\label{sec:lattice}

As discussed in Sect.~\ref{sec:pdffitting.DIS}, parton distributions arise from
 non-perturbative QCD dynamics.
 %
 Therefore,
since we are currently not able
 to  analytically solve strongly-coupled non-Abelian gauge theories,
 it is very challenging to compute PDFs from first principles.
 %
 Perhaps the only possibility,
 beyond model calculations, consists in
 exploiting recent progress in lattice QCD~\cite{Gupta:1997nd}.
 %
 This method is based on discretizing the QCD Lagrangian in
 a finite-volume Euclidean lattice, which naturally introduces 
 an ultraviolet cutoff, and then computing directly non-perturbative QCD quantities on this lattice, before taking the continuum limit.
 %
 Such lattice QCD calculations require minimal external input,
 in particular only the hadronic mass scale $\Lambda_\text{QCD}$ and the values of the quark masses, or
 alternatively, the physical pion and kaon masses.

Here we briefly review some of these recent developments in lattice QCD calculations of PDFs.
%
For a more detailed overview of this progress,
together with the study of their interplay with state-of-the-art global analysis, see
Ref.~\cite{PDFlattice2017}, a white paper that will be
produced as the outcome of the dedicated workshop
{\it ``Parton Distributions and Lattice QCD calculation in the LHC era''},\footnote{\url{http://www.physics.ox.ac.uk/confs/PDFlattice2017}} which took place in
Oxford in March 2017.
%
This workshop
 brought together experts from the two communities (lattice QCD and global PDF fitting)
 to explore the synergies
and complementary aspects between the two approaches.
%
The discussion and results shown in this section represent
an brief excerpt of the material contained
in that white paper (see also~\cite{Nocera:2017war}).

Given that parton distributions have a formal definition in terms of the nucleon matrix elements
of certain non-local operators
(see the discussion in
Sect.~\ref{sec:pdffitting.DIS}), it is in principle 
possible to compute PDFs using lattice QCD.
From the practical point of view, however, given the extremely
CPU intensive nature of these calculations,
most lattice QCD results on PDFs have been limited for a long time to the first two moments of non-singlet flavour
combinations for large (unphysical) quark masses.
%
These restrictions have been overcome in the recent years, with several groups providing now
results of PDF moments for physical pion and kaon masses.

Moreover, the development of novel
strategies to overcome the limitations in computing the first few 
moments~\cite{Lin:2012ev,Constantinou:2014tga,Syritsyn:2014saa} has allowed the determination of more challenging quantities, 
such as gluon and flavor-singlet matrix elements as well as higher PDF moments.
%
More recently, both conceptual and numerical breakthroughs in lattice QCD computations have allowed
even greater progress, providing the first attempts to evaluate PDFs
and related quantities directly in Bjorken-$x$ space~\cite{Lin:2014zya,Alexandrou:2015rja,Chen:2016utp,Alexandrou:2016jqi}.
%
These developments have pushed lattice QCD calculations
to the point where, for the first time, it is possible to provide information on the PDF shape
of specific flavour combinations, both for quarks and for antiquarks, and meaningful comparison with 
global fits can start to be made.

Recent progress in lattice QCD calculations of PDFs and related
quantities has also been partly driven
by a greatly improved control on the systematic uncertainties
that enter the calculation
of relatively simple quantities such as nucleon matrix elements, which correspond to
low moments of the PDFs.
%
These systematic uncertainties, among others, include using
physical pion masses, reducing the excited-state contamination, and using large lattices
to remove finite-size effects.
%
Moreover, to make contact with the physical world and experimental
data, the numerical results are extrapolated to the continuum 
and infinite-volume limits.
%
In addition, the past decade has seen significant progress in
the development of efficient algorithms for the generation of
ensembles of gauge field configurations, which represent the QCD
vacuum, and tools for extracting the relevant information from lattice QCD
correlation functions.

In order to be reliably used in phenomenological
applications, lattice QCD calculations must demonstrate control over all relevant
sources of
systematic uncertainty introduced by the discretisation of QCD on the
lattice, such as those discussed above.
%
For a coherent assessment of the present state of lattice QCD
calculations of various quantities, the degree to which each
systematic uncertainty has been controlled in a given calculation should be an important
consideration.
%
The quality of individual lattice calculations can be quantitative
assessed based on criteria such as those from the
FLAG analysis of flavour physics on the lattice~\cite{Aoki:2016frl}.

The traditional approach for lattice QCD calculations of
parton distributions
has been to determine the matrix elements of local twist-two operators, where twist is the dimension minus the spin, that can be related to the Mellin moments of PDFs. In principle, given a sufficient number of Mellin moments, PDFs can be reconstructed from the inverse Mellin transform.
%
In practice, however, the calculation is limited to the lowest three moments, because power-divergent mixing occurs between twist-two operators on the lattice.
%
Three moments are unfortunately insufficient to reconstruct the full
$x$-dependence of the PDFs without significant model dependence~\cite{Detmold:2003rq}. The lowest three moments do provide, however, useful information, both as benchmarks of lattice QCD calculations and as constraints in global extractions of PDFs.
%
For instance, provided systematic uncertainties are kept under control, one can envisaging adding
these lattice QCD calculations of PDF moments as an additional theoretical
constrain to the global fit, on the same footing as the momentum
and valence sum rules.

Now we briefly summarize the state-of-the-art of lattice QCD calculations of the first moment
of unpolarized PDFs, which are those for which systematic uncertainties
are under better control. See Ref.~\cite{PDFlattice2017} for a more exhaustive set of
comparisons, including those of lattice QCD calculations with global fits.
%
The observables that are discussed here are defined as follows:
\begin{enumerate}

\item The first moment of the flavour triplet combination, $T_3=u^+-d^+$,
\begin{equation}
\left.\langle x\rangle_{u^+-d^+}(\mu^2)\right|_{\mu^2=Q^2}
\equiv
\int_0^1 dx\, x\left\{u(x,Q^2)+\bar{u}(x,Q^2)-d(x,Q^2)-\bar{d}(x,Q^2)\right\} \, .
\label{eq:unpfmumdtot}
\end{equation}

\item The first moment of the individual quark $q^+=q+\bar{q}$ PDFs,
\begin{equation}
\left.\langle x\rangle_{q^+=u^+,d^+,s^+,c^+}(\mu^2)\right|_{\mu^2=Q^2}
\equiv
\int_0^1 dx\, x\left\{q(x,Q^2)+\bar{q}(x,Q^2)\right\} \, .
\label{eq:unpfmiqtot}
\end{equation}

\item The first moment of the gluon PDF,
\begin{equation}
\left.\langle x \rangle_g(\mu^2)\right|_{\mu^2=Q^2}
\equiv
\int_0^1 dx\, x\, g(x,Q^2) \, .
\label{eq:unpfmg}
\end{equation}

\end{enumerate}
%
In Table~\ref{tab:unpolLQCDstatus1} we show a selection of recent results
for the  moments defined in Eqns.~(\ref{eq:unpfmumdtot}-\ref{eq:unpfmg}).
%
As can be seen, for $\langle x\rangle_{u^+-d^+}$ the lattice QCD uncertainties
vary between 5\% and 15\%, with the quoted results not agreeing among themselves within
errors.
%
For the first moment of the gluon, $\langle x \rangle_g$, the uncertainties
are around 10\%, and for the individual total quark combinations they vary between
 10\% and 20\%.
 %
 Therefore, while current determinations of the first moments are unlikely to provide
 constraints on global PDF fits (where uncertainties are the few-percent level), future
 calculations with improved systematic errors might be able to make a difference.
 %
 On the other hand,  existing calculations can already be used to provide meaningful
 constraints on polarized PDFs, where uncertainties are rather larger than in
 the unpolarized case due to the scarcer dataset.

%-------------------------------------------------------------------------------
\begin{table}[t] 
\renewcommand{\arraystretch}{1.2} 
\centering 
\begin{threeparttable}
\begin{tabular}{llclll}
Mom. & Collab. & Ref. & $N_f$ & Value\\
\toprule
$\langle x\rangle_{u^+-d^+}$ 
& LHPC\,14  
  & \cite{Green:2012ud} 
  & 2+1 
  & 0.140(21)\\
& ETMC 17  
  & \cite{Alexandrou:2017oeh} 
  & 2   
  & 0.194(9)(11)\\
& RQCD 14  
  & \cite{Bali:2014gha} 
  & 2   
  & 0.217(9)\\
\midrule
$\langle x\rangle_{u^+}$
&  ETMC 17  
  & \cite{Alexandrou:2017oeh} 
  & 2 
  & $0.453(57)(48)$\\
\midrule
$\langle x\rangle_{d^+}$
& ETMC 17  
  & \cite{Alexandrou:2017oeh} 
  & 2 
  & $0.259(57)(47)$\\
\midrule
$\langle x\rangle_{s^+}$
& ETMC 17  
  & \cite{Alexandrou:2017oeh} 
  & 2 
& $0.092(41)(0)$\\
\midrule
$\langle x\rangle_{g}$
& ETMC 17  
  & \cite{Alexandrou:2017oeh} 
  & 2 
  & 0.267(22)(27)\\
\bottomrule
\end{tabular}
\end{threeparttable}
\caption{\small Selected
recent lattice QCD calculations of the
first moments of unpolarized PDFs, defined in
Eqns.~(\ref{eq:unpfmumdtot}-\ref{eq:unpfmg}), evaluated at
 $\mu^2=4$ GeV$^2$.
%
See Ref.~\cite{PDFlattice2017} for more details.
}
\label{tab:unpolLQCDstatus1}
\end{table}
%------------------------------------------------------------------------------

While the lowest moments of the PDFs provide crucial benchmarks to
assess the reliability of lattice QCD calculations
of the nucleon structure, as well as potentially
useful information for global PDF fits,
they do not allow the complete $x$-dependence of the PDFs to be reconstructed.
%
In particular, the calculation of PDF moments is mostly insensitive to the small-$x$
region.
%
To bypass these limitations, recently a number of approaches have been
developed, aiming  to determine the $x$-dependence of PDFs directly from lattice QCD.
%
One of the most important approaches goes under the name of
{\it quasi-PDFs}, first formulated by Ji in Refs.~\cite{Ji:2013dva,Ji:2014gla}.
%
For simplicity, we focus in the following on the flavor nonsinglet case
so that we can neglect the mixing with the gluons.

In this approach, the unpolarized quark quasi-PDF $\widetilde{q}(x,\Lambda,p_z)$ is defined as a momentum-dependent
nonlocal static matrix element:
\begin{align}
\label{eq:qPDF}
\widetilde{q}(x,\Lambda,p_z)  \equiv \int \frac{dz}{4\pi} e^{-i x z p_z} 
\frac{1}{2}\sum_{s=1}^2\left\langle p,s\right\vert \bar{\psi}(z)\gamma_\alpha e^{ig\int_0^z
A_z(z^\prime) dz^\prime} \psi(0) \left\vert p,s\right\rangle \, ,
\end{align}
where $\Lambda$ is an UV cut-off scale,
typically chosen to be the inverse lattice spacing $1/a$.
%
Note that, because $p$ is finite, the momentum fraction $x$ can be larger than unity.
%
As we see from Eq.~(\ref{eq:qPDF}), these quasi-PDFs are defined for
nucleon states at finite momentum.
%
Therefore, in order to make contact with the standard collinear PDFs and thus with
phenomenology, they
must be related to the corresponding light-front PDF, for which the nucleon
momentum is taken to infinity.

In the  large-momentum  effective field theory (LaMET) approach, the
quasi-PDF $\widetilde{q}(x,\Lambda,p_z)$ can be related to the $p_z$-independent
light-front PDF $q(x,Q^2)$ through the following relation~\cite{Ji:2013dva,Ji:2014gla}
\begin{equation} \label{eq:qPDFmatching}
\widetilde{q}(x,\Lambda ,p_z) = 
  \int_{-1}^1 \frac{dy}{\left\vert y\right\vert} 
    Z\left( \frac{x}{y}, \frac{\mu}{p_z}, \frac{\Lambda}{p_z}\right)_{\mu^2 = Q^2} q(y,Q^2) +
  \mathcal{O}\left( \frac{\Lambda_\text{QCD}^2}{p_z^2},\frac{m^2}{p_z^2}\right) \, ,
\end{equation}
where $\mu$ is the renormalisation scale;
$Z$ is a matching kernel; and $m$ is the nucleon mass.
%
Here the $\mathcal{O}\left(m^2/p_z^2\right)$ terms are target-mass corrections and the $ \mathcal{O}\left(\Lambda_\text{QCD}^2/p_z^2\right)$ terms are higher twist effects, both of which are suppressed at large nucleon momentum.
%
A complementary approach to the LaMET methods views instead the quasi-PDF as a ``lattice cross-section'' from which the light-front PDF can be factorized~\cite{Ma:2014jla, Ma:2014jga}. 

Preliminary results from lattice calculations of quasi-PDFs have been rather encouraging~\cite{Lin:2014zya,Alexandrou:2015rja,Chen:2016utp,Alexandrou:2016jqi}, although a number of important limitations still need to be overcome.
%
To illustrate this progress in lattice calculations of $x$-space
PDFs, in Fig.~\ref{fig:xspaceLatticePDFs} we show
the renormalized unpolarized isovector quark distribution, $u-d$,
   after one-loop matching and mass correction at the renormalization scale $\mu=2.4$~GeV.
%
The red band shows the extrapolation to infinite momentum, together with the associated
uncertainties.
%
The negative-$x$ part of this figure
is related to the antiquark distribution by means
of the following relation:
\be
\label{eq:minusPDF}
\bar{u}(x)-\bar{d}(x) = - u(-x)+ d(-x)\, \, \quad {\rm for}~~x>0.
\ee
Although these calculations are still in its infancy, they represent a promising
approach to be able to complement global PDF fits with non-trivial
information about the $x$ dependence.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.45]{9-future/plots/xspaceLatticePDFs.pdf}
   \caption{\small Left: 
the renormalized unpolarized isovector
   quark distribution, $u-d$, at $\mu=2.4$~GeV, where
the red band shows the extrapolation to infinite momentum.
%
The negative-$x$ part of this figure
is related to the antiquark distribution by means of Eq.~(\ref{eq:minusPDF}).
   %
   Right: comparison of polarized quark triplet,
$\Delta u - \Delta d$,
obtained from quasi-PDF computed with either fully renormalized matrix elements (blue) 
or with bare matrix elements multiplied by the local
axial current $Z$-factor (magenta).
%
We also show the results of two recent
polarized PDF fits, namely DSSV08 and JAM15.
    \label{fig:xspaceLatticePDFs}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The same methodology can be also applied to other nucleon matrix elements,
including the polarized PDFs.
%
To show this, in Fig.~\ref{fig:xspaceLatticePDFs} we also display
the
comparison of matched helicity PDFs for the same quark flavor
combination, namely $\Delta u - \Delta q$ (and where also
the convention in  Eq.~(\ref{eq:minusPDF}) is adopted).
%
These results have been obtained from quasi-PDF computed with either fully renormalized matrix elements 
or with bare matrix elements multiplied by the local axial current $Z$-factor.
%
For illustration purposes, 
we also show the results of two recent
polarized PDF fits, namely DSSV08~\cite{deFlorian:2009vb}
and JAM15~\cite{Sato:2016tuz}).
%
The small-$x$ region, where the current lattice QCD calculations are not reliable,
is not shown.
%
From this comparison, we see that the lattice QCD results are still far from the global
fits (and thus also from the experimental data), but as in the unpolarized case there
is still ample room for further progress here.

Despite these remarkable developments, there still remain a number of important challenges that must be overcome
before one can achieve a complete  determination of the $x$-dependence of PDFs directly from lattice QCD
that is competitive with the global PDF fits.
%
In particular, excellent control over the various sources of systematic uncertainties that affect
the calculation must be reached.
%
Some of these are common to the calculations of PDF moments, as discussed above,
but there are also a number of additional systematic errors specific to quasi-PDFs, such as those
associated with the finite nucleon momentum of the lattice calculations and with the
renormalisation of quasi-PDFs.
%
Once this is achieved, and given the recent fast progress, it is thus conceivable that in the future
lattice QCD calculations of $x$-space PDFs can be used to constrain global analyses.
