The theoretical calculations that enter PDF determinations depend on a number
of external parameters, such as the value of the strong coupling constant
$\alpha_s(m_Z)$ and that of the heavy quark masses.
%
The choice of these input parameters therefore represents as an additional component
of the total PDF uncertainty, known as theoretical parametric
uncertainty.
%
As there can be strong correlations between the fitted PDFs and the values of these
inputs, a robust
evaluation of the PDF errors requires a consistent
combination of these with the PDF parametric uncertainties.
%
On the another hand, the global analysis can also provide an independent
determination of those QCD parameters, which can in turn contribute to the
world average values.


\subsubsection{The strong coupling constant $\alpha_S$}
\label{sec:fitmeth.theoryunc.alphas}
%
The current world average value for the
strong coupling constant is $\alpha_s(m_Z)=0.1181\pm 0.0011$.
%
This average is
extracted from six families of determinations, namely $\tau$ decay, lattice
results, $e^+e^-$ jets and event shapes, structure functions, EW precision fits,
and $t\bar t$ cross sections at LHC.
%
These are combined with theoretical predictions at
NNLO or higher orders using the $\chi^2$ averaging method~\cite{Olive:2016xmw}.
%
The 2015 PDF4LHC combined PDF sets are based on a slightly different value of
$\alpha_s(m_Z)=0.1180\pm 0.0015$~\cite{Butterworth:2015oua}, that is,
rounded to a value that is often
used in global fits and with a
 somewhat more conservative uncertainty band. 
%
Individual PDF groups also extract values of $\alpha_s(m_Z)$ including the
uncertainties that are solely from their global analyses.

The choice of strong coupling constant affects a global PDF analysis in two main ways, through
 the DGLAP evolution of the PDFs themselves, and the perturbative QCD predictions for the processes that enter the fit.
%
To study these effects, a scan over different values of $\alpha_s(m_Z)$ value is typically performed.
%
For each choice
of $\alpha_s$, the best-fit of PDFs is found and then the $\chi^2$ profile
is constructed. 
%
The best--fit value of $\alpha_s(m_Z)$ is then identified and the uncertainty on this can be determined in a similar way to the standard PDF uncertainties, using either a `$\Delta\chi^2=1$' or a tolerance criteria.

As an illustrative example, Fig.~\ref{fig:as1} shows the $\chi^2$ profiles from MMHT and NNPDF NNLO global analyses. 
The extracted $\alpha_s(m_Z)$ values at NNLO are $0.1172\pm 0.0013$~\cite{Harland-Lang:2015nxa}
and $0.1173\pm 0.0007$~\cite{Ball:2011us,Lionetti:2011pw} respectively. 
%
The CT and ABMP groups have also extracted values of the strong coupling,
finding at NNLO the values $0.115\pm 0.003$~\cite{Dulat:2015mca}
and $0.1147\pm 0.0008$~\cite{Alekhin:2017kpj}, that is, with rather lower central values than the MMHT and NNPDF
determinations.
%
The error reported by the CT group is much larger than the other groups due to 
the stronger tolerance condition used, in a way that is consistent within uncertainties
with the MMHT and NNPDF results.
%
There is therefore a large
spread in the best-fit values from the different PDF groups, and so
the combined $0.1156 \pm 0.0021$ which enters the world average has a much larger error than
those reported by the individual groups.
%
At NLO. the global analyses prefers a $\alpha_s(m_Z)$
value that is about $0.002\sim 0.003$ higher than at NNLO,
compensating for the missing higher--order corrections.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.65]{4-fitting/plots/globalNNLO.pdf}
  \includegraphics[scale=0.6]{4-fitting/plots/alphasfit-21-nnlo-global.pdf}
   \caption{\small
  Left plot: the profile of global $\chi^2$ as in a scan of $\alpha_s(m_Z)$
  in MMHT2014 NNLO analysis~\cite{Harland-Lang:2015nxa}.
  %
  Right plot: the profile of global $\chi^2$ as in a scan of $\alpha_s(m_Z)$
  in NNPDF2.1 NNLO analysis~\cite{Ball:2011us}, where the error bars indicate fluctuations of
  the $\chi^2$ due to finite number of MC replicas used for each $\alpha_s$ value. 
    \label{fig:as1}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The choice of a value for the strong coupling constant $\alpha_s(m_Z)$ obviously
has a significant impact on the predictions for various important processes at
hadron colliders, such as the Higgs boson production via gluon fusion and
top quark pair production, both of which are proportional to
$\alpha_s^2$ at LO.
%
As mentioned above, here it is crucial to account for the correlations
between $\alpha_s$ and PDFs when evaluating the full uncertainties
of observables at hadron colliders.
%
For example, it is well known that the gluon PDF is anti-correlated
with $\alpha_s$ in the small and intermediate $x$ regions due to the
constraints from scaling violations of inclusive structure functions,
which can partly compensate the change of cross sections due to change of
$\alpha_s$ in the matrix elements.

In principle, it is possible within a global PDF analysis to treat $\alpha_s(m_Z)$ in
exactly the same way as other PDF parameters, {\it e.g.}, in the Hessian method,
by calculating the full Hessian matrix and then determining the eigenvector directions
and the uncertainties along each direction. 
%
In this way the PDF+$\alpha_s$ uncertainty on any observable can
be evaluated using the Hessian error PDFs, in exactly the same way as for the standard case
where only PDF uncertainties are included. 
%
The ABM and later ABMP group follows
precisely this procedure. 
%
The downside of this approach is that it is
not possible to separate the PDF and $\alpha_s$
uncertainties, and each error PDF will be associated with a different
value of $\alpha_s$.

A more convenient but completely equivalent method has been proposed in~\cite{Lai:2010nw}.
%
There, it has been shown that, under the quadratic approximation for the
$\chi^2$, the full PDF+$\alpha_s$ uncertainty can be calculated by
simply adding the usual PDF uncertainty and the $\alpha_s$ uncertainty in
quadrature, with the eigenvectors for PDF uncertainties 
constructed with $\alpha_s$ fixed to its best--fit value.
%       
The $\alpha_s$ uncertainty is then calculated through one additional
eigenvector (with two directions) constructed by fixing $\alpha_s(m_Z)$ to
its upper and lower limits and then fitting the remaining PDF parameters in the usual way.
%
The equivalence of the above two approaches is shown in Fig.~\ref{fig:as2} for the gluon and charm quark PDFs. 
%
This latter approach is now adopted by CT, MMHT, and NNPDF collaborations
due to its simple form and ease of use.   
%
Note that the upper and lower
limits on $\alpha_s(m_Z)$ can come either from the fit itself, as in
the case of MMHT 2014, or can be chosen according to the world average, as 
in CT14 and NNPDF3.1. 
%
Modified ranges for the $\alpha_s$ uncertainty as compared
to the initially assumed range $\delta\alpha_S$
can be easily obtained by means of a linear rescaling~\cite{Butterworth:2015oua}.  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.38]{4-fitting/plots/pA224a_vs_ct66_0_mu2.pdf}
  \includegraphics[scale=0.38]{4-fitting/plots/pA224a_vs_ct66_4_mu2.pdf}
   \caption{\small
   Comparison of the combined PDF+$\alpha_s$ uncertainties for the gluon and charm quark
   PDFs using the full eigenvectors including $\alpha_s(m_Z)$ in the Hessian
   matrix (filled error band with dotted borders) and with the separate PDF and
   $\alpha_s$ uncertainties added in quadrature
   (hatched band with solid borders) from~\cite{Lai:2010nw}.
%
Here the dashed lines
   indicate the PDF uncertainties only.
    \label{fig:as2}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Heavy quark masses}
\label{sec:fitmeth.theoryunc.heavyq}
%
Global PDF fits also rely on the input values of the heavy quark (charm, bottom and top) masses
as external parameters.
%
For instance, in an analysis based on the GM-VFNS (see Sect.~\ref{sec:global.heavyq}), the
charm and bottom quark masses enter through the running of $\alpha_s$,
the boundary conditions for the switching of active flavors, as well as the predictions for the
inclusive DIS structure functions and for open charm/bottom production
in DIS or hadron--hadron collisions.
%
On the other hand, the dependence on the top quark mass is less pronounced unless top quark
production data itself are included in the analysis.
%
In this case,  since some cross-sections such as  $gg\to t\overline{t}$ are strongly dependent on both
$\alpha_s$, the top quark mass, and the gluon PDF, it is important
to account for the non-trivial correlation
between the values of $m_t$ and $\alpha_S$ used in the fit.

Concerning the charm and bottom masses,
the world averages and the corresponding
uncertainties for $m_c$ and $m_b$ in $\overline {\rm MS}$ scheme are~\cite{Olive:2016xmw}
\begin{equation}
m_c(m_c)=1.27\pm 0.03\,{\rm GeV}\,\,{\rm and}\,\,m_b(m_b)=4.18\pm 0.035\,{\rm GeV} \,,
\end{equation}
which can be translated into following pole masses as described in~\cite{Harland-Lang:2015qea}      
\begin{equation}
m_c(m_c)=1.5\pm 0.2\,{\rm GeV}\,\,{\rm and}\,\,m_b(m_b)=4.9\pm 0.2\,{\rm GeV} \,,
\end{equation}
by using the 3--loop conversion between pole and  $\overline {\rm MS}$
values for the bottom quark mass, together with a known relation between
the bottom and charm masses~\cite{Hoang:2005zw}.
%
The larger uncertainties in the pole mass case arise
due to the fact that the pole mass is in fact not well defined due to the diverging series, {\it i.e.} there is a renormalon ambiguity of order $\sim 0.1-0.2$ GeV.
%
On the other hand, this effect largely cancels in the difference of the two masses, and therefore the above uncertainties are highly correlated.

The majority of PDF groups use the pole mass as input, as
the relevant coefficient functions and matrix elements are calculated in
the on--shell scheme.
%
In particular, CT14 takes a default value of $m_{c~(b)}=1.3~(4.75)$ GeV,
MMHT14 takes $1.4~(4.75)$ GeV and NNPDF3.1 takes $1.51~(4.92)$ GeV.
%
Both the CT14 and MMHT 2014 NNLO analyses prefer a lower charm quark mass of
about 1.3 GeV~\cite{Dulat:2015mca,Harland-Lang:2015qea} if it is treated as
a free parameter and fitted to data, which is
consistent with the converted value from the $\overline {\rm MS}$ world average value.
%
In ABMP16 the $\overline {\rm MS}$ masses are extracted directly from the
fit, giving $m_c(m_c)=1.252\pm 0.018$ GeV and $m_b(m_b)=3.84\pm 0.12$ GeV~\cite{Alekhin:2017kpj}.   

As in the case of the strong coupling constant,
the uncertainty due to the heavy quark masses
can be calculated by constructing an additional eigenvector from fits with
alternative mass values.
%
The full uncertainty  can then be obtained by adding it in quadrature to the
PDF uncertainty obtained with the default choice of heavy quark masses.
%
For example, the CT14, MMHT 2014 and NNPDF3.1 analyses all provide a series of best--fit
PDFs with $m_c$ or $m_b$ fixed to alternative values around their default choices.
%
However, there has so far not been an agreement on a common choice of the
heavy quark masses and their errors in global PDF analyses, although this is foreseen for the next PDF4LHC recommendation.

Fig.~\ref{fig:as3} shows the dependence of the predicted total cross sections for electroweak gauge
boson
and Higgs boson production at the 13 TeV LHC on the choice of charm quark pole mass
used in CT14 NNLO analysis~\cite{Hou:2017khm}. 
%
It is found that varying $m_c$ by 0.2 GeV has a negligible effect on the Higgs boson cross section
and induces at most a 2\% change in the weak boson cross sections. This is well
within the PDF uncertainties.
%
Similar conclusion has been reached in the MMHT 2014 analysis~\cite{Harland-Lang:2015qea}.
%
In addition, in both the CT14 and NNPDF3.1 analyses, it is observed that the
effect of varying $m_c$ can be partly canceled by changes of the
non--perturbative component of the charm PDFs, since it is re-absorbed into
the fitted charm boundary condition.
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.6]{4-fitting/plots/CT14ICSEAmcs_WpWm_kf2_Vrap.pdf}
  \hspace{0.2in}
  \includegraphics[scale=0.6]{4-fitting/plots/CT14ICSEAmcs_ZH_kf2_Vrap.pdf}
   \caption{\small
   Dependence of the total cross sections for weak boson and Higgs boson production
   at the 13 TeV LHC 13 on the choice of charm quark mass in the CT14 NNLO analysis~\cite{Hou:2017khm}.
   The ellipse indicates the PDF uncertainties at 90\% C.L. Also shown is the dependence
   of the cross sections on the assumed momentum fraction carried by non--perturbative
   charm PDF at the initial scale. 
    \label{fig:as3}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

     
