The LHC has provided precision measurements of the inclusive transverse
momentum spectra of the $Z$ boson, which may be exploited for the purposes of PDF fitting.
%
In this respect,
there are three distinct regions of the $p_T$ spectrum.
%
At small $p_T\ll m_Z$, the fixed--order predictions
diverge due to the higher--order logarithms generated by soft gluon radiation.
%
Here, QCD resummation techniques are needed in order
to achieve reliable predictions, see~\cite{Collins:1984kg,
Davies:1984hs,Ellis:1997ii,Qiu:2000ga,Landry:2002ix,Mantry:2010mk,Becher:2010tm,Catani:2013tia}.
%
Such predictions require additional non--perturbative input that cannot be calculated from first principles~\cite{Qiu:2000hf,Landry:2002ix}, and  therefore the $Z$ $p_T$ distribution cannot be reliably used for the extraction of the collinear PDFs in this region.

On the other hand, at large $p_T\gg m_Z$, the fixed--order predictions
can also receive large logarithmic contributions due to soft gluon radiation
at the partonic threshold of the $Z$ boson and the recoiling jet~\cite{Becher:2013vva,Kidonakis:2014zva}.
It has been shown that those contributions can increase the
integrated cross sections with $p_T>200\,{\rm GeV}$ by $\sim$$5\%$ compared to
the NLO prediction at the LHC~\cite{Becher:2013vva}.
%
The third region is defined by intermediate $p_T\sim m_Z$ values, where
the standard fixed--order predictions can be trusted.
%
Therefore, is in this region where the $p_T$ distribution of $Z$ bosons
can provide additional constraints on the PDFs, in particular on the gluon.

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

At leading order,
$Z$ boson production with finite transverse momentum includes the following  partonic
subprocesses
\begin{equation}
q\bar q\rightarrow Zg,\,\, gq\rightarrow Zq,\,\, g\bar q\rightarrow Z\bar q.
\end{equation}
In the leptonic channel where experimental measurements are the cleanest,
the kinematics of the $Z$ boson, namely the transverse momentum $p_{T}$ and rapidity
$y_Z$, can be reconstructed from the momenta of the lepton pair produced in the $Z$ decay. The momentum fractions of the initial--state partons are given by
\begin{equation}
x_1=\frac{m_T}{\sqrt s}e^{y_Z}+\frac{p_{T}}{\sqrt s}e^{y_j},\, 
\,\,\,x_2=\frac{m_T}{\sqrt s}e^{-y_Z}+\frac{p_{T}}{\sqrt s}e^{-y_j},\, 
\end{equation}\
where $\sqrt s$ is the center of mass energy of the two incoming hadrons,
$m_T$ is the transverse mass of the $Z$ boson and $y_j$ is the
rapidity of the recoiling parton. For inclusive production with respect to the hadronic recoil, that is integrating over $y_j$, these momentum fractions are therefore not uniquely determined, although for LO kinematics lower limits can be derived from the above equation.
%
Typically, the LHC experiments
measure the double differential cross sections in $p_T$ and $y_Z$ at the
$Z$ peak, although the
off-shell region, where the contributions from virtual photon can be important,
can also be considered.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
  \includegraphics[scale=0.5]{3-datatheory/plots/NNPDF30_nlo_as_0118_ATLASZPT8TEVYDIST-BIN1_ptZ_vs_gdu.pdf}
   \caption{\small 
   Correlations coefficients
   between the cross sections in various $p_{T}$ bins and the gluon, down- and
   up-quark PDFs as a function of $x$, from Ref.~\cite{Boughezal:2017nla}.
   %
The binning corresponds to the ATLAS 8 TeV
measurement~\cite{Aad:2015auj} within the rapidity interval $0<|y_Z|<0.4$. 
    \label{fig:ptz4}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The cross sections at moderate and large transverse momentum are dominated
by contributions from the gluon and quark scattering and are strongly
correlated with the gluon PDF in the
region relevant for Higgs boson production at the LHC.
%
That is 
illustrated in Fig.~\ref{fig:ptz4}, which shows the PDF-induced correlations
between the cross sections in different $p_T$ bin, in the rapidity interval $0<|y_Z|<0.4$,
and the gluon, down- and up-quark
PDFs at various $x$ values~\cite{Boughezal:2017nla}.
%
We can see that indeed the correlations with the gluon at $x\sim 10^{-2}$
almost reach 0.9.
%
Moderate correlations with quark PDFs at $x\sim 10^{-3}$
are also observed.
%
This highlights the fact that the $Z$ $p_T$ distributions provides
in principle a handle on the gluon in a region of $x$ that lies
between that covered by HERA structure functions, at smaller
values of $x$, and that
covered by inclusive jets and $t\bar{t}$ production, at larger $x$.

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.3]{3-datatheory/plots/ZmumuptSyst1.pdf}
  \hspace{0.5in}
  \includegraphics[scale=0.3]{3-datatheory/plots/uncert_normed_xsec_dpt_y0-2.pdf}
   \caption{\small 
   Left plot: the decomposition of the various experimental statistical
   and systematic uncertainties on the normalized $p_T$ spectra
   of the $Z$ boson in dimuon channel from ATLAS at 8 TeV~\cite{Aad:2015auj} .
   %
   Right plot: the corresponding experimental uncertainties
   for the normalized $p_T$ spectra
   of the $Z$ boson in dimuon channel from CMS at 8 TeV~\cite{Khachatryan:2015oaa} .
    \label{fig:ptz3}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The experimental measurements of the $p_T$ distribution of the $Z$ boson
have reached the percent level 
for both ATLAS~\cite{Aad:2015auj}
and CMS~\cite{Khachatryan:2015oaa} at the LHC Run I,
due to both the clean dilepton
final state as well as the high statistics of the signal.
%
The ATLAS measurement extracts the cross sections at three different (`Born', `bare' and `dressed') particle
levels when considering the effect of final--state photon radiation. The Born and bare
levels are defined from the lepton kinematics before and after final--state QED
radiation, while the dressed
level is defined by further combining the momentum of the lepton with photons
radiated within a certain cone.
%
These distributions can be presented with  the $Z$ boson rapidity integrated over, or separated
into different rapidity intervals, and can be on or off the $Z$--peak.
In addition, measurements of the distributions with respect to the angular variable
$\phi^*_{\eta}$~\cite{Aad:2015auj}, which is proportional to $p_{T,Z}$ at small transverse momentum, are also available.
%
As $\phi^*_{\eta}$ only depends on the direction of the lepton momenta, which are better measured than
the momenta themselves, this allows  the experimental systematics to be reduced.

 We summarise the available measurements on $p_T$
spectra of the $Z$ boson relevant to constraining the PDFs below:

\begin{itemize}

\item
The normalized $Z$ $p_T$ distribution in different rapidity
intervals by the ATLAS collaboration~\cite{Aad:2011gj,Aad:2014xaa} at LHC Run I (7 TeV). 

\item
The normalized and unnormalized distributions
of lepton pairs with respect to $p_T$ or $\phi^*_{\eta}$ in different rapidity intervals by the ATLAS
collaboration~\cite{Aad:2015auj} at LHC Run I (8 TeV). 

\item
The normalized $Z$ $p_T$ distribution integrated over rapidity
by the CMS collaboration~\cite{Chatrchyan:2011wt} at LHC Run I (7 TeV). 

\item
The normalized and unnormalized double differential $Z$  distribution in $p_T$ and rapidity by the CMS
collaboration~\cite{Khachatryan:2015oaa} at LHC Run I (8 TeV). 

\item
The normalized 
 $Z$  $p_T$ distribution and the ratio to the  $W$ $p_T$ distribution by the CMS
collaboration~\cite{Khachatryan:2016nbe} at LHC Run I (8 TeV). 

\item
The  unnormalized $Z$ distribution in $\phi^*_{\eta}$ in the forward
region by the LHCb collaboration~\cite{Aaij:2012mda,Aaij:2015vua} at LHC Run I (7 and 8 TeV). 

\item
The normalized  $Z$ distribution in $p_T$ or 
$\phi^*_{\eta}$ in the forward region by the LHCb collaboration~\cite{Aaij:2015gna} at LHC Run I (7 TeV). 

\item
The  unnormalized distribution of the lepton pair with respect to $\phi^*_{\eta}$ in different rapidity intervals by the D0
collaboration~\cite{Abazov:2014mza} at Tevatron Run II (1.96 TeV). 

\end{itemize} 

A summary of the experimental uncertainties for
the ATLAS and CMS 8 TeV measurements~\cite{Aad:2015auj,Khachatryan:2015oaa} of the normalized $Z$ $p_T$ distribution is shown in Fig.~\ref{fig:ptz3}. The luminosity uncertainty and some of the systematic errors largely cancel
in the normalized distributions.
%
Both ATLAS and CMS have measured the $Z$ $p_T$ up to about 1 TeV, 
while ATLAS has a finer binning at small $p_T$, motivated for the usefulness
of low $p_T$ measurements to tune Monte Carlo event generators.

For ATLAS, the total systematic uncertainty is well within 1\% for $p_T$ smaller than 200 GeV and
then $O(10\%)$ at the higher $p_T$ tail.
%
The statistical
uncertainties are tiny, starting at $\sim$$0.2\%$ for $p_T$$\sim$10 GeV, and are within 1\%
in most of the relevant $p_T$ region for both ATLAS and CMS.
%
The PDF uncertainties from
individual PDF groups are about 2\%, which is already larger than the experimental
errors in general, even before considering  the  spread between different PDFs; such data can therefore provide valuable PDF constraints.
%
We emphasize that the high-precision of these measurements is an unprecedented
challenge for QCD calculations, which need to achieve a per-mile level accuracy
to match the experimental precision.


    

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

The NLO QCD corrections to the $Z$ $p_T$ distribution were calculated a long time
ago~\cite{Gonsalves:1989ar,Baer:1991qf,Arnold:1990yk}, while more recently the EW corrections have been studied extensively~\cite{Maina:2004rb,Kuhn:2004em,Kuhn:2005az,Becher:2013zua}.
%
The NLO QCD corrections are found to be sizeable at LHC energies, and large QCD scale variations are found in the predicted $p_T$ spectra, rendering such predictions not suitable for
PDF determination.
%
However, the NNLO QCD corrections have recently
been calculated by two independent groups, in one case using the antenna subtraction method~\cite{Ridder:2015dxa,Ridder:2016nkl,Gehrmann-DeRidder:2016jns}
and in the other the $N$-jettiness subtraction method~\cite{Boughezal:2015ded,
Boughezal:2016isb}; these are found to be in good agreement.
%
While the original calculations are for $Z$+jet production, these
can readily be translated to the case of inclusive production of $Z$ boson
for not too small values of $p_T$.

At NNLO, the theoretical uncertainty due to the QCD scale variation is fount to be greatly reduced, allowing 
the measurements of the $Z$ boson $p_T$ spectra to be included for the first time in precision
PDF determination.
%
Moreover, these calculation include the leptonic decays of the $Z$
boson and thus the parton--level selection cuts may be applied to the theoretical predictions, allowing a direct comparison
with the measured fiducial cross sections without relying any experimental phase space
extrapolation.
%  
To illustrate this, Fig.~\ref{fig:ptz2} (taken
from~\cite{Gehrmann-DeRidder:2016jns}) shows the NLO and NNLO predictions for the unnormalized and
normalized $Z$ boson $p_T$ spectra at the 8 TeV LHC. 
%
The central values of the renormalization and factorization scale are set to the
transverse mass of the $Z$ boson, with scale variations calculated by varying
these simultaneously by a factor of 2 up and down.
%
The NNLO corrections are moderate for the unnormalized distribution, being about
5\% at low $p_T$ and 9\% at high $p_T$. The remaining scale variations range
from 1\% to 6\% depending on value of $p_T$. The EW corrections are small
at moderate transverse momentum but can be sizeable in the tail region,
reaching around $-10$\% for $p_T$ values greater than 600 GeV.
%
However,
as the statistical errors in the tail region are currently quite large,
and this prevents a direct probe of these EW effects.
%
For the normalized distribution, the denominator used is the inclusive
$Z$ production cross section at NNLO in same fiducial region and
with independent scale variations.
%
The size of the QCD corrections are found to be similar to the unnormalized case, since
the scale variations of the NNLO fiducial cross-section are very small.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.43]{3-datatheory/plots/ATLAS_ptz_bin_ratNLO_NNLOFull.pdf}
\includegraphics[scale=0.43]{3-datatheory/plots/ATLASnorm_ptz_bin_ratNLO_NNLOFull.pdf}
 \caption{\small 
   Left (right): the absolute (normalized) transverse momentum distribution
   of the inclusive $Z$ boson production at LHC 8 TeV~\cite{Gehrmann-DeRidder:2016jns}
   computed with NNPDF3.0, compared with the corresponding ATLAS measurements.
   %
   The green and blue bands indicate the NLO and NNLO predictions with the
   corresponding theoretical uncertainties from scale variations.
   %
    The fiducial cuts on the final-state charged
   leptons are $p_{T,l}>20\,{\rm GeV}$ and $|\eta_l|<2.4$.  
    \label{fig:ptz2}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

A detailed phenomenological study and comparison of the 
NNLO calculation to the ATLAS and CMS 8 TeV measurements has been presented in~\cite{Ridder:2016nkl,Gehrmann-DeRidder:2016jns}.
%
Good agreement between the NNLO theory and data for the normalized distribution
in ranges from 20 GeV $<p_T< 900$ GeV, in all rapidity intervals, is observed.
%
The conclusions are similar for the CMS data.
%
However, there is some discrepancy in the comparison to the ATLAS unnormalized distributions, see Fig.~\ref{fig:ptz2}, with the data tending to overshoot the theory over a wide $p_T$ range.
%
On the other hand, the NNLO prediction for the shape of the $p_T$ distribution is in good agreement with the data down to a $p_T$ value of 4 GeV, and is largely improved in comparison to the NLO predictions. 

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

The impact of the $Z$ boson $p_T$ data at LHC Run I
has been studied
recently within a global analysis framework~\cite{Boughezal:2017nla,Ball:2017nwa}.
%
In these studies, an additional uncorrelated theoretical uncertainty of $\sim$$1\%$ has been
added to all $p_T$ bins to account for the theoretical uncertainty due to
the residual MC integration error in the NNLO calculations.
%
Without including these errors, it was found that NNLO
predictions can not describe the data well, especially in the case of the
normalized distributions. Some tension is also found 
between the ATLAS 7 TeV normalized $p_T$ distribution~\cite{Aad:2014xaa} and the 8 TeV
$p_T$ distribution from both ATLAS and CMS~\cite{Aad:2015auj,Khachatryan:2015oaa}.
The ATLAS 7 TeV data also pulls the PDFs in a very different direction
with respect to the HERA inclusive DIS data~\cite{Boughezal:2017nla}.
In~\cite{Ball:2017nwa} it is concluded that the inclusion of the ATLAS 7 TeV
normalized data in the global analyses does not appear to be justified.

In Fig.~\ref{fig:ptimp} the impact of the ATLAS and CMS 8 TeV data on the NNPDF3.1
global analyses~\cite{Ball:2017nwa} is shown, by comparing the changes of the PDFs in the
analysis with and without the $Z$ $p_T$ data sets included. The uncertainty in the gluon
PDF is seen to be slightly reduced in the $x$ region of $10^{-2}\sim 10^{-1}$.
In the same region the gluon PDF receives constraints from the precision
measurements on top-quark pair production and HERA inclusive DIS, both of which are
present in the same analysis. It is also found that the 8 TeV data
lead to a moderate reduction in the PDF uncertainty on the total strangeness.
  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
  \includegraphics[scale=0.4]{3-datatheory/plots/xg-31-nnlo-ZpT.pdf}
  \includegraphics[scale=0.4]{3-datatheory/plots/xsp-31-nnlo-ZpT.pdf}
   \caption{\small
   Impact of the $Z$ boson transverse momentum measurements from
   ATLAS and CMS 8 TeV on the gluon PDF and the total strangeness in the
   NNPDF3.1 global analyses~\cite{Ball:2017nwa}.  
    \label{fig:ptimp}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

It is important to highlight that in Fig.~\ref{fig:ptimp} the impact of the
ATLAS and CMS $Z$ $p_T$ data is rather moderate since many other datasets
do constrain the PDFs (including the gluon).
%
As shown in~\cite{Boughezal:2017nla}, the effects of these measurements
is rather more marked if the input dataset does not contain other data such
as jets of top-quark pair production data that also pin down the gluon,
see Sect.~\ref{sec:structure.gluon} for a discussion in this respect.
