\label{sec:theoreticalunc}

The development of sophisticated
methodologies for PDF fits, as well as the
availability of a wealth of high-precision data,
have reduced the PDF uncertainties in global
analysis, arising mostly from experimental data and from procedural
choices, to the few percent-level in the most constrained regions.
%
At this level of accuracy, various
theoretical uncertainties become more and more important, representing
a major limitation for present and (even more) for future studies.
%
Therefore, robustly accounting for these theoretical uncertainties
is of the main goals of PDF fitters for the near future.

In this section, we focus specifically on
the role of the theoretical uncertainties due to missing higher orders (MHOU)
in the
QCD coupling constant, namely those arising
from the truncation of the asymptotic perturbative
expansion.
%
In this respect, there have been a number studies recently
on how to estimate MHOU, although we are still
far from a conclusive answer. In the following we first review progresses
on MHOU of calculations of non-hadronic and hadronic processes and then
several recent studies related to PDF determination.

We emphasize that these theoretical uncertainties from MHO
should not be confused with the parametric theoretical
uncertainties, that is, 
those arising from the choice of the values of
input parameters such as $\alpha_s$ and $m_c$.
%
These have been reviewed in Sect.~\ref{sec:fitmeth.theoryunc},
and it is well established how to estimate its impact on the
PDF fit and to propagate these parametric
uncertainties into collider cross-sections.

\subsubsection{MHOU on matrix element calculation}
%
The most frequently used and probably also the simplest method of estimating
the missing higher order corrections is the variation of the
renormalization
and factorization scales in a given fixed-order
calculation.
%
In the case of the total inclusive cross sections or decay rates
with a single hard scale $Q$, usually one
varies the QCD renormalization scale $\mu_R$ within the interval $[Q/r,\,rQ]$.
The induced changes on the physical observable,
either from three-point evaluations or from a scan over the entire range,
are taken as the {\it uncertainty} of the MHO, assuming then
typically either a Gaussian (or
two half-Gaussians) or a flat distribution.
%
The conventional choice is
$r=2$, which is found to work well in most cases, but that underestimates
the true higher order corrections in certain cases, especially if
the fixed-order calculation is carried out at leading order.

At hadron
colliders, there exist in
addition the factorization scale $\mu_F$, arising from the 
factorization of collider QCD divergences
due to initial state hadrons.
%
The two scales $\mu_R$
and $\mu_F$ can be varied either simultaneously or independently within
above range, with the later case usually further restricted to
$1/r\leq \mu_F /\mu_R\leq r$.
%
However, even for a single scale problem, there
can still be different choices of the central or nominal scale,
e.g., $Q/2$ or $2Q$, motivated by either QCD resummation or speed
of convergence of the series~\cite{Czakon:2016dgf}, which leads to further
ambiguities in the estimation of
theoretical uncertainties from scale variations.
%
There are also some studies on utilizing the so--called principal of maximum
conformality to determine the QCD renormalization scale at different
orders, which claims a much smaller MHOU~\cite{Mojaza:2012mf} than traditional
scale variations. 

Determining a suitable prescription for scale variations becomes more
complicated
when moving to differential observables, since here
more hard scales, including
those related to the kinematics, are involved.
%
This therefore usually
requires a dynamical choice for the central scale, which often also depends
on the specific distribution considered.
%
For example, in a recent study on the hadronic
production of top-quark pairs~\cite{Czakon:2016dgf} it was shown that the
preferable scale is half of the transverse mass of the top quark when
studying the transverse momentum distribution of the top quark, and
one fourth of the sum of transverse mass of top quark and anti-quark
when studying rapidity distribution of the top quark.
%
Starting from a given choice of the nominal scale,
the scale variations
can then be evaluated in a similar way to the inclusive case and serve as
an estimate of MHOU.
%
One further complexity arises concerning the correlations
of the MHOU or scale variations in different regions of the
distribution.
%
Typically, they are assumed to be fully (anti-)correlated
in the entire region which leads to very small theoretical uncertainties
in case of a normalized distribution.
%
There have however been attempts to
decorrelate these scale variations based on consideration of the
kinematic dependence of the QCD corrections~\cite{Olness:2009qd}.      

Alternative proposals for estimating MHOU based on
results at known orders exists, such a the so-called
Cacciari-Houdeau (CH) approach~\cite{Cacciari:2011ze}.
The basic idea is to express the full perturbation series in terms of
the expansion parameter $\alpha_s(Q)$ and assume that
all the expansion coefficients
follow the same uniform bounded probability density distribution, in the Bayesian
sense.
Bayesian inference can be used to calculate the probability density of
the unknown higher-order coefficients given those known coefficients
at lower orders. Thus the MHOU, including its probability density distribution, which will be non-Gaussian in general,
can be constructed.
%
The original CH method was developed
for the study of non-hadronic
processes.
%
Subsequently,
in the modified CH ($\overline{\rm CH}$)~\cite{Bagnaschi:2014wea}
approach, it was generalized to include hadronic processes.
%
There the
expansion parameter has been adjusted to $\alpha_s(Q)/\lambda$, with the
parameter $\lambda$ determined from a global survey of selected
processes with known higher-order corrections.
%
To be specific, the best value
of $\lambda$ computed from
the predicted probability density of higher orders is required
to match the distribution from a frequency count in the survey.
%
For hadronic
processes, the optimal value of $\lambda$ is found to be about 0.6,
meaning that the {\it true} perturbative expansion parameter
is actually around $1.7\alpha_s(Q)$ rather than $\alpha_s(Q)$.

To give another example,
the series acceleration method~\cite{David:2013gaa} can also be applied to
approximate the
full result for physical
observables based on the available
information from a finite number
terms of the asymptotic series, {\it e.g.},
using Levin-Weniger sequence transforms.
%
In the `Passarino--David' method of Ref.~\cite{David:2013gaa} it is assumed a uniform
distribution (in the Bayesian sense) for the theory prediction is assumed between the last known partial sum of the perturbation series
and its approximated value from Weniger $\delta$-transform.

To illustrate how these various methods compare
to each other, Fig.~\ref{fig:thunc1}
shows the predictions for the
production cross sections of the Higgs boson via gluon
fusion at the LHC 8 TeV, calculated at LO, NLO, NNLO and approximate N$^3$LO
with a nominal scale of $\mu_R=\mu_F=m_H$~\cite{Forte:2013mda}. The MHOU as estimated
from different approaches are shown, including scale variations, CH, $\overline {\rm CH}$
and the series acceleration method of~\cite{David:2013gaa} at various
perturbative orders.
%
Note that different approaches may have different
interpretations on the uncertainties.
%
In the case of the $\overline{\rm CH}$ method,
the $\lambda$ value has been adjusted to
give almost equal expansion coefficients for the known orders~\cite{Forte:2013mda}.
The $\overline{\rm CH}$ method predicts a larger uncertainty, while the scale variation uncertainty is larger than the $CH$ result.
%
Encouragingly, the difference in the MHOU bands is observed to decrease with increasing perturbative order.
Indeed, the different
approaches turn out to give similar sizes
for the MHOU associated to the N$^3$LO calculation, with
the exception of the original CH.
In addition, we note that
the series acceleration method also induces a shift in the
central value of the prediction.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
  \includegraphics[scale=0.45]{9-future/plots/Higgs_Fig2.pdf}
   \caption{\small
   The cross section for Higgs boson production in gluon fusion calculated
   at increasing perturbative orders~\cite{Forte:2013mda}.
   At each order the theoretical uncertainty is shown for 
   using scale variation (red circles), the CH method (blue crosses),
   and the $\overline {\rm CH}$ method (green squares); at N$^3$LO the
   Passarino-David uncertainty based on series acceleration method
   is also shown (purple diamonds).
    \label{fig:thunc1}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
              
\subsubsection{MHOU on parton distributions}

Global determinations of PDFs are based on
perturbative calculations of matrix elements and DGLAP splitting kernels,
suitably combined to predict a variety of physical cross-sections.
%
In these perturbative calculations, in principle one should
account for their associated  MHOU, which then  propagates into the
resulting PDFs via the fitting of the
theoretical predictions to the experimental
data.
%
Therefore,
sophisticated treatments on the MHOU from different sources
are required in order
to study the impact on the PDFs, not unlike the treatment
of the experimental
systematic uncertainties.
%
Crucially, the correlations between theoretical predictions of
different experimental bins of one process and between different
processes must be accounted for.
%
Furthermore, when making any theoretical prediction,
one should also take care of the correlations between the
MHOU of the PDFs and
of those coming from
the MHOU of the process studied, since they may rely on the same
perturbative expansion for the relevant matrix elements.

Due to the significant complexity of this problem,
there is still no satisfactory solution, 
and the MHOU have not been included in any of the public PDFs from
global determinations. 
%
However, it is possible
to restrict ourselves to a region where a single process is most likely
dominant in the MHOU, such as inclusive
jet cross sections and the gluon PDF at large-$x$. In such a case, it may still be possible to construct a
simple prescriptions like using scale variations.
%
To illustrate this, in Fig.~\ref{fig:thunc2} (Left)
we show the impact of the choice of the QCD scales in calculations of 
the inclusive jet cross sections on the gluon PDFs at $Q=2$ GeV for
alternative CT10 NNLO fits~\cite{Gao:2013xoa}, with two different $\chi^2$ definitions
(see Sect.~\ref{sec:fitmeth.min.chi2def}).
%
Note that the theoretical predictions
here are only at NLO, though the PDFs are determined
at NNLO.
   

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.37]{9-future/plots/gluon_jcc21x_to_cy29c_1.pdf}
  \hspace{0.1in}
  \includegraphics[scale=0.38]{9-future/plots/cteq-jets3.pdf}
   \caption{\small 
   Left plot: dependence of the gluon PDF on the choice of QCD scales
   used in the
   calculation of inclusive jet cross sections in
   CT10 NNLO fits~\cite{Gao:2013xoa}, normalized to the central value
   of the NLO gluon.
   %
   Right plot: impact of the theoretical uncertainties from
   the inclusive jet
   cross sections included in the CT10 NNLO fits~\cite{Rojo:2015acz}
   in the resulting gluon PDFs.
    \label{fig:thunc2}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

From Fig.~\ref{fig:thunc2} we can see that in
the higher $x>0.2$ region, the spread of the gluon PDFs
by using scales of $0.5$, 1 and $2$ times the central
scale (in this case the jet $p_T$) is of the order of the nominal PDF uncertainties. This illustrates the potential
significance of the MHOU in PDF determination.
 %
In a related study, also based on alternative CT10 NNLO fits, scale variations
of the NLO inclusive jet cross sections are further decomposed into
several correlated systematics described by five nuisance parameters~\cite{Rojo:2015acz}.
%
By treating those systematics in a similar way to the experimental
correlated systematic errors, it is possible to include the MHOU
in the standard PDF uncertainty on the same footing as the experimental
systematic uncertainties.
%
As shown in the right plot of
Fig.~\ref{fig:thunc2}, the inclusion of theory errors from the
jet cross-sections in the CT10 NNLO fit results in an increase of
the gluon PDF uncertainty
at large-$x$, consistent with  the left panel of
Fig.~\ref{fig:thunc2}.   

Another possibility to provide a rough estimate of the MHOU
consists in checking the convergence of the fitted PDFs with
increasing orders.
%
Fig.~\ref{fig:thunc3} shows the comparison of the nominal
PDF uncertainties
with the difference of the central PDFs fitted at NLO and NNLO for
gluon and total singlet PDFs at $Q=100$ GeV
in the NNPDF3.0 fits~\cite{Ball:2014uwa}.
%
This difference between the PDF central values at NLO and NNLO
provides a conservative upper bound 
of the MHOU associated to the NNLO PDFs.
%
From this comparison, we can observe that
there are regions where the shifts of
NLO to NNLO PDFs are comparable or even larger than the conventional
PDF uncertainties.
%
Here, one could also apply the CH or $\overline{\rm CH}$
approach based on the fitted PDFs at LO, NLO and NNLO.
%
For example
Ref.~\cite{Forte:2013mda} found that the resulting MHOU of the NNLO PDFs have a
negligible impact on the Higgs production cross section through gluon
fusion, but on the other hand
it could be relevant for top-quark pair production.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
  \includegraphics[scale=0.37]{9-future/plots/xg-th_nnlo_vs_nlo.pdf}
  \hspace{0.1in}
  \includegraphics[scale=0.37]{9-future/plots/xsinglet-th_nnlo_vs_nlo.pdf}
   \caption{\small
   Left  plot: comparisons between the nominal PDF uncertainties with
   the difference of the central PDFs determined at NLO and NNLO for the
   gluon  PDF at $Q=100$ GeV in the NNPDF3.0 fit.
   %
   Right plot: same for the total quark singlet, $\Sigma(x,Q^2)$.
    \label{fig:thunc3}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
