Now we turn to discuss the inclusive production of electroweak
gauge bosons, the so-called Drell-Yan process~\cite{Drell:1970wh}.
%
This process has been of enormous historical importance since it
provided a first window on the quark flavour separation in the proton
beyond the information contained on DIS structure functions.
%
Nowadays, Drell-Yan cross-sections provide the backbone of global
PDF fits together with the fixed-target and HERA structure function data.

\subsubsection*{PDF sensitivity}\label{sec:datatheory.gauge.sensitivity}
%
The lowest order contributions to $W$ and $Z/\gamma^*$ production proceed via
the following partonic subprocesses:
\begin{align}
u\overline{d}, \,c\overline{s} \quad (u\overline{s}, \,c\overline{d}) &\to W^+ \;, \\
d\overline{u},\, s\overline{c} \quad (s\overline{u}, \,d\overline{c}) &\to W^- \;, \\
q\overline{q} &\to Z/\gamma^* \;,
\end{align}
where we show the Cabibbo suppressed contributions in brackets and
where $q$ corresponds to all active flavours.
%
These processes can therefore tell us about the flavour decomposition of the proton, given  that each flavour subprocesses carries a different weight
in the total cross-section.
%
To examine the dominant PDF sensitivity we can approximate the CKM matrix as diagonal, and thus ignore the bracketed contributions.
%
In this case it is informative to consider the ratio of $W^+$ to $W^-$ production, differential
in the rapidity $y_W$ of the produced boson,
\begin{equation}\label{wpwm}
R_\pm = \frac{{\rm d}\sigma(W^+)/{\rm d}y_W}{{\rm d}\sigma(W^-)/{\rm d}y_W}=\frac{u(x_1)\overline{d}(x_2)+c(x_1)\overline{s}(x_2) + 1\leftrightarrow 2}{d(x_1)\overline{u}(x_2)+s(x_1)\overline{c}(x_2) + 1\leftrightarrow 2}\;.
\end{equation}
and the corresponding $W$ asymmetry 
\begin{equation}\label{wasym}
A_W = \frac{{ d}\sigma(W^+)/{ d}y_W-{ d}\sigma(W^-)/{\rm d}y_W}{{ d}\sigma(W^+)/{\rm d}y_W+{ d}\sigma(W^-)/{ d}y_W}=\frac{u(x_1)\overline{d}(x_2)+c(x_1)\overline{s}(x_2)-d(x_1)\overline{u}(x_2)-s(x_1)\overline{c}(x_2) + 1\leftrightarrow 2}{u(x_1)\overline{d}(x_2)+c(x_1)\overline{s}(x_2)+d(x_1)\overline{u}(x_2)+s(x_1)\overline{c}(x_2) + 1\leftrightarrow 2}\;.
\end{equation}
We will for simplicity consider the $W$ rapidity, rather than the experimentally observable rapidity of the charged lepton from the $W$ decay, in what follows. These variables are clearly correlated; we will comment further on this at the end.

Thus these ratios are in general sensitive to a fairly non--trivial combination of quark and anti--quark PDFs evaluated at the following values of $x$:
\be
x_{1}=\frac{M_W}{\sqrt{s}}e^{+ y_W}\, ,\qquad
x_{2}=\frac{M_W}{\sqrt{s}}e^{- y_W}\, .
\ee
While these expression completely define the PDF sensitivity of these observables at LO, it is informative to consider various kinematic limits, where these expressions simplify and more straightforward  approximate dependences become apparent. Including only the (dominant) $u$ and $d$ contributions, we can in particular consider the cases of central and forward $W$ production
\begin{align}\label{wcent}
{\rm Central}&:\qquad y_W\sim 0\qquad x_1\sim x_2 = x_0,\qquad \overline{u}(x_{1,2})\sim \overline{d}(x_{1,2})\;,\\ \label{wfor}
{\rm Forward}&: \qquad y_W \gtrsim 2, \qquad x_1 \gg x_2,  \qquad q(x_1)\sim q_V(x_1),\,\overline{u}(x_2)\sim \overline{d}(x_2)\; ,
\end{align}
where $x_0=M_W/\sqrt{s}$ and $q=u,d$. At the LHC we have $ x_0  = 0.005-0.01$, while in the forward region $x_2 \ll 1$, and therefore the $\overline{d}\sim\overline{u}$ approximation is a very good one. For the case of negative $W$ rapidity we can of course simply interchange $x_1 \leftrightarrow x_2$.

In the central region, applying the simplification of Eq.~(\ref{wcent}) and dropping the $c,s$ contributions we find
\begin{align}
R_{\pm}&\sim \frac{u(x_0)}{d(x_0)}\;,\\
A_W&\sim  \frac{u_V(x_0)-d_V(x_0)}{u(x_0)+d(x_0)}\;.
\end{align}
Thus $A_W$ is sensitive to the valence difference, while $R_{\pm}$ is sensitive to the ratio of $u$ to $d$ at $x_1\sim x_2 \sim x_0$. For these reasonably low $x$ values, the valence $u$ and $d$ quarks are fairly small, and so we roughly expect $R_{\pm}\sim 1$ and $A_W \sim 0$, with the departures from these values being due to the precise flavour content of the proton, in particular the fact that the valence distributions are not completely negligible in this region.

In the forward region, applying the simplification of Eq.~(\ref{wfor}) and again dropping the $c,s$ contributions we find that the following
relations hold
\begin{align}
R_{\pm}&\sim \frac{u_V(x_1)}{d_V(x_1)}\;,\\
A_W&\sim  \frac{u_V(x_1)-d_V(x_1)}{u_V(x_1)+d_V(x_1)}\;.
\end{align}
Thus these provide (equivalent) sensitive constraints on the $u/d$ ratio at high $x$

Considering now the case of $Z$ production, then in the forward case
we find,
using the same approximations as before,
that the differential ratio of $W$ to $Z$ production is given by
\begin{equation}\label{wzrat}
\frac{{ d}\sigma(W^+)/{ d}y_W+{\rm d}\sigma(W^-)/{\rm d}y_W}{{\rm d}\sigma^Z/{\rm d y_Z}}\approx \frac{u_V(x_1)+d_V(x_1)}{0.29 u_V(x_1)+0.37 d_V(x_1)}\;,
\end{equation}
where the factors in the denominator arise from the electroweak $Z$--quark couplings.
%
For the central region a similar result. evaluated at $x_0$, is found
up to an overall factor of 2.
%
Therefore, we can see that the $W^{\pm}$ and $Z$ cross sections provide very similar information about the $u$ and $d$ quarks. 

Up to this point we have omitted the contribution from the strange quarks to $W$ and $Z$ production. Generally speaking this is washed out when considering ratio observables, justifying their omission above, although the $W$ asymmetry displays some sensitivity to the strange difference $s-\overline{s}$.
%
On the other hand, the contribution to the absolute cross sections is not negligible, in particular at lower $x$. Thus for example the $Z$ cross section at central rapidity becomes, for five active quark flavours,
\begin{equation}
\label{zcross}
\frac{  d\sigma^Z}{{ d y_{ll}}}\sim 0.29\Big(u(x_0)\overline{u}(x_0)+c(x_0)\overline{c}(x_0)\Big)+0.37\Big(d(x_0)\overline{d}(x_0)+s(x_0)\overline{s}(x_0)+b(x_0)\overline{b}(x_0)\Big)\, .
\end{equation}
Therefore, provided the absolute cross section data are sufficiently accurate and the other quark flavours are sufficiently well determined, this may for example be sensitive to the currently less well determined strange quark distribution.
%
Moreover, this is not a case of a simple overall normalization; as the $Z$ rapidity increases the valence $u,d$ contributions will become increasingly dominant, and the contribution from the strange (and the heavy flavours) will decrease. Thus the shape of the $Z$ rapidity distribution is sensitive to the proton strangeness, as well as well as to that from the heavy flavour PDFs.
%
Similar considerations also apply for the absolute $W^{\pm}$ cross sections.

Moving away from the $Z$ peak region, the Drell--Yan process is dominated by an off--shell intermediate photon, with cross-section given by
\begin{equation}\label{dy}
\frac{{ d}\sigma^{\rm DY}}{{ d y_{ll}}}\sim \sum_i
e_i^2\Big(q(x_1)\overline{q}(x_2)+q(x_2)\overline{q}(x_1)\Big)\, ,
\end{equation}
where the sum runs over all active quark flavours.
%
Thus in comparison to Eq.~(\ref{zcross}), a different combination of the quark and anti--quark PDFs is probed, due to the differing electromagnetic couplings.
%
In particular, the relative $u\overline{u}$ to $d\overline{d}$ contribution is now a factor of $\sim 5$ higher in comparison to the $Z$ cross section.
%
At the LHC, low mass Drell--Yan production therefore provides complementary flavour information in the low to intermediate $x$ region. In addition, as the cuts on the final--state lepton transverse momenta tend to increase the relative important of the higher order contributions, for which the $Z$ $p_\perp$ can be non--zero, this can be sensitive to the gluon PDF at lower $x$, which contributes through the NLO $g\to q\overline{q}$ splitting.
%
On the other hand,
high mass Drell-Yan production (with $M_{ll} \gg m_Z$) is sensitive to the $q, \overline{q}$ PDFs at high $x$, in particular the anti--quarks, which are rather less well determined in this region.

A further constraint is provided by considering the Drell--Yan process on fixed proton and neutron (in practice, deuteron) targets. By using isospin symmetry the PDFs between the proton and the neutron can be related
\be
u^p=d^n \qquad d^p=u^n\;,
\ee
allowing an extra handle on the proton flavour decomposition.
%
In particular, such fixed target experiments generally have larger acceptance in the $x_1\gg x_2$ region (where $x_1$ is defined with respect to the proton beam) for which the first term in Eq.~(\ref{dy}) is dominant, with $q(x_1)\sim q_V(x_1)$. It is then straightforward to show that
\be
\frac{\sigma^{pn}}{\sigma^{pp}}\sim \frac{\overline{d}(x_2)}{\overline{u}(x_2)}\;.
\ee 
That is, they are sensitive to quark sea decomposition in the intermediate to high $x_2\sim 0.01-0.3$ region probed by these fixed target experiments~\cite{Ellis:1990ti}.
%
This however comes with the added complication that the nuclear corrections accounting for the fact that the neutron is bound in a deuteron nucleus, and therefore the `free' neutron PDF is not directly probed.
%
Fixed target $pp$ scattering alone does not suffer from this issue, and is sensitive to the quark sea (dominantly, the $\overline{u}$) in the same $x$ region, but is much less directly sensitive to the $\overline{d}/\overline{u}$ decomposition.

Turning now to the case of $W,Z$ production at the Tevatron, the fact that we have $p\overline{p}$ collisions affects the flavours probed. In particular, we can use charge--conjugation symmetry to write
\begin{equation}\label{ppbar}
q^p=\overline{q}^{\overline{p}}\;.
\end{equation}
In fact, it is straightforward to show that in the region of valence quark dominance, the cross section ratio $R^\pm$ and the asymmetry $A_W$ are again sensitive to the $u/d$ ratio and the valence difference $u_V-d_V$, while the $Z$ cross section again provides similar information to $W^\pm$ cross section sum. 
%
Nonetheless, these conclusions are only approximately true, and the presence of a $\overline{p}$ beam provides complementary flavour information.

Finally, we emphasize that we have considered here the distributions with respect to the (unobservable) rapidity of the $W$ boson to simplify the discussion.  In general we should correctly account for the kinematics, as well as weight the corresponding $q\overline{q}$ contributions by the appropriate $W$ decay distributions. This is in fact provides a further handle on the flavour sensitivity of this observable, as by changing the $p_\perp$ cut on the charged lepton, different weights of the different quark contributions are achieved, see
{\it e.g.}~\cite{Martin:2009iq,Martin:2012da} for further details.
%
Nonetheless, the forward and central $W$ rapidity regions are certainly correlated with the equivalent lepton rapidity regions that are measured experimentally, and so the above discussion provides a qualitative guide for the PDF sensitivity of $W$ boson production.
%
However, as we will discuss below, the current simulation codes for $W$ and $Z$ production include the full kinematics of the leptonic decays, and therefore there is no need to explicitly correct back to the $W$ rapidity - the purpose of the discussion in this section being purely illustrative.

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

A non--exhaustive list of the available Drell-Yan measurements is as follows:

\begin{itemize}

\item The most precise fixed target Drell--Yan data come from the E866/NuSea~\cite{Towell:2001nh} experiment at Fermilab, while the E906/SeaQuest experiment~\cite{seaquest} will extend out to higher $x$, and is currently taking data. 

\item The Tevatron collider has produced a range of data on $W$ and $Z$ production, including measurements of the $Z$ rapidity distribution~\cite{Abazov:2007jy,Aaltonen:2010zza} and in $W$ production both the lepton~\cite{Abazov:2013rja,D0:2014kma} and the $W$~\cite{Aaltonen:2009ta,Abazov:2013dsa} asymmetries. 

\item Early LHC measurements of the the $Z$ rapidity distribution presented by CMS~\cite{Chatrchyan:2011wt} and ATLAS~\cite{Aad:2011dm}.

\item CMS Drell--Yan data at 7 TeV~\cite{CMSDY}, for $15 < M_{ll} < 1500$ GeV and at 8 TeV~\cite{CMS:2014jea}, which increased the upper mass limit to 2000 GeV. These are presented double differentially in the rapidity and invariant mass of the lepton pair.

\item ATLAS 7 TeV Drell--Yan invariant mass distribution (integrated over rapidity) at high~\cite{Aad:2013iua} ($116<M_{ll}<1500$ GeV) and low~\cite{Aad:2014qja} ($26<M_{ll}<66$) invariant masses.

\item CMS~\cite{Chatrchyan:2012xt,Chatrchyan:2013mza} 7 TeV $W$ asymmetry, and ATLAS~\cite{Aad:2011dm} $W^+$ and $W^-$ cross section data.

\item ATLAS high precision $W$ and $Z,\gamma^*$ data~\cite{Aaboud:2016btc}, using the full $4.6\,{\rm fb}^{-1}$ data set at 7 TeV. The Drell--Yan rapidity distribution is presented double differentially in three intervals of lepton pair mass, over the $45<M_{ll}<150$ GeV range. In the $Z$ peak and higher mass regions the measurement was also extended out to $|y_{ll}|=3.6$.

\item LHCb $Z$ rapidity distributions at 7~\cite{Aaij:2012mda,Aaij:2015gna}, 8~\cite{Aaij:2015vua} and 13~\cite{Aaij:2016mgv} TeV.

\item LHCb lepton rapidity distributions for $W^+$ and $W^-$ production at 7~\cite{Aaij:2014wba}, and 8~\cite{Aaij:2015zlq,Aaij:2016qqz} TeV.

\item In Ref.~\cite{Aaij:2015zlq},
cross section ratios between the 7 and 8 TeV LHCb $W$ and $Z$  measurements are presented, with the cancellation in various systematic uncertainties providing a more precise PDF sensitivity.

\end{itemize}

Thus at the LHC several measurements have already been presented, with many others
in preparation.
%
The $Z/\gamma^*$ data are available over a wide range of invariant masses, providing extensive coverage in $x$.
%
The $W$ data are increasingly presented as individual cross sections, including the correlated error information, to provide the maximum possible constraints.
%
While the majority of cases, such as with the ATLAS and CMS measurements, are limited to the central rapidity region, that is, a lepton pseudorapidity of $|\eta_l|<2.4$, this reach is extended by exploiting the forward acceptance of the LHCb detector, for which $2 <\eta_l < 4.5$ is accessed.
%
This allows the high and low $x$ region to be probed.

The most recent ATLAS $W$ and $Z,\gamma^*$ data~\cite{Aaboud:2016btc}, which uses the full $4.6\,{\rm fb}^{-1}$ data set at 7 TeV demonstrates the level of precision that is now being achieved. The $Z$ rapidity distribution and $W$ asymmetry are shown in Fig.~\ref{fig:ATLASWZdata} (in the latter case the individual $W$ measurements are available).
%
The high experimental precision is clear, in particular in the $Z$ distribution where excluding the luminosity uncertainty it is as low as $\sim 0.3\%$ at central rapidity, while the error on the PDF predictions, as well as the spread between sets, is significantly larger. The impact of such data can therefore be sizeable,
as will be discussed in short.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{figure}
  \begin{center}
 \includegraphics[width=0.37\textwidth]{3-datatheory/plots/Zpeak_cen_NNLO_combined}\qquad\qquad
 \includegraphics[width=0.37\textwidth]{3-datatheory/plots/Wasym25_NNLO_combined}
  \end{center}
\caption{\small The ATLAS 7 TeV measurements of $Z$ rapidity distribution (left) and $W$ asymmetry (right)
based on $\mathcal{L}=4.6\,{\rm fb}^{-1}$, compared
with different NNLO PDF sets. Taken from~\cite{Aaboud:2016btc}.}
\label{fig:ATLASWZdata}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5%%%%%%%%%%%%%%%%%%%%%%%

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

$W$ and $Z$ boson production is arguably the simplest process one can consider at a hadron collider, and indeed it was the first hadroproduction process for which the NNLO calculation became available, with the total cross sections being calculated in the early 90s~\cite{Hamberg:1990np}\footnote{A mistake in the one--loop real emission contribution was reported in~\cite{Harlander:2002wh}.}.
%
A decade later, in~\cite{Anastasiou:2003ds,Anastasiou:2003yy} the NNLO corrections to the differential $W$ and $Z$ rapidity distributions was presented for the first time.
%
A more direct comparison with experimental observables was provided in~\cite{Melnikov:2006di,Melnikov:2006kv} which presented the NNLO calculation fully differential in the final--state leptons, including in addition spin correlations, finite width effects and $\gamma - Z$ interference.
%
This was accompanied by the public release of the \texttt{FEWZ} simulation code, with subsequent improvements reported in~\cite{Gavin:2010az,Gavin:2012sy} and~\cite{Li:2012wna}, where NLO EW corrections (first calculated in~\cite{Berends:1984qa,Berends:1984xv,Baur:1997wa,Baur:2001ze,CarloniCalame:2007cd,Dittmaier:2009cr}) were included.
%
The \texttt{DYNNLO}~\cite{dynnlo2} parton level MC provides an alternative tool for generating $W$ and $Z$ production, again including spin correlations, finite width effects, and $\gamma - Z$ interference, but currently without EW corrections.
%
This code allows for arbitrary user-defined cuts on the final--state partons and leptons to be imposed and histograms to be made, in contrast to \texttt{FEWZ}, where a selection of pre-determined cuts and histograms may be applied. 

These two codes differ in their theoretical treatment of the processes, in particular in the method that is applied to achieve the (non--trivial) cancellation of IR singularities at intermediates steps in the calculation. While \texttt{FEWZ} uses the local `sector decomposition' method~\cite{Melnikov:2006di,Melnikov:2006kv} that provides an automated method for extracting and canceling the IR poles, \texttt{DYNNLO} applies an alternative non--local `$q_T$-subtraction' approach~\cite{dynnlo1} which uses the transverse momentum $q_T$ of the produced $W$ or $Z$ as a cut variable, treating the calculation in a different way above and below some $q_T^{\rm cut}$.

It is unfortunately now quite well established that these two codes can give non--negligible differences in their predictions for the same cross-sections even for identical input parameters.
%
For example, in the recent ATLAS high precision $W$ and $Z/\gamma^*$ analysis~\cite{Aaij:2015zlq} the difference in the fiducial cross section predictions can be as high as $\sim 1\%$, that is larger than the experimental uncertainties. This is due to the differing  subtraction procedures, which affects the predicted boson $p_\perp$ distributions.
%
The differences between the predicted cross sections are generally more significant when more restrictive cuts on the final--state leptons are imposed; for the total $W$, $Z$ cross sections the codes agree to within $0.2\%$~\cite{Aaij:2015zlq}. A closer investigation of this issue and its impact on PDF determination will clearly be essential, given the that the experimental uncertainties of existing and
future measurements is at the permille level.

More recently the \texttt{MCFM} event generator~\cite{Campbell:2011bn} has extended the NLO simulation of $W$ and $Z$ production to NNLO~\cite{Boughezal:2016wmq}. This takes a similar non--local approach to \texttt{DYNNLO}, but using the $N$--jettiness variable rather than the $q_T$. Here, it is shown that a careful and process--dependent choice of the cut on the 0--jettiness variable, $\tau_0^{\rm cut}$ (the equivalent of $q_T^{\rm cut}$ above) is required in order to balance the requirements of sufficient statistical precision and control over systematic power corrections that increase in importance as this cut is increased. 

Finally, event generators including transverse momentum resummation are also available. The \texttt{DYRes}~\cite{Bozzi:2010xn} code combines NNLO fixed--order with NNLL resummation, while \texttt{ResBos}~\cite{Balazs:1995nz} combines NLO fixed--order with NNLL resummation. However, typical observables that are used in PDF fits are chosen to be largely insensitive to such resummation effects, which are most important as the $W$, $Z$ transverse momentum becomes small, and so these codes are in general not used in PDF analyses.

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

As described in Sect.~\ref{sec:datatheory.gauge.sensitivity}, inclusive gauge boson production has played a crucial role in determining the quark flavour decomposition of the proton. Indeed, these have been included in all major PDF analyses for some time, from earlier fixed target data through to measurements at the Tevatron and increasingly at the LHC. Two recent LHC results are show in Fig.~\ref{fig:WZpdf}. In the left panel we show the CMS fit~\cite{Khachatryan:2016pev} to the down valence quark distribution. The baseline fit is to the HERA I+II data only, which is compared to the result including the CMS 8 TeV W boson production data. The change in shape and sizeable reduction in the PDF uncertainty over a wide range of $x$ is clear. 

In the right panel we show the impact on the strange quark fraction relative to the light quark sea
\be
R_s=\frac{s+\overline{s}}{\overline{u}+\overline{d}}\;,
\ee
of the ATLAS high precision $W$ and $Z/\gamma^*$ data~\cite{Aaboud:2016btc}. As described in Section~\ref{sec:datatheory.gauge.sensitivity}, provided the light quark flavours are sufficiently well determined, and the data are sufficiently precise, the size and shape of the $W, Z$ rapidity distributions can provide constraints on the strange quark PDFs. This is clear from the figure, where the fit to the ATLAS data predict a significantly higher value of $R_s$ in comparison to previous PDF fits, which do not include the ATLAS data. A hint of this effect is seen in the earlier `ATLAS-epWZ12' result~\cite{Aad:2012sb}, but it is only with the more recent high precision data that a clear effect becomes apparent.
%
The size and shape of the strange PDF will be further discussed in Sect.~\ref{sec:structure.strange}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
 \begin{center}
 \includegraphics[width=0.38\textwidth]{3-datatheory/plots/Figure_004-b}
  \includegraphics[width=0.53\textwidth]{3-datatheory/plots/Rs_com_new}
  \end{center}
\caption{(Left) The $d_V(x,Q^2)$ PDF, with the CMS fit to HERA I+II data only, and including the CMS 8 TeV W production data. Taken from~\cite{Khachatryan:2016pev}. (Right) The ratio of the strange quark to the light quark sea, $R_s$, with different PDF predictions and the result of the ATLAS fit to HERA I+II data and the high precision $W$ and $Z/\gamma^*$ data shown. Taken from~\cite{Aaboud:2016btc}.}\label{fig:WZpdf}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

As mentioned above, forward weak gauge boson production data from the LHCb experiment allows
obtaining important information on the flavour separation between the quarks and antiquarks in the proton
in the large-$x$ region, beyond the kinematic coverage of the ATLAS and CMS
measurements.
%
The role of the latest LHCb data on $W^\pm$ and $Z$ production at $\sqrt{s}=7$ and 8 TeV
has been studied in the context of the NNPDF3.1 analysis~\cite{Rojo:2017xpe}, finding that they can lead to improved determinations of the the up and down quark PDFs in the region $x\gsim 0.1$,
with an uncertainty reduction that can be as large as a factor 2.
%
To illustrate this effect,
in Fig.~\ref{fig:lhcb-nnpdf31} we show
the down quark  PDF at $Q = 100$ GeV, comparing the
results of the NNPDF3.1 baseline with those of the same fit but
without any LHCb data included~\cite{Rojo:2017xpe}.
%
We find that the LHCb data tends to enhance the value of $d(x,Q^2)$ by almost one-sigma
in the region $x\simeq 0.3$, with a marked decrease of PDF
uncertainties.
%
This comparison highlights the important role that LHCb data plays
in determining the quark flavour separation in the large-$x$ region.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
 \begin{center}
 \includegraphics[width=0.32\textwidth,angle=-90]{3-datatheory/plots/lhcb-nnpdf31}
  \end{center}
\caption{\small The down quark  $d(x,Q^2)$ at $Q = 100$ GeV, comparing the
results of an NNPDF3.1 baseline fit with those of the same
fit without LHCb data~\cite{Rojo:2017xpe}.
%
We show the PDF
ratio normalized to the central value of NNPDF3.1 (left) and the relative PDF uncertainties (right).
}\label{fig:lhcb-nnpdf31}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
