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\title{
\begin{flushright}
{\normalsize   TTP/13-015\\
  HU-EP-13/26\\
  May  2013\\}
\end{flushright}
\vspace{1.0cm}
Weak Interactions in Top-Quark Pair Production\\ 
at Hadron Colliders: An Update} 
\author{
   {\large J.H.~K\"uhn} $^a$,\,
  {\large A.~Scharf} $^{b}$\, and  
  {\large P.~Uwer} $^c$\\[3mm]
  $^a${\small {\em Institut f{\"u}r Theoretische Teilchenphysik,
  Karlsruhe Instititut of Technology (KIT)}}\\
  {\small {\em 76128 Karlsruhe, Germany}}\\[2mm]
  $^b${\small {\em 
Institut f\"ur Theoretische Physik und Astrophysik, 
Universit\"at W\"urzburg}}\\ 
{\small {\em D-97074 W\"urzburg, Germany}}\\[2mm]
  $^c${\small {\em Institut f\"ur Physik,
  Humboldt-Universit\"at zu Berlin}}\\
  {\small {\em 12489 Berlin, Germany}}
}
\date{}
\maketitle
\begin{abstract}
Weak corrections for top-quark pair production at hadron
colliders are revisited. Predictions for collider energies of 8~TeV,
adopted to the present LHC run, and for 14~TeV, presumably relevant
for the next round of LHC experiments, are presented. Kinematic regions
with large momentum transfer are identified, where the corrections 
become large and may lead to strong distortions of differential 
distributions, thus mimicking anomalous top quark couplings. 
As a complementary case we investigate the threshold region, 
corresponding to configurations with small relative
velocity between top and antitop quark, which is particularly 
sensitive to the top-quark Yukawa coupling. We demonstrate, that
nontrivial upper limits on this coupling are well within reach of
ongoing experiments.
\\[2mm]
%PACS numbers: 
\end{abstract}

%\thispagestyle{empty}
%\newpage

\section{Introduction}
\label{intr}
During the past years the determination of the top quark mass, 
its couplings, production and decay 
rates has been pursued successfully at the Tevatron. Based on an integrated
luminosity of more than  $5~{\rm fb}^{-1}$ per experiment collected by 
both CDF and D0 at 1.96~TeV, a sample of nearly 100000 top quark pairs has been 
produced. The analysis of these events 
has lead, for example, to a top mass determination  
 of $M_t= 173.18 \pm 0.94 {\rm GeV}$ \cite{Aaltonen:2012ra}, 
corresponding to a relative error of about half percent. 
The total production cross section $\sigma_{t\bar t} = 7.65 \pm
0.42$~pb \cite{D0confnote6363} determined at Tevatron 
is in very good  agreement with the theory predictions
\cite{Moch:2008qy,Aliev:2010zk,Moch:2012mk,Cacciari:2011hy,Czakon:2012pz,Czakon:2012zr,Baernreuther:2012ws,Ahrens:2011px,Kidonakis:2011ca,Kidonakis:2008mu,Beneke:2012wb,Czakon:2013goa}.
The same is true for the cross section measurements performed at the LHC.
Very recently also the $t\bar t$ invariant mass distribution has
been measured at LHC over a wide kinematical range
\cite{:2012qa,:2012rq}. Similar to the cross section measurements the
results are in agreement with the Standard Model (SM) predictions.  
In contrast,
surprising deviations from the theory predictions  
have been observed in the Tevatron experiments \cite{Abazov:2012oxa,Abazov:2011rq,Aaltonen:2012it,Aaltonen:2011kc} by investigating the
so-called charge asymmetry predicted originally fifteen years ago
\cite{Kuhn:1998jr,Kuhn:1998kw}. (For recent discussions of theoretical
predictions in the context of the SM see for example 
Refs.~\cite{Kuhn:2011ri,Hollik:2011ps,Bernreuther:2012sx,Ahrens:2011uf,Almeida:2008ug}).

Although these are impressive achievements already now, expectations for
top quark physics at the LHC fly even higher. Based on integrated
luminosities close to $5~{\rm fb}^{-1}$ per experiment at 7~TeV,
the top mass has been determined in a combined analysis 
to $M_t= 173.3 \pm 1.4~{\rm GeV}$ \cite{ATLAS:2012coa} already now. 
With an integrated luminosity of more than $20~{\rm fb}^{-1}$ per experiment
collected recently at 8~TeV, several million top-quark pairs per
experiment have been produced. 
The high energy run at 14~TeV with its expected integrated luminosity of 
$100~{\rm fb}^{-1}$, will deliver about $10^8$ top quark
pairs per experiment during the coming years. The LHC is,
obviously, a factory of top quarks, allowing for a precise determination
of their properties and their production dynamics in a large kinematic region. 
The large center of mass energy available at the LHC will thus be used to
investigate top production with partonic subenergies of several TeV and
thus explore the point-like nature of the heaviest of the fundamental
particles. On the theoretical side precise predictions valid at
highest accessible energies are required. With the recently completed 
next-to-next-to leading order QCD predictions 
\cite{Czakon:2012pz,Czakon:2012zr,Baernreuther:2012ws,Czakon:2013goa} a major step has
been taken. However when it comes to ultimate precision or highest
energies weak corrections significantly affect predictions within the
Standard Model. Two kinematic regions are of particular interest: 
\begin{enumerate}
\item[{\it i.})] Hard scattering events with partonic subenergies $\hat s$ and
  momentum transfers  $|\hat t|$  and $|\hat u|$ ($\hat s, \hat
  u$ and $\hat t$ denote the partonic Mandelstam variables) far larger than $M_t$ are
  affected by large negative corrections. These may reach
  nearly twenty percent, affecting transverse-momentum and angular
  distributions, and might well mimic anomalous top quark
  couplings. These negative corrections---if not taken into account in
  the theoretical predictions---could also hide a possible rise of the
  cross section due to a heavy resonance.
\item[{\it ii.})] The rate for events very close to the production threshold, 
  with relative top-antitop velocity $\beta\leq M_H/M_t$ is enhanced
  by the exchange of the relatively light Higgs boson. This effect 
  can be approximately described by a Yukawa potential and is 
  reminiscent of Sommerfeld rescattering corrections.
\end{enumerate}

Weak corrections to top quark pair production have first been
been studied twenty years ago \cite{Beenakker:1993yr}.  The complete
results, where some deficiencies were corrected and the result
given in closed analytical form, can be found in
Refs.~\cite{Kuhn:2005it,Bernreuther:2005is} and
Refs.~\cite{Bernreuther:2006vg,Kuhn:2006vh} for quark- and
gluon-induced processes, respectively. Numerical results (which,
however, differed from those presented in
Refs.~\cite{Bernreuther:2006vg,Kuhn:2006vh} and were corrected later) have
been published in Ref.~\cite{Moretti:2006nf}. Purely electromagnetic
corrections, which can be handled separately from the weak
corrections, are evaluated in Ref.~\cite{Hollik:2007sw}.  
As a consequence of cancellations between the positive contributions
from $\gamma$g-fusion and negative corrections to $q\bar q$-annihilation
the combined effect amounts at most to -4\%, if one considers
$\pt$-values as high as 1.5~TeV. The impact on the $\sqrt{\hat s}$
distribution remains below 1\%. The details of these corrections are
strongly cut-dependent and we refer to Ref.~\cite{Hollik:2007sw} for details. 

In the present paper we refrain from repeating the somewhat lengthy 
analytical formulae for the weak corrections 
and concentrate on the physics implications. In particular we also
update results previously obtained using modern parton distribution functions
(PDF's) and the most recent values for the input parameters.


\section{Large momentum transfers}
\label{sec-large-momenta}
Before entering the detailed numerical discussion, let us recall the
basic qualitative aspects of weak corrections for the present
case.
\begin{figure}
  \begin{center}
    \includegraphics[height=2.5cm]{lo-qcd}\vspace*{0.5cm}

    \includegraphics[height=2.5cm]{qq_tt_w}
    \caption{Lowest order QCD (a--d) and weak (e) amplitudes}
    \label{fig:Born}
  \end{center}
\end{figure}
%
With the Born amplitudes being of order
$\alpha_s$ (Figs.~\ref{fig:Born} a)-d) both for quark and gluon induced
QCD processes, and  of order $\alpha_{\rm weak}$ for
the lowest order weak process 
(Fig.~\ref{fig:Born} e), weak corrections start entering the cross
section at loop-induced order 
$\alpha_s^2\alpha_{\rm weak}$ only. The absence of an interference term
between the lowest order strong and neutral current amplitudes in the
quark induced process, which would be of order $\alpha_s\alpha_{\rm weak}$,
follows trivially from the different colour flow in the 
two relevant amplitudes Fig.~\ref{fig:Born}a and e, respectively. 
 

Sample diagrams for weak corrections to quark- and gluon-induced
amplitudes are shown in Figs.~\ref{fig:VirtualCor-qq} and 
\ref{fig:VirtualCor-gg}.
For gluon fusion weak effects start as
corrections to the QCD induced amplitudes. 
%
\begin{figure}[!htbp]
  \begin{center}
    \leavevmode
    \includegraphics[height=6cm]{VirtualCor-qq}
    \caption{\label{fig:VirtualCor-qq}
      Sample diagrams for the virtual corrections.}
  \end{center}
\end{figure}

%
\begin{figure}[!htbp]
  \begin{center}
    \leavevmode
    \includegraphics[height=10cm]{VirtualCor-gg}
    \caption{
          \label{fig:VirtualCor-gg}
          Sample diagrams for the virtual corrections. $\Gamma$
          stands for all contributions from gauge boson, Goldstone boson
          and Higgs exchange.}
  \end{center}
\end{figure}
%
\label{sec:real}
\begin{figure}
  \begin{center}
    \includegraphics[height=6cm]{RealCor}
    \caption{Sample diagrams for the real corrections to the
      quark-induced process.}
    \label{fig:RealCor}
  \end{center}
\end{figure}
For quark-antiquark annihilation the situation is more involved in view of
a specific class of order $\alpha_s^2\alpha_{\rm weak}$ 
contributions to the quark induced processes,  
which must be considered separately. In this case weak
and strong interaction are intimately intertwined, and corrections 
with virtual and real (Fig.~\ref{fig:RealCor}) 
gluon emission must be combined to arrive at an infrared finite result. 
The proper combination of real and virtual contributions is illustrated
in Fig.~\ref{fig:IR-cancellation}.
\begin{figure}
   \begin{center}
     \begin{picture}(100,80)
       \put(-140,-40){\includegraphics[width=9.cm]{cuta}}
       \put(0,-40){\includegraphics[width=9.cm]{cutb}}
     \end{picture}
%     \includegraphics[width=8.5cm]{cuta}\hspace{1cm}%
%     \includegraphics[width=8.5cm]{cutb}
     \caption{Sample diagrams for the proper combination of virtual and real
       corrections to the quark-induced process.}
     \label{fig:IR-cancellation}
   \end{center}
\end{figure}
This issue is discussed in more detail in Ref.~\cite{Kuhn:2005it}.
Only a specific combination of couplings is present in this case: The
top quark triangle in Fig.~\ref{fig:IR-cancellation} is attached to
two gluons with vector coupling. 
% hence the neutral current coupling receives contributions from 
% the axial coupling $g_A^t$ only. 
As long as we are interested in parity-even observables (like cross
section or $\pt$-distributions), the light quark coupling to the $Z$
boson is therefore restricted to its axial coupling $g_A^q$
proportional to its isospin $I_3^q$.  This, in turn, leads to a strong
cancellation of this specific type of correction between u- and
d-quark induced processes. Since, furthermore, these contributions are
small (see Fig IV.3 of Ref.~\cite{Kuhn:2005it}) for one species of
quarks already, (less than one percent at threshold and about two
percent at very high energies), this group of corrections could be
neglected in the following discussion.  This observation might,
eventually, facilitate the combination of strong and weak corrections
discussed at the end of this paper.

For large parton energies the total corrections are negative, for quark- as
well as for gluon-induced processes. However, as a consequence of the
non-vanishing weak charge both in the initial as well as in the final
state, the corrections for quark induced top production are about twice 
those of the gluon induced process, with important consequences for the 
energy dependence of the corrections.

As discussed in Ref.~\cite{Kuhn:2006vh} for proton-proton collisions at 14~TeV, 
(denoted by LHC14 in the following), the total cross section for top production
is dominated by gluon fusion. In contrast, production of top quarks with
large invariant mass  of the $t\bar t$-system or at large transverse momenta 
is mainly induced by quark-antiquark annihilation, a consequence of the
different parton luminosities (see Fig.~\ref{fig:LHCdifplot14}
for LHC operating at 14 TeV and Fig.~\ref{fig:LHCdifplot} for LHC
running at 8 TeV).
The relative increase of the the quark-induced processes in combination
with the different strength of
the weak corrections for the two reactions thus leads to an
additional increase of weak corrections for very large transverse 
momenta.
\begin{figure}
  \begin{center}
    \scalebox{0.8}{\input{./figures/figLO_Mttb_LHC14}} 
    \vspace*{0.5cm}

    \scalebox{0.8}{\input{./figures/figLO_pTtop_LHC14}} 
    \caption{\label{fig:LHCdifplot14}%
      Leading-order differential cross section for the LHC (14 TeV) 
      as a function of
      $\pt$ and $M_{t\bar t}$. Shown is the sum (full) and the contributions 
      from gluon
      fusion (dashed) and quark--antiquark annihilation (dotted).
      }
  \end{center}
\end{figure}
\begin{figure}
  \begin{center}
    \scalebox{0.8}{\input{./figures/figLO_Mttb_LHC8}} 
    \vspace*{0.5cm}

    \scalebox{0.8}{\input{./figures/figLO_pTtop_LHC8}} 
    \caption{\label{fig:LHCdifplot}%
      Leading-order differential cross section for the LHC (8TeV) 
      as a function of
      $\pt$ and $M_{t\bar t}$. Shown is the sum (full) and the contributions 
      from gluon
      fusion (dashed) and quark--antiquark annihilation (dotted).
      }
  \end{center}
\end{figure}

For the numerical results presented
in this paper we use the parton distribution function 
MSTW2008NNLO PDF set\footnote{We follow closely the setup used for
  the NNLO QCD corrections so that the results presented
  here can be combined with the QCD corrections.} \cite{Martin:2009iq},
evaluated at a factorization scale $\mu_F=M_t$, and the coupling constants
\begin{displaymath}
  \alpha(2 M_t) = {1\over 126.3},\quad
  \alpha_s = 0.1,\quad
  \sin^2\theta_W  = 0.231.
\end{displaymath}
For the masses we use
\begin{displaymath}
  M_Z = 91.1876~\GeV,\quad
  M_W = 80.425~\GeV,\quad
  M_b = 4.82~\GeV,\quad
  M_t = 173.2~\GeV,
\end{displaymath}
and, if not stated otherwise, $M_H = 126~\GeV$.
The weak mixing angle and the masses of W- and Z-boson are, of course,
interdependent. Nevertheless, we expect that, using the $\overline{\mbox{MS}}$ 
value for
the weak mixing angle, some of the (uncalculated) higher-order
corrections are included and, therefore, a better phenomenological
description is achieved.
For the details of the renormalisation we refer to Ref.~\cite{Kuhn:2006vh}.
 
Another important aspect is the nontrivial angular dependence of the
weak corrections. As is well known, the leading Sudakov
logarithms proportional $\log^2 (s/M_W^2)$ are only dependent on the
(weak) charge of the incoming and outgoing particles, 
subleading terms may exhibit a nontrivial angular dependence (see
e.g.\ \cite{Kuhn:1999nn,Kuhn:2001hz}). 
This is reflected in characteristic angular dependent 
virtual corrections which affect the rapidity distributions of top
quarks at the LHC and might well mimic anomalous couplings of the
particles involved.
\begin{figure}
  \begin{center}
\scalebox{0.56}{\input{./figures/figqqb_partonic.tex}} 
\hspace{0.8cm}
\scalebox{0.56}{\input{./figures/figgg_partonic.tex}} 
\caption{\label{fig:partoncor}Relative weak corrections at
  parton level for the quark- and gluon-induced reactions as functions
  of the squared parton energy $\hat s$ for two characteristic masses
  of the Higgs boson.}
     \end{center}
\end{figure}   
%
%
%
\begin{figure}
  \begin{center}
    \vspace{1cm}
    \scalebox{0.56}{\input{./figures/figqqb_angle.tex}} 
    \hspace*{6mm}
    \scalebox{0.56}{\input{./figures/figgg_angle.tex}} 
    \caption{\label{fig:angularcor}%
      Relative weak corrections for the quark-  and gluon-induced
      reactions as functions of the scattering angle close to threshold
      ($\sqrt{\hat s}=370\, {\rm GeV}$) and at high 
      energies ($\sqrt{\hat s}=3\,{\rm TeV}$}
  \end{center}
\end{figure}   

Let us now enter the description of the corrections in more detail.
The corrections at the partonic level are shown in
Fig.~\ref{fig:partoncor} for quark and gluon induced processes as
functions of $\hat s$. For the quark--anti-quark channel we include
only the infrared finite vertex corrections which are responsible for
the Sudakow suppression at large momentum transfer. The box
contributions for the $q\bar q$ process are important only for the
charge-asymmetric piece
\cite{Kuhn:2011ri,Hollik:2011ps,Bernreuther:2012sx,%
  Ahrens:2011uf,Almeida:2008ug}, and can be neglected in the present
context.  As expected, away from very small $\hat s$ the corrections
are negative and about twice as large for quark- compared to
gluon-induced processes. Only very close to threshold one observes
corrections which become positive for a light Higgs boson and will be
discussed in section 3. For the ficticious case of $M_H=1~{\rm TeV}$
two pronounced structures are visible in the gluon-fusion channel: The
interference between the Born amplitude and the $s$-channel Higgs
boson contribution (last diagram of Fig.3) is visible as slight
depletion around 1~TeV, the interference with the $Z$ plus $\chi_Z$
contribution arising from the same diagram is responsible for the dip
close to threshold. For $M_H=126~{\rm GeV}$ this dip is
overcompensated by the positive contribution of roughly 5\% from the
Yukawa interaction discussed in more detail in section 3. This same
difference of 5\% between $M_H= 126~{\rm GeV}$ and $1~{\rm TeV}$ is
also visible in the threshold behaviour of the $q\bar q$-initiated
reaction. 

The angular dependence of the corrections is shown in
Fig.\,\ref{fig:angularcor} separately for quark and gluon induced
processes close to threshold at 370 GeV (upper blue curve) and for 3
TeV (lower red line). 
Let us, in a first step, discuss the results for the quark-induced
reaction (Fig.~\ref{fig:angularcor} left).
Again we restrict the analysis to the vertex
correction. Close to threshold the process is dominated by (isotropic)
$S$-waves, at high energies ($3~{\rm TeV}$) the Dirac form factor
dominates both for Born amplitude and correction.
This leads to a constant ratio as function of the scattering angle. At
low energies ($\sqrt{\hat s}=370$~GeV) we find a positive correction
of about 2 \%. At large energies the Sudakow suprression leads to
negative corrections of about $-18$ \%. Note that the box diagrams
while not particularly enhanced would lead to sizable
asymmetric and small symmetric corrections. 
For details we refer to Ref.~\cite{Hollik:2011ps}. 
The gluon induced part, in contrast, is markedly angular dependent.
For large $\hat s$ and small scattering angle the corrections are
small, since the Sudakov-like behaviour cannot be expected in this
case. At ninety degrees, in contrast, the Sudakov limit is applicable
and the corrections become large.
%%
\begin{figure}
  \begin{center}
  \scalebox{0.7}{\input{./figures/figTotalXSection}} 
 \caption{\label{fig:sigtot}%
Relative weak corrections for the total cross section
functions of the total cms energy for three different masses of the
Higgs boson.}
     \end{center}
\end{figure}   
%%
\begin{figure}
  \begin{center}
\scalebox{0.7}{\input{./figures/figTopMassXSection.tex}} 
    \hspace*{1mm}
\caption{\label{fig:TopMassDependence}
  Relative corrections as function of the top-quark mass.}
     \end{center}
\end{figure}   
%%
%%

Let us now discuss observables at the hadron level. In difference to
the discussion at parton level we include in the analysis now also
box-contributions and real corrections, thus the full set of
corrections are investigated.
The corrections for the total cross section are shown in
Fig.~\ref{fig:sigtot} as function of $\sqrt{s}$, for two
characteristic choices of the Higgs mass, $M_H=126~{\rm GeV}$ and
$1000~{\rm GeV}$. We allow $M_H$ to move away from its recently
determined value \cite{:2012gk,:2012gu} to illustrate the effect of
the Higgs-top Yukawa coupling. The corrections are evidently small, of
order minus two percent for all LHC energies and only moderately
sensitive to $M_H$.  In addition we demonstrate the impact of an
enhanced Yukawa coupling with $g_Y=2g_Y^{SM}$ discussed in more detail
in section 3.
In this case the negative corrections from the large transverse
momentum region are overcompensated by the positive ones for small
$t\bar t$ masses.  Given the recent progress concerning the NNLO QCD
calculations the theoretical uncertainties will eventually reach
3--4\%.  At this level of accuracy the weak corrections become
important and need to be taken into account. As reference we show in
Fig.~\ref{fig:TopMassDependence} the relative weak correction as
function of the top-quark mass. At 8 TeV centre of mass energy the
corrections are about $-1.7$\%. At 14 TeV the high-energy regime of
the cross section becomes more accessible leading to slightly larger
corrections of order of $-1.9$\%.
\begin{figure}
  \begin{center}
    \vspace{1cm}
    \scalebox{0.55}{\input{./figures/figMttb_rel_LHC8}} 
    \hspace*{8mm}
    \scalebox{0.55}{\input{./figures/figpTtop_rel_LHC8}}\\[12mm]
    \scalebox{0.55}{\input{./figures/figMttb_rel_LHC14}} 
    \hspace*{8mm}
    \scalebox{0.55}{\input{./figures/figpTtop_rel_LHC14}} 
 \caption{\label{fig:distr}%
Relative weak corrections for the invariant $t\bar t$ mass (left) and 
transverse momentum (right) distribution for LHC8 (upper) and LHC14 
(lower plots) and for Higgs masses of 126~GeV and 1~TeV. 
}
     \end{center}
\end{figure}   
The situation is drastically different, once we consider differential
distributions in the region of large transverse momenta $\pt$ or large
masses $M_{t\bar t}$ of the $t\bar t$ system.  The corrections are
shown in Fig.~\ref{fig:distr} for proton-proton collisions with center
of mass energies of 8~TeV and 14~TeV both for the $\pt$- and the
$M_{t\bar t}$-distributions. For illustration we again present the
relative corrections for Higgs masses of 126~GeV and 1~TeV. The strong
increase with increasing $\pt$ is evident.  Based on the present data
sample, corresponding to to more than 20~${\rm fb}^{-1}$, corrections
close to -10\% could be observed at 8~TeV.

To investigate the angular dependence of the $t\bar t$
system in its center of mass frame one could consider the
distribution in the rapidity difference 
$\Delta y_{t\bar  t}=y_{t}-y_{\bar t} $ 
which, for fixed
$M_{t\bar t}$ can be directly translated into the angular distribution.
To illustrate the distributions and the size of the corrections, the
differential distributions $d\sigma/d\Delta y_{t\bar t}$ are shown in
Fig.\ref{fig:rapidity-distr}  for 8 (left) and 14~TeV (right), considering only 
events with $M_{t\bar t}$ larger than 1~TeV in the former and 2~TeV in 
the latter case. The corresponding corrections are also displayed in
Fig.\ref{fig:rapidity-distr}. 
The pronounced peaking of the cross section for large
rapidity differences in Fig.\ref{fig:rapidity-distr} (top) is an obvious consequence of the
$t$-channel singularity, the enhanced negative corrections around
$\Delta y_{t\bar t}= 0$ in Fig.\ref{fig:rapidity-distr} (bottom) are a consequence of the
Sudakov condition $\hat s$ and $ |{\hat t}|\gg M_W^2$. Since the
distribution in $\Delta y_{t\bar t}$ is at the same time sensitive to
anomalous couplings, these could well be masked by the large radiative
corrections. 
\begin{figure}
  \begin{center}
    \vspace{1cm}
    \scalebox{0.55}{\input{./figures/figLO_DEta_Mttb1_LHC8}} 
    \hspace*{8mm}
    \scalebox{0.55}{\input{./figures/figLO_DEta_Mttb2_LHC14}} \\[12mm]
    \scalebox{0.55}{\input{./figures/figDEta_MttbCut_LHC8}} 
    \hspace*{8mm}
    \scalebox{0.55}{\input{./figures/figDEta_MttbCut_LHC14}} 

 \caption{\label{fig:rapidity-distr}%
Rapidity distributions with invariant mass cuts at leading order (upper plots) and relative weak corrections to
these distributions (lower plots) for LHC8 (left) and LHC14 (right).
}
     \end{center}
\end{figure}   

Let us at this point speculate about the combination of weak and
QCD corrections. Clearly, the evaluation of corrections of ${\cal
O}(\alpha_s\alpha)$ is out of reach in the foreseeable future. 
Thus, strictly speaking, both a multiplicative 
(of the form $(1+\delta_{QCD})(1+\delta_{W}))$ and an additive
(of the form $(1+\delta_{QCD}+\delta_{W}))$ treatment is equally
justified. The difference between the two assumptions can be
considered as an estimate of the theory uncertainty. It may be
usefull to devise a strategy, how to implement eventually the major part of
the combined corrections.
As mentioned in the beginning, QED and purely weak corrections can be
treated seperately in the present case. Furthermore, QED corrections are
small and the resulting uncertainty of combined QCD and QED terms even
smaller. In principle, by adjusting color coefficients, the recently
available two-loop QCD corrections could be employed to arrive at the
full combined QED and QCD results. Concerning the weak corrections, we
observe that a major part of the QCD corrections originates from
configurations involving soft and/or collinear emission. Let us then 
reconstruct the effective two-body kinematics by using the $t\bar t$ 
invariant mass as $\hat s$ and the scattering angle with respect to 
the beam direction, as defined in the $t\bar t$ rest frame as 
partonic scattering angle. Using this information would allow to apply
the weak correction factor which also depends on $\hat s$ 
and $\hat t$ only. It remains to be seen, to which extent this approach
can be implemented in currently available event generators. 

\section{The top-pair threshold region and the Yukawa coupling}
\label{sec-threshold}

As illustrated in Fig.~\ref{fig:partoncor}, the corrections for
top-pair production very close to threshold exhibit a significant
dependence on the mass of the Higgs boson.  In fact, both for quark
and gluon induced process the difference in the correction between a
light ($M_H=126~{\rm GeV}$) and a heavy ($M_H=1000~{\rm GeV}$) Higgs
boson amounts to about 5\%. This effect has been discussed in some
detail for pair production at an electron-positron collider
\cite{Grzadkowski:1986pm,Strassler:1990nw,Fadin:1990nz,Jezabek:1993tj,Harlander:1995dp}
and for quark-antiquark collisions \cite{Harlander:1995dp} and is
closely related to the well-known Sommerfeld rescattering corrections,
originally obtained in the framework of QED.  Similar considerations
are also applicable to gluon fusion \cite{Kuhn:2006vh}.

For a Yukawa potential induced by the Higgs exchange,
\be
V_Y(r)= - \kappa\; {1\over r}\,e^{-r/r_Y}\,\,{\rm with}\,\,
\kappa=\frac{g_Y^2}{4\pi}=\frac{\sqrt{2} G_F M^2_t}{4\pi} \approx 0.0337
\,\,{\rm and}\,\, r_Y=1/M_H,
\ee
the dominant correction evaluated directly at threshold 
is given by the factor $1+\kappa\frac{M_t}{M_H}$.
(The full result can be found in \cite{Grzadkowski:1986pm,Jezabek:1993tj}.) 
Indeed the difference of 5\% 
between the heavy and the light Higgs boson is well
consistent with this simple approximation. For quark-antiquark annihilation
the positive offset is shown in Fig.\ref{fig:partoncor} (left).
For gluon fusion the Yukawa
enhancement is partially masked by a negative contribution originating 
from the interference of the tree-level amplitude with the amplitude from
the triangle diagrams with $Z$ and $\chi_Z$ in the $s$-channel
(Fig.~\ref{fig:VirtualCor-gg}). 
The difference, however, between a heavy and a light Higgs boson
of about 5\% remains unchanged.

As evident from Fig.\ref{fig:partoncor}, the Yukawa enhancement is located in
the region close to threshold, with relative $t\bar t$-velocity 
$\beta$ less than $M_H/M_t$.
For the moment we consider the weak corrections as an overall
$\beta$-dependent factor which multiplies the complicated threshold
behaviour induced by the partly attractive, partly repulsive QCD
potential. (For QCD effects see e.g. \cite{Kiyo:2008bv} and 
references therein.) 
In principle the effect of a light Higgs exchange could be split into a
short range piece, which leads to a  $\beta$-independent correction
term, and a long-range piece, which can be absorbed by adding 
Yukawa and QCD potential. The energy dependence can then be obtained from a
Green's function treatment. This approach has been discussed in more
detail in \cite{Harlander:1995dp} for the cases of top production in 
electron-positron
and quark-antiquark annihilation. Since $r_Y$,
the characteristic lenght of the Yukawa potential, is still 
significantly smaller than $r_B$, the Bohr radius of the would-be toponium
ground state, 
\be
r_Y/r_B=(\frac{4}{3}\alpha_s \frac{M_t}{2})/M_H\approx 1/6,
\ee
the simple multiplicative treatment advocated above is sufficient for
the presently required level of precision.

As discussed above,
the impact on the total cross section from the variation of $M_H$ is
relatively small, less than one percent, both for the Tevatron and the
LHC. Differential distributions, however, are significantly
more sensitive to the Yukawa coupling. This is demonstrated in 
Figs.~\ref{fig:threshold-distr}, \ref{fig:threshold-distr-tev},
where the correction factors for the distribution with respect to 
$M_{t\bar t}$ are evaluated for the Tevatron, LHC8 and LHC14 in the
region close to threshold.
 
As expected from the previous discussion, differences 
around 5\% between the cases $M_H= 126~{\rm GeV}$ and 1~TeV are visible.
\begin{figure}
  \begin{center}
\vspace{1cm}
    \scalebox{0.55}{\input{./figures/figMttb_rel_LHC8_Thresh}} 
\hspace*{8mm}
    \scalebox{0.55}{\input{./figures/figMttb_rel_LHC14_Thresh}} 
 \caption{\label{fig:threshold-distr}%
Relative weak corrections for the mass distribution in the
framework of the SM assuming  $M_H=126~{\rm GeV}$ (solid blue curve) 
and 1000~GeV (dashed red curve), and for the case of an enhanced Yukawa
coupling   $g_Y=2 g^{SM}_Y$ with  $M_H=126~{\rm GeV}$ (dotted black curve). 
The two plots represent LHC8 and LHC14. 
}
     \end{center}
\end{figure}   
\begin{figure}
  \begin{center}
\vspace{1cm}
    \scalebox{0.7}{\input{./figures/figMttb_rel_Tevatron_Thresh}} 
 \caption{\label{fig:threshold-distr-tev}%
Same as Fig.\ref{fig:threshold-distr} but for the Tevatron.
}
     \end{center}
\end{figure}   
It remains to be seen, whether the experimental mass resolution and
normalization of the cross section will be sufficiently precise to pin
down the 5\%-effect and thus determine directly the Yukawa coupling
$g_Y$.  At the same time this approach requires a detailed theoretical
understanding of the QCD predictions for the threshold behaviour,
governed by the remnants of the bound states, as discussed in
\cite{Kiyo:2008bv}.  However, in any case this approach should allow
to provide an upper limit on modifications of $g_Y$ that might be
postulated in theories beyond the Standard Model.  Let us 
assume, for example, the case of an enhanced Yukawa coupling
$g_Y=2g_Y^{SM}$. This magnifies the Yukawa correction by a factor four
and implies an enhancement of the cross section close to threshold by
about 20\%. (See dashed curves in
Figs.~\ref{fig:threshold-distr}, \ref{fig:threshold-distr-tev}. Such an
energy dependent offset relative to the SM prediction should be
visible in Tevatron or LHC analyses.

\section{Outlook and conclusions}
\label{outlook}
A sizable data sample has been collected by LHC experiments at a
center-of-mass energy of 8~TeV, and the Higgs boson has been discovered
with a mass of about 126~GeV. In view of these developments an update of
the weak corrections to top quark pair production has been
presented. We demonstrate that these corrections start to become
important already for the 8~TeV run, if an experimental precision of 5
percent can be reached. This observation applies both for large
transverse momenta, say above 500~GeV, where negative corrections around
5\% are observed, and for top quark production close to threshold which
is enhanced by about 5\% due to the attractive Yukawa interaction. A
detailed study of the top-antitop spectrum close to threshold could,
therefore, determine the strength of the Yukawa coupling or, at
least provide interesting upper limits. We also investigate the
distribution of top and antitop with respect to their rapidity
difference $\Delta y_{t\bar t}$ for the subsample with large invariant
mass and observe marked distortions of order 8\% (LHC8) and 12\%
(LHC14). Clearly these effects might be misinterpreted as evidence for 
anomalous couplings and thus have to be well under control. Last not
least we indicate a possible approach for combining QCD and weak
corrections in the framework of a Monte Carlo generator.


{\bf Acknowledgments}\\[3mm]
The work of J.H.K. was supported by BMBF Project 05H12VKE.
Discussions with Jeannine Wagner-Kuhr on experimental issues are
gratefully acknowledged. The work of A.S. has been partly supported by
the German Ministry of Education and Research (BMBF) under contract
no. 05H12WWE.





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\end{document}


