increase in uncertainties to other sea quarks at low $x$. These increased
uncertainties at low $x$ in the light sea all result from the same lack of
constraint previously having been supplied directly by the HERA data at low
$x$ and $Q^{2}$.

In order to illustrate the effect of removal of the HERA data in a slightly
different manner, in Fig. 79 we also display the the ratio of the up quark
(left) and gluon (right) at $Q^{2}=10^{4}~{}\rm GeV^{2}$, a scale more
relevant for LHC physics. The up quark at high $x$ displays the same change
already evident in the $u_{V}$ at lower $Q^{2}$, i.e. it is a little higher in
normalisation, with largely unchanged uncertainty. Also, as at low $Q^{2}$,
the gluon distributions above $x=0.01$ in MSHT20 and in the fit without HERA
data are very similar, both in terms of central values and uncertainty. Below
$x=0.01$ we see the gluon in the fit without HERA data falling significantly
below that of MSHT20, and the relative size of the uncertainties increases
until at $x=10^{-5}$ it has become more than 5 times bigger. Clearly this
alternative gluon distribution would give a rather different prediction for
Higgs boson production, particularly as a function of rapidity, with much
bigger uncertainties. Below $x=0.01$ the relative shape of the ratio of the up
quark from the fit without HERA data to that in MSHT20 follows that of the
gluon rather closely, as does the increase in the uncertainty. This is simply
a consequence of the fact that at very small $x$ the evolution of quarks, and
hence their values at high $Q^{2}$, is driven very largely by the gluon, and
the plots would look similar in this region for all quarks and antiquarks.

Figure 77: (Left) $\bar{u}$ PDF and (right) $\bar{d}$ PDF percentage errors at
$Q^{2}=1.9~{}\rm GeV^{2}$ at NNLO showing the effect of removing the HERA data
from the MSHT20 default global fit.

Figure 78: (Left) $s+\bar{s}$ PDF and (right) $g$ PDF percentage errors at
$Q^{2}=1.9~{}\rm GeV^{2}$ at NNLO showing the effect of removing the HERA data
from the MSHT20 default global fit.

Figure 79: (Left) $u$ PDF ratio and (right) $g$ PDF ratio at
$Q^{2}=10^{4}~{}\rm GeV^{2}$ at NNLO showing the effect of removing the HERA
data from the MSHT20 default global fit.

At this stage, now we know the HERA data have a direct impact on the
uncertainties of some of the PDFs at small $x$, we can examine the
eigenvectors of the full fit in more detail to look for signs of their
sensitivity to the HERA data. This goes beyond the simplified analysis of
which data set most constrains each eigenvector in Section 5.3.2 and Table 10.
Specifically, given we know which eigenvectors contribute to which PDFs in
which regions of $x$ from Table 11, we can identify which eigenvectors are
relevant to the low $x$ uncertainties and analyse their dependence on the HERA
data. However, as discussed previously, with the HERA data significantly
altering the central values of some of the PDFs, the eigenvectors alter once
the HERA data sets are removed, and so this analysis still has some
limitations. In any case, from Table 11 we can see that the eigenvector with
the largest contribution to the gluon uncertainties at low $x$ in our default
MSHT20 fit is eigenvector 21 (which is sensible as this eigenvector is largely
made up of the low $x$ gluon power $\delta_{g}$). Whilst this eigenvector is
constrained by the ATLAS 8 TeV $Z$ $p_{T}$ and ATLAS 8 TeV single differential
$t\bar{t}$ dilepton data sets in each direction, examining the changes in
$\chi^{2}$ as we move along the eigenvector in each direction reveals that the
HERA data (specifically the HERA $e^{+}p$ NC $820~{}\rm GeV$ data set) is the
next most constraining data set in the ‘+’ direction. This illustrates the
sensitivity of the HERA data set to the low $x$ gluon, and hence its pull on
the central fit and effect in reducing the uncertainties. It is also
interesting to note that this eigenvector demonstrates how a direction
previously most constrained by the HERA data is now most constrained by new
LHC data. In the ‘-’ direction along this eigenvector however the HERA data
are less sensitive, with many of the new LHC data sets providing more
stringent constraints. Performing a similar analysis for the up and down
antiquarks or total strangeness, which also have notably increased error bands
at low $x$ in the absence of the HERA data, is more difficult, as in the case
of the antiquarks they are not one of the basis PDFs. For the strangeness
several different eigenvectors at low $x$ contribute.

We may also use such a method to understand why the HERA data have an effect
on the central values of the valence quarks at low $x$, without affecting
their uncertainties. For the up valence quark it is eigenvector 28 which is
most relevant at low $x$. Studying its variations in $\chi^{2}$ as we move
from the central fit along either eigenvector direction indicates that the
HERA data are indeed sensitive to moving away from the central PDFs,
particularly in the ‘+’ direction. However, whilst the HERA data therefore
favour moving in one direction away from the central fit here, several of the
other data sets in the fit, including the ATLAS 8 TeV $Z$ $p_{T}$, ATLAS 8 TeV
$W^{\pm}$ and CMS 8 TeV $W$ data sets, all constrain the uncertainties in both
directions, preventing the uncertainties from showing any impact of the effect
of the HERA data. Again, in the absence of these newer LHC data sets the HERA
data would be one of the most constraining data sets on this eigenvector in
the ‘+’ direction. A similar picture also emerges for the down valence PDF and
eigenvector 22, although it is less clear-cut, with more data sets
constraining the uncertainties and the effects of fixed target data sets also
relevant, whilst eigenvector 20 also contributes to the down valence at low
$x$ too. It is at least interesting to note that the latter is constrained by
the NMC $d$ data in the ‘-’ direction, indicating the effect of the fixed
target data sets on the PDFs in this region and reinforcing the suggestion
that it is these which drive some of the PDF changes in the fit with the HERA
data sets removed.

Finally, as we suggested that much of the improvement in the global fit
quality when the HERA data are removed is due to extra momentum that can be
given to PDFs when less momentum is required at small $x$ (particularly in the
gluon), we also investigate a NNLO fit where the total momentum at input is
not required to be exactly 1. On releasing this constraint the global fit
quality improves by about 15 units and the input momentum is 1.011. Hence,
given our typical tolerance value we can conclude that the fit is consistent
with the momentum sum rule being preserved at the one sigma level, with some
small preference for a slightly larger value of the total momentum. The data
sets which improve most in $\chi^{2}$ are similar to those when the HERA data
are omitted, i.e. BCDMS and NMC fixed-target DIS data and ATLAS 8 TeV
$Z~{}p_{T}$ data improve most, followed by ATLAS 7 TeV $W$, $Z$ data. There is
an increase in momentum mainly in the gluon distribution, and rather less so
in the quarks. Hence, a general increase in the gluon, i.e. some increase at
high $x$ without a compensating drop at low $x$ is able to resolve some of the
tension between fixed target DIS data and ATLAS 8 TeV $Z$ $p_{T}$ data without
a very significant change in quark and antiquark PDFs. On this basis we
conclude that there is generally consistency with the conclusions drawn above,
and that some of the tension between HERA data and other sets can be
alleviated by artificially increased input momentum, which can translate into
a modification of quark and antiquark evolution resulting from a change in the
gluon or to an increase in gluon initiated cross sections. However, clearly
much of the tension remains. This may be due to further theoretical
corrections beyond NNLO which are not well mimicked by an increase in
momentum, due to genuine data tensions, or most likely some combination of
both.

###  8.8 Further tensions between data sets

In any global fit with the breadth and detail of the MSHT20 PDFs there will be
tensions between data sets. In addition to those already discussed, such as
that between the E866 Drell-Yan ratio data and the high precision ATLAS 7 TeV
$W$, $Z$ data in Section 8.1, or the small tension between some of the jet
data sets in Section 4.4, there are a number of further, sometimes larger
tensions. We comment on the most significant of these in this section.

One of the most prevalent tensions within the global fit is that between
BCDMS, DØ $W$ asymmetry and ATLAS 8 TeV $Z$ $p_{T}$ data and other data sets
in the fit, in particular with the ATLAS Drell-Yan data sets. In order to
demonstrate and investigate this, the changes in the fit quality of many of
the individual data sets in the fit, and the overall fit quality, when the
former three data sets are removed, are given in Table 16. In Fig 80 we then
show the impact their removal has on the gluon and strangeness, respectively.

We first consider the BCDMS data, which comprise both the proton and deuteron
data [49]. The first column of Table 16 presents the effect of removing just
this pair of data sets. Upon its removal the overall $\chi^{2}$ of the other
60 data sets improves significantly by 53 points, demonstrating a very clear
tension between the BCDMS data and other data sets in the global fit. In
particular, analysing the changes in fit quality of the individual data sets,
the three NMC data sets (proton, deuteron and $d/p$ ratio) all improve by
several points in $\chi^{2}$ with the ratio improving by 7 points. Similarly,
the SLAC $p$ and $d$ structure function data improve by a total of nearly 10
points. Therefore there is a notable tension between the BCDMS data and other
structure function data sets. Beyond this, the E866 Drell-Yan data improve by
11 points and the combined HERA data by approximately $\Delta\chi^{2}=20$,
though in the latter case this absolute improvement is enlarged by the high
number of points the HERA data includes. Other data sets also show significant
sensitivity to the presence or absence of the BCDMS data, such as the DØ II
$W\rightarrow\nu\mu$ asymmetry data and the LHCb $Z\rightarrow e^{+}e^{-}$
[157], which both improve by more than 2 points upon removal of the BCDMS data
despite only having 10 and 9 data points respectively. Whilst this analysis is
done at fixed $\alpha_{S}(M_{Z}^{2})=0.118$, some of the tensions of the BCDMS
data with other data sets may be related in essence to the differences in pull
on $\alpha_{S}$, which can be compensated for elsewhere, such as in the gluon
due to effects on the DGLAP evolution. In particular, the BCDMS data prefer a
low value of $\alpha_{S}(M_{Z}^{2})$, but when this is fixed at
$\alpha_{S}(M_{Z}^{2})=0.118$, it instead favours a generally larger high $x$
gluon than the global fit prefers, in order to slow the fall of the structure
function with $Q^{2}$ by increasing the positive contribution from gluon to
quark-antiquark splitting. This is seen clearly in in Fig. 80 (left). Hence,
the BCDMS data may be in tension with other data sets, which favour a smaller
high $x$ gluon and/or, in some cases, larger values of the strong coupling.
The pulls on the strong couplings of different data sets may therefore
indicate potential tensions with the BCDMS data if they prefer a larger value
of the strong coupling. The pulls on $\alpha_{S}(M_{Z}^{2})$ will indeed be
analysed in a future publication [166], but we already see some consequences
of it even in the fixed $\alpha_{S}(M_{Z}^{2})$ used in this article.

The ATLAS 7 TeV high precision Drell-Yan $W$, $Z$ data set and the 8 TeV
$W^{\pm}$ data set also both display tensions with the BCDMS data, with each
improving by 4.7 and 1.5 points in $\chi^{2}$ respectively, again they both
favour larger values of $\alpha_{S}(M_{Z}^{2})$ in contrast to the lower
values favoured by the BCDMS data. However, the ATLAS 8 TeV double
differential $Z$ data set also favours larger values of $\alpha_{S}$ but shows
very little change in $\chi^{2}$ upon the removal of the BCDMS data. This
demonstrates that the tension can clearly be more complicated than this and
may be due to, for example, the shape of the valence quark distributions
preferred by BCDMS data, and the separation into up and down quarks. Despite
these numerous tensions, other data sets nonetheless favour the inclusion of
the BCDMS data in their fit quality, with some of the jet data sets (although
not all) such as the DØ II jets, ATLAS 7 TeV jets and the CMS 7 TeV jet data
sets worsening a little by $\Delta\chi^{2}=1.4,1.2,4.1$ respectively when the
BCDMS data are removed. On the other hand the CMS 8 TeV jets shows very little
impact of the BCDMS data removal on its $\chi^{2}$. Given the 7 TeV jet data
(from both ATLAS and CMS) favour lower values of the strong coupling (like the
BCDMS data) whilst the 8 TeV data do not, again perhaps this has an impact.
The exact reasons for this agreement with the LHC jet data is unclear as the
latter favours a lower high $x$ gluon (see for example Fig. 12 where the
effect of the removal of the LHC jet data is to raise the gluon here) in
contrast to the BCDMS data. A further data set which worsens noticeably upon
the removal of the BCDMS data sets is the LHCb 2015 $W$, $Z$ data set which
deteriorates by 4.9 points for 67 points, implying compatibility with the
valence quarks preferred by BCDMS data.

The DØ $W$ asymmetry data [21] have already been analysed in section 8.4 and
shown to have significant effects on the global fit, particularly on its
uncertainties, despite being a small data set. Therefore perhaps it is
unsurprising that it also has both tensions and consonances with several of
the other data sets in the fit. Nonetheless its global effect across all the
remaining data sets is small with only a marginal improvement of less than 5
points across the other global fit data sets, perhaps as a result of its small
number of points. Its effects on individual data sets are however larger, and
in particular it affects several of the other electroweak data sets. As
pointed out in section 8.4, replacing this data set with the same data
regarded as an electron asymmetry improved the $\chi^{2}$ of the remaining
data sets by $\Delta\chi^{2}=8.8$, with the older DØ electron asymmetry data
in the fit showing a notable improvement in $\chi^{2}$ of 2.6. Therefore, as
one might anticipate, once the DØ $W$ asymmetry data are removed completely
the older DØ II electron asymmetry data [151] improve substantially by
$\Delta\chi^{2}=5.8$ for just 12 points. On the other hand, the similar DØ II
muon asymmetry data set [152] worsens in $\chi^{2}$ by 3.1 so some of this
improvement is lost in the overall global fit. At the same time, the LHCb 2015
$W$, $Z$ data, ATLAS 7 TeV high precision $W$, $Z$ data set and ATLAS 8 TeV
$W^{\pm}$ data also improve by 2.8, 1.0 and 1.6, respectively, when this data
set is taken out, but the ATLAS 8 TeV double differential $Z$ data worsens by
1.4; this reflects the effects the DØ $W$ asymmetry data has on the up and
down quarks and antiquarks. The effect of the DØ $W$ asymmetry data on the
BCDMS data are also given in Table 16. The DØ $W$ asymmetry worsens notably
when the BCDMS data is left out, implying that they have similar pulls, most
likely on the quarks.

Data set |  $N_{pts}$ |  $\Delta\chi^{2}$ relative to MSHT20
---|---|---
|  |  no BCDMS |  no DØ $W$ asym |  no $Z$ $p_{T}$ |  All 3 removed
BCDMS $\mu p$ $F_{2}$ [49] |  163 |  - |  +2.3 |  +1.1 |  -
BCDMS $\mu d$ $F_{2}$ [49] |  151 |  - |  -1.9 |  -0.3 |  -
NMC $\mu p$ $F_{2}$ [50] |  123 |  -3.2 |  -0.4 |  -1.2 |  -3.6
NMC $\mu d$ $F_{2}$ [50] |  123 |  -3.2 |  +0.2 |  -4.6 |  -7.0
NMC $\mu n/\mu p$ [51] |  148 |  -7.4 |  -0.2 |  -1.2 |  -7.6
E665 $\mu p$ $F_{2}$ [52] |  53 |  -0.1 |  -0.2 |  +1.6 |  +1.2
E665 $\mu d$ $F_{2}$ [52] |  53 |  -0.3 |  -0.2 |  +2.5 |  +2.2
SLAC $ep$ $F_{2}$ [53, 54] |  37 |  -3.6 |  -0.2 |  -0.2 |  -4.7
SLAC $ed$ $F_{2}$ [53, 54] |  38 |  -5.9 |  +0.4 |  +0.6 |  -5.8
E866/NuSea $pp$ DY [149] |  184 |  -11.1 |  -0.7 |  +0.6 |  -13.7
E866/NuSea $pd/pp$ DY [150] |  15 |  -0.5 |  -0.8 |  -1.6 |  -2.1
CCFR $\nu N\rightarrow\mu\mu X$ [57] |  86 |  +0.7 |  +0.3 |  -1.7 |  -1.0
NuTeV $\nu N\rightarrow\mu\mu X$ [57] |  84 |  -1.9 |  -0.5 |  -1.4 |  -7.2
HERA $e^{+}p$ NC $920$ GeV [84] |  402 |  -8.5 |  +0.4 |  -2.6 |  -4.3
HERA $e^{-}p$ NC $460$ GeV [84] |  209 |  -2.0 |  -0.5 |  -5.4 |  -12.2
HERA $e^{-}p$ NC $575$ GeV [84] |  259 |  -1.4 |  -0.3 |  -2.9 |  -4.5
HERA $e^{-}p$ NC $920$ GeV [84] |  159 |  -6.7 |  -0.4 |  -2.2 |  -10.3
DØ II $p\bar{p}$ incl. jets [125] |  110 |  +1.4 |  +0.2 |  +1.9 |  +3.4
CDF II $p\bar{p}$ incl. jets [124] |  76 |  +0.1 |  -0.7 |  +3.1 |  +2.7
CDF II $W$ asym. [90] |  13 |  -1.4 |  -0.3 |  -0.3 |  +2.0
DØ II $W\rightarrow\nu e$ asym. [151] |  12 |  -0.1 |  -5.8 |  +0.3 |  -5.2
DØ II $W\rightarrow\nu\mu$ asym. [152] |  10 |  -2.7 |  +3.1 |  -1.6 |  -1.4
DØ $W$ asym. [21] |  14 |  +4.5 |  - |  -1.6 |  -
LHCb $Z\rightarrow e^{+}e^{-}$ [157] |  9 |  -2.1 |  -0.7 |  +1.0 |  -2.1
LHCb $W$ asym. $p_{T}>20$ GeV [158] |  10 |  +0.3 |  +0.2 |  +1.2 |  +1.5
CMS double diff. Drell-Yan [72] |  132 |  -0.5 |  -0.5 |  +2.6 |  +2.3
LHCb 8 TeV $Z\rightarrow ee$ [97] |  17 |  +1.1 |  -0.8 |  +1.2 |  +1.6
LHCb 2015 $W$, $Z$ [95, 96] |  67 |  +4.9 |  -2.8 |  +1.6 |  +0.1
ATLAS 7 TeV jets [18] |  140 |  +1.2 |  -0.6 |  -2.3 |  -1.9
CMS 7 TeV $W+c$ [99] |  10 |  -0.4 |  0.0 |  +1.6 |  +2.1
ATLAS 7 TeV high precision $W$, $Z$ [20] |  61 |  -4.7 |  -1.0 |  -7.5 |  -11.8
CMS 7 TeV jets [100] |  158 |  +4.1 |  -0.9 |  +0.1 |  +1.8
CMS 8 TeV jets [101] |  174 |  -0.3 |  -1.6 |  -7.0 |  -7.2
CMS 2.76 TeV jet [107] |  81 |  -0.7 |  +0.3 |  -1.6 |  -1.8
ATLAS 8 TeV $Z$ $p_{T}$ [75] |  104 |  -0.7 |  +8.1 |  - |  -
ATLAS 8 TeV single diff $t\bar{t}$ [102] |  25 |  -0.6 |  0.0 |  +3.2 |  +2.4
ATLAS 8 TeV single diff $t\bar{t}$ dilepton [103] |  5 |  -0.1 |  -0.2 |  -0.6 |  -0.9
CMS 8 TeV double differential $t\bar{t}$ [105] |  15 |  +0.3 |  0.0 |  -2.0 |  -1.6
CMS 8 TeV single differential $t\bar{t}$ [108] |  9 |  -0.3 |  -0.4 |  -2.7 |  -3.6
ATLAS 8 TeV High-mass Drell-Yan [73] |  48 |  +0.6 |  +0.7 |  -3.4 |  -2.2
ATLAS 8 TeV $W$ [106] |  22 |  -1.5 |  -1.6 |  -4.6 |  -6.5
ATLAS 8 TeV double differential $Z$ [74] |  59 |  -0.1 |  +1.4 |  -5.4 |  -6.6
Total common data sets |  4363 |  -53.0 |  -4.7 |  -41.3 |  -116.2
Table 16: The change in $\chi^{2}$ (with negative indicating an improvement in
the fit quality) for a selection of data sets once the BCDMS [49], DØ $W$
asymmetry [21] and ATLAS 8 TeV $Z$ $p_{T}$ [75] data sets are removed,
illustrating the tensions of these data sets with several of the other data
sets in the global fit.

Figure 80: The MSHT20 (left) gluon and (right) $s+\bar{s}$ PDFs at NNLO and
$Q^{2}=10^{4}~{}\rm GeV^{2}$, compared to the same fit upon removal of the
BCDMS data, removal of the ATLAS 8 TeV $Z$ $p_{T}$ data and removal of the LHC
jets, top and $Z$ $p_{T}$ data.

Finally, the ATLAS 8 TeV $Z$ $p_{T}$ data show significant impacts on the fit
qualities of other data sets with the overall fit improving by
$\Delta\chi^{2}=41.3$ when this data set is not included. This is also a
reflection of its overall poorer fit quality and relatively large number of
points. This overall improvement in the $\chi^{2}$ masks significant variation
between the data sets with several improving significantly and several
worsening notably, demonstrating both tensions and consonances with the ATLAS
$Z$ $p_{T}$ data and other data sets in the fit. This is true within the
structure function data sets, which vary noticeably in $\chi^{2}$ when it is
removed, with the NMC data sets all improving whilst the E665 data fit quality
worsens. This again may partly reflect the fact that the NMC $p$ and $d$ data
both favour higher values of $\alpha_{S}$, in contrast to the ATLAS 8 TeV $Z$
$p_{T}$ data which favour lower values of the strong coupling, as does the
E665 data. The ATLAS $Z$ $p_{T}$ data are also in tension with the dimuon data
sets, which improve by 3.1 across the NuTeV and CCFR data sets, whereas the
CMS 7 TeV $W+c$ data set favours its inclusion as it worsens by 1.6. The
sensitivity of the $Z$ to the strangeness results in the $Z$ $p_{T}$ data
altering its shape and size, as seen in Fig. 80 (right). Here, the absence of
the $Z$ $p_{T}$ data increases the strangeness, with the largest effect being
near $x=0.3$. In general a lower strangeness is favoured by the older dimuon
data sets but disfavoured by the strangeness enhancements preferred in newer
LHC Drell-Yan data. However, the details in the change in shape on omission of
the $Z$ $p_{T}$ data seem to improve the Dimuon data fit, despite the increase
in magnitude. The HERA combined data in the table also improve by 13.1, albeit
for a very large number of data points.

The impact of the ATLAS $Z$ $p_{T}$ data is also clear on the electroweak
boson, top and jet data sets. In regard to the first of these, there is an
appreciable difference in how the slightly older LHC electroweak data sets
tend to worsen a little once it is removed (including all 4 of the LHCb
electroweak data sets included as well as the CMS double differential Drell-
Yan data), whilst the new ATLAS Drell-Yan data sets all improve significantly
when the ATLAS $Z$ $p_{T}$ data are removed. The 7 TeV $W$, $Z$ and 8 TeV high
mass Drell-Yan, $W^{\pm}$ and double differential $Z$ data sets improve by
$\Delta\chi^{2}=-7.5,-3.4,-4.6$ and $-5.4$ respectively, showing a noteworthy
tension with the 8 TeV $Z$ $p_{T}$ data. At least part of this tension or
improvement is due to the change in total strangeness, with the ATLAS 7 and 8
TeV precision Drell-Yan data indeed favouring strangeness enhancement in the
$10^{-2}<x<10^{-1}$ region. The effect on the jets data is also somewhat split
between the older and newer data with the DØ jets and CDF jets data each
worsening once it is removed whilst the newer ATLAS and CMS jet data sets
across 2.76, 7 and 8 TeV improve by a total of $\Delta\chi^{2}=-10.8$. The
tension with the new LHC jet data is a reflection of the different pulls of
the $Z$ $p_{T}$ and new LHC jet data on the high $x$ gluon, as seen in Fig.
18, with the LHC jet data preferring a lower gluon in this region in contrast
to the $Z$ $p_{T}$ data. The same effects on the high $x$ gluon also apply for
the top data with the new LHC top data also favouring a lower gluon and
improving by 2.1 once the ATLAS $Z$ $p_{T}$ data are removed. However the
detailed picture is a little more varied, with the ATLAS 8 TeV single
differential $t\bar{t}$ data worsening by 3.2 points in $\chi^{2}$ for 25 data
points, whilst the other top data, including the ATLAS 8 TeV single
differential $t\bar{t}$ dilepton data and the CMS 8 TeV single and double
differential top data improve by 5.3 across 29 data points. One interesting
detail to notice on the topic of the ATLAS $Z$ $p_{T}$ data and its tensions
with the newer LHC data sets is that in Table 2 the ATLAS 8 TeV $Z$ $p_{T}$
data set is one of the few data sets fit worse in the MSHT20 global fit than
it would be using the MMHT14 global fit, again in contrast to the ATLAS Drell-
Yan data sets which were all poorly fit in predictions using MMHT14. Finally,
the effects of removing the $Z$ $p_{T}$ data on the $\chi^{2}$ of the BCDMS
and DØ $W$ asymmetry data are notable. The BCDMS data show little change,
perhaps indicating the data sets have either compatible or orthogonal effects,
whilst the DØ $W$ asymmetry actually improves slightly. In contrast the $Z$
$p_{T}$ fit quality worsens by 8.1 points in $\chi^{2}$ when the DØ $W$
asymmetry is removed, clearly favouring its inclusion.

If all 3 data sets are removed together, then the effects seen when the data
sets are individually removed are also apparent, as given in the final column
of Table 16. In some places, such as for the E866 Drell-Yan $pp$ data, some of
the HERA data, NuTeV dimuon data or the ATLAS 8 TeV double differential $Z$
data, the improvement upon removing all 3 data sets is notably larger than the
sum of removing the three individually. This is also true of the overall
$\Delta\chi^{2}~{}=~{}-116.2$ improvement and indicates the magnification of
some of the tensions inherent by the inclusion of subsequent data sets. This
reflects the 3 data sets having similar pulls to one another in a number of
respects. In other places, in contrast, the tensions are more similar to, or
even less than the sum of the individual $\chi^{2}$ changes upon removal of
the three data sets, perhaps indicating these represent separate tensions.
This is for example true for the NMC $n/p$, CDF $W$ asymmetry, LHCb 2015 $W$,
$Z$ and others. In the case of the improvements in the fit quality of the LHC
jets and LHC top data sets, this is perhaps expected given that both the BCDMS
and ATLAS $Z$ $p_{T}$ data favour a larger high $x$ gluon and lower
$\alpha_{S}$ value, which is in tension with these LHC data sets. The removal
of all three data sets also makes clear the tensions of these with the new LHC
Drell-Yan data sets, with the fit quality of the ATLAS 7 TeV $W$, $Z$
improving significantly by 11.8 (i.e. 0.19 per data point); similarly the
ATLAS 8 TeV $W^{\pm}$ and double differential $Z$ also improve noticeably by
$\Delta\chi^{2}=-6.5,-6.6$ (or equivalently 0.30 and 0.12 per data point).

##  9 Predictions for benchmark processes

In Figures 81 and 82 we show predictions for various benchmark processes at
the Tevatron (1.96 TeV), LHC (8 and 13 TeV) and a FCC–pp (100 TeV) for the
MMHT14 and MSHT20 PDF sets, at NLO and NNLO. In all cases we show the ratio to
the MMHT14 central prediction, for ease of comparison. We use LO electroweak
perturbation theory, with the $qqW$ and $qqZ$ couplings defined by

$g_{W}^{2}=G_{F}M_{W}^{2}/\sqrt{2},\qquad g_{Z}^{2}=G_{F}M_{Z}^{2}\sqrt{2},$
(33)

and other electroweak parameters as in [9]. We take the Higgs mass to be
$m_{H}=125~{}\rm GeV$, and the top pole mass $m_{t}=172.5~{}\rm GeV$. The
Higgs cross section corresponds to the gluon fusion channel only. For the
$t\overline{t}$ cross section we use top++ [76].

Figure 81: Benchmark cross sections obtained with the NLO MMHT14 [1] and the
NLO MSHT20 PDFs. Results are normalized to the central value of the MMHT14
prediction, and PDF uncertainties only are shown.

Figure 82: Benchmark cross sections obtained with the NNLO MMHT14 [1] and the
NNLO MSHT20 PDFs. Results are normalized to the central value of the MMHT14
prediction, and PDF uncertainties only are shown.

The main purpose of the presentation is to investigate how both the central
values and the uncertainties of the predictions have changed in going from
MMHT14 to the MSHT20 PDFs. We therefore provide results for the Tevatron, and
the LHC at 8 and 13 TeV, as well as for a 100 TeV $pp$ FCC, to give a
representative spread in energies. We do not intend to present definite
predictions or compare in detail to other PDF sets, as both these results are
frequently provided in the literature with very specific choices of codes,
scales and parameters which may differ from those used here.

Considering the NNLO case first, we can see that the central value of the
$W^{\pm}$ cross sections are generally lower in the MSHT20 case in comparison
to the MMHT14, by $\sim 1\sigma$ or less with the exception of the Tevatron,
where the change is $\sim 1.5\sigma$. The $Z$ boson cross section is generally
rather unchanged, and hence the ratio of $Z$ to $W$ is higher by $\sim 1\%$.
This is principally due to the large range of precision LHC data on $W$, $Z$
production and the resultant changes these induce in the quark flavour
decomposition. The uncertainties are somewhat lower in the MSHT case, in
particular at larger energies. For the $t\overline{t}$ and Higgs cross
sections, these are slightly lower in the MSHT case due to the smaller gluon,
with a non–negligible reduction in uncertainty, due principally to the impact
of new LHC data and their constraint on the gluon at intermediate to high $x$.
It is interesting to note though that the central value of the 13 TeV Higgs
cross section remains relatively stable.

At NLO, the $W$ and $Z$ cross sections generally change by less, being even
slightly larger in the MSHT case, with the exception of the 100 TeV case,
where the trend matches the NNLO comparison. The resulting ratio of $Z$ to $W$
cross sections is roughly unchanged. We may expect a qualitatively different
behaviour in comparison to the NNLO fit, due e.g. to the difficulty the NLO
fit has in accounting for the range of precision LHC DY data. We can see for
example in Section 7 that the strangeness in the NLO fit is rather different
from the NNLO case, potentially as a result of this. For the $t\overline{t}$
and Higgs cross sections, the trend of lower cross sections and smaller
uncertainties for the MSHT PDF set is rather similar, with the exception of
the Tevatron $t\overline{t}$ cross section, which is higher in the MSHT case.

##  10 Predictions for a selection of LHC processes

Figure 83: Data and theory comparison for the CMS 13 TeV $W+c$ data set,
including both shifted and unshifted data, without refitting.

Figure 84: MMHT14 and MSHT20 NNLO predictions for the cross section ratios of
$Z\to ll$ production (with $p_{\perp}^{l}>25$ GeV, $|\eta_{l}|<2.5$) and the
$t\overline{t}$ total cross section, compared to data from ATLAS [30]. PDF
uncertainties alone shown in the theory, while the experimental uncertainties
are added in quadrature.

In this section we present a brief selection of predictions for LHC
measurements not currently included in the fit. Namely we consider: CMS data
on $W+c$ jet at 13 TeV; ratios of $Z$ and $t\overline{t}$ cross sections at 8
and 13 TeV measured by ATLAS; CMS measurements of differential and total
$t$–channel single top production; and the gluon at low $x$ and $Q^{2}$
compared to extractions from fits to exclusive $J/\psi$ and open charm
production

The CMS $W$ \+ charm jet data are presented differentially in the muon
pseudorapidity at 13 TeV [167]. We achieve a good description of the data with
$\chi^{2}/N_{pts}=13.0/10$ without refitting. This shows that these data are
already well predicted by using the MSHT20 PDFs, albeit only using NLO theory
(for recent results at NNLO see [123]). After refitting, this $\chi^{2}$ is
unchanged, and indeed changes little even if the data set is given a larger
weight in the fit. The $\chi^{2}$ predicted for this data set using the MMHT14
PDFs is 14.3. Therefore it perhaps shows a slight preference for the
moderately increased strangeness included in MSHT20, although it is clearly
consistent with both the MMHT14 and MSHT20 global fits. In particular,
regarding the strangeness content of the proton, given this data set is
already well predicted by the MSHT20 default PDFs it suggests the amount of
strangeness in MSHT20, which is intermediate between the lower strangeness
favoured by the dimuon data and the unsuppressed strangeness favoured by the
ATLAS 7 TeV $W$, $Z$ data [20], is favoured by this 13 TeV data set. Indeed,
as described in [167], the CMS 13 TeV $W+c$ data do not favour unsuppressed
high $x$ strangeness. The comparison of the data and theory before refitting
is shown in Fig. 83. There is both a difference in normalisation and perhaps
shape in the theory compared to the data, with the total $\chi^{2}$ coming
dominantly (10.7 out of 13.0) from the lowest rapidity point for the $W^{+}$
and the fourth rapidity point for both the $W^{+}$ and $W^{-}$, though these
largely appear to be due to fluctuations. This does not change upon refitting.

Figure 85: MMHT14 and MSHT20 NNLO predictions for the $t$–channel single top
and anti–top production. We compare to the sum $t+\overline{t}$ (upper plots)
and the ratio of $t$ to the total $t+\overline{t}$ (lower plots) at 13 TeV,
differential in the top/antitop $p_{\perp}$ (left-hand plots) and rapidity
(right-hand plots), compared to the data from CMS [31]. PDF uncertainties
alone are shown in the theory, while the experimental uncertainties are added
in quadrature.

Next, in Fig. 84 we compare the MMHT14 and MSHT20 NNLO predictions for the
cross section ratios involving $Z\to ll$ production (with $p_{\perp}^{l}>25$
GeV, $|\eta_{l}|<2.5$) and the $t\overline{t}$ total cross section, to data
from ATLAS [30]. Here, when ratios of different processes at the same beam
energy or ratios of the same process at different energies are taken, certain
experimental systematics cancel, and hence a cleaner result can be obtained.
For $Z$ and $t\overline{t}$ predictions we use MCFM 8.3 [114] and top++ [76],
respectively. The ratio of $Z$ to $t\overline{t}$ at 13 TeV is shown in the
left figure, and we can see that the agreement is good for both PDF sets, with
some reduction in PDF error seen in the MSHT20 case. Clearly the PDF
uncertainties are somewhat larger than the experimental error, indicating that
this may be a useful measurement to include in future fits. The ratio of the
$t\overline{t}$ cross sections at 13 to 8 TeV are shown in the right figure,
and we can observe that both MMHT14 and MSHT20 lie somewhat below the data,
though within uncertainties. The MSHT20 prediction has smaller PDF
uncertainties and lies a little closer to the data, due to the impact of other
LHC data on the gluon PDF presumably. On the other hand, the PDF uncertainties
are significantly smaller than the data uncertainty, and hence we may expect
it to play less of a role in any future fit.

We also compare to CMS 13 TeV data on $t$–channel single top production,
considering in particular the sum $t+\overline{t}$ and the ratio of $t$ to
$\overline{t}$, differential in the top/antitop $p_{\perp}$ and rapidity [31],
as well as the ratio of the total $t$ to $\overline{t}$ production, $R_{t}$,
presented in [32]. We use the NNLO calculations in [168, 169]. The
differential spectra are shown in Fig. 85 with the upper two figures
containing the spectra for the sum of the top and antitop and the lower two
figures containing the ratio. The ratio of the total cross sections for $t$
and $\bar{t}$ is given in Fig. 86. We can see that the agreement is good in
all cases, albeit within rather large experimental uncertainties. There is in
particular no sign of any similar significant disagreement to that observed
[170] in certain distributions in the ATLAS 7 TeV measurement (though not the
8 TeV). We show the corresponding comparison to the 7 TeV top and anti–top
rapidity and $p_{\perp}$ distributions [171] in Fig. 87 and observe a similar
undershooting of the $\overline{t}$ $p_{\perp}$ distribution in particular, at
higher $p_{\perp}$. It is worth commenting here that the differential
distributions are generally dominated by systematics, and therefore a full
evaluation of the fit quality would require a careful treatment of their
correlations, though this is beyond the scope of the current comparison.
Nonetheless, the PDF uncertainties on these predictions are small, generally
at the sub percent level, and hence given the rather large experimental
uncertainties these particular data sets would not be expected to have a
significant impact on the fit.

Figure 86: MMHT14 and MSHT20 NNLO predictions for the ratio of $t$ to
$\overline{t}$ total cross sections at 13 TeV, compared to the CMS measurement
[32].

Figure 87: MMHT14 and MSHT20 NNLO predictions for $t$–channel single top
(upper plots) and anti–top (bottom plots) production, differential in the
top/antitop $p_{\perp}$ (left-hand plots) and rapidity (right-hand plots),
compared to the data from ATLAS [171]. PDF uncertainties alone are shown in
the theory, while the experimental uncertainties are added in quadrature.

Finally, we compare the NLO gluon PDF at low $x$ and $Q^{2}$ to specific
extractions that are dedicated to constraining this region. There is an
absence of precise data in global fits which are directly sensitive to the
gluon in this regime, and hence the resulting uncertainties are rather large.
In particular, we compare to the extraction of [35], based on a fit to LHCb
exclusive $J/\psi$ production data at 7 [33] and 13 TeV [34], and the
extraction of [38] based on a fit to LHCb data on $D$ meson production at 5
[36], 7 [33] and 13 TeV [37]. In more detail, the former study applies a two
parameter power–law fit in the low $x$ region, while the intermediate to high
$x$ region is constrained to follow the MMHT14NLO set, while the latter
applies Bayesian reweighting to the NNPDF3.1 set.

We note that the interpretation of both of these data sets is not completely
straightforward. For the LHCb exclusive data, the theoretical predictions are
given in terms of generalized PDFs, rather than the collinear ones directly,
and the scale variation uncertainty is very large. However, a recent study
[35] uses a known method for relating the generalized PDFs to the collinear
ones, and argues that the scale variation uncertainty is due to a double
counting which can be controlled by a $Q_{0}\simeq m_{c}$ subtraction. For the
$D$ meson data, the scale variation uncertainties are again very large, but it
is argued in [38] that this can be effectively overcome by considering
normalized distributions.

Figure 88: Comparison of the MMHT14 (left) and MSHT20 (right) NLO gluon PDFs
at $Q^{2}=2.4~{}\rm GeV^{2}$ with a power–law fit to exclusive $J/\psi$ data
[35] and a NNPDF3.0 fit with the addition of LHCb $D$ meson data [38].

Comparisons of the MMHT14 and MSHT20 NLO gluon PDFs at $Q^{2}=2.4~{}\rm
GeV^{2}$ to the above extractions are shown in the left and right plots of
Fig. 88, respectively. We can first see that indeed the uncertainty on the
global PDFs is very large in this region. The uncertainty resulting from the
$D$ meson fit and in particular the exclusive $J/\psi$ fit is on the other
hand much smaller. This reflects the significant potential constraining power
of these data sets, though in the latter case the precise size of the
uncertainty band will be driven by the more restrictive parameterisation of
the power–law fit, and the manner of determining theoretical uncertainties is
a potential source of uncertainty in both cases. The MSHT20 NLO gluon is
somewhat lower at low $x$ than MMHT14, which in the latter case does not
overlap entirely within uncertainties with the dedicated fits. However, given
the lack of directly constraining data in this region in the fits, there is no
strong reason to expect this improved agreement in the MSHT20 case, and
clearly for either sets the PDF uncertainties are very large. A fit which
additionally constrains by hand the gluon to be similar to that required by
the $J/\psi$ analyses, by introducing high precision gluon pseudo-data at low
$x$ and $Q^{2}$, finds that only a small deterioration in the global fit
quality is required to obtain a gluon of the suggested form at NLO, as the
agreement within the uncertainty bands in Fig. 88 (right) would suggest.
However, imposing the same constraint on the gluon at NNLO leads to a
deterioration of about $\Delta\chi^{2}\sim 200$ in the fit quality to HERA
data, implying that the NNLO corrections to the cross sections for these heavy
flavour processes would have to be significant in order for them to be
successfully incorporated in a global fit at NNLO.

##  11 Comparison of MSHT20 with other PDF sets

In this section we compare our PDFs with other representative available PDF
sets, focussing on the most phenomenologically relevant NNLO case for brevity.
The most direct comparison is with the CT18A [3] and NNPDF3.1 [2] sets, which
both result from global fits that include a significant amount of LHC Run–I
data, much of which we also fit, though we do fit more recent data sets not
included in either of these. We will for brevity refer to these as the
‘global’ sets in what follows.

Both CT and NNPDF also use a GM–VFNS, which have been shown to converge with
that used in our analysis as the order increases [172]. There are nevertheless
some significant differences in the theoretical approaches. For example,
NNPDF3.1 does not apply deuteron and heavy nuclear target corrections by
default, though the effect has been discussed in subsequent publications [173,
174], and NNPDF3.1 uses as default a fitted charm at the input scale rather
than generating all heavy flavour perturbatively, as is done by ourselves and
CT18. Moreover, the MSHT and NNPDF collaborations use quite a different
procedure for the analysis. The NNPDF collaboration combine a Monte Carlo
representation of the probability measure in the space of PDFs, with the use
of neural networks to give a set of unbiased input distributions. We instead
use parameterisations of the input distributions based on Chebyshev
polynomials, where the optimum order of the polynomials for the various PDFs
has been explored in detail in the fit. As with MSHT20, CT18 use a tolerance
criterion to evaluate the experimental PDF uncertainties. The CT18 fit is also
based on a polynomial parameterisation (in their case Bernstein rather than
Chebyshev), albeit with a rather more restrictive parameterisation in general,
for example with respect to the $s-\bar{s}$ distribution (which is taken to be
zero) and the light quarks in certain regimes. On the other hand, the CT18
tolerance criterion is now shown to be consistent with taking account of the
differences that can potentially arise from fitting with a range of modified
input parameterisations - i.e. the total uncertainty is, in some sense,
designed in order to incorporate the contribution from parameterisation
uncertainty.

Figure 89: Comparison of the MSHT20 NNLO gluon PDF at $Q^{2}=10^{4}~{}\rm
GeV^{2}$ with other representative PDF sets, presented as a ratio to the MSHT
set. These correspond to $\alpha_{S}(M_{Z}^{2})=0.118$ in all cases, though
the set corresponding to the preferred value of $\alpha_{S}(M_{Z}^{2})=0.1147$
is also shown in the ABMP case.

Figure 90: Comparison of the MSHT20 NNLO light sea PDF at $Q^{2}=10^{4}~{}\rm
GeV^{2}$ with other representative PDF sets, presented as a ratio to the MSHT
set. These correspond to $\alpha_{S}(M_{Z}^{2})=0.118$ in all cases.

We also compare against the ABMP16 [4] and HERAPDF2.0 [5] sets. In the former
case a global fit is performed, including some LHC data, though rather less
than is included in the global sets above, and crucially a FFNS is used in the
calculation of DIS cross sections. Also, rather lower $Q^{2}$ and $W^{2}$ cuts
than other analyses are used in the fits to DIS data, with higher–twist
corrections included, which have a significant impact in this additional
region of parameter space. It is also the case that no deuteron DIS data are
fit. The upshot of these differences in procedure is that a rather low value
of $\alpha_{S}(M_{Z}^{2})=0.1147$ is preferred by the fit, and the
corresponding PDFs are often rather different, as we will see. A further point
to note is the use of the $\Delta\chi^{2}=1$ criterion to evaluate the errors.
This is also applied by the HERAPDF2.0, albeit with additional model and
parameterisation uncertainties, which we include in our comparisons, following
the prescription of [5]. This set results from a fit to HERA DIS data alone,
with a parametric form with many fewer free parameters than ours.

In all cases we consider the sets defined at $\alpha_{S}(M_{Z}^{2})=0.118$.
While this is not the value preferred by ABMP, we argue it will give the most
direct comparison, given any observable quantities will often depend on the
value of the coupling. However for the gluon (which depends on the strong
coupling the most) we also show the ABMP result for their best fit
$\alpha_{S}(M_{Z}^{2})=0.1147$. For the CT set we take the ‘A’ PDF, for which
the ATLAS high precision $W$, $Z$ data at 7 TeV [20] are included, as these
are fit in both the MSHT20 and NNPDF3.1 cases.

Figure 91: Comparison of the MSHT20 NNLO up and down quark PDFs at
$Q^{2}=10^{4}~{}\rm GeV^{2}$ with other representative PDF sets, presented as
a ratio to the MSHT set. These correspond to $\alpha_{S}(M_{Z}^{2})=0.118$ in
all cases.

We first show results for the gluon PDF in Fig. 89. The PDF uncertainty for
NNPDF3.1 is comparable in size to MSHT20, albeit a little lower at higher
$x\gtrsim 0.1$, while the CT18 PDF uncertainty is larger than both of these.
This difference is also seen in the earlier CT14 set in comparison to NNPDF3.0
and MMHT14, which entered the PDF4LHC15 combination [143]. We believe this
larger CT18 uncertainty is due to a slightly more conservative tolerance
criterion than used in our dynamical tolerance procedure. In all cases the
corresponding updated PDF sets show a marked improvement in the error bands
with respect to these older sets. However, in terms of the absolute values of
the PDFs, we can see that the NNPDF3.1 gluon is systemically higher (lower)
than MSHT20 and CT18 at low to intermediate (high) $x$, even lying outside the
corresponding uncertainty bands in some regions. This is in contrast to the
earlier sets, where the agreement was rather better; thus we are in a slightly
problematic situation that while the PDF uncertainties of the newer sets are
improved with respect to the PDF4LHC15 sets, their spread has not necessarily
decreased and has even increased in some regions. The cause of this
enhancement at intermediate $x\sim 10^{-3}-10^{-2}$ is in part due to the fact
that NNPDF3.1 now fit the charm quark PDF, see [2] for a detailed discussion,
while at larger $x\gtrsim 0.1$ the impact of new LHC data on the central value
of the high $x$ gluon region appears to be somewhat larger in the NNPDF case,
where it reduces from versions NNPDF3.0 to NNPDF3.1, in comparison to that
seen in MSHT20 (see Fig. 17 (left)). At low $x$ the CT18 gluon is larger than
MSHT20, outside their respective uncertainty bands. This effect was not seen
as clearly in the PDF4LHC15 sets, but historically the CT/CTEQ gluons have
been larger at very small $x$ due to the positive definite nature of their
input gluon at the starting scale for evolution. We should note, however, that
the agreement between the MMHT14, NNPDF3.0 and CT14 gluons in the PDF4LHC15
combination was rather better than one might expect, and indeed better than
any quark or antiquark distribution, and may be regarded as at least partially
coincidental.

For the ABMP16 set we can see that the agreement is poor for the default value
of $\alpha_{S}(M_{Z}^{2})=0.1147$, with the gluon lying well outside the
quoted uncertainty bands across a wide range of $x$. This is in part due to
the differing value of $\alpha_{S}$, however we can see that while taking a
consistent value of $\alpha_{S}(M_{Z}^{2})=0.118$ does improve the agreement,
it is still not good. This is a known result from previous comparisons, and in
[175] it was argued that this is primarily due to their use of a FFNS as
opposed to our use of a GM-VFNS for DIS data. The uncertainty for the gluon
and all other PDF sets shown is smaller than ours, due to the use of a simple
$\Delta\chi^{2}=1$ criterion for the calculation of PDF uncertainties.
Interestingly, the ABMP16 gluon with $\alpha_{S}(M_{Z}^{2})=0.118$ is rather
more similar to the NNPDF3.1 case, though given the significant differences in
their methodologies and data included in the fit, this is possibly largely
coincidental. In the HERAPDF2.0 case the gluon is lower (higher) at high (low)
$x$ than MSHT20, lying outside the quoted uncertainty bands.

In Fig. 90 the result for light quark sea
$S=2(\overline{u}+\overline{d})+s+\overline{s}$ is shown, and the agreement
between the global sets is reasonable, though not entirely within errors, and
with some reduction (enhancement) at high (low) $x$ for CT18 relative to
MSHT20. This combination however masks some of the true differences seen in
the quark flavour decomposition, as demonstrated in Fig. 91, where the up and
down quark PDFs are shown. As for the gluon, there is some enhancement in the
NNPDF3.1 up and in particular down quarks PDFs at intermediate $x\sim
10^{-3}-10^{-2}$, with the latter lying outside uncertainty bands. This is in
contrast to the PDF4LHC sets, where the agreement was better, albeit with the
NNPDF3.0 up quark PDF actually lying below the MMHT14 PDF across a wide range
of $x$. A major factor in this difference can again be traced to the updated
methodology in the NNPDF3.1 case, which is seen in [2] to enhance the up and
down in this region, due to some extent but not entirely to their fitting of
charm. The trend for the $\overline{u}$ and $\overline{d}$ (not shown) is
rather similar, while as we will see below the strangeness in the NNPDF case
is rather lower, compensating the contribution from this enhancement to the
light sea. As for the gluon, at low $x$ all three quark distributions are
larger in the CT18 case in comparison to MSHT, lying outside the uncertainty
bands. In general, again as for the gluon the MSHT20 and NNPDF3.1 uncertainty
bands are rather comparable in size, with CT18 being somewhat larger, though
this depends on the precise $x$ region being considered.

In the ABMP16 and HERAPDF2.0 cases, we can see that level of agreement with
MSHT is rather poor. The HERAPDF2.0 light sea, up and down all lie above
MSHT20 at low to intermediate $x$, outside the error bands, while the up quark
enhancement persists to high $x$. A rather similar, though generally less
pronounced, trend is seen in the ABMP16 case for the light sea and the up,
while the down is in somewhat better agreement, though still not within errors
across the whole $x$ region. An enhancement of the ABM-type up quark was also
noticed and commented on in [175]. The effect was deduced to be mainly due to
the use of the FFNS scheme, with somewhat slower evolution of the charm over
much of the $x$ range being compensated for by an increase in the up-quark in
fitting inclusive DIS data, but at highest $x$ the use of the lower $Q^{2}$
and $W^{2}$ cuts was also seen to play some role.

The up and down valence distributions are shown in Fig. 92. The agreement
between NNPDF3.1 and MSHT20 is in both cases rather good across all $x$. This
is a rather interesting development, as previously [1] both the up and down
valence were found to be rather larger in the NNPDF3.0 case in comparison to
MMHT14. This is probably due to a combination of both new LHC data placing
some constraint in this region on both sets and hence tending to bring them
together, as well as the more flexible parameterisation now used in MSHT.
Indeed, the difference between the CT18 and MSHT20 down valence is rather
striking in the intermediate $x$ region, and closely follows the trend seen in
Fig. 45 when comparing our fit to the MSHT20 data set, but using the previous
less flexible parameterisation. This may suggest some issues with
parameterisation flexibility in the CT18 case, though in [3] it is claimed
that the PDF uncertainty accounts for this source of uncertainty. It is
however noticeable that the uncertainty on the CT18 down valence distribution
is smaller in comparison to MSHT20 and NNPDF3.1, in contrast to most other
cases. The NNPDF3.1 uncertainty is generally rather larger at low to
intermediate $x$ in both cases, perhaps due to the greater parameterisation
flexibility there. For the ABMP16 and HERAPDF2.0 sets, we can see that the
agreement is poor for the up valence, with both distributions undershooting
with respect to MSHT20 at low $x$ and overshooting at intermediate to high
$x$, often well outside uncertainty bands. For the down valence, the agreement
between HERAPDF2.0 and MSHT20 is equally poor, while for ABMP16 it is somewhat
better. In the former case the PDF in the intermediate $x$ region is rather
similar to CT18, indicating the role of parameterisation flexibility here and
also the lack of LHC data in the fit.

Figure 92: Comparison of the MSHT20 NNLO up and down quark valence PDFs at
$Q^{2}=10^{4}~{}\rm GeV^{2}$ with other representative PDF sets, presented as
a ratio to the MSHT set. These correspond to $\alpha_{S}(M_{Z}^{2})=0.118$ in
all cases.

Figure 93: Comparison of the MSHT20 NNLO $\overline{d}-\overline{u}$ PDF at
$Q^{2}=10^{4}~{}\rm GeV^{2}$ with other representative PDF sets. These
correspond to $\alpha_{S}(M_{Z}^{2})=0.118$ in all cases.

The $\overline{d}-\overline{u}$ distribution is shown in Fig. 93. The global
sets are broadly in agreement across the entire $x$ region, while the CT18
uncertainty is somewhat larger at intermediate $x\sim 0.01-0.1$ and lower at
low $x$, than MSHT20 and NNPDF3.1, perhaps due to parameterisation. The PDF
uncertainties in the NNPDF3.1 case at high $x$ are clearly larger, due to the
greater parameterisation flexibility in this less constrained region, and
interestingly the central value somewhat higher than the other sets. The
ABMP16 distribution displays a strong preference to be negative in the $x\sim
10^{-4}-10^{-2}$ region, and lies below the MSHT20 distribution, outside
uncertainty bands. We emphasise here that the parameterisation is sufficiently
flexible to allow the MSHT20 distribution to also be negative in this region,
and indeed it is consistent with this within error bands, but that the fit
does not prefer such a negative distribution. Indeed, we note that some of the
fits where we remove some data sets can move further in the direction of the
ABMP16 distribution, see e.g. Fig. 67. This is consistent with both NNPDF3.1
and CT18, which are also in disagreement with ABMP16 here. For HERAPDF2.0 the
parameterisation at low $x$ is fixed to give zero, while at intermediate $x$ a
completely opposite behaviour to all other PDFs is preferred, with the
distribution being negative. This highlights the fact that HERA data have no
effective constraint on this distribution, and an improvement is noted when
also including LHC data, see [176].

Figure 94: Comparison of the MSHT20 NNLO $s+\overline{s}$ PDF at
$Q^{2}=10^{4}~{}\rm GeV^{2}$ with other representative PDF sets, presented as
a ratio to the MSHT set. These correspond to $\alpha_{S}(M_{Z}^{2})=0.118$ in
all cases.

We now consider the strangeness, shown in Fig. 94. The global sets are broadly
in agreement and indeed in all cases have higher strangeness, with smaller PDF
uncertainties, in comparison to the PDF4LHC sets, due to the impact of new LHC
data and in particular the ATLAS 7 TeV high precision $W$, $Z$ data. However,
in more detail we can see that NNPDF3.1 and to a lesser extent CT18 lie below
the MSHT20 distribution in the $x\sim 10^{-3}-10^{-1}$ region, in the former
case even slightly outside the PDF uncertainty band. From Fig. 65 (bottom) we
have seen that the ATLAS 8 TeV data on $W$ and $Z$ production also prefer a
larger strangeness, consistent with the 7 TeV, and that when both are fit
together the impact is more significant than when just the 7 TeV is fit. Given
NNPDF3.1 and CT18 do not include these data sets, this may well explain the
difference. In addition, these analyses use NLO theory for the dimuon cross
sections, which may have some impact, in particular given the $D\to\mu$
branching ratio is fixed in these cases. The ABMP16 and HERAPDF2.0 strangeness
distribution lie below MSHT20 in the $x\sim 0.01$ region, though are still
roughly consistent within uncertainties, while at low $x$ ABMP16 lies above
MSHT20, and at high $x$ HERAPDF2.0 lies below MSHT20. We note that ABMP16 are
reasonably consistent with MSHT20 in the region where there are direct data
constraints on the strange distribution, while at very low $x$ the difference
is driven by that in the gluon distribution, which in turn drives the strange
quark evolution in this region.

Figure 95: Comparison of the MSHT20 NNLO $s-\overline{s}$ and charm PDFs at
$Q^{2}=10^{4}~{}\rm GeV^{2}$ with other representative PDF sets. These
correspond to $\alpha_{S}(M_{Z}^{2})=0.118$ in all cases. The charm case is
presented as a ratio to the MSHT set while for the $s-\overline{s}$ the PDF
values are shown.

Next, we consider the strangeness asymmetry for the global PDF sets, shown in
Fig. 95 (left). We can see that NNPDF3.1 and MSHT20 are consistent across the
entire $x$ region, and in particular show a clear preference for a non–zero
and positive asymmetry in the $x\sim 0.1$ region. Interestingly, while the
NNPDF3.1 uncertainties are larger in the unconstrained low $x$ region, in this
$x\sim 0.1$ region they are actually somewhat smaller than MSHT20. Generally
the NNPDF3.1 central value undergoes somewhat more variation at higher $x$.
The CT18 asymmetry is fixed to be zero by construction, as is evident in the
plot, though we note that at NNLO DGLAP evolution itself will generate an
asymmetry [165]. We do not show the corresponding ABMP16 and HERAPDF2.0
distributions, as these are similarly constrained to be zero at input.

We show the charm quark PDF in Fig. 95 (right) for the global sets. These
broadly follow the trend of the gluon, but the NNPDF3.1 PDF at $x\gtrsim 0.01$
lies significantly below MSHT20 and even more so relative to CT18, and with
rather larger uncertainties. To some extent this is due to the lower gluon PDF
at high $x$ observed above, and its contribution to the charm through
evolution, but in addition the effect of fitting charm is seen in [2] to
reduce the charm PDF in this region.

Figure 96: As in Figs. 89 (left) and 95 (right), but with the NNPDF3.1 set
corresponding to perturbative charm shown.

Finally, as we have discussed the role that the fitting of charm appears to
play in some regions in increasing the differences seen between NNPDF3.1 and
MSHT20/CT18, we show in Fig. 96 the same comparisons as in Figs. 89 (left) and
95 (right), for the gluon and charm quark PDFs, but now replacing the NNPDF3.1
default set with that corresponding to perturbative charm. For the gluon, we
can see that the agreement at intermediate $x\sim 10^{-3}-10^{-2}$ is indeed
improved, as expected from the discussion above. The agreement at high $x$ is
also somewhat better, though the NNPDF gluon remains lower than the other
sets. On the other hand, at low $x$ we can see that the NNPDF gluon now lies
above MSHT20, outside error bands, consistent with CT18 and reflecting the
default NNPDF gluon in this region. For the charm PDF we can see that clearly
the PDF uncertainty on the charm is greatly reduced, in particular at high
$x$, as we expect. As with the gluon the low $x$ distribution lies somewhat
above MSHT20, consistent with CT18. More significantly, at high $x$ the NNPDF
charm PDF lies below CT18 and MSHT20, outside of error bands, as a result of
feed down from the gluon at high $x$, which as mentioned above follows a
similar, though milder, trend. The charm distributions do not mirror the gluon
quite as closely as they might since the values of charm mass are different,
with pole masses $m_{c}=1.4~{}\rm GeV$ for MSHT20 but $m_{c}=1.3~{}\rm GeV$
for CT18 and $m_{c}=1.51~{}\rm GeV$ for NNPDF3.1. This automatically raises
the charm distribution for CT18 a little compared to MSHT20 and lowers it for
NNPDF3.1.

##  12 Availability of MSHT20 PDFs

We provide the MSHT20 PDFs in the LHAPDF format [177]:

http://lhapdf.hepforge.org/

as well as on the repository:

http://www.hep.ucl.ac.uk/msht/

We present NLO and NNLO eigenvector sets of PDFs at the default value of
$\alpha_{S}(M_{Z}^{2})=0.118$:

MSHT20nnlo_as118
MSHT20nlo_as118

as well as the NLO set at the best fit value of $\alpha_{S}(M_{Z}^{2})=0.120$
and a LO set at $\alpha_{S}(M_{Z}^{2})=0.130$:

MSHT20nlo_as120
MSHT20lo_as130

In addition to the above, we provide best fit NLO and NNLO sets for a short
range of $\alpha_{S}(M_{Z}^{2})$ values:

MSHT20nlo_as_smallrange
MSHT20nnlo_as_smallrange

in order to allow the $\alpha_{s}$ \+ PDF uncertainty to be evaluated,
according to the procedure outlined in [178]. In particular, PDFs from
$\alpha_{S}(M_{Z}^{2})=0.115$ to 0.121 (0.122) at NNLO (NLO) are provided, in
increments of 0.001.

##  13 Conclusions

We have performed fits to the available global hard scattering data in order
to determine the PDFs of the proton at NLO and NNLO, as well as at LO. These
PDF sets, denoted MSHT20, supersede the MMHT14 sets [1], that were obtained
using a similar framework. They include a very significant update in terms of
data included, and also a number of important improvements in the framework of
the PDF fit and in terms of more precise cross section calculations for a
number of processes. The MSHT20 PDF sets may be accessed, as functions of
$x,Q^{2}$ in computer retrievable form, as described in Section 12.

First we summarize the improvements to the theoretical framework. As in MMHT14
we base the parameterisation of the input distributions on Chebyshev
polynomials. Previously we typically used 4 polynomials for each PDF. However,
it was shown in [47] that in order to be able to obtain comfortably sub-
percent precision 6 polynomials are needed. Hence, with the exception of the
still poorly constrained strange asymmetry, our parameterisations for each set
include 6 Chebyshev polynomials (or equivalent for the gluon). In particular,
we have a much improved parameterisation for the $\bar{d},\bar{u}$ difference
and now parameterise the ratio directly rather than $\bar{d}-\bar{u}$. Hence,
we use many more free parameters than in MMHT, 52 in total, and this results
in an improvement of around 75 units in $\chi^{2}$ at NNLO. As there is still
some redundancy in parameters, these are not all allowed to be free in the PDF
error evaluation, and hence we have 32 eigenvector pairs. This is to be
compared with 25 in MMHT14. Our treatment of deuteron and heavy nucleus
corrections are the same as in MMHT14, and the deuteron correction obtained
from the fit is similar to before. Our treatment of correlated systematic
uncertainties is the same as in MMHT14, but we now additionally include
information on statistical correlations. Our treatment of heavy flavour via a
general-mass variable flavour scheme is identical to MMHT14, i.e. we use the
“optimal” version of the Thorne-Roberts GM-VFNS [59]. We do now, however, use
full NNLO cross sections for the NNLO fit to dimuon data in neutrino DIS [25],
and for all LHC inclusive jet data [68], as well as for all the new types of
collider processes in our analysis (with the exception of the small amount of
$W+c$ data [99], where the NNLO calculation is only now becoming available).

Second, we briefly discuss the new data sets included, and their effects on
the PDFs. We are now able to use the final combined H1 and ZEUS Runs I + II
HERA data for the neutral and charged current [5], and for the heavy flavour
cross sections [26]. Neither of these has a huge impact compared to previous
similar data, but they do make slight changes to the best-fit PDFs and a
little more so to the uncertainties. We include final Tevatron data, in
particular the $W$-asymmetry data from DØ [21]. This has a large impact on
both the central value and uncertainty of the high-$x$ down quark, and also
now have a surprisingly large effect on related PDFs. We also have very much
more precise LHC data: for $W,Z$ rapidity distributions from ATLAS, CMS and
LHCb; on high and low mass Drell-Yan production; for inclusive top-quark-pair
production; and on inclusive jet production. The $W,Z$ data has a very
significant impact on quark flavour separation, in particular the strange
quark and antiquark, and on the valence quarks at small $x$. The jet data adds
constraints to the high-$x$ gluon. We also include a large amount of data of
types that were not included in the MMHT14 analysis. This includes single and
double differential data on top pair production, and also on the $p_{T}$
distribution of $Z$ bosons. Both of these constrain the high-$x$ gluon, but
there is some tension between them, as well as tensions with jet data and
older DIS data. We include new data on $W+$ jets and $W+$ charm jets, the
latter providing essentially a consistency check on the strange quark,
obtained as a compromise between that obtained from inclusive $W,Z$ data and
dimuon data. Overall, it is still the case that the constraints on PDFs come
from a very wide variety of data sets, both old and new. For the first time we
make a detailed investigation into which data sets are most responsible for
determining the central PDFs, i.e. the best fit, and also from this central
point, which PDFs have most constraint on the uncertainty eigenvectors. We
find that the best fit is, unsurprisingly, largely driven by those data sets
with a large number of data points, particularly if the measurements are
precise, e.g the inclusive HERA data. Then, it turns out that eigenvectors are
often constrained by smaller data sets which are often not fit optimally
within the best fit, due to their small weight, and which can therefore
deteriorate quite quickly in certain directions away from the best fit.
Nonetheless, in most cases there are multiple data sets that place rather
similar constraints on these eigenvectors, highlighting the constraining power
and relative stability of a global PDF fit. Compared to MMHT14 we find that
LHC data are providing many more important constraints, these still being most
clearly evident for quark flavour decomposition.

Some LHC data are not included in the present fits; for example single top
quark production data and data on low scale heavy quark production at
colliders. These choices are either due to the so-far limited impact these
data would provide, or due to the lack of a calculation of the NNLO cross
section and/or uncertainty in the existing calculations. However, these types
of data are found to be consistent with MSHT20 predictions, and may add
important constraints in future fits. We also avoid fitting LHC data at 13
TeV. This is partially because there is still a limited amount of data of the
form which would constrain PDFs at a similar level of precision to the 7 and 8
TeV data. However, it is also a conscious decision to avoid the highest energy
LHC data in order for the PDFs to be free of influence from these data, should
the PDFs be used to provide evidence for any discrepancies in the Standard
Model seen in future analyses.

The new MSHT20 PDFs only significantly differ from the MMHT14 PDF sets in the
details of flavour decomposition. This is seen for $u_{V}$, mainly at small
$x$ and particularly $d_{V}$ for more general $x$. It is also seen for the
$\bar{u},\bar{d}$ difference. The main new data probing valence quarks and
light flavour asymmetry are the $W$ rapidity measurements at the LHC and
Tevatron. The MMHT14 partons indeed give a poor description of some of these
data, particularly the extremely precise recent LHC data. A much better fit is
obtained with MSHT20 PDFs, partially facilitated by the new parameterisation,
particularly in the case of $\bar{d}/\bar{u}$. The only other significant
change is in the total strange quark distribution, with a significant increase
in magnitude, of the order of the size of the MMHT14 uncertainty. This is
driven very largely by the precise $W,Z$ data at the LHC, and particularly the
relative difference in $Z$ and $W$ production. The overall light quark
distributions, and most particularly the gluon distribution are not changed
significantly from the MMHT14 PDFs, or even from MSTW08 [9]. Hence, one is
unlikely to obtain a dramatic change in the prediction for the many processes
of phenomenological relevance when using MSHT20 PDFs instead of MMHT14 PDFs,
though the uncertainty does decrease, sometimes quite significantly. We
demonstrate this for a variety of benchmark processes. In summary, we most
definitely recommend the use of MSHT20 PDFs for the most reliable prediction
for the central values and for both the best and, in most cases, reduced
uncertainties.

It is becoming increasingly necessary for LHC studies for us to have PDFs
determined as precisely as possible. This means examining not only individual
PDF sets and their uncertainties, but also the comparison between sets
obtained from similar data and using similar frameworks. Hence, we compare to
other recent global PDF sets, in particular NNPDF3.1 [2] and CT18A [3], which
are most directly comparable to MSHT20, but also to the ABMP16 set [4] and the
final HERAPDF2.0 set [5]. We note that while MSHT20, CT18A and NNPDF3.1 in
general agree well with each other, and uncertainties have generally decreased
in each compared to the previous PDF releases, in some respects there is more
relative difference in the central values than there were for the MMHT14, CT14
and NNPDF3.0 sets. This is particularly the case for the gluon distribution,
where NNPDF3.1 now has a rather different shape to MSHT20 and CT18A, being
significantly softer at high-$x$, and also having the smallest uncertainty
despite in general fitting less data than MSHT20. Some other PDF comparisons
show some notable differences, both in central values and uncertainties. Some
of these differences are due to the choice of data fitted or the treatment of
these, but others are undoubtedly due to methodology, e.g parameterisation
choices, or other inherent assumptions in the fitting procedure. For example
NNPDF3.1 now chooses to fit the charm by default, and this is responsible for
some systematic difference compared to MSHT20 and CT18A. It will be
interesting to continue to investigate the comparison between the PDFs, and
try to understand the reasons for differences.

In this article we have made PDFs available at NNLO, which has become the
standard, but also NLO and even LO, should these be useful. However we note
that the most recent precision data from the LHC has finally made the inherent
superiority of the NNLO analysis very clear, with the NLO fit struggling badly
with the detailed description of some $W,Z$ data and the LO fit now being
unable to give a truly quantitatively satisfactory description of the most
precise data. As well as the NLO and NNLO PDFs at the default
$\alpha_{S}(M_{Z}^{2})=0.118$ value we also make PDFs available at nearby
values, in order that the PDF + $\alpha_{S}(M_{Z}^{2})$ uncertainty can be
calculated by treating the PDFs defined at the upper and lower
$\alpha_{S}(M_{Z}^{2})$ values as additional eigenvectors [178]. In subsequent
studies we will return to some additional issues, such as the dependence of
the fit on the value of $\alpha_{S}$ and on the heavy quark ($c,b$) masses,
and make PDFs available for a wide variety of values in each case. We will
also provide a PDF set analogous to MSHT20 but including QED corrections, in
particular with a photon PDF, using the same approach as in [24]. We will in
addition look into the theoretical uncertainties associated with the PDFs,
noting that this is a further source of uncertainty to that presented here for
PDFs defined at a fixed perturbative order.

However, we conclude by noting that the MSHT20 PDFs presented in this article
are obtained from the most up-to-date set of data currently available and
using the most complete and sophisticated global analysis we have so far
performed. Hence, we present these sets as our best possible PDFs for
analysing data from any experiment where hadrons are at least one of the
colliding particles, or for making predictions for all such processes.

## Acknowledgements

We would like to thank Mandy Cooper-Sarkar, Stefano Forte, Sasha Glazov,
Claire Gwenlan, Jun Gao, Joey Huston, Pavel Nadolsky, Juan Rojo, Maria Ubiali
and the wider PDF4LHC working group for many discussions on PDFs and related
issues. We would also like to thank, in no particular order: Maria Ubiali for
providing EW corrections for ATLAS $Z$ boson $p_{\perp}$ distribution; Claire
Gwenlan, and Mandy Cooper-Sarkar and Jan Kretzschmar for helpful information
related to various ATLAS data sets; Engin Eren and Katerina Lipka for help
with CMS data; Chris Hays for discussions on Tevatron data and the DØ $W$
asymmetry in particular; Matthias Komm for clarification on the details of the
CMS single top analysis; Jun Gao for providing theoretical calculations for
single top and dimuon production; Sasha Glazov and Simone Amoroso for
providing grids for and significant discussion relating to the ATLAS 8 TeV $Z$
production data; Alexander Huss and Joao Pires for providing NNLOJET
predictions and helpful discussion on their implementation; Valerie Lang and
Gavin Pownall for help relating to the ATLAS $W$ \+ jets data; Pavel
Starovoitov and in particular Mark Sutton for help relating to APPLgrid;
Bogdan Malaescu for clarification and discussion relating to the ATLAS jet
data; Francesco Giuli for helpful information related to various ATLAS data
sets and in particular the ATLAS 8 TeV $W$ data set; Emanuele Nocera for
useful discussion relating to the treatment of top quark pair production data.

T. C. and R. S. T. thank the Science and Technology Facilities Council (STFC)
for support via grant awards ST/P000274/1 and ST/T000856/1. S. B. acknowledges
financial support from STFC. L. H. L. thanks STFC for support via grant award
ST/L000377/1.

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