In this section, we discuss the PDF information that can be obtained from top quark
pair production measurements, and at the end we also review the constraints
can could potentially be derived from single top quark production.

\subsubsection*{PDF sensitivity}\label{sec:datatheory.top.sensitivity}

The production of top quark pairs at the LHC is dominated by the
gluon-gluon luminosity, which represents around 85\% of the total cross-section.\footnote{Note
that this statement is not true at the lower values of $\sqrt{s}$ of the Tevatron,
where top quark pair production is dominated instead by quark-antiquark annihilation.}
%
Therefore, provided that other sources of theoretical uncertainties
such as missing higher orders and the values of the top mass $m_t$ can
be kept under control, including top quark production data into the global
PDF fit has the potential to constrain the gluon in the large-$x$ region,
which is currently affected by large uncertainties.

To illustrate the kinematical sensitivity of
top quark pair production to the gluon,
in Fig.~ \ref{fig:topdiffcorr}
we show the correlation coefficient $\rho\lc g(x,Q), d\sigma\rc$
between the gluon
   PDF at $Q=100$ GeV and the theory predictions for
   the differential distributions in $y_{t\bar{t}}$
    and $m_{t\bar{t}}$ at $\sqrt{s}=8$ TeV,
   as a function of $x$.
   %
   In this comparison, each curve corresponds to specific measurement bin.
   %
   The interpretation of Fig.~\ref{fig:topdiffcorr} is that the higher the absolute value of the correlation coefficient, the
   bigger the sensitivity to the gluon for those specific values of $x$.
   %
   We observe that this sensitivity is high for values of $x$ up to $x\simeq 0.6-0.7$, beyond
   the reach of other processes sensitive to the gluon, such as inclusive jet production.
   %
   Moreover, it is important
   to emphasize that the availability of differential distributions significantly extends
   the kinematical coverage beyond that provided by the total inclusive cross sections.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.80]{3-datatheory/plots/topdiffcorrelation.pdf}
   \caption{\small The correlation coefficient between the gluon
   PDF at $Q=100$ GeV and the theory predictions for
   the absolute differential distributions in $y_{t\bar{t}}$
   (left) and $m_{t\bar{t}}$ (right plot) at $\sqrt{s}=8$ TeV,
   as a function of $x$.
   %
   Each curve corresponds to a specific measurement bin.
   %
   The higher the absolute value of the correlation coefficient, the
   bigger the sensitivity to the gluon in those specific values of $x$.
    \label{fig:topdiffcorr}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\subsubsection*{Experimental data}\label{sec:datatheory.top.data}

The available data on top quark pair production are summarised below:

\begin{itemize}
\item Earlier measurements, presented at the total cross section level, have been performed first at the Tevatron~\cite{Aaltonen:2013wca}
and then by ATLAS and CMS~\cite{ATLAS:2012aa,ATLAS:2011xha,TheATLAScollaboration:2013dja,
Chatrchyan:2013faa,Chatrchyan:2012bra,Chatrchyan:2012ria} at 7, 8, and 13 TeV.

\item Single-inclusive differential distributions of top quark pair production
have been presented by ATLAS~\cite{Aad:2015mbv} and CMS~\cite{Khachatryan:2015oqa}.
%
These include measurements both at the level of top-level observables extrapolated
to the full phase space ($p_T^t$, $y_{t\bar{t}}$, $m_{t\bar{t}}$), as well as at the level
of observables contributed in terms of directly observable quantities
(charged lepton $p_T$ and rapidity, $b$-tagged jet kinematics etc).

\item Double differential distributions for top quark pair production may also be performed, as illustrated by the recent CMS measurement~\cite{Sirunyan:2017azo}
of normalized double differential distributions, {\it i.e.}, as a function
of $p_{t\bar{t}}$ and $m_{t\bar{t}}$.

\end{itemize}

The differential measurements are often presented both
as absolute distributions as well as normalized to the total fiducial cross-section,
in order to benefit from a number of cancellations between experimental
systematic uncertainties.

\subsubsection*{Theoretical calculations and tools}\label{sec:datatheory.top.theory}

The NNLO QCD calculation of the total $t\bar{t}$ production cross section has been available
since 2013~\cite{Czakon:2013goa,Czakon:2012pz,Baernreuther:2012ws,Czakon:2011xx}, including
the resummation of logarithmically enhanced threshold corrections up to NNLL~\cite{Cacciari:2011hy}.
%
More recently, the full NNLO corrections to the single inclusive distributions
in top quark pair production have been computed~\cite{Czakon:2016dgf,Czakon:2015owf,Czakon:2014xsa}.
%
Differential NNLO results are available for the rapidity of the top quark and the top-pair 
system, $y_t$ and $y_{t\bar{t}}$, the transverse 
momentum of the top quark, $p_T^t$, and the invariant mass of the top-pair
pair $m_{t\bar{t}}$, though not for other variables such as $p_T^{t\bar{t}}$ since
these vanish at leading order.

Moreover, when differential distributions probe the TeV region, electroweak corrections
(including taking into account photon-initiated processes,
see Sect.~\ref{sec:QED.photon}) also become relevant, and need to be included
in the theoretical calculations.
%
In~\cite{Czakon:2017wor} (see also~\cite{Pagani:2016caq}),
the NNLO QCD calculations were combined with the state--of--the--art NLO
EW corrections, in the latter case including not only the $\mathcal{O}\lp \alpha_s^2
\alpha\rp$ but also the $\mathcal{O}\lp \alpha_s\alpha^2\rp$
and $\mathcal{O}\lp \alpha^3\rp$ contributions.
%
This study showed that an accurate description of the tails of the kinematical
distributions, such as the high-$p_{T}^t$ and high-$m_{t\bar{t}}$ regions,
must include NLO EW corrections.

An important limitation of the calculations discussed above is that they are restricted to stable
top quarks.
%
On the other hand, when experimental measurements
are presented at the top quark level, they are  extrapolated from the fiducial cross-sections
using some theoretical model, thus possibly biasing the result by an amount
which is difficult to quantify.
%
Ideally, one would like a fully differential calculation with NNLO corrections
included both for production and decay, in order to directly compare
with experimentally observable quantities.
%
An important milestone in this respect
was the recent calculation of top-quark pair-production and decay~\cite{Gao:2017goi} which
allows providing predictions for observables constructed from top-quark leptonic and b-tagged jet final states,
based on an approximation to the exact NNLO corrections to  production and exact NNLO corrections
to the decay.

Concerning the tools for the inclusion of top quark differential data into PDF fits,
there exist two basic approaches.
%
The first one is based on computing {\tt APPLgrids} for the NLO calculation using
either {\tt MCFM} or {\tt Sherpa} (see also Sect.~\ref{subsec:fast}), and then supplementing these with the
NNLO/NLO bin-by-bin $K$-factors from~\cite{Czakon:2016dgf,Czakon:2015owf}.
%
An improved strategy has been made feasible by the recent availability
of {\tt FastNLO} tables~\cite{Czakon:2017dip} that allow the efficient calculation
of NNLO top quark pair distributions for arbitrary PDF sets and input $\alpha_s(m_Z)$ values.
%
The latter option provides a more precise evaluation of the PDF--dependent NNLO corrections, although as shown explicitly in~\cite{Czakon:2016olj}
the dependence of these $K$-factors on the PDF set is very small.

\subsubsection*{Impact on PDFs}\label{sec:datatheory.top.impact}

The availability of the NNLO calculation of the total cross-sections
for top quark pair production 
has made it possible to include top quark data
from the Tevatron and the LHC consistently into a
NNLO PDF fit.
%
By applying Bayesian reweighting to NNPDF2.3, it was shown in Ref.~\cite{Czakon:2013tha}
that top quark data could reduce the PDF uncertainties in the large-$x$
gluon by up to 20\% for $x\simeq 0.2$ (see also previous related work in~\cite{Beneke:2012wb}).
%
Several other global fits, such as ABMP16 and MMHT14, also include total
$t\bar{t}$ cross-sections in their baseline fits.
%
While these results provided an encouraging indication of the PDF constraining potential
of $t\bar{t}$ production, the full exploitation of this potential clearly required
the use of differential distributions.

The impact of the $\sqrt{s}=8$ TeV top quark
pair differential data from ATLAS and CMS at 8 TeV
on the NNPDF3.0 analysis has been quantified in~\cite{Czakon:2016olj}.
%
Here, it was shown that the constraints on the large-$x$ gluon were at this point competitive with
those provided by inclusive jet production, despite the
smaller number of experimental data points. See also~\cite{Guzzi:2014wia} for
related work based on approximate NNLO calculations.
%
An important result of the investigations of~\cite{Czakon:2016olj} was that the constraints from the
normalized distributions were in general superior to those from their absolute
counterparts, most likely because of the cancellation of systematic uncertainties
that takes place in this case.
%
In addition, top quark differential distributions at 8 TeV from the LHC have been included
in the recent NNPDF3.1 global analysis, and other groups have also studied the impact
of this data into their PDF fits in a preliminary form.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.45]{3-datatheory/plots/ttbar-impact.pdf}
   \caption{\small Left plot: the impact of the LHC 7 and 8 TeV inclusive
   top-quark pair cross-section data on the large-$x$ gluon of
   NNPDF2.3~\cite{Czakon:2013tha}.
   %
   Right plot: the impact of the LHC 8 TeV differential distributions in
    top-quark pair production on the $gg$ luminosity~\cite{Czakon:2016olj}, compared with a baseline fit
    based on the NNPDF3.0 global analysis without the jet data.
    \label{fig:ttbar-impact}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

A challenge in the study of~\cite{Czakon:2016olj} was the observed tension between some of the
ATLAS and CMS distributions, such as $m_{t\bar{t}}$, which prevented their simultaneous inclusion
in the global fit.
%
While the underlying cause of these discrepancies is still under investigation,
this limitation was bypassed by identifying pairs of distributions which could be fitted
with good quality at the same time and that exhibited comparable constraining power.
%
Further investigations of this issue, including 13 TeV data and comparisons between data
and theory in terms of lepton and jets observables, should be able to
shed more light on the origin of such tension.

In order to illustrate the impact of the top quark pair production data
on the large-$x$ gluon, in Fig.~\ref{fig:ttbar-impact} we show
how the PDF uncertainties of the NNPDF2.3 gluon are reduced
once the Tevatron and the  LHC 7 and 8 TeV inclusive
   top-quark pair cross-section data are included by means of Bayesian
   reweighting~\cite{Czakon:2013tha}.
   %
   As can be seen, at the level of total cross section data the impact is still moderate,
   with PDF error reduction being at most $\simeq 20\%$.
   %
   On the other hand, in Fig.~\ref{fig:ttbar-impact} we also show
   the impact on the $gg$ luminosity of the normalized 8 TeV ATLAS and CMS
   $t\bar{t}$ differential distributions, compared to a baseline
   fit based on the NNPDF3.0 settings but without the jet data (since the NNLO
   corrections to jet production were not available at the time).
   %
   One clearly sees here how the impact is much more significant, highlighting the
   increase in constraining power of the differential distributions are compared
   to the total cross section data, specially in the large-$x$ region, where
   PDF uncertainties can be reduced by more than a factor of 2.

\subsubsection*{Single top production}\label{sec:datatheory.top.single}

In addition to top quark pair production, single top
production can also provide in principle useful PDF--sensitive information.
%
Such a process can proceed via the scattering of a bottom
quark with a light quark, see Fig.~\ref{fig:singletop} (left) for a typical diagram,
and will therefore provide information about the
$b$--quark PDF.
%
In addition, due to the presence of the $b$-quark in the initial state, it provides
an important testing ground for the different heavy quark flavour
schemes used in the calculation, analogously to those described in Sect.~\ref{sec:global.heavyq}
for the case of DIS structure functions.
%
That is, it is possible to use
a $n_f=4$ massive scheme, a $n_f=5$ massless scheme, or a matched scheme interpolating between
the two, see the discussion in Refs.~\cite{Maltoni:2012pa,Forte:2015hba}.

State or the art calculations of this process are based on NNLO QCD theory both
for the total cross sections and for differential
distributions~\cite{Brucherseifer:2014ama,Berger:2016oht}, and
LHC measurements at 8 TeV and 13 TeV of total cross sections (including ratios
of top to anti--top production) as well as
single inclusive distributions are already available~\cite{Aaboud:2017pdi,Aaboud:2016ymp}, although
some of them only in preliminary form.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.40]{3-datatheory/plots/singletop}
   \caption{\small Left plot: one of the Feynman diagrams for single-top production
   at leading order, illustrating its sensitivity to the $b$-quark
   PDF.
   %
   Right plot: comparison of the theoretical predictions for the ratio $R_t=\sigma_t/\sigma_{\bar{t}}$
   from different PDF sets and the corresponding ATLAS measurements at $\sqrt{s}=13$ TeV
   from~\cite{Aaboud:2016ymp}.
    \label{fig:singletop}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Moreover, since the production of top and anti--top quarks  is generated by different initial
state partons, cross section ratios such as $R_t\equiv \sigma_t/\sigma_{\bar{t}}$
can provide important information on the quark flavour separation of the proton.
%
To illustrate this point, we show in Fig.~\ref{fig:singletop} (right)
a comparison of the theoretical predictions for the $R_t$ ratio
   from different PDF sets and the corresponding ATLAS measurements at $\sqrt{s}=13$ TeV
   from~\cite{Aaboud:2016ymp}.
 %
 While experimental uncertainties are still large, due to the limited statistics, we can see that
 the measurement may eventually become sensitive to differences between PDF sets.

In addition, similar comparisons could also be performed for differential distributions,
 either at the level of top kinematic variables or at the level of observable quantities
 constructed from leptons and $b$-jets.
 %
 In the case of the ATLAS 8 TeV measurements~\cite{Aaboud:2016ymp}, these distributions
 are available including the full experimental covariance matrix, and therefore
 all the ingredients are available in order to quantify for the first time
 the impact of the LHC
 single top production data on the PDFs.

