%Title: Hard Scattering and Electroweak Corrections at High Energies 
%Author: J.H. K&uuml;hn
%Published: <EM> ACTA PHYSICA POLONICA B </EM> <B> 39 </B> (2008) (1639-1654). <EM> Proceedings of the Cracow Epiphany Conference on LHC Physics</EM> Cracow, Poland, January 4-6, 2008. 
%Title: Hard Scattering and Electroweak Corrections at High Energies 
%Author: Johann H. K&uuml;hn, 
%hep-ph/xxx
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\begin{document}
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\title{
\begin{flushleft}
\normalsize TTP08-13 \\
\end{flushleft}
\vspace{1.0cm}
Hard Scattering and Electroweak Corrections at High Energies }
\author{
  {\large J.H.~K\"uhn}\\[3mm]
  {\small {\it Institut f{\"u}r Theoretische Teilchenphysik,
  Universit{\"a}t Karlsruhe}}\\
  {\small {\it 76128 Karlsruhe, Germany}}\\[2mm]
}
\date{}
\maketitle
\begin{abstract}
After a brief recollection of joint scientific work with Staszek Jadach, recent
results on electroweak radiative corrections for scattering processes in the TeV region
are presented. The status of the four-fermion scattering amplitudes is discussed, with
emphasis on logarithmically enhanced contributions in two-loop approximation.
Predictions for the production of $\gamma$, $Z$ and $W$ with large transverse momenta
together with a jet are presented. For $p_T$ above 1TeV the electroweak corrections 
may well reach several tens of percent. A similar situation is observed 
for top-antitop  quark producition at large invariant mass of the $t\bar{t}$ system.
\end{abstract}

\thispagestyle{empty}
\newpage

\section{Introduction}
\label{intr}

This year's Cracow Epiphany Conference serves a twofold purpose: Like every
year it brings together colleagues from Cracow and abroad to stimulate
discussions on recent developments in theoretical an experimental particle
physics. However, in addition, this year we are celebrating Staszek Jadach's 
sixtieth anniversary, which gives us the opportunity to look back in time 
into our common past and  to honour his scientific achievements.

My personal recollections go back to the Cracow-Munich meetings of the years
1976 and later, which were at that time initiated by Leo van Hove, 
Andre Bia\l as and Kacper Zaleski.
This opened for us fascinating new possibilties, at the
scientific level, for new projects and collaborations, but also to get
aquainted with the situation at the other side of the ``iron curtain'' and to
get to know new colleagues and form new friendships. My friendship and
collaboration with Staszek dates back to this time. Several years later we
were both involved in the preparations for LEP. That is where our
joint scientific papers start, the first ones documenting the MonteCarlo 
generator TIPTOP,
constructed to simulate top quark production in electron-positron
annihilation,  a reaction which 
at that time was of course still considered as one of the Standard Model
processes of LEP \cite{Jadach:1986gz,Jadach:1986xp,IgoKemenes:1987vr}.

Later,
during Staszek's stay at the Max-Planck Institute in the years 1986 and 1987 
our attention shifted more and more to precision physics that started to
become one of the many interesting options for LEP and SLC. In
particular, the possibility for measurements with polarized beams offered the
possibility to determine the weak mixing angle with unprecedented precision,
and complementary precision calculations were called for, to be able to
fully exploit this unique possibility \cite{Jadach:1987ws}.

Of course, instead of polarized beams, the study of tau polarization in the 
final state was another interesting possibility and indeed turned out to 
be a nearly  equally powerful option. The study of semileptonic decays
was not only particularly suited for the analysis of tau polarization, at the same
time a wealth of hadron physics was just waiting to be uncovered. To provide
an adequate tool to experimentalists, the Monte Carlo generator TAUOLA was
constructed (together with Zbyszek Wa\.s), simulation the decay of polarized tau
leptons into a multitude of final states \cite{Jadach:1990mz}. Both the
conceptual and the phenomenological aspects where quite exciting, and 
\cite{Jadach:1990mz}, together with the subsequent developments 
\cite{Jezabek:1991qp,Jadach:1993hs}
has turned out to be a gold mine ever since then, as far as physics results are
concerned and in terms of citations. Clearly, I have profited a lot from
Staszek's enormous expertise in MonteCarlo programs and am grateful for this
experience.

Time goes on, and our interests have shifted. Nevertheless, we are
all urgently waiting for the start of the LHC and have invested a lot of
effort into its preparation. During the rest of the talk I will therefore 
briefly describe some topics in electroweak physics which have been 
investigated with my colleagues in Karlsruhe, and which are specifically devoted 
to LHC experiments.

Historically, weak phenomena have manifested themselves at low energies
through effective four-fermion interactions, as expected for reactions with
energy transfer far smaller than the mass of the exchanged bosons. Although
much information about the structure of the charged and neutral currents was
collected in the course of time, the description through this effective 
four-fermion coupling remained valid and the existence of the 
$W$  and $Z$ bosons was
only demonstrated in the early eighties at the CERN proton-antiproton collider.

From then on a second period started, where the characteristic energies 
were comparable to the masses of the weak gauge bosons. Unification of
electromagnetic and weak interactions became manifest and electroweak
precision physics culminated in the LEP and SLC experiments. 

The picture is about to change again, and with the 
turn-on of LHC a third period will soon emerge 
. Compared to characteristic energies in the TeV region, 
the masses of $W$  and $Z$  can be considered as small, 
``infrared'' parameters and powers of $\cal{L}$,  
the large logarithm of the ratio $s/m_W^2$, 
may appear in the evaluation of exclusive and inclusive
cross sections. These large logarithms may 
at least partially compensate the smallness of the weak coupling constant 
and lead to (negative) enhanced corrections. 

In exclusive reactions the dominant ``leading''
logarithms are given by $(\alpha_W/\pi {\cal L}^2)^n$, where 
$\alpha_W/\pi\approx 0.01$ and ${\cal L}^2\approx 20 \;{\rm to}\; 40$ and $n$
denotes the loop order. In addition these corrections are multiplied by
the sum of the eigenvalues of the quadratic Casimir operator characterizing the
representations of the external particles.
All these leading terms are negative, and for the LHC and in one-loop 
approximation the effects may easily accumulate to ten to thirty
percent, which makes the need for a suffiently precise  control of the 
two-loop terms evident.  

These considerations have triggered a wave of interest in this 
so-called Sudakov asymptotic regime 
\cite{Sud,Jac,Kur,DegSir,Bec,CiaCom,KuhPen,Fad,KPS,KMPS,Bec2,BRV,Mel,DenPoz,
HKK,BeeWet,FKM,DMP,Bec3,Bec4,Poz,FKPS,JKPS3,JKPS}.
In Karlsruhe we have, on one hand, explored
four-fermion processes (recently also $W$-pair production) in great detail, 
demonstrating the importance of subleading logarithms
and evaluating these up to the N${}^3$LL-approximation. On the other hand we
have, in one-loop approximation (including only a limited part of subleading
terms), studied a number of processes of phenomenological importance, like $W$
or $Z$ plus jet production at large transverse momenta, and top quark pair
production. Results for all these reactions will be reviewed in the following.

\section{Form Factors and Four-Fermion Scattering at Two Loops}
\label{FormFac}

Let us start with the analysis of four-fermion scattering where the
investigation of the subleading terms has been performed at the
most detailled level.
In Refs.~\cite{KPS,KMPS} we have extended the leading logarithmic (LL)
analysis of \cite{Fad}. The next-to-leading logarithmic (NLL) and
next-to-next-to-leading logarithmic (NNLL) corrections to the high
energy asymptotic behavior of the neutral current four-fermion processes
have been resummed to all orders using the {\it evolution equation}
approach discussed below. Only the light quark case was considered and the mass
difference between the neutral and charged gauge bosons was neglected.
On the basis of this result the logarithmically enhanced part of the
phenomenologically important two-loop corrections to the total cross
section and to various asymmetries was obtained including the
$\ln^n(s/M_{W,Z}^2)$ terms with $n=2,~3,~4$.  The results up to NLL have
been confirmed by the explicit one-loop \cite{Bec,BRV,DenPoz} and
two-loop \cite{HKK,BeeWet,FKM,DMP,Poz} calculations.  The subleading
logarithms in the TeV region are comparable to the leading terms due to
their large numerical coefficients.  Thus, the calculation of the
remaining two-loop quadratic and linear logarithms was necessary to control the
convergence of the logarithmic expansion.  
These are of special interest both from the phenomenological and
conceptual point of view, because, in contrast to the higher powers of
the logarithm, the ${\rm N^3LL}$ terms are sensitive to the details of the gauge boson mass
generation.  The first results beyond the NNLL approximation have been
obtained in \cite{FKM,FKPS,JKPS}.\\
The complete result can be found
in \cite{JKPS3}, where the calculation of the two-loop logarithmic corrections 
to the neutral current four-fermion processes was completed and
the previously neglected effects of the gauge boson mass difference
incorporated.  The two-loop logarithmic
terms were derived within the {\it expansion by regions} approach
\cite{BenSmi,SmiRak,Smi1,Smi2} by inspecting the structure of
singularities of the contributions of different regions.  The
calculation was significantly simplified by taking the exponentiation of
the logarithmic corrections in the Sudakov limit into account.  This
property naturally appears and can be fully elaborated within the
evolution equation approach \cite{Mue1,Col,Sen1}.  To identify the pure
QED infrared logarithms which are compensated by soft real photon
radiation, the {\it hard} evolution equation which governs the dependence 
of the amplitudes on $s$ was combined with the {\it infrared} evolution
equation \cite{Fad} which describes the dependence of the amplitude on
an infrared regulator. \\
Let us now recall the strategy and the main results of these investigations in more detail:\\

{\em\large\quad The Form Factor and Evolution Equations}\\[3mm]
\begin{figure}
  \centering
%\psfrag{Q [GeV]}{$Q$ [GeV]}
\epsfig{figure=figures/F2R-logs-color.eps,height=6cm}
\caption{\label{fig2} 
Leading and subleading logarithmic contributions to the two-loop
form factor for the $SU(2)$ toy model.}
\end{figure}
Starting from the form factor in Born approximation
\be
{\cal F}_{Born}=\bar{\psi}(p_2)\gm_\mu\psi(p_1)
\ee
the all-order resummation of leading and subleading logarithms is based on the evolution equation %\cite{Col,Sen1}
\be
{\partial\over\partial\ln{Q^2}}{\cal F}=
\left[\int_{M^2}^{Q^2}{\dd x\over x}\gm(\al(x))+\zeta(\al(Q^2))
+\xi(\al(M^2)) \right] {\cal F}
\label{evoleqf}
\ee
with the solution
\be
{\cal F}=F_0(\al(M^2))\exp \left\{\int_{M^2}^{Q^2}{\dd x\over x}
\left[\int_{M^2}^{x}{\dd x'\over x'}\gm(\al(x'))+\zeta(\al(x))
+\xi(\al(M^2))\right]\right\} \,. 
\label{evolsolf} 
\ee
To obtain all non-power suppressed terms up to two loops corresponds to the reconstruction of all terms of
the form $\al^2{\cal L}^i$ with $i=0,\ldots,4$. In arbitrary orders this is equivalent to the ${\rm N^4LL}$ 
approxmation, collecting all terms of the form $\al^n{\cal L}^{2n-i}$, again with $i=0,\ldots,4$.
From eqn.~(\ref{evolsolf}) it is evident that the NNLL approximation can be obtained from the evaluation of 
the anomalous dimensions $\zeta(\al), \xi(\al)$ 
to one loop, $\gm(\al)$ to two loop (from a massless two-loop calculation), $F_0(\al)$ to
one loop and the lowest two coefficients of the $\beta$-function. 
To derive the ${\rm N^3LL}$ approximation, the linear logarithms have to be calculated 
in an explicit evaluation of the two-loop form factor and, finally, ${\rm N^4LL}$ requires even the 
constant two-loop term, which is available for the form factor in the Abelian theory only.
Since these considerations are equally applicable to the four-fermion process (which are 
governed by the same soft and collinear logarithms) the two-loop form factor has been evaluated
in \cite{JKPS3,Jan,FKPS} in the high energy limit.
The set of fermionic, Abelian and non-Abelian two-loop vertex corrections is shown in
Fig.~\ref{fig:feynnf},~\ref{fig:feynAb},~\ref{fig:feynNA} respectivaly.\\
Defining
\be
{\cal F}(M,Q)={\cal F}_{Born}\left[1+{\al\over 4\pi}f^{(1)}+\left({\al\over 4\pi}\right)^2f^{(2)}
+\ldots\right]
\ee
and ${\cal L}=\log(Q^2/M^2)$, the high energy behaviour in the Abelian and $SU(2)$ case respectively
is given by
\bea
f^{(2)}_{Abelian}&=&{1\over 2}{\cal L}^4
-3{\cal L}^3
+\left(8+{2\over 3}\pi^2\right){\cal L}^2
-\left(-24\zeta_3+9+4\pi^2\right){\cal L}\cr
&&+256\, {\rm Li}_4\left({1\over 2}\right)+{32\over 3}\ln^42-{32\over 3}\pi^2\ln^22
-{52\over 15}\pi^4+80\zeta_3+{52\over 3}\pi^2+{25\over 2}\,,
\cr
f^{(2)}_{SU(2)}&=&{9\over 32}{\cal L}^4
-{19\over 48}{\cal L}^3
-\left({463\over 48}-{7\pi^2\over 8}\right){\cal L}^2
+\left(29-{11\over 24}\pi^2\right){\cal L}\,,
\eea
where $M_{\rm Higgs}=M_{\rm gauge\, boson}\equiv M$ was adopted for the $SU(2)$ case and three doublets of
fermions were assumed. The relative size of the corrections for this $SU(2)$ "toy model" is 
shown in Fig.~\ref{fig2}. The gauge boson induced terms are clearly dominant, the Higgs and light fermion
terms small. Comparing the relative importance of LL versus ${\rm NLL}$ terms and so on, one finds
an oscillation behaviour and considerable compensations between the different powers of the
logarithms.\\ %(Fig.~\ref{fig2}).\\

\begin{figure}
  \centering
  \includegraphics[scale=0.8]{figures/vertex-Wcorr.eps}
  \caption{Fermionic vertex correction}
  \label{fig:feynnf}
\end{figure}%
\begin{figure}
  \centering
      %%[scale=\fscale]
  %\valignbox[b]
\includegraphics[scale=0.8]{figures/vertex-LA.eps}%\\(a)}
  \hspace{2mm}
  %\valignbox[b]
\includegraphics[scale=0.8]{figures/vertex-NP.eps}%\\(b)}
  \hspace{2mm}
  %\valignbox[b]
\includegraphics[scale=1]{figures/vertex-BE.eps}%\\(c)}
  \hspace{2mm}
  %\valignbox[b]
\includegraphics[scale=0.8]{figures/vertex-fcorr.eps}%\\(d)}
  \caption{Abelian vertex corrections}
  \label{fig:feynAb}
\end{figure}%
\begin{figure}
  \centering
  \includegraphics[scale=0.8]{figures/vertex-BE-CA.eps}%\\(a)}
  \hspace{8mm}
  \includegraphics[scale=0.8]{figures/vertex-Wcorr-CA.eps}%\\(b)}
  \hspace{8mm}
  \includegraphics[scale=0.8]{figures/vertex-Wcorr-cc.eps}\\[4mm]%\\(c)}
  %\hspace{8mm}
  \includegraphics[scale=0.8]{figures/vertex-Wcorr-Higgs.eps}%\\(d)}
  \hspace{8mm}
  \includegraphics[scale=0.8]{figures/vertex-Wcorr-HiggsGoldstone.eps}%\\(e)}
  \hspace{8mm}
  \includegraphics[scale=0.8]{figures/vertex-Wcorr-Goldstones.eps}%\\(f)}
  \caption{Non-Abelian vertex corrections}
  \label{fig:feynNA}
\end{figure}%

{\em\large\quad Four Fermion scattering}\\[3mm]
The behaviour of four-fermion amplitudes is again best discussed in the toy model of a massive
$SU(2)$ theory. The isospin and  chiral decomposition needed to describe this process is 
characterized by the amplitudes
\bea
{\cal A}^\lm&=&
\bar{\psi_2}t^a\gm_\mu{\psi_1}
\bar{\psi_4}t^a\gm_\mu{\psi_3}\, , \nn \\
{\cal A}^\lm_{LL}&=&
\bar{\psi_2}_Lt^a\gm_\mu{\psi_1}_L
\bar{\psi_4}_Lt^a\gm_\mu{\psi_3}_L\, ,  
\label{basis}\\
{\cal A}_{LR}^d&=&
\bar{\psi_2}_L\gm_\mu{\psi_1}_L
\bar{\psi_4}_R\gm_\mu{\psi_3}_R\,. \nn
\eea
Adopting this basis and defining a reduced amplitude $\tilde{A}$ (actually a vector in the
isospin/chiral basis) by splitting off the collinear logarithms
\be
{\cal A}={ig^2\over s}{\cal F}^2\tilde{\cal A}\,,
\label{evolamp}
\ee
it is possible to define a matrix evolution equation
\be
{\partial \over \partial \ln{Q^2}}\tilde{\cal A}=
{\bf \chi}(\al(Q^2))\tilde {\cal A} \, ,
\label{evoleqa}
\ee
which can be solved iteratively.
The matrix $\chi$ of the ``soft'' anomalous dimensions is obtained from the 
one-loop result for four-fermion scattering
in the high energy limit, if one is interested in the NNLL terms only.
To obtain the four-fermion cross section in ${\rm N^3LL}$ approximation, the form factor is 
required up to ${\rm N^3LL}$ and the matrix $\chi$ up to two loops. The latter can be obtained from 
the hard contribution to the single pole part of four fermion scattering \cite{JKPS3}. For the
$SU(2)$ toy model discussed already above and for fermions with identical isospin in initial
and final state ($u\bar{u}\rightarrow u'\bar{u}'$) the two loop part of the cross section is given by
\be
%% same isospin
\sigma^{(2)}=\left[{9\over 2}{\cal L}^4
-{449\over 6}{\cal L}^3
+\left({4855\over 18}+{37\pi^2\over 3}\right){\cal L}^2
\right]\sigma_B\,,
\ee
Again large cancelations between different powers of the logarithm are observed.\\

\begin{figure}
  \begin{center}
    \begin{tabular}{cc}
      \hspace*{-9mm}
      \epsfxsize=8.5cm
      \epsffile{figures/sudpqqol.eps}&
      \hspace*{-9mm}
      \epsfxsize=8.5cm
      \epsffile{figures/sudpqq.eps}\\
      \hspace*{-9mm}  
      (a)&\hspace*{-9mm}(b)
    \end{tabular}
  \end{center}
  \caption{\label{figC1} \small Separate logarithmic contributions
    to $R(e^+e^-\to q\bar q)$ in \% to the Born approximation: (a) the
    one-loop LL $(\ln^2(s/M^2)$, long-dashed line), NLL $(\ln^1(s/M^2)$,
    dot-dashed line) and N$^2$LL $(\ln^0(s/M^2)$, solid line) terms; (b)
    the two-loop LL $(\ln^4(s/M^2)$, short-dashed line), NLL
    $(\ln^3(s/M^2)$, long-dashed line), NNLL $(\ln^2(s/M^2)$, dot-dashed
    line) and N$^3$LL $(\ln^1(s/M^2)$, solid line) terms.}
\end{figure}

{\em\large\quad Electroweak Theory}\\[3mm]
Two important aspects have to be taken into account, when moving from $SU(2)$
to the full electroweak theory:\\[1mm]
i) The appearance of the massless photon leads to infrared singularities which must be seperated
from the complete result and eventually cancelled against those from real radiation.
 The matrix $\chi$ and the collinear factor is modified accordingly,
following the prescription of \cite{Fad}.\\[1mm]
ii) The mass difference between $M_W$ and $M_Z$ must be taken into account and, starting from
${\rm N^3LL}$, details of the gauge boson mass generation start to matter.\\[1mm] 
The relative size of the logarithmically enhanced terms of the corrections 
for $e^+e^-\rightarrow q\bar{q}$ is shown
in Fig.~\ref{figC1} for one- and two-loop contributions, respectively.
Similar techniques have been applied to $W$-pair production in electron-positron annihilation
\cite{KMP}. In this case a strikingly different behaviour of longitudinal vs. transverse polarized
$W$'s is observed, which, using the equivalence-theorem, is easily traced to the different
isospin-representation of the Goldstone modes ($I={1\over 2}$) and the gauge bosons ($I=1$).
Furthermore, as a consequence of the direct and strong coupling of the longitudinal $W$ to
the top quark, the proper treatment of the mixing of the $W-tb$ system is important for 
this interesting case.

\section{Z, photon and $W$ production at large transverse momenta}

\begin{figure}
\begin{center}
\epsfig{figure=figures/Fig2b.Z.eps,height=4.7cm}\hspace*{0.7cm}
\epsfig{figure=figures/Fig2b.A.eps,height=4.7cm}
\caption{\label{fig3AZ} 
Relative NLO (solid) and NNLO (dotted) corrections 
wrt. the LO prediction  and statistical
  error (shaded area) for the unpolarized integrated cross section for 
$pp\rightarrow Z j$ (left) and $pp\rightarrow \gamma j$ (right)
at $\sqrt{s}=14$TeV as a function of $p_T^{cut}$.}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
%\epsfig{figure=figures/Fig2b.W+.eps,height=6cm}
\epsfig{figure=figures/Fig2-1.eps,height=8.6cm}
\caption{\label{fig3W} 
Unpolarized integrated cross section as a function of
$p_T^{cut}(W)$ for $W^+$ (a) and $W^-$ (b) production: estimated
statistical error (shaded area) and relative electroweak corrections
in NLO (dotted) and NNLO (solid) approximation.}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\epsfig{figure=figures/Fig3a.eps,height=7cm}
\hspace{0.8cm}
\epsfig{figure=figures/Fig3b.eps,height=7cm}
\caption{\label{fig4}
Ratio of the transverse momentum distributions for $W^+/W^-$, $W^+/\gamma$, $W^+/Z$ and
$W^-/Z$ at the LHC: LO (thin solid), NLO (dotterd) and NNLOO (thick solid) predictions.}
\end{center}
\end{figure}

In view of the enormous luminosity expected for the LHC, the production of $Z$, $W$ or
photons at large transverse momenta, recoiling against a quark or gluon jet will be observed
with sizable event rates. Indeed, gauge bosons with $p_T$ up to 2TeV will be observed, 
corresponding to scattering energies of up to 4TeV. In this region large (negative)
electroweak corrections are expected. This has motivated a series of investigations of these
reactions, moving systematically from $Z$ and photon \cite{MMR,KKPS1,KKPS2,KKPS3} 
to $W$ production \cite{KKPS4,KKPS5,HKK1}.
In addition to the full evaluation of the one-loop terms the NLL and NNLL two-loop
corrections are presented in \cite{KKPS1,KKPS2,KKPS3,KKPS4,KKPS5}.\\
Compact analytical formulae were obtained in \cite{KKPS1,KKPS2,KKPS3,KKPS4,KKPS5}. The virtual
corrections can be decomposed into "Abelian" and "non-Abelian" parts which can be traced back
to the original $SU(2)\times U(1)$ structure of the theory. For $W$  production furthermore,
the photonic corrections do not form a seperate gauge invariant subset and must be included
in the analysis, together with real radiation, i.e. $W+\gamma+$jet final states. Their
contribution can only be obtained from a Monte Carlo generator.\\
 Let us, in a first step,
consider $Z$-production. For small $\sqrt{\hat{s}}$ up to 200GeV, the corrections are practically
irrelevant, below 0.3\%. For $\sqrt{\hat{s}}=4$TeV, however, they amount up to 40\%.
In this region the dominant logarithmic terms are given by
\bea
{H_1^{\mathrm{ A}}}&\stackrel{NLL}{\sim}&
-\left[\log^2\left(\frac{|\hat{s}|}{M_W^2}\right)
-3\log\left(\frac{|\hat{s}|}{M_W^2}\right)
\right] H_0^A\,, \\
{H_1^{\mathrm{ N}}}&\stackrel{NLL}{\sim}&
-\left[
\log^2\left(\frac{|\hat{t}|}{M_W^2}\right)
+\log^2\left(\frac{|\hat{u}|}{M_W^2}\right)
-\log^2\left(\frac{|\hat{s}|}{M_W^2}\right)
\right] H_0^N\,,
\eea
with the remaining subleading terms below 2.5\%. Using arguments discussed in \cite{Mel,DenPoz,DMP} also
the dominant two-loop terms can be predicted, which lead to a slight reduction of the
corrections by about 5\% at the highest energies. The relative one-loop and dominant two-loop
corrections for $Z$ and $\gamma$ production 
are shown in Fig.~\ref{fig3AZ} and compared to the expected statistical precision at LHC, 
assuming an integrated luminosity of $200fb^{-1}$, and full efficency for leptonic decay 
modes. The corresponding results for $W^+$ and $W^-$ production are shown in 
Fig.~\ref{fig3W}.\\ 
The theory uncertanty in predicting the absolute cross section originats from
uncalculated higher orders in QCD and uncertanties in the parton distribution functions. On the
experimental side the measured boson transverse momentum may be smeared, 
which leads to a corresponding smearing of the steeply
falling $p_T$ distribution. QCD uncertanties could be significantly reduced by considering
ratios of $W^+, W^-,Z$ and $\gamma$ distributions (Fig.~\ref{fig4}). These ratios are shown again
in Born approximation, including the full one-loop and as a third option, including the
dominant two-loop terms. The $W^+/W^-$ ratio remains practically unchanged, the $W/Z$ and
$W/\gamma$ ratios are shifted by 10 to 25\% for the largest transverse momenta.

\section{Top produciton}

\begin{figure}
\begin{center}
\epsfig{figure=figures/ttbarBoxInterf1.eps,height=2cm} %
\epsfig{figure=figures/ttbarBoxInterf2.eps,height=2cm}
\caption{\label{figTop1} Mixed QCD and electroweak corrections to $q\bar{q}$ induced 
top quark production}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\epsfig{figure=figures/ttbarVertexCorr1.eps,height=2cm} %
\epsfig{figure=figures/ttbarVertexCorr2.eps,height=2cm}
\caption{\label{figTop2} Generic weak corrections to $q\bar{q}$ induced top quark production}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\epsfig{figure=figures/gg_ttbarVertexCorr1.eps,height=1.6cm} %
\epsfig{figure=figures/gg_ttbarSelfEnergy.eps,height=1.6cm}
\epsfig{figure=figures/gg_ttbarBox.eps,height=1.6cm}\\[5mm]
\epsfig{figure=figures/gg_ttbarVertexCorr2.eps,height=1.6cm}
\epsfig{figure=figures/gg_ttbarTriangle.eps,height=1.6cm}
\caption{\label{figTop3} Representative weak corrections to 
$gg\rightarrow t\bar{t}$}
\end{center}
\end{figure}

As a third topic, clearly connected to the start of the LHC in the course of this year, let us
consider the impact of electroweak corrections on top-quark pair production. An early 
discussion can be found in \cite{BDHW,KW} where the main issues were adressed for the first time.
Later, with the advent of LHC, it was pointed out \cite{BMS,KSU1} that the results for the quark 
induced process had been incomplete, with mixed QCD-electroweak box contributions connected
to real and virtual gluon emission (Fig.~\ref{figTop1})
being of the same order as the generic electroweak corrections (Fig.~\ref{figTop2}).
The corrections to the gluon-induced process were analysed in \cite{Ber,KSU2} where
some discrepancies were observed relative to earlier work \cite{Smi1,BDHW,Mor}. Representative 
diagrams are shown in Fig.~\ref{figTop3}. Of specific interest are the s-channel contributions from
$Z$, $\chi$ and the Higgs boson. If kinematically allowed, the Higgs propagator would lead
to a resonant behaviour, damped, of course, by the finite width of the Higgs boson. The $Z$
and $\chi$ amplitudes exhibit a smooth behaviour in the full kinematic range. The relative size 
of the corrections are shown in Fig.~\ref{fig6} for quark and gluon induced processes as functions of
$\sqrt{\hat{s}}$ from threshold up to 1TeV. Not surprisingly, again large
negative corrections are observed for large energies. Close to threshold, however, one also
obtains sizable corrections which are strongly dependent on $m_H$, with a difference
between large $m_H$ and $m_H=120$GeV of 5 to 10\%. This is the consequence of the
attractive Yukawa potential resulting from a light Higgs boson exchange.\\
To arrive at a prediction for the LHC the partonic cross sections must be convoluted
with the parton distribution functions. As shown in Fig.~\ref{fig7}, the 
gluon induced process is dominant at low $p_T$, the $q\bar{q}$ induced processes
for $p_T$ above 700GeV. This is reflected in the size of the corrections which are
shown in Fig.~\ref{fig7} and compared to the anticipated statistical error, assuming an integrated
luminosity of $200fb^{-1}$. The corresponding results for the distribution on the
invariant mass of the $t\bar{t}$ system are shown in Fig.~\ref{fig8}.

\begin{figure}
\epsfig{figure=figures/qqmh.eps,height=7cm}
\epsfig{figure=figures/ggmh.eps,height=7cm}
\caption{\label{fig6} Relative corrections for $q\bar{q}$ (left)
and $gg$ (right) induced processes as functions of the partonic 
energy $\sqrt{\hat{s}}$ for different Higgs boson masses}
\end{figure}

\begin{figure}
\epsfig{figure=figures/ptcutlo.eps,height=7cm}
\epsfig{figure=figures/ptcutnlo.eps,height=7cm}
\caption{\label{fig7}
Left: Fraction of events from $q\bar{q}$ and $gg$ induced processes
with $p_T>p_{Tcut}$ as functions of $p_{Tcut}$, compared to the total
$t\bar{t}$ production cross section.
Right: Relative correction for the cross section for $t\bar{t}$ production
with $p_T>p_{Tcut}$. Also shown is the anticipated statistical error
at the LHC.}
\end{figure}

\begin{figure}
\epsfig{figure=figures/mttcutnlo.eps,height=7cm}
\caption{\label{fig8} Relative correction for the cross section for $t\bar{t}$
production with $M_{t\bar{t}}>M_{t\bar{t}cut}$. Also shown is the anticipated 
statistical error at the LHC.}
\end{figure}

\section{Summary}
\label{summ} 
The LHC will be the first collider exploring the energy region where parton scattering
energies will be significantly larger than the masses of the gauge bosons. This leads us
into a region where, depending on the choice of the observables, electroweak corrections may become
large, of the order 10 to 30\%. For precise predictions the logarithmically enhanced 
terms are required in one and two-loop approximation. At present, detailed studies of
two loop ${\rm N^3LL}$ effects are available for the fermion form factor, for four-fermion
processes and in ${\rm N^2LL}$ for $W$-pair production. Studies involving the full one-loop
corrections and, partially, the dominant two-loop terms, have been performed for $Z,W$
and $\gamma$ production at large transverse momenta and, similarly, for top quark pair
production. It will be exciting to see the first of these reactions soon.
\vspace{4mm}

{\bf Acknowledgments}\\[3mm]
I would like to thank the organizers of the Cracow Epiphany Conference for their
generous hospitality.
I am grateful to B. Feucht, A. Kulesza, F. Metzler, A. Penin, A. Scharf, M. Schulze,
V. Smirnov and P. Uwer for their collaboration on the topics discussed on this review
and F. Metzler for help in preparing this manuscript.
This work has been supported by BMBF grant No. 05HT4VKA/3, by SFB-TR09 and Graduiertenkolleg
"Hochenergiephysik und Teilchenastrophysik".

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%\cite{Hollik:2007sq}
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%\cite{Beenakker:1993yr}
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%\cite{Kao:1999kj}
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%\cite{Bernreuther:2005is}
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%\cite{Kuhn:2005it}
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%\cite{Moretti:2006nf}
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%\cite{Bernreuther:2006vg}
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%\cite{Kuhn:2006vh}
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\end{thebibliography}

\end{document}
