CERN-TH-2024-080

TIF-UNIMI-2024-4

LO, NLO, and NNLO Parton Distributions for LHC Event Generators

Juan Cruz-Martinez1, Stefano Forte2, Niccolò Laurenti2, Tanjona R.
Rabemananjara3,4, and Juan Rojo3,4

1CERN, Theoretical Physics Department, CH-1211 Geneva 23, Switzerland
2Tif Lab, Dipartimento di Fisica, Università di Milano and
INFN, Sezione di Milano, Via Celoria 16, I-20133 Milano, Italy
3Department of Physics and Astronomy, Vrije Universiteit, NL-1081 HV Amsterdam
4Nikhef Theory Group, Science Park 105, 1098 XG Amsterdam, The Netherlands

Abstract

We present NNPDF4.0MC, a variant of the NNPDF4.0 set of parton distributions
(PDFs) at LO, NLO and NNLO, with and without inclusion of the photon PDF,
suitable for use with Monte Carlo (MC) event generators, which require PDFs to
satisfy additional constraints in comparison to standard PDF sets. These
requirements include PDF positivity down to a low scale $Q\sim 1$ GeV, smooth
extrapolation in the very small and large $x$ regions, and numerically stable
results even in extreme regions of phase space for all PDFs. We compare the
NNPDF4.0MC PDFs to their baseline NNPDF4.0 counterparts, and to the NNPDF2.3LO
set entering the Monash tune of the Pythia8 event generator. We briefly assess
the phenomenological impact of these PDFs on the cross-sections for hard and
soft QCD processes at the LHC.

###### Contents

  1. 1 Introduction
  2. 2 Methodology
  3. 3 The NNPDF4.0MC PDFs
  4. 4 Impact on LHC physics
  5. 5 Summary and outlook

##  1 Introduction

Monte Carlo (MC) event generators [1, 2, 3, 4] provide a complete description
of the final state in high-energy particle collisions, and, as such, are an
essential ingredient in the interpretation of particle physics experiments.
Widely used event generators for LHC physics include Pythia8 [5, 6], HERWIG7
[7, 8], SHERPA [9, 10], POWHEG [11], mg5_aMC@NLO [12], and more recently
PanScales [13, 14, 15, 16].

Within a MC event generator, parton distributions (PDFs) [17, 18] are used not
only in the evaluation of hadronic cross-section through their convolution
with partonic matrix elements, but also for the initial-state backwards parton
shower, and as inputs to the modeling of non-perturbative phenomena [19] such
as the underlying event (UE), multiple parton interactions (MPI), and related
soft QCD processes. For these latter aspects, PDFs should respect some
additional constraints in comparison to default PDFs. First, their usage in
initial-state showers requires that they be non-negative down to the
perturbative cutoff of $Q\simeq 1$ GeV. Furthermore, their application to
models of the UE, MPI, and other low-energy QCD phenomena demands a very
smooth extrapolation down to very small $x$ and very small $Q^{2}$ values, and
a gluon PDF that grows sufficiently fast in the small $x$ region. In order to
prevent numerical problems associated to Monte Carlo integration and sampling,
PDFs should be numerically stable even in extreme regions of phase space which
may be irrelevant for phenomenology. Finally, in order to match to standard
parton showers, the charm PDF must be generated perturbatively (i.e. an
intrinsic component is not allowed), and in order to account for electroweak
corrections, the possibility of including a photon PDF $\gamma(x,Q^{2})$ and
QED splittings in perturbative evolution should be allowed.

Several groups [20, 21, 22, 23, 24] have presented variants of their LO PDF
sets, aimed to usage in MC event generators. For instance, the NNPDF2.3QED LO
PDFs developed in [25, 26, 27] were integrated in Pythia8, and used as one of
the inputs for its popular Monash tune [28] of non-perturbative QCD physics.
Beyond LO, BFKL-resummed variants of the NNPDF3.1 PDF set including the
constraints on the small-$x$ gluon from $D$-meson production at LHCb presented
in [29, 30, 31] also satisfy the above requirements, and are available in
Pythia8 as a stand-alone PDF set.

Here we present variants of NNPDF4.0 [32, 33, 34, 35] at LO and, for the first
time, NLO and NNLO, tailored to their usage in modern MC event generators. The
main goal of these NNPDF4.0MC sets is to satisfy the requirements discussed
above, while at the same time providing the best possible description of the
NNPDF4.0 dataset, in particular at NLO and NNLO.

##  2 Methodology

Unless otherwise specified, we adopt the same experimental dataset, theory
calculations, and methodology used in the construction of the recent MHOU,
QED, and aN3LO NNPDF4.0 PDF sets [34, 35, 36]. In particular, we exploit the
new NNPDF theory pipeline [37] built upon the EKO [38] evolution code, YADISM
DIS module [39], and PineAPPL fast grid interface [40]. The same values of the
input SM parameters are used, in particular $\alpha_{s}(m_{Z})=0.118$ for the
LO, NLO, and NNLO fits. We only provide a central PDF, instead of a set of PDF
replicas representing the PDF probability distribution, because in the
presence of extra constraints uncertainties might become unreliable, and they
are anyway not relevant for applications to MC event generators.

#### Positivity and perturbative charm.

Positivity of MC PDFs is required both for their usage in the initial-state
shower as well as for the modeling of soft QCD phenomena. At LO, PDFs can be
identified with physical cross-sections and hence are positive-definite. This
is not necessarily true at NLO and beyond, where PDFs become scheme dependent
and may be negative in certain regions of the phase space. Whereas in the
commonly used $\overline{\rm MS}$ scheme PDFs are positive also at NLO and
beyond, this only holds in the perturbative region, i.e. at high enough scale
[41, 42, 43], and correspondingly PDF positivity may fail when extrapolating
to low $Q$ values.

In the baseline NNPDF4.0 analysis, PDF positivity is imposed at the initial
parametrization scale ($Q_{0}=1.65$ GeV) at LO and at a higher scale,
$Q^{2}_{\rm pos}=5$ GeV2, at NLO and beyond, following the prescription of
[41, 43]. In addition, positivity of a set of physical observables at
$Q^{2}_{\rm pos}$ is also imposed. Therefore, within the NNPDF4.0 methodology,
the NLO and NNLO PDFs may be negative at low values of $Q^{2}$ as long as,
upon evolution, they become positive at $Q^{2}\geq Q^{2}_{\rm pos}$. Even
though this may happen in regions of phase space for which there are no direct
experimental constraints, or such that a fixed-order leading-twist
approximation breaks down, positivity is nevertheless required by MC
generators. Furthermore, in the default NNPDF4.0 sets the charm PDF is
parametrized and determined from the data on the same footing as all other
PDFs [44], with its behavior for $Q<Q_{0}$ determined by backwards QCD
evolution together with the matching from the $n_{f}=4$ to the $n_{f}=3$
flavor scheme [45]. However, a variant of NNPDF4.0 in which charm vanishes in
the $n_{f}=3$ flavor scheme and is determined by perturbative matching
conditions in the $n_{f}=4$ scheme is also available; in this case PDFs are
parametrized at $Q_{0}=1$ GeV, hence below the matching scale, set at
$\mu_{c}=m_{c}=1.51$ GeV.

We consequently start from this perturbative charm variant of NNPDF4.0, with
perturbative matching conditions used to determine charm at the matching scale
$\mu_{c}=m_{c}$. We then impose the positivity of $g(x,Q_{0})$ and
$\Sigma(x,Q_{0})$ at $Q_{0}=1$ GeV by squaring the corresponding neural
network outputs. This ensures positivity of the gluon and the quark singlet
PDFs at $Q_{0}=1$ GeV and consequently also for $Q>Q_{0}$ thanks to their rise
at small $x$ induced by perturbative QCD evolution as the scale is increased.
Positivity of individual quark and antiquark PDFs is imposed at $Q^{2}_{\rm
pos}=5$ GeV2 as in the default. This is sufficient to guarantee positivity
down to $Q_{0}$ both at large $x$, where perturbative evolution is moderate
even at low scale, and also at small $x$, where nonsinglet PDFs vanish. This
strategy leads to positive-definite PDFs in the full range of $(x,Q^{2})$
probed by MC generators at LO and NLO.

At NNLO, the perturbative matching conditions lead to a charm PDF that at
$Q=m_{c}$ is negative at small $x\lesssim 10^{-2}$, though it is already
positive at all $x$ for $Q^{2}\gtrsim 5$ GeV2. Hence at NNLO it is not
possible to simultaneously satisfy at $\mu_{c}=m_{c}$ the requirements that
charm be positive and determined by perturbative matching. As we will discuss
in Sect. 3, the low-scale positivity of the gluon at small $x$ is disfavored
by the data and consequently imposing it leads to some deterioration of the
fit quality.

#### Extrapolation in $x$ and $Q^{2}$.

General-purpose MC event generators should provide reliable results for the
broadest possible region of phase space. This requires input PDFs with a
smooth behavior in a wide $Q$ range, from $Q\simeq 1$ GeV (initial-state
showers, non-perturbative QCD modeling) up to $Q\sim 100$ TeV (relevant for
future particle colliders and for applications to astroparticle physics) and
from $x\simeq 10^{-9}$ (forward particle production) all the way up to
large-$x$ values close to the elastic limit $x=1$ (required for high-mass new
physics searches). Since these regions extend beyond the coverage of available
data, a robust extrapolation procedure is necessary.

While PDF extrapolation in $Q^{2}$ is fixed by perturbative QCD evolution,
extrapolation in $x$ depends on assumptions. In the NNPDF4.0 approach,
extrapolation to the small $x$ and large $x$ regions is provided by the output
of a preprocessed neural network, and thus controlled by the behavior of both
the neural net and the preprocessing function. This extrapolation to low
$Q^{2}$ and large $x$ values might be affected by numerical instabilities,
both native, and related to their storage as LHAPDF grids. Specifically, the
low $Q^{2}$ behavior is controlled by evolution from higher scales, that may
amplify small differences in the initial condition, due to the growing value
of $\alpha_{s}(Q)$, while at large $x$ PDFs become very small and thus
particularly sensitive to numerical instabilities. These two issues are
intertwined, since even small $\mathcal{O}\left(10^{-5}\right)$ numerical
differences in the solution of evolution equations may be enough to distort
the PDFs in the large $x$ region where they are almost vanishing. While such
instabilities are innocuous for phenomenological applications, they may lead
to numerical issues when PDFs are used in MC generators.

In order to prevent these instabilities and ensure that the MC PDFs are
everywhere smooth and well-behaved, the NNPDF4.0MC PDFs are delivered as an
LHAPDF grid with a finer coverage in $x$ for the the region
$x\in\left[0.7,0.95\right]$. For $x\gtrsim 0.95$, PDFs essentially vanish and
any residual oscillations can be safely set to zero. In addition,
instabilities of the order of the accuracy of the LHAPDF interpolation are
averaged out by means of a dedicated Gaussian filter. Possible issues related
to backward evolution are prevented by parametrizing PDFs at $Q_{0}=1$ GeV, so
no backward evolution is needed. We thus deliver LHAPDF grids that provides an
interpolated output for all $x\in\left[10^{-9},1\right]$ and
$Q\in\left[1,10^{6}\right]$ GeV.

#### QED evolution and the photon PDF.

As shown in [31, 34, 46, 47, 48] and related studies, the impact of inclusion
of a photon PDF alongside quark and gluon PDFs is moderate, its main effect
being a reduction of the gluon momentum fraction by up to around $0.5\%$ in
favor of the photon. Here we take the photon PDF $\gamma(x,Q^{2})$ at $Q=1$
GeV from the NNPDF4.0 QED NNLO PDF set [34], we include it as boundary
condition to the QCD$\otimes$QED evolution of the LO, NLO, and NNLO NNPDF4.0MC
PDFs, and impose a momentum sum rule that now also includes a photon
contribution. We adopt the so-called exact-iterated (EXA) solution of the
QCD$\otimes$QED evolution equations, as implemented in EKO [38], as in Ref.
[34] to which we refer for more details. For pure QCD evolution we use instead
the truncated (TRN) solution as in Ref. [32], so that in each case the PDF
sets presented here are based on the same form of the solution of the
evolution equations as their default counterparts.

#### NNPDF4.0MC overview.

In Table 1 we summarize the settings adopted for the NNPDF4.0MC PDFs, compared
to those of their baseline counterparts: LHAPDF naming ID, publication
reference, PDF parametrization scale and solution of the evolution equations,
positivity scale, value of $\alpha_{s}(m_{Z})$, and treatment of charm (data-
driven, or determined from perturbative matching). In this table
$q_{i},\bar{q}_{i}$ denote light (up, down, and strange) quarks and antiquark
PDFs, as, following [41, 43], positivity of the charm PDF is never imposed.

ID | Ref. |  evolution ($Q_{0}$) |  Positivity ($Q_{\rm pos}$) | $\alpha_{s}(m_{Z})$ | Charm
---|---|---|---|---|---
NNPDF23_lo_as_0130_qed | [27] |  QCD${}_{\rm LO}\otimes$QEDLO TRN (1.0 GeV) |  $g,q_{i},\bar{q}_{i}>0$ ($1$ GeV) | 0.130 | pert.
NNPDF40_lo_as_01180 | [32] |  QCDLO TRN (1.65 GeV) |  $g,q_{i},\bar{q}_{i}>0$ ($1.65$ GeV) | 0.118 | fitted
NNPDF40_lo_pch_as_01180 | [32] |  QCDLO TRN (1.65 GeV) |  $g,q_{i},\bar{q}_{i}>0$ ($1$ GeV) | 0.118 | pert.
NNPDF40MC_lo_as_01180 | t.w. |  QCDLO TRN (1.0 GeV) |  $g,q_{i},\bar{q}_{i}>0$ ($1$ GeV) | 0.118 | pert.
NNPDF40MC_lo_as_01180_qed | t.w. |  QCD${}_{\rm LO}\otimes$QEDLO EXA (1.0 GeV) |  $g,q_{i},\bar{q}_{i}>0$ ($1$ GeV) | 0.118 | pert.
NNPDF40_nlo_as_01180 | [32] |  QCDNLO TRN (1.65 GeV) |  $g,q_{i},\bar{q}_{i}>0$ ($\sqrt{5}$ GeV) | 0.118 | fitted
NNPDF40_nlo_pch_as_01180 | [32] |  QCDNLO TRN (1 GeV) |  $g,q_{i},\bar{q}_{i}>0$ ($\sqrt{5}$ GeV) | 0.118 | pert.
NNPDF40MC_nlo_as_01180 | t.w. | QCDNLO TRN (1 GeV) |  $g,\Sigma>0$ ($1$ GeV) | 0.118 | pert.
$q_{i},\bar{q}_{i}>0$ ($\sqrt{5}$ GeV)
NNPDF40_nlo_as_01180_qed | [34] |  QCD${}_{\rm NLO}\otimes$QEDNLO EXA (1.65 GeV) |  $g,q_{i},\bar{q}_{i}>0$ ($\sqrt{5}$ GeV) | 0.118 | fitted
NNPDF40MC_nlo_as_01180_qed | t.w. | QCD${}_{\rm NLO}\otimes$QEDNLO EXA (1 GeV) |  $g,\Sigma>0$ ($1$ GeV) | 0.118 | pert.
$q_{i},\bar{q}_{i}>0$ ($\sqrt{5}$ GeV)
NNPDF40_nnlo_as_01180 | [32] |  QCDNNLO TRN (1.65 GeV) |  $g,q_{i},\bar{q}_{i}>0$ ($\sqrt{5}$ GeV) | 0.118 | fitted
NNPDF40_nnlo_pch_as_01180 | [32] |  QCDNNLO TRN (1 GeV) |  $g,q_{i},\bar{q}_{i}>0$ ($\sqrt{5}$ GeV) | 0.118 | pert.
NNPDF40MC_nnlo_as_01180 | t.w. | QCDNNLO TRN (1 GeV) |  $g,\Sigma>0$ ($1$ GeV) | 0.118 | pert.
$q_{i},\bar{q}_{i}>0$ ($\sqrt{5}$ GeV)
NNPDF40_nnlo_as_01180_qed | [34] |  QCD${}_{\rm NNLO}\otimes$QEDNLO EXA (1.65 GeV) |  $g,q_{i},\bar{q}_{i}>0$ ($\sqrt{5}$ GeV) | 0.118 | fitted
NNPDF40MC_nnlo_as_01180_qed | t.w. | QCD${}_{\rm NNLO}\otimes$QEDNLO EXA (1 GeV) |  $g,\Sigma>0$ ($1$ GeV) | 0.118 | pert.
$q_{i},\bar{q}_{i}>0$ ($\sqrt{5}$ GeV)
Table 1: The NNPDF4.0MC PDFs presented in this work (t.w.) and their baseline
counterparts.

##  3 The NNPDF4.0MC PDFs

We now compare the NNPDF4.0MC PDF sets to the baseline NNPDF4.0 fits and to
the NNPDF2.3QED LO PDFs used for the Monash tune [28] of Pythia8. Here we only
present some representative results; an extensive set of comparisons is
available online.111https://data.nnpdf.science/vp-
public/NNPDF40MC_comparisons/

The fit quality for the NLO and NNLO PDF sets of Table 1 is summarized in
Table 2, where we show the number of data points and the $\chi^{2}$ per data
point; LO $\chi^{2}$ values are not shown since fit quality at LO is generally
poor and the specific value of the $\chi^{2}$ is not significant. For pure QCD
PDFs, we find that at NLO (NNLO) the total $\chi^{2}$ per data point of the
baseline fit increases from 1.28 (1.16) to 1.30 (1.22), an effect of about
1$\sigma$ (3$\sigma$) in units of the statistical variance of the $\chi^{2}$
distribution for $n_{\rm dat}=4443$ (4626) data points. Therefore, imposing
the MC PDF conditions has no statistically significant effect at NLO, while at
NNLO it only leads to a mild deterioration of fit quality, related to the fact
that the rapid rise of the gluon at small $x$ as the scale increases tends to
lead in turn to a negative gluon at scales $Q^{2}\lesssim$ few GeV2 [43], so a
low-scale positive gluon is difficult to accommodate. For the QCD$\otimes$QED
sets, the same behavior is observed at NNLO, while now at NLO a more
significant deterioration of fit quality is seen. This can be traced to the
fact that subleading terms included in the EXA solution of the evolution
equations lead to perturbative evolution that is faster than for the TRN
solution, especially when the anomalous dimension is large, which then makes
the problem with low-scale gluon positivity more serious at NLO. The
difference between the pure QCD and QCD$\otimes$QED cases at NLO therefore is
a measure of the size of missing NNLO corrections.

Dataset by process group | NLO | NNLO
---|---|---
$n_{\rm dat}$ | QCD | QCD+QED | $n_{\rm dat}$ | QCD | QCD+QED
BL | MC | BL | MC | BL | MC | BL | MC
DIS NC | 1953 | 1.35 | 1.37 | 1.38 | 1.54 | 2110 | 1.22 | 1.30 | 1.22 | 1.29
DIS CC | 988 | 0.91 | 0.92 | 0.94 | 0.95 | 989 | 0.90 | 0.89 | 0.90 | 0.89
DY NC | 669 | 1.58 | 1.84 | 1.67 | 2.04 | 736 | 1.20 | 1.30 | 1.22 | 1.33
DY CC | 197 | 1.38 | 1.56 | 1.40 | 1.61 | 157 | 1.45 | 1.55 | 1.47 | 1.57
Top pairs | 66 | 2.40 | 2.14 | 2.51 | 2.47 | 64 | 1.27 | 1.16 | 1.31 | 1.27
Single-inclusive jets | 356 | 0.82 | 0.88 | 0.83 | 0.93 | 356 | 0.94 | 1.01 | 0.93 | 1.00
Dijets | 144 | 1.51 | 1.55 | 1.56 | 1.62 | 144 | 2.01 | 2.01 | 1.94 | 1.93
Photon | 53 | 0.57 | 0.60 | 0.64 | 0.74 | 53 | 0.76 | 0.67 | 0.74 | 0.68
Single top | 17 | 0.36 | 0.36 | 0.38 | 0.36 | 17 | 0.37 | 0.38 | 0.39 | 0.40
Total | 4443 | 1.28 | 1.30 | 1.30 | 1.44 | 4626 | 1.16 | 1.22 | 1.17 | 1.22
Table 2: The number of data points and the $\chi^{2}$ per data point for the
NLO and NNLO baseline NNPDF4.0 fits (BL), compared to their NNPDF4.0MC
counterparts (MC), with the same process categorisation as in Ref. [36]. The
$\chi^{2}$ values are provided for the QCD-only (NNPDF40(MC)_<order>_as_01180)
and for the QCD$\otimes$QED (NNPDF40(MC)_<order>_as_01180_qed) fits of Table
1.

The MC and baseline LO and NLO PDFs are compared in Fig. 1, where we display
the gluon, up and antidown PDFs at $Q=1$ GeV, $2$ GeV, and $1$ TeV. Recall
that the small-$x$ behavior of all quark and antiquark PDFs is the same, and
dominated by that of the singlet quark distribution. We show the full $x$
region in which the NNPDF4.0MC PDFs are provided via the LHAPDF interpolation,
i.e. $10^{-9}\leq x\leq 1$. Note that the NNPDF2.3LO set was only provided for
$x\geq 10^{-7}$, while for smaller $x$ values PDFs are frozen to their value
at $x=10^{-7}$. Apart from this trivial difference, the main difference
between the 2.3 and 4.0 LO sets is that for NNPDF4.0MC the rise of the
small-$x$ gluon is qualitatively similar at LO and NLO, a feature facilitating
the tuning of soft QCD models in MC event generators. The main difference
between the MC and default NLO PDFs is related to the small $x$ positivity of
the gluon at low scale. As the scale $Q$ is increased, relative differences
between the various PDF sets are washed out by perturbative evolution.

Figure 1: The NNPDF4.0MC LO and NLO gluon, up, and antidown PDFs (from left to
right) compared to NNPDF2.3LO and NNPDF4.0 NLO, at three scales: $Q=1$ GeV, 2
GeV, and $1$ TeV (from top to bottom). Only central values are shown, in the
region for which PDFs are provided via LHAPDF.

In order to demonstrate smoothness of the NNPDF4.0MC sets in the large-$x$
extrapolation region, we display in Fig. 2 the NLO and NNLO NNPDF4.0MC PDFs
for $x=0.85$ as a function of scale, compared to the central value of their
baseline counterparts. The $Q$ range shown corresponds to the full
interpolation range in the LHAPDF grids that we provide. All PDFs displayed
exhibit a satisfactory level of smoothness.

Figure 2: The NNPDF4.0MC NLO and NNLO gluon, up, antiup, antidown, strange and
total charm PDFs (from left to right and from top to bottom), compared to
their baseline counterparts as a function of scale for a fixed large $x=0.85$
value. The range $1\leq Q\leq 10^{6}$ GeV shown corresponds to the full
interpolation range provided by the LHAPDF grids that we deliver.

In order to fully assess the difference between the MC sets and their baseline
counterparts, in Fig. 3 we display the ratio of the NNPDF4.0MC NLO PDFs to the
baseline, also showing the 68% CL PDF uncertainties on the latter. In order to
trace the origin of differences, the NNPDF4.0 NLO set with perturbative charm
of Table 1 is also shown. In the region $x\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}10^{-3}$, where the
bulk of experimental data is located, the quark MC PDFs are mostly contained
within the uncertainty band of the baseline. Larger differences, that can be
traced to the requirement of low-scale positivity, are observed for the gluon
PDF, especially at small $x\lesssim 10^{-2}$. These in turn propagate onto the
other PDFs at small $x$, all of which display a stronger small-$x$ rise in
comparison to the baseline in the extrapolation region $x\lesssim 10^{-3}$.

The results of Fig. 3 imply that the additional model assumptions entering the
MC PDFs do not distort the baseline PDFs in the bulk of the data region beyond
the $1\sigma$ level, indicating that most LHC cross-sections obtained with the
NNPDF4.0MC sets will be consistent with those derived using the baseline PDFs.
In fact, it is clear from Fig. 3 that for most PDFs, especially for the sea
quark PDFs, a large part of the difference between the MC PDFs and the default
is due to having adopted perturbative charm.

Figure 3: The NLO NNPDF4.0MC gluon, up, down, antiup, strange and charm PDFs
at $Q=100$ GeV (from left to right and from top to bottom), shown as a ratio
to their baseline counterpart. The uncertainty shown is the 68% CL on the
baseline. The baseline variant with perturbative charm is also shown.

##  4 Impact on LHC physics

We now carry out a brief assessment of the phenomenological impact of
similarities and differences between the MC PDFs and their baseline
counterparts shown in Figs. 1–3.

First, in Fig. 4 we display the gluon-gluon, quark-antiquark, and quark-quark
parton luminosities at the LHC with $\sqrt{s}=13.6$ TeV as a function of the
invariant mass of the final state $m_{X}$, computed from the same PDFs shown
in Fig. 3, and shown as a ratio to the NNPDF4.0 baseline. The luminosities are
integrated over the full rapidity range and are thus dominated by the PDF
behavior in the central rapidity region, where $x_{1}\sim x_{2}\sim
m_{X}/\sqrt{s}$. For $50~{}{\rm GeV}\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}m_{X}\mathrel{\hbox
to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}1~{}{\rm TeV}$ this is a medium-small $x$ region, where
differences between the MC PDFs and the baseline are generally moderate and
only noticeable for the gluon. Indeed, in the case of the gluon-gluon
luminosity MC PDFs lead to a suppression of around 2% in comparison to the
baseline for $100~{}{\rm GeV}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}m_{X}\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}3~{}{\rm TeV}$,
while otherwise differences between NNPDF4.0 NLO and its MC variant are at the
1% level, and only become larger, though well within uncertainties, for
$m_{X}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}100$ GeV due to stronger small $x$ rise of the MC PDFs.

Figure 4: The gluon-gluon, quark-antiquark, and quark-quark parton
luminosities at the LHC with $\sqrt{s}=13.6$ TeV as a function of the
invariant mass $m_{X}$ for the same PDFs as in Fig. 3, shown as a ratio to the
NNPDF4.0 baseline.

We then consider representative inclusive hard cross-sections: Higgs and gauge
boson production at the LHC with $\sqrt{s}=13.6$ TeV, computed using the
ggHiggs [49] and n3loxs [50] codes. In Fig. 5 we compare results obtained at
NLO and NNLO (both for PDF and the matrix element) with the MC sets and their
baseline counterparts, and for the latter also aN3LO, using the settings of
Ref. [35]. The uncertainty shown is for the MC sets only that related to
missing higher orders in the matrix element, evaluated from standard 7-point
scale variation, while for the baseline sets it also includes the PDF
uncertainty, combined in quadrature with it. The moderate differences observed
reflect those seen in parton luminosities.

Figure 5: The inclusive NLO and NNLO cross-sections for Higgs production in
gluon fusion, in association with a $Z$ boson, and in vector boson fusion
(top), and on-shell and high-mass $W$ and on-shell $Z$ production at the LHC
$\sqrt{s}=13.6$ TeV (bottom), comparing NNPDF4.0MC PDFs and the baseline. For
the baseline NNPDF4.0, the aN3LO result is also shown. The uncertainty shown
is scale variation with 7-point prescription only for the MC PDFs, combined in
quadrature with the PDF uncertainty for the baseline sets.

We turn next to processes that are also sensitive to soft physics. We show
results for LHC differential distributions at leading order obtained from
Pythia8 simulations interfaced to the Rivet analysis toolkit [51]. We neglect
PDF uncertainties and only display the central values found using NNPDF2.3LO,
NNPDF4.0 NLO, and NNPDF4.0MC NLO PDFs. We first consider the normalized $Z$
boson transverse momentum distribution, reconstructed from bare dilepton
events, either electrons or muons, which is sensitive to both soft and hard
QCD. In Fig. 6 the Pythia8 LO predictions for $1~{}{\rm GeV}\leq
p_{\perp}(\ell\ell)\leq 300~{}{\rm GeV}$ are compared to ATLAS data at 7 TeV
from Ref. [52]. The low and high $p_{\perp}$ regions respectively probe soft
and hard QCD radiation. For the normalized distributions shown, higher-order
QCD corrections partially cancel out. The difference between PDF sets is
negligible, and good agreement with the data is found using all PDF sets
except at very small $p_{\perp}$ in the electron channel.

Figure 6: The normalized $Z$ boson $p_{T}$ distribution computed at LO using
Pythia8 and Rivet using NNPDF2.3LO, NNPDF4.0 NLO, and NNPDF4.0MC NLO PDFs.
Predictions are compared to the ATLAS [52] data at $\sqrt{s}=7~{}\mathrm{TeV}$
using bare electron (left) or muon (right) pairs; error bars on the data
include statistical and systematic uncertainties. Both the absolute
distribution (top) and the ratio of the theory prediction to the data (bottom)
are shown.

We next consider the fiducial cross-sections for Higgs production in the $H\to
ZZ^{\star}\to 4\ell~{}(\ell=e,\mu)$ decay channel. In Fig. 7 we compare
predictions to the ATLAS data collected at $\sqrt{s}=13~{}\mathrm{TeV}$ with
an integrated luminosity of $\mathcal{L}=139~{}\mathrm{fb}^{-1}$ [53]. Results
are shown for the the transverse momentum distribution of the four hardest
leptons in the event, $p_{T}^{4\ell}$, in the rapidity range
$1.0<|y_{4\ell}|<1.5$, and for the transverse momentum of the leading jet in
the invariant mass range $115<m_{4\ell}<130~{}\mathrm{GeV}$. Also in this
case, differences between different PDF sets are negligible, and good
agreement with the data is found.

Figure 7: Same as Fig. 6 for the fiducial cross-section for Higgs production
at $\sqrt{s}=13~{}\mathrm{TeV}$ in the $H\to ZZ^{\star}\to
4\ell~{}(\ell=e,\mu)$ decay channel. The four-lepton $p_{T}$ distribution for
$1.0<|y_{4\ell}|<1.5$ (left) and the $p_{T}$ of the leading jet in events with
$\geq 1$ jet for $115<m_{4\ell}<130~{}\mathrm{GeV}$ (right) are shown,
compared to ATLAS data [53].

We then turn to the energy flow, defined as

$\frac{dE}{d\eta}=\frac{1}{|\eta_{\rm max}-\eta_{\rm
min}|}\left(\frac{1}{N_{\rm inel}}\sum_{i=1}^{n_{\rm
part}}E_{i}\theta(\eta_{i}>\eta_{\rm min})\theta(\eta_{i}<\eta_{\rm
max})\right)\,,$ (1)

where $\eta$ is the midpoint of the rapidity interval, $\left[\eta_{\rm
min},\eta_{\rm max}\right]$, $N_{\rm inel}$ is the number of inelastic pp
collisions, and $n_{\rm part}$ is the number of stable particles in the event
whose energy is equal to $E_{i}$. The energy flow in dijet events and in
minimum-bias events at $\sqrt{s}=7~{}\mathrm{TeV}$ in the forward
$3.2\leq\eta\leq 4.9$ ranges is shown in Fig. 8, compared to the CMS data of
[54]. For the dijet sample, a $p_{\perp}^{\rm jet}>20$ GeV cut is imposed. For
both dijet and minimum-bias events, the simulations based on NNPDF2.3LO
display good agreement with the data, while those obtained using NNPDF4.0 NLO
sets (both MC and baseline) tend to undershoot the experimental measurements,
which suggests the need for a dedicated tune of soft QCD physics.

Figure 8: Same as Fig. 6, now for the energy flow in dijet (left) and minimum-
bias events with $\sqrt{s}=7~{}\mathrm{TeV}$ and $3.2\leq\eta\leq 4.9$
compared to CMS data [54].

Finally, in Fig. 9 we show the charged-hadron multiplicity distribution,
differential in pseudorapidity and in transverse momentum, $d^{2}N_{\rm
ch}/d\eta dp_{\perp}$, as a function of $p_{\perp}$ at fixed rapidity
$\lvert\eta\lvert=0.3$ and as a function of $\eta$ integrated over the full
$p_{\perp}$ range. Predictions are compared to the CMS measurements of [55],
for events that satisfy both $p_{\perp}\leq 2~{}\mathrm{GeV}$ and
$\lvert\eta\lvert<2.5$ in order to highlight the sensitivity to the modeling
of nonperturbative QCD dynamics. As in the case of the energy flow, the
NNPDF2.3LO set provides the best description of the experimental data, while
the NNPDF4.0 sets undershoot the CMS measurements. Indeed, both the energy
flow of Fig. 8 and the charged-hadron differential distributions of Fig. 9 are
sensitive to non-perturbative QCD processes, hence achieving a good
description requires a dedicated tune of soft QCD. The Monash 2013 tune of
Pythia8 used here is based on NNPDF2.3LO, explaining the good agreement found
for this set.

Figure 9: Same as Fig. 8, now for the charged-hadron transverse momentum
(left) and pseudorapidity (right) distributions in proton-proton collisions at
$\sqrt{s}=7~{}\mathrm{TeV}$, comparing to the CMS measurements of [55].

##  5 Summary and outlook

The NNPDF4.0MC PDFs presented in this work satisfy the requirements of event
generators not only at LO but also at NLO and NNLO accuracy, while the NLO and
NNLO sets provide a satisfactory description of the global dataset and
minimize differences in comparison to the baseline sets, ensuring their
reliability to evaluate hard cross-sections at the LHC and elsewhere. It thus
becomes possible to combine the precision and accuracy enjoyed by global PDF
sets at NLO and NNLO without compromising the usability of these PDFs in
generators for initial-state radiation and the modeling of soft QCD processes.

In order to also achieve agreement with the data for non-perturbative
processes such as the underlying event, pileup, and low-$p_{T}$ radiation, the
soft QCD models specific to each event generator will need to be tuned to the
data using as input these new NNPDF4.0MC PDFs, since their behavior,
especially for low-$x$ physics, becomes a component of the tuning model. Such
dedicated tunes will be needed in order for the NNPDF4.0MC PDF sets to become
instrumental in the development of a next generation of Monte Carlo codes that
reaches higher perturbative accuracy. To this purpose, we aim to collaborate
with event generator developers in order to integrate NNPDF4.0MC in their
frameworks and produce dedicated tunes of soft QCD physics such that the whole
palette of LHC processes, from the soft to the perturbative region, can be
satisfactory described within a single physics simulation.

The NNPDF4.0 MC sets are made available through the LHAPDF interface [56] and
the NNPDF Collaboration website.222https://nnpdf.mi.infn.it/nnpdf4-0-mc/

### Acknowledgments

We are very grateful to Gavin Salam and Peter Skands for many useful
discussions, productive feedback, and extensive benchmarking concerning the MC
PDF sets presented in this work. We also thank Melissa van Beekveld, Silvia
Ferrario Ravasio, Christian Gutschow, Max Knobbe, Frank Krauss, and Oliver
Mattelaer for assistance with the validation of the NNPDF4.0 MC PDFs. J. R.
and T. R. are partially supported by NWO, the Dutch Research Council, and by
the Netherlands eScience Center (NLeSC).

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