%Title: Two-loop electroweak corrections at high energies
%Author: Ansgar Denner, Michael Melles and Stefano Pozzorini
%Published: <EM> Nucl. Phys. Proc. Suppl. </EM> <B> 116 </B> (2003) 18-22. 
%hep-ph/0211196
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%%%%%%%%%%%%%%%%%%%%%%%%%%  appendix.tex %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


% declarations for front matter
\title{Two-loop electroweak corrections at high energies}

\author{A.~Denner%
\address{Paul Scherrer Institut (PSI), CH-5232 Villigen, Switzerland}%
%\thanks{Ansgar.Denner@psi.ch}%
, 
M.~Melles$^{\mathrm{a}}$
%${}^\mathrm{a}$
%%\thanks{Michael.Melles@psi.ch}
and
S.~Pozzorini%
\address{Institut f\"ur Theoretische Teilchenphysik,
Universit\"at Karlsruhe,
D-76128 Karlsruhe, Germany}%
%%\thanks{pozzorin@particle.uni-karlsruhe.de}
%\thanks{Talk presented  by S.~P. 
%at the International Symposium on Radiative Corrections RADCOR 2002, September 8--13, Kloster Banz, Germany.
%}
}
\begin{document}


\thispagestyle{empty}
\def\thefootnote{\fnsymbol{footnote}}
\setcounter{footnote}{1}
\null
%\draftdate
\hfill  PSI-PR-02-20
\\
\strut\hfill TTP02-36\\
\strut\hfill hep-ph/0211196
\vskip 0cm
\vfill
\begin{center}
{\Large \bf
Two-loop electroweak corrections at high energies%
\footnote{
Talk presented  by S.~P. 
at the International Symposium on Radiative Corrections RADCOR 2002, September 8--13, Kloster Banz, Germany.
}
\par} \vskip 2.5em
{\large
{\sc A.~Denner$^1$, M.~Melles$^1$ and S.~Pozzorini$^2$
}\\[1ex]
{\normalsize $^1$ \it Paul Scherrer Institut\\
CH-5232 Villigen PSI, Switzerland}\\[2ex]
{\normalsize $^2$ \it
Institut f\"ur Theoretische Teilchenphysik,
Universit\"at Karlsruhe\\
D-76128 Karlsruhe, Germany
}}

\par \vskip 1em
\end{center}
\par
\vskip .0cm 
\vfill {\bf Abstract:} \par 
We discuss  two-loop leading and angular-dependent next-to-leading
logarithmic  electroweak virtual corrections to arbitrary processes 
at energies above the electroweak scale.
The relevant Feynman diagrams involving soft-collinear gauge 
bosons $\gamma,\PZ,\PW^\pm$ have been evaluated in eikonal approximation.
We present results %for the leading- and the angular-dependent next-to-leading logarithms, 
obtained from the analytic evaluation of massive loop integrals.
To isolate mass singularities we used  the Sudakov method and alternatively the  sector decomposition method in the Feynman-parameter representation.
\par
\vskip 1cm
\noindent
November 2002 
\par
\null
\setcounter{page}{0}
\clearpage
\def\thefootnote{\arabic{footnote}}
\setcounter{footnote}{0}



\begin{abstract}
We discuss  two-loop leading and angular-dependent next-to-leading
logarithmic  electroweak virtual corrections to arbitrary processes 
at energies above the electroweak scale.
The relevant Feynman diagrams involving soft-collinear gauge 
bosons $\gamma,\PZ,\PW^\pm$ have been evaluated in eikonal approximation.
We present results %for the leading- and the angular-dependent next-to-leading logarithms, 
obtained from the analytic evaluation of massive loop integrals.
To isolate mass singularities we used  the Sudakov method and alternatively the  sector decomposition method in the Feynman-parameter representation.
\end{abstract}

% typeset front matter (including abstract)
\maketitle

%\include{./appendix}
%\include{./sumdiag}

\section{Introduction}
The main task of future colliders such as the LHC
%\cite{Haywood:1999qg} 
or an $\Pep\Pem$ Linear Collider (LC)
%\cite{Accomando:1998wt,Aguilar-Saavedra:2001rg,Abe:2001wn} 
is  the investigation of the origin of electroweak symmetry breaking and the
exploration of the limits of the Electroweak Standard Model.  In order
to disentangle effects of physics beyond the Standard Model, the
inclusion of QCD and electroweak radiative corrections into the
theoretical predictions is crucial.

In the energy range of future colliders, \ie at energies above the
electroweak scale, $\sqrt{s}\gg\MW$, the electroweak
corrections are enhanced by large logarithmic contributions \cite{Kuroda:1991wn} of the type
%$\alpha^N\log^{2N-k}{\left(s/\MW^2\right)}$
\beq\label{logform}%\refeq{logform}
\alpha^N\log^{M}{\left(\frac{s}{\MW^2}\right)},\quad M>0.
\eeq
The leading logarithms (LL), also known as Sudakov logarithms
\cite{Sudakov:1954sw}, correspond to $M=2N$,
the next-to-leading logarithms (NLL) to $M=2N-1$, etc.
%
The logarithmic dependence on various kinematic invariants 
$r=s,t,u,\dots$  gives rise to
subleading logarithms that involve ratios of invariants,
and  which we denote as angular-dependent logarithms
\beq\label{anglogform}%\refeq{anglogform}
\alpha^N\log^{M-L}{\left(\frac{s}{\MW^2}\right)}\log^{L}{\left(\frac{|r|}{s}\right)},\quad M-L>0.
\eeq
We will consider the kinematical region $|r|\gg \MW^2$, where all invariants are much larger than the electroweak scale.
The general form  of electroweak logarithmic corrections 
is complicated by the 
hierarchy of mass scales 
%\beq
$
\Mt\sim \MH\sim \MZ\sim \MW 
%M_{\Pt,\PH} \sim M_{\PW,\PZ}
\gg m_f\gg \la,
$
%\eeq
%M_{\Pt,\PH}^2 \sim M_{\PW,\PZ}^2
%with various invariants $r_{kl}=(p_k+p_l)^2$, 
with heavy masses %$\Mt,\MH,\MZ,\MW$ 
at the electroweak scale, light-fermion masses $m_f$, and the photon mass $\la$  as infrared regulator.
%Note that we consider the limit where all invariants are much larger than the electroweak scale.
As a consequence all logarithms of the large ratios ${\MW}/ m_f$ and ${\MW}/\la$  have  to be taken into account.

All the above logarithmic terms 
%can be predicted in a  process-independent way, basing on the fact that they 
constitute the singular part of the corrections in the massless limit. 
%$M^2/s \to 0$. 
They result  either 
as mass singularities
from soft/collinear  emission of virtual or real particles off initial or final-state particles,
or as remnant of ultraviolet singularities after parameter renormalization. 

At the one-loop level, it has been proven that for processes that are not mass-suppressed at high energies 
the electroweak logarithms are universal, and 
general results have been given \cite{Denner:2001jv} and applied to gauge-boson pair production at the LHC \cite{Accomando:2001fn}.
These results are in agreement with various explicit diagrammatic calculations for many $2\to2$ scattering processes 
%in the Standard Model and the Minimal Supersymmetric Standard Model
\cite{Beenakker:1993tt,Ciafaloni:1999xg,Beccaria:2000fk,Beccaria:2001yf,Layssac:2001ur}.
%In \citere{Beccaria:2001yf} the importance of the angular-dependent
%leading logarithms has been stressed. 
The approximate size of the one-loop electroweak LL and NLL for a  typical 
$2\to 2$  cross section is %
\beqar\label{oneloopestimate}%\refeq{oneloopestimate}
\frac{\de \si_{1,\LL}}{\si_0} &\simeq&
-\frac{\alpha}{\pi\sw^2}\log^2\left(\frac{s}{\MW^2}\right)\simeq -26\%
,\nl
\frac{\de \si_{1,\NLL}}{\si_0} &\simeq&
+\frac{3\alpha}{\pi\sw^2}\log\left(\frac{s}{\MW^2}\right)\simeq 16\%,
\eeqar
at $\sqrt{s}=1\TeV$, with $1-\sw^2=\cw^2=\MW^2/\MZ^2$.
The LL and NLL have similar size and opposite sign resulting in large cancellations.

Assuming that at high energies the symmetric phase of the electroweak
theory can be used, 
resummations of the electroweak logarithms have
been proposed based on techniques and results known from QCD. 
Fadin \etal\ \cite{Fadin:2000bq} have resummed the LL by
means of the infrared evolution equation.
% and matching conditions
%between the symmetric electroweak and purely electromagnetic regime.
K\"uhn \etal\ have applied results from QCD to resum the logarithmic
corrections to massless 4-fermion processes, $\Pep\Pem\to\Pf\bar\Pf$
up to  the NLL \cite{Kuhn:2000nn} and even to the NNLL \cite{Kuhn:2001hz}.
It was found that at $1\TeV$ there is no
clear hierarchy between LL, NLL and NNLL,
%leading, sub-leading, and sub-sub-leading corrections 
and that the angular-dependent logarithms are important.
%
Melles has proposed a resummation of the NLL for arbitrary processes
 \cite{Melles:2001gw}, which relies on the prescription of matching a 
symmetric $\SUtwo\times\Uone$ theory with QED at the electroweak scale.
Recently also an extension of this resummation to  the angular-dependent 
NLL has been proposed \cite{Melles:2001dh}. 
%A first attempt to include
%also real corrections was performed in \citere{Ciafaloni:2001mu}.

All these resummations amount to exponentiations
of the electroweak logarithms. The approximate size of the 
resulting two-loop LL and NLL for typical $2\to 2$ processes 
at $\sqrt{s}=1\TeV$ is
\beqar\label{twoloopestimate}%\refeq{twoloopestimate}
\frac{\de \si_{2,\LL}}{\si_0} &\simeq&
+\frac{\alpha^2}{2\pi^2\sw^4}\log^4\left(\frac{s}{\MW^2}\right)\simeq 3.5\%
,\nl
\frac{\de \si_{2,\NLL}}{\si_0} &\simeq&
-\frac{3\alpha^2}{\pi^2\sw^4}\log^3\left(\frac{s}{\MW^2}\right)\simeq -4.2\%
,
\eeqar
and it is clear that in view of the precision
objectives of a LC below the per-cent level these two-loop
%electroweak 
logarithms must be under control.

All the above resummation prescriptions result from 
matching a {\em symmetric} $\SUtwo\times\Uone$ theory and 
QED at the electroweak scale, assuming that other effects 
related to  spontaneous symmetry breaking may be neglected at high energies. 
This assumption needs to be checked by 
explicit diagrammatic two-loop calculations based on the  
electroweak Lagrangian, where  all nontrivial effects 
related to  {\em spontaneous symmetry breaking} are taken into account, 
in particular 
(i) the large gap $\MW\sim\MZ\gg\la$ in the gauge sector,
%two-scale nature of the gauge sector with a 
%large gap $\MW\sim\MZ\gg\la$,
(ii) the mixing between the gauge-group eigenstates $B,W^3$ resulting into the mass eigenstates $\gamma,\PZ$,
and 
(iii) the presence of longitudinal gauge bosons as physical asymptotic states.

The resummation of the two-loop LL has been checked for the
massless fermionic form factor in \citeres{Melles:2000ed,Hori:2000tm} and for
arbitrary processes in the massive Coulomb gauge in \citere{Beenakker:2000kb}.
The resummation of the NLL has so far not been
confirmed by explicit electroweak two-loop calculations.

A subset of the NLL is furnished by the
angular-dependent logarithms of type \refeq{anglogform} with $M=2N$, $L=1$.
These contributions are numerically important 
%as has  been stressed in 
\cite{Beccaria:2001yf,Kuhn:2000nn,Kuhn:2001hz}. At
one-loop order, in the t'~Hooft--Feynman gauge,  they result only from diagrams where a gauge boson is
exchanged between two external lines. Similarly, the angular-dependent
NLL  at two-loop order can be traced back to a
relatively small set of Feynman diagrams. This allows us to perform 
a diagrammatic calculation of the two-loop angular-dependent
NLL for arbitrary processes. 
%The calculation is
%based on the eikonal approximation. 
The relevant massive two-loop
integrals have been evaluated in  eikonal approximation, and the 
logarithms have been obtained analytically using
two independent methods: the first one goes back to Sudakov
\cite{Sudakov:1954sw}, the other one uses sector decomposition
 of Feynman-parameter integrals \cite{Hepp:1966eg,Roth:1996pd,Binoth:2000ps}.
A detailed description of this calculation can be found in \citere{nextpaper}.
Here we summarize the main ingredients and results.
%In the following we restrict ourselves to virtual corrections. 
%These contain infrared divergences that have to be  compensated 
%%in physical observables by
%by real photon emission. 

%Nevertheless, the remaining electroweak virtual logarithms  
%are of physical significance since 
%%Although they correspond to the well-known soft and collinear
%%singularities observed in QCD, they do not cancel in observables. This
%%is on the one hand due to the fact that 
%the masses of the electroweak
%gauge bosons provide a physical cut-off and there is no need to
%include real Z-boson and W-boson bremsstrahlung.  





\begin{figure*}
\beqar\label{diagtwolegs}%\refeq{diagtwolegs}
%\hspace{-7mm}
\diagtwoL{jk}{ab}&=&
\vcenter{\hbox{\smalldiagramtwoL}}
\, ,\qquad
\diagtwoC{jk}{ab}=%&=&
\vcenter{\hbox{\smalldiagramtwoC}}
\, ,\qquad
\diagtwoY{jk}{abc}=%&=&
\vcenter{\hbox{\smalldiagramtwoY}}\, ,\nl
\diagthreeL{jkl}{ab}&=&
\vcenter{\hbox{\smalldiagramthreeLinv}}
\quad,\qquad
\diagthreeY{jkl}{abc}=%&=&
\vcenter{\hbox{\smalldiagramthreeY}}
\quad,\qquad
\diagfourL{jklm}{ab}=%&=&
\vcenter{\hbox{\smalldiagramfourL}}
\quad.\nn
\eeqar
\vspace{3mm}

{Figure 1. Two-loop  diagrams with soft-collinear gauge bosons $a,b,c=\gamma,\PZ,\PWpm$ exchanged between external legs $j,k,l,m=1,\dots,n$.}
%\caption{Two-loop  diagrams with soft-collinear gauge bosons $a,b,c=\gamma,\PZ,\PWpm$ exchanged between external legs $j,k,l,m=1,\dots,n$.}
\label{diagrams}%\reffi{diagrams}
\end{figure*}


\section{Feynman diagrams}
In the following we consider electroweak processes%
\footnote{As a convention, all particles $\varphi_{i_k}$  and their momenta $p_k$ are assumed to be incoming. The corresponding $2\to n-2$ processes are easily obtained by crossing symmetry.}
%\beq\label{process}%\refeq{process}
$
\varphi_{i_1}(p_1)\ldots \varphi_{i_n}(p_n)\to 0
$,
%\eeq
involving $n$
arbitrary mass-eigenstate  particles.   
%(fermions, transverse or longitudinal gauge bosons, Higgs bosons).
The kinematical invariants 
are denoted as $r_{kl}=(p_k+p_l)^2\simeq 2p_kp_l$, and the  matrix elements as
\beq\label{mel}%\refeq{mel}
\M \equiv \M^{\varphi_{i_1}\ldots \varphi_{i_n}}(p_1,\dots,p_n).
\eeq
We restrict ourselves to matrix elements that are not mass-suppressed at high energies.  In this case global gauge invariance implies
\beq\label{chargecons}%\refeq{chargecons}
\sum_{k=1}^n  \M\, I^a (k) = \O\left(\frac{M^2}{s}%^{\varphi_{i_1}\dots\varphi_{i_n}}
\right)\, \M,\quad a=\gamma,\PZ,\PWpm,
\eeq
where the gauge couplings $I^a(k)$ act as (transposed) matrices on the 
external-legs 
%index $\varphi_{i_k}$ 
$k$ of the matrix element% 
\footnote{Details concerning our notation can be found in \citere{Denner:2001jv}.}.

In the t'~Hooft--Feynman  gauge, the leading mass singularities  originate 
from diagrams with soft-collinear  virtual gauge bosons  coupling to external particles.
The relevant two-loop diagrams are depicted in \reffi{diagrams}, where 
the  soft-collinear gauge bosons %($a,b,c=\gamma,\PZ,\PWpm$)
are  exchanged between two, 
%($\diagtwoL{}{}$--$\diagtwoY{}{}$), 
three, 
%($\diagthreeL{}{}$--$\diagthreeY{}{}$) 
or four 
%($\diagfourL{}{}$) 
of the $n$ on-shell external legs.

Each loop integral has to be evaluated for all different mass assignments 
that  occur in the electroweak model. For the internal lines we have the cases $a,b,c=\gamma,\PZ,\PWpm$ and 
for the external masses we assume $\MW\gsim m_{\mathrm{ext}}\gg \la$.

The Feynman diagrams are evaluated in  eikonal approximation, \ie by 
neglecting mass terms and  the momenta of the soft gauge bosons everywhere in the numerators, apart from the momenta 
%in the trilinear gauge-boson couplings. 
in the couplings of three soft gauge bosons.
In the massless limit, longitudinal gauge bosons have to be  substituted by the corresponding would-be Goldstone Bosons using the 
Goldstone-Boson equivalence theorem.


\section{Loop integrals in logarithmic approximation}

In the  evaluation of the loop integrals we include the LL and the  angular-dependent NLL, and we use  $\Mt\simeq \MH \simeq \MZ \simeq \MW$, \ie we neglect logarithms of ratios of heavy masses.
Two analytical methods have been used:
the Sudakov method \cite{Sudakov:1954sw} 
 and sector decomposition \cite{Hepp:1966eg,Roth:1996pd,Binoth:2000ps}, which 
permits to factorize  overlapping ultraviolet or mass singularities in Feynman-parameter integrals.
% detailed description of these methods is postponed to a forthcoming publication \cite{nextpaper}, 
Here we only sketch the main steps of the second method         applied to a generic two-loop massive integral.

{\bf Step 1:} The integral
%\beq
%I=\int \frac{\rd^d q_1}{(2\pi)^d}\int\frac{\rd^d q_2}{(2\pi)^d}
%\frac{N(\{q_j\},\{p_l\})}{\prod_{i=1}^{n+1}
%(k_i^2-m_i^2+\ri \epsilon)}
%\eeq
is written in Feynman parametrization
and the denominator is split into polynomials  according to the hierarchy of scales $s\gg r \gg M^2\gg\dots\gg\la^2$ in the diagram\footnote{To extract the angular-dependent logarithms $\log{(s/r)}$ with  $r=t,u,\dots$, we compute the integrals 
in the euclidean region 
in various limits of the type $s\gg t=u$, $s=t\gg u$, etc., where we separate the energy scales in various ways.}
\beqar
I&=&%\int_0^1 
\int_{[0,1]^n} \rd \vec{x}\,
%\prod_{i=1}^n \rd x_i 
\frac{
%\delta (1-\sum_i x_i )
f(\vec{x})}{\left[
D(\vec{x})
%{s} P_{s}(\vec{x})+{M^2} P_{M}(\vec{x})+\ldots +{\la^2} P_{\la}(\vec{x})
\right]^e}
,\\
D(\vec{x})&=&{s} P_{s}(\vec{x})+{r} P_{r}(\vec{x})+\ldots +{\la^2} P_{\la}(\vec{x}).\nn
\eeqar
These polynomials have various zeros 
of the form 
\beq\label{polyone}%\refeq{polyone}
P(\vec{x})=\sum_{i=1}^m x_i P_i(\vec{x})=0,
%\quad\mbox{at}\,\,
%x_1=\dots=x_m=0.
\eeq
at $x_1=\dots=x_m=0$, which give rise to  mass singularities.

{\bf Step 2:} In order to separate  overlapping singularities, we  decompose  the sector  $[0,1]^m$ into $m$ subsectors $\Omega_j$ with $x_j>x_{i\neq j}$, and in each subsector $\Omega_j$ we perform variable transformations $x_i\to x_j x'_i$, which remap $\Omega_j\to[0,1]^m$ and permit to factorize the variable  $x_j$ in
\beq\label{polytwo}%\refeq{polytwo}
P(\vec{x})= \left[ P_j(\vec{x})+\sum_{i\neq j} x'_i P_i(\vec{x})\right]{x_j}.%=0
\eeq

{\bf Step 3:} Recursive application of step 2 permits to factorize all zeros
at all scales, until  the 
%\beq
%P(\vec{x})= \hat{P}(\vec{x})\,\prod_{k} { x_k},\quad\mbox{with}\quad \hat{P}(\vec{0})\neq 0.
%\eeq
denominator assumes the form 
\beqar
D(\vec{x})&=&
\left\{\left[{s} \hat{P}_s(\vec{x})\prod_k {x_k}
+{r} \hat{P}_r(\vec{x})\right]\prod_l {x_l}
\right.\nl&&\left.{}
+\ldots +{\la^2} \hat{P}_\la(\vec{x})\right\}\prod_m {x_m},
\eeqar
where all Feynman parameters that give rise to mass singularities  are factorized and the polynomials $\hat{P}$ are non-vanishing. This allows for 
a simple power counting of the logarithmically divergent integrations.


{\bf Step 4:} 
In leading-logarithmic approximation  the polynomials $\hat{P}$  can be treated as constants $\hat{P}(\vec{x})\simeq \hat{P}(\vec{0})$, and 
all logarithms of ratios of scales can be extracted by analytical integration of the singular parameters. 

\section{Results}
As a basis for the  presentation of our two-loop results, we recall the one-loop results 
for LL and angular-dependent NLL given in \citere{Denner:2001jv}.
\subsection{One-loop results}
At one-loop level we have
\beq
\M_1= \M_{0}(1+ \de_\EW ).%\quad \de_\EW=\de_\sew+\de_\sem, 
%\qquad
%\de_\EW  =\frac{\alpha}{4 \pi} 
%\sum_{k=1}^n 
%\de_\EW (k)
\eeq
%external-leg ``factors''
The most symmetric form to write the   
electroweak logarithmic corrections consists in splitting them  into 
$\de_\EW=\de_\sew+\de_\sem$, with
\beqar\label{splitting}%\refeq{splitting}
\de_\sew
&=&
%=
\left.\de_\EW\right|_{\la=\MW}
,\nl
\de_\sem
&=&
%=
\left. \de_{\EW} -\de_{\EW}\right|_{\la=\MW}.
\eeqar
The symmetric electroweak part (sew) corresponds to the case when the photon mass $\la$ equals the electroweak scale and reads
\beqar\label{sewpart}%\refeq{sewpart}
\lefteqn{
\de_\sew=%\left.\de_\EW\right|_{\la=\MW}=
\frac{\alpha}{4\pi}\sum_{k=1}^n\left\{
-\frac{1}{2}\cew(k) \log^2\frac{s}{\MW^2}
\right.}%\quad
&&\\
&&\left.
\hspace{-7mm}{}
+\sum_{l\neq k} \sum_{a=\gamma,\PZ,\PWpm} 
I^a(k) I^{\bar{a}}(l)
\log\frac{|r_{kl}|}{s}
\log\frac{s}{\MW^2}
\right\},\nn
\eeqar
%$U(1)_{\mathrm{em}}$
%$\la\ll \MW$ \imp 
where 
%the gauge couplings $I^a$ and the electroweak Casimir operator  
$\cew=Y^2/(4\cw^2)+C^{\SUtwo}/\sw^2$ represents the electroweak Casimir operator.
The remaining part is a subtracted electromagnetic (sem) contribution 
originating from the fact that $\la\ll\MW$,
\beqar\label{sempart}%\refeq{sempart}
\lefteqn{
\de_\sem
=\frac{\alpha}{4\pi}\sum_{k=1}^n 
\left\{
-\frac{1}{2}Q^2(k)
\left[
2\log\frac{s}{m_k^2}\log{\frac{\MW^2}{\la^2}}
\right.\right.}
\quad&&\nl
&&\left.\left.
\hspace{-10mm}
-\log^2{\frac{\MW^2}{m_k^2}}
\right]
%\right.\nonumber\\&& \left.
+\sum_{l\neq k}
Q(k) Q(l)
\log\frac{|r_{kl}|}{s}
\log\frac{\MW^2}{\la^2}
\right\}.\nln
\eeqar
%originating from the fact that $\la\ll\MW$.

\subsection{Two-loop results}
Detailed results for the individual two-loop diagrams in
\reffi{diagrams}
are presented in 
%a forthcoming publication 
\citere{nextpaper}. 
These diagrams have to be combined as follows
\beqar\label{combination}%\refeq{combination}
\lefteqn{
\de\M_2=\sum_{a,b}\left\{\sum_{j,k}
\left[\frac{1}{2}\left(\diagtwoL{jk}{ab}
+\diagtwoC{jk}{ab}\right)
+\sum_c \diagtwoY{jk}{abc}\right]
\right.
}&&
\nl&&\left.\hspace{-6mm}
{}+\sum_{j,k,l}%\sum_{a,b}
\left[\diagthreeL{jkl}{ab}
%+(j\leftrightarrow l)+(j\leftrightarrow k)
+\frac{1}{6}\sum_c\diagthreeY{jkl}{abc}
\right]%\nl&&{}
{}+\frac{1}{8}\sum_{j,k,l,m}%\sum_{a,b}
\diagfourL{jklm}{ab}\right\},\nn
\eeqar
taking into account all sums over virtual gauge bosons $a,b,c=\gamma,\PZ,\PWpm$ and external legs%
\footnote{In the sums only the contributions from different external legs  $j\neq k$, etc. have to be considered.}
$j,k,l,m=1,\dots,n$, with appropriate symmetry factors.
These sums can be simplified by means of \refeq{chargecons}, and it turns out
that the result corresponds to the exponentiation of the  one-loop corrections \refeq{sewpart},\refeq{sempart} in the form 
\beq\label{result}%\refeq{result}
\M_2=
\M_{0}
\exp\left(\de_\sew\right) 
\exp\left(\de_\sem\right), 
%\qquad
%\de_\EW  =\frac{\alpha}{4 \pi} 
%\sum_{k=1}^n 
%\de_\EW (k)
\eeq
where the symmetric electroweak and the subtracted electromagnetic parts exponentiate separately. 

\section{Discussion}
Our result confirms the exponentiation of the electroweak LL obtained with the infrared evolution equation \cite{Fadin:2000bq} and in the Coulomb gauge \cite{Beenakker:2000kb}.
The subset of diagrams (A-C) has been evaluated also in the special case of massless external particles and agreement has been found with the form-factor calculation of \citeres{Melles:2000ed,Hori:2000tm}. The exponentiation of the electroweak angular-dependent NLL 
agrees with the results of \citere{Kuhn:2000nn} %\citere{Kuhn:2001hz}
for massless fermionic processes, % $\Pep\Pem\to\Pf\bar\Pf$, 
and \citere{Melles:2001dh} for  arbitrary processes. 
These results were obtained using  matching conditions at the electroweak scale. 

In the following we discuss the idea of matching by
applying it to our result. If we set $\la=\MW$ or $s=\MW^2$,  we obtain  
\beqar\label{matching}%\refeq{matching}
&&\hspace{-6mm}
\la=\MW \Rightarrow \de_\sem =0,\,\,
\M_2=
\exp\left[\de_\sew\right] \M_{0},\nl
&&\hspace{-6mm}
s=\MW^2 \Rightarrow\de_\sew =0,\,\,
\M_2=\exp\left[\de_\sem\right] \M_{0},
\eeqar
\ie we observe  exponentiation 
within a  symmetric $\SUtwo\times\Uone$ theory ($\la=\MW$)
and QED ($s=\MW^2$). This provides a simple consistency check of our result. 
However, we stress that the matching conditions \refeq{matching}
are not sufficient in order to determine %give no information about 
the  interference terms $\de_\sew \de_\sem$  between electroweak and electromagnetic contributions, which vanish at both matching points.
These two-loop terms are fixed by our 
full electroweak calculation ($s\gg\MW^2\gg\la^2$) and 
are crucial in order to predict the ordering of the two exponentials in \refeq{result},
which is non-trivial at the level of the 
angular-dependent NLL, since
\beq
[\de_\sew, \de_\sem]=\O\left[\log\frac{r_{kl}}{s}
\log^2\frac{s}{\MW^2}
\log\frac{\MW^2}{\la^2}
\right].
\eeq
The fact that $\de_\sem$ appears on the right-hand side in  \refeq{result}
%corresponds to the intuitive picture, 
means that the subtracted electromagnetic 
corrections %(``below the electroweak scale'') 
are only sensitive to the electromagnetic charges of the external particles.

\begin{thebibliography}{99}

 \newcommand{\vj}[4]{{\sl #1 }{\bf #2 }\ifnum#3<100 (19#3) \else (#3) \fi #4}
 \newcommand{\epjc}[3]{\vj{Eur. Phys. J. C}{#1}{#2}{#3}}
 \newcommand{\npb}[3]{\vj{Nucl. Phys. B}{#1}{#2}{#3}}
 \newcommand{\plb}[3]{\vj{Phys. Lett. B}{#1}{#2}{#3}}
 \newcommand{\prd}[3]{\vj{Phys. Rev. D}{#1}{#2}{#3}}
 \newcommand{\prl}[3]{\vj{Phys. Rev. Lett.}{#1}{#2}{#3}}
 \providecommand{\etal}{{\it et al.}}
 \newcommand{\hepph}[1]{{hep-ph/#1}}

%%\cite{Haywood:1999qg}
%\bibitem{Haywood:1999qg}
%S. Haywood, P.R. Hobson, W. Hollik, Z. Kunszt et al.,
%hep-ph/0003275, in {\sl Standard Model Physics (and more) at the LHC},
%eds.~G. Altarelli and M.L. Mangano, (CERN-2000-004, Gen\`eve, 2000) p.~117.
%%%CITATION = HEP-PH 0003275;%%

%%\cite{Accomando:1998wt}
%\bibitem{Accomando:1998wt}
%E.~Accomando {\it et al.},  
%%[ECFA/DESY LC Physics Working Group Collaboration],
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%Phys.\ Rept.\  {\bf 299} (1998) 1
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%%\cite{Aguilar-Saavedra:2001rg}
%\bibitem{Aguilar-Saavedra:2001rg}
%J.~A.~Aguilar-Saavedra {\it et al.},
%TESLA Technical Design Report Part III: Physics at an $\mathrm{e^+e^-}$
%Linear Collider,
%hep-ph/0106315.
%%%CITATION = HEP-PH 0106315;%%

%%\cite{Abe:2001wn}
%\bibitem{Abe:2001wn}
%T.~Abe {\it et al.},  %[American Linear Collider Working Group Collaboration],
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%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].


% literature on electroweak logarithms

% appearance of electroweak logarithms
%\cite{Kuroda:1991wn}
\bibitem{Kuroda:1991wn}
M.~Kuroda, G.~Moultaka and D.~Schildknecht,
%``Direct one loop renormalization of SU(2)-L x U(1)-Y four fermion processes and running coupling constants,''
\npb{350}{1991}{25};\\
%%CITATION = NUPHA,B350,25;%%
%
%\cite{Degrassi:1992ue}
%\bibitem{Degrassi:1992ue}
G.~Degrassi and A.~Sirlin,
%``Gauge invariant selfenergies and vertex parts of the Standard Model in the pinch technique framework,''
\prd{46}{1992}{3104};\\
%%CITATION = PHRVA,D46,3104;%%
%
%\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,''
\npb{452}{1995}{80};\\
%%CITATION = HEP-PH 9503442;%%
%
%\cite{Denner:1997ug}
%\bibitem{Denner:1997ug}
A.~Denner, S.~Dittmaier and T.~Hahn,
%``Radiative corrections to Z Z $\to$ Z Z in the electroweak standard model,''
\prd{56}{1997}{117};\\
%%CITATION = HEP-PH 9612390;%%
%
%\cite{Denner:1998kq}
%\bibitem{Denner:1998kq}
A.~Denner and T.~Hahn,
%``Radiative corrections to W+ W- $\to$ W+ W- in the electroweak standard  model,''
\npb{525}{1998}{27}.
%%CITATION = HEP-PH 9711302;%%



%\cite{Sudakov:1954sw}
\bibitem{Sudakov:1954sw}
V.~V.~Sudakov,
%``Vertex Parts At Very High-Energies In Quantum Electrodynamics,''
Sov.\ Phys.\ JETP {\bf 3} (1956) 65
[Zh.\ Eksp.\ Teor.\ Fiz.\  {\bf 30} (1956) 87].
%%CITATION = SPHJA,3,65;%%

% general one-loop results

%\cite{Denner:2001jv}
\bibitem{Denner:2001jv}
A.~Denner and S.~Pozzorini,
%``One-loop leading logarithms in electroweak radiative corrections.  I: Results,''
\epjc{18}{2001}{461}; 
%[\hepph{0010201}];
%%CITATION = HEP-PH 0010201;%%
%\cite{Denner:2001gw}
%\bibitem{Denner:2001gw}
%A.~Denner and S.~Pozzorini,
%``One-loop leading logarithms in electroweak radiative corrections.  II: Factorization of collinear singularities,''
\epjc{21}{2001}{63};\\
%[\hepph{0104127}];\\
%%CITATION = HEP-PH 0104127;%%
%\cite{Pozzorini:rs}
%\bibitem{Pozzorini:rs}
S.~Pozzorini, {\em Dissertation}, Universit\"at Z\"urich, 2001,
%``Electroweak Radiative Corrections At High Energies,''
hep-ph/0201077.
%%CITATION = HEP-PH 0201077;%%

%\cite{Accomando:2001fn}
\bibitem{Accomando:2001fn}
E.~Accomando, A.~Denner and S.~Pozzorini,
%``Electroweak-correction effects in gauge boson pair production at the  LHC,''
Phys.\ Rev.\ D {\bf 65} (2002) 073003.
%[hep-ph/0110114].
%%CITATION = HEP-PH 0110114;%%


% explicit one-loop calculations of electroweak logarithms

%\cite{Beenakker:1993tt}
\bibitem{Beenakker:1993tt}
W.~Beenakker \etal,
%W.~Beenakker, A.~Denner, S.~Dittmaier, R.~Mertig and T.~Sack,
%``High-energy approximation for on-shell W pair production,''
\npb{410}{1993}{245};
%%CITATION = NUPHA,B410,245;%%
%\cite{Beenakker:1993rm}
%\bibitem{Beenakker:1993rm}
%W.~Beenakker, A.~Denner, S.~Dittmaier and R.~Mertig,
%``On shell W pair production in the TeV range,''
\plb{317}{1993}{622}.
%%CITATION = PHLTA,B317,622;%%

%\cite{Ciafaloni:1999xg}
\bibitem{Ciafaloni:1999xg}
P.~Ciafaloni and D.~Comelli,
%``Sudakov enhancement of electroweak corrections,''
\plb{446}{1999}{278}.
%[\hepph{9809321}].
%%CITATION = HEP-PH 9809321;%%

%\cite{Beccaria:2000fk}
\bibitem{Beccaria:2000fk}
M.~Beccaria \etal,
%M.~Beccaria, P.~Ciafaloni, D.~Comelli, F.~M.~Renard and C.~Verzegnassi,
%``Logarithmic expansion of electroweak corrections to four-fermion  processes in the TeV region,''
\prd{61}{2000}{073005};
%[\hepph{9906319}];
%%CITATION = HEP-PH 9906319;%%
%
%\cite{Beccaria:2000xd}
%\bibitem{Beccaria:2000xd}
%M.~Beccaria \etal,
%M.~Beccaria, P.~Ciafaloni, D.~Comelli, F.~M.~Renard and C.~Verzegnassi,
%``The role of the top mass in b production at future lepton colliders,''
\prd{61}{2000}{011301};
%[\hepph{9907389}];\\
%%CITATION = HEP-PH 9907389;%%
%
%\cite{Beccaria:2001an}
%\bibitem{Beccaria:2001an}
%M.~Beccaria \etal,
%M.~Beccaria, S.~Prelovsek, F.~M.~Renard and C.~Verzegnassi,
%``Top quark production at TeV energies as a potential SUSY detector,''
\prd{64}{2001}{053016};
%[\hepph{0104245}];\\
%%CITATION = HEP-PH 0104245;%%
%
%\cite{Beccaria:2001fu}
%\bibitem{Beccaria:2001fu}
%M.~Beccaria \etal,
%M.~Beccaria, M.~Melles, F.~M.~Renard and C.~Verzegnassi,
%``SUSY scalar production in the electroweak Sudakov regime of lepton  colliders,''
Phys.\ Rev.\ D {\bf 65} (2002) 093007;\\
%[hep-ph/0112273].
%%CITATION = HEP-PH 0112273;%%
%\cite{Beccaria:2001jz}
%\bibitem{Beccaria:2001jz}
M.~Beccaria, F.~M.~Renard and C.~Verzegnassi,
%``Top quark production at future lepton colliders in the asymptotic  regime,''
\prd{63}{2001}{053013};
%[\hepph{0010205}];
%%CITATION = HEP-PH 0010205;%%
%
%\cite{Beccaria:2001vb}
%\bibitem{Beccaria:2001vb}
%M.~Beccaria, F.~M.~Renard and C.~Verzegnassi,
%``Logarithmic SUSY electroweak effects on four-fermion processes at TeV  energies,''
\prd{63}{2001}{095010};
%[\hepph{0007224}];
%%CITATION = HEP-PH 0007224;%%
%
%\cite{Beccaria:2001yf}
%\bibitem{Beccaria:2001yf}
%M.~Beccaria, F.~M.~Renard and C.~Verzegnassi,
%``The role of universal and non universal Sudakov logarithms in four
%fermion processes at TeV energies: The one-loop approximation revisited,''
%Phys.\ Rev.\ D {\bf 64} (2001) 073008
%[hep-ph/0103335];\\
%%CITATION = HEP-PH 0103335;%%
%
%\cite{Beccaria:2002cf}
%\bibitem{Beccaria:2002cf}
%M.~Beccaria, F.~M.~Renard and C.~Verzegnassi,
%``The reality of virtual supersymmetry,''
hep-ph/0203254.
%%CITATION = HEP-PH 0203254;%%
%

%\cite{Beccaria:2001yf}
\bibitem{Beccaria:2001yf}
M.~Beccaria, F.~M.~Renard and C.~Verzegnassi,
%``The role of universal and non universal Sudakov logarithms in four
%fermion processes at TeV energies: The one-loop approximation revisited,''
Phys.\ Rev.\ D {\bf 64} (2001) 073008.
%[hep-ph/0103335].

%\cite{Layssac:2001ur}
\bibitem{Layssac:2001ur}
J.~Layssac and F.~M.~Renard,
%``High energy behaviour of gamma gamma $\to$ f anti-f processes in SM and  MSSM,''
\prd{64}{2001}{053018};\\
%[\hepph{0104205}];\\
%%CITATION = HEP-PH 0104205;%%
%
%\cite{Gounaris:2002fa}
%\bibitem{Gounaris:2002fa}
G.~J.~Gounaris, J.~Layssac and F.~M.~Renard,
%``Description of e- e+ $\to$ gamma gamma, Z gamma, Z Z in SM and MSSM,''
hep-ph/0207273.
%%CITATION = HEP-PH 0207273;%%


% resummation of logarithms

%\cite{Fadin:2000bq}
\bibitem{Fadin:2000bq}
V.~S.~Fadin \etal, 
%V.~S.~Fadin, L.~N.~Lipatov, A.~D.~Martin and M.~Melles,
%``Resummation of double logarithms in electroweak high energy processes,''
\prd{61}{2000}{094002}. 
%[\hepph{9910338}].
%%CITATION = HEP-PH 9910338;%%

%\cite{Ciafaloni:2000ub}
%\bibitem{Ciafaloni:2000ub}
%P.~Ciafaloni and D.~Comelli,
%``Electroweak Sudakov form factors and nonfactorizable soft QED effects  at NLC energies,''
%\plb{476}{2000}{49}
%[\hepph{9910278}].
%%CITATION = HEP-PH 9910278;%%

%\cite{Kuhn:1999de}
%\bibitem{Kuhn:1999de}
%J.~H.~Kuhn and A.~A.~Penin,
%``Sudakov logarithms in electroweak processes,''
%hep-ph/9906545.
%%CITATION = HEP-PH 9906545;%%

%\cite{Kuhn:2000nn}
\bibitem{Kuhn:2000nn}
J.~H.~K\"uhn, A.~A.~Penin and V.~A.~Smirnov,
%``Summing up subleading Sudakov logarithms,''
\epjc{17}{2000}{97}.
%[\hepph{9912503}].
%%CITATION = HEP-PH 9912503;%%
%
%\cite{Kuhn:2000hx}
%\bibitem{Kuhn:2000hx}
%J.~H.~Kuhn, A.~A.~Penin and V.~A.~Smirnov,
%``Subleading Sudakov logarithms in electroweak processes,''
%Nucl.\ Phys.\ Proc.\ Suppl.\  {\bf 89} (2000) 94
%[hep-ph/0005301].
%%CITATION = HEP-PH 0005301;%%
%
%\cite{Kuhn:2001hz}
\bibitem{Kuhn:2001hz}
J.~H.~K\"uhn, S.~Moch, A.~A.~Penin and V.~A.~Smirnov,
%``Next-to-next-to-leading logarithms in four-fermion electroweak  processes at high energy,''
Nucl.\ Phys.\ B {\bf 616} (2001) 286.
%[hep-ph/0106298].
%%CITATION = HEP-PH 0106298;%%

%\cite{Melles:2001gw}
\bibitem{Melles:2001gw}
M.~Melles,
%``Subleading Sudakov logarithms in electroweak high energy processes to  all orders,''
\prd{63}{2001}{034003};
%[\hepph{0004056}];
%%CITATION = HEP-PH 0004056;%%
%\cite{Melles:2001ia}
%\bibitem{Melles:2001ia}
%M.~Melles,
%``Resummation of Yukawa enhanced and subleading Sudakov logarithms in  longitudinal gauge boson and Higgs production,''
\prd{64}{2001}{014011};
%[\hepph{0012157}].
%%CITATION = HEP-PH 0012157;%%
%\cite{Melles:2001mr}
%\bibitem{Melles:2001mr}
%M.~Melles,
%``Electroweak renormalization group corrections in high energy
%processes,''
\prd{64}{2001}{054003};
%[\hepph{0102097}].
%%CITATION = HEP-PH 0102097;%%
%
%\cite{Melles:2001ye}
%\bibitem{Melles:2001ye}
%M.~Melles,
%``Electroweak radiative corrections in high energy processes,''
\hepph{0104232}.
%%CITATION = HEP-PH 0104232;%%


%\cite{Melles:2001dh}
\bibitem{Melles:2001dh}
M.~Melles,
%``Resummation of angular dependent corrections in spontaneously broken  gauge theories,''
Eur.\ Phys.\ J.\ C {\bf 24} (2002) 193.
%[hep-ph/0108221].
%%CITATION = HEP-PH 0108221;%%

% explicit two-loop calculations

%\cite{Melles:2000ed}
\bibitem{Melles:2000ed}
M.~Melles,
%``Mass gap effects and higher order electroweak Sudakov logarithms,''
\plb{495}{2000}{81}.
%[\hepph{0006077}].
%%CITATION = HEP-PH 0006077;%%

%\cite{Hori:2000tm}
\bibitem{Hori:2000tm}
M.~Hori, H.~Kawamura and J.~Kodaira,
%``Electroweak Sudakov at two loop level,''
\plb{491}{2000}{275}.
%[\hepph{0007329}].
%%CITATION = HEP-PH 0007329;%%

%\cite{Beenakker:2000na}
%\bibitem{Beenakker:2000na}
%W.~Beenakker and A.~Werthenbach,
%``Electroweak Sudakov logarithms in the Coulomb gauge,''
%Nucl.\ Phys.\ Proc.\ Suppl.\  {\bf 89} (2000) 88
%[hep-ph/0006009].
%%CITATION = HEP-PH 0006009;%%

%\cite{Beenakker:2000kb}
\bibitem{Beenakker:2000kb}
W.~Beenakker and A.~Werthenbach,
%``New insights into the perturbative structure of electroweak Sudakov logarithms: Breakdown of conventional exponentiation,''
\plb{489}{2000}{148};
%[\hepph{0005316}];
%%CITATION = HEP-PH 0005316;%%
%
%\cite{Beenakker:2001kf}
%\bibitem{Beenakker:2001kf}
%W.~Beenakker and A.~Werthenbach,
%``Electroweak two-loop Sudakov logarithms for on-shell fermions and  bosons,''
Nucl.\ Phys.\ B {\bf 630} (2002) 3.
%[hep-ph/0112030].
%%CITATION = HEP-PH 0112030;%%


% sector decomposition


%\cite{Hepp:1966eg}
\bibitem{Hepp:1966eg}
K.~Hepp,
%``Proof Of The Bogolyubov-Parasiuk Theorem On Renormalization,''
Commun.\ Math.\ Phys.\  {\bf 2} (1966) 301.
%%CITATION = CMPHA,2,301;%%

%\cite{Roth:1996pd}
\bibitem{Roth:1996pd}
M.~Roth and A.~Denner,
%``High-energy approximation of one-loop Feynman integrals,''
Nucl.\ Phys.\ B {\bf 479} (1996) 495.
%[hep-ph/9605420].
%%CITATION = HEP-PH 9605420;%%

%\cite{Binoth:2000ps}
\bibitem{Binoth:2000ps}
T.~Binoth and G.~Heinrich,
%``An automatized algorithm to compute infrared divergent multi-loop  integrals,''
Nucl.\ Phys.\ B {\bf 585} (2000) 741.
%[hep-ph/0004013].
%%CITATION = HEP-PH 0004013;%%



\bibitem{nextpaper}%\cite{nextpaper}
A.~Denner, M.~Melles  and S.~Pozzorini, in preparation.



\end{thebibliography}


\end{document}



% violation of Bloch Nordsieck theorem

%\cite{Ciafaloni:2000df}
\bibitem{Ciafaloni:2000df}
M.~Ciafaloni, P.~Ciafaloni and D.~Comelli,
%``Bloch-Nordsieck violating electroweak corrections to inclusive TeV  scale hard processes,''
\prl{84}{2000}{4810};
%[\hepph{0001142}];
%%CITATION = HEP-PH 0001142;%%
%
%\cite{Ciafaloni:2000rp}
%\bibitem{Ciafaloni:2000rp}
%M.~Ciafaloni, P.~Ciafaloni and D.~Comelli,
%``Electroweak Bloch-Nordsieck violation at the TeV scale: 'Strong' weak
%interactions?,''
\npb{589}{2000}{359};
%[\hepph{0004071}];
%CITATION = HEP-PH 0004071;%%
%
%\cite{Ciafaloni:2001gm}
%\bibitem{Ciafaloni:2001gm}
%M.~Ciafaloni, P.~Ciafaloni and D.~Comelli,
%``Electroweak double logarithms in inclusive observables for a generic  initial state,''
\plb{501}{2001}{216};
%[\hepph{0007096}];
%%CITATION = HEP-PH 0007096;%%
%
%\cite{Ciafaloni:2001vt}
%\bibitem{Ciafaloni:2001vt}
%M.~Ciafaloni, P.~Ciafaloni and D.~Comelli,
%``Bloch-Nordsieck violation in spontaneously broken Abelian theories,''
Phys.\ Rev.\ Lett.\  {\bf 87} (2001) 211802;
%[hep-ph/0103315];
%%CITATION = HEP-PH 0103315;%%
%
%\cite{Ciafaloni:2001vu}
%\bibitem{Ciafaloni:2001vu}
%M.~Ciafaloni, P.~Ciafaloni and D.~Comelli,
%``Enhanced electroweak corrections to inclusive boson fusion processes at  the TeV scale,''
Nucl.\ Phys.\ B {\bf 613} (2001) 382.
%[hep-ph/0103316].
%%CITATION = HEP-PH 0103316;%%

% reviews


%%\cite{Denner:2001mn}
%\bibitem{Denner:2001mn}
%%A.~Denner,
%%``Electroweak radiative corrections at high energies,''
%in {\it Proc. of the International Europhysics Conference on
%High-Energy Physics (HEP 2001)}, [hep-ph/0110155].
%%%CITATION = HEP-PH 0110155;%%

% real logarithms

%\cite{Ciafaloni:2001mu}
\bibitem{Ciafaloni:2001mu}
M.~Ciafaloni, P.~Ciafaloni and D.~Comelli,
%``Towards collinear evolution equations in electroweak theory,''
Phys.\ Rev.\ Lett.\  {\bf 88} (2002) 102001.
%[hep-ph/0111109].
%%CITATION = HEP-PH 0111109;%%



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