The CTEQ-TEA global analysis was established by Wu-Ki Tung and collaborators
in the early 1990s with
the CTEQ1 PDFs~\cite{Morfin:1990ck}.
%
The most recent release of general purpose PDFs from this
 collaboration are the CT14 PDF sets~\cite{Dulat:2015mca}, which include the nominal sets as well as
alternative sets with different choices of $\alpha_s$ and the maximum number of active flavors $n_f^{\rm max}$.
%
In the CT approach, the PDFs are parameterised at the starting scale $Q_0=1.3\, {\rm GeV}$ using the
functional form
Eq.~(\ref{pdffunc}) described in Sect.~\ref{sec:fitmeth.PDFpara.func}.
%
In pre--CT14 analyses, the interpolating function $I_f(x)$ was
chosen as an exponential of a polynomial in $x$ or $\sqrt x$, such that positivity
conditions on the PDFs at the initial scale were enforced.
%
In the recent CT14 analysis, on the other hand, an improved  parametrization choice
was introduced, with for example for the $u$-valence quark
\begin{equation}
I_{u_v}(y)=d_0p_0(y)+d_1p_1(y)+d_2p_2(y)+d_3p_3(y)+d_4p_4(y),
\end{equation}   
where $y=\sqrt x$ and $p_n$ are the fourth order Bernstein polynomials, given by
\begin{equation}
p_0(y)=(1-y)^4, \,\,p_1(y)=4y(1-y)^3,\,\,p_2(y)=6y^2(1-y)^2,\,\,p_3(y)=4y^3(1-y),\,\,p_4(y)=y^4 \, .
\end{equation}
Namely, the interpolating function is chosen as a fourth-order polynomial in $y$
with an expansion in the basis of Bernstein polynomials. 
%
As discussed in Sect.~\ref{sec:fitmeth.PDFpara.func}, this greatly increases the stability of the fit within the Hessian approach.
%
Moreover, in the CT14
case the positivity of PDFs at $Q_0=1.3\,{\rm GeV}$ is not imposed {\it a priori} but rather
emerges automatically as a
consequence of the fit to data.
%
The CT14 PDFs have a total number of 28 free parameters; using a yet
more flexible parametrizations, by adding higher-order
polynomials, is found to have a small effect on both the best--fit and the estimated
PDF uncertainties in the region that is well constrained by the experimental data.

The CT14 global analysis includes a wide variety of experimental measurements.
%
The majority comes from the inclusive DIS and semi--inclusive DIS measurements of the structure functions
and the reduced cross section measurements from fixed-target experiments (BCDMS~\cite{Benvenuti:1989rh,Benvenuti:1989fm},
NMC~\cite{Arneodo:1996qe}, CCFR~\cite{Yang:2000ju,Seligman:1997mc,Goncharov:2001qe}, NuTeV~\cite{MasonPhD},
CDHSW~\cite{Berge:1989hr}) or
HERA experiments~\cite{Aktas:2004az,Abramowicz:1900rp,Aaron:2009aa,Collaboration:2010ry}.
A $Q$ cut of 2 GeV and $W$ cut of 3.5 GeV are
adopted in the selection of DIS data to minimize non--perturbative effects
from either nuclear corrections or higher--twists corrections.
%
Thus no further nuclear
or higher twists corrections are included in theory predictions in CT14 except for those
already applied in the unfolding of experimental data. For the NC DIS process, the CT14 analysis uses a
treatment of heavy-quark mass effects up to NNLO, through a variant of the GM-VFN scheme
known as S-ACOT-$\chi$~\cite{Guzzi:2011ew}. 
%
For CC DIS, the theoretical calculations are
only implement at NLO, which is judged to be sufficient given the relatively small number of data points and their large experimental errors.
Drell-Yan production data from
fixed-target experiments (E605~\cite{Moreno:1990sf}, E866~\cite{Towell:2001nh})
and $W/Z$ boson production data from Tevatron~\cite{Abe:1996us,Acosta:2005ud,Abazov:2007pm,
Abazov:2006gs,Aaltonen:2010zza} including
the new D0 electron charge asymmetry data~\cite{D0:2014kma}, are also fit.

The Tevatron $W,Z$ data provide further discriminations on
quark flavors in large-$x$ region, with the $W$ asymmetry data
probing the average slope of $d/u$ ratio at $x\gtrsim 0.1$, see Sect.~\ref{sec:datatheory.gauge.sensitivity}
for a more detailed discussion.
%
NNLO predictions from ResBos~\cite{Balazs:1995nz,
Balazs:1997xd,Landry:2002ix,Guzzi:2013aja} are used for the $W/Z$ boson
production data, with a $p_T$ cut imposed on the charged leptons, and incorporating soft gluon
resummation effects at small $p_T$ of the vector boson.
%
These resummed predictions provide
a better description of the $p_T$ spectrum of the charged leptons. The updated
D0 electron charge asymmetry data~\cite{D0:2014kma} shows a large impact on the $d/u$ PDF ratio at
large-$x$ comparing to CT10 and CT10W~\cite{Lai:2010vv,Gao:2013xoa}.
%
In the CT10 fits
the D0 lepton charge asymmetry data resulted in larger asymptotic value of $d/u$ though
tensions were found between different subsets of the data or the D0 data and other
DIS experiments.
%
As shown in Fig.~\ref{fig:ct1}, for CT14 the updated D0 electron charge asymmetry data shows better
agreements with other data sets in the global analysis and drives the $d/u$ ratio
to a lower value close to CTEQ6.6~\cite{Nadolsky:2008zw} at large-$x$.
%
The $d/u$ ratio in CT14 also shows good agreement with the extraction from CJ12~\cite{Owens:2012bv}, which is
based on independent large--$x$ and low--$W$ DIS data, including power corrections and deuteron corrections.    

Similar data on $W/Z$ boson production from LHC Run I are also included from the ATLAS~\cite{Aad:2011dm},
CMS~\cite{Chatrchyan:2013mza,Chatrchyan:2012xt} and LHCb~\cite{Aaij:2012vn}
experiments, which further extend the coverage to the intermediate and small $x$ region.
In addition, single inclusive jet production from the Tevatron~\cite{Aaltonen:2008eq,Abazov:2008ae}
and the LHC~\cite{Aad:2011fc,Chatrchyan:2012bja} are fitted, providing the dominant constraint
on the gluon PDF at large $x$, with the latter data also extending the coverage to the intermediate
$x$ region.
%
For inclusive jet production at hadron colliders only the NLO predictions were available
at the time of the CT14 fit, and therefore this is used in the NNLO fit. This will be updated with the recent NNLO
calculations~\cite{Currie:2016bfm} in future installments of the CTEQ-TEA global analysis.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.6]{5-pdfgroups/plots/ct14dou.pdf}
   \caption{\small
   A comparison of the  $d/u$ ratio at $Q=10$ GeV for
   the CT14 NNLO (solid blue), CT10 NNLO (dashed red), and
    CJ12 NLO (green lines) analyses, together with the
    corresponding
90\% CL PDF uncertainties. From Ref.~\cite{Dulat:2015mca}. 
    \label{fig:ct1}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The CTEQ-TEA group uses the Hessian method with certain tolerance conditions for the
nominal fits in the determination of PDF uncertainties at 90\% CL, see Sect.~\ref{sec:pdfuncertainties}.
%
In addition, the Lagrange Multiplier (LM) scan method is also adopted 
 for specific important observables or for PDFs in region poorly constrained
by data. 
%
In the pre-CT10 analyses, it
was found that within a global $\chi^2$ tolerance of $\Delta \chi^2=100$ (for more than 2000 data
points) the fits agree with all experiments at the 90\% CL.
%
In latter CTEQ-TEA analyses, a more
efficient dynamic tolerance criteria is adopted to account for the agreement with individual
data set.
%
It is constructed from an equivalent Gaussian variable, e.g.,
\begin{equation}
\label{sec:equivalent}
S_n=\sqrt{2\chi^2(N_n)}-\sqrt{2N_n-1},
\end{equation}
where $N_n$ is the total number of data points in data set $n$ and $\chi^2(N_n)$
represents the $\chi^2$ of the fit to that data set. $S_n$ follows a normal distribution
given the number of data points is large enough.
%
Thus a value of $S_n$ greater than
1.3 will be excluded at 90\% CL.
%
Subsequently,
a second layer of penalty is added to the global
$\chi^2$ when determining the boundaries of confidence intervals, called a Tier-2 penalty,
and defined as
\begin{equation}
P=\sum_{n=1}^{N_{\rm exp}}(S_n/S_{n,\rm best})^{16},
\end{equation}
where the sum runs over all data sets included and we normalize $S_n$ to its value in
the best-fit $S_{n,\rm best}$ to account for poor fit to certain experiments.
%
The power of 16 is
introduced so that the penalty will reach the tolerance of 100 as soon as any data set
shows disagreement at 90\% CL.
%
The tolerance criteria then changes to $\Delta\chi^2$+P=100.
Fig.~\ref{fig:ct2} shows the distribution of $S_{n,\rm best}$ for all 33 experiments included
in CT14 analysis.
%
The
distribution is wider than a normal distribution, indicating the presence of disagreement,
or tensions, between some of the included experiments, a well-established fact in global
PDF fits.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.60]{5-pdfgroups/plots/ct14sn.pdf}
   \caption{\small
   Best-fit values of the equivalent Gaussian variable $S_n$, Eq.~(\ref{sec:equivalent}),
   for all 33 experiments in the CT14 NNLO global analysis~\cite{Dulat:2015mca}. 
    \label{fig:ct2}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

With the best-fit and $2N_{\rm eig}$ uncertainty eigenvectors sets, the asymmetric errors for any QCD
observable $\mathcal{F}$ can be calculated through the master formula:
\bea
(\delta \mathcal{F})_+&=&\sqrt{\sum_{i=1}^{N_{\rm eig}}[{\rm max}(\mathcal{F}_{+i}-\mathcal{F}_0,\mathcal{F}_{-i}-\mathcal{F}_0,0)]^2}\, ,\,\,\qquad \nonumber \\
(\delta \mathcal{F})_-&=&-\sqrt{\sum_{i=1}^{N_{\rm eig}}[{\rm max}(\mathcal{F}_0-\mathcal{F}_{+i},\mathcal{F}_0-\mathcal{F}_{-i},0)]^2}\, ,
\eea
where $\mathcal{F}_0$ is the prediction from central set, and
$\mathcal{F}_{+i}$ and $\mathcal{F}_{-i}$ are from
two error sets in the direction of $i$-th eigenvector.
%
These PDF errors can be scaled
down to 68\% CL with a factor of 1.64 assuming Gaussian distributions.


As mentioned above, the CTEQ-TEA analysis also uses the Lagrange multiplier method~\cite{Stump:2001gu} to
cross-check the error estimation from nominal Hessian sets. In the CT14 analysis,
Lagrange multiplier scans have been performed for the cross sections of Higgs boson production
via gluon fusion and of the top quark pair production at the LHC. In such scans the
best-fits and the associated $\chi^2$ are found for each fixed value of the observable studied.
%
Then the PDF uncertainties on the observable are determined from the $\chi^2$ profile obtained
using the same tolerance criteria as in the Hessian method.
%
Fig.~\ref{fig:cthiggs} shows the
good agreement of the 90\% CL uncertainties for the Higgs cross sections from the CT14
Hessian PDFs and the CT14 LM scans.
%
The latter can be read off from the intersection of
the horizontal line $\Delta \chi^2=100$ and the various curves. The LM method does not rely on the linear approximation, and
therefore serves as a robust check of the Hessian results.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.55]{5-pdfgroups/plots/ct14ha.pdf}
  \hspace{0.3in}
  \includegraphics[scale=0.55]{5-pdfgroups/plots/ct14hb.pdf}
   \caption{\small
   Dependence of the increase in $\chi^2$ with respect to the global
   minimum $\chi^2_{\rm min}$, defined as
   $\Delta\chi^2 =\chi^2- \chi^2_{\rm min}$,
   for the constrained CT14 fit on the expected
   cross section $\sigma_H$ at the LHC 8 and 13 TeV~\cite{Dulat:2015mca}, for $\alpha_s(m_Z)$=0.118.
%
The solid
   and dashed curves are for the constrained fits without and with the Tier-2 penalties,
   respectively. The red dots correspond to the upper and lower 90\% CL limits calculated
   by means of the Hessian method.  
    \label{fig:cthiggs}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

There are a few other dedicated applications of the CTEQ-TEA global analysis that have
been presented recently.
%
The CTEQ-TEA analyses uses the
world average of strong coupling constant $\alpha_s(m_Z)$ as an input. Usually the nominal fit
is performed with $\alpha_s(m_Z)=0.118$ at both NLO and NNLO, but additional fits with alternative
$\alpha_s$ choices are also provided, which can be used to compute the combined
PDF+$\alpha_s$ uncertainties.
%
The fit itself  provides a much weaker constraint on $\alpha_s$ than the world average,
see also the discussion in Sect.~\ref{sec:fitmeth.theoryunc.alphas}.
%
Similarly, the pole mass of the charm and bottom quarks are chosen to be close to the
world average values, with $m_c=1.3\,{\rm GeV}$ and $m_b=4.75\,{\rm GeV}$.

The CTEQ-TEA group also
provides specialized fits with non-perturbative charm quark PDFs~\cite{Hou:2017khm,Dulat:2013hea}.
%
For these studies, a sea-like or
valence-like charm distribution is added to the nominal parametrization and then
fitted to the data.
%
Limits on the momentum fraction carried by the fitted charm at
the initial scale $Q_0=1.3$ GeV are then derived. In the most recent CT14 analysis, the limits
are 1.6\% for the SEA model and 2.1\% for the BHPS model both at 90\% CL~\cite{CT14IC}.
Finally, there
are also CT14 QED PDFs~\cite{Schmidt:2015zda} based on a radiative ansatz for the inelastic
component of the photon PDFs.
%
The 90\% CL limit on the momentum fraction of the proton
carried by the photon is found to be 0.11\% at $Q_0=1.3$ GeV,
as derived from a fit to the ZEUS measurement
on isolated photon production~\cite{Chekanov:2009dq}.
%
Recent progress in PDF fits with QED corrections and the role of the photon
PDF is discussed in Sect.~\ref{sec:QED}.

