The CTEQ-Jefferson Lab (CJ) collaboration has performed a series of global
PDF analyses~\cite{Accardi:2009br,Accardi:2011fa} with
the latest PDF set being CJ15~\cite{Accardi:2016qay}, following the
previous CJ12 set~\cite{Owens:2012bv}. The analyses are carried out at NLO in QCD only, and
focus on utilising DIS data at the highest-$x$ values applicable to a perturbative QCD
treatment. The kinematic selection cuts are chosen to be $Q^2>1.69\,{\rm GeV}^2$
and $W^2>3\,{\rm GeV}^2$ so as to keep data points at low-$Q$ and high-$x$,
unlike the more restrictive cuts used by other analyses.
%
This results in about 1300 more data points from proton and
deuteron targets, roughly a 50\% increase as comparing to standard cuts.
These
additional data points provide useful information on the PDFs at large-$x$, into the
$x\gtrsim 0.7$ region where the constraints for most global analyses are indirect
or purely from extrapolation. In particular, the deuterium data can improve on the
determination of $d$ quark at large-$x$, for which the proton DIS data are less
sensitive.     

For the treatment of heavy-quark mass effects in DIS structure functions, CJ12 uses
a ZM-VFN scheme with heavy-quark masses implemented as the flavor thresholds.
%
CJ15 uses a more adequate GM-VFN scheme S-ACOT~\cite{Kramer:2000hn} to better describe
data over a wide kinematic range, including the threshold regions.
%
It is found that
the implementation of the GM-VFN scheme leads to large changes in the gluon PDF at large-$x$.  
Going to low-$Q$ and large-$x$ involves further complications to the theoretical predictions
for the DIS structure functions, as finite $Q^2$ corrections to the leading-twist
calculation, {\it i.e.} power corrections of $\mathcal O(1/Q^2)$, must be taken into account.
%
In this respect, the
CJ analyses adopt the standard Operator Product Expansion (OPE) expression for the target mass corrections (TMCs)
which allows structure functions at finite $Q^2$ be expressed in terms of their
massless ($M^2/Q^2\sim 0$) values through the scaling variable
$\rho^2=1+4x^2M^2/Q^2$~\cite{Georgi:1976ve,Brady:2011uy}.
%
Other subleading $1/Q^2$ corrections, including
higher twists, they are parametrized by a phenomenological function form~\cite{Accardi:2016qay}.

Another important aspect of the CJ analyses concerns the nuclear corrections for processes
with deuteron targets, which become significant in the intermediate and large-$x$ region
and are equally important for low and high $Q$ values.
%
The nuclear corrections
account for Fermi motion, binding, and nucleon off-shell effects can be implemented
as convolutions with nuclear smearing functions.
%
In the CJ12 analysis, three PDF fits are
provided with different models of deuteron corrections, CJ12min, CJ12mid and CJ12max,
corresponding to mild, medium, and strong corrections, respectively.
%
These corrections are only applied at the level of
structure functions.

The CJ15 analysis employs a phenomenological parametrization
for part of the deuteron corrections with free parameters fitted to data, reducing
the model dependence and increasing the flexibility of the fit.
%
Moreover, these deuteron corrections
are formulated at the parton level and can therefore also be applied to non DIS processes.    
For example, the total quark PDF in deuteron can be written as $q^d=q^{d({\rm on})}
+q^{d({\rm off})}$, with the on-shell and off-shell components given by~\cite{Melnitchouk:1993nk,Kulagin:1994fz},
\begin{align}\nonumber
q^{d({\rm on})}(x,Q^2)&=\int{dz \over z}f^{({\rm on})}(z)q^N(x/z,Q^2)\;,\\ \label{eq:cjq}
q^{d({\rm off})}(x,Q^2)&=\int{dz \over z}f^{({\rm off})}(z)\delta f^N(x/z,Q^2)q^N(x/z,Q^2),
\end{align}
where $q^N(x,Q^2)$ corresponds to the quark PDF in a free nucleon.
%
The on-shell and off-shell smearing
functions $f^{({\rm on})}$ and $f^{({\rm off})}$ can be calculated systematically within
the weak binding approximation, using the deuteron wave functions~\cite{Kulagin:2004ie,Kahn:2008nq}.
The nominal CJ15 PDF fit is based on AV18 wave functions~\cite{Wiringa:1994wb}, but alternative fits with
CD-Bonn~\cite{Machleidt:2000ge}, WJC-1 and WJC-2~\cite{Gross:2008ps} wave functions are also provided. 
The off-shell correction $\delta f^N(x)$ in Eq.~(\ref{eq:cjq}) is parametrized as~\cite{Kulagin:2004ie}
\begin{equation}
\delta f^N(x)=C(x-x_0)(x-x_1)(1+x_0-x).
\end{equation}
The two zeros $x_0$, $x_1$ and the normalization $C$ are free parameters fitted to data
with the constraint of maintaining the total number of valence quarks in the nucleon.
%
It was found that different wave function models give similar quality
of fits to the global data set and result in changes in the PDFs that are well within the uncertainties,
since their differences can be largely compensated by the parametrization of
the off-shell corrections.  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.49]{5-pdfgroups/plots/cj_du_err.pdf}
  \includegraphics[scale=0.49]{5-pdfgroups/plots/cj_du_pdfs.pdf}
   \caption{\small 
   Left plot: relative error (90\% C.L.) on the $d/u$ PDF ratio as a function of $x$ at
   $Q^2=10\,{\rm GeV}^2$ from the CJ15 fit, compared with uncertainties obtained in fits excluding
   certain data sets~\cite{Accardi:2016qay}.
   %
   Right plot: comparison of the $d/u$ ratio at $Q^2=10\,{\rm GeV}^2$ for different PDF
   sets, including CJ15, MMHT14, CT14, and JR14, with PDF uncertainties shown for 90\%
   C.L. except for MMHT14 at 68\% C.L.~\cite{Accardi:2016qay}
    \label{fig:cj2}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In the CJ15 analysis two new data sets are found to have significant effects on
constraining the $d$-quark PDF at large-$x$, besides the deuteron data.
%
These are the
measurement of the $F_2$ structure function of a nearly free neutron inside a deuteron nucleus
from the BONuS experiment~\cite{Baillie:2011za,Tkachenko:2014byy} at Jefferson Lab using a
spectator tagging technique, and
the lepton and reconstructed $W$ boson charge asymmetry measurements from
D0 Run 2 with full luminosities~\cite{Abazov:2013rja,D0:2014kma}.
%
In Fig.~\ref{fig:cj2} the plot on the left shows the
impact of different data sets on PDF uncertainty (90\% C.L.) of $d/u$ ratio in CJ15 fits.
It was found that at $x\lesssim 0.3$ the DIS data from deuteron target can reduce
the PDF uncertainty on $d/u$ by almost 50\%.
%
For $x\gtrsim 0.3$ the $W$ asymmetry data provides the
dominant constraint. Besides, the constraint from deuteron DIS data
dies out for $x\gtrsim 0.6$ which turns into fit to the deuteron off-shell corrections.
%
In Fig.~\ref{fig:cj2}, the comparison of the $d/u$ ratios from
CJ15, MMHT14, CT14 and JR14 PDFs is shown.
%
They are in good agreements within PDF uncertainties as $x$ goes to 1.
%
The CJ15 PDF set has smaller PDF error on $d/u$ in general, with
an extrapolated value
\begin{equation}
d/u \longrightarrow_{x\rightarrow 1} 0.09\pm 0.03,
\end{equation}
at the 90\% C.L., due to the new data sets on constraining $d$-quark at large-$x$.
%
Note that the fact that the $d/u$ ratio has a finite $x\to 1$ limit is built-in in the CJ15
parametrization~\cite{Ball:2016spl}.

On the other hand, with the additional data sets that are less sensitive to the nuclear corrections,
{\it i.e.}, the D0 $W$ asymmetry data and BONuS data, the CJ15 analysis is able to
pin down the deuteron corrections through the interplay of these measurements and the deuteron DIS
structure function data.
%
This is illustrated in Fig.~\ref{fig:cj3}, which gives the deuteron to isoscalar nucleon ratio $F_2^d/F_2^N$ from
the CJ15 fits with different input wave functions together with the 90\% C.L. uncertainty
of the CJ15 nominal fit shown in the coloured bands. Significant corrections are found for
$x\gtrsim 0.7$.
%
The ratio turns out to be insensitive to the choice of wave functions, since it
is only the combination of the wave function and the off-shell corrections that are
constrained by current data.               

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.49]{5-pdfgroups/plots/cj_F2dN_wfn.pdf}
   \caption{\small 
   Ratio of deuteron to isoscalar nucleon structure functions $F_2^d/F_2^N$ at
   $Q^2=10\,{\rm GeV}$ for the CJ15 fits with different models of wave functions~\cite{Accardi:2016qay};
   the colored band is the 90\% C.L. error for CJ15 nominal fit, obtained
   using the AV18 nuclear wave functions. 
    \label{fig:cj3}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
