140} (2001) 475
[hep-ph/0104049].
%%CITATION = HEP-PH 0104049;%%

%\cite{Beenakker:1998gr}
\bibitem{Beenakker:1998gr}
W.~Beenakker, F.~A.~Berends and A.~P.~Chapovsky,
%``Radiative corrections to pair production of unstable particles: Results  for
%e+ e- $\to$ 4fermions,''
Nucl.\ Phys.\ B {\bf 548} (1999) 3
[hep-ph/9811481].
%%CITATION = HEP-PH 9811481;%%

%\cite{Kurihara:2001um}
\bibitem{Kurihara:2001um}
Y.~Kurihara, M.~Kuroda and D.~Schildknecht,
%``On e+ e- $\to$ W+ W- $\to$ 4f (+ gamma) at LEP2,''
Phys.\ Lett.\ B {\bf 509} (2001) 87
[hep-ph/0104201].
%%CITATION = HEP-PH 0104201;%%

%\cite{Denner:1994ca}
\bibitem{Denner:1994ca}
A.~Denner, S.~Dittmaier and R.~Schuster,
%``Probing the Higgs mechanism via gamma gamma $\to$ W+ W-,''
Phys.\ Rev.\ D {\bf 51} (1995) 4738
[hep-ph/9411268] and
%%CITATION = HEP-PH 9411268;%%
%
%\cite{Denner:1995jv}
%\bibitem{Denner:1995jv}
%A.~Denner, S.~Dittmaier and R.~Schuster,
%``Radiative corrections to gamma gamma $\to$ W+ W- in the electroweak standard model,''
Nucl.\ Phys.\ B {\bf 452} (1995) 80
[hep-ph/9503442].
%%CITATION = HEP-PH 9503442;%%

%\cite{Jikia:1996uu}
\bibitem{Jikia:1996uu}
G.~Jikia,
%``Electroweak O(alpha) corrections to W+ W- pair production in polarized
%gamma gamma collisions,''
Nucl.\ Phys.\ B {\bf 494} (1997) 19
[hep-ph/9612380].
%%CITATION = HEP-PH 9612380;%%

%\cite{Bardin:1986fi}
\bibitem{Bardin:1986fi}
D.~Y.~Bardin, S.~Riemann and T.~Riemann,
%``Electroweak One Loop Corrections To The Decay Of The Charged Vector Boson,''
Z.\ Phys.\ C {\bf 32} (1986) 121;\\
%%CITATION = ZEPYA,C32,121;%%
%
%\cite{Jegerlehner:1986vs}
%\bibitem{Jegerlehner:1986vs}
F.~Jegerlehner,
%``Vector Boson Parameters: Scheme Dependence And Theoretical Uncertainties,''
Z.\ Phys.\ C {\bf 32} (1986) 425
[Erratum-ibid.\ C {\bf 38} (1988) 519];\\
%%CITATION = ZEPYA,C32,425;%%
%
%\cite{Denner:tx}
%\bibitem{Denner:tx}
A.~Denner and T.~Sack,
%``The W Boson Width,''
Z.\ Phys.\ C {\bf 46} (1990) 653.
%%CITATION = ZEPYA,C46,653;%%

%\cite{Melnikov:1995fx}
\bibitem{Melnikov:1995fx}
K.~Melnikov and O.~I.~Yakovlev,
%``Final state interaction in the production of heavy unstable particles,''
Nucl.\ Phys.\ B {\bf 471} (1996) 90
[hep-ph/9501358];\\
%%CITATION = HEP-PH 9501358;%%
%
%\cite{Beenakker:1997bp}
%\bibitem{Beenakker:1997bp}
W.~Beenakker, A.~P.~Chapovsky and F.~A.~Berends,
%``Non-factorizable corrections to W pair production,''
Phys.\ Lett.\ B {\bf 411} (1997) 203
[hep-ph/9706339] and
%%CITATION = HEP-PH 9706339;%%
%
%\cite{Beenakker:1997ir}
%\bibitem{Beenakker:1997ir}
%W.~Beenakker, A.~P.~Chapovsky and F.~A.~Berends,
 %``Non-factorizable corrections to W-pair production: Methods and analytic
%results,''
Nucl.\ Phys.\ B {\bf 508} (1997) 17
[hep-ph/9707326];\\
%%CITATION = HEP-PH 9707326;%%
%
%\cite{Denner:1997ia}
%\bibitem{Denner:1997ia}
A.~Denner, S.~Dittmaier and M.~Roth,
%``Non-factorizable photonic corrections to e+ e- $\to$ W W $\to$ 4fermions,''
Nucl.\ Phys.\ B {\bf 519} (1998) 39
[hep-ph/9710521].
%%CITATION = HEP-PH 9710521;%%

%\cite{Dittmaier:1999mb}
\bibitem{Dittmaier:1999mb}
S.~Dittmaier,
%``A general approach to photon radiation off fermions,''
Nucl.\ Phys.\ B {\bf 565} (2000) 69
[hep-ph/9904440];\\
%%CITATION = HEP-PH 9904440;%%
%
%\cite{Roth:1999kk}
%\bibitem{Roth:1999kk}
M.~Roth,
%``Precise predictions for four-fermion production in electron positron
%annihilation,''
PhD thesis, ETH Z\"urich No. 13363 (1999),
hep-ph/0008033.
%%CITATION = HEP-PH 0008033;%%



\end{thebibliography}