% Please use the skeleton file you have received in the 
% invitation-to-submit email, where your data are already
% filled in. Otherwise please make sure you insert your 
% data according to the instructions in PoSauthmanual.pdf
\documentclass{PoS}

\usepackage{colordvi}
\usepackage{epsfig}
\usepackage{axodraw}
\usepackage{epsfig}
\usepackage{graphicx}
\usepackage{rotate}
\usepackage{latexsym}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{multirow}
\usepackage{dsfont}

\title{NNLL Electroweak Corrections to Gauge Boson Pair Production at LHC}

\ShortTitle{
NNLL Electroweak Corrections to Gauge Boson Pair Production at LHC}

\author{\speaker{Sandro Uccirati}
\thanks{In collaboration with J.H.K\"uhn, F.Metzler, A.A.Penin}\\
        Institut f\"ur Theoretische Teilchenphysik,
           Universit\"at Karlsruhe,\\ 
           76128 Karlsruhe, Germany\\
        E-mail: \email{uccirati@particle.uni-karlsruhe.de}}

%\author{Another Author\\
%        Affiliation\\
%        E-mail: \email{...}}

\abstract{
The high energy behaviour of electroweak corrections to gauge boson pair 
production at LHC can be derived in the framework of evolution 
equations up to NNLL contributions to all order in perturbation 
theory.
On the basis of this general result we derive the one- and two-loop 
cross section in NNLL approximation for W-pair production at partonic 
and hadronic level.
}

\FullConference{
European Physical Society Europhysics Conference on High Energy Physics,
EPS-HEP 2009,\\
		 July 16 - 22 2009\\
		 Krakow, Poland}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  personal abbreviations and macros
%    the following package contains macros used in this document:
\input macro.tex
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

\section{Introduction}
%--
The most important goal of the LHC is the search for the Higgs 
Boson and one of its discovery channels is the reaction 
$p\,p\to H \to W^+\,W^-$. 
The production of a pair of $W$ bosons constitutes therefore an 
important background for Higgs production and decay at LHC. 
In addition, the comparison of precise measurements of the production 
of gauge boson pairs with theoretical predictions is a good test of the 
gauge trilinear couplings predicted by the Standard Model. 
The LHC will exploit the high energy region where the W bosons can be 
considered as light and the main contribution to the cross section 
comes from large {\em Sudakow} logarithms \cite{Sudakov:1954sw}. 
%These corrections are the topic of our study.
%--
\section{Evolution equations for $SU(N)$ and 
            for the electroweak SM}
%--
\vspace{-0.1cm}
In a $SU(N)$ gauge theory in the high energy limit, where all the 
kinematical invariants are of the same order and far 
larger than the gauge boson mass, $|s|\sim |t| \sim |u| \gg M^2$, 
the asymptotic energy dependence of the amplitude is dominated by 
{\em Sudakov} logarithms $\log(Q^2/M^2)$ ($Q^2=-s$) and easily studied 
within the evolution equations framework.

The logarithms of collinear and soft-collinear origin depend only on 
the properties of the external particles and are known to factorize. 
They do not depend on the specific process and can be determined studying
the asymptotic behaviour of the scattering form factors depicted in 
\fig{Z}.
%--
%\vspace{-0.2cm}
\begin{figure}[h]
$$
F_\psi
\;
=
\;
\raisebox{0.1cm}{\scalebox{0.8}{
\begin{picture}(80,30)(0,0)
 \ArrowLine(0,20)(30,10)               
 \ArrowLine(30,-10)(0,-20)             
 \GCirc(40,0){15}{0.8}            
 \Photon(55,0)(80,0){2}{6}
 \LongArrow(78,7)(68,7)        \Text(85,4)[cb]{$p$}
\end{picture}
}}
\qquad\qquad
F_\phi
\;
=
\;
\raisebox{0.1cm}{\scalebox{0.8}{
\begin{picture}(80,30)(0,0)
 \DashLine(0,20)(30,10){3}               
 \DashLine(0,-20)(30,-10){3}             
 \GCirc(40,0){15}{0.8}            
 \Photon(55,0)(80,0){2}{6}        
 \LongArrow(78,7)(68,7)        \Text(85,4)[cb]{$p$}
\end{picture}
}}
\qquad\qquad
F_A
\;
=
\;
\raisebox{0.1cm}{\scalebox{0.8}{
\begin{picture}(80,30)(0,0)
 \Photon(0,20)(30,10){2}{6}               
 \Photon(0,-20)(30,-10){2}{6}             
 \GCirc(40,0){15}{0.8}            
 \Line(55,0)(80,0)        
 \LongArrow(78,7)(68,7)        \Text(85,4)[cb]{$p$}
\end{picture}
}}
$$
\\[-0.8cm]
\caption{Fermion/scalar 
scattering in an external singlet vector 
field and scattering of a gauge boson in an external scalar field.
The momentum of external field satisfies $p^2=s=-Q^2$.
}
\label{Z}
\end{figure}
%--

\noindent
At leading order in $M^2/Q^2$ they 
are known to satisfy the following evolution equation 
\cite{Mueller:1979ih}
%--
\bqa
F_i &=& Z_i\,F_i^{\rm Born},
\qquad
{\partial Z_i \over\partial\ln{Q^2}} =
\bigg[
 \int_{M^2}^{Q^2}{dx\over x}\gm_i(g^2(x))
+ \zeta_i(g^2(Q^2))
+ \xi_i(g^2(M^2)) 
\bigg]\,Z_i,
\qquad
i= \psi,\phi,A
\nl
Z_i &=& 
Z_{i,0}(g^2(M^2))\,\exp \bigg\{
\int_{M^2}^{Q^2}{d x\over x}\bigg[
  \int_{M^2}^{x}{d x'\over x'}\gm_i(g^2(x'))
+ \zeta_i(g^2(x))
+ \xi_i(g^2(M^2))
\bigg]
\bigg\}
\,.\
\label{evoleqz}
\eqa
%--
%The indices $i= \psi,\phi,A$ refer to fermions, scalars and gauge 
%bosons respectively.
%The perturbative functions 
%$f= Z_i, Z_{i,0},\gm_i, \zeta_i, \xi_i$ are given by 
%the series in the coupling constant $g^2(M^2)$, 
%$f(g^2)=\sum_{m=0}^\infty\,g^{2m}\,f^{(m)}$, 
%whose coefficients $f^{(m)}$ can be determined by 
%comparing the asymptotic fixed order result with the expansion 
%in the coupling constant of \eqn{evoleqz}, which at NNLL reads:
%%--
%\bqa
%Z^{(0)}_i\!
%&=&
%1,
%\qquad\qquad
%Z^{(1)}_i\!
%=
%  {1 \over 2}\,\gamma_i^{(1)}\,L^2 
%+ \big( \zeta_i^{(1)}+\xi_i^{(1)} \big)\,L
%+ Z_{i,0}^{(1)},
%\qquad\qquad\qquad
%L= \log\frac{Q^2}{M^2}
%\\
%Z_i^{(2)}\!
%&=&
%  {1 \over 8} \big[ \gamma_i^{(1)} \big]^2 L^4
%+ \frac{\gamma_i^{\!(1)}}{2}
%  \Big[ \zeta_i^{(1)} + \xi_i^{(1)}\! - \frac{\beta_0}{3} \Big] L^3
%+ {1 \over 2} \Big[ 
%    \gamma_i^{\!(2)} + \big( \zeta_i^{(1)}\! + \xi_i^{(1)} \big)^2 
%  - \beta_0\,\zeta_i^{(1)} + Z_{i,0}^{(1)}\gamma^{(1)}
%  \Big] L^2,
%%+ {\cal O}(L)
%\!\!\!
%\nn
%\label{expsolz}
%\eqa
%%--
%where we have used $g^2(x)=g^2(M^2)\,[1-g^2(M^2)\,\beta_0\,\log(x/M^2)]$.
%The one-loop coefficients $\gamma_i^{(1)}$, $\zeta_i^{(1)}$,$\xi_i^{(1)}$ 
%and $Z_{i,0}^{(1)}$ are enough to determine the leading and 
%next-to-leading logarithms at two-loop level, while to compute the 
%two-loop NNLL one needs in addition $\gamma_i^{(2)}$, 
%which is also known (\cite{Kodaira:1981nh}).

The remaining soft logarithms (not contained in $Z_i$) depend on 
the process under consideration.
We concentrate now on $2\to 2$ processes, where 
the incoming particles transform under the fundamental representation 
of the group (as happens for processes with incoming fermions).
If $T_a$ are the generators of the algebra, for final states in the 
fundamental and adjoint representation the structure of the 
amplitude is 
$
A_F= 
  A_1\,\bfm{T_a}\times \bfm{T_a} 
+ A_2\,\bfm{\mathds{1}}\times \bfm{\mathds{1}}
$
and
$
A_A= 
  A_1\,\bfm{T_a}\bfm{T_b} + A_2\,\bfm{T_b}\bfm{T_a} 
+ A_3\,\delta_{ab}\bfm{\mathds{1}}
$
respectively.
The coefficients $A_j$ can be seen as the components of a 
vector $\vec{A}$.
Factorizing out of $\vec{A}$ the $Z_i$ factors 
described above, the remaining reduced amplitude satisfies a 
renormalization group like equation \cite{Sen:1982bt} whose 
solution is given by a path-ordered exponent:
%--
\bq
\vec{A}= 
\prod_{n=1}^{4}Z_{i_n}^{1/2}\,\vec{A}'\!,
\quad\;\;\;
{\partial \vec{A}' \over \partial \ln{Q^2}}=
\bfm{\chi}(g^2(Q^2))\,\vec{A}'\!,
\quad\;\;\;
\vec{A}'\! =
{\rm P}\,\exp{\!
\bigg[\int_{M^2}^{Q^2}\!{\dd x\over x}\bfm{\chi}(g^2(x)) \bigg]
}
\vec{A}_0(g^2(M^2)),\;
\label{evoleqa}
\eq
%--
where $i_n$ takes the values $\psi$, $\phi$ or $A$ depending on 
whether the $n^{\rm th}$ external particle is a fermion, a scalar or 
a vector boson and $\bfm{\chi}$ is the soft anomalous dimension matrix.
%Expanding \eqn{evoleqa} in $g^2(\mu^2)$ and using the expansion for 
%$Z_i$ given in \eqn{expsolz}, we get:
%%--
%\bqa
%\vec{A}^{(1)}
%&=&
%  {1 \over 2}\,\gamma^{(1)}\vec{A}^{\rm Born}\,L^2 
%+ \big( \zeta^{(1)} + \xi^{(1)} + \bfm{\chi}^{(1)} \big)
%  \vec{A}^{\rm Born}\,L
%+ \vec{A}_{0}^{(1)},
%\nl
%\vec{A}^{(2)}
%&=&
%  {1 \over 8}\,\big[ \gamma^{(1)} \big]^2 
%  \vec{A}^{\rm Born}\,L^4
%+ \frac{\gamma^{\!(1)}}{2}
%  \Big[ 
%  \zeta^{(1)} + \xi^{(1)}\! + \bfm{\chi}^{(1)} - \frac{\beta_0}{3} 
%  \Big] 
%  \vec{A}^{\rm Born}\,L^3
%\nl
%&&
%+ \,{1 \over 2}\,\bigg\{
%    \Big[ 
%      \gamma^{\!(2)} + \big( \zeta^{(1)}\! + \xi^{(1)} \big)^2 
%    - \beta_0\,\zeta^{(1)} 
%    + (\bfm{\chi}^{(1)}\big)^2 - \beta_0 \bfm{\chi}^{(1)} 
%    \Big] \vec{A}^{\rm Born}
%  + \vec{A}_{0}^{(1)}
%  \bigg\}\,L^2
%+ {\cal O}(L)
%\quad
%\label{expsola}
%\eqa
%%--
%where the coefficients $f=\gm^{(m)},\zeta^{(m)},\xi^{(m)}$ 
%are given by $f=1/2\,\sum_{n=1}^4\,f_{i_n}$.
%The one-loop matrix $\bfm{\chi}^{(1)}$ and the vector 
%$\vec{A}_{0}^{(1)}$ can again be read comparing 
%$\vec{A}^{(1)}$ in the above equation with the direct one-loop 
%computation.
Expanding all perturbative functions of \eqn{evoleqz} and \eqn{evoleqa} 
in $g^2(\mu^2)$ and absorbing $Z_{i,0}$ in $\vec{A}_{0}$, we get:
%--
\bqa
\vec{A}^{(1)}
&=&
  {1 \over 2}\,\gamma^{(1)}\vec{A}^{\rm Born}\,L^2 
+ \big( \zeta^{(1)} + \xi^{(1)} + \bfm{\chi}^{(1)} \big)
  \vec{A}^{\rm Born}\,L
+ \vec{A}_{0}^{(1)},
\qquad\qquad\qquad
L= \log\frac{Q^2}{M^2}
\nl
\vec{A}^{(2)}
&=&
  {1 \over 8}\,\big[ \gamma^{(1)} \big]^2 
  \vec{A}^{\rm Born}\,L^4
+ \frac{\gamma^{\!(1)}}{2}
  \Big[ 
  \zeta^{(1)} + \xi^{(1)}\! + \bfm{\chi}^{(1)} - \frac{\beta_0}{3} 
  \Big] 
  \vec{A}^{\rm Born}\,L^3
\nl
&&
+ \,{1 \over 2}\,\bigg\{
    \Big[ 
      \gamma^{\!(2)} + \big( \zeta^{(1)}\! + \xi^{(1)} \big)^2 
    - \beta_0\,\zeta^{(1)} 
    + (\bfm{\chi}^{(1)}\big)^2 - \beta_0 \bfm{\chi}^{(1)} 
    \Big] \vec{A}^{\rm Born}
  + \vec{A}_{0}^{(1)}
  \bigg\}\,L^2
+ {\cal O}(L)
\quad
\label{expsola}.
\eqa
%--
where 
$\{\gm,\zeta,\xi\}=1/2\,\sum_{n=1}^4\,\{\gm_{i_n},\zeta_{i_n},\xi_{i_n}\}$.
For the running of the coupling constant we have used 
$g^2(x)=g^2(M^2)\,[1-g^2(M^2)\,\beta_0\,\log(x/M^2)]$.
The one-loop coefficients can be fixed by comparing the first expression
in \eqn{expsola} with the asymptotic one-loop result (see 
\cite{Kuhn:1999nn} for explicit values).
They are sufficient to determine throught the second equation in 
\eqn{expsola} the leading and next-to-leading logarithms at two-loop 
level, while to compute the two-loop NNLL one needs in addition 
$\gamma_i^{(2)}$, which is also known (\cite{Kodaira:1981nh}).

In the electroweak Standard Model, which is a $SU(2)_L\times U(1)_Y$ 
theory, the situation is complicated by the mixing of the two gauge 
groups as a consequence of the symmetry breaking and by the presence 
of two irreducible scales: the electroweak scale 
$M\sim M_W\sim M_Z\sim M_H$ and the infrared scale $\lambda$.
In order to proceed without breaking gauge invariance 
\cite{Fadin:1999bq}, one considers in a first step the case where the 
SM obeys the evolution equations of an unbroken $SU(2)_L\times U(1)_Y$ 
theory and all gauge bosons have the same mass ($\lambda=M$).
The amplitude takes the simple form $U_{_Y}(M)\,\vec{A}_{_L}(M)$
where $\vec{A}_{_L}$ is obtained from \eqn{expsola} 
for the group $SU(2)_L$ (with $g^2\to \alpha/(4\pi s_W^2)$) and 
$U_{_Y}$ is the hypercharge $U(1)$ factor.
All logarithms are here of the form $\ln(Q^2/M^2)$.
Then we move to the (physical) case where $\lambda \ll M$ and 
the infrared logarithms $\ln(Q^2/\lambda^2)$ are described by 
the $U(1)$ QED factor $U_{_{QED}}$ up to power suppressed terms 
in $\lambda/M$.
Requiring the matching of the two regions at $M=\lambda$ 
leads to the result:
%--
\bq
\vec{A}(M,\lambda)\, 
=\;
C(M)\;U_{_{QED}}\;U_{_{QED}}^{-1}\big|_{\lambda=M}\;U_{_Y}\;\vec{A}_{_L}
\label{matching}.
\eq
%--
The matching coefficient $C(M)$ describes the effect of the 
neglected terms in $\lambda/M$.
This strategy has been used at NNLL level in many similar studies
\cite{Kuhn:1999nn}.

The chiral structure of the SM can be taken into account enlarging 
the basis in which the vector $\vec{A}$ is defined, while the mass 
splitting between the bosons at the electroweak scale can be 
efficiently included in the parameter $\xi$ \cite{Kuhn:1999nn}.
The effect of the top Yukawa coupling are also considered
following the approach described in the last reference of 
\cite{Kuhn:1999nn}.
%--
\section{W-pair production at hadron colliders}
%--
\vspace{-0.1cm}
The partonic processes relevant for the W-pair production at hadron 
colliders are the gluon fusion and quark-antiquark annihilation. 
The contribution of the former to the total cross section is about 
$5\%$, therefore we concentrate our analysis on 
the process $q\,\bar{q}\to W^+\,W^-$ which is described at leading 
order by the first two diagrams depicted in \fig{fig:born}.
In the high energy limit only final states where both $W$'s have the 
same polarization are not mass suppressed.
In addition, the case where both $W$'s are longitundinal can be reduced 
by means of the Goldstone theorem to the production of a pair of 
charged Goldstone bosons, reducing the study of the process to the 
last two diagrams of \fig{fig:born}.
%--
\vspace{0.4cm}
\begin{figure}[h]
$$
\scalebox{0.8}{
\begin{picture}(60,20)(0,-3)
 \ArrowLine(30,20)(0,20)        \Text(-5,17)[cb]{$q$}
 \ArrowLine(0,-20)(30,-20)      \Text(-5,-23)[cb]{$\bar{q}$}
 \ArrowLine(30,-20)(30,20)     
 \Photon(60,20)(30,20){2}{6}    \Text(73,17)[cb]{$W^-$}
 \Photon(30,-20)(60,-20){2}{6}  \Text(73,-23)[cb]{$W^+$}
\end{picture}
}
\qquad\quad
\scalebox{0.8}{
\begin{picture}(60,20)(0,-3)
 \ArrowLine(15,0)(0,20)        \Text(-5,17)[cb]{$q$}
 \ArrowLine(0,-20)(15,0)       \Text(-5,-23)[cb]{$\bar{q}$}
 \Photon(15,0)(45,0){2}{7}     \Text(30,5)[cb]{$\gamma,Z$}
 \Photon(60,20)(45,0){2}{5}    \Text(73,17)[cb]{$W^-$}
 \Photon(45,0)(60,-20){2}{5}   \Text(73,-23)[cb]{$W^+$}
\end{picture}
}
\qquad\qquad
\scalebox{0.9}{
\begin{picture}(30,20)(0,-3)
\LongArrow(0,0)(30,0)
\Text(15,-15)[cb]{\small high energy}
\Text(15,-22)[cb]{\small limit}
\end{picture}
}
\qquad\quad
\scalebox{0.8}{
\begin{picture}(60,20)(0,-3)
 \ArrowLine(30,20)(0,20)        \Text(-5,17)[cb]{$q$}
 \ArrowLine(0,-20)(30,-20)      \Text(-5,-23)[cb]{$\bar{q}$}
 \ArrowLine(30,-20)(30,20)     
 \Photon(60,20)(30,20){2}{6}    \Text(73,17)[cb]{$W_{_T}^-$}
 \Photon(30,-20)(60,-20){2}{6}  \Text(73,-23)[cb]{$W_{_T}^+$}
\end{picture}
}
\qquad\quad
\scalebox{0.8}{
\begin{picture}(60,20)(0,-3)
 \ArrowLine(15,0)(0,20)        \Text(-5,17)[cb]{$q$}
 \ArrowLine(0,-20)(15,0)       \Text(-5,-23)[cb]{$\bar{q}$}
 \Photon(15,0)(45,0){2}{7}     \Text(30,5)[cb]{$\gamma,Z$}
 \DashLine(60,20)(45,0){3}     \Text(73,17)[cb]{$\phi^-$}
 \DashLine(45,0)(60,-20){3}    \Text(73,-23)[cb]{$\phi^+$}
\end{picture}
}
$$
\\[-0.6cm]
\caption{Tree level diagrams contributing to the partonic process}
\label{fig:born}
\end{figure}
%--
We have explicitly computed the one-loop corrections to 
these diagrams and fixed the one-loop coefficients of the evolution 
equations by which we have determined the two-loop leading 
logarithms (up to NNLL).
%in the $\overline{\rm MS}$ renormalization 
%scheme for the couplings and the Weinberg angle and using the standard 
%on-shell mass renormalization for the massive bosons. 
%Of course in the longitudinal case also the renormalization of the 
%Goldstone theorem has been considered.
In \fig{fig:delta} the one- and two-loop corrections for left-handed 
$u$ quarks in the initial state are plotted showing an important
contribution at NNLL which partially cancels the NLL.
The behaviour for the $d$ quarks is similar, while the right-handed 
contribution is negligible.
%--
\begin{figure}[h]
 \begin{center}
  \begin{tabular}{cc}
   \includegraphics[scale=0.55]{delta_ul_T_NLO.eps}   &
   \includegraphics[scale=0.55]{delta_ul_L_NLO.eps}   \\
   \includegraphics[scale=0.55]{delta_ul_T_NNLO.eps}   &
   \includegraphics[scale=0.55]{delta_ul_L_NNLO.eps}   \\
  \end{tabular}
  \caption[]
  {One- and two-loop corrections to the partonic cross section 
   for left-handed $u$ quarks with energy $\sqrt{\hat{s}}= 1$ TeV 
   ($\delta^{i}= d\sigma_{i}/d\sigma_{^{\rm LO}}$, $i=NLO,\;NNLO$). 
  }
  \label{fig:delta}
 \end{center}
\vspace{-0.5cm}
\end{figure}
%--
%\begin{figure}[ht]
% \begin{center}
%  \begin{tabular}{cc}
%   \includegraphics[scale=0.55]{delta_ul_T_NLO.eps}   &
%   \includegraphics[scale=0.55]{delta_ul_L_NLO.eps}   \\
%   \includegraphics[scale=0.55]{delta_dl_T_NLO.eps}    &
%   \includegraphics[scale=0.55]{delta_dl_L_NLO.eps}
%  \end{tabular}
%  \caption[]{}
%  \label{fig:deltaNLO}
% \end{center}
%%\end{figure}
%%\begin{figure}[ht]
% \begin{center}
%  \begin{tabular}{cc}
%   \includegraphics[scale=0.55]{delta_ul_T_NNLO.eps}   &
%   \includegraphics[scale=0.55]{delta_ul_L_NNLO.eps}   \\
%   \includegraphics[scale=0.55]{delta_dl_T_NNLO.eps}    &
%   \includegraphics[scale=0.55]{delta_dl_L_NNLO.eps}
%  \end{tabular}
%  \caption[]{}
%  \label{fig:deltaNNLO}
% \end{center}
%\end{figure}
%--

The hadronic distributions are then obtained folding the partonic 
cross section with the parton distribution functions.
For the $p_T$-distribution we have:
%--
\bq
{d\sigma\over dp_T}=
{1\over N_c^2}\,\sum_{ij}
\int_0^1\!\!dx_1\int_0^1\!\!dx_2\,
f_{h_1,i}(x_1,\mu_F^2)\,f_{h_2,j}(x_2,\mu_F^2)\,
\theta(x_1x_2-\tau_{\rm min})\,
{d\hat{\sigma}_{ij}\over dp_T}
\nn
\eq
%--
\vspace{-0.2cm}
\bq
{d\hat{\sigma}_{ij}\over dp_T}=
{4p_T\over \sqrt{\hat{s}-4M_W^2}\sqrt{\hat{s}-s\,\tau_{\rm min}}}\,
\Big[\,
  {d\hat{\sigma}_{ij}\over d\cos\theta} + (\hat{t}\leftrightarrow\hat{u})
\,\Big]
\qquad
\hat{s}= x_1\,x_2\,s
\qquad
\tau_{\rm min}={4(p_T^2+M_W^2)\over s}
\eq
%--
\vspace{0.1cm}
Restricting the analysis to high $p_T$ values, the partonic process 
is in the range of validity of the high energy limit.
The results, plotted in \fig{fig:pt}, show that the NNLO corrections 
at high energies are between $1\%$ and $25\%$ for transversal $W$'s and 
between $-1\%$ and $3\%$ in the longitudinal case.
%--

%--
\begin{figure}[h]
 \begin{center}
  \begin{tabular}{cc}
   \includegraphics[scale=0.55]{dsigdpt_T.eps}   &
   \includegraphics[scale=0.55]{dsigdpt_L.eps}   \\
   \includegraphics[scale=0.55]{deltapt_T.eps}    &
   \includegraphics[scale=0.55]{deltapt_L.eps}
  \end{tabular}
  \caption[]{
  Hadronic $p_T$ distributions at LHC for beem energy of $14$ TeV
  ($\delta^{i}= d\sigma_{i}/d\sigma_{^{\rm LO}}$, $i=NLO,\;NNLO$).
  }
  \label{fig:pt}
 \end{center}
\vspace{-0.5cm}
\end{figure}
%\begin{figure}[ht]
% \begin{center}
%  \begin{tabular}{cc}
%   \includegraphics[scale=0.55]{dsigdM_T.eps}   &
%   \includegraphics[scale=0.55]{dsigdM_L.eps}   \\
%   \includegraphics[scale=0.55]{deltaM_T.eps}    &
%   \includegraphics[scale=0.55]{deltaM_L.eps}
%  \end{tabular}
%  \caption[]{}
%  \label{fig:M}
% \end{center}
%\end{figure}

%\section{Conclusions}
%--
\begin{thebibliography}{99}

%\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 = ZETFA,30,87;%%

%\cite{Mueller:1979ih}
\bibitem{Mueller:1979ih}
  A.~H.~Mueller,
  %``On The Asymptotic Behavior Of The Sudakov Form-Factor,''
  Phys.\ Rev.\  D {\bf 20} (1979) 2037.
  %%CITATION = PHRVA,D20,2037;%%
\;
%\cite{Collins:1980ih}
%\bibitem{Collins:1980ih}
  J.~C.~Collins,
  %``Algorithm To Compute Corrections To The Sudakov Form-Factor,''
  Phys.\ Rev.\  D {\bf 22} (1980) 1478, 
  %%CITATION = PHRVA,D22,1478;%%
%\cite{Collins:1989bt}
%\bibitem{Collins:1989bt}
%  J.~C.~Collins,
  %``Sudakov form factors,''
  Adv.\ Ser.\ Direct.\ High Energy Phys.\  {\bf 5} (1989) 573
  %[arXiv:hep-ph/0312336].
  %%CITATION = 00319,5,573;%%
\;
%\cite{Sen:1981sd}
%\bibitem{Sen:1981sd}
  A.~Sen,
  %``Asymptotic Behavior Of The Sudakov Form-Factor In QCD,''
  Phys.\ Rev.\  D {\bf 24} (1981) 3281.
  %%CITATION = PHRVA,D24,3281;%%

%\cite{Sen:1982bt}
\bibitem{Sen:1982bt}
  A.~Sen,
  %``Asymptotic Behavior Of The Wide Angle On-Shell Quark Scattering 
  %Amplitudes In Nonabelian Gauge Theories,''
  Phys.\ Rev.\  D {\bf 28} (1983) 860.
  %%CITATION = PHRVA,D28,860;%%
\;
%\cite{Sterman:1986aj}
%\bibitem{Sterman:1986aj}
  G.~Sterman,
  %``Summation of Large Corrections to Short Distance Hadronic 
  %Cross-Sections,''
  Nucl.\ Phys.\  B {\bf 281} (1987) 310.
  %%CITATION = NUPHA,B281,310;%%
\;
%\cite{Botts:1989kf}
%\bibitem{Botts:1989kf}
  J.~Botts and G.~Sterman,
  %``Hard Elastic Scattering In QCD: Leading Behavior,''
  Nucl.\ Phys.\  B {\bf 325} (1989) 62.
  %%CITATION = NUPHA,B325,62;%%

%\cite{Kuhn:1999nn}
\bibitem{Kuhn:1999nn}
  J.~H.~Kuhn, A.~A.~Penin and V.~A.~Smirnov,
  %``Summing up subleading Sudakov logarithms,''
  Eur.\ Phys.\ J.\  C {\bf 17} (2000) 97
  %[arXiv:hep-ph/9912503].
  %%CITATION = EPHJA,C17,97;%%
%\cite{Kuhn:2001hz}
%\bibitem{Kuhn:2001hz}
  J.~H.~Kuhn, 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
  [Erratum-ibid.\  B {\bf 648} (2003) 455]
  %[arXiv:hep-ph/0106298].
  %%CITATION = NUPHA,B616,286;%%
\;
%\cite{Feucht:2004rp}
%\bibitem{Feucht:2004rp}
  B.~Feucht, J.~H.~Kuhn, A.~A.~Penin and V.~A.~Smirnov,
  %``Two-loop Sudakov form factor in a theory with mass gap,''
  Phys.\ Rev.\ Lett.\  {\bf 93} (2004) 101802
  %[arXiv:hep-ph/0404082].
  %%CITATION = PRLTA,93,101802;%%
\;
%\cite{Jantzen:2005az}
%\bibitem{Jantzen:2005az}
  B.~Jantzen, J.~H.~Kuhn, A.~A.~Penin and V.~A.~Smirnov,
  %``Two-loop electroweak logarithms in four-fermion processes at high
  %energy,''
  Nucl.\ Phys.\  B {\bf 731} (2005) 188
  [Erratum-ibid.\  B {\bf 752} (2006) 327]
  %[arXiv:hep-ph/0509157].
  %%CITATION = NUPHA,B731,188;%%
\;
%\cite{Kuhn:2007ca}
%\bibitem{Kuhn:2007ca}
  J.~H.~Kuhn, F.~Metzler and A.~A.~Penin,
  %``Next-to-Next-to-Leading Electroweak Logarithms in W-pair Production at
  %ILC,''
  Nucl.\ Phys.\  B {\bf 795} (2008) 277
  [Erratum-ibid.\  {\bf 818} (2009) 135]
  %[arXiv:0709.4055 [hep-ph]].
  %%CITATION = NUPHA,B795,277;%%

%\cite{Kodaira:1981nh}
\bibitem{Kodaira:1981nh}
  J.~Kodaira and L.~Trentadue,
  %``Summing Soft Emission In QCD,''
  Phys.\ Lett.\  B {\bf 112} (1982) 66.
  %%CITATION = PHLTA,B112,66;%%
\;
%\cite{Davies:1984hs}
%\bibitem{Davies:1984hs}
  C.~T.~H.~Davies and W.~J.~Stirling,
  %``Nonleading Corrections To The Drell-Yan Cross-Section 
  %At Small Transverse Momentum,''
  Nucl.\ Phys.\  B {\bf 244} (1984) 337.
  %%CITATION = NUPHA,B244,337;%%
\;
%\cite{Korchemsky:1987wg}
%\bibitem{Korchemsky:1987wg}
  G.~P.~Korchemsky and A.~V.~Radyushkin,
  %``Renormalization of the Wilson Loops Beyond the Leading Order,''
  Nucl.\ Phys.\  B {\bf 283} (1987) 342.
  %%CITATION = NUPHA,B283,342;%%


%\cite{Fadin:1999bq}
\bibitem{Fadin:1999bq}
  V.~S.~Fadin, L.~N.~Lipatov, A.~D.~Martin and M.~Melles,
  %``Resummation of double logarithms in electroweak high energy 
  %processes,''
  Phys.\ Rev.\  D {\bf 61} (2000) 094002
  %[arXiv:hep-ph/9910338].
  %%CITATION = PHRVA,D61,094002;%%
\;
%\cite{Melles:2000gw}
%\bibitem{Melles:2000gw}
  M.~Melles,
  %``Subleading Sudakov logarithms in electroweak high energy processes 
  %to  all orders,''
  Phys.\ Rev.\  D {\bf 63} (2001) 034003
  %[arXiv:hep-ph/0004056].
  %%CITATION = PHRVA,D63,034003;%%

\end{thebibliography}

\end{document}


