%Title: Next-to-Next-to-Leading Electroweak Logarithms in W-Pair Production at ILC
%Author: J.H. K&uuml;hn, F. Metzler, A.A.Penin
%Published: <EM> Nucl.Phys </EM> <B> B795: </B> (2008) .277-290. 
%arxiv/0709.4055

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\begin{document}
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\title{
\begin{flushright}
{\normalsize  ALBERTA-THY-12-07\\[0mm]}
{\normalsize TTP07-22\\[10mm] }
\end{flushright}
\vspace{1.0cm}
Next-to-Next-to-Leading Electroweak Logarithms in $W$-pair Production
at ILC}


\author{
  {\large J.H.~K\"uhn} $^a$,\,
  {\large F.~Metzler} $^a$,\,
  {\large A.A.~Penin} $^{b,a,c}$\\[2mm]
  $^a${\small {\it Institut f{\"u}r Theoretische Teilchenphysik,
  Universit{\"a}t Karlsruhe}}\\
  {\small {\it 76128 Karlsruhe, Germany}}\\[2mm]
  {\small\it $^b$ Department of Physics,
  University Of Alberta}\\
  {\small \it Edmonton, AB T6G 2J1, Canada}\\[2mm]
  {\small\it $^c$ Institute for Nuclear Research,
    Russian Academy of Sciences,}\\
  {\small\it  117312 Moscow, Russia}
        }
\date{}
\maketitle
\begin{abstract}
  We derive the high energy asymptotic behavior of gauge boson
  production cross section in a spontaneously broken $SU(2)$ gauge theory
  in the next-to-next-to-leading logarithmic approximation.  On the
  basis of this result we obtain the logarithmically enhanced two-loop
  electroweak corrections to the differential cross section of $W$-pair
  production at ILC/CLIC up to the second power of the large logarithm.
  \\[2mm]
  PACS numbers: 12.15.Lk
\end{abstract}

%\thispagestyle{empty}
%\newpage


\section{Introduction}
The $W$-pair production at $e^+e^-$ colliders plays a crucial role for
testing the Standard Model of electroweak interactions.  At
LEP2 this process has been used for the determination of the $W$-boson mass
$M_W$, a fundamental parameter of the standard model, through
$W$-boson reconstruction with an
uncertainty of 40~MeV \cite{LEP}. Furthermore, the triple gauge boson coupling
as predicted by the non-Abelian gauge theory has been verified within a few percent.
The experimental study of the $W$-pair production at the International Linear Collider (ILC) is
expected to improve the accuracy of the mass determination to 7~MeV due
to much higher luminosity \cite{Agu}. Moreover, the advent of ILC will
give access to the new high energy domain where the cross section is
increasingly sensitive to the triple gauge boson coupling and  $W$-pair production
could be used as a probe of the non-Abelian structure of the electroweak
interactions and of possible  gauge boson anomalous couplings.
To match the experimental accuracy, the theoretical
analysis has to take into account the electroweak radiative corrections.
The one-loop corrections to the cross section of the on-shell $W$-pair
production have been evaluated by different groups \cite{LemVel,Boh,FJZ,Bee}
already decades ago.  The calculation of the one-loop  corrections
to the $W$-boson mediated $e^+e^-\to 4f$ processes has been performed
in the double pole approximation in Ref.~\cite{Bee2} and
recently the full analysis has been completed  in Ref.~\cite{Den}. These results
ensure  an accuracy significantly better than one percent when the characteristic energy $\sqrt{s}$ of
the process is about the gauge boson mass.  However, once $\sqrt{s}$ is far
larger than $M_W$, the cross section receives  virtual corrections
enhanced by  powers of electroweak ``Sudakov'' logarithm $\ln\bigl({s/
  M_{W,Z}^2}\bigr)$, which at the energies of about one TeV have to be
controlled to two loops in order to keep the theoretical error below one
percent.  This is even more valid for energies of 3 TeV anticipated
for the CLIC project \cite{Ass}.
In the case of  light fermion pair production these
corrections are already available through the
next-to-next-to-next-to-leading logarithmic  (${\rm N}^3 {\rm LL}$)
approximation, {\it i.e.}
including all the two-loop logarithmic terms
\cite{Fad,KPS,KMPS,FKPS,JKPS}.  For $e^+e^-\to W^+W^-$ production only
the leading-logarithmic (LL) and the next-to-leading logarithms (NLL)
are known so far \cite{Fad,Mel1,DMP,BRV}.

In the present paper we extend the analysis of $W$-pair production to
the next-to-next-to-leading logarithmic (NNLL) approximation following
the approach developed in Refs.~\cite{KPS,KMPS,FKPS,JKPS} for the
four-fermion processes.  The limit of the small-angle production, which
could be interesting in the case of the transverse gauge bosons because
the corresponding cross section is peaked in the forward direction,
remains beyond the scope of the present paper. In the next section we
outline the approach and derive the NNLL corrections to the differential
cross section of the gauge boson pair production in a spontaneously
broken $SU(2)$ model which emulates the massive gauge boson sector of
the Standard Model of electroweak interactions.  We generalize the
result to the $SU(2)\times U(1)$ gauge theory with a heavy top quark in Section~\ref{sec3}.
A brief summary and conclusions are given in Section~\ref{sum}.



\section{High energy asymptotic of the massive gauge boson production}
\label{sec2} Let us  consider as a toy model  the spontaneously broken
$SU(2)$ gauge theory with
the Higgs mechanism for gauge boson mass generation and with  massless
left-handed fermion doublets. The model retains the main features of
the massive gauge boson sector of the Standard Model.
In this case the result can be presented in a simple
analytical form and constitutes the basis for the further extension
to the full electroweak theory. We study the process of  gauge
boson pair production in  fermion-antifermion annihilation at high
energy and fixed angle when all the kinematical invariants are of
the same order and  far larger than the gauge boson mass,
$|s|\sim |t| \sim |u| \gg M^2$. In this limit the asymptotic energy
dependence of the field amplitudes is dominated by {\em Sudakov}
logarithms  \cite{Sud,Jac} and  governed by the evolution
equations.  The method of the evolution equations in the context of
the electroweak corrections is described in detail for  the fermion
pair production in Ref.~\cite{JKPS}. It can be directly extended to
the gauge boson production as  briefly described below.

\begin{figure}[t]
\begin{center}
\hspace*{10mm}
\begin{picture}(100,60)(0,0)
\SetScale{0.8} \SetWidth{1} \ArrowLine(0,60)(50,60)
\ArrowLine(50,0)(0,0) \ArrowLine(50,60)(50,0)
\Photon(50,60)(100,60){4}{6.5} \Photon(50,00)(100,00){4}{6.5}
\Text(95,-10)[cc]{$(a)$}
\end{picture}
\hspace*{0mm}
\begin{picture}(100,60)(0,0)
\SetScale{0.8} \SetWidth{1} \ArrowLine(0,60)(50,60)
\ArrowLine(50,0)(0,0) \ArrowLine(50,60)(50,0)
\Photon(50,60)(100,00){4}{8.5} \Photon(100,60)(50,0){4}{8.5}
\end{picture}
\hspace*{5mm}
\begin{picture}(150,60)(0,0)
\SetScale{0.8} \SetWidth{1} \Photon(50,30)(100,30){4}{6.5}
\ArrowLine(50,30)(0,0) \ArrowLine(0,60)(50,30)
\DashLine(100,30)(150,0){5} \DashLine(100,30)(150,60){5}
\Text(62,-10)[cc]{$(b)$}
\end{picture}
\end{center}
\caption{\label{fig1} \small  The diagrams represent $(a)$
transverse and $(b)$ longitudinal gauge boson pair production in
fermion-antifermion annihilation at high energy in the Born
approximation.}
\end{figure}
Due to helicity conservation a pair of either transverse or longitudinal
gauge bosons can be produced in the high energy limit.
The transverse gauge bosons
behave like vector particles in the adjoint representation
while the longitudinal gauge bosons, as a consequence  of the equivalence
theorem, behave like scalar particles in the fundamental representation.
The structure of the Sudakov logarithms in these cases is significantly different
and we consider them separately.


\subsection{Transverse polarization}
\label{sec21} The transverse gauge bosons are the true vector
particles and the Born amplitude in this case is given by the
$t$-channel and $u$-channel fermion exchange diagrams, Fig.~\ref{fig1}$a$. It is
convenient to introduce the functions ${\cal Z}_\psi$ and ${\cal
Z}_A$ which describe the asymptotic dependence on the large momentum
transfer $Q$ of the fermion scattering amplitude in an external
singlet vector field and of the vector boson in an external scalar
singlet field, {\it i.e.} of the respective form factors in the  Euclidean region.
In  leading order in $M^2/Q^2$ these
functions are known to satisfy the following linear evolution equation  \cite{Mue,Col,Sen1}
\begin{eqnarray}
{\partial\over\partial\ln{Q^2}}{\cal Z}_i&=&
\left[\int_{M^2}^{Q^2}{\dd x\over x}\gm_i(\al(x))+\zeta_i(\al(Q^2))
+\xi_i(\al(M^2)) \right] {\cal Z}_i \,,
\label{evoleqz}
\end{eqnarray}
with the solution
\begin{eqnarray}
{\cal Z}_i&=&\exp \left\{\int_{M^2}^{Q^2}{\dd x\over x}
\left[\int_{M^2}^{x}{\dd x'\over x'}\gm_i(\al(x'))+\zeta_i(\al(x))
+\xi_i(\al(M^2))\right]\right\}
\,,
\label{evolsolz}
\end{eqnarray}
which satisfies the initial condition ${\cal Z}_i\big|_{Q^2=M^2}=1$.  Here the
perturbative functions $\gm_i(\al)$ {\it etc.} are given by the series
in the coupling constant $\alpha(\mu^2)$, {\it e.g.}
$\gm_i(\al)=\sum_{n=1}^\infty (\al/4\pi)^n\gm_i^{(n)} $. For the
amplitude of transverse boson production ${\cal A}_T$ let us
introduce the reduced amplitude $\tilde{\cal A}_T$ so that
\begin{equation} {\cal
  A}_T={\cal Z}_\psi{\cal Z}_A\tilde{\cal A}_T\,.
\label{ampred}
\end{equation}
Due to the factorization property of the Sudakov logarithms associated
with the {\it collinear} divergences of the massless theory
\cite{FreTay} the reduced amplitude satisfies the simple renormalization
group like equation \cite{Sen2,Ste,BotSte}
\begin{equation}
{\partial \over \partial \ln{Q^2}}\tilde {\cal A}_T=
\bfm{\chi}_T(\al(Q^2))\tilde {\cal A}_T \,,
\label{evoleqa}
\end{equation}
where $Q^2=-s$ and  $\bfm{\chi}_T$ is the soft anomalous dimension matrix
acting in the space of the isospin amplitudes.  The solution of the
above equation is given by the path-ordered exponent
\begin{equation}
\tilde {\cal A}_T=
{\rm P}\!\exp{\left[\int_{M^2}^{Q^2}
{\dd x\over x}\bfm{\chi}_T(\al(x))\right]}{{\cal A}_0}_T(\al(M^2))\,,
\label{evolsola}
\end{equation}
where ${{\cal A}_T}_0$ determines the initial conditions for the
evolution equation at $Q=M$.  By calculating the functions entering the
evolution equations order by order in $\al$ one gets the logarithmic
approximations for the amplitude.  For example, the LL approximation
includes all the terms of the form $\al^n{\cal L}^{2n}$ and is
determined by the one-loop coefficients $\gm_i^{(1)}$. The NLL
approximation includes all the terms of the form $\al^n{\cal L}^{2n-m}$
with $m=0,~1$ and requires the one-loop coefficients $\zeta_i^{(1)}$,
$\xi_i^{(1)}$, and $\bfm{\chi}^{(1)}$ as well as the one-loop running of
$\al$ in $\gm_i(\al)$, and so on.  To get the NNLL terms $\al^n{\cal
  L}^{2n-2}$ one needs in addition the two-loop coefficient
$\gm_i^{(2)}$,  the two-loop running of $\al$ in $\gm_i(\al)$ and
the one-loop contribution to ${{\cal A}_T}_0$.

The anomalous dimensions $\gm(\alpha)$, $\zeta(\alpha)$ and
$\bfm{\chi}(\al)$ are mass-independent and can be associated with
the infrared divergences of the massless (unbroken) theory.  From
the QCD result (see {\it e.g.} \cite{Cat} and references therein)
adopted for the specific case of $SU(2)$ gauge group, $n_f$ chiral
quarks, and one scalar in the fundamental representation we get
\begin{equation}
\gm_\psi^{(1)}=-{3/2}\,,\qquad \gm_\psi^{(2)}=-{65\over
3}+{\pi^2}+{5\over 6}n_f\,,\qquad \qquad\zeta_\psi^{(1)}={9\over
4}\,,\qquad \zeta_A^{(1)}=0\,, \label{1lgam}
\end{equation}
and $\gm_A^{(n)}=8\gm_\psi^{(n)}/3$. The matrix
$\bfm{\chi}_T^{(1)}$ can be extracted from the results of Refs.~
\cite{BFD,GloTej}. In the isospin basis
$(\bfm{\sigma}^a\bfm{\sigma}^b,~\bfm{\sigma}^b\bfm{\sigma}^a,
~\delta^{ab}\cdot\bfm{1})$ it takes the form
\begin{equation}
\bfm{\chi}^{(1)}_T=\pmatrix{ -2 (\ln(x_-)+i\pi) & 0 & \ln({x_+ \over
x_-})) \cr
          0 & -2 (\ln(x_+)+i\pi)  & \ln({x_- \over x_+}) \cr
    (\ln(x_+)+i\pi) & (\ln(x_-)+i\pi) & 0}\,,
\label{1lchit}
\end{equation}
where $x_\pm=(1\pm\cos\theta)/2$ and $\theta$ is the production
angle. Note that in this basis the Born amplitude up to a common factor
is given by the vector $(1/\xm,1/\xp,0)$.  At the same time the functions $\xi_i(\al)$
and ${\cal A}_T(\al)$ do depend on the infrared structure of the
model and require the calculation in the spontaneously broken phase.
For example $\xi_\psi^{(1)}=0$ \cite{KMPS} and from the result of
Ref.~\cite{Bee} we obtain $\xi_A^{(1)}=0$. To emulate the $e^+e^-\to
W^+_TW^-_T$ process one has to project the amplitude on the relevant
initial and final isospin states, which is straightforward.  The
Born cross section in the high energy limit reads
\begin{equation}
 {{\rm d} \sigma^B_T \over
  {\rm d} \Omega}={\alpha^2(M^2)\over 4s}{\xp(\xp^2+\xm^2)\over \xm}\,,
\label{bsigt}
\end{equation}
and  is peaked in the forward direction.
We define the perturbative expansion for the differential cross section
in the $\overline{\rm MS}$ renormalized coupling
constant $ \alpha\equiv\alpha(M^2)$  as follows
\begin{equation}
{{\rm d} \sigma \over {\rm d} \Omega}=
\left[1+\left( {\alpha \over 4\pi} \right)\dl^{(1)}
+\left( {\alpha\over 4\pi} \right)^2
\dl^{(2)}+\ldots\right]
{{\rm d} \sigma^B \over {\rm d} \Omega} \,.
\label{sersig}
\end{equation}
Expanding the Sudakov
exponents to NNLL order for the one- and two-loop corrections we get
\begin{eqnarray}
\dl_T^{(1)}&=& \left(\gamma^{(1)}_\psi+\gamma^{(1)}_A\right){\cal
L}^2(s)+2\left[\zeta^{(1)}_\psi+\zeta^{(1)}_A+\xi^{(1)}_\psi+\xi^{(1)}_A
+t_{11}^{(1)}+t_{31}^{(1)}+ {x_-\over x_+}
\left(t_{12}^{(1)}+t_{32}^{(1)}\right)\right]
\nn\\
&& \times{\cal L}(s) + \dl_{0T}^{(1)} \label{1lsigta}
\\
\dl_T^{(2)}&=& {1\over
2}\left(\gamma^{(1)}_\psi+\gamma^{(1)}_A\right)^2{\cal
L}^4(s)+2\Bigg[\zeta^{(1)}_\psi+\zeta^{(1)}_A+\xi^{(1)}_\psi+\xi^{(1)}_A
+t_{11}^{(1)}+t_{31}^{(1)}+ {x_-\over x_+}
\left(t_{12}^{(1)}+t_{32}^{(1)}\right)
\nn\\
&& -{1\over
6}\beta_0\Bigg]\left(\gamma^{(1)}_\psi+\gamma^{(1)}_A\right){\cal
L}^3(s) + \Bigg[\gamma^{(2)}_\psi+\gamma^{(2)}_A
+2\left(\zeta^{(1)}_\psi+\zeta^{(1)}_A+\xi^{(1)}_\psi+\xi^{(1)}_A
\right)\Bigg(\zeta^{(1)}_\psi+\zeta^{(1)}_A
\nn\\
&&\left.+\xi^{(1)}_\psi+\xi^{(1)}_A+2\left(t_{11}^{(1)}+t_{31}^{(1)}+ {x_-\over x_+}
\left(t_{12}^{(1)}+t_{32}^{(1)}\right)\right)\right)
-\beta_0\left(t_{11}^{(1)}+t_{31}^{(1)}+ {x_-\over x_+}
\left(t_{12}^{(1)}+t_{32}^{(1)}\right)\right.
\nn\\
&& \zeta^{(1)}_\psi+\zeta^{(1)}_A\Bigg)+
{t_{11}^{(1)}}^2+t_{12}^{(1)}t_{21}^{(1)}+t_{11}^{(1)}t_{31}^{(1)}+
t_{13}^{(1)}t_{31}^{(1)}+t_{21}^{(1)}t_{32}^{(1)}+t_{31}^{(1)}t_{33}^{(1)}
+ {x_-\over
x_+}\nn\\
&&
\times\left(t_{12}^{(1)}\left(t_{11}^{(1)}+t_{22}^{(1)}+t_{31}^{(1)}\right)
+t_{32}^{(1)}\left(t_{13}^{(1)}+t_{22}^{(1)}+t_{33}^{(1)}\right)\right)
+\left(t_{11}^{(1)}+t_{31}^{(1)}+ {x_-\over x_+}
\left(t_{12}^{(1)}+t_{32}^{(1)}\right)\right)^2\nn\\
&&
+ \dl_{0T}^{(1)}\left(\gamma^{(1)}_\psi+\gamma^{(1)}_A\right)\Bigg]
{\cal L}^2(s) \label{2lsigta}
\end{eqnarray}
where $\beta_0=43/6-n_f/3$ is the one-loop beta-function, $t_{ij}^{(1)}\equiv {\rm
Re}[\bfm{\chi}^{(1)}_T]_{ij}$, ${\cal L}(s)\equiv\ln(s/M^2)$, and $\dl_{0T}^{(1)}$ is the
nonlogarithmic part of the one-loop corrections which can be extracted from the result of
Ref.~\cite{Bee}. For the Higgs boson mass $M_H=M$ it takes a simple form
\begin{eqnarray}
\dl_{0T}^{(1)}&=&\left({5+3\xm \over 2(\xm^2+\xp^2)}
-{5 \over \xp}\right)\ln^2(\xm)
+{3\xm \over \xm^2+\xp^2}\ln(\xp^2)+{4 \over \xp}\ln(\xm)\ln(\xp)
\nn\\
&&+{9-19\xm \over 2(\xm^2+\xp^2)}\ln(\xm) -{5\xp \over
2(\xm^2+\xp^2)} -{7\pi^2 \over 18}-{13\pi \over 3 \sqrt{3}}+{455
\over 36}\,-{10 \over 9}n_f. \label{del0t}
\end{eqnarray}
Substituting the values of the coefficients for $n_f=12$ we obtain
\begin{eqnarray}
\dl_T^{(1)}&=&-{11 \over 2} {\cal L}^2(s)
+\left[\left(-8+{4 \xm \over \xp}\right)\ln(\xm)+4 \ln(\xp)
+{9\over 2}\right]{\cal L}(s)
\nn\\
&&+\left({5+3\xm \over 2(\xm^2+\xp^2)}
-{5 \over \xp}\right)\ln^2(\xm)
+{3\xm \over \xm^2+\xp^2}\ln(\xp^2)+{4 \over \xp}\ln(\xm)\ln(\xp)
\nn\\
&&+{9-19\xm \over 2(\xm^2+\xp^2)}\ln(\xm)
-{5\xp \over 2(\xm^2+\xp^2)}
-{7\pi^2 \over 18}-{13\pi \over 3 \sqrt{3}}-{25 \over 36}\,,
\label{1lsigt}\\
\dl_T^{(2)}&=&{121 \over 8} {\cal L}^4(s)
+\left[\left(44-{22\xm \over \xp}\right)\ln(\xm)-22\ln(\xp)
-{341 \over 18}\right]{\cal L}^3(s)
\nn\\
&&+\left[\left(32+{4\xm^2 \over \xp^2}
-{55+33\xm \over 4 (\xm^2+\xp^2)}
+{55-40\xm \over 2\xp}\right)\ln^2(\xm)
+\left(8-{33\xm \over 2(\xm^2+\xp^2)}\right)\ln^2(\xp)
\right.
\nn\\
&&\left.-\left(28
+{22 -4\xm\over \xp}\right)\ln(\xm)\ln(\xp)
+\left(-{70 \over 3}+{35\xm \over 3\xp}
-{99-209\xm \over 4(\xm^2+\xp^2)}\right)\ln(\xm)
\right.
\nn\\
&&\left.+{35 \over 3}\ln(\xp)
+{55\xp \over 4(\xm^2+\xp^2)}+{209\pi^2 \over 36}
+{143\pi \over 6\sqrt{3}}-{863 \over 24}\right]{\cal L}^2(s)\,.
\label{2lsigt}
\end{eqnarray}

\subsection{Longitudinal polarization}
\label{sec22} The equivalence theorem relates the cross section of
the longitudinal gauge boson production $e^+e^-\to W^+_LW^-_L$ to
the production of  the Goldstone bosons $e^+e^-\to \phi^+\phi^-$.
The Born amplitude is now  given by the $s$-channel annihilation
diagram, Fig.~\ref{fig1}$b$. The analysis of the high energy asymptotic
for the longitudinal gauge boson production goes along the
line described in the previous section and is very similar
to the one for fermion pair production \cite{KMPS}.
Instead of ${\cal Z}_A$ one should use the function
${\cal Z}_\phi$ which correspond to the scalar particle scattering
in an external singlet vector field. The necessary parameters of the
evolution equation read $\gm_\phi^{(n)}=\gm_\psi^{(n)}$,
$\zeta_\phi^{(1)}=3$, and $\xi_\phi^{(1)}=0$. The structure of the
reduced amplitude is also different.  In the isospin basis
$(\bfm{\sigma}^a\otimes\bfm{\sigma}^a,\bfm{1}\otimes\bfm{1})$ the
one-loop matrix of the soft anomalous dimensions takes the form
\begin{equation}
\bfm{\chi}^{(1)}_L= \pmatrix{
-4\left(\ln\left({x_+}\right)+i\pi\right)+
2\ln\left({x_+\over x_-}\right) \,\,&
4\ln\left({x_+\over x_-}\right) \cr
{3\over 4}\ln\left({x_+\over x_-}\right) \label{china} &
0}\,.
\label{1lchil}
\end{equation}
Note that in this basis the Born amplitude is given by the vector $(1,0)$.
The corresponding Born cross section  reads
\begin{equation}
 {{\rm d} \sigma^B_L \over
  {\rm d} \Omega}={\alpha^2(s)\over 4s}{\xp\xm\over 4}
\,.
\label{bsigl}
\end{equation}
It has a maximum at $\theta=90^o$.  We proceed as in the case of
transverse polarization and obtain the one- and two-loop NNLL
corrections to the differential cross section
\begin{eqnarray}
\dl_L^{(1)}&=& \left(\gamma^{(1)}_\psi+\gamma^{(1)}_\phi\right){\cal
L}^2(s)+2\left[\zeta^{(1)}_\psi+\zeta^{(1)}_\phi+\xi^{(1)}_\psi+\xi^{(1)}_\phi
+l_{11}^{(1)}+4l_{21}^{(1)}\right]{\cal L}(s) + \dl_{0L}^{(1)} \label{1lsigla}
\\
\dl_L^{(2)}&=& {1\over
2}\left(\gamma^{(1)}_\psi+\gamma^{(1)}_\phi\right)^2{\cal
L}^4(s)+2\left[\zeta^{(1)}_\psi+\zeta^{(1)}_A+\xi^{(1)}_\psi+\xi^{(1)}_\phi
+l_{11}^{(1)}+4l_{21}^{(1)}-{1\over 6}\beta_0\right]\left(\gamma^{(1)}_\psi+\gamma^{(1)}_\phi\right)
\nn\\
&& \times{\cal
L}^3(s) + \Bigg[\gamma^{(2)}_\psi+\gamma^{(2)}_\phi
+2\left(\zeta^{(1)}_\psi+\zeta^{(1)}_\phi+\xi^{(1)}_\psi+\xi^{(1)}_\phi
\right)\left(\zeta^{(1)}_\psi+\zeta^{(1)}_\phi+\xi^{(1)}_\psi+\xi^{(1)}_\phi
\right.\nn\\
&&
\left.+2l_{11}^{(1)}+8l_{21}^{(1)}\right)-\beta_0\left(l_{11}^{(1)}+4l_{21}^{(1)}+\zeta^{(1)}_\psi+\zeta^{(1)}_\phi\right)
+{l_{11}^{(1)}}^2+l_{21}^{(1)}\left(4l_{11}^{(1)}+l_{12}^{(1)}+4l_{22}^{(1)}\right)
\nn\\
&&
+\left(l_{11}^{(1)}+4l_{21}^{(1)}\right)^2+ \dl_{0L}^{(1)}\left(\gamma^{(1)}_\psi+\gamma^{(1)}_\phi\right)\Bigg]
{\cal L}^2(s) \nn\\
&&\label{2lsigla}
\end{eqnarray}
where  $l_{ij}^{(1)}\equiv {\rm
Re}[\bfm{\chi}^{(1)}_L]_{ij}$. From the result of Ref.~\cite{Bee}  for $M_H=M$
we obtain
\begin{eqnarray}
\dl_{0L}^{(1)}&=&-{5 \over 2\xp}\ln^2(\xm) +{1 \over 2\xm}\ln^2(\xp)
-{7\pi^2 \over 3}+{32\pi \over 3 \sqrt{3}}-{25 \over 36}-{10\over
9}n_f\,. \label{del0l}
\end{eqnarray}
For $n_f=12$ this gives
\begin{eqnarray}
\dl_L^{(1)}&=&-3{\cal L}^2(s)
+\left[-10\ln(\xm)+2\ln(\xp)+{21\over 2}\right]{\cal L}(s)^2
\nn\\
&&-{5 \over 2\xp}\ln^2(\xm)
+{1 \over 2\xm}\ln^2(\xp)
-{7\pi^2 \over 3}+{32\pi \over 3 \sqrt{3}}-{505 \over 36}\,,
\label{1lsigl} \\
\dl_L^{(2)}&=&{9 \over 2} {\cal L}^2(s)
+\left[30\ln(\xm)-6\ln(\xp)-{85\over 3}\right]{\cal L}^3(s)
\nn\\
&&+\left[\left(38+{15 \over 2\xp}\right)\ln^2(\xm)
+\left(2-{3 \over 2\xm}\right)\ln^2(\xp)
-8\ln(\xm)\ln(\xp)
\right.
\nn\\
&&\left.-{535 \over 6}\ln(\xm)+{107 \over 6}\ln(\xp)
+9\pi^2 -{32\pi \over \sqrt{3}}+{229 \over 4}\right]{\cal L}^2(s)\,.
\label{2lsigl}
\end{eqnarray}


\section{$W$-pair production in $e^+e^-$ annihilation}
\label{sec3}
In the standard electroweak model the perturbative expansion involves
the $SU_L(2)$ coupling constant $\al_{ew}$ and the $U(1)$ hypercharge
coupling constant $\al_Y$.  We eliminate the latter by means of the
relation $\al_Y=\tan^2\theta_W\,\al_{ew}$, where $\theta_W$ is the weak mixing
angle, and consider the one-parameter series for the cross section in
$\al_{ew}$ of the form of Eq.~(\ref{sersig}).  In the high energy limit the
transverse gauge bosons are produced only in annihilation of the
left-handed electron-positron pair.  The corresponding Born cross
section is given by Eq.~(\ref{bsigt}) with $\al$ replaced by
$\al_{ew}$.  The longitudinal gauge bosons
can be produced in the annihilation of the electron-positron pair of
both chiralities.  In the case of the left-handed initial state fermions
the Born cross section gets the contribution from the $SU_L(2)$ and the
hypercharge virtual gauge bosons and reads
\begin{equation}
 {{\rm d} \sigma^B_{-L} \over
  {\rm d} \Omega}={1\over \cos^4\theta_W}{\alpha_{ew}^2(s)\over 4s}{\xp\xm\over 4}
\,.
\label{bsiglm}
\end{equation}
For the right-handed initial state fermions the Born cross section
is saturated by the hypercharge gauge boson
\begin{equation}
{{\rm d} \sigma^B_{+L}\over {\rm d} \Omega}=4\sin^4\theta_W\,{{\rm d}
\sigma^B_{-L}\over{\rm d} \Omega}\,.
\label{bsiglp}
\end{equation}
The analysis of the radiative corrections  in the Standard Model
is complicated by the presence of the mass gap and mixing in the gauge sector
and by the large top quark Yukawa coupling.  In next section we use the
method of Ref.~\cite{Fad,JKPS} to separate the electroweak and QED
Sudakov logarithms. In Sect.~\ref{sec32} we extend the evolution
equation approach to the Yukawa enhanced contributions.  In
Sect.~\ref{sec33} we present the final numerical results for the
two-loop NNLL corrections to the differential cross sections.


\subsection{Separating QED Sudakov logarithms}
\label{sec31} The electroweak Standard Model with the spontaneously
broken $SU_L(2)\times U(1)$ gauge group involves both the massive
$W$ and $Z$ bosons and the massless photon. The corrections to the
fully exclusive cross sections due to the virtual photon exchange
are infrared divergent and should be combined with soft real
photon emission to obtain infrared finite physical observables. We
regularize the infrared divergences by giving the photon a small
mass $\lm$. Thus besides the electroweak Sudakov logarithms discussed
above the radiative corrections contain the QED Sudakov logarithms
of the form $\ln(Q^2/\lm^2)$. To disentangle the electroweak and QED
logarithms we use the approach of Ref.~\cite{Fad,KMPS,JKPS}. While
the dependence of the amplitudes on the large momentum transfer is
governed by the {\it hard} evolution equations ({\it  c.f.}
Eqs.~(\ref{evoleqz},~\ref{evoleqa})), their dependence on the photon
mass is governed by the {\it infrared} evolution equations
\cite{Fad}.  In the limit $\lm^2\ll M_W^2,~m_t^2\ll Q^2$ the infrared
evolution equations in the full theory are the same as in QED and the
solution to the NNLL accuracy in the massless fermion approximation
$m_f=0~(f\ne t)$ is given by the factor
\begin{eqnarray}
{\cal U}&=&U_0(\al_e)\exp{\left\{ -{\alpha_e(\lm^2)\over
4\pi}\left[\left(2 -\left( {290\over 27}+{40\over
9}\ln\left({x_+\over x_-}\right)\right){\alpha_e\over \pi}
\right)\ln^2\left({Q^2\over \lm^2}\right)\right.\right.}
\nn\\
& &-\left(3+ 4\ln\left({x_+\over x_-}\right)\right)
\ln\left({Q^2\over\lm^2}\right) +{40\over 27}{\alpha_e\over \pi}
\ln^3\left({Q^2\over\lm^2}\right)-
\left(\ln\left({M_W^2\over\lm^2}\right)-1\right)^2\Bigg]
\nn\\
&&+{\cal O}(\al_e^3)\Bigg\},
\label{QED}
\end{eqnarray}
where $\al_e$ is the $\overline{\rm MS}$ QED coupling constant. The NNLL approximation
for ${\cal U}$ can be obtained from the result for the
fermion-antifermion production  \cite{KMPS} by proper modification
of the QED anomalous dimensions. It is convenient to normalize
the QED factor so that ${\cal U}(\al_e)\big|_{s=\lm^2=M_W^2}=1$.
In order to cancel  the singular
dependence on the photon mass, the QED Sudakov
exponent~(\ref{QED}) should be combined with the the real photon
emission, which is also of pure QED nature if the  energy of emitted
photons  is much smaller than $M_{W}$.

Two sets of equations completely determine the dependence of the
amplitudes on two dimensionless variables $Q/M_W$ and $Q/\lm$ up to
the initial conditions which are fixed through the matching
procedure. To get the purely weak logarithms one subtracts the QED
exponent~(\ref{QED}) from the
exponent given by the solution of the hard evolution equation. This
can naturally be formulated in terms of the functions parameterizing
the solution. The functions  $\gm$, $\zeta$, and
$\bfm{\chi}$ are  mass-independent. Therefore the anomalous dimensions
parameterizing the purely  weak logarithms can be obtained by
subtracting the QED contribution  from the result of the unbroken symmetry
phase calculation to all orders in the coupling constants. In the order of
interest they can be found in or easily derived from the result of
Ref.~\cite{KMPS}. For example, we have
\begin{eqnarray}
%\bfm{\chi}^{(1)}&=&\bfm{\chi}^{(1)}|_{SU(2)}+4\ln\left({x_+\over
%x_-}\right)\bfm{1}\,,
%\nn\\
\gm_A^{(2)}&=&\gm_A^{(2)}|_{SU(2)}-{800\over 27}\sin^2\theta_W\,,
\\
\gm_\phi^{(2)}&=&\gm_\phi^{(2)}|_{SU(2)}+{52\over
9} \tan^2\theta_W-{800\over
27}\sin^2\theta_W\,,\label{g2sub}
\end{eqnarray}
and so on. Here the  $SU(2)$ contributions are given by the results
of Sect.~\ref{sec2} with $n_f=12$.
The only new ingredient in comparison with the light fermion pair production
\cite{KMPS} is the effect of the  large Yukawa coupling of the
third generation quarks on the  longitudinal gauge boson production
which is considered in the next section.

On the other hand the  functions
$\xi$ and ${\cal A}_0$ are infrared sensitive and require the use of
the  true mass eigenstates of the Standard Model in the perturbative
calculation. In the NNLL approximation one needs the one-loop
contribution to these quantities which can be extracted from the
result of Ref.~\cite{Bee}. For example, for the left-handed initial state fermions we find
\begin{eqnarray}
\xi_\psi^{(1)}+\xi_A^{(1)}&=&{1-4\cos^2\theta_W+8\cos^4\theta_W\over
2\cos^2\theta_W}\ln\left({M_Z^2\over M_W^2}\right)\,,
\nn\\
\xi_\psi^{(1)}+\xi_\phi^{(1)}&=&{(1-2\cos^2\theta_W)^2\over
\cos^2\theta_W}\ln\left({M_Z^2\over M_W^2}\right)\,. \label{xi1ew}
\end{eqnarray}
At the same time the ${\cal A}_0^{(1)}$ term results in the one-loop
nonlogarithmic contribution to the cross section. The corresponding
expression directly follows from Ref.~\cite{Bee} and is rather
lengthy so we do not give it explicitly. Note that in
Ref.~\cite{Bee} the result is presented in the on-shell
renormalization scheme and in the limit $\lm\ll m_e$. We convert
it to $\overline{\rm MS}$ scheme and to the massless
electron approximation using the formulae of Refs.~\cite{FKS,Pen}.


\subsection{Top quark Yukawa coupling effects}
\label{sec32}
The large Yukawa coupling of the third generation quarks
to the scalar (Higgs and Goldstone) bosons results in  specific
logarithmic corrections proportional to  $m_t^2/M_W^2$.  These
corrections are determined by the ultraviolet quark and scalar boson
field renormalization and can be taken into account through the
modification of the evolution equations for the corresponding $\cal
Z$-functions. The  analysis is straightforward but
complicated because the Yukawa interaction mixes  quark and
scalar boson evolution and in general does not commute with the
$SU(2)$ and hypercharge couplings. However, due to the factorization
property of the collinear logarithms the Yukawa enhanced
contribution to NLL approximation is given simply by the product of
the one-loop Yukawa corrections and the double logarithmic exponent
as observed in Ref.~\cite{Mel2}. The structure of the
NNLL contribution is much more complicated and we restrict the
analysis to a simplified model with $\sin\theta_W=0$, {\it i.e.}
with no hypercharge interaction.  Let us introduce the following
five-component vector in the space of $\cal Z$-functions $\bfm{\cal
Z}=({\cal Z}_\phi,~{\cal Z}_\chi,~{\cal Z}_{b-},~{\cal
Z}_{t-},~{\cal Z}_{t+})$, where the subscript $+$ $(-)$ stand for
the right (left) quark  fields and ${\cal Z}_\chi$ corresponds to
the transition of the Higgs boson into the neutral Goldstone boson in the external
singlet vector field. The evolution equation for this vector takes
the form
\begin{figure}[t]
\begin{center}
\hspace*{10mm}
\begin{picture}(100,60)(0,0)
\SetScale{0.8} \SetWidth{1} \ArrowLine(0,30)(50,60)
\ArrowLine(50,0)(0,30) \ArrowLine(50,60)(50,0)
\DashLine(50,60)(100,60){5} \DashLine(50,00)(100,00){5}
\CBoxc(0,30)(6,6){Black}{Black}
\end{picture}
\hspace*{10mm}
\begin{picture}(100,60)(0,0)
\SetScale{0.8} \SetWidth{1} \ArrowLine(0,30)(50,60)
\ArrowLine(50,0)(0,30) \DashLine(50,60)(50,0){5}
\ArrowLine(50,60)(100,60) \ArrowLine(100,00)(50,00)
\CBoxc(0,30)(6,6){Black}{Black}
%\Text(45,-15)[cc]{$(a)$}
\end{picture}
\hspace*{10mm}
\begin{picture}(100,60)(0,0)
\SetScale{0.8} \SetWidth{1} \DashLine(0,30)(50,60){5}
\DashLine(50,0)(0,30){5} \ArrowLine(50,60)(50,0)
\ArrowLine(50,60)(100,60) \ArrowLine(50,00)(100,00)
\CBoxc(0,30)(6,6){Black}{Black}
\end{picture}
\end{center}
\caption{\label{fig2} \small  The one-loop diagrams contributing to
the anomalous dimension matrix  $\bfm{\zeta}$. The arrow lines
correspond to the third generation quarks. The dashed lines
correspond to the Higgs, neutral or charged Goldstone bosons. The
black square represent an external singlet vector field}
\end{figure}
\begin{equation}
{\partial\over\partial\ln{Q^2}}\bfm{\cal Z}=
\Bigg[\int_{M_W^2}^{Q^2}{\dd x\over
x}\bfm{\gm}(\al_{ew}(x))+\bfm{\zeta}(\al_{ew}(Q^2),\alpha_{Y\!uk}(Q^2))
+\bfm{\xi}(\al_{ew}(M_W^2)) \Bigg] \bfm{\cal Z} \,,
\label{evoleqzvec}
\end{equation}
with the solution
\begin{equation}
\bfm{\cal Z}={\rm P}\!\exp \left\{\int_{M_W^2}^{Q^2}{\dd x\over x}
\left[\int_{M_W^2}^{x}{\dd x'\over
x'}\bfm{\gm}(\al_{ew}(x'))+\bfm{\zeta}(\al_{ew}(x),\alpha_{Y\!uk}(x))
+\bfm{\xi}(\al_{ew}(M_W^2))\right]\right\}\bfm{\cal Z}_0 \,,
\label{evolsolzvec}
\end{equation}
where  $\bfm\gamma^{(1)}=(-3/2)\cdot\bfm{1}$ and $\bfm{\xi}=0$. The
anomalous dimension matrix $\bfm{\zeta}$ includes all the dependence
on the Yukawa coupling $\alpha_{Y\!uk}$. We eliminate the latter by
means of the relation $\alpha_{Y\!uk}={m_t^2\over 2M_W^2}\al_{ew}$,
and consider  the one-parameter series for the anomalous dimension
in $\al_{ew}$. The one-loop coefficient reads
\begin{equation}
\zeta^{(1)}={1\over 4}\pmatrix{ 12 & 0 & 0 & 0 & 0\cr
           0 & 12 & 0 & 0 &  0 \cr
           0& 0 & {9} & 0 & 0\cr
           0 & 0& 0 & {9} &  0 \cr
           0& 0 & 0&0 &{9} \cr
           }\,+{m_t^2\over 4M_W^2}\pmatrix{ 0 & 0 & {6} & 0 & -{6}\cr
           0 & 0 & 0 & {6} & -{6}\cr
           {1}& 0 & 0 & 0 & -{1}\cr
           0 & {1}& 0 & 0& -{1} \cr
           -{1} & -{1}  & -{1} &-{1}  &0 \cr
           }\,,
\label{1lzetayuk}
\end{equation}
where the first term representing the  pure $SU_L(2)$ contribution
follows from the result of Sect.~\ref{sec21} and the second term
represents the Yukawa contribution, Fig.~\ref{fig2}. It can be
extracted from the known one-loop result (see {\it e.g.}
Ref.~\cite{Mel2,Bec}). The proper initial condition which corresponds to
the Born amplitudes of the quark and scalar boson production in
$e^+e^-$ annihilation is given by the vector $\bfm{\cal
Z}_0=(1,-1,-1,1,0)$. In NNLL approximation one needs also the
one-loop running of the Yukawa coupling with the corresponding
beta-function $\beta_0^{Y\!uk}={9\over 4}-{3m_t^2\over 4M_W^2}$.
By expanding the solution for  the component ${\cal Z}_\phi$ we obtain the
two-loop corrections enhanced by the second or fourth power of the
top quark mass. Note that in the production amplitude one  has to
take into account also the interference between the one-loop Yukawa
contribution to ${\cal Z}_\phi$  and the one-loop logarithmic term
in the reduced amplitude and the electron ${\cal Z}_\psi$ function.
The two-loop NNLL Yukawa enhanced contribution to the cross section
reads
\begin{equation}
\delta^{(2)}_{NNLL}\bigg|_{Y\!uk}=\left[-{3\over 4}{m_t^4\over
M_W^4}+{m_t^2\over M_W^2}\left(30 \ln(\xm)-6\ln(\xp)-{45\over
4}\right) \right] {\cal L}^2(s)\,. \label{2ldelyuk}
\end{equation}
This expression  approximates the full result up to the terms
suppressed by  $\sin^2\theta_W\sim 0.2$.


\begin{figure}
\begin{center}
\begin{tabular}{cc}
\hspace*{-25pt}\epsfig{figure=wt1l1t.eps,height=4.5cm}&
\hspace*{25pt}\epsfig{figure=wl1l1t.eps,height=4.5cm}\\
\hspace*{-2pt}$(a)$ & \hspace*{47pt}$(b)$
\end{tabular}
\caption{\label{fig3} \small
  The one-loop logarithmic corrections to the differential cross section
  relative to the Born approximation at $\sqrt{s}=1$~TeV as functions of
  the production angle for $(a)$ transverse and $(b)$ longitudinal
  polarization of the gauge bosons.}
\end{center}
\end{figure}

\subsection{Numerical results}
\label{sec33} We adopt the following input values \cite{PDG} $M=M_W=80.41$~GeV,
$M_H=117$~GeV, $m_t=174$~GeV for the masses and
$\sin^2\theta_W=0.231$, $\al_{ew}=3.38\cdot 10^{-2}$ for the
$\overline{\rm MS}$ coupling constants renormalized at the scale of
the gauge boson mass. Note that the coupling constants
in the Born cross section of the longitudinal gauge boson production
are renormalized at the scale $\sqrt{s}$.
\vspace*{-3mm}
\paragraph{\bf\em  Transverse polarization.}
We obtain the following one and two-loop NNLL corrections to the
cross section
\begin{eqnarray}
\delta^{(1)}_{T}&=&-4.73\,{\cal L}^2(s) +\left[\left(-6.15 +
4.00\, {\xm \over \xp} \right)\ln(\xm) + 2.15\, \ln(\xp) + 4.43\right]
{\cal L}(s)
\nn\\
&&+\left(-{4.70 \over \xp}+{2.35+1.95\,\xm \over \xm^2+\xp^2} \right)
\ln^2(\xm)+{4.00 \over \xp}\ln(\xm)\ln(\xp)
+{3.00\,\xm \over \xm^2+\xp^2}\ln^2(\xp)
\nn\\
&&+\left(0.54+{4.95-9.65\,\xm\over \xm^2+\xp^2}\right)
\ln(\xm)-0.54\ln(\xp)-{2.35\,\xp\over \xm^2+\xp^2}-6.59\,,
\label{1lrest}
\\
\delta^{(2)}_{T}&=&11.17\,{\cal L}^4(s)
+\left[\left(29.08-{18.91\, \xm \over \xp}\right)\ln(\xm)
-10.17\,\ln(\xp)-14.49\right]{\cal L}^3(s) \nn\\&&
+\Bigg[\left(18.92+{22.21 -12.61\,\xm\over \xp}+{4.00\, \xm^2 \over
\xp^2} -{ 11.11+9.22\,\xm\over \xm^2+\xp^2}\right)\ln^2(\xm)
\nn\\
&&-\left(9.24+{18.91 \over \xp}
+{3.39\,\xm \over\xp}\right)\ln(\xm)\ln(\xp)
+\left(2.32-{14.18\,\xm \over \xm^2+\xp^2}\right)\ln^2(\xp)
\nn\\
&&-\left(15.26-{11.40\,\xm \over \xp}+ {23.40-45.61\,\xm \over
\xm^2+\xp^2}\right)\ln(\xm) +3.87\,\ln(\xp)+{11.11\,\xp\over \xm^2+\xp^2}
\nn\\
&&
+24.92\Bigg]{\cal
L}^2(s)\,.\label{2lrest}
\end{eqnarray}
The one-loop result is well known \cite{Bee} and
is given here to show the structure of the logarithmic expansion,
see Figs.~\ref{fig3}$a$ and \ref{fig4}$a$.
In Figs.~\ref{fig5}$a$ and \ref{fig6}$a$ the
values of different logarithmic two-loop contributions  as well as their
sum are plotted as functions of the production angle
at the center of mass energy of $1$~TeV and $3$~TeV, respectively.
The two-loop subleading contributions  exceed the LL one in absolute value in
the small angle region. However, due to the partial cancellation between
the NLL and NNLL terms the total NNLL approximation is  close to
the LL one. It has a fairly flat angular dependence
and amount to about 5\% for $\sqrt{s}= 1$~TeV
and 20\% for $\sqrt{s}= 3$~TeV.
\vspace*{-3mm}

\begin{figure}
\begin{center}
\begin{tabular}{cc}
\hspace*{-25pt}\epsfig{figure=wt1l3t.eps,height=4.5cm}&
\hspace*{25pt}\epsfig{figure=wl1l3t.eps,height=4.5cm}\\
\hspace*{-2pt}$(a)$ & \hspace*{47pt}$(b)$
\end{tabular}
\caption{\label{fig4} \small
  The same as Fig.~\ref{fig3} but for $\sqrt{s}=3$~TeV.}
\end{center}
\end{figure}

\paragraph{\bf\em  Longitudinal polarization.}
For the left-handed initial state fermions the one and two-loop NNLL
corrections to the cross section read
\begin{eqnarray}
\delta^{(1)}_{L}&=&-2.38\, {\cal L}^2(s)+
\left[-6.91\,\ln(\xm)+0.75\,\ln(\xp)-3.48\right]{\cal L}(s)
\nn\\
&&-{2.19 \over \xp} \ln^2(\xm)+{0.65 \over \xm} \ln^2(\xp)
+0.19\,\left(\ln(\xm)-\ln(\xp)\right)+31.07\,,
\label{1lresl}
\\
\delta^{(2)}_{L}&=&2.82\,{\cal L}^4(s)
+\left[16.41\,\ln(\xm)-1.79\,\ln(\xp)+11.87\right]{\cal L}^3(s)
\nn\\
&&+\Bigg[\left(18.38+{5.20 \over \xp}\right) \ln^2(\xm)+
\left(0.28-{1.55 \over \xm}\right)\ln^2(\xp)
\nn\\
&&+3.11\,\ln(\xm)\ln(\xp)+49.(10.)\,\ln(\xm)-18.(4.)\,\ln(\xp)-95.(20.)\Bigg]
{\cal  L}^2(s)\,. \label{2lresl}
\end{eqnarray}
In the
two-loop NNLL contribution the error bars indicate the uncertainty
due to our approximation of the Yukawa enhanced contribution. The
structure of the logarithmic corrections differs from  the case
of the transverse polarization as one can see on
Figs.~\ref{fig3}$b$,  \ref{fig4}$b$ and \ref{fig5}$b$.
The sum of the  two-loop logarithmic terms is strongly angular dependent and
varies between -2\% and 2\% for $\sqrt{s}= 1$~TeV and
between -2\% and 8\% for $\sqrt{s}= 3$~TeV.
\begin{figure}
\begin{center}
\begin{tabular}{cc}
\hspace*{-25pt}\epsfig{figure=wt2l1t.eps,height=4.5cm}&
\hspace*{25pt}\epsfig{figure=wl2l1t.eps,height=4.5cm}\\
\hspace*{-2pt}$(a)$ & \hspace*{47pt}$(b)$
\end{tabular}
\caption{\label{fig5} \small
  The two-loop logarithmic corrections to the differential cross section
  relative to the Born approximation at $\sqrt{s}=1$~TeV as functions of
  the production angle for $(a)$ transverse and $(b)$ longitudinal
  polarization of the gauge bosons.}
\end{center}
\end{figure}


For the right-handed initial state fermions the Born cross section is
suppressed by the factor $4\sin^4\theta_W\approx 0.2$ in comparison to
the left-handed case. Moreover the two-loop logarithmic corrections
turned out to be about $3\cdot 10^{-3}$ for all the scattering angles.
Thus they are of no phenomenological importance and are not presented
here.




\section{Summary}
\label{sum}
In the present paper we employed the evolution equation approach to
analyze the high energy asymptotic behavior of the gauge boson
production through the annihilation of fermion-antifermion pair in the
spontaneously broken $SU(2)$ gauge model.  The result has been used to
compute the two-loop NNLL electroweak corrections to the differential
cross section of $W$-pair production in $e^+e^-$ annihilation.  The
corrections are comparable and even exceed the LL terms depending on the
production angle. The structure of the corrections is different for the
transverse and longitudinal boson production.  In the first case we
observe the cancellation between the huge NLL and NNLL contributions so
that the sum is dominated by the LL term and amounts of about 5\% at
$\sqrt{s}\sim 1$~TeV and 20\% at $\sqrt{s}\sim 3$~TeV.  For the longitudinal bosons the corrections
exhibit significant cancellation between the LL, NLL and NNLL terms so
that the sum does not exceed 2\% in absolute value for $\sqrt{s}\sim
1$~TeV. The cancellation becomes less pronounced at
higher energy. The uncertainty of the
theoretical prediction for the on-shell $W$-pair production at ILC is now
determined by the unknown two-loop linear logarithmic terms. For the
fermion pair production such terms are know to contribute about 1--2\%
of the cross section \cite{JKPS}.  This value can be used as an estimate
of the accuracy of our approximation.  We should emphasize that our
approximation breaks down at small production angles where the {\it
  Regge} logarithms $\ln(-t/s)$ become large.

\begin{figure}
\begin{center}
\begin{tabular}{cc}
\hspace*{-25pt}\epsfig{figure=wt2l3t.eps,height=4.5cm}&
\hspace*{25pt}\epsfig{figure=wl2l3t.eps,height=4.5cm}\\
\hspace*{-2pt}$(a)$ & \hspace*{47pt}$(b)$
\end{tabular}
\caption{\label{fig6} \small
  The same as Fig.~\ref{fig5} but for $\sqrt{s}=3$~TeV.}
\end{center}
\end{figure}



\vspace{5mm}
\noindent
{\bf Acknowledgments}\\[3mm]
We thank S. Dittmaier for usefull discussion.
This work is supported by the DFG Sonder\-forschungsbereich/Transregio~9
``Computergest\"utzte Theoretische Teilchenphysik'' SFB/TR 9
and by BMBF grant 05HT4VKA/3. The work of F.M. has been supported by
the Graduiertenkolleg  "Hochenergiephysik und Teilchenastrophysik".

\begin{thebibliography}{99}

%\cite{unknown:2004qh}
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\end{document}
