%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.3]{5-pdfgroups/plots/htwist.pdf}\qquad
   \includegraphics[scale=0.3]{5-pdfgroups/plots/gluhigh.pdf}
   \caption{\small Left: the higher twist coefficient for the $F_2$ structure function from the ABMP16 fit, including the associated
     1--$\sigma$ uncertainty.
%
     Right: percentage difference in ABMP $n_f=3$ gluon distribution between the default result and fits performed with higher $W^2$ cuts but
     without the  higher twist corrections.
%
The 1--$\sigma$ uncertainty bands are also shown.
%
     Taken from~\cite{Alekhin:2017kpj}.
    \label{fig:ABMhtwist}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The ABMP16~\cite{Alekhin:2017kpj} set is the latest PDF fit from the ABM collaboration,
following on from the previous
ABM11~\cite{Alekhin:2012ig}, ABM12~\cite{Alekhin:2013nda}, and ABKM09~\cite{Alekhin:2009ni} sets.
%
In turn, these PDF sets were based on the earlier fits of
Refs.~\cite{Alekhin:1996za,Alekhin:2000ch,Alekhin:2005gq} to HERA and fixed proton and deuteron target DIS data, with the ABKM09 fit~\cite{Alekhin:2009ni} and those that follow it including in addition fixed target Drell--Yan and dimuon production data from neutrino DIS on fixed nuclear targets.
%
The ABM sets of parton distributions are parameterised in terms of polynomials in $x$,
see Sect.~\ref{sec:fitmeth.PDFpara},
with the latest fits including 25 free parameters.
%
In the context of this fit to a somewhat reduced dataset,
the use of the classical `$\Delta\chi^2=1$' criteria for determination of the PDF errors
is applied. All sets from ABKM09 onwards go to NNLO in the strong coupling.

Two notable features of these fits are the use of a purely FFNS for the charm and bottom quark contributions to DIS structure functions using the $\overline{\rm MS}$
running masses~\cite{Alekhin:2010sv},
 and the treatment of higher--twist effects.
%
 In the latter case, no attempt is made to impose a cut to remove the region of sensitivity to such effects. Rather, a lower cut of $W>1.8$ GeV is imposed for the DIS data, lower that what
 is typically applied in other PDF fits.
%
 The structure functions are then given by
\be\label{eq:abmf}
F_i(x,Q^2)=F_i^{\rm TMC}(x,Q^2)+\frac{H_i^{\tau=4}}{Q^2}\;,
\ee
where $i=2,T$. Thus $x$ dependent and $Q^2$ independent twist--4 corrections $H^4_i$ are included. While the effect of these dies off with increasing $Q^2$, at lower scales they can have a significant effect. These are then parameterised in terms of cubic splines defined at $x_{k}$ ($k=1...7$) points roughly linearly spaced between $x=$ 0 and 1, which are then determined from the fit. The result for the $F_2$ correction is shown in Fig.~\ref{fig:ABMhtwist} (left), and is found to be inconsistent with zero, in particular at higher $x$.
%
The effect of these corrections, and of conversely omitting them and including a more stringent $W^2$ cut on the DIS data is shown in Fig.~\ref{fig:ABMhtwist} (right) for the extracted gluon PDF.
%
The fit with the cut of $W^2>12.5$ GeV$^2$
and no higher twist corrections is found to prefer a somewhat larger gluon at higher $x$, and in some regions lies outside the 1--$\sigma$ uncertainty band of the default fit.

In addition to the higher twist corrections included in Eq.~(\ref{eq:abmf}),
the structure function functions also include target mass corrections, that is the impact of terms $\sim M_N^2/Q^2$, where $M_N$ is the nucleon mass. These are taken into account according the Georgi--Politzer prescription~\cite{tmc} (see also~\cite{Alekhin:2012ig}), with
\be
F_2^{\rm TMC}(x,Q^2)=\frac{x^2}{\xi^2\gamma^3}F_2(x,Q^2)+6\frac{x^3M_N^2}{Q^2\gamma^4}\int_\xi^1\frac{{\rm d}\xi'}{\xi'}F_2(\xi',Q^2)\;,
\ee
where $\xi=2x/(1+\gamma)$ and $\gamma=(1+4x^2M_N^2/Q^2)^{1/2}$, and a similar result holds for $F_T$. Thus, as $Q^2 \to \infty$ the corrected $F_i^{\rm TMC}$ reduce to the regular $F_i$.
%
The same TMC are also included in other PDF analysis, such as in the NNPDF family~\cite{Ball:2008by}.

As mentioned above, the ABMP fit in  uses a purely fixed flavour scheme to describe the DIS
structure function data.
%
That is, this is fit with $n_f=3$ light quark PDFs with the heavy $c,b$ treated as massive final--state partons which can be produced at order NLO and higher,
regardless of the value of $Q^2$.
%
It is then
argued that the bulk of the DIS data can be described within this scheme.
%
The Tevatron and LHC collider as well as fixed--target DY data, on the other hand, for which $\mu_F^2\gg m_{c,b}^2$, is treated using a $n_f=5$
flavour set evolved from the same input by means of the NNLO matching conditions~\cite{Alekhin:2009ni}.
%
The use of a FFN scheme as opposed to a GM-VFN scheme for the DIS
structure functions has been shown to be one of the
dominant reasons for the differences between the ABM family and the
global fits~\cite{Ball:2013gsa,Thorne:2014toa}.
%
PDF sets for $n_f=3,4$ and 5 active flavours are made publicly available, with the caveat that
each set should be used within its specified validity regime, for instance the ABMP16
$n_f$ set can only be used in applications such as $Q \ge m_b$.

A further feature of note in the ABM family of PDF sets
is that the strong coupling $\alpha_S$  is always
determined from the fit together with the PDFs.
%
In the ABM11 analysis, this was found to be $\alpha_S(m_Z^2)=0.1134\pm 0.0011$ at NNLO, that is in some tension with the PDG world average
value~\cite{Olive:2016xmw} of $\alpha_S(m_Z^2)=0.1181\pm 0.0013$ (the dominant uncertainty in which is determined by the lattice QCD input)
used by the CT and NNPDF collaborations, and the value extracted by MMHT.
%
While in~\cite{Alekhin:2012ig,Alekhin:2017kpj} the omission of higher twist corrections is found to lead to a sizeable increase in $\alpha_s$, in contrast in~\cite{Thorne:2014toa} the use of the FFNS within the MSTW framework is found to lead to a smaller extracted value, consistent with that seen by ABM(P), while higher twist effects are found to have less of an impact.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\begin{center}
  \includegraphics[scale=0.36]{5-pdfgroups/plots/scanalp.pdf}
  \caption{\small The $\overline{{\rm MS}}$ value of the top quark mass $m_t(m_t)$ obtained in the ABMP16 fit for variants of $\alpha_S(m_Z^2)$ (data points),
    compared to the corresponding best-fit values (hatched bands).
    %
    Taken from~\cite{Alekhin:2017kpj}.
    \label{fig:ABMmt}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In the ABM11 fit~\cite{Alekhin:2012ig}, the heavy quark masses $m_{c,b}$ were included in the $\overline{\rm MS}$ scheme for the first time, in contrast to other PDF fits which adopt instead the pole scheme.
%
This allows the heavy quark mass values to be constrained directly from the PDG results without relying on the perturbative transformation between the $\overline{\rm MS}$ and pole masses, which is known to be poorly convergent.
%
Thus, the quark masses are left free in the fit but with the PDG values added in as pseudo--data points.
%
Specifically, for the charm mass $m_c(m_c)$, the DIS data included in this fit are then found to give a comparable error to the PDG average.

The ABM12 fit~\cite{Alekhin:2013nda} included hadron collider data for the first time,  with a range of LHC $W$ and $Z$ boson measurements,
as well as and top quark pair production data from the LHC and the
Tevatron.
%
The latest ABMP16 fit~\cite{Alekhin:2017kpj} includes an increased LHC set, including single top for the first time, as well as Tevatron lepton asymmetry data.
%
In addition, the HERA I+II combined data set and updated NOMAD and CHORUS data on dimuon production are fit.
%
For the $t\overline{t}$ data, the mass $m_t$ is treated in the $\overline{\rm
  MS}$ scheme and is determined from the fit, giving $m_t(m_t)=160.9\pm 1.1$ GeV. This is consistent with the PDG value of $160.0^{+4.8}_{-4.3}$ GeV, although this clearly has quite a large uncertainty, as it based on a single Tevatron measurement.
%
The result is shown in Fig.~\ref{fig:ABMmt}, with the masses extracted at different $\alpha_S(m_Z^2)$ are also given. The correlation between $m_t$ and $\alpha_s$ is clear; as discussed in~\cite{Alekhin:2017kpj},
further information on this correlation
can be provided by considering single top production data. 

Interestingly, the extracted value of the strong coupling from the ABMP16
analysis, $\alpha_S(m_Z^2)=0.1147\pm 0.0008$, is somewhat larger in this fit than in the previous sets, due dominantly to the HERA I+II combined data,
as well as by the increased number of LHC experiments,
although this is still lower than the world average value.
%
The ABMP16 PDFs are available at NNLO for 3,4 and 5 fixed flavours, and for a range of $\alpha_S(m_Z^2)$ values, from 0.112--0.120 in steps of 0.001, in the $n_f=5$ flavour scheme,
including the corresponding Hessian eigenvectors.
%
Note that, as explained, in Sect.~\ref{sec:fitmeth.theoryunc}, in the
ABM fits the uncertainties due to the value of $\alpha_S(m_Z^2)$
and the heavy quark masses is included as an additional Hessian
eigenvector.

 




