\documentclass[a4paper]{PoS}
\usepackage{cite}
\title{Electroweak corrections in top-quark physics (1)}

\ShortTitle{Electroweak corrections in top-quark physics (1)}

\author{Johann K\"uhn$^a$, \speaker{Peter Uwer}$^{b}$\\%
\llap{$^a$}Institut f\"ur Theoretische Teilchenphysik,
Karlsruhe Institute of Technology (KIT)\\
76128 Karlsruhe, Germany\\\\
\llap{$^b$}Institut f\"ur Physik, Humboldt-Universit\"at zu Berlin,\\
12489 Berlin, Germany\\
E-mail: \email{Johann.Kuehn@KIT.edu}, \email{Peter.Uwer@physik.HU-Berlin.de}}

\abstract{Electroweak corrections for inclusive top-quark pair
  production are reviewed and updated results for the weak corrections
  taking into account the recently measured Higgs mass are
  presented.}

\FullConference{8th International Workshop on Top Quark Physics\\
  14-18 September, 2015\\
  Ischia, Italy}

\graphicspath{{figs/}}
\def\as{\ensuremath{\alpha_s}}
\def\Fig#1{Fig.~\ref{#1}}
\def\Ref#1{Ref.~\cite{#1}}
\def\Refs#1{Refs.~\cite{#1}}
\begin{document}

\section{Introduction}
With the start of the LHC significant progress has been made
concerning top-quark physics. The experimental precision of the
inclusive cross section for top-quark pair production is now at the
level of a few per cent. At the same time also theoretical predictions
within QCD have been significantly improved by the calculation of the
next-to-next-to-leading order QCD corrections
\cite{Czakon:2013goa,Czakon:2015owf}. The theoretical uncertainties
due to uncalculated higher orders, uncertainties of the parton
distribution functions (PDFs), and uncertainties of the QCD coupling
constant \as\ as well as the top-quark mass lead to a total
uncertainty of about 3\%. At this level of accuracy electroweak
corrections need to be considered as well. From a naive power counting
one expects that $\alpha\as$ corrections, with $\alpha$ denoting the
electromagnetic coupling, can lead to corrections of the order of
10\%, while the pure electroweak corrections of order $\alpha^2$ and
the mixed corrections of order $\alpha\as^2$ may give percentage
corrections. A more detailed analysis shows that because of colour
only very few and kinematically suppressed partonic channels
contribute at order $\alpha\alpha_s$ (e.g. $\gamma g\to t\bar t$,
$b\bar b \to t\bar t$) leading to a further PDF suppression. In case
of the pure electroweak corrections ($\alpha^2$) the quark charges and
the PDF's lead to a further suppression leaving the mixed corrections
of order $\alpha\as^2$ as the dominant contribution. They can be
further `enhanced' by large Sudakov logarithms. Weak Sudakov
logarithms can lead to large negative contributions at large momentum
transfer.  Inclusive quantities are typically dominated by the
threshold region, and are thus less sensitive to Sudakov type
corrections. However, differential cross sections at large momentum
transfer can be significantly modified. This can affect new physics
searches if not taken into account: the possible raise of the cross
section due to a (hypothetical) new resonance could be shadowed by the
negative weak corrections if not considered in the analysis.
Similarly, using spin observables for new physics searches, weak
corrections can play an important role since spin observables which
are zero in QCD due to parity invariance are not protected to receive
weak corrections.  At the production threshold of the top-quark pair
the attractive top-Higgs Yukawa interaction leads to an enhancement of
the cross section. A detailed study of this effect may provide useful
information to constrain the top-quark Yukawa coupling.  For future
experimental investigations it is thus very important to take
electroweak corrections into account. Since QED corrections form a
gauge invariant contribution,
traditionally weak \cite{Beenakker:1993yr,Kuhn:2005it,%
  Bernreuther:2005is,Kuhn:2006vh,Bernreuther:2006vg,%
  Kuhn:2013zoa} and QED \cite{Hollik:2007sw} 
corrections were studied separately. 
In the next section we will briefly discuss the impact of the weak
corrections using up to date input for the top-quark mass and the
Higgs boson mass. In section \ref{sec:QED} we review the status of the QED
corrections. It is worth noting that similar to QCD corrections
also  electroweak corrections have recently witnessed tremendous progress
concerning the automated evaluation (see for example 
\Refs{Chiesa:2015mya,Pagani:2012xg,Kallweit:2014xda,Frixione:2014qaa,%
Campbell:2015vua}). 

\section{Impact of weak corrections}
\begin{figure}[htbp]
  \begin{center}
    \leavevmode
    \includegraphics[width=0.49\textwidth]{Fig10}
    \includegraphics[width=0.49\textwidth]{Fig14c}
    \caption{Impact of the weak correction on the inclusive cross
    section for top-quark pair production as a function of the center of
    mass energy (left). Threshold effects due to attractive top--Higgs
    Yukawa interaction (right).}
    \label{fig:CrossSection}
  \end{center}
\end{figure}
For details on the calculation of the weak corrections we refer to 
\Refs{Beenakker:1993yr,Kuhn:2005it,%
  Bernreuther:2005is,Kuhn:2006vh,Bernreuther:2006vg,%
  Kuhn:2013zoa}. 
In \Ref{Kuhn:2013zoa} the numerical results have been updated
using the recently measured value of the Higgs mass.
In \Fig{fig:CrossSection} (left) the corrections to the inclusive
cross section is shown as function of the cms energy. Consistent with
the naive expectations the relative corrections are of the order of
$-1.8$\% at 7 TeV reaching slightly less than $-2.1$\% at 30 TeV. For
comparison, results using a Higgs mass of 1 TeV are shown, illustrating
that the Higgs contribution leads to a positive correction to the cross
section of about 0.5\%. Indeed a naive rescaling of the top-quark Yukawa
coupling by a factor of 4 leads to a cross section which is about
2 \% larger. The right plot in \Fig{fig:CrossSection} shows the
sensitivity of the threshold region on the top-quark Yukawa
interaction. At the threshold, the Higgs corrections lead to a positive
correction to the cross section of about 5\%. A two times larger
top-quark Yukawa coupling would thus lead to 20\% larger
corrections. Experimental studies of the threshold region can thus provide
important constraints on the top-quark Yukawa coupling. We also note that
the size of the effects is perfectly consistent with the
theoretical estimate based on the Sommerfeld enhancement, for details
we refer to \Ref{Kuhn:2013zoa}.
\begin{figure}[htbp]
  \begin{center}
    \leavevmode
    \includegraphics[width=0.49\textwidth]{Fig12e}
    \includegraphics[width=0.49\textwidth]{Fig12f}
    \caption{Sudakov suppression at large momentum transfer.}
    \label{fig:Distributions}
  \end{center}
\end{figure}
In \Fig{fig:Distributions} the Sudakov suppression at large momentum
transfer is illustrated.  At a transverse momentum of 1 TeV the
Sudakov logarithms lead to a relative suppression of about 10\%. Since
the structure of the Sudakov logarithms is known one may approximate
the weak corrections by the known Sudakov logarithms. Extending the
Monte Carlo for FeMto barn processes (MCFM) to incorporate also weak
corrections for top-quark pair production, this has been studied
recently in \Ref{Campbell:2015vua}. While the approximation works
reasonably well for the $q\bar q$ partonic channel, the naive
approximation fails for the $gg$ channel. Consistent with the findings
of \Ref{Kuhn:2013zoa} this has been traced back in
\Ref{Campbell:2015vua} to the more complicated angular dependence in
the $gg$ channel. As pointed out in \Ref{Kuhn:2013zoa} the Sudakov
limit in gluon fusion requires the partonic Mandelstam variable $\hat
t$ or $\hat u$ to be large. Indeed it has been shown in
\Ref{Campbell:2015vua} that the Sudakov
approximation can be improved by an additional cut on the top-quark
rapidity.  The origin of the Sudakov logarithms are soft- and
collinear configurations which are regulated by the gauge boson
masses. For massless gauge bosons the corresponding singularities
would require the inclusion of real emission processes to cancel the
singularities. In case of massive gauge bosons it is argued that real
emission processes lead to a distinguishable finale state. The
combination of the real and virtual corrections which could lead to a
partial cancellation of the would-be singularities is thus not
performed. In \Ref{Baur:2006sn} it has been pointed out that at very
high energies, the clear separation between states with and without
real emission might no longer be possible thus requiring at least a
partial combination of the two. The positive contribution from the
real corrections can than partially compensate the negative Sudakov
logarithms. With the high energetic events available at the LHC Run II
the analysis of \Ref{Baur:2006sn} should be updated using an
up-to-date setup. To facilitate the experimental analysis the Sudakov
enhanced corrections have been implemented recently in an updated
version of the HATHOR program \cite{Aliev:2010zk}. In
addition, \Ref{Kuhn:2013zoa} also describes a prescription to combine
the weak corrections with parton shower results.

\section{QED corrections}
\label{sec:QED}
The QED corrections were first calculated in \Ref{Hollik:2007sw}. The
results for the inclusive cross section as evaluated in \Ref{Hollik:2007sw}
are shown in Tab.~\ref{tab:QED}. In \Ref{Hollik:2007sw} the MRST2004
PDF set has been used and the factorization scale $\mu$ is set to twice the
top-quark mass $m_t$. While the overall size of the corrections agree well
with the naive estimates given in the introduction, the contribution
from $g\gamma$ turns out to be quite large, taking into account that
the photon PDF should be supressed by an additional factor $\alpha$.
Furthermore, a significant cancellation between the quark--anti-quark
initiated processes and the $g\gamma, gg$ channels can be observed. In
case of the $g\gamma$ channel the price of an additional factor
$\alpha$ in the photon PDF seems to be compensated by the large $qg$
luminosity.
\begin{table}[htbp]
  \begin{center}
    \leavevmode
    \begin{tabular}[h]{lll}
      \hline\hline \multicolumn{3}{c}{$\sigma_{\rm tot}$ without cuts
      [pb]}\\
      Process & Born & Correction\\ \hline
      $u\bar u$ & \phantom{0}34.25 & $-$1.41 \\
      $d\bar d$ & \phantom{0}21.61 & $-$0.228 \\
      $s\bar s$ & \phantom{00}4.682 & $-$0.0410 \\
      $c\bar c$ & \phantom{00}2.075 & $-$0.0762 \\
      $gg$      & 407.8 & \phantom{$-$}2.08 \\
      $g\gamma$ &       & \phantom{$-$}4.45 \\
      $pp$      & 470.4 & \phantom{$-$}4.78 \\ \hline\hline
    \end{tabular}
    \caption{QED corrections to inclusive top-quark pair production 
      \cite{Hollik:2007sw}.}
    \label{tab:QED}
  \end{center}
\end{table}
In \Ref{Hollik:2007sw} also distributions are
investigated. For the transverse momentum distribution, the corrections
are positive below 500 GeV. Above 500 GeV the corrections are negative and
amount to $-$6\% at a transverse momentum of 3~TeV. In
\Ref{Hollik:2007sw} it is argued that the negative corrections at
large transverse momentum are due to logarithmically enhanced final
state corrections. Given the progress concerning the PDF extraction it
will be important for future analysis to study the impact of the
recently published NNPDF and CTEQ14QED PDF sets. In addition, in view
of a possible combination of the QED corrections with the QCD results,
it would be useful to update the results to the commonly used scale
setting $\mu=m_t$. First results in this direction have been prepared
by Pagani and Zaro using \Ref{Frixione:2014qaa} and are shown in \Fig{fig:QED}. 
\begin{figure}[htbp]
  \begin{center}
    \leavevmode
    \includegraphics[width=0.55\textwidth]{TT_top_pt_7tevtev}
    \caption{Comparison of QED and weak corrections, 
      prepared by Davide Pagani and Marco Zaro using 
      MadGraph5\char`_aMC@NLO
      \cite{Frixione:2014qaa}.}
    \label{fig:QED}
  \end{center}
\end{figure}
As can be seen from \Fig{fig:QED} the QED corrections lead to a
partial compensation of the weak corrections, such that the sum of
the two is almost zero, however, at the price of introducing a large
PDF uncertainty which is shown as a band in the lowest plot. As PDF
set NNPDF2.3QED has been used. Further studies are required to compare
with CTEQ14QED and investigate the behavior at significantly larger
momentum transfer.

\section{Conclusion} 
The experimental precision of cross section measurements for top-quark
pair production currently achieved requires the inclusion of
electroweak corrections.The weak corrections to the inclusive cross
section amount to about $-$2\% which is of the order of the
theoretical uncertainty quoted for the NNLO QCD corrections. At
threshold the enhancement due to the attractive Yukawa interaction may
provide a mean to constrain the top-quark Yukawa coupling. At large
momentum transfer the corrections---if not taken into account---may
hide a possible raise of the cross section due to new physics
resonances at large invariant mass. The QED corrections can also lead
to percentage corrections. However, recent studies indicate that the
PDF's introduce a significant uncertainty.  \bibliographystyle{JHEP}

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