xFitter Collaboration ††thanks: xFitter Contact: Sasha Glazov:
<EMAIL_ADDRESS>
Webpage: www.xFitter.org

# xFitter: An Open Source QCD Analysis Framework A resource and reference
document for the Snowmass study

The xFitter Developers’ Team  H. Abdolmaleki  S. Amoroso  V. Bertone  M. Botje
D. Britzger  S. Camarda  A. Cooper-Sarkar  J. Fiaschi  F. Giuli  A. Glazov  C.
Gwenlan  F. Hautmann  H. Jung  A. Kusina  A. Luszczak  T. Mäkelä  I. Novikov
F. Olness  R. Sadykov  P. Starovoitov  M. Sutton  O. Zenaiev

( 15 March 2022)

###### Abstract

We provide an overview of the xFitter open-source software package, review the
general capabilities of the program, and highlight applications relevant to
the Snowmass study. An updated version of the program (2.2.0) is available on
CERN GitLab,111xFitter version 2.2.0: https://gitlab.cern.ch/fitters/xfitter
and this has been updated to a C++ codebase with enhanced and extended
features. We also discuss some of the ongoing and future code developments
that may be useful for precision studies. We survey recent analyses performed
by the xFitter developers’ team including: W and Z production, photon PDFs,
Drell-Yan forward-backward asymmetry studies, resummation of small-$x$
contributions, heavy quark production, constraints on the strange PDF,
determination of the pion PDF, and determination of the pion Fragmentation
Functions. Finally, we briefly summarize selected applications of xFitter in
the literature. The xFitter program is a versatile, flexible, modular, and
comprehensive tool that can provide impact studies for possible future
facilities. We encourage the use of xFitter, and welcome new contributions
from the community.

###### Contents

  1. I Introduction to xFitter
    1. I.1 xFitter Overview
    2. I.2 xFitter Capabilities
    3. I.3 xFitter and Snowmass2021
  2. II Future Developments with xFitter
  3. III Overview of recent xFitter results
    1. III.1 W and Z Boson Production
    2. III.2 The Photon PDF
    3. III.3 The Forward-Backward Asymmetry in Neutral Current Drell-Yan Production
    4. III.4 Impact of low-x resummation on QCD analysis of HERA data
    5. III.5 Heavy quark matching scales: Unifying the FFNS and VFNS
    6. III.6 Constraining the Strange PDF
    7. III.7 The Pion PDF
    8. III.8 The Pion Fragmentation Function
  4. IV Applications of xFitter
    1. IV.1 Usage of xFitter by LHC Collaborations
    2. IV.2 Nuclear PDFs with xFitter
    3. IV.3 Higgs Physics with xFitter
    4. IV.4 BSM Physics with xFitter
      1. IV.4.1 Enhancing the sensitivity of BSM searches with improved PDFs
      2. IV.4.2 Applications with SMEFT
  5. V Conclusion

##  I Introduction to xFitter

###  I.1 xFitter Overview

xFitter Alekhin:2014irh  (formerly HERAfitter) is an open-source software
package that provides a framework for the determination of the parton
distribution functions (PDFs) of the proton and related subjects.222The
xFitter program is available on the web at: www.xFitter.org xFitter version
2.0.1 is currently available, and version 2.2.0 will be released imminently
and offers an expanded set of tools and options. It incorporates experimental
data from a wide range of experiments including fixed-target, Tevatron, HERA,
and LHC data sets. xFitter can analyze this data using predictions up to next-
to-next-to-leading-order (NNLO) in perturbation theory with a variety of
theoretical calculations including numerous methodological options for
carrying out PDF fits and plotting tools which help visualize the results.
While primarily based on the collinear factorization foundation, xFitter also
provides facilities for fits of dipole models and transverse-momentum
dependent (TMD) distributions. The package can be used to study the impact of
new precise measurements from hadron colliders, and also assess the impact of
future colliders. This paper provides a brief overview of xFitter with
emphasis of the features relevant for the Snowmass2021 study.

The need for precision PDFs: The PDFs are the essential components that allow
us to make theoretical predictions for experimental measurements of collider
experiments with initial state protons and hadrons. Despite the recent
progress of PDF analyses (including NLO and NNLO calculations), the
uncertainty for many precision measurements at the LHC stems nowadays
primarily from the PDFs HERAFitterdevelopersTeam:2015cre ; Accomando:2019vqt ;
Amoroso:2020fjw . Hence, our ability to fully characterize the Higgs boson and
constrain Beyond-Standard-Model (BSM) physics signatures ultimately comes down
to how accurately we determine the underlying PDFs; this is the focus of the
xFitter project.

Open Source Code: The xFitter package is provided at www.xFitter.org, and a
write-up of the program can be found in Ref. Alekhin:2014irh , and an overview
of available tutorials in Ref. Bertone:2017tig  including some presented at
MCnet-CTEQ schools. The xFitter framework has already been used for more than
100+ analyses including many LHC studies. The code structure of the xFitter
package is modular, and it allows for various theoretical and methodological
options. Currently it contains interfaces to QCDNUM Botje:2010ay , APFEL
Bertone:2013vaa ; Bertone:2017gds , LHAPDF Buckley:2014ana , APPLGRID
Carli:2010rw , APFELGRID Bertone:2016lga , FastNLO Stober:2015nlg , HATHOR
Aliev:2010zk , among other packages.

xFitter also has a large number of data sets available, including a variety of
fixed target experiments, HERA, Tevatron, and LHC. It is also possible to add
new custom data sets such as LHeC LHeC:2020van  and EIC Accardi:2012qut ;
Proceedings:2020eah ) pseudo-data.

###  I.2 xFitter Capabilities

PDF Fits & Analysis: First and foremost, xFitter provides a flexible open-
source framework for performing PDF fits to data. The PDFs are the fundamental
object that xFitter works with, and it has a variety of utilities to read,
write, and manipulate the standardized PDF file format and associated
uncertainties. For example, xFitter is able to read and write PDFs in the
LHAPDF6 format Buckley:2014ana .

xFitter-draw: xFitter can also automatically generate comparison plots of data
vs. theory. There are a variety of options for the definition of the
$\chi^{2}$ function and the treatment of experimental uncertainties. Examples
are presented in Ref. Alekhin:2014irh .

Nuclear PDFs: xFitter has also been extended to produce nuclear PDFs; this was
used to produce the TUJU19 nPDF set of Ref. Walt:2019slu .

Pion PDFs: xFitter can also produce meson PDFs, and Ref. Novikov:2020snp
illustrates this for the case of pion PDFs.

Pseudo-Data: An important application of xFitter is to understand how a
particular data set or experiment will impact the PDFs. A typical study might
be to use pseudo-data from a proposed experiment (e.g. LHeC or EIC) to
constrain the relative uncertainty on the underlying PDFs. For example, Ref.
Abdolmaleki:2019acd  used LHeC pseudo-data to constrain the strange PDF with
charged-current DIS charm production data. Additionally, forward-backward
Drell-Yan asymmetry pseudo-data were prepared to simulate the end of Run-II
LHC (i.e. $300~{}fb^{-1}$), and also the HL-LHC; these pseudo-data have been
used for PDF profiling in Refs. Accomando:2019vqt  and Abdolmaleki:2019ubu .

Profiling & Reweighting: xFitter is able to perform PDF profiling and
reweighting studies. The reweighting method allows xFitter to update the
probability distribution of a PDF uncertainty set (such as a set of NNPDF
replicas) to reflect the influence of new data. For the PDF profiling, xFitter
compares data and MC predictions based on the $\chi^{2}$-minimization, and
then constrains the individual PDF eigenvector sets taking into account the
data uncertainties. For example, Ref. HERAFitterdevelopersTeam:2015cre  used
the Tevatron $W$-boson charge asymmetry and of the $Z$-boson production cross
sections data to study the impact on the PDFs using Hessian profiling and
Bayesian reweighting techniques.

In a separate study Accomando:2019vqt , the forward-backward asymmetry in
neutral current Drell-Yan production provides powerful constraints on the
valence quark PDFs, and this in turn can impact both SM Accomando:2018nig ;
Accomando:2017scx  and BSM Accomando:2016tah ; Accomando:2016ehi  physics.

NNLO & QED PDFs: As many PDF analyses are now extended out to NNLO, the NLO
QED effects can also become important. For example, including QED processes in
the parton evolution will break the isospin symmetry as the up and down quarks
have different couplings to the photon. xFitter is able to include NLO QED
effects, and this is illustrated in Ref. xFitterDevelopersTeam:2017fxf  which
computes the photon PDF as determined using a NNLO QCD and NLO QED analysis.

Transverse-momentum-dependent distributions: Transverse-momentum-dependent
(TMD) parton distribution functions Angeles-Martinez:2015sea  encode
nonperturbative information on hadron structure, extending to the transverse
plane the one-dimensional picture given by collinear PDFs, and providing a 3D
imaging of hadron structure. Similarly to collinear PDFs, TMDs can be
parameterised and fitted to experimental data. Within the xFitter framework,
the extraction of TMDs from fits to experimental data has been carried out in
the cases of CCFM evolution Hautmann:2013tba ; Dooling:2014kia ;
Hautmann:2014uua  and Parton Branching evolution Martinez:2018jxt ;
Hautmann:2017fcj ; Hautmann:2017xtx . xFitter is able to write and manipulate
TMDs in the TMDlib format Hautmann:2014kza ; Abdulov:2021ivr .

Small-$x$ resummation: xFitter can also study the impact of the
$\ln(1/x)$-resummation corrections to the DGLAP splitting functions and DIS
coefficient functions. The resummation formalism for both the splitting
functions and the coefficient functions is developed in Catani:1994sq . The
Leading-Log $\ln(1/x)$ (LLx) Kuraev:1976ge ; Kuraev:1977fs ; Balitsky:1978ic ;
Jaroszewicz:1982gr  and Next-to-Leading-Log $\ln(1/x)$ (NLLx) Catani:1994sq ;
Catani:1993rn ; Fadin:1998py ; Ciafaloni:1998gs  resummed calculations are
implemented in the public code HELL Bonvini:2016wki ; Bonvini:2017ogt . The
phenomenological effects of the $\ln(1/x)$ resummation are investigated in
Ref. xFitterDevelopersTeam:2018hym . In a related study Bonvini:2019wxf , a
more flexible PDF parameterisation is used with xFitter which provides a
better description of the combined inclusive HERA I+II data, expecially at low
$x$.

Pole & $\overline{MS}$ running masses: Another feature of xFitter is the
ability to handle both pole masses and $\overline{MS}$ running masses. While
the pole mass is more closely connected to what is measured in experiments,
the $\overline{MS}$ mass has advantages on the theoretical side of improved
perturbative convergence. xFitter was used to perform a high precision
determination of the $\overline{MS}$ charm mass in this new framework
Bertone:2016ywq .

Dipole models: We have several dipole models GolecBiernat:1998js ;
Bartels:2002cj ; Iancu:2003ge  implemented in xFitter which describe HERA
inclusive and diffractive DIS cross sections very well and are a natural
description of QCD reaction in the low x and low $Q^{2}$ region, fits to HERA
data are shown in Luszczak:2013rxa ; Luszczak:2016bxd . The gluon distribution
Luszczak:2016bxd  determined from dipole model in xFitter can be applied to
the description of selected LHC processes such as gamma-proton and nucleus-
nucleus collisions Luszczak:2017dwf ; Luszczak:2019vdc .

###  I.3 xFitter and Snowmass2021

We briefly discuss how xFitter might contribute to some of the future projects
being studied in the Snowmass2021 planning process.

LHC & HL-LHC: The xFitter package has been used for more than 100 analyses
including many LHC studies; a more complete list is available at
www.xFitter.org. Applying this work to data taken at HL-LHC is a natural
extension.

To highlight just one LHC example, the strange quark PDF has generated
considerable attention in the recent literature. There is a comprehensive
study Cooper-Sarkar:2018ufj  that examines the compatibility of both the ATLAS
Aaboud:2016btc ; Aad:2014xca  and CMS Chatrchyan:2013uja  data in a uniform
framework using the xFitter program.

EIC & LHeC: The EIC AbdulKhalek:2021gbh  and LHeC LHeC:2020van  facilities
will provide lepton–nucleon scattering in a collider configuration with a
variety of beams.

In addition to exploring the proton PDFs, these colliders can also study
nuclear PDFs with nuclear beams, and also meson (pion & kaon) structure via
leading neutron production. xFitter is capable of studying both nuclear PDFs
Walt:2019slu  and meson PDFs Novikov:2020snp . Additionally, xFitter can also
compute the transverse momentum dependent (TMD) distributions Alekhin:2014irh
.

DUNE: The Deep Underground Neutrino Experiment (DUNE) will use an intense
neutrino beam generated at Fermilab to study open questions about neutrino
oscillations. The massive DUNE detectors will also contribute to the study of
proton decay and Grand Unified Theories, as well as observe neutrino signals
from supernova core-collapse Arguelles:2019xgp .

In particular, the NuSTEC white paper Alvarez-Ruso:2017oui  outlines the
status and challenges of neutrino-nucleus interactions, with special attention
to DUNE. Improvements in PDF nuclear correction factors and the generation of
nuclear PDFs fit, specifically to neutrino–nucleus interactions in the
relevant energy range, can help minimize systematic errors for the +30%
fraction of DUNE events coming from the DIS region. This would enhance
analyses in both the near and far detectors. Thus, improvements by xFitter on
both proton and nuclear PDFs Walt:2019slu  can contribute to the DUNE project.

UHE Cosmic Rays Recent advances in neutrino astronomy have enabled us to study
ultra-high energy cosmic rays (UHECR) by studying atmospheric neutrinos. For
example, IceCube Aartsen:2016xlq ; Aartsen:2013jdh  has isolated more than 100
high-energy cosmic neutrinos, with energies between 100 TeV and 10 PeV.

Interpretation of these measurements would benefit from accurate PDFs in the
low-$x$ region. An example application is the evaluation of the prompt flux of
atmospheric neutrinos originating from the semileptonic decays of heavy-
flavored hadrons produced in the interactions of UHECR with nuclei in the
atmosphere Bhattacharya:2016jce ; Zenaiev:2019ktw . The prompt atmospheric-
neutrino flux represents a relevant background for searches of highly
energetic cosmic neutrinos. Thus, increased precision of both PDFs and nuclear
corrections in the very low-$x$ region would improve theoretical predictions
in this extreme kinematic region.

##  II Future Developments with xFitter

The latest xFitter release, 2.2.0, represents a significant restructuring of
the code. The new version provides significantly improved modularity for PDF
parameterization, evolution, theory predictions and minimization. This
additional flexibility simplifies further developments of the code.

The planned developments contain improvements of the existing functionality
and expansion of the code capabilities. Both goals will be achieved in a
modular manner, without increase of the requirements for the core package. An
emphasis is given on usage of modern, industry-standard libraries.

There are several ideas on improvement of the execution time required for
minimization. This is important in particular in view of large amount of data
samples which will come from LHC, EIC, and other experiments. xFitter already
contains an interface to the FK-tables (“apfelgrid”) via interface to APFEL.
These tables are based on APPLGRID predictions and combine evolution and
convolution in one step. The interface to modern minimization packages, such
as CERES ceres-solver , which is included in version 2.2.0, provides
additional opportunity to employ automatic differentiation. Introduction of
this capability is planned for the next release. Combined with improved
minimization code, this should improve convergence time for specific problems
by more that order of magnitude.

xFitter already contains highly efficient code for likelihood function
calculation. It will be improved further, by more extensive use of BLAS and
EIGEN libraries.

The state-of-the-art PDF fits include NNLO QCD plus NLO electroweak
corrections. However, in many cases these corrections are provided as fixed
k-factor tables. It is planned to include them using grid-based methods or
with semi-analytic routines H1:2018mkk ; Britzger:2020kgg .

xFitter plans further developments beyond conventional PDFs. In particular,
the TMDs which have already been introduced through the parton branching
method Hautmann:2017fcj  will be developed by including branching scale-
dependent resolution parameters Hautmann:2019biw  as a new functionality.
Furthermore, TMDs will also be introduced independently of the parton
branching method using the existing interface to the DYTURBO Camarda:2019zyx
package. It is also planned to introduce functionality for simultaneous PDF
and fragmentation function fits.

The inclusion of theoretical uncertainties in PDFs is currently one of the
main targets of state-of-the-art PDF global fits. xFitter plans to provide PDF
sets incorporating theoretical uncertainties by applying the resummation-scale
technique Bertone:2022sso ; Bertone:2022ope .

NLO EW corrections are important at the precision level of the Drell-Yan
measurements at the LHC and the Tevatron. Complete one-loop EW corrections for
NC and CC Drell-Yan processes are realized in ReneSANCe Monte Carlo event
generator that uses the SANC modules for differential cross-sections. The
predictions obtained by ReneSANCe were thoroughly cross-checked with the
results of MCSANC integrator. The complete one-loop EW corrections can be
separated in the code into weak, QED FSR, QED ISR and interference (IFI)
contributions. The dominant part of the higher-order EW corrections is QED FSR
which can be modelled by PHOTOS program. The effects of QED FSR calculated by
PHOTOS and ReneSANCe agree very well. The K-factors for the remaining parts of
NLO EW can be computed with ReneSANCe for various differential cross sections,
$A_{FB}$ and charge asymmetries: $K^{\text{EW}}=\frac{d\sigma_{\text{LO
QCD}}^{\text{NLO EW}}}{d\sigma_{\text{LO QCD}}^{\text{LO EW}}}.$ In factorized
approach this K-factors applies to NNLO QCD cross-sections:
$d\sigma_{\text{NNLO QCD}}^{\text{NLO EW}}=d\sigma_{\text{NNLO QCD}}^{\text{LO
EW}}\cdot K^{\text{EW}}.$

In the case of NC DY the separation of the FSR contribution is straightforward
and appears at the level of Feynman diagrams. But the corresponding separation
in the case of CC DY is not trivial and it is even not gauge invariant. For
the sake of agreement with PHOTOS, a special prescription for this separation
was introduced in SANC modules.

A formal separation of the pure weak (PW) and QED contributions $\delta^{PW}$
and $\delta^{QED}$ to the total $W^{+}\to u+\bar{d}$ decay width
$\Gamma_{W}^{PW+QED}=\Gamma_{W}^{LO}(\delta^{PW}+\delta^{QED})$ is depend on
’t Hooft scale parameter $\mu_{PW}$ and
$\delta^{QED}=\frac{\alpha}{pi}\left[Q_{W}^{2}(\frac{11}{6}-\frac{\pi^{2}}{3})+(Q_{u}^{2}+Q_{d}^{2})(\frac{11}{8}-\frac{3}{4}\log\frac{M_{W}^{2}}{\mu_{PW}^{2}})\right]$.
In order to separate the FSR QED we can choose
$\mu_{PW}=M_{W}\exp(-\frac{11}{12})$. This is in agreement with the
corresponding treatment in PHOTOS.

##  III Overview of recent xFitter results

###  III.1 W and Z Boson Production

Figure 1:  We show the d-valence PDF relative uncertainty at $Q^{2}=1.7~{}{\rm
GeV}^{2}$ as a function of the Bjorken-$x$ determined with a fit to the HERA I
data (blue), HERA I and Tevatron W-boson asymmetry and Z-boson data (yellow),
and HERA I and Tevatron W-boson lepton asymmetry and Z-boson data (green).
Figure from Ref. HERAFitterdevelopersTeam:2015cre .

In Reference HERAFitterdevelopersTeam:2015cre , the xFitter collaboration
(formerly HERAFitter) analyzed measurements of the W-boson charge asymmetry
and of the Z-boson production cross sections, performed at the Tevatron
collider in Run II by the D0 and CDF collaborations. Figure 1 shows the
comparison of the relative uncertainty of the PDFs using the HERA I data
alone, and together with the Tevatron $W/Z$ data. This new data significantly
reduces the PDF uncertainty, and this is most noticeable for the case of the
$d$-valence, as shown.

To study the possible model dependence due to the W-boson rapidity
reconstruction, an alternative fit was performed in which the W-boson charge
asymmetries measured by CDF and D0 were excluded, and the latest D0
measurement of the electron asymmetry was included. In Fig. 1, the fit to the
lepton asymmetry data (green band) yields very compatible results to the
W-boson asymmetry data (yellow band), but the uncertainties on the $d_{v}$ PDF
are up to twice as large.

These findings highlight the importance of the Tevatron W- and Z-boson
production data to constrain the PDFs, and illustrate the utility of xFitter
to easily demonstrate the impact of individual data sets. All the supporting
material to allow fits of the Tevatron data, including the updated correlation
model and the grid files for fast theory calculations, are publicly available
on the xFitter web page.

###  III.2 The Photon PDF

Figure 2: Comparison between the photon PDF $x\gamma(x,Q^{2})$ at
$Q^{2}{=}10^{4}\,{\rm GeV}^{2}$ from the present NNLO analysis
(xFitter_epHMDY) with the corresponding results from NNPDF3.0qed NNPDF:2014otw
, LUXqed Manohar:2016nzj  and HKR16 Harland-Lang:2016apc . The PDF
uncertainties are shown at the 68% CL obtained from the MC method, while model
and parametrisation uncertainties are addressed separately. For HKR16 only the
central value is shown, while for LUXqed the associated PDF uncertainty band
is included. Figure from Ref. xFitterDevelopersTeam:2017fxf .

Figure 3: A sample profiling comparison illustrating the impact of the
$A_{FB}$ data on the valence quark PDFs and their weighted sum for the
original (red) and profiled (blue, green, orange) CT14nnlo Dulat:2015mca  set.
Figure from Ref. Accomando:2019vqt .

Achieving the highest precision for theoretical predictions requires that the
calculations include perturbative QCD corrections up to (N)NNLO, and
electroweak (EW) corrections up to NLO. Working at this level of precision
demands the QED effects are fully included, and this requires the introduction
of the photon parton distribution of the proton, $x\gamma(x,Q^{2})$.

To demonstrate these capabilities, Ref. xFitterDevelopersTeam:2017fxf
analyzed recent ATLAS measurements of high-mass Drell-Yan dilepton production
at $\sqrt{s}=8$ TeV to determine the photon PDF, and to compare it with some
of the existing determinations from the literature.

To include the photon PDF, xFitter links to the APFEL program Bertone:2013vaa
to incorporate NLO QED effects. The inclusion of NLO QED evolution effects are
cross checked using the independent QEDEVOL Sadykov:2014aua  code based on the
QCDNUM Botje:2010ay  evolution program. Additionally, the aMCfast interface
Bertone:2014zva  is used to include the photon-initiated contributions in the
EW calculations within MadGraph5_aMC@NLO Alwall:2014hca .

The resulting photon PDF determination represents an important validation of
our understanding on the nature and implications of the photon PDF. Fig. 2
shows the results as compared with other recent QED fits. For $x\leq 0.1$, the
four determinations of the photon PDF are consistent within uncertainties. For
smaller values of $x$, the photon PDF from LUXqed and HKR16 is somewhat
smaller than xFitter_epHMDY, but still in agreement at the 2-$\sigma$ level.
This agreement is further improved if the PDF uncertainties in xFitter_epHMDY
arising from variations of the input parametrisation are added to experimental
uncertainties. Moreover, the results of this work and NNPDF3.0QED agree at the
68% CL for $x\leq 0.03$, and the agreement extends to smaller values of $x$
once the parametrisation uncertainties in xFitter_epHMDY are accounted for.
The LUXqed and the HKR16 calculations of $x\gamma(x,Q^{2})$ are very close to
each other across the entire range of $x$.

The results of this study have been made possible by a number of technical
developments that should be of direct application for future PDF fits
accounting for QED corrections. These technical improvements will certainly be
helpful for future studies of the photon PDF of the proton.

###  III.3 The Forward-Backward Asymmetry in Neutral Current Drell-Yan
Production

The DY-induced lepton charge asymmetry in charged current (CC) processes has
been an effective way to constrain PDFs. Ref. Accomando:2019vqt  examines the
use of neutral current (NC) lepton charge asymmetry measurements, which are
traditionally used in the context of precision determinations of the weak
mixing angle $\theta_{W}$, to obtain improved PDF constraints.

Specifically, a forward-backward asymmetry $A^{*}_{FB}$ is computed at NLO
using the MadGraph5_aMC@NLO program Alwall:2014hca  interfaced to APPLgrid
Carli:2010rw  through aMCfast Bertone:2014zva . The $A^{*}_{FB}$ is sensitive
to valence quark PDFs through the combination of chiral couplings
$(2/3u_{V}+1/3d_{V})$. The valence quark PDFs in turn also influence the sea
PDF constraints via the sum rule relations.

For this study, three sets of pseudodata relevant for LHC Run 2, 3, and HL-LHC
were considered. Fig 3 displays the impact of the data on the valence quark
PDFs and their combination weighted by their electric charges, for 30, 300 and
3000$\,{\rm fb}^{-1}$. Comparing those profiled error bands, we note a visible
improvement in the distribution of the valence quark PDFs, especially in the
region of small $x$. Some improvement in the high $x$ region of quark PDFs can
be obtained employing sufficiently large data samples and applying suitable
rapidity cuts on the $A^{*}_{FB}$ observable.

This analysis illustrates how high-statistics measurements, both cross section
and asymmetry distributions, from the LHC Runs 2, 3 and the HL-LHC stage can
be exploited to place constraints on the PDFs. The PDF profiling calculations
in the xFitter framework can flexibly be used to investigate the impact of
pseudodata on PDF determinations.

###  III.4 Impact of low-x resummation on QCD analysis of HERA data

Figure 4: The H1 extraction of $F_{L}$ compared to the predictions with and
without the $\ln(1/x)$ resummation (NLLx) contributions. Figure from Ref.
xFitterDevelopersTeam:2018hym .

In Ref. xFitterDevelopersTeam:2018hym , xFitter explored the impact of
resumming the Next-to-Leading-Log $\ln(1/x)$ (NLLx) contributions in the small
$x$ region using a QCD analysis of HERA data. This work was prompted, in part,
by the observation that the $\chi^{2}$ of the NNLO fits was not improved as
compared with the NLO fit for low values of $Q^{2}$.

This analysis used the final combined $e^{\pm}p$ cross-section measurements of
H1 and ZEUS H1:2015ubc  which covered the kinematic range of $Q^{2}$ from
0.045 GeV2 to 50000 GeV2 and of Bjorken $x$ from 0.65 down to
$6{\times}10^{-7}$. Fits to this final combined HERA DIS cross-section data
within the conventional DGLAP framework of QCD have shown some tension at low
$x$ and low $Q^{2}$. Our goal was to determine whether incorporating the
$\ln(1/x)$-resummation terms into the HERAPDF fits might help resolve these
tensions.

While the details are presented in Ref. xFitterDevelopersTeam:2018hym , in
brief it was observed that the total $\chi^{2}$ for the fits improved from
1468/1207 for the NNLO fit, to 1394/1207 for the NNLO+NNLLx fit. The nominal
improvement of the total $\chi^{2}$ reflects the fact that the small $x$ data
is only a portion of the full data set; however, the impact in the small $x$
region is significant as indicated in Fig. 4 where we see the $\ln(1/x)$
contributions yield an improved fit for the longitudinal structure function in
the small $x$ and low $Q^{2}$ region. The precise kinematic region where this
resummation is important is delineated in Fig. 11 of Ref.
xFitterDevelopersTeam:2018hym .

Additionally, the resummation yields a larger gluon PDF in the small $x$ and
low $Q^{2}$ region, and this avoids potential problems due to a negative gluon
PDF in forward physics. These features make PDFs with ln(1/x) resummation much
more suitable for use in MC generators, such as Sherpa Gleisberg:2008ta .

In conclusion, $\ln(1/x)$ resummation provides substantial improvement in the
description of the precise HERA1+2 combined data. The NLLx contributions yield
an improved description of the data, especially the longitudinal structure
function $F_{L}$ in the small $x$ region, as compared to both the (un-
resummed) NLO and NNLO analysis, and it also avoids potential problems of the
negative gluon PDF at low $x$ and $Q^{2}$.

###  III.5 Heavy quark matching scales: Unifying the FFNS and VFNS

Figure 5: The b-quark structure function $F_{2}^{b}(x,\mu)$ for different
choices of the matching scales $\mu_{m}{=}\\{m_{b}/2,m_{b},2m_{b}\\}$
(indicated by the vertical lines) computed at NLO (Fig.-a) and NNLO (Fig.-b).
Figure from Ref. xFitterDevelopersTeam:2017fzy .

High precision phenomenological analysis of DIS data requires a proper
treatment of the heavy quarks mass scales. The Variable Flavor Number Scheme
(VFNS) incorporates the heavy quark mass scale across the full kinematic range
by varying the number of active flavors ($N_{F}$) in the DGLAP QCD evolution.
In the VFNS, the matching scale $\mu_{m}$ determines the transition from
$N_{F}$ to $N_{F}+1$ active flavors. Historically, $\mu_{m}$ was taken to be
the heavy quark mass $m_{H}$; however, this choice is not required, and there
are advantages to using a flexible matching scale. This feature is implemented
in xFitter via the APFEL code, and we demonstrate the benefits in Ref.
xFitterDevelopersTeam:2017fzy .

An essential element of this analysis is to implement the correction matching
of the PDFs across the flavor threshold. The matching conditions have been
computed to NLO and NNLO within the xFitter code, and the impact of this
choice on the physical structure function $F_{2}^{b}(x,Q)$ is illustrated in
Fig. 5. We emphasize some key features illustrated in the matching conditions.

At NLO (Fig. 5-a), we observe that there are sizeable differences for the
three choices of matching scales, $\mu_{b}=\\{m_{b}/2,m_{b},2m_{b}\\}$. As the
NLO matching conditions incorporate the ${\cal O}(\alpha_{S})$ DGLAP evolution
contributions, the differences of these curves simply reflects the NNLO ${\cal
O}(\alpha_{S}^{2})$ correction. Thus, when we compute the NNLO result (Fig.
5-b), we find the differences due to the matching scale are now significantly
reduced as they are N3LO ${\cal O}(\alpha_{S}^{3})$.

Ref. xFitterDevelopersTeam:2017fzy  illustrates the use of the variable
matching scale for the case of heavy quark (charm and bottom) production at
HERA. The flexibility of choosing the $\mu_{m}$ matching scale provides a
number of advantages. By adjusting the matching scale, we can effectively
transition between a VFNS and a FFNS in a seamless manner. For example,
$\mu_{m}{=}m_{b}$ would correspond to the traditional VFNS, and
$\mu_{m}{\to}\infty$ would correspond to the traditional FFNS. Furthermore we
can choose any scale in between. Thus, the variable heavy flavor matching
scale $\mu_{m}$ generalizes the transition between the FFNS and the VFNS.

On a more practical level, this flexibility allows one to shift the matching
scale so that the discontinuities associated with the $N_{F}$ to $N_{F}+1$
transition do not lie in the middle of a specific data set.

In summary, the flexibility of choosing the heavy flavor matching scale
$\mu_{m}$ generalizes the transition between a FFNS and a VFNS, and provides a
theoretical “laboratory” which can quantitatively test proposed
implementations. The ability to vary the heavy flavor matching scales not only
provides new insights into the intricacies of QCD, but also has practical
advantages for PDF fits

###  III.6 Constraining the Strange PDF

Figure 6:  The relative strange PDF uncertainty at $\mu_{f}^{2}{=}100$ GeV2 of
the original and profiled ABMP16 PDF set. Figure from Ref.
xFitterDevelopersTeam:2019ygc .

The study of heavy quark production plays a critical role enabling us to fully
characterize the properties of the SM. The charm quark is especially useful in
this respect as the combination of fixed-target and collider data allow us to
investigate the full kinematic spectrum from the threshold region ($m_{c}\sim
Q$) to the asymptotic high energy limit ($m_{c}\ll Q$). Additionally, the
charged-current charm production provides direct access to the strange quark
distribution. The strange PDF is of particular interest because it still has
large uncertainties despite being extensively investigated in a number of
experiments such as inclusive $W/Z$ production and $W+c$ associated
production. Looking to the future, it is clearly important to reduce the
uncertainty of the PDFs in general, and the strange-quark in particular, as we
strive to make increasingly precise tests of the SM.

In Ref. xFitterDevelopersTeam:2019ygc , we used the xFitter framework to study
charm production in charged-current deep-inelastic scattering (DIS) using LHeC
pseudodata to estimate the potential improvement from a future DIS facility.
As xFitter implements both fixed-flavor- and variable-flavor-number schemes,
we also compared the impact of these different theoretical choices to
highlight several interesting aspects of multi-scale calculations.

For the LHeC parameters, we assumed a 7 TeV proton beam on a 60 GeV electron
beam yielding $\sqrt{s}\sim 1.3$ TeV with a nominal design luminosity of
$10^{33}{\rm cm}^{-2}\,{\rm s}^{-1}$. Compared to HERA, this extends the
covered kinematic range by an order of magnitude in both $x_{Bj}$ and $Q^{2}$.
The predictions are provided for unpolarized beams in the kinematic range
$10^{2}<Q^{2}<10^{5}\,{\rm GeV}^{2}$, $10^{-4}<x_{Bj}<0.25$, and
$0.0024<y<0.76$.

Using the high-statistics pseudodata with the xFitter analysis, we find strong
constraints on the strange-quark PDF, especially in the previously unexplored
small-$x_{Bj}$ region. Figure 6 illustrates the relative improvement of the
strange PDF uncertainty using the ABMP16 PDFs Alekhin:2018pai . The general
reduction of the PDF uncertainties is evident across the full $x$ range. We
also performed calculations using the NNPDF3.1 PDFs NNPDF:2017mvq  and
obtained similar conclusions.

As xFitter can compute in both the Fixed Flavor Number Scheme (FFNS) Variable
Flavor Number Scheme (VFNS), we investigated the differences due the the
scheme choice. We computed the reduced cross section using both the FFNS and
VFNS, and then cut out those pseudodata where the scheme uncertainty was
larger than the PDF uncertainty. In Fig. 6 this result (in blue) is labeled as
“profiled with cuts.” If we include all the data (labeled, “profiled all
data”) we obtain the uncertainty band labeled in green. Comparing these two
results, we find that the pseudodata can impose very strong constraints on the
PDFs and this is independent of the particular heavy-flavor scheme.

###  III.7 The Pion PDF

Figure 7: Comparison between the pion PDFs obtained in this work, a
determination by the JAM collaboration Barry:2018ort , and the GRVPI1 pion PDF
set Gluck:1991ey . Figure from Ref. Novikov:2020snp .

To further demonstrate the versatility of xFitter, Ref. Novikov:2020snp
presented the first open-source analysis of parton distribution functions
(PDFs) of charged pions. The pion is the simplest $q\bar{q}$ state in the
quark-parton model of hadrons. However, despite this apparent simplicity, the
pion structure is currently poorly understood, especially compared to the
proton.

Experimentally, the pion PDF is known mostly from QCD analyses of Drell-Yan
(DY) and prompt photon production data. This analysis used a combination of
Drell-Yan (E615 and NA10) and prompt photon (WA70) data to provide constraints
on both the quarks and gluons across the kinematic range.

The calculations are implemented at next-to-leading order (NLO) using
APPLgrids generated by MCFM generator which allows for efficient numerical
computations; additionally, modifications were made to APPLgrid which
facilitate both meson and hadron PDFs in the initial state.

Fig. 7 presents the fitted pion PDFs and compares with results from JAM
Barry:2018ort  and GRVPI1 Gluck:1991ey . In general, the results obtained from
xFitter compare favorably to JAM within uncertainties, but do show differences
with the older GRV analysis. For the uncertainty bands, the $\mu_{R}$ and
$\mu_{F}$ renormalization scales were varied, as well as the PDF parameter
values. For the valence distribution, the dominant uncertainty is the scale
variation, and this is rather well constrained. For the sea and gluon
distributions, the calculation only included the direct photons in the model;
hence, the missing fragmentation contribution to the prompt photon production
process is the dominant uncertainty source. This is certainly an issue to be
improved in a future analysis.

To further illustrate the flexibility of the xFitter framework, both a
3-parameter and 4-parameter form for the initial valence distribution was
used. This type of parameter flexibility is crucial as it allows, for example,
the investigation of the PDF slope in the large $x$ limit.

While this study demonstrated the versatility of xFitter to perform meson PDF
analysis, there are numerous directions this work can be extended. For
example, new $J/\Psi$ data could play an important role in constraining the
gluon Chang:2020rdy , and data from future experiments such as COMPASS++/AMBER
Adams:2018pwt , may allow for more flexible parameterizations and improved
constraints.

###  III.8 The Pion Fragmentation Function

Figure 8:  Comparison of the preferred Fit E [IPMx] as well as Fit B for
charged pion FFs ($\pi^{+}{+}\pi^{-}$) at NNLO with NNFF1 [4] at NNLO, JAM19
[13] at NLO, DSEHS [5] at NLO for $Q^{2}{=}100\,{\rm GeV}^{2}$. Note,
discretion is necessary when interpreting the very low z region; see text.
Figure from Ref. Abdolmaleki:2021yjf .

In addition to pion PDFs, xFitter can also compute pion fragmentation
functions. Reference Abdolmaleki:2021yjf  presents the first open-source
analysis of fragmentation functions of charged pions (entitled IPM-xFitter)
computed at next-to-leading order (NLO) and next-to-next-to-leading order
(NNLO) accuracy in perturbative QCD using the xFitter framework. This study
incorporated a comprehensive set of pion production data from single-inclusive
annihilation (SIA) processes, as well as the most recent measurements of
inclusive cross-sections of single pion by the BELLE collaboration.

A primary goal of this analysis was to investigate the influence of the
BELLE13 Belle:2013lfg , BELLE20 Belle:2020pvy , and BaBar BaBar:2013yrg  data
sets on the resulting fragmentation functions. A total of five fits were
generated using different data combinations and different kinematic cuts. 1)
Fit A focused on the impact of the BELLE13 data set without BELLE20; 2) Fit B
used the BELLE20 data set without BELLE13; 3) Fit C used the BELLE20 data set
without BaBar; 4) Fit D focused the impact of the BELLE20 data set without
either BELLE13 or BaBar, and imposed a $z{>}0.2$ cut on the BELLE20; 5) Fit E
excluded BELLE13 but included BELLE20 with a $z{>}0.2$ cut, and BaBar with a
$z{>}0.1$ cut.

Comparisons of selected fits are displayed in Fig. 8 along with results from
the literature. The different fits are in reasonable agreement for larger $z$
values, but begin to diverge for smaller $z$. Note, discretion is necessary
when interpreting the very low z region as the extrapolation of the
fragmentation function grids extends beyond the region fitted in the
individual analyses. For example, the JAM19 focus was on SIDIS in the region
$z\gtrsim 0.2$, and NNFF1 used a lower kinematic cut of $z_{min}{=}0.02$ for
$Q=M_{Z}$ and 0.075 for $Q{<}M_{Z}$. While Fit E is our preferred fit, we also
display Fit B to highlight the impact of the low z cuts.

This study generally found good quality for the fits across most of the $z$
range, but the description of the data in the low-z region remains an
unresolved puzzle. The resulting NLO and NNLO pion FFs provide valuable
insights for applications in present and future high-energy analysis of pion
final state processes.

Contemporaneous with the above study, a related investigation was reported in
Ref. Khalek:2021gxf  using a complementary neural-network approach with
different data sets; this opens new avenues for future study and may help
resolve some of the issues in the small $z$ region.

##  IV Applications of xFitter

###  IV.1 Usage of xFitter by LHC Collaborations

xFitter is the standard tool for QCD analyses by the ATLAS and CMS
Collaborations at the LHC, resulting in a number of different studies.

xFitter has been used to quantify the sensitivity to PDFs of differential W
and Z bosons ATLAS:2012sjl ; CMS:2013pzl ; ATLAS:2016nqi ; CMS:2016qqr  and of
W+charm quark CMS:2018dxg ; CMS:2021oxn  cross-section measurements, showing
how they provide important constraints on the strangeness content of the
proton. Other studies have illustrated the gluon PDF sensitivity of $t\bar{t}$
cross-sections measurements CMS:2017iqf ; CMS:2017zpm ; ATL-PHYS-PUB-2018-017
. An ATLAS study ATLAS:2016oxs  highlighted cross-section ratios, such as for
$t\bar{t}/Z$ measured at different energies, as a powerful quantity to probe
the gluon/sea PDF ratio free of large experimental uncertainties.

The xFitter capabilities to perform coherent analysis of PDFs plus additional
parameters of interest, has been exploited to measure SM parameters which
correlate with PDFs. The $t\bar{t}$ cross-sections have been used for
extractions of the top quark mass ATLAS:2017dhr ; CMS:2018fks ; CMS:2019esx ,
which for the CMS analyses has been done simultaneously with a determination
of the strong coupling and of the PDFs. Simultaneous extractions of the strong
coupling with the PDFs have been performed by CMS also from inclusive jets
CMS:2016lna ; CMS:2021yzl  and dijet CMS:2017jfq  cross-sections, recently
extended to determinations of the contribution of SMEFT operators introducing
new four-fermion interactions simultaneously with the PDFs CMS:2021yzl .

In Ref. ATLAS:2016nqi , high precision measurements of the inclusive
differential $W^{\pm}$ and $Z/\gamma^{*}$ boson cross sections at 7 TeV were
added to the HERA data, resulting in the PDF set ATLASepWZ16, which improved
on the HERAPDF2.0 set in various respects. Firstly, the strange content of the
sea was determined rather than assumed to be a fixed fraction of the light
sea. Indeed, compared to previous determinations, the strange sea was found to
be enhanced at low $x$. Secondly, the accuracy of the valence quark
distributions for $x<$ 0.1 was improved.
In Ref. ATLAS:2021qnl , the ATLAS Collaboration performed a PDF fit including
for the first time measurements for the production of W and Z boson in
association with a jet, resulting in the ATLASepWZVjets20 PDF set. The $V$ \+
jets data are sensitive to partons at higher $x$ than can be accessed by
inclusive $W,Z/\gamma^{*}$ data and, in particular, they constrain the
$\bar{d}$ and $\bar{s}$ quarks at higher $x$.
And more recently an ATLAS PDF fit ATLAS:2021vod  was performed including a
variety of measurements of different production processes (Drell-Yan, V+jets,
direct photon production, $t\bar{t}$ and inclusive jets data) at different
center-of-mass energies. The resulting set of PDFs is called ATLASpdf21.

###  IV.2 Nuclear PDFs with xFitter

Figure 9: Selected nuclear PDF ratios for lead as compared to proton at NLO
for TUJU19 Walt:2019slu , nCTEQ15 Kovarik:2015cma , EPPS16 Eskola:2016oht ,
and DSSZ deFlorian:2011fp  at $Q^{2}=100~{}{\rm GeV}^{2}$. Figures from Ref.
Walt:2019slu .

Reference Walt:2019slu  extended xFitter to include the capability of
performing fits to nuclear PDFs, and generated a new set of nPDFs (TUJU19) at
both NLO and NNLO. While this extension is capable of performing a
simultaneous fit to both the proton and nuclear degrees of freedom, to ensure
stability they first generated a proton baseline derived with a very similar
setup as for the HERA2.0 PDFs H1:2015ubc , and then used this as the starting
point for the nPDF investigation. Both charged-lepton and neutrino DIS data
sets wer included, and and isoscalar corrections and correlated uncertainties
were also incorporated.

The necessary modifications to the xFitter code were extensive. The PDF
parameterization was extended to include the nuclear A dimension; this
increased the number of fitting parameters, and these changes were propagated
throughout the code (e.g., in the steering file and the MINUIT interface). The
nuclear data sets depend on the nuclear A and Z, so this information was
provided within the data files and the input/output routines were modified
accordingly. Finally, the DGLAP evolution code was modified to evolve nuclear
PDFs covering different combinations of A and Z individually. These
modifications included a new numerical integration routine to accommodate the
flexibility of an A-dependent normalization.

This work highlights the versatility and flexibility of the xFitter program,
and this study serves as a foundation for a wide variety other investigations
involving nuclear interactions. The details of this work are presented in Ref.
Walt:2019slu , and the code is available on GitLab.333 The xFitter code for
the TUJU19 nPDFs is available at:
https://gitlab.com/fitters/xfitter/-/tree/NuclearPDFs_2.0.1

###  IV.3 Higgs Physics with xFitter

Precision studies in the Higgs sector of the Standard Model (SM) are central
to current LHCHiggsCrossSectionWorkingGroup:2016ypw  and forthcoming
Cepeda:2019klc  physics programs at the LHC. The dominant mechanism for the
production of Higgs bosons in $pp$ collisions at the LHC is given by gluon
fusion. With the very high accuracy reached in perturbative QCD calculations
of Higgs production cross sections, currently of next-to-next-to-next-to-
leading order (N3LO) Chen:2021isd  in the QCD coupling $\alpha_{s}$, the
theoretical systematic uncertainties on the predictions for Higgs boson
production are strongly influenced by the gluon PDF, as well as the sea-quark
PDFs coupled to gluons through initial-state QCD evolution. The PDF
contribution is estimated Cepeda:2019klc  to be about 30$\%$ of the total
uncertainty, including $\alpha_{s}$ and scale variations.

The primary source of knowledge of the gluon PDF is provided at present by
HERA deep inelastic scattering (DIS) experimental measurements. While future
DIS experiments LHeC:2020van ; Accardi:2012qut ; Proceedings:2020eah  are
proposed to extend the range and accuracy of our current knowledge of the
gluon PDF, substantial progress can also come from measurements at the LHC
itself, particularly in the forthcoming high-luminosity phase HL-LHC
Azzi:2019yne . Gluon PDF studies have been considered so far from the analysis
of light-quark jets AbdulKhalek:2020jut , open Zenaiev:2019ktw ;
Cacciari:2015fta  and bound-state Flett:2019pux  charm and bottom quark
production, top quark production Czakon:2016olj . These studies rely on
parton-level processes with colored particles in the lowest-order final state,
which are influenced by large radiative corrections. Ref. Amoroso:2020fjw
proposes an alternative approach, based on considering color-singlet
production at the LHC, and (analogously to the case of DIS at HERA) achieving
sensitivity to the gluon PDF through ${\cal O}(\alpha_{s})$ contributions,
guided by criteria of perturbative stability and experimental precision.

Drell-Yan (DY) lepton pair production via electroweak vector boson exchange is
one of the most precisely measured processes at the LHC. The DY cross section
summed over the vector-boson polarizations is sensitive to the gluon PDF for
finite vector-boson transverse momenta $p_{T}$. However, in the $p_{T}$ region
where the cross section is largest, it is affected by large perturbative
corrections to all orders in $\alpha_{s}$ (see e.g. Angeles-Martinez:2015sea ,
and references therein). Ref. Amoroso:2020fjw  therefore turns to the
contributions of the individual polarizations of the vector boson. It exploits
the sensitivity of the DY angular coefficient $A_{0}$ associated with the
longitudinal vector-boson polarization to the gluon PDF in order to constrain
the Higgs boson cross section from gluon fusion. The coefficient $A_{0}$ is
perturbatively stable, as illustrated by the smallness of its next-to-leading-
order (NLO) and next-to-next-to-leading-order (NNLO) Gauld:2017tww  radiative
corrections for finite $p_{T}$, and precisely measured at the LHC CMS:2015cyj
; ATLAS:2016rnf . With xFitter, we can investigate in detail the impact of
precision measurements of the longitudinally-polarized vector-boson
coefficient $A_{0}$ on the theoretical predictions for the Higgs boson
production cross section. These studies can further be extended, using
xFitter, to other DY angular coefficients, since additional sensitivity may be
gained from longitudinal-transverse polarization interferences, as in the
parity-conserving $A_{1}$ and parity-violating $A_{3}$ coefficients.

An example illustrating the implications of the longitudinally polarized
$A_{0}$ coefficient on the Higgs boson production is shown in Fig. 10
Amoroso:2020fjw  versus the Higgs boson rapidity $y_{H}$. The SM Higgs boson
production is computed in the gluon fusion mode for $\sqrt{s}=13$ TeV $pp$
collisions, using the MCFM code Campbell:2019dru  at NLO in QCD perturbation
theory. We evaluate PDF uncertainties on the Higgs cross section including
constraints obtained with xFitter from $A_{0}$ profiling near the $Z$-boson
mass. We see that in the region $-2{\raisebox{-2.58334pt}{\hbox
to0.0pt{$\,\sim\,$\hss}}\raisebox{1.72218pt}{$\,<\,$}}y_{H}{\raisebox{-2.58334pt}{\hbox
to0.0pt{$\,\sim\,$\hss}}\raisebox{1.72218pt}{$\,<\,$}}2$ the uncertainty is
reduced by about 30 - 40 $\%$ in the Run III scenario, and a further reduction
to about 50 $\%$ takes place in the HL-LHC scenario.

Figure 10: Ratio of PDF uncertainties for the gluon-gluon fusion SM Higgs
boson cross-section in $pp$ collisions at $\sqrt{s}=$13 TeV as a function of
the Higgs rapidity Amoroso:2020fjw . The blue band shows the uncertainties of
the CT18nnlo PDF set Hou:2019efy , reduced to 68% CL coverage. The red and
green bands show the uncertainties of the CT18nnlo including constraints from
the $A_{0}$ measurement and assuming 300 fb-1 and 3 ab-1, respectively.

In Fig. 11 Amoroso:2020fjw  we perform a higher-order N3LO calculation for the
Higgs boson total cross section using the code ggHiggs Bonvini:2014jma ;
Bonvini:2018ixe . We report the result for the cross section and its
uncertainty in the cases of the current CT18nnlo Hou:2019efy , NNPDF3.1nnlo
NNPDF:2017mvq  and MSHT20nnlo Bailey:2020ooq  global sets as well as projected
sets, based on complete LHC data sample AbdulKhalek:2018rok . The
PDF4LHC15scen1/2 sets, which are PDF projections including HL-LHC pseudodata,
also show a smaller, but not negligible, reduction in uncertainties.
Notwithstanding the numerical differences, the behavior is qualitatively
similar for the different sets, and provides further support at N3LO to the
picture given in Fig. 10 for the NLO Higgs boson rapidity cross section.

Figure 11: The gluon-gluon fusion Higgs boson production cross-section
Amoroso:2020fjw  at N3LO for different PDFs, showing the uncertainty from PDFs
and their expected reduction including constraints from the $A_{0}$
measurement assuming 300 fb-1 and 3 ab-1, respectively.

The approach illustrated above can be extended to mass regions away from the
$Z$ peak, where it has the potential to provide complementary physics
information. High-mass DY angular distributions allow the region of larger $x$
momentum fractions to be accessed and will be relevant for associated Higgs
boson production with a gauge/Higgs boson or heavy-flavour quarks.
Measurements of $A_{0}$ at low masses may be used to probe $p_{T}$ dependent
gluon PDF effects, influencing the Higgs boson $p_{T}$ spectrum for low
transverse momenta and ratios of Higgs to DY $p_{T}$ spectra Cipriano:2013ooa
, as well as the small-$x$ regime Bonvini:2018ixe ; Hautmann:2002tu  of Higgs
boson production relevant to the highest energy frontier Contino:2016spe .

###  IV.4 BSM Physics with xFitter

The capabilities of xFitter can be exploited for studies of new physics beyond
the Standard Model (BSM).

Consider for instance BSM experimental searches at the LHC in
dielectron/dimuon channels and charged lepton plus missing transverse energy
channels. These are classic methods to search for new $Z^{\prime}/W^{\prime}$
gauge bosons: see for example the analyses by ATLAS in Refs. ATLAS:2019erb ;
ATLAS:2019lsy  and by CMS in Refs. CMS:2018ipm ; CMS:2021ctt . Such searches
will be further pursued at the High-Luminosity LHC (HL-LHC) CidVidal:2018eel .

In the case of BSM scenarios with narrow vector resonances one can rely on
traditional “bump search” analyses based on the Breit-Wigner (BW) lineshape.
In the case of BSM scenarios featuring vector resonances with large width, on
the other hand, the default bump searches are likely not sufficient: instead
of an easily observable narrow BW lineshape, the resonance appears as a broad
shoulder spreading over the SM background. Alternative experimental approaches
can be applied to the case of wide resonances, in which one has to exploit the
tails of the measured mass distributions. Examples of such approaches include
for instance “counting strategy” analyses. These approaches are much more
dependent than bump searches on the modeling of the dominant production
process Accomando:2019ahs , for both signal and background.

Then, in the case of wide-resonance searches the role of PDFs becomes
prominent as one of the main sources of theoretical systematic uncertainties,
because it directly affects our ability to test BSM scenarios and the
experimental sensitivity to new physics. This thus influences the potential of
experimental searches for discovering or setting exclusion bounds on heavy BSM
states.

In the multi-TeV region of $W^{\prime}/Z^{\prime}$ masses defined by the
current LHC exclusion limits ATLAS:2019erb ; ATLAS:2019lsy ; CMS:2018ipm ;
CMS:2021ctt , quark distributions in the valence sector dominate the PDF
systematics. With an xFitter profiling analysis, Ref. Fiaschi:2021okg  finds
that the valence quark systematics can be improved by combining high-precision
measurements of Charged Current (CC) and Neutral Current (NC) Drell-Yan
asymmetries in the mass region near the SM weak boson poles. This exploits the
sensitivity of the NC forward-backward asymmetry $A_{\rm{FB}}$ to the charge-
weighted linear combination $(2/3)u_{V}+(1/3)d_{V}$ of up-quark and down-quark
valence distributions Accomando:2019vqt  and the sensitivity of the CC lepton-
charge asymmetry $A_{W}$ to the difference $u_{V}-d_{V}$. The constraints from
the $A_{\rm{FB}}$ and $A_{W}$ combination at the SM weak boson mass scale,
examined in the two projected luminosity scenarios of 300 fb-1 (for the LHC
Run 3) and of 3000 fb-1 (for the HL-LHC CidVidal:2018eel ), turn out to
improve the relative PDF uncertainties by up to around 20% Fiaschi:2021okg  in
the region of the invariant and transverse mass spectra between 2 TeV and 6
TeV, in which evidence for $W^{\prime}$ and/or $Z^{\prime}$ states with large
widths could first be observed.

We can then analyze quantitatively how the “improved PDFs”, obtained from the
reduction of the valence PDF uncertainty due to $A_{\rm{FB}}$ and $A_{W}$
precision measurements at the SM weak boson mass scale, result into an
enhancement of the experimental sensitivity to BSM searches at the TeV scale.
In the next subsection we illustrate this with a specific example.

In this kind of analysis, an important feature, stemming from multi-resonant
profiles of mass spectra and influencing the experimental search strategies,
is the following. Strong interference effects between the BSM resonances
themselves and between the BSM and SM states can give rise to a statistically
significant depletion of events below the BW peak in the invariant mass
distribution, leading to the appearance of a pronounced dip Accomando:2019ahs
; Fiaschi:2021okg . It is possible to define the significance of the depletion
of events in a manner similar to that for the excess of events of the peak,
which can be used to extract model dependent exclusion and discovery limits in
the model’s parameter space. One can then present such limits resulting from
the analysis of the spectra for either the peak or the dip. The xFitter
framework enables one to investigate whether the peak or the dip analyses
especially benefit from the “improved PDFs”.

We next turn to the specific example of BSM $W^{\prime}/Z^{\prime}$ states in
composite Higgs models, considering broad-resonance searches in the leptonic
channels. For sensitivity studies comparing leptonic with heavy-quark channels
in composite Higgs models see e.g. Liu:2019bua .

####  IV.4.1 Enhancing the sensitivity of BSM searches with improved PDFs

New gauge sectors in strongly-coupled models of electroweak symmetry breaking
Panico:2015jxa  based on composite Nambu-Goldstone Higgs Kaplan:1983fs ;
Kaplan:1983sm  feature multiple $W^{\prime}$ and $Z^{\prime}$ broad
resonances, and interference effects of the heavy bosons with each other and
with SM gauge bosons. An example is the 4-Dimensional Composite Higgs Model
(4DCHM) realization DeCurtis:2011yx  of the minimal composite Higgs model of
Ref. Agashe:2004rs . The parameter space of the model can be characterized in
terms of two parameters, the compositeness scale $f$ and the coupling
$g_{\rho}$ of the new resonances, with the resonances mass scale being of
order $M\sim fg_{\rho}$ Panico:2015jxa ; Giudice:2007fh . Ref. Fiaschi:2021sin
selects two 4DCHM benchmarks, denoted by A and B, each characterized by
specific values of $f$, $g_{\rho}$ and the resonance masses. We next present
an example of the results for exclusion and discovery limits in the parameter
space of these two 4DCHM benchmarks which can be obtained by using the
“improved PDFs” Fiaschi:2021okg .

In Fig. 12 Fiaschi:2021sin  we show the limits from NC Drell-Yan on the model
parameter space, described in terms of $f$ and $g_{\rho}$, for the HL-LHC
stage with center-of-mass energy of 14 TeV and an integrated luminosity of
3000 fb-1. In the background we give contour plots for the masses of the new
gauge bosons. The blue and red curves are obtained respectively with the
baseline CT18NNLO PDF set Hou:2019efy  and with the profiled PDFs
Fiaschi:2021okg  using the $A_{\rm{FB}}$ and $A_{W}$ combination with 3000
fb-1 of integrated luminosity. The top panel in Fig. 12 is for the peak
analysis, the bottom panel is for the dip analysis. In this setup, the peak of
benchmark A would still be below the experimental sensitivity, while the peak
of benchmark B, if the improved PDFs are employed, would be right below the
2$\sigma$ exclusion. When exploiting the depletion of events in the dip below
the peak, the sensitivity on the model increases remarkably. Furthermore, as
visible from the bottom plot of Fig. 12, the improvement on the PDF also has a
very large impact particularly in the region of small $f$ and large
$g_{\rho}$, where the sensitivity on the dip can overtake the LHC reach in the
peak region once the profiled PDFs are employed. Taking into account the
reduction of PDF uncertainty, benchmark A would be now at the edge of the
5$\sigma$ discovery, while the sensitivity on benchmark B would almost reach
3$\sigma$.

Figure 12: Exclusion and discovery limits at 3000 fb-1 for the peak (top) and
for the dip (bottom) for $Z^{\prime}$ resonances with $\Gamma/M$ = 20%
Fiaschi:2021sin . The short (long) dashed contours give the BSM boson mass
$M_{Z_{2}}$ ($M_{Z_{3}}\simeq M_{W_{2}}$) in TeV.

Tab. 1 Fiaschi:2021sin  lists, in the case of the two benchmarks A and B, the
integration intervals in invariant mass for the dip region as well as the
resulting cross sections for the SM background and the complete 4DCHM,
together with the associated PDF uncertainty using the baseline CT18NNLO PDF
set Hou:2019efy  and the profiled version Fiaschi:2021okg  using the
$A_{\rm{FB}}$ and $A_{W}$ combination with 3000 fb-1 of integrated luminosity.
It reports also the obtained significances for an integrated luminosity of
3000 fb-1 employing the two PDF errors. The improvement in significance due to
the profiled PDFs is sizeable, as significances grow by an amount from 40% to
90% (i.e., by a factor of about 10 in comparison) for the benchmarks A and B,
respectively. A more modest improvement in significance is found
Fiaschi:2021sin  in the case of the peak.

Benchmark A
---
inf [TeV] | sup [TeV] |  $\sigma_{\rm SM}$ [fb] |  $\sigma_{\rm SM+BSM}$ [fb]
2.06 | 4.99 | 1.69 $\cdot$ 10-1 | 1.42 $\cdot$ 10-1
$\Delta_{\rm PDF}$ base [fb] |  $\Delta_{\rm PDF}$ profiled [fb] |  $\alpha$ (base) |  $\alpha$ (profiled)
9.5 $\cdot$ 10-3 | 4.6 $\cdot$ 10-3 | 3.34 | 4.82

Benchmark B
---
inf [TeV] | sup [TeV] |  $\sigma_{\rm SM}$ [fb] |  $\sigma_{\rm SM+BSM}$ [fb]
1.36 | 3.36 | 1.53 | 1.45
$\Delta_{\rm PDF}$ base [fb] |  $\Delta_{\rm PDF}$ profiled [fb] |  $\alpha$ (base) |  $\alpha$ (profiled)
6.8 $\cdot$ 10-2 | 3.1 $\cdot$ 10-2 | 1.53 | 2.91

Table 1: Integration limits for the dip region, integrated cross section for
the SM background and the complete model and its PDF uncertainty with the
baseline CT18NNLO PDF set and the profiled PDF set using $A_{\rm FB}+A_{W}$
pseudodata as well as the significances $\alpha$ employing the two PDF errors
for an integrated luminosity of 3000 fb-1 for the benchmarks A and B
Fiaschi:2021sin .

The above studies can be extended to CC Drell-Yan. In fact, it is noted in
Fiaschi:2021sin  that, for cases in which both neutral and charged new states
appear and are correlated by theory, the reduction of systematic PDF error can
profitably be applied to combined $W^{\prime}$ and $Z^{\prime}$ searches. For
example, in the 4DCHM a direct $W^{\prime}$ exclusion (or indeed discovery)
achieved in the CC channel can be used indirectly to probe the existence of a
$Z^{\prime}$ better than this can be done with direct searches in the NC
channel. The analysis of the dip often provides more stringent limits than the
signal coming from the peak, with the reduction of systematic PDF errors
playing a crucial part in this conclusion.

####  IV.4.2 Applications with SMEFT

In the discussion of the previous subsection, we study the impact of
“improved” PDFs on BSM physics at the multi-TeV scale, while the improvement
itself in the PDFs comes from processes at the SM weak boson mass scale. The
asymmetry pseudodata employed in the profiling Fiaschi:2021okg  are centered
around the SM vector boson peaks and take into account only SM contributions.
Since we consider TeV scale BSM resonances, the new physics contribution to
the forward-backward asymmetry $A_{FB}$ at the weak scale, which is due to the
interference between SM and BSM process, is small, and a similar conclusion
holds for the charged-current differential cross section distribution.

On the other hand, a different scenario arises if one includes data or
pseudodata in the PDF determination which do contain contamination from BSM
contributions. This is the case, for example, of data or pseudodata at high
mass scales in the multi-TeV region. In this case, assuming purely SM
predictions would lead to biased estimations on new physics constraints: see
for instance the recent studies Greljo:2021kvv ; CMS:2021yzl .

This bias can be avoided if the PDFs are extracted simultaneously with the BSM
parameters. Such non-biased approaches are feasible with xFitter via
interfaces to BSM computations, for instance by using SM effective field
theory (SMEFT) expansions. For example, the CIJET Gao:2012qpa ; Gao:2013kp
interface allows extending jet cross sections with dimension-6 SMEFT operators
for purely left-handed, vector-like or axial vector-like color-singlet
exchanges. An xFitter analysis in this direction is underway.

##  V Conclusion

The xFitter program is a versatile, flexible, modular, and comprehensive tool
that can facilitate analyses of the experimental data and theoretical
calculations.

It is a valuable framework for bench-marking and understanding differences
between PDF fits, and it can provide impact studies for possible future
facilities including HL-LHC, EIC, LHeC, DUNE, and UHE Cosmic Ray experiments.
We encourage use of xFitter, and welcome new contributions from the community
to ensure xFitter continues to incorporate the latest theoretical advances and
precision experimental data.

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