\documentclass{PoS}

\title{%
Top-quark pair production close to threshold at hadron colliders: an update
}

\ShortTitle{%
Top-quark pair production close to threshold
}


\author{\speaker{Johann K\"uhn}\\%
        Institut f\"ur Theoretische Teilchenphysik, 
        Karlsruhe Institute of Technology (KIT)\\
        76128 Karlsruhe, Germany\\
        E-Mail: \email{Johann.Kuehn@KIT.edu}}

\author{Peter Uwer\\
        Institut f\"ur Physik, Humboldt-Universit\"at zu Berlin,\\
        12489 Berlin, Germany\\
        E-Mail: \email{Peter.Uwer@physik.HU-Berlin.de} }

\abstract{%
QCD and weak corrections for top-antitop production at the LHC 
are presented with emphasis on their behaviour in the threshold region.
}

\FullConference{12th International Symposium on Radiative Corrections
(Radcor 2015) and Loopfest XIV (Radiative Corrections for the LHC and 
Future Colliders)\\
15-19 June 2015\\
UCLA Department of Physics \& Astronomy, Los Angeles, CA, USA}

\def\mur{\mu_r}
\def\muf{\mu_f}
\def\Im{\mbox{Im}}
\def\cA{{\cal A}}
\def\cB{{\cal B}}
\def\cAc{{\cal A}_{c}}
\def\cAnc{{\cal A}_{nc}}
\def\cBc{{\cal B}_{c}}
\def\cBnc{{\cal B}_{nc}}
\def\cCh{{{\cal C}_h}}
\def\GeV{\mbox{GeV}}
\def\TeV{\mbox{TeV}}
\def\pb{\mbox{pb}}
\begin{document}  

\section{Introduction}
%
Top-quark pair production close to threshold is considered as ideal option
for an electron-positron collider. However, although technically more 
difficult, a number of closely related questions can also be investigated at
a hadron collider like the LHC. Concerning QCD, it may be possible to
exploit the subtle differences between production of $t\bar t$ colour 
singlet and colour octet configurations, respectively, and thus investigate
the subtleties of top-antitop interactions. In principle this might allow 
to disentangle attractive and thus closely bound from repulsive
$t\bar t$ systems. The impact of these effects on various differential
distributions is investigated. 

A second, equally important issue are the
weak corrections to $t\bar t$ production at a hadron
collider, and their impact on the total cross section 
for top-quark pair production and on differential distributions.
In particular, we investigate the influence of weak corrections
on transverse momentum and rapidity distributions and the interplay
between Higgs boson exchange and the Yukawa potential between top 
and antitop. We observe a pronounced sensitivity of the threshold 
cross section on an enhanced Yukawa coupling.

\section{QCD corrections in the $t\bar t$ threshold region}
%
QCD corrections for $t\bar t$ production in the threshold region have 
reached a remarkable level of precision, as far as electron-positron
annihilation is concerned. The first papers based on leading order 
Greens functions were written more than twenty years ago
\cite{fadinkhoze,peskin,jezabekkuhn1,sumino}.
In the meantime these calculations were pushed to 
next-to-next-to-next-to leading order (N$^3$LO) \cite{Beneke:2015kwa},
leading to a precision which will allow to even get access to subtle effects 
like those originating from the Yukawa interaction between the top
and the antitop quark and which is particularly strong in the threshold 
region \cite{harlanderkuhn, beneke}. In total a precision of the top-mass
determination around 50~MeV may well be within reach. 

For the time being, however, experiment is limited to hadron colliders.
Top-mass determinations have been made originally by the Tevatron 
and more recently by the LHC with a cms-energy of up to 13~TeV.
Abundant top-quark production has been observed.
However, the precision of the top-mass determination is still limited
to about 1~GeV. This is a natural consequence of the fact 
that the kinematical reconstruction is based on decays of colour
triplet quarks with a mass, which is presumably close, but not identical
to the pole mass.

The lowest order processes for hadronic top-pair production are
\begin{equation}
q+\bar{q}\to g^*\to t+\bar t
\end{equation}
with $t\bar t$ in a repulsive colour octet configuration, and
\begin{equation}
g+g\to t + \bar t
\end{equation}
with $t\bar t$ either in a (repulsive) color octet or an (attractive)
colour singlet configuration. In next-to-leading order (NLO) the
QCD potential can be cast into the following form 
\begin{equation}
\tilde V^{[1,8]}(\vec q) = -\frac{ 4\pi\alpha_s(\mur)\, C^{[1,8]}}{ {\vec q}\,^2}\,
  \bigg[1+\frac{\alpha_s(\mur)}{4\pi}
    \bigg( \beta_0 \, \ln\frac{\mur^2}{{\vec q}\,^2} +a_1
    \bigg)
    \bigg]\,,
\end{equation}
with $C^{[1]}=C_F=4/3$ and $C^{[8]}=C_F-C_A/2 = -1/6$ as colour 
coefficients and $a_1=(31/9) C_A - (20/9)T_F n_f$.

It is well known that the relatively large width of the top quark 
of around 1.36~GeV \cite{jezabekkuhn2} is comparable to the Rydberg constant
of the $t\bar t$-system, $(C^{[1]}\alpha_s)^2 m_t/4 \approx 1.5$~GeV.
This leads to a merging of the higher S-states and a moderate 
enhancement of the 1S peak as far as the $t\bar t$ colour singlet
configuration is concerned. The repulsive color octet combination,
on the other hand, is significantly suppressed (Fig.1).
\begin{figure}[htbp]
  \begin{center}
    \leavevmode
    \includegraphics[width=0.55\textwidth]{xs1lhc14}
    \caption{Invariant mass distribution for the leading subprocesses.}
    \label{fig:ThresholdLeadingChannels}
  \end{center}
\end{figure}
Including QCD corrections of order $\alpha_s$, the cross section of the
reaction $i+j\to t+\bar t (+X)$ can, at parton level, be cast
into the following form  \cite{Kiyo:2008bv}
\begin{equation}
  M\frac{d\hat \sigma_{ij\to t\bar t}}{d M}(\hat s, M^2,\muf^2) 
  = F_{ij\to T}(\hat{s},M^2,\muf^2)\,
\, \frac{1}{m_t^2}\,{\rm Im} \,G^{[1,8]}(M+i\Gamma_t)\, .
\end{equation}
The first factor, $F_{ij\to t\bar t}$, can be evaluated purely 
perturbatively 
\begin{eqnarray}
  F_{ij \rightarrow T}(\hat s, M^2,\muf^2)
  &=&
  {\cal N}_{ij\to t\bar t} \,
  \frac{\pi^2\,\alpha_s^2(\mur)}{3\hat s}\,
  \left( 1 + \frac{\alpha_s(\mur)}{\pi}\, {\cCh} \right)
\nonumber \\
&& \hspace{-0.4cm}
  \times
  \left[\, \delta_{ij \rightarrow t\bar t} \, \delta (1-z)
    +
    \frac{\alpha_s(\mur)}{\pi}\,
    \bigg( \cAc(z) + \cAnc(z) \bigg)\,
  \right]\,.
\label{eq:F}
\end{eqnarray}
and depends obviously on the specific initial state configuration, 
characterized 
by $i$ and $j$. The second factor, $\Im G^{[1]}$ or $\Im G^{[8]}$,
characterizes the final state, which may appear in a color singlet
or octet configuration. Note, that also reactions 
like $qg\to t\bar t q$ contribute in NLO. Let us also note that
the factor ${\cal N}_{ij\to t\bar t}$ characterizes the normalisation, $\cCh$ 
the hard virtual correction, ${\cal A}_c$ the collinear part of the
real corrections
and ${\cal A}_{nc}$ non-collinear real emission. The explicite
analytic results for all these quantities can be found in \cite{Kiyo:2008bv}.
The complete prediction for producing the dominant $t\bar t$ states
in the threshold region are shown in Fig.~\ref{fig:ThresholdLeadingChannels}.
Here ${}^3S^{[8]}_1$ stands for the spin triplet, color octet $S$-wave state,
and ${}^1S^{[1]}_0$ for the spin singlet, color singlet $S$-wave state.
\begin{figure}[htbp]
  \begin{center}
    \leavevmode
    \includegraphics[width=0.55\textwidth]{xs2lhc14}
    \caption{Invariant mass distribution close to threshold including
      all partonic production channels.}
    \label{fig:ThresholdAllChannels}
  \end{center}
\end{figure}
The corresponding result, including, however, all production channels,
is shown in Fig.~\ref{fig:ThresholdAllChannels}.
This analysis is strictly speaking, only valid in the threshold region.
Nevertheless it can be compared to a NLO calculation which completely
ignores bound-state effects (Fig.~\ref{fig:ResummedVsFixedOrder})
\begin{figure}[htbp]
  \begin{center}
    \leavevmode
     \includegraphics[width=0.55\textwidth]{mtt_threshold}
    \caption{Comparison of threshold resummed results with fixed 
      order QCD predictions.}
    \label{fig:ResummedVsFixedOrder}
  \end{center}
\end{figure}
Good qualitative agreement is observed from threshold up to several
tens of GeV above. However, the parton cross section without
bound-state effects obviously vanishes below the nominal threshold $2
m_t= 344.8 \GeV \pm {\cal O}(1 \GeV)$, and the subtle effects leading to
a difference between color singlet and and color octet production are
obviously absent.

To summarize: The differential distribution for top-quark 
production has been calculated which differs significantly from the
one based on stable top quarks and which ignores the differences 
between color singlet and color octet bound-states. The additional 
threshold enhancement of around $10 \pb$ is small compared to the
total cross section of roughly $800 \pb$ (for 14~TeV) but may, 
nevertheless, be noticeable, if one investigates specifically
the threshold region. Note, that these subtle threshold effects 
are ignored in present theory predictions, even those extending to
NNLO.

\section{Weak corrections to $t\bar t$ production}
%
With increasing precision of $t\bar t$ measurements at hadron 
colliders weak corrections will start to become relevant.
The leading QCD Born amplitude for $q\bar q \to t\bar t$
is of order $g_s^2$, the leading weak amplitude of order $g_w^2$.
Hence one might expect the dominant weak correction to arise 
from the interference term and thus to be of order $g_s^2 g_w^2$.
However,  upon closer inspection one finds that the first correction is 
in fact of order $g_s^4 g_w^2$ and thus remains to be relatively small,
i.e. typically about one percent. There are, however, two regions where
an enhancement can be expected \cite{uwerscharf,oldpaper}:

i.) The region of very large $\hat s$, say above 1~TeV, and similarly 
large momentum transfer $-\hat t$, where sizable negative corrections   
can be observed, a consequence of the large negative ``Sudakov'' 
logarithms. For quark-antiquark induced processes these are about a 
factor two larger than those for gluon-induced processes, a consequence
of the fact that for quark induced processes weak charges are present
both in the initial and the final state.

ii.) Very close to the $t\bar t$  threshold, say between 340~GeV
and 400~GeV the attractive force between $t$ and $\bar t$, induced
by the Higgs-boson exchange, plays an important role. This might 
offer the unique possibility of measuring the Higgs coupling to
top quarks, at least if a precision of six percent or better can
be achieved.

Let us demonstrate these effects in the following. The total 
cross section for $t\bar t$ production at the LHC depends only marginally
on $m_H$ with a relative weak correction varying between $-1.8\%$ and
$-2.3\%$ for $m_H$ between 120~GeV and 1000~GeV. The effect is 
significantly enhanced, once events are selected with large
$t\bar t$ masses (e.g. $M_{t\bar t}> 2 \TeV$ and large transverse
momenta. Let us therefore discuss the implementations of these corrections
on individual distributions. The total cross section receives a correction
of around 1.8\% for a Higgs boson mass of roughly 120~GeV. The 
size of the corrections to the $M_{t\bar t}$- and $p_t$-distributions,
that can be determined by experiment, increases with increasing transverse
momenta. The corrections are always negative and amount to ten percent 
and more, if the invariant mass of the $t\bar t$ system reaches 
2~TeV and if events with large transverse momenta of the top quark are
required (Fig.~\ref{fig:EWCorrections}).
\begin{figure}[htbp]
  \begin{center}
    \leavevmode
    \includegraphics[width=0.49\textwidth]{Fig12e}
    \includegraphics[width=0.49\textwidth]{Fig12f}
    \caption{Impact of weak corrections on differential distributions at large momentum transfer.}
    \label{fig:EWCorrections}
  \end{center}
\end{figure}

The other kinematic configuration of interest  is the region close to
threshold, say between $2m_t\approx 350$~GeV and 400~GeV. In this case 
the dominant effect is induced by the Yukawa potential
\begin{equation}  
  V_Y(r) = - \frac{\kappa}{r} e^{-r/r_Y}
\end{equation} 
with 
\begin{equation}
\kappa = \frac{g_Y^2}{4\pi} = \frac{\sqrt{2} G_F m_t^2}{4\pi} \approx 0.034
\end{equation}
and $r_Y= 1/M_H$. The range of the potential $r_Y$ 
is still small compared to the size of the bound state
$r_{\rm Bohr} = \langle \frac{4}{3}\alpha_s \frac{M_t}{2} \rangle^{-1}$ with
$r_Y/r_{\rm Bohr} \approx  1/6$. This leads to an enhancement of roughly 
$(1+\kappa \frac{m_t}{m_H}) \approx 1.05$. This effect vanishes quickly
with increasing $\hat s$ and is practically absent for $\hat s = 400 \GeV$.
\begin{figure}[htbp]
  \begin{center}
    \leavevmode
    \includegraphics[width=0.49\textwidth]{Fig14c}
    \caption{Threshold corrections due to attractive Yukawa interaction.}
    \label{fig:EWThreshold}
  \end{center}
\end{figure}
(Fig.~\ref{fig:EWThreshold})

The Yukawa coupling that is employed in this context is fixed to the 
Standard model value $g_Y^2 = \sqrt{2} G_F m_t^2$.
It remains to be seen, whether the experimental mass resolution and
the normalization of the cross section will be sufficiently precise
to pin down this 5\% effect and thus determine directly the 
Yukawa coupling. 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 \cite{uwerscharf}.
The relevance of such a limit can be best seen from Fig.~\ref{fig:EWThreshold}
where an enhancement of the top-Higgs Yukawa coupling
by a factor two has been assumed, leading to an increase 
of the correction by a factor four and thus to a significant enhancement
of the cross section in the threshold region.

\section{Summary}

Top-quark pair production in the threshold region will be a unique
laboratory to pin down subtle effects related to mass generation
and spontaneous symmetry breaking. 
This holds true for hadronic interactions,
where the difference between attractive and repulsive 
top-antitop interaction may lead to visible modifications of the 
cross section. Weak interactions, on the other hand, in particular
the exchange of Higgs bosons between top and antitop quarks,
will lead to a modification of the cross section amounting to 
several percent in the Standard Model. These effects are particularly
sensitive to the Yukawa coupling between the top quark and the Higgs boson
and could lead to a non-trivial limit on the strength of this until
now quite elusive coupling.
%
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%
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\end{document}
