The MMHT14 PDFs~\cite{Harland-Lang:2014zoa} are the successor to the MSTW08~\cite{Martin:2009iq} set, which
in turn derives from the earlier MRST and MRS studies.
%
The first NLO fit~\cite{Martin:1987vw} to DIS data was performed in the late 80s, while in the mid 90s the MRS(A)~\cite{Martin:1994kn} fit was released, including data from HERA and the Tevatron for the first time. This corresponded to a truly global analysis, fitting to fixed target, DIS and hadroproduction data to constrain the PDFs as precisely as possible. Subsequent releases have all built on this approach, but with significant advances achieved over the years due to improvements in both theory and experiment, as
well as with significant methodological developments.
%
The MRST98 release~\cite{Martin:1998sq} was the first set to include a full treatment of heavy flavours within the GM--VFNS developed in~\cite{Thorne:1997ga}, and discussed further in Sect.~\ref{sec:global.heavyq}. This was motivated by the new HERA measurements of the charm structure function, which demonstrated the importance of a consistent treatment of charm production and low and high scales;  indeed, the introduction of this flavour scheme resulted in an improved description of such data.
%
The MRST02 release~\cite{Martin:2002aw} included a full treatment of PDF errors for the first time, described further below
(see also Sect.~\ref{sec:pdfuncertainties}), while the MRST04~\cite{Martin:2004ir} set went to NNLO for the first time.

These elements were all incorporated in the major MSTW08~\cite{Martin:2009iq} release. This presented a global fit to a range of DIS data from HERA and fixed proton and nuclear targets, fixed target Drell--Yan and dimuon production and $W$, $Z$ and jet production at the Tevatron, with $O(2500)$ data points in total. Fits were performed up to NNLO in the strong coupling, with an improved dynamical error treatment, and with an up to date heavy flavour scheme applied.
%
This release aimed to provide a general-purpose PDF set for use at the LHC, which
began operation soon after the release, and was subsequently very widely used in LHC phenomenological studies and experimental analyses.
%
This fit was updated in the latest MMHT14~\cite{Harland-Lang:2014zoa} set, which includes a number of theoretical and experimental updates.
%
In particular for the first time LHC data on $W$, $Z$, $t\overline{t}$ and jet production are included, as well as updated HERA data on the charged, neutral, charm and longitudinal structure functions, and updated Tevatron $W$ and $Z$ measurements. As in earlier fits, for DIS data a $Q^2$ cut of 2 ${\rm GeV}^2$ and $W^2$ cut of 15 ${\rm GeV}^2$ is imposed to avoid sensitivity to higher twist corrections.

In the case of Tevatron jet production, in the absence of a full NNLO calculation at the time, an approximation to the NNLO corrections based on the threshold corrections of~\cite{Kidonakis:2000gi} was applied in the NNLO MSTW08 fit, with the judgment being made that the difference between this and the full NNLO result would be expected to be smaller than the systematic uncertainties on the data, which itself provided the only direct constraint on the gluon at high $x$.
%
At the LHC much of the jet data are quite far from threshold, while those that do not probe a kinematically similar region to the Tevatron data, and so at NNLO these are not included in the MMHT14 fit. For the $t\overline{t}$ data the top mass is allowed to be determined from the fit, with the pole mass value of $m_t=172.5\pm 1$ GeV taken as an input.
%
This gives a value at NNLO that is consistent with the world average, while at NLO it is somewhat lower.

The MSTW PDFs were parameterised in terms of simple polynomials in $x$, with 29 free parameters. However, in~\cite{Martin:2012da} it was shown that this parameterisation was not sufficiently adaptive to describe the Tevatron and LHC $W$ asymmetry data. In particular, it was necessary to introduce a more flexible basis for the interpolating function described in Sect.~\ref{sec:fitmeth.PDFpara.func}, with
\be
\label{eq:chebmmht}
I_f(x)=\sum_i^n \alpha_{f,i}T_i(y(x))\;,
\ee 
where $T_i$ is a Chebyshev polynomial of order $i$ and $y(x)=1-2\sqrt{x}$ is chosen so as to sample a wide range of $x$, and has the additional advantage that this provides a half--integer separation in powers of $x$, as expected on Regge theory grounds.

In order to determine how many parameters $n$ were needed in Eq.~(\ref{eq:chebmmht}),
in~\cite{Martin:2012da} pseudo--data points with a constant percentage error were generated for the required distributions, in terms of a very large order polynomial with additional smoothness constraints applied.
%
The fractional deviation from the true PDF, as well as the decrease in $\chi^2$, were then determined as the number of parameters were increased, until no further significant improvement was observed and the level of agreement was well below the PDF uncertainty for the set.
%
In this way $n=4$ was arrived at as a good choice with which to parameterise the $u_V$, $d_V$, $s+\overline{s}$ and light quark sea $S$ distributions.
%
Fitting to the same MSTW08 data set, these resulted in some improvement in the fit quality, but with the only significant change in the PDF being in the $u_V$ at lower $x$.
%
This was found to lie outside the previous PDF uncertainty band, and the additional flexibility provided a greatly improved description of $W$ asymmetry data from the LHC Run I. 

In the MMHT14 set, this Chebyshev parameterisation is used at $Q_0^2=1\,{\rm GeV}^2$  for the $u_V$, $d_V$, $s+\overline{s}$ and light quark sea $S$ distributions, while for the gluon a term with $n=2$ Chebyshev is included, but with a second term still present, as in MSTW08, which has a different low $x$ power and provides the additional flexibility at low $x$ that is required by the HERA data; this has the effect that the gluon at NLO and higher becomes negative at low $x$ and $Q^2$. Standard polynomial parameterisations are taken for the less constrained $s-\overline{s}$ and $\overline{d}-\overline{u}$ distributions, although as the data becomes more precise we can expect this to change. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.70]{5-pdfgroups/plots/deutCJcomp.pdf}
   \caption{\small Fitted MMHT14 deuteron correction factors with uncertainty, compared to the CJ12~\cite{Owens:2012bv} predictions. Taken from~\cite{Harland-Lang:2014zoa}.
    \label{fig:deutcorr}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%

A further improvement described in~\cite{Martin:2012da} that is included in the MMHT14 set is in the treatment of the non--perturbative corrections that should in general be applied when considering DIS data on deuteron targets, to account for the binding of the proton and neutron within the deuteron. While in MSTW08 and earlier fits, a fixed shadowing correction at small $x$ was applied, a more flexible approach is now taken. In particular the deuteron corrections are freely parameterised in terms of a function $c(x)$, which is determined along with its corresponding uncertainties from the PDF fit. This resulted in a significantly improved description of the BCDMS deuteron structure function data, the E866 Drell--Yan asymmetry and the Tevatron lepton asymmetry data, with some significant changes in $d_V$. The result of the MMHT14 fit is shown in Fig.~\ref{fig:deutcorr} and compared against different model predictions used in the CJ12~\cite{Owens:2012bv} analysis. Interestingly, very good agreement is found with the CJ12mid prediction, demonstrating the power of global PDF fits to extract additional physical information beyond the PDFs themselves.

In MMHT14, the Hessian approach is applied to calculate the PDF errors, with the `dynamical' tolerance criteria described in Sect.~\ref{sec:pdfuncertainties.hess} taken. 
%
For MMHT14 the 68\% uncertainties are calculated using this procedure. In the fit there are 37 free PDF parameters in total, however in the error determination certain parameter directions are found to be largely degenerate, leading to departures from quadratic $\chi^2$ behaviour. This is corrected by fixing some parameters when calculating the error eigenvectors, reducing the number of 25, that is 50 directions.

Other theoretical updates in the MMHT14 include the treatment of the $D\to \mu$ branching ratio, which is required in the fit to dimuon production in DIS. This is now determined from the fit but with the measurement of~\cite{Bolton:1997pq}, which is not determined from dimuon production data, included as a data point. The result is somewhat lower than that taken in MSTW08, corresponding to a larger strangeness, but the most dramatic effect is that the $\sim 10\%$ uncertainty on the branching ratio allows for a much larger strangeness uncertainty when fitting to the same data. Other smaller improvements include an updated treatment of nuclear corrections and a multiplicative, rather than additive, treatment of systematic uncertainties where appropriate. 

In contrast to other global fits, where it is taken as an input, in MMHT the value of the strong coupling is allowed to be determined by the fit, it being argued that valuable information can be provided from global PDF fits about this object. This in addition serves as a consistency test on the overall fit; if the extracted value is in strong tension with the world average then this might indicate that further work is needed. In the fit the preferred values at NLO and NNLO are indeed found to be consistent with the world average, and including this as an additional data point is not found to affect the fit significantly.
%
In particular, the detailed study of~\cite{Harland-Lang:2015nxa} found best fit values of $\alpha_S(m_Z^2)=0.1201\pm 0.0015$ at NLO and $\alpha_S(m_Z^2)=0.1172\pm 0.0013$, to be compared with the world average value of $\alpha_S(m_Z^2)=0.1181\pm 0.0013$. The NNLO $\chi^2$ profile for $\alpha_S(m_Z^2)$ and the corresponding individual constraints from the most constraining data sets are shown in Fig.~\ref{fig:MMHTalphas}. The corresponding PDF sets for a range of $\alpha_S$ values, from 0.108 to 0.128 in steps of 0.001, are publicly available.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.63,trim={0 -2cm 0 0},clip]{5-pdfgroups/plots/globalNNLO.pdf}\quad
   \includegraphics[scale=0.83]{5-pdfgroups/plots/NNLOlimits.pdf}
   \caption{\small Left: global $\chi^2$ values as a function of $\alpha_S(m_Z^2)$. Right:
   best fit $\alpha_S(m_Z^2)$ together with the upper and lower 1$\sigma$ constraints from the most constraining data sets. Both figures correspond to NNLO fits and are taken from~\cite{Harland-Lang:2015nxa}.
    \label{fig:MMHTalphas}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5

Concerning the treatment of the heavy quarks, the GM-VFNS scheme is taken, applying the `optimal'
TR scheme of~\cite{Thorne:2012az} which improves the smoothness of the transition region where the number of active flavours is increased by one. The charm and bottom pole masses are fixed to
$m_c~(m_b)=1.4~(4.75)$ GeV, but a detailed study of the heavy quark mass dependence was
performed in~\cite{Harland-Lang:2015qea}.
%
In particular, when $m_c$ and $m_b$ were taken as free parameters in the fit, the preferred values were found to be somewhat lower than these defaults, possibly suggesting a $\overline{\rm MS}$ scheme may be more appropriate, although the impact of such a choice is expected to be fairly small. Sets with a range of charm and bottom masses, as well as with a maximum number of  active flavours $n_f^{\rm max}=3$ or 4, are also made available.

A subsequent study within the MMHT framework on the PDF impact of the final HERA I + II combined data~\cite{Abramowicz:2015mha}, which was released following MMHT14, was performed in~\cite{Thorne:2015caa}. This was found to lead to some reduction in the PDF uncertainties, principally in the gluon, with little change in the central values.
%
It was therefore decided not to release an updated set but rather to wait until more precise and varied LHC data became available, as well as theoretical calculations such as NNLO jet production. Subsequent work towards a new PDF set has been presented in for example~\cite{Harland-Lang:2017dzr}, where a first fit at NNLO to jet data is presented, while the impact of new LHC data in the fit is seen to be significant. In addition, the first steps towards the inclusion of the photon PDF within the MMHT framework are presented, see Sect.~\ref{sec:QED} for more discussion. A further public release is therefore anticipated in the near future.

