Since the first run of the Tevatron at Fermilab, inclusive jet production at hadron colliders has provided the dominant 
constraints on the gluon PDF at large--$x$.
%
The
definition of jet cross sections starts from a well-defined jet algorithm~\cite{Salam:2009jx}, which
is usually chosen to be infrared and collinear safe so that the corresponding parton--level cross section can be calculated in
perturbative QCD for hard scattering at high energies.
%
The most commonly used jet algorithm at the LHC is the anti--$k_T$ algorithm~\cite{Cacciari:2008gp}, provided with
the 4--vector recombination scheme.
%
Other common choices include the $k_T$ algorithm~\cite{Catani:1993hr,
Ellis:1993tq}, the Cambridge-Aachen algorithm~\cite{Dokshitzer:1997in}, as well as the Midpoint algorithm~\cite{Blazey:2000qt}, which was sometimes used at the Tevatron.
%
The main parameter entering a jet algorithm is the jet radius $R$, which roughly defines
how large is the jet in the $\lp \eta,\phi\rp$ plane, and whose optimal value
depends on the specific application~\cite{Cacciari:2008gd}.

When comparing to the calculated parton--level cross section  to the experimentally measured jet cross section, it is essential to correct these to the hadron level.
%
That is, additional  non--perturbative corrections due, for example, to the underlying event and
hadronization effects, must be accounted for. These are usually provided by the experimental collaborations as
multiplicative factors derived from leading--order event generators.
%
The size of such corrections can be significant at low-$p_T$, as high as $\sim 20\%$, while at high-$p_T$ they are generally small, at the percent level~\cite{Khachatryan:2016mlc}. Variations of these non--perturbative corrections, by considering for example difference generator predictions, are then treated as an additional source of correlated systematic
error.
%
Although PDF fits typically use parton--level predictions, results also exist which include the matching of NLO calculations to parton shower and 
hadronization~\cite{Alioli:2010xa}, which can be directly compared with the data at hadron--level.          

\subsubsection*{PDF sensitivity}\label{sec:datatheory.jets.sensitivity}
%
At leading order,
jet production at hadron colliders includes the following subprocesses
\begin{align}
&gg\rightarrow gg,\, gg\rightarrow q\bar q,\, gq\rightarrow gq,\, q\bar q\rightarrow gg,\,\nonumber\\
&q\bar q \rightarrow q\bar q,\, q\bar q \rightarrow q'\bar q',\, q\bar q'\rightarrow q\bar q',\, qq\rightarrow qq,\,
qq'\rightarrow qq', 
\end{align}
along with the corresponding
charge conjugate processes.
%
Therefore, jet production is sensitive to both the gluon and quark PDFs,
with the dominant partonic subprocess depending on the specific jet $p_T$.
%
The kinematics of the two leading jets in the final state
can be characterized by their rapidities $y_{(1,2)}$ and their transverse momenta $p_{T,{(1,2)}}$. At LO we have $p_{T,1}=p_{T,2}=p_T$, and the momentum fractions carried by the
two incoming partons are given by
\begin{equation}
x_1=\frac{p_T}{\sqrt s}(e^{y_1}+e^{y_2}),\, \,\,\,x_2=\frac{p_T}{\sqrt s}(e^{-y_1}+e^{-y_2}),
\end{equation}
where $\sqrt s$ is the center of mass energy of the two incoming hadrons. If we instead consider the rapidity
of the jet in the centre--of--mass frame of the dijet system, $y^*\equiv (y_1-y_2)/2$, and the boost of the dijet $y_b\equiv(y_1+y_2)/2$,
we have
\begin{equation}
x_1x_2=\frac{4p_T^2\cosh^2 y^*}{s},\, \,\,\,x_1/x_2=e^{2y_b}.
\end{equation}
Note that beyond LO there will be multiple jets in the final state from additional QCD radiation,
and in general the $p_T$ balance of the two leading jets will be lost.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.19]{3-datatheory/plots/autoATLAS7R4fracLO.pdf}
  \includegraphics[scale=1.35]{3-datatheory/plots/jetcorr.pdf}
   \caption{\small
   Left: Fractional contributions from different partonic channels to the single inclusive jet
   production at the LHC 7 TeV at LO in the central rapidity region, computed
   with NNPDF3.0~\cite{Currie:2017ctp}.
   %
   Right: Correlations between binning cross sections from ATLAS on the single inclusive jet
   production at the LHC 7 TeV and the gluon PDF; the  dashed curves correspond to
   experiment bins at low $p_T$. 
    \label{fig:jet3}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


Experimentally, jet production can be measured in various ways. The most commonly used type for PDF fits is the single inclusive jet cross section, double differential in the jet $p_T$ and
rapidity $y$.
%
Here, one count all jets in a single event and includes them in the same distribution.
%
Such a double differential cross section can be sensitive to different flavor combinations, depending on the kinematic region considered. In Fig.~\ref{fig:jet3} (left) the fractional contributions from the different parton--level subprocesses to the inclusive jet cross section
in central rapidity region at the LHC is shown, as a function of the jet $p_T$. We can see that at low $p_T$ the channels involving initial--state gluons are dominant, while at higher $p_T$ the $q\overline{q}$ contribution increases, but nonetheless with a sizeable gluon--induced fraction. As the quark PDFs are generally already well constrained by DIS data in these kinematic regions, jet data is therefore dominantly sensitive to the gluon PDF.
%
This is illustrated in Fig.~\ref{fig:jet3}, which show the correlations between the inclusive jet cross section and the gluon PDF at various $x$ values.
%
This follows the  ATLAS binning~\cite{Aad:2011fc}, with each curve corresponding to one bin.
%
From this we can see that the inclusive jet production can further constrain the gluon PDF in a wide range of
$x$ between $x\simeq 10^{-3}$ and $x\simeq 0.7$.

In addition to the single inclusive case, there are also measurements of the
double differential cross sections for inclusive dijet production, that is with
respect to $y^*$ and invariant mass of the two leading jets, or even triple differential cross
sections, e.g., with respect to $y_b$, $y^*$, and average $p_T$ of the two leading jets.
Through such refined binning one can probe different initial states more efficiently. The large $y_b$ region usually receives more contributions from gluon initial states, while at large $y^*$ and $p_{T}$  initial states with two valence quarks dominate, allowing the $d$-valence PDF at high-$x$ to be further constrained.
 
\subsubsection*{Experimental data}\label{sec:datatheory.jets.data}

The currently available measurements on jet production at hadron collider which are relevant for constraining the PDFs are as follows:

\begin{itemize}

\item
The double differential single inclusive jet production cross section data
from the CDF~\cite{Abulencia:2007ez,Aaltonen:2008eq} and D0~\cite{Abazov:2008ae,Abazov:2011vi} collaboration, at Tevatron Run II (1.96 TeV).

\item
The double differential single inclusive jet production cross section data
from the ATLAS~\cite{Aad:2010ad,Aad:2011fc,Aad:2013lpa,Aad:2014vwa} and CMS~\cite{Chatrchyan:2012bja,Chatrchyan:2014gia,Khachatryan:2016mlc} collaborations at LHC Run I
(7 and 8 TeV).

\item
The double differential inclusive dijet production cross section data
from the ATLAS~\cite{Aad:2010ad,Aad:2011fc,Aad:2013tea} and CMS~\cite{Chatrchyan:2011qta,Chatrchyan:2012bja} collaborations at LHC Run I
(7 and 8 TeV).

\item
The triple differential inclusive dijet production cross section data
from the CMS collaboration~\cite{Sirunyan:2017skj} at LHC Run I
(8 TeV).

\item
The measurements of the ratio of double differential cross sections in single inclusive jet
production at different centre--of--mass energies, 2.76, 7 and 8 TeV,
from the ATLAS~\cite{Aad:2013lpa} and CMS~\cite{Khachatryan:2016mlc} collaborations, at LHC Run I.

\item
The double differential single inclusive jet production cross section data
from the CMS collaboration~\cite{Khachatryan:2016wdh} at LHC Run II (13 TeV).

\item More recently, measurements of triple differential dijet cross sections are becoming available, see e.g. the recent CMS analysis~\cite{CMS:2016wpr} at 8 TeV.

\end{itemize} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
  \includegraphics[scale=0.3]{3-datatheory/plots/SystSummaryR06_Y0.pdf}
  \includegraphics[scale=0.45]{3-datatheory/plots/InclusiveExperimentalUnc_Ybin0.pdf}
   \caption{\small
    Representative systematic errors in single inclusive jet measurement at
    LHC 7 TeV in the central rapidity region, from ATLAS (left)~\cite{Aad:2014vwa}
    and CMS (right)~\cite{Chatrchyan:2014gia}. 
    The luminosity error is not included in the ATLAS plot.
    %
    We note that in both cases the jet energy scale (JES) is the dominant source
    of experimental uncertainties.
   \label{fig:jet4}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Due to the complexity of jet reconstruction and calibration there are a large number of experimental systematic uncertainties, with typically $N_{\rm sys}\sim 50 -100$ correlated systematic errors for $N_{\rm dat}\sim 100  - 200$ data points, in the case of both ATLAS and CMS.
%
In the most recent ATLAS and CMS 7 TeV measurements~\cite{Aad:2014vwa,Chatrchyan:2014gia}, the total correlated experimental uncertainties are at a level of about $5\sim 20$\% in most regions. On the other hand, the uncorrelated systematic errors and statistical errors are at one percent level or less in general, and therefore the uncertainty on such data is generally completely systematics dominated.
%
The typical experimental systematics from both ATLAS and CMS are shown in Fig.~\ref{fig:jet4}, and are seen to be dominated by the jet energy scale~\cite{Aad:2014vwa,Chatrchyan:2014gia}. The increasing precision of the LHC jet data, and the generally small uncorrelated errors, makes it rather challenging to fit the jet data well across the entire kinematic region in {\it e.g.}
the case of the ATLAS 7 TeV measurement from the 2011 dataset~\cite{Aad:2014vwa}.
%
A full account of these issues will almost certainly require a better understanding of both the experimental systematics and sources of theoretical errors that have not generally been included in PDFs fits previously.

In addition, for measurements of the ratios of the double differential
inclusive jet production cross sections at different centre--of--mass the experimental
systematic errors largely cancel out, although the statistical uncertainties are somewhat larger, see e.g. the CMS 2.76, 7 and 8 TeV~\cite{Khachatryan:2016mlc} and ATLAS 2.76 and 7 TeV~\cite{Aad:2013lpa} measurements.
%
Such measurements of cross-section rations between different center-of-mass energies
$\sqrt{s}$ are potentially useful for PDF fits~\cite{Mangano:2012mh}, due to the
cancellation not only of experimental systematics but also of theory errors such
as scale variations.

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

The NLO QCD corrections to single inclusive jet and inclusive dijet production
were first calculated in the early 90's~\cite{Ellis:1992en,Kunszt:1992tn}, and has been implemented
in two numerical programs, {\tt NLOjet++}~\cite{Nagy:2001fj,Nagy:2003tz} and
{\tt MEKS}~\cite{Gao:2012he}. Recently, the
NNLO QCD corrections to the same process have been completed for all partonic
channels~\cite{Currie:2016bfm,Ridder:2013mf,Currie:2014upa,Currie:2017eqf},
with the exception of some sub--leading colour contributions.
%
The calculation is based on the Antenna subtraction
method~\cite{Ridder:2015dxa,Ridder:2016rzm} for isolating the
infrared singularities in QCD real radiations, and is part of the {\tt NNLOJET} program
for the evaluation of NNLO hadron-collider cross-sections.

Fig.~\ref{fig:jet1} (left) shows
the NNLO QCD corrections to inclusive jet production at the 7 TeV LHC, 
with the anti--$k_T$ algorithm and a central scale choice of the leading jet $p_T$.
The NNLO QCD corrections are seen to be significant at low-$p_T$, leading to a
10\% increase with respect to NLO, while at high-$p_T$ the NNLO corrections are small.
%
The NLO scale variations bands are asymmetric at low-$p_T$
and, interestingly, largely underestimate the perturbative uncertainties.
%
Electroweak (EW) corrections
can be significant at high-$p_T$ for central rapidities due to the presence of
large EW Sudakov logarithms, but are well below 1\% for a rapidity greater than
unity~\cite{Dittmaier:2012kx}.

There are ambiguities in choosing the appropriate 
QCD scale even in the simplest case of single inclusive jet production~\cite{Carrazza:2014hra}.
%
In particular, one can take either the $p_T$
of the individual jet or the leading jet in the event. While these variables are the same at LO, where the two jets are produced back--to--back, at higher orders there exist more than two jets which can have large differences in $p_T$. The NNLO predictions using these two choices
for the central scales are studied in~\cite{Currie:2017ctp}, and are found they lead to vary significantly.
%
This is shown in Fig.~\ref{fig:jet1} (Right), where at high-$p_T$ the two predictions converge as expected, but  at low and intermediate $p_T$, there are
significant differences of the central values in comparison to the size of scale
variations.
%
Indeed, the two error bands (from scale variations) do not even overlap.
%
Although it seems that the 
NNLO predictions using the individual jet $p_T$ as the central scale tend to follow the
trends of ATLAS data better, clearly further investigations
are needed to resolve the ambiguity of scale choice in the NNLO predictions, including
specifically PDF fits based on exact NNLO theory for the jet data.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.2]{3-datatheory/plots/ATLAS7R4ratiotoNLO.pdf}
  \hspace{0.1in}
  \includegraphics[scale=0.2]{3-datatheory/plots/ATLAS7R4NNLOratiotodata.pdf}
   \caption{\small
   Left: Predictions on single inclusive jet production at the LHC 7 TeV
   using ATLAS binning and anti-$k_T$ algorithm with $R=0.4$, with a central scale choice of leading
   jet $p_T$ and scale variations by varying renormalization and factorization
   scales simultaneously by a factor of 2~\cite{Currie:2017ctp}. 
   %
   Right: For the same setup comparing the NNLO predictions using a central
   scale choice of the leading jet $p_T$ (green) and the individual jet $p_T$ (red)~\cite{Currie:2017ctp}.
    \label{fig:jet1}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

As well as fixed--order predictions, there are various theoretical calculations including analytic QCD resummation~\cite{Kidonakis:2000gi,
Kumar:2013hia,Klasen:2013cba,deFlorian:2013qia,Carrazza:2014hra}.
It has been shown in~\cite{deFlorian:2013qia} for the case of inclusive jet production at the LHC, 
that the approximate NNLO predictions from the expansion of threshold resummation
agree well with the exact NNLO predictions for the all--gluon channel at large $p_T$. Over the full rapidity range the threshold
expansion reproduces the fixed--order results down to a $p_T$ of about 400 GeV with
the same value shifted to lower $p_T$ for large rapidity region. Such approximate NNLO
predictions have been used in previous global analysis involving jet data~\cite{Harland-Lang:2014zoa,
Ball:2014uwa}.
The jet cross sections are also sensitive to the jet algorithm used, in particular on the value
of the jet radius $R$.
%
A larger jet radius usually leads to
a larger inclusive cross section and better convergence in the perturbative expansion.
%
While this also reduces the non-perturbative corrections from QCD hadronization, it
increases the correction from underlying events. At the LHC, ATLAS uses $R=0.4$
and $R=0.6$, while CMS uses instead 0.5
and 0.7.

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

Jet data from the Tevatron and the LHC Run I have already played an important role in global 
analyses~\cite{Watt:2013oha,Ball:2014uwa,Dulat:2015mca,Harland-Lang:2014zoa}, although
in the NNLO fits these only currently apply NLO or approximate NNLO theoretical predictions.
Indeed, it was found that removing all jet data from the global analyses can
lead to an increase of the gluon PDF uncertainties at large-$x$ by at least a factor of two~\cite{Ball:2014uwa}.
%
See Ref.~\cite{Rojo:2014kta} for an overview of such studies.
%
There exist also independent studies from the ATLAS and CMS
collaborations~\cite{Khachatryan:2014waa,Khachatryan:2016mlc,Sirunyan:2017skj,Aad:2013lpa}
on the effects of jet data on the PDFs, based on NLO fits and using the {\tt xFitter}
program~\cite{Alekhin:2014irh}.

To illustrate the impact on PDFs of LHC inclusive jet data,
in Fig.~\ref{fig:jet5}, taken from~\cite{Sirunyan:2017skj}, we
shows the impact of the CMS 8 TeV jet data
on the gluon PDF, by adding the data into a base fit with HERA DIS data only~\cite{Abramowicz:2015mha}.
The inclusion of both the single inclusive jet data and the inclusive
dijet data leads to a sizeable reduction in the gluon PDF uncertainty at large $x$.
Meanwhile in the same fit a reduction of the PDF uncertainty in
the valence quark at high $x$ is also observed, providing a complementary constraint
to Drell-Yan
and fixed-target DIS data.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.23]{3-datatheory/plots/CMSdijet1.pdf}
  \hspace{0.3in}
  \includegraphics[scale=0.23]{3-datatheory/plots/CMSdijet2.pdf}
   \caption{\small
   Effects of the CMS 8 TeV jet data on the gluon PDF when adding up to a
   PDF fit with HERA DIS data only. The left plot compares the fit with
   only DIS data and the fit with CMS dijet data in addition. The right
   plot compares the fit to HERA DIS plus CMS dijet data and fit to HERA
   DIS plus CMS inclusive jet measurements~\cite{Sirunyan:2017skj}.
   %
   This study is performed using {\tt xFitter}, with PDF uncertainties obtained
   with the Hessian method.
   \label{fig:jet5}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

With the full NNLO predictions on inclusive jet
and dijet production now available, we can foresee significant advances in
 pinning down the gluon PDF at large $x$, in particular using the increasingly precise
inclusive jet data from LHC Run I and Run II.
%
Moreover, in the latter case, the increased luminosity will allow to probe jet production
deep into the TeV region, thus accessing PDFs at larger values of $x$ where
available constrains are rather scarcer.
