∎

e1corresponding author<EMAIL_ADDRESS>
11institutetext: Hampton University, Hampton, VA 23668, U.S.A.
22institutetext: Jefferson Lab, Newport News, VA 23606, U.S.A.
33institutetext: Department of Physics, Southern Methodist University, Dallas,
TX 75275, U.S.A. 44institutetext: Fermi National Accelerator Laboratory,
Batavia, IL 60510, U.S.A. 55institutetext: Department of Physics, Illinois
Institute of Technology, Chicago, IL 60616, U.S.A.

# Deuterium scattering experiments in CTEQ global QCD analyses: a comparative
investigation

A. Accardiaddr1,addr2  T. J. Hobbsaddr3,addr2,addr4,addr5,e1  X.
Jingaddr3,addr2  P. M. Nadolskyaddr3

(Received: date / Accepted: date)

###### Abstract

Experimental measurements in deep-inelastic scattering and lepton-pair
production on deuterium targets play an important role in the flavor
separation of $u$ and $d$ (anti)quarks in global QCD analyses of the parton
distribution functions (PDFs) of the nucleon. We investigate the impact of
theoretical corrections accounting for the light-nuclear structure of the
deuteron upon the fitted $u,d$-quark, gluon, and other PDFs in the CJ15 and
CT18 families of next-to-leading order CTEQ global analyses. The investigation
is done using the $L_{2}$ sensitivity statistical method, which provides a
common metric to quantify the strength of experimental constraints on various
PDFs and ratios of PDFs in the two distinct fitting frameworks. Using the
$L_{2}$ sensitivity and other approaches, we examine the compatibility of
deuteron data sets with other fitted experiments under varied implementations
of the deuteron corrections. We find that freely-fitted deuteron corrections
modify the PDF uncertainty at large momentum fractions and will be relevant
for future PDFs affecting electroweak precision measurements.

††journal: Eur. Phys. J. C

Author ORCIDs:
0000-0002-2077-6557 — A. Accardi
0000-0002-2729-0015 — T. J. Hobbs
0000-0003-2679-5262 — X. Jing
0000-0003-3732-0860 — P. M. Nadolsky

###### Contents

  1. 1 Introduction
    1. 1.1 Light-parton structure of the nucleon in electroweak precision measurements
    2. 1.2 The role of nuclear-medium effects
    3. 1.3 Paper organization
  2. 2 Low-energy QCD effects
    1. 2.1 Deuteron-structure effects
    2. 2.2 Power-suppressed effects
  3. 3 Methods
    1. 3.1 Selections of experimental data sets
    2. 3.2 Modifications in the fitting methodologies
    3. 3.3 The $L_{2}$ sensitivity technique
  4. 4 Comparison of deuteron data impact in the CJ15 and CT18 fits
    1. 4.1 Overall agreement of theory and data
    2. 4.2 Impact of deuteron corrections on the $d/u$ PDF ratio
    3. 4.3 Impact on the valence PDFs in the LHC EW precision region
    4. 4.4 Impact of deuteron corrections on the gluon PDF
    5. 4.5 Valence-sector PDF pulls: the $d/u$ ratio
    6. 4.6 PDF pulls in the gluon and light-quark sea sectors
  5. 5 Conclusion
  6. 6 Appendix: Comparison procedure and technical details

Declarations
Funding: This work was funded by the U.S. Department of Energy under Grant No.
DE-SC0010129 at SMU, and T. J. Hobbs received financial support from a JLab
EIC Center Fellowship. The work of A. Accardi was supported by the U.S.
Department of Energy contract DE-AC05-06OR23177, under which Jefferson Science
Associates LLC manages and operates Jefferson Lab. A. Accardi and and X. Jing
were additionally supported by DOE contract DE-SC0008791.
Conflicts of interest: The authors have no relevant financial or non-financial
interests to disclose.
Availability of data and material: The data sets generated during and/or
analysed during the current study are available from the corresponding author
on reasonable request.
Code availability: The $L_{2}$ sensitivity analysis code and supporting files
are available at the following location: https://tinyurl.com/CJCT2021. This or
another link will be continuously maintained.
Authors’ contributions: We affirm that all authors made important and
substantial contributions to the conception, execution, and analysis contained
in this study.

##  1 Introduction

###  1.1 Light-parton structure of the nucleon in electroweak precision
measurements

Figure 1: Hessian correlations Stump:2001gu ; Nadolsky:2001yg ;
Nadolsky:2008zw  for the values of $\sin\theta_{W}$ extracted from $Z$ boson
production at the LHC 8 TeV Armbruster:2018 . Left: correlations with valence
PDFs and PDF ratios at $Q=$81.45 GeV, plotted as a function of $x$ for CT14
NNLO PDFs. Right: the same, for correlations with PDFs of individual flavors.

Electroweak precision tests of the Standard Model (SM) at hadron colliders are
nontrivially sensitive to the parton flavor composition of initial-state
hadrons. For spin-independent inclusive observables at the CERN Large Hadron
Collider (LHC) – or, indeed, at any high-enough energy facility such as the
Relativistic Heavy Ion Collider, the Jefferson Lab CEBAF accelerator, or the
future Electron-Ion Collider – this flavor composition is typically specified
by helicity-averaged parton distribution functions (PDFs) of the proton. The
PDFs, $f(x,Q)$, have long been of strong interest from the perspective of both
fundamental Quantum Chromo-Dynamics (QCD) as well as particle phenomenology,
given that they quantify the probability of resolving a quark or gluon
constituent of flavor $f$ carrying a fraction $x$ of the proton’s longitudinal
momentum in a scattering process with a squared energy scale $Q^{2}\gtrsim
1\mbox{ GeV}^{2}$. For this reason, the PDFs play a central role in predicting
cross sections for $pp$ collisions at the LHC, and, in particular, their
accuracy influences the ability of LHC measurements or other high-energy data
to constrain the SM parameters, including in the electroweak sector.

Due to the challenge of reducing their uncertainties and empirically
distinguishing among their parton flavors, PDFs have historically been
determined most robustly through “global QCD fits” Jimenez-Delgado:2013sma ;
Gao:2017yyd ; Kovarik:2019xvh ; Ethier:2020way , now increasingly performed at
next-to-next-to-leading order (NNLO) accuracy in $\alpha_{s}$, and drawing
upon large collections of experimental measurements sensitive to QCD and
different underlying PDF combinations. In spite of the growing number of LHC
measurements, deeply-inelastic scattering (DIS) experiments involving fixed
hadronic or nuclear targets at BCDMS, NMC, SLAC, and JLab continue to provide
key information to disentangle the PDFs in recent global QCD analyses such as
CJ15 Accardi:2016qay , ABMP16 Alekhin:2017kpj , CT18 Hou:2019efy , NNPDF3.1
Ball:2017nwa , and MSHT20 Bailey:2020ooq . The fixed-target experiments
complement analogous DIS collisions at the HERA $ep$ collider by extending the
momentum fraction coverage to larger $x$ values and adding measurements on
deuterium targets. In fact, such experiments provide the leading constraints
on the (anti)quark PDFs at low scales $Q$ and large momentum fractions
$x\gtrsim 0.05$, as well as on the gluon PDF by observing scaling violations
over the same kinematic region Accardi:2016muk ; Wang:2018heo ; Hobbs:2019gob
.

Figure 2: Lagrange Multiplier scans on $d_{val}(x=0.03,Q=85\,\mathrm{GeV})$
(left) and $u_{val}(x=0.03,Q=85\,\mathrm{GeV})$ (right), showing the changes
in the $\chi^{2}$ for all data sets and most sensitive experimental data sets
in the CT18Z NNLO global QCD analysis Hou:2019efy .

In precision tests of the electroweak sector, the substantial PDF dependence
of the involved theoretical calculations affects experimental determinations
of SM parameters, such as the weak-mixing angle $\theta_{W}$ extracted from
the $A_{\mathrm{FB}}$ forward-backward asymmetry measured in the production of
$Z$ bosons during Runs 1 and 2 of the LHC. Fig. 1 illustrates typical Hessian
correlations Stump:2001gu ; Nadolsky:2001yg ; Nadolsky:2008zw  of PDF
combinations (left) and PDFs (right) with the $\sin\theta_{W}$ values
extracted from 8 TeV $A_{\mathrm{FB}}$ measurements at the LHC. Here, the
correlations are computed using the preliminary (unpublished) ATLAS Run-1 data
Armbruster:2018  on $\sin\theta_{W}$ extracted with individual PDF eigenvector
sets of the CT14 NNLO ensemble Dulat:2015mca .

In the left subfigure, we see that the values of the extracted
$\sin\theta_{W}$ are strongly correlated with the valence combinations of
light-quark PDFs at $Q=81.45$ GeV, $d_{\mathit{val}}(x,Q)\equiv
d(x,Q)-\bar{d}(x,Q)$ at $x\approx 0.008-0.05$ and $u_{\mathit{val}}(x,Q)\equiv
u(x,Q)-\bar{u}(x,Q)$ at $x\approx 0.008-0.1$. In addition, significant
correlations with the extracted $\sin\theta_{W}$ exist at higher $x\gtrsim
0.3$ as well, especially for the PDF ratio $d/u$, and again for
$d_{\mathit{val}}$. Remarkably, the correlations are weaker with the PDFs of
individual parton flavors (shown at right) than with the valence combinations.
An anti-correlation with the $d$-quark (green dashed line) at $x\sim 0.3$,
affecting $A_{\mathrm{FB}}$ at smaller $x$ via the valence-quark sum rule, is
evident in this case, though not exceptionally strong.

The sizable correlations between fitted PDFs and $\sin\theta_{W}$ in Fig. 1
are consistent with the significant PDF uncertainties on these and similar
BSM-sensitive quantities, including the $W$ boson mass, $M_{W}$, and Higgs
cross section, $\sigma_{H}$. For this reason, the realization of next-
generation precision in the determination of these electroweak parameters is
critically dependent on the reduction of their associated PDF uncertainties,
including the high-$x$ uncertainty of the $d$-quark and gluon ($g$) PDFs, as
well as of $d_{\mathit{val}}$ and $d/u$.

We might ask where the experimental constraints on the relevant PDF
combinations for LHC electroweak precision tests arise from. While direct
measurements at the LHC will supply increasing information on the PDFs
affecting $A_{\mathrm{FB}}$ Fu:2020mxl  and other observables Khalek:2018mdn ,
recent CTEQ studies Accardi:2016qay ; Hou:2019efy ; Wang:2018heo ;
Hobbs:2019gob ; Hou:2019gfw  find that deep-inelastic scattering experiments
on nuclear targets will continue to provide strong constraints on the down-
quark PDFs in the nucleon in the near future. In a global QCD analysis,
experimental measurements made solely on proton targets are at present
insufficient for full separation of parton distributions for $d$, $s$, $g$,
and anti-quark flavors. Assuming parton-level charge symmetry,
$d^{p}(x,Q)\approx u^{n}(x,Q)$, between the PDFs in the proton $p$ and the
neutron $n$, and correcting for low-energy nuclear effects Accardi:2016muk ,
one can then use scattering processes on the deuteron or heavier nuclei to
constrain the down-type PDFs in the proton.

We illustrate the importance of fixed-target data in the determination of the
weak mixing angle with the help of Lagrange Multiple (LM) scans Pumplin:2001ct
in Fig. 2, in which we examine the dependence of the figure-of-merit function
$\chi^{2}$ in the CT18Z NNLO analysis Hou:2019efy  on the values of the
valence $u_{val}$ (left) and $d_{val}$ (right) quarks at $x=0.03$ and $Q=85$
GeV. We plot the change in $\chi^{2}$, as compared to the value in the best
fit, for all data sets (labeled as “Total” in the figure) and for individual
experiments with the highest sensitivity to this PDF combination. The curves
for the experimental data sets are labeled according to the convention in
Table 1. The LM scans show that a small group of DIS experiments – NMC ratio
of $d$ and $p$ DIS cross sections Arneodo:1996qe , inclusive HERA I+II DIS
Abramowicz:2015mha , BCDMS $p$ and $d$ reduced cross sections Virchaux:1991jc
; virchaux1992measurement ; Benvenuti:1989rh ; Benvenuti:1989fm , CCFR $F_{3}$
structure function Seligman:1997mc  – contribute the largest variations in
$\Delta\chi^{2}$ when the valence PDFs are varied, together with several
lepton pair production experiments by ATLAS, LHCb, E605, and E866. In the case
of $d_{val}$ in the right Fig. 2, the BCDMS measurements on $p$ and $d$ show
somewhat different preferences, with the deuteron data clearly preferring a
higher $d_{val}(0.03,85\mbox{ GeV})$. In these and other cases, the deuteron
DIS measurements, with their large numbers of precise data points, have large
statistical significance in the global fit. As this LM scan also illustrates,
the proton and deuteron data sets in the CT18 fit sometimes prefer somewhat
different values of $d_{val}(x,Q)$, which in turn may hamper the efforts for
reducing the PDF uncertainty in the relevant EW precision measurements.

In this article, we will employ a statistical technique called the $L_{2}$
sensitivity Hobbs:2019gob  that can be viewed as a fast approximation to the
LM scans that are usually very computing-intensive. With this technique, we
will survey agreement between the constraints on the PDFs from the deuteron
and other data sets in a wide range of $x$ and for various treatments of
deuteron corrections. While we have investigated sensitivities for various PDF
flavors, our presentation will focus on the sensitivities to $u$, $d$, and $g$
PDFs that are most affected.

Apart from its significance for the LHC and SM phenomenology, the physics of
large-$x$ quark PDFs is interesting in its own right. Nonperturbative QCD
approaches Jimenez-Delgado:2013sma ; Holt:2010vj ; Roberts:2013mja  and
lattice QCD Lin:2017snn ; Lin:2020rut  provide increasingly complete
predictions for the flavor composition of unpolarized protons at $x\to 1$,
which in turn inform one on the role of color confinement in the binding of
the valence quarks. These predictions can then be confronted with
phenomenological determinations of Mellin moments and PDFs at large $x$. For
example, Ref. Courtoy:2020fex  observes that the proton (BCDMS $F_{2}^{p}$,
E866 $pp$ DY, HERAI+II $ep$ DIS) and deuteron (BCDMS $F_{2}^{d}$, NMC $p/d$
ratio) experiments in the CT18 NNLO analysis have somewhat different
preferences for the effective exponents $\beta$ controlling the
$(1-x)^{\beta}$ falloff of the valence up and down quark PDFs at $x\to 1$. In
turn, these differences impact comparisons of phenomenological PDFs against
large-$x$ predictions from quark counting rules Brodsky:1973kr ;
Brodsky:1974vy ; Farrar:1975yb  and other nonperturbative approaches
Holt:2010vj ; Roberts:2013mja . The analysis methods utilized in this paper
can shed light on these issues.

###  1.2 The role of nuclear-medium effects

Extracting parton-level information from nuclear data sets involving the
deuteron or heavier targets requires an understanding of the effects of the
nuclear environment Ethier:2020way ; Accardi:2016muk . The trivial dependence
on the nuclear atomic number $A$ and charge $Z$ is normally implemented by
constructing a nuclear PDF as a linear combination of the PDFs on free protons
and neutrons, as reviewed, e.g., in Sec. 5. A of Ref. Kovarik:2019xvh . On top
of this trivial $(A,Z)$ dependence, low-energy interactions in the nuclear
medium may modify the quark and gluon distributions relatively to those in
free nucleons. The nuclear corrections that account for these deviations can
be computed with increasing sophistication and connection to the formal theory
describing low-energy dynamics. One can, for example, utilize
phenomenological, data-driven ratios to convert nuclear-target cross sections
to free-nucleon ones Hou:2019efy ; Ball:2020xqw ; parametrize and fit the
nuclear deformation of the PDFs either inside the deuteron in a nucleon PDF
fit Bailey:2020ooq ; Martin:2009iq  or in a heavy nucleus in a nuclear PDF fit
Eskola:2016oht ; Kovarik:2015cma ; Deflorian:2011fp ; Hirai:2007sx ;
Khanpour:2016pph ; AbdulKhalek:2019mzd ; or, finally, adopt a dynamical model
of the low-energy nucleon-nucleon interactions and calculate the hard cross-
section as a double convolution of parton distributions inside the nucleons
and nucleon wave functions inside the nuclear targets Accardi:2016qay ;
Alekhin:2017fpf ; Kulagin:2004ie . The resulting extractions of the nucleon
PDFs from nuclear data then have a dependence on the assumed nuclear
corrections.

The specific methods used typically differ when analyzing light nuclear
targets (e.g., the deuteron) or heavy nuclei (e.g., Fe). Here, we concentrate
on and summarize the techniques utilized in the CT18 and CJ15 fits which form
the starting point for the study presented in this paper, and then briefly
mention other approaches:

  1. 1.

Deuteron corrections. A dynamic deuteron correction in the CJ15 next-to-
leading order (NLO) PDF fit Accardi:2016qay  was applied to any process
involving interaction with a deuterium target, including both DIS and Drell-
Yan experiments, as detailed at greater length in Sec. 2.1. This correction
allows the CJ15 NLO fit to include the fixed-target DIS data from SLAC and
JLab at the largest $x$ not admitted by other groups. The correction can be
understood as arising from several dynamical effects, such as the relativistic
Fermi motion of bound nucleons, binding corrections, and nucleon off-shellness
effects. In practice these mechanisms are taken into account via convolutions
of free-hadron cross sections with nuclear smearing functions calculated
starting from bound nucleon wave-functions. Nuclear correction mostly affects
the intermediate and large regions of $x$.

The CJ15 analysis also applies a phenomenological parametrization for the off-
shell deformation of the scattered nucleon’s structure function (in short,
“off-shell corrections”) with parameters fitted to data to increase the model
flexibility. Care is taken that the valence quark number inside the nucleon is
not modified; since the off-shell function is flavor independent and has no
significant dependence on $Q^{2}$, it must then change the sign in the
interval of $[0,1]$, meaning that it is essentially given by a polynomial with
one or more roots in this range.

An analysis similar in spirit to the CJ15 fit is available from the AKP group
Alekhin:2017fpf , and differences in the extracted off-shell functions are
currently being investigated by the two groups jointly. Alternatively, the
deuteron correction can be fitted using a purely phenomenological
parametrization as in the MSHT20 analysis Bailey:2020ooq , or the additional
uncertainty associated with the deuteron effects can be learned from the
global analysis data themselves, as done in the NNPDF study Ball:2020xqw .
Other groups do not include deuteron corrections by selecting the fitted data
in a $\\{x,Q\\}$ region where the deuteron corrections are small compared to
the precision of data, as is done, e.g., in the CT18 analysis Hou:2019efy .

  2. 2.

Heavy-nucleus effects. Nucleon PDF fits may include DIS experiments performed
on heavy nuclear targets, such as CCFR osti_879078  and NuTeV Goncharov:2001qe
, involving 56Fe, and CHORUS Kayis_Topaksu_2008 , with 82Pb (which is not
presently fitted in either CT or CJ packages studied here). It has been known
empirically for some time that the structure functions of these heavier
nuclear targets exhibit $x$\- and $A$-dependent deviations from the structure
function of the physical deuteron, owing to a variety of physical processes
characterizing the nuclear medium Frankfurt:1988nt ; Arneodo:1992wf ;
Kopeliovich:2012kw ; Malace:2014uea , including the heavy-nucleus analogue of
the EMC and Fermi-motion effects discussed for the deuteron at high $x$, and
nuclear (anti)shadowing phenomena at lower $x$.

To address these effects, the CT group corrects DIS cross sections on iron
(CCFR osti_879078 , CDHSW Berge:1989hr , and NuTeV Goncharov:2001qe ) and
proton-copper Drell-Yan (E605 Moreno:1990sf ) to the corresponding cross
sections on deuterium using a phenomenological parametrization of the nuclear-
to-deuteron cross section ratios based on results in Arneodo:1992wf  (see also
Rondio:1993mf , Fig. 2a), which depends on $x$ but not on $Q^{2}$ in the
fitted region. To include the heavy-nuclear data in the MSHT20 Bailey:2020ooq
and earlier MMHT fits, a nuclear correction factor, $R_{f}$, Martin:2009iq
having the form $f^{A}(x,Q^{2})=R_{f}(x,Q^{2},A)f(x,Q^{2})$, where
$f^{A}(x,Q^{2})$ is defined to be the PDF of a proton bound in a nucleus of
mass number $A$, was determined. This was assessed using the de Florian et al.
nuclear PDF (nPDF) of Ref. Deflorian:2011fp , then weighted by a 3-parameter
modification factor as in Ref. Martin:2009iq , which is actively refitted
along with the PDF-associated parameters. NNPDF Ball:2018twp  determines the
uncertainty due to heavy-nuclear effects using a similar statistical procedure
as for the deuteron.

As can be seen from this summary, global analyses vary in their treatments of
the nuclear effects, but with the frequent conclusion that the resulting
differences are marginal in comparison to contemporary PDF uncertainties.
Indeed, the experiments in question are fitted reasonably well, while the
higher-order QCD and parametrization uncertainties on the PDFs are comparable
for the most part to the nuclear ones at NLO or even NNLO. In the present
study, however, we identify several areas where the deuteron corrections will
play a prominent role in the near future, as the field advances toward higher
accuracy in the determination of nucleon PDFs. We compare, in particular, the
effects of deuteron corrections in two independent PDF global fits by the
CTEQ-JLab (CJ) and CTEQ-TEA (CT groups which differ in their phenomenological
focus, data selection and, crucially, the treatment of scattering process in
nuclear targets. We find that the deuteron effects will have pronounced
consequences for both the fitted PDFs in the large-$x$ region and the
correlations among the PDFs and quantities derived from them in an extended
$x$ range.

More specifically, this paper will elucidate constraints on the $d$-quark,
gluon, and other PDFs arising in the CT18 and CJ15 global fits. We will
accomplish this by analyzing the $L_{2}$ sensitivity to various PDFs
Hobbs:2019gob , a simple informative figure of merit that allows us to look
inside the CJ and CT fits and understand the constraints from the fitted
experiments on various parton flavors in an expansive region of $x$ and $Q$.

###  1.3 Paper organization

After this introduction, the remainder of the article is as follows. In Sec.
2, we briefly present the deuteron-structure corrections (Sec. 2.1) with which
this investigation is primarily concerned, as well as power-suppressed QCD
effects, also relevant to fits involving nuclear data and/or at lower $Q$ and
$W$ (in Sec. 2.2). In Sec. 3, we summarize the relevant features of the CT and
CJ fitting frameworks and the special modifications we made in the two
analyses to allow their direct comparisons. In Sec. 3.3, we review the $L_{2}$
sensitivity method. In Sec. 4, we apply this method to analyze the impact of
deuteron-structure corrections on the fit results, and examine the patterns of
PDF pulls obtained in the several iterations of CT/CJ under different assumed
treatments of the deuteron corrections. As representative cases, we will
concentrate on the $d/u$ ratio in Secs. 4.2 and 4.5, and on the gluon in Secs.
4.4 and 4.6. In Section 4.3 we also explore the implications of our analysis
to precision measurements of the weak mixing angle. The conclusions in Sec. 5
are followed by a technical appendix describing a numerical procedure to
reconcile the error analyses in the CJ and CT approaches.

##  2 Low-energy QCD effects

###  2.1 Deuteron-structure effects

The critical low-energy effect considered in this study, which arises due to
MeV-scale dynamics characterizing nuclear bound states, is the modification of
the parton-level substructure of nucleons embedded in the nuclear medium — in
particular, those effects arising in the most weakly bound nuclear system, the
two-body deuteron.

Figure 3:  We plot the nuclear correction ratio, $F^{N}_{2}/F^{d}_{2}$,
calculated using the central CJ15 fit results for several selections of the
$Q^{2}$ scale. Each of the four panels above highlights a given value of
$Q^{2}$, while graying out the curves for other scales in order to retain
visual information on the scale dependence of the correction factor at large
$x$. In the upper two panels, which focus on lower scales, $Q^{2}=5,10$ GeV2,
the dotted lines indicate the range of $x$ that is only accessible to CJ
($W^{2}\\!>\\!3$ GeV2) but not CT ($W^{2}\\!>\\!12.25$ GeV2), due to the more
conservative cut of the latter.

In the CJ framework, these corrections are treated as nuclear wave-function
effects, and the deuteron parton distributions $f^{d}$ are calculated as a
convolution of the bound nucleon’s parton distributions, $\widetilde{f}^{N}$,
with a suitable nucleonic “smearing function,” ${\cal S}^{N/d}$:

$\displaystyle f^{d}(x,Q^{2})$ $\displaystyle=\int\frac{dz}{z}\,\int
dp^{2}_{N}$ (1) $\displaystyle\hskip 56.9055pt{\cal
S}^{N/d}(z,p^{2}_{N})\,\widetilde{f}^{N}(x/z,p^{2}_{N},Q^{2})\ .$

Here, $z$ represents the momentum fraction of the (isoscalar) nucleon within
the deuteron, defined as $z\equiv(M_{d}/M_{N})(p_{N}\cdot q/p_{d}\cdot q)$;
$p_{d,N}$ are the deuteron and nucleon four-momenta; and $M_{d,N}$ are their
respective on-shell masses. This representation is founded on the so-called
Weak Binding Approximation (WBA) to the calculation of nuclear structure
functions Kulagin:2004ie ; Kulagin:1989mu , where the ${\cal S}^{N/d}$
smearing function is calculable based on an assumed nuclear potential; as in
Ref. Accardi:2016qay , we assume the AV18 potential. Since $p_{N}$ is
generically off-shell for a bound nucleon, but typically by only a small
amount, one can further expand the bound-nucleon PDF, $\widetilde{f}^{N}$, in
powers of its off-shellness, $\omega=(p_{N}^{2}-M_{N}^{2})/M_{N}^{2}$, as

$\displaystyle\widetilde{f}^{q/N}(y,p^{2}_{N},Q^{2})$
$\displaystyle=f^{N}(y,Q^{2})$ (2)
$\displaystyle+\frac{p_{N}^{2}-M_{N}^{2}}{M_{N}^{2}}\,\delta
f^{N}(y,Q^{2})+O(\omega^{2})\ .$

The first term, corresponding to $p_{N}^{2}=M_{N}^{2}$, gives the PDF of the
free, on-shell nucleon. In the second term, the $\mathcal{O}(\omega)$
coefficient (also known as “off-shell function”) can be phenomenologically
parametrized and determined in a global fit from the interplay of data
involving deuterium targets and information involving free-nucleon-based
observables like $W$ boson production at the Tevatron, the Relativistic Heavy
Ion Collider (RHIC) or the LHC. Like in Ref. Accardi:2016qay , we assume the
flavor-independent 3-parameter shape function

$\delta f^{N}(x)=C(x-x_{0})(x-x_{1})(1+x_{0}-x)\ ,$ (3)

with $x_{1}$ fixed by requiring the off-shell PDFs to satisfy the quark-number
sum rule. Further technical details and a discussion of the fit results can be
found in Ref. Accardi:2016qay .

Figure 4:  Kinematics of the DIS data included in the fits discussed in this
paper. The HERA DIS collider data were taken on proton targets; the fixed-
target SLAC, JLab, BCDMS and NMC experiments include both proton and deuterium
target data at approximately the same kinematics. The $W^{2}=12.25$ GeV2 and
$W^{2}=3$ GeV2 cuts adopted, respectively, by the CT and CJ fits are shown by
dashed and dot-dashed lines, respectively. The figure is taken from Ref.
Owens:2012bv .

Section 4 considers three main scenarios for implementing the deuteron
corrections (d.c.) discussed above:

  1. (i)

an uncorrected scenario for which no nuclear effects are included for the
deuteron;

  2. (ii)

a fixed scenario in which the nuclear wave-function effects (on- and off-
shell) are frozen to the AV18-informed choice of Ref. Accardi:2016qay , and
the off-shellness correction, $\delta f^{N}(x)$, is set to the CJ15 central
fit; and

  3. (iii)

a free scenario particular to CJ, in which the parameters in Eq. (3) for the
off-shell nucleon are allowed to vary.

The dynamical deuteron corrections are natively implemented in the CJ
framework, and the off-shell parameters can be simultaneously fitted with the
PDFs. So far, however, the CT code only supports deuteron corrections given in
the form of analytic interpolations, such as the one obtained from the
correction in Melnitchouk:1994rv . To implement the fixed CJ15 deuteron
correction in the CT framework and render it more directly comparable to CJ
with respect to its treatment of deuteron target data, we instead multiply the
experimental DIS deuteron structure function by the $F_{2}^{N}/F_{2}^{D}$
nucleon-to-deuteron ratio plotted in Fig. 3:

$\displaystyle F_{2}^{N}\equiv
F_{2,\text{exp}}^{d}\left(\frac{F_{2}^{N}}{F_{2}^{d}}\right)_{\text{CJ15}},$
(4)

with $F_{2}^{N}=F_{2}^{p}+F_{2}^{n}$. The effective isoscalar combination of
proton and neutron structure functions thus defined can then be directly
compared to uncorrected theoretical calculations of the isoscalar deuteron DIS
structure function. On this logic, the CT and CJ fits with a fixed CJ15
correction are placed on similar theoretical footing regarding the
implementation of the deuteron effects, with the main difference being whether
the correction is imposed within the theoretical structure function
calculation or in the $F^{d}_{2}$ experimental data – a fact which is
immaterial for the sake of evaluating the $\chi^{2}$-function and allows us to
compare the impact of the same fixed correction on the CJ and CT frameworks.
While a full analysis of the nuclear correction uncertainties is outside the
scope of this article, the effect of letting the nuclear off-shell parameters
free to vary in the present analysis can be appreciated by comparing the CJ
fits in the fixed and free nuclear corrections scenarios.

The size and $x$ dependence of the deuteron corrections, as quantified by the
isoscalar nucleon-to-deuteron structure function ratio $F^{N}_{2}/F^{d}_{2}$,
are shown in Fig. 3 for several representative choices of $Q^{2}$. One
immediately notices that deuteron corrections depend on the DIS scale and, at
large $x$, increase with $Q^{2}$ toward a fixed point in the
$Q^{2}\\!\to\\!\infty$ Bjorken limit; as such, deuteron corrections become
effectively scale independent for $Q^{2}\gtrsim 50$ GeV2. For each plotted
value of $Q^{2}$, the figure also indicates the maximum $x$ values below which
data are accepted in the CJ and CT fits according to their $W^{2}>3\mbox{\
and\ }W^{2}>12.25$ GeV2 kinematic cuts, respectively. For CJ, which extends
the analyzed DIS data set to the low-$Q^{2}$ and large-$x$ values shown in
Fig. 4, it is imperative to correctly account for the $Q^{2}$ dependence of
the deuteron correction in order to avoid conflicts with the leading-twist
logarithmic $Q^{2}$ evolution that constrains the fitted gluon distribution in
DIS experiments. For CT, with its larger $W^{2}$ cut, the deuteron corrections
are small and nearly scale independent, as seen in Fig. 3, except for the less
precise BCDMS deuteron points with $x\gtrsim 0.6$ (see the kinematical map in
Fig. 4), where some influence from the deuteron correction is expected and
indeed quantified in Sec. 4.

###  2.2 Power-suppressed effects

Due to their less conservative kinematical restrictions on $Q^{2}$ and
$W^{2}$, the CJ global fits extend into a region for which power-suppressed
corrections are non-negligible, as depicted in Fig. 4. On the one hand,
dynamical higher-twist corrections of $\mathcal{O}(\Lambda^{2}/Q^{2})$ emerge
because of the presence of multi-parton correlations within the soft portion
of the factorized DIS process, for which the first subleading contribution to
the twist expansion for unpolarized scattering are matrix elements of
twist-$4$ operators Ellis:1982cd ; Qiu:1990xxa . As in CJ, these are often
determined phenomenologically using forms like
$F_{2}(x,Q^{2})=F_{2}^{\text{LT}}(x,Q^{2})\left[1+C(x)\big{/}Q^{2}\right]$,
where $F_{2}^{\text{LT}}$ represents the leading-twist structure function, and
a fitted coefficient, $C(x)=\alpha x^{\beta}(1+\gamma x)$, parametrizes the
power-suppressed twist-4 corrections. On the other hand, target-mass
corrections of $\mathcal{O}(M^{2}_{N}/Q^{2})$ are due to the non-negligible
mass, $M_{N}$, of the struck nucleon, and are implemented via the operator
product expansion of Georgi and Politzer Georgi:1976ve ; DeRujula:1976baf  or
related prescriptions, as extensively reviewed in Schienbein:2007gr ;
Brady:2011uy . Both corrections are natively implemented in the CJ framework.

In contrast, CT imposes more restrictive kinematical cuts in $W^{2}$, such
that the standard CT data sets lie beyond the region for which the finite
$\sim\\!1/Q^{2}$ corrections are significant. In the past CT studies it
mattered little whether the deuteron correction was included according to a
specific model or not included at all (the default choice). While we expect
some interplay between the deuteron and power-suppressed effects, we do not
systematically isolate the latter and leave their investigation to future
studies.

##  3 Methods

###  3.1 Selections of experimental data sets

The CT18 and CJ15 global data sets consist of both high-energy measurements as
well as data down to the few-GeV region. Despite their somewhat differing
phenomenological emphases — with CT generally aimed toward high-energy
processes, and CJ toward the low-$Q$ and large-$x$ region probed at facilities
like JLab and SLAC — there is a substantial overlap with respect to the key
experiments they include. In Table 1, we provide a complete listing of the
experiments included in each global analysis. Of particular importance for
interpreting our results in Sec. 4, the leftmost column of Table 1 designates
a process-based category label for each experiment, placing, e.g., the BCDMS
$F^{d}_{2}$ inclusive structure function data in Group 4: DIS Deuterium. We
will investigate the agreement between these groups of experiments with the
help of $L_{2}$ sensitivities. In contrast, previous studies Wang:2018heo ;
Hobbs:2019gob ; Courtoy:2020fex  employing the same technique focused
primarily on the individual experiments.

While the CJ and CT analyses share 10 experiments, Table 1 shows that the CJ
fits include additional DIS data at fixed-target energies from SLAC, HERMES,
JLab, and NMC. They also include Tevatron measurements of charge asymmetries
reconstructed to the level of $W$ bosons and cross sections for photon plus
leading jet production. The CT PDF fits more extensively cover collider
observables. They include HERA heavy-quark production, an assortment of cross
sections and cross section asymmetries in electroweak boson production at the
LHC, as well as LHC cross sections for inclusive jet and $t\bar{t}$
production. The CT fits include CCFR and NuTeV cross sections for both
inclusive (Group #8) and semi-inclusive (Group #10, for opposite-sign dimuon
production) charge-current DIS on iron. The CT fits, however, implement only
the most direct measurements of Tevatron and LHC charge asymmetries in
$W\to\ell$ lepton decay, presented as a function of the rapidity and
transverse momentum of the charged lepton. They do not include the CDF and DØ
boson-level charge asymmetries fitted by CJ, which directly probe the
large-$x$ PDF ratios, while they also involve additional recursive unfolding
of the data that utilizes a PDF-dependent calculation to reconstruct the weak
boson’s rapidity.

Group # | Experiment Group | Experiment Name | CT, $N_{\mathit{pt}}$ | CJ, $N_{\mathit{pt}}$ | Ref.
---|---|---|---|---|---
1 |  $\gamma$+jet | DØ $\gamma$+jet |  | 56 | Abazov:2008er
2 | Jets Tevatron | CDF Run-2 inclusive jet production | 72 | 72 | Aaltonen:2008eq
|  | DØ Run-2 inclusive jet production | 110 | 110 | Abazov:2008ae
3 | DIS Proton | HERA Run I+II | 1120 | 1185 | Abramowicz:2015mha
|  | H1 $F_{L}$ | 9 |  | Collaboration:2010ry
|  | H1 $\sigma_{r}^{b}$ | 10 |  | Aktas:2004az
|  | Combined HERA $\sigma_{r}^{c}$ | 47 |  | Abramowicz:1900rp
|  | JLab proton |  | 136 | Malace:2009kw
|  | SLAC proton |  | 564 | Whitlow:1991uw
|  | HERMES proton |  | 37 | airapetian2011inclusive
|  | NMC $F_{2}$ |  | 275 | Rondio:1997se
|  | BCDMS $F_{2}^{p}$ | 337 | 351 | virchaux1992measurement ; Benvenuti:1989rh
4 | DIS Deuterium | BoNuS $F^{n}_{2}/F^{d}_{2}$ |  | 191 | tkachenko2014measurement
|  | JLab deuteron |  | 136 | Malace:2009kw
|  | SLAC deuteron |  | 582 | Whitlow:1991uw
|  | HERMES deuteron |  | 37 | airapetian2011inclusive
|  | NMC $F_{2}^{d}$/$F_{2}^{p}$ | 123 | 189 | Arneodo:1996qe
|  | BCDMS $F_{2}^{d}$ | 250 | 254 | Virchaux:1991jc ; Benvenuti:1989fm
5 | WZ Tevatron | CDF Run-1 lepton $A_{ch}$ , $p_{Tl}>$ 25 GeV | 11 |  | Abe:1996us ; Abe:1998rv
|  | CDF Run-2 electron $A_{ch}$ , $p_{Tl}>$ 25 GeV | 11 | 11 | Acosta:2005ud
|  | CDF Run-2 Z rapidity | 29 | 28 | Aaltonen:2010zza
|  | DØ Run-2 9.7 $\mathrm{fb}^{-1}$ electron $A_{ch}$ , $p_{Tl}>$ 25 GeV | 13 | 13 | D0:2014kma
|  | DØ Run-2 muon $A_{ch}$ , $p_{Tl}>$ 20 GeV | 9 |  | Abazov:2007pm
|  | DØ Run-2 $Z$ rapidity | 28 | 28 | Abazov:2007jy
|  | CDF $W$ asymmetry |  | 13 | Aaltonen:2009ta
|  | DØ $W$ asymmetry |  | 14 | Abazov:2013dsa
|  | DØ muon charge asymmetry |  | 10 | Abazov:2013rja
6 | WZ LHC | LHCb 7 TeV 1.0 $\mathrm{fb}^{-1}$ $W/Z$ forward rapidity cross sec. | 33 |  | Aaij:2015gna
|  | LHCb 8 TeV 2.0 $\mathrm{fb}^{-1}$ $Z\to e^{-}e^{+}$ forward rapidity cross sec. | 17 |  | Aaij:2015vua
|  | CMS 8 TeV 18.8 $\mathrm{fb}^{-1}$ muon charge asymmetry $A_{ch}$ | 11 |  | Khachatryan:2016pev
|  | LHCb 8 TeV 2.0 $\mathrm{fb}^{-1}$ $W/Z$ cross sec. | 34 |  | Aaij:2015zlq
|  | ATLAS 8 TeV 20.3 $\mathrm{fb}^{-1}$, $Z$ $p_{T}$ cross sec. | 27 |  | Aad:2015auj
|  | CMS 7 TeV 4.7 $\mathrm{fb}^{-1}$ muon $A_{ch}$ , $p_{Tl}>$ 35 GeV | 11 |  | Chatrchyan:2013mza
|  | CMS 7 TeV 840 $\mathrm{pb}^{-1}$ electron $A_{ch}$ , $p_{Tl}>$ 35 GeV | 11 |  | Chatrchyan:2012xt
|  | ATLAS 7 TeV 35 $\mathrm{pb}^{-1}$ $W/Z$ cross sec., $A_{ch}$ | 41 |  | Aad:2011dm
7 | Drell-Yan | E605 p+Cu | 119 |  | Moreno:1990sf
|  | E866, $\sigma^{pd}/\sigma^{pp}$ | 15 |  | Towell:2001nh
|  | E866, $\sigma^{pp}$ | 184 | 121 | E866lanl ; Webb:2003ps
|  | E866, $\sigma^{pd}$ |  | 129 | E866lanl ; Webb:2003ps
8 |  $\nu$-A incl. DIS | CDHSW $F_{2}$ | 85 |  | Berge:1989hr
|  | CDHSW $xF_{3}$ | 96 |  | Berge:1989hr
|  | CCFR $F_{2}$ | 69 |  | Yang:2000ju
|  | CCFR $xF_{3}$ | 86 |  | Seligman:1997mc
9 |  $t\bar{t}$ production | CMS 8 TeV 19.7 $\mathrm{fb}^{-1}$ , $t\bar{t}$ norm. top $p_{T}$ and $y$ cross sec. | 16 |  | Sirunyan:2017azo
|  | ATLAS 8 TeV 20.3 $\mathrm{fb}^{-1}$, $t\bar{t}$ $p^{t}_{T}$, $m_{t\bar{t}}$ abs. spectrum | 15 |  | Aad:2015mbv
10 |  $\nu$-A dimuon SIDIS | NuTeV $\nu\mu\mu$ SIDIS | 38 |  | osti_879078
|  | NuTeV $\nu\bar{\mu}\mu$ SIDIS | 33 |  | osti_879078
|  | CCFR $\nu\mu\mu$ SIDIS | 40 |  | Goncharov:2001qe
|  | CCFR $\nu\bar{\mu}\mu$ SIDIS | 38 |  | Goncharov:2001qe
11 | Jets LHC | CMS 7 TeV 5 $\mathrm{fb}^{-1}$, single incl. jet cross sec., R = 0.7 | 158 |  | Chatrchyan:2014gia
|  | ATLAS 7 TeV 4.5 $\mathrm{fb}^{-1}$, single incl. jet cross sec., R = 0.6 | 140 |  | Aad:2014vwa
|  | CMS 8 TeV 19.7 $\mathrm{fb}^{-1}$, single incl. jet cross sec., R = 0.7 | 185 |  | Khachatryan:2016mlc
Table 1: A comprehensive listing of experiments included within the CT and CJ
frameworks for this study, grouped according to the experimental process they
represent. For each experiment, we give the number of experimental data
points, $N_{\mathit{pt}}$, included into the CT and CJ fits presented in this
study, with blank entries indicating that the given experiment is omitted from
the respective global fit.

###  3.2 Modifications in the fitting methodologies

For the study presented in this article, we modified some default settings of
the CJ15 and CT18 fits, fully described respectively in Ref. Accardi:2016qay
and Hou:2019efy , to place the two fitting frameworks on a common footing and
isolate the impact of various assumed treatments of the deuteron structure.

  1. 1.

We match perturbative orders between the two fits at NLO in $\alpha_{s}$. In
practice, this means that in the CT fits, performed by default at NNLO, we
instead compute the hard cross sections, perturbative PDF evolution, and
running of $\alpha_{s}$ at $\mathcal{O}(\alpha_{s})$ accuracy to agree with
the default NLO settings used in CJ.

  2. 2.

We perform supplementary fits by excluding some data sets that appear in one
fit only. While both CJ and CT fits include Tevatron lepton charge asymmetry
measurements presented as a function of the charged lepton’s rapidity, the CJ
fit also includes the fixed-target low $W^{2}$ and $Q^{2}$ DIS data from SLAC
Whitlow:1991uw  and JLab Malace:2009kw , as well as the CDF Aaltonen:2009ta
and DØ Abazov:2013dsa  $W$ boson charge asymmetry with reconstructed weak
boson kinematics. On the other hand, CT makes use of neutrino-initiated DIS
data sets on heavy nuclear targets (both inclusive and semi-inclusive DIS
[SIDIS] di-muon production in $\nu$-A scattering). In CT, data on heavy-
nuclear targets are fitted at the isoscalar level after being corrected in the
fit using a phenomenological parametrization of the
$F^{A}(x,Q^{2})/F^{d}(x,Q^{2})$ ratio from Ref. Rondio:1993mf . To isolate the
impact of these extra experiments, we performed CJ fits without the $W$
asymmetry and SLAC DIS data sets, and CT fits without the inclusive $\nu$-A
DIS data.

  3. 3.

As in the original CJ and CT publications, we estimate the final PDF
uncertainties using the Hessian method Stump:2001gu , but in this paper we fix
the tolerance to be $T^{2}\\!=\\!10$ for both global analyses, in between the
nominal $T^{2}=2.71$ in the CJ15 fit and the $T^{2}=37$ value (at the 68%
probability level) used in the CT18 fits. Furthermore, we do not include the
additional “Tier-2” tolerance contribution Hou:2019efy ; Lai:2010nw  that is
applied in the CT18 fits to prevent the error PDFs from running into strong
disagreements with individual experiments, but content ourselves with the
“Tier-1” tolerance as defined in Pumplin:2002vw .

Regarding the lattermost point, in the CJ analysis it is additionally
necessary to implement a numerical prescription at the level of individual
eigenvector directions of the diagonalized Hessian matrix to guarantee
$\Delta\chi^{2}=T^{2}$ to the needed accuracy and to ensure the validity of
the analysis methods utilized in this article. These technical details are
reviewed in the appendix.

###  3.3 The $L_{2}$ sensitivity technique

In the next two sections, we will investigate the impact of the DIS deuteron
data on the PDFs with the help of the “$L_{2}$ sensitivity” Hobbs:2019gob .
The method is easy to use and has already been applied to clarify the
sensitivity of the global data sets to the CT18 NNLO PDFs Hou:2019efy , LHC
parton luminosities Amoroso:2020lgh  (Sec. II.2), and effective exponents of
the high-$x$ PDF falloff Courtoy:2020fex . Here we give a quick summary of the
$L_{2}$ technique and refer the interested reader to Ref. Hobbs:2019gob  for
additional details.

Figure 5:  A comparison of the PDF pulls of the deuterium DIS data computed
according to the $L_{2}$ method at 2 GeV for the CT fixed d.c. (left) and CJ
fixed d.c. (right) fits; both cases are for the scenario with fixed deuteron
corrections. Here and elsewhere, we place on the vertical axis a self-
explanatory label of $\Delta\chi^{2}$ as we plot the $L_{2}$ sensitivity which
approximates this quantity.

The $L_{2}$ sensitivity provides a fast approximation to the information
contained in the LM scans typified by Fig. 2. It does so by quantifying the
extent to which variations in the fitted PDFs drive the shifts in the log-
likelihood $\chi^{2}_{E}$ for each experiment $E$. For example, the $L_{2}$
plots discussed below supply the change in an experiment’s value of $\chi^{2}$
given a defined increase in the value of a PDF of interest, as a function of
$x$ for fixed $Q$. That being the case, if two experiments are in relative
tension with respect to the behavior of a given PDF — say, one favors a larger
PDF value in a given $x$ range, the other favors a smaller value — an increase
in the PDF would result in a negative change,
$\Delta\chi^{2}_{E_{1}}\\!<\\!0$, in the $\chi^{2}$ for the first experiment,
and a positive shift, $\Delta\chi^{2}_{E_{2}}\\!>\\!0$, for the second
experiment. As shall be seen below, these competing pulls are visualized by
the $L_{2}$ method as opposing deviations from zero in the negative/positive
directions. In this fashion, the $L_{2}$ sensitivity can encapsulate, on a
flavor-by-flavor basis, the patterns of pulls exerted on the PDFs (often
called “PDF pulls” below for short) by various experiments or groups of
experiments as a function of $x$ and $Q$. In fact, the method is not limited
to the influence of data sets on the PDFs, but can be extended to any
observable that can be calculated starting from those.

More formally, the $L_{2}$ sensitivity yields an approximation of the shift
$\Delta\chi^{2}_{E}$ in the value of the log-likelihood for experiment $E$
caused by an upward 1$\sigma$ variation of the chosen PDF or PDF-dependent
theoretical prediction. We evaluate the $L_{2}$ sensitivity in the Hessian
approximation as

$\displaystyle S_{f,L2}(E)\,$
$\displaystyle\equiv\,\vec{\nabla}\chi^{2}_{E}\cdot\frac{\vec{\nabla}f}{|\vec{\nabla}f|}$
$\displaystyle=\Delta\chi^{2}_{E}\,\cos{\varphi(f,\chi^{2}_{E})}\ ,$ (5)

where $\cos{\varphi(f,\chi^{2}_{E})}$ represents the cosine of the correlation
angle between a PDF of flavor $f$ (or, indeed, any PDF derived quantity) and
the experimental $\chi^{2}_{E}$, evaluated over the $2N$ Hessian eigenvector
sets. In the Hessian formalism adopted both by the CT and CJ frameworks to
estimate PDF uncertainties, this correlation cosine can be computed as
indicated in Ref. Wang:2018heo :

$\displaystyle\cos{\varphi(f,\chi^{2}_{E})}$ $\displaystyle=\frac{1}{4\Delta
f\Delta\chi^{2}_{E}}\sum_{j=1}^{N}(f_{j}^{+}-f_{j}^{-})([\chi^{2}_{E}]_{j}^{+}-[\chi^{2}_{E}]_{j}^{-})\
,$ (6)

where

$\Delta
f=\left|\vec{\nabla}f\right|=\frac{1}{2}\sqrt{\sum_{j=1}^{N}\left(f_{j}^{+}-f_{j}^{-}\right)^{2}}\
,$ (7)

and “$\pm$” denote the PDF variations along the positive and negative
direction of the $j$-th Hessian eigenvector.

Group # | Experiment | CJ no d.c. | CJ fixed d.c. | CJ no-W_slac | CT no d.c. | CT fixed d.c. | CT no nu-A
---|---|---|---|---|---|---|---
1 |  $\gamma$+jet | 62/56 | 61/56 | 60/56 | – | – | –
2 | Jets Tevatron | 37/182 | 36/182 | 36/182 | 225/182 | 225/182 | 229/182
3 | DIS proton | 3007/2548 | 2973/2548 | 2330/1848 | 1818/1523 | 1812/1523 | 1806/1523
4 | DIS deuteron | 1363/1389 | 1214/1389 | 704/671 | 401/373 | 387/373 | 381/373
| SLAC deuteron (only) | 507/582 | 376/582 | – | – | – | –
5 | WZ Tevatron | 193/117 | 133/117 | 99/90 | 113/101 | 129/101 | 137/101
| CDF $W$ asym. (only) | 14/13 | 17/13 | – | – | – | –
| DØ $W$ asym. (only) | 82/14 | 12/14 | – | – | – | –
6 | WZ LHC | – | – | – | 267/185 | 347/185 | 315/185
7 | Drell-Yan | 302/250 | 284/250 | 272/250 | 454/318 | 348/318 | 344/318
8 |  $\nu$-A incl. DIS | – | – | – | 269/336 | 279/336 | –
9 |  $t\bar{t}$ production | – | – | – | 44/31 | 45/31 | 46/31
10 |  $\nu$-A dimuon SIDIS | – | – | – | 103/149 | 104/149 | 103/149
11 | Jets LHC | – | – | – | 594/483 | 595/483 | 598/483
| TOTAL | 4963/4542 | 4699/4542 | 3501/3097 | 4289/3681 | 4271/3681 | 3959/3345
Table 2: For each fit, we report the total $\chi^{2}$ per point,
$\chi^{2}/N_{\mathit{pt}}$, as well as its breakdown according to the
experiment categories listed in Table 1. The data sets removed in the CJ no-
W_slac fit are singled out and emboldened in their respective categories. The
$\nu$-A inclusive data excluded in the CT no nu-A fit are also emboldened. We
note that the $\chi^{2}/N_{\mathit{pt}}$ values for CJ free d.c. are the same
as those reported above for CJ fixed d.c. given that the central fit of CJ
fixed d.c. corresponds to the best fit obtained when freely varying all
parameters in CJ free d.c.

When $L_{2}$ sensitivities are summed over all experiments, the resulting sum
should be close to zero by construction, assuming deviations from a symmetric
Gaussian probability distribution are negligible (see 6 for further
discussion). As an example, Fig. 5 shows the combined $L_{2}$ sensitivities,
$S_{f,L2}(E)$, of the experiments in the DIS deuteron group (#4 in Table 1) to
each parton flavor separately, calculated according to Eq. (5) for the CT and
CJ fixed d.c. fits, where the deuteron corrections were fixed to the central
value determined in the CJ15 analysis. These figures can be interpreted as the
statistical pulls at fixed $Q=2$ GeV from this group of experiments on each
PDF flavor, $f(x,Q)$, at the $x$ values specified on the horizontal axis. One
can observe quite large deviations from zero, with $S_{f,L2}$ values nearly
reaching $\pm 10$ units in some regions of $x$. This non-negligible pull by
the deuteron DIS experiments is ultimately offset by contributions from other
groups of experiments to obtain a zero result (within about one unit of
$\chi^{2}$) when summing over all of these. It is therefore interesting to
investigate which experimental groups pull significantly against the DIS
deuteron data sets, as we do below in Sec. 4.5 and 4.6. Here and elsewhere, we
compute $L_{2}$ sensitivities at a default scale of $Q=2$ GeV as this is close
to the initial scale, 1.3 GeV, for DGLAP evolution, as well as to the $Q$
values accessed in DIS experiments and typical scales used to present
predictions for PDFs in nonperturbative QCD models and lattice QCD
Hobbs:2019gob . We typically do not observe pronounced differences in the
$L_{2}$ sensitivity plots for distinct scales, $Q_{1,2}$, provided
$|Q_{2}-Q_{1}|\lesssim\mathcal{O}(100\,\mathrm{GeV})$. A set of companion
plots generated at a higher scale, $Q=100$ GeV, may be viewed at Ref.
CT18website .

Fig. 5 contains a substantial amount of information. For example, looking at
the left panel for CT fixed d.c., the negative $S_{f,L2}$ for the $d$ quark at
$x\approx 0.25$ indicates that the deuteron DIS data prefer a higher $d$ quark
at $x\approx 0.25$ than the nominal $d$-quark PDF in the full fit. Similarly,
the positive $S_{f,L2}$ for the $u$ quark at the same $x$ indicates a
preference for a relatively lower $u$-quark PDF. In totality, the deuteron DIS
data prefer a higher $d/u$ at $x=0.1\\!-\\!0.4$ than that obtained in the full
CT fit. (This preference for an enhanced $d/u$ in this $x$ interval is further
confirmed in Fig. 9 of Sec. 4.5.)

From the right panel of Fig. 5, we also read that the deuteron DIS data in the
CJ fixed d.c. fit prefer an enhanced $d/u$ over a slightly higher interval
$x=0.3\\!-\\!0.7$. Regarding other flavors and $x$ ranges, in the left panel
of CT fixed d.c. we observe a significant preference of the deuteron DIS group
for lower $u$\- and $d$-quark PDFs at $x=0.01\\!-\\!0.1$, in the region
relevant for LHC $W$-boson production. One also notices a preference for a
larger gluon PDF in the interval, $x=0.02\\!-\\!0.1$, relevant for Higgs-boson
production at the LHC, and, at slightly lower $x$, for a larger perturbative
charm-quark PDF, which is radiatively generated from the gluon.

Finally, we remark that the Hessian sensitivity is most effective in
identifying the top 5-10 experiments or groups of experiments sensitive to
variations in the chosen PDF, as has been verified by comparing the rankings
obtained from Hessian sensitivities and LM scans Wang:2018heo ; Hobbs:2019gob
. However, especially for subleading experiments, detailed rankings depend on
the chosen definition of the sensitivity indicator and deviations from the
simple linear approximation that we assumed when using symmetric finite
derivatives in $S_{f,L2}(E)$ as in Eqs. (6) and (7).

Experiment | $N_{\mathit{pt}}$ | no d.c. NLO | no d.c. NNLO | no d.c. NNLO-X | fixed d.c. NLO | $\delta^{\mathrm{NNLO}}$ | $\delta^{\mathrm{NNLO-X}}$ | $\delta^{\mathrm{fixed\,d.c.}}$
---|---|---|---|---|---|---|---|---
$\gamma$+jet | 0 | – | – | – | – | – | – | –
Jets Tevatron | 182 | 225 | 236 | 233 | 225 | 11 | 8 | 0
DIS proton | 1523 | 1818 | 1865 | 1839 | 1812 | 47 | 21 | -6
DIS deuteron | 373 | 401 | 406 | 405 | 387 | 5 | 4 | -14
WZ Tevatron | 101 | 113 | 120 | 110 | 129 | 7 | -3 | 16
WZ LHC | 185 | 267 | 259 | 185 | 347 | -8 | -82 | 80
Drell-Yan | 318 | 454 | 364 | 432 | 348 | -90 | -22 | -106
$\nu$-A incl. DIS | 336 | 269 | 285 | 260 | 279 | 16 | -9 | 10
$t\bar{t}$ production | 31 | 44 | 28 | 29 | 45 | -16 | -15 | 1
$\nu$-A dimuon SIDIS | 149 | 103 | 107 | 111 | 104 | 4 | 8 | 1
Jets LHC | 483 | 594 | 608 | 603 | 595 | 14 | 9 | 1
TOTAL | 3681 | 4289 | 4277 | 4208 | 4271 | -12 | -81 | -18
Table 3: Complementary to Table 2, we compare the values of $\chi^{2}$
obtained by CT in the no d.c. fit at NLO as well as at NNLO with both default
CT18 settings and those of the alternative, CT18X global fit (first three
columns, respectively) with CT fixed d.c. at NLO. The corresponding shifts
away from the $\chi^{2}$ obtained under CT no d.c. NLO for each experimental
group, $\delta^{\mathrm{NNLO}}$, $\delta^{\mathrm{NNLO-X}}$, and
$\delta^{\mathrm{fixed\,d.c.}}$, are given in the lattermost columns.

##  4 Comparison of deuteron data impact in the CJ15 and CT18 fits

In accordance with the preceding discussion in Sec. 2 and 3, our analysis will
be based on a series of fits named according to the following convention:

  1. 1.

Fits without deuteron corrections: CT no d.c., CJ no d.c.;

  2. 2.

Fits with the fixed CJ15 correction: CT fixed d.c., CJ fixed d.c.;

  3. 3.

A CJ fit in which the off-shellness correction is freely varied: CJ free d.c.;

  4. 4.

Fits with the fixed CJ15 deuteron correction and variations in the fitted data
sets: CT no nu-A (removing inclusive $\nu$-A DIS experiments from CT), and CJ
no-W_slac (removing the CDF [131] and DØ [132] $W$ boson asymmetry data and
SLAC DIS [proton and deuteron] sets from CJ).

In each enumerated case, the CT and CJ fits are methodologically comparable to
each other. The differences between the fits in the first two categories will
highlight the nontrivial effects of deuteron corrections. We consider several
variations of the fixed d.c. fits. Firstly, the comparison of the CJ
sensitivity plots in categories 2 and 3 demonstrates that freeing the d.c.
parameters in the CJ fit tangibly improves the agreement among all categories
of experiments. Secondly, we try to make the CT and CJ fits comparable not
only methodologically but also (partially) with respect to data selections. We
do this in category 4 by removing the indicated data sets from each fit.

###  4.1 Overall agreement of theory and data

We first review the overall quality of these fits by referring to Table 2,
which lists values for the total $\chi^{2}$ per point
($\chi^{2}/N_{\mathit{pt}}$) according to the experimental groups listed in
Table 1. The $\chi^{2}$ values for SLAC DIS, Tevatron $W$-boson asymmetry, and
$\nu$-A DIS data sets are shown in Table 2 in separate lines.

The $\chi^{2}$ values in Table 2 indicate that the deuteron data agree
globally with the published fits of both groups. Deuteron corrections are
essential to the CJ analysis, which includes deuteron DIS data at the largest
values of $x$ from SLAC as well as the very sensitive DØ data on the
reconstructed $W$-charge asymmetry Accardi:2016qay . Even if the total
$\chi^{2}/N_{\mathit{pt}}$ may seem only marginally improved, the highlighted
data sets require deuteron corrections to reconcile the SLAC DIS deuteron data
with the deuteron-independent DØ $W$-asymmetry data.

In CT, the inclusion of deuteron corrections also improves the description of
the DIS deuteron data (especially the BCDMS $F^{d}_{2}$ measurements),
producing a $14$-unit reduction in $\chi^{2}$ for the
$N_{\mathit{pt}}\\!=\\!373$ points fitted in Group #4, with an additional
$6$-unit reduction once the inclusive $\nu$-A data are removed. Mainly due to
the absence of the SLAC DIS data in CT, this is a smaller relative reduction
than that observed for CJ in the left columns of Table 2.

It is also interesting to note far more substantial shifts in $\chi^{2}$
within Table 2 among the other CT experimental groups: the introduction of the
fixed deuteron correction in CT improves the $\chi^{2}$ of the DY data (#7) by
more than $100$ units, while at the same time, increasing the $\chi^{2}$ of
the LHC weak-boson production (#6) by an opposing $\Delta\chi^{2}$ change of
80 units. The $\chi^{2}$ for group 5, “WZ Tevatron”, also increases by 16
units. The inclusive $\nu$-A DIS data set (#8) is fitted well globally, with
$\chi^{2}/N_{pt}=0.80\ (0.83)$ in the CT no d.c. (CT fixed d.c.) fit. In sum,
the fixed deuteron correction improves the CT total $\chi^{2}$ by 18 units for
3670 data points. Since the $\nu$-A DIS data are well-described, removing them
from the CT fit actually increases the total $\chi^{2}/N_{pt}$, but also seems
to release some tension with the WZ LHC data, whose $\chi^{2}$ value decreases
by 32 units. While these look like modest changes, overall, we will see next
that they do influence some PDF flavors and the local compatibility among
select groups of experiments.

It is instructive to evaluate the shifts in $\chi^{2}$ discussed above in the
context of corresponding variations associated with NNLO effects. For this
purpose, we compare in Table 3 values of $\chi^{2}$ obtained with CT for the
default NLO fits explored in this paper, CT no d.c. and CT fixed d.c., against
two NNLO fits. These latter fits are the NNLO counterpart of CT no d.c., which
we call no d.c. NNLO, and an alternative NNLO fit adopting the modified DIS
scale choice of the CT18X NNLO fit discussed in Ref. Hou:2019efy , which we
denote no d.c. NNLO-X. These additional fits allow us to quantify the impact
on $\chi^{2}$ of the NNLO correction itself, as well as the effect of
perturbative scale variations at NNLO, respectively. In addition, in the
trailing columns of Table 3 we also show $\chi^{2}$ differences of each of
these fits with respect to our base CT no d.c. fit at NLO. For example,

$\delta^{\mathrm{NNLO}}\equiv\chi^{2}(\mathrm{no\,d.c.\,NNLO})-\chi^{2}(\mathrm{no\,d.c.\,NLO})\
.$ (8)

These comparisons illustrate that inclusion of the fixed deuteron correction
at NLO may impact the theoretical description of select experimental groups at
a level comparable to NNLO corrections and scale variations. This is
especially evident for the DIS deuteron data, for which the 14-unit reduction
in $\chi^{2}$ discussed above for CT fixed d.c. can be compared to small 5 and
4-unit increases with no d.c. NNLO and no d.c. NNLO-X, respectively. The
above-mentioned $\chi^{2}$ shift for the Drell-Yan data with CT fixed d.c.
similarly surpasses the corresponding shifts obtained with the NNLO fits. In
contrast, the $t\bar{t}$ data, for instance, are largely insensitive to the
deuteron correction at NLO, but are significantly better-fitted with NNLO
corrections. Ultimately, as future generations of QCD fits pursue higher
accuracy at NNLO and beyond, it may therefore also be appropriate to consider
deuteron corrections.

###  4.2 Impact of deuteron corrections on the $d/u$ PDF ratio

As discussed in Sec. 1.1, experimental information involving deuterium
targets, especially measurements of the DIS structure function of the deuteron
(Group #4, “DIS Deuterium,” in Table 1), has been pivotal for separating the
$d$-quark content of the proton from other parton flavors. The leading impact
of deuteron corrections thus primarily influences the extracted $d$-PDF at
high $x$, where the deuteron most prominently differs from a superposition of
a free proton and a free neutron. In contrast, the effect of the deuteron
corrections on the $u$-type PDFs is comparatively mild, as these are most
directly constrained by measurements of the proton’s structure function (Group
#3, “DIS Proton”).

To gauge the leading impact of deuteron corrections, in this subsection we now
examine the $x$ dependence of the $d/u$ ratio within the CT/CJ frameworks,
before proceeding in Sec. 4.3 to an examination of the indirect effects on the
lower-$x$ $d_{\mathit{val}}$ PDF relevant for $\sin^{2}\theta_{W}$ and on the
gluon PDF in Sec. 4.4 (the PDF pulls will be considered in Secs. 4.5 and 4.6).

Figure 6:  Upper row: The PDF ratios $d/u$ and their asymmetric error bands
for $T^{2}=10$ at scale $Q=2$ GeV within the CT (left) and CJ (right) fitting
frameworks. We normalize all $d/u$ error bands to the ratio from the central
no d.c. fit (without any assumed deuteron correction). The left panel shows
the CT no d.c., CT fixed d.c., and CT no nu-A error bands. The right panel
shows the analogous CJ no d.c., CJ fixed d.c., CJ no-W_slac, and the CJ free
d.c. fits. The abscissas are scaled to highlight the impact of the deuteron
corrections at large $x$, where the impact is most pronounced, as well as the
modest enhancement in the $d/u$ ratio for $x\lesssim 0.01$ in CT at left.
Lower row: now showing the absolute $d/u$ ratios on a linear $x$-axis scale to
highlight the behavior at high $x$.

Fig. 6 illustrates the impact of the $F^{d}_{2}$ corrections, as introduced in
Sec. 2.1, on $d/u$. The upper row shows the $d/u$ ratio and its error band
obtained in the discussed series of fits, normalized to the central value
obtained in the fits with no deuteron corrections. The lower row shows the
unnormalized $d/u$ ratios themselves, using a linear horizontal scale to
better visualize the $x>0.1$ interval, and in particular the $x\to 1$ behavior
of this quantity. In both rows, the left and right panels give results for CT
and CJ fits, respectively. We see that the deuteron corrections have a
qualitatively similar impact on $d/u$ in both CT and CJ, especially at
$x\\!\gtrsim\\!0.1$, with evidence of a mild, few-percent enhancement of the
fitted $d/u$ ratio over the no d.c. fits for $0.1\\!\lesssim x\\!\lesssim 0.5$
once the fixed deuteron correction is included. This enhancement turns into a
suppression at still higher $x\\!\gtrsim\\!0.5$, beyond which $d/u$ is
strongly affected by the $2$-body nucleon-nucleon corrections included in the
$F^{d}_{2}$ calculation. For CJ, this suppression is larger than in the CT
case, but compatible with the latter within the respective uncertainties of
each fit.

The qualitative $x$ dependence of the deuteron-corrected fit of $d/u$ in the
top rows of Fig. 6 closely follows the $F^{N}_{2}/F^{d}_{2}$ ratio plotted in
Fig. 3, in which $F^{N}_{2}$ and $F^{d}_{2}$ represent the deuteron structure
function computed using the isoscalar and full nuclear predictions,
respectively. Indeed, in the no d.c. fits, $F^{d}_{2}$ is effectively fitted
as an isoscalar target, but in the CT fixed d.c., for example, the $F^{d}_{2}$
data for the physical deuteron are corrected to $F^{N}_{2}$, which leads to a
relative suppression of the fitted $d/u$ PDF ratio for $x\\!\gtrsim\\!0.5$.

In the CJ and CT fixed d.c. fits (red curves), the d.c. parameters are held
constant at their values obtained from the central CJ15 fit. If, on the other
hand, the d.c. parameters in Eq. (3) are actively fitted with the PDF shape
parameters, we obtain the same central PDFs but a narrower uncertainty band on
$d/u$, as shown by the CJ free d.c. error band in the right panels of Fig. 6.
This reduction in the PDF uncertainty of $d/u$, which at first sight is
paradoxical because we have increased the number of fit parameters, is
actually a consequence of the correlation between the treatment of $F^{d}_{2}$
structure function data and extracted $d$-PDF. More specifically, when the
nucleon off-shellness parameters are freed, the variations in the $d$-quark
parameters that were necessary to encompass the $F^{d}_{2}$ data in the CJ
fixed d.c. fit are partly absorbed by the parameters of Eq. (3). In other
words, releasing the off-shell parameters reduces tensions in parameter space,
and ultimately also diminishes the overall experiment-by-experiment
$\chi^{2}_{E}$ variations in the PDF analysis, as we shall note again below in
Sec. 4.5 and 4.6. We have verified that, over the same $x$ range, the relative
uncertainty in the determination of other flavors, such as, e.g., the gluon,
does correspondingly increase. This is an instance of the fact that,
typically, the constraining power of a fit can be enhanced by a greater number
of free parameters only in a limited sector of parameter space.

The different choices of data sets in the CT and CJ fits also affect the
fitted $d/u$ PDF ratio. For example, unlike CJ, the CT fit includes DIS data
from inclusive $\nu$-A collisions, multiplied by a phenomenological
parametrization of the heavy-nuclear structure function relative to the
physical deuteron. The green bands in Fig. 6 (left) are for the CT no nu-A
variant of the CT fit. The removal of these data augments the shifts in the CT
$d/u$ ratio induced by including the fixed deuteron correction, which now is
comparable to the CJ result. The reason for this may simply lie in the lack of
further deuteron-to-isoscalar proton plus neutron corrections, or may also be
related to possible discrepancies between neutral- and charge-current DIS data
or interactions Kopeliovich:2012kw ; Kovarik:2010uv ; Paukkunen:2013grz .

Conversely, the CT fits do not include the low-$W$ SLAC data and the
reconstructed $W$-boson asymmetry from the DØ experiment that are influential
upon the large-$x$ $d$-quark fit in CJ Accardi:2016qay . When these are
removed from the CJ fit as well — see the green CJ no-W_slac bands in the
right column of Fig. 6 — we obtain an enlarged uncertainty that includes both
the deuteron corrected and uncorrected bands. The uncertainty on $d/u$ at
$x\to 1$ in the CJ no-W_slac fit is wider than in the CT no nu-A fit in part
in reflection of different parametrization forms and selection of experiments
between the CJ and CT analyses.

Figure 7:  The valence $d$-quark PDFs and their uncertainties, normalized to
the central values obtained in the no d.c. fits. The fits and conventions are
the same as for Fig. 6.

###  4.3 Impact on the valence PDFs in the LHC EW precision region

The effects of including deuteron structure corrections to $F^{d}_{2}$, while
most pronounced at $x>0.1$, have some consequences in the low-$x$ region as
well. In the CT result shown in Fig. 6(left), modest enhancements in $d/u$ at
about $x\\!\sim\\!10^{-3}$ can be seen for the CT fixed d.c. fit. While the
sea-quark PDFs are relatively unaffected in this kinematic region, the valence
component of the $d$-quark and, to a lesser extent, the $u$-quark PDFs in this
small-$x$ region are sensitive to the inclusion and theoretical treatment of
both neutrino-nucleus and deuterium data at large $x$. This sensitivity is
mainly a consequence of corresponding valence sum rules.

Fig. 7 illustrates this feature. In the left panel, the fixed d.c. CT best-fit
valence PDF (red dotted line), shown in this case at $Q=81$ GeV $\approx
M_{W}$, prefers a slightly lower $d_{val}$ at $x\approx 0.008-0.13$ than in
the no d.c. fit, and a slightly higher one at $x\approx 0.13-0.45$. This
deviation becomes still more pronounced if the neutrino-nucleus DIS data are
removed (leading to the green dot-dashed line). In the right panel, we observe
for CJ a qualitatively similar trend and associated $x$ dependence over
slightly shifted $x$ regions of $0.01-0.1$ and $0.1-0.53$.

In Sec. 1.1, we illustrated in Figs. 1 and 2 that the weak-mixing angle
measurements at the LHC using the forward-backward asymmetry,
$A_{\mathrm{FB}}$, are sensitive to the valence $d$\- and $u$-PDFs in an
$x$-region about $\sim\\!0.03$, i.e., where Fig. 7 indicates a dependence of
these PDFs upon the treatment of the deuteron/heavy-nucleus data at higher-$x$
values.

Figure 8:  Same as Fig. 7, but now for the gluon PDF.

###  4.4 Impact of deuteron corrections on the gluon PDF

The deuteron data sets can also impact the gluon density through
$Q^{2}$-scaling violations; this is particularly true when these measurements
cover a large range of the four-momentum squared, $Q^{2}$, of the exchanged
boson. Similarly to the case for $d_{\mathit{val}}$ in Sec. 1.1, the Lagrange
Multiplier scans Hou:2019efy  and PDF sensitivity techniques Wang:2018heo ;
Hobbs:2019gob  in the CT18 analysis collectively demonstrate that the gluon at
$x>0.1$ receives significant constraints from the DIS deuterium data.
Constraints from the extensive fixed-target DIS data are in fact competitive
with the HERA DIS data, which probe the lower-$x$ region, and also with LHC
and Tevatron inclusive jet production, which cover a wide $x$ range but
involve complex arrays of systematic effects which remain under active study.

To address this point, in Fig. 8 we plot the error bands for the gluon PDF at
$Q\\!=\\!2$ GeV as determined in the series of fits discussed at the beginning
of this Section, again normalized to the central value obtained in the no d.c.
fits. As seen in the left panel of Fig. 8, the gluon PDF in the CT fits
exhibits a modest sensitivity to the chosen deuteron correction treatment,
with a dependence that is somewhat moderated by the adopted $W^{2}>12.25$ GeV2
cut. Still, as with $d/u$, there is a qualitative tendency for the fixed
deuteron correction to reduce the high-$x$ gluon PDF, with this modification
being enhanced by the exclusion of the inclusive $\nu$-A DIS data. Like CT,
the CJ fits seen in Fig. 8 (right), which include the SLAC DIS data, similarly
display a relative suppression of the gluon for $x\\!\gtrsim\\!0.3$ once
deuteron corrections are taken into account. While this effect is of moderate
size, it is nonetheless statistically significant in the context of the
$T^{2}=10$ tolerance used to determine the uncertainty bands. For CJ, it will
be interesting to confirm this effect by fitting the full JLab 6 GeV inclusive
data set Jlab6-inprep , and, even more so, the JLab 12 GeV data which will
augment the precision of the available DIS measurements over a wide $Q^{2}$
range at large $x$, once available.

###  4.5 Valence-sector PDF pulls: the $d/u$ ratio

In Fig. 9, we plot the $L_{2}$ sensitivities of the groups of experiments to
the $d/u$ PDF ratio for varied implementations of the deuteron corrections.
$L_{2}$ sensitivities for fits exploring data-set variations are shown in Fig.
10, which we also discuss below.

Figure 9:  The $L_{2}$ sensitivities computed according to Eq. (5) for $Q=2$
GeV, giving the pulls on the $d/u$ PDF ratio of the process-dependent data
sets fitted by CT (left) and CJ (right). Upper, middle, lower rows: results
for the no d.c., fixed d.c., and free d.c. fits discussed at the beginning of
Sec. 4.

Starting with the no d.c. fits that either do not include (CT) or remove (CJ)
deuteron corrections, we notice that, in both cases, the landscape of PDF
pulls tends to be dominated by a few competing groups of experiments, which
differ between the two fitting frameworks. For CT in the left panel, these are
the LHC $W/Z$ production (Group #6) and inclusive nuclear DIS (Group #8),
which possess the sharpest opposing pulls at $x\\!\sim\\!0.2$, for example, in
the direction of either favoring or disfavoring a larger $d/u$ ratio,
respectively. At slightly higher $x\\!>\\!0.3$ values, which are of particular
interest from the perspective of QCD-informed models of the $d/u$ behavior as
$x\to 1$, these are joined by the DIS-deuterium (#4) and Drell-Yan (#7) groups
of experiments. Turning to the CJ case, displayed in the top-right panel, it
is the DIS deuterium (#4) and gamma-jet (#1) groups that dominate the
landscape of PDF pulls — and quite strikingly at large $x$ — with lesser but
also significant pulls from the Tevatron $W/Z$ production data (#5). The
large-$x$ pulls are expected, since the SLAC data are quite sensitive to
nuclear dynamics in the deuteron target, as already noted.

Once fixed deuteron-structure corrections are introduced into the respective
fixed d.c. fits, the relative patterns of PDF pulls experience an intriguing
series of shifts, which we display in the middle row of Fig. 9. For CJ, the
deuteron corrections largely resolve the huge tensions between the photon+jet
and DIS deuteron data, because dynamical nuclear effects in the latter are now
included in the theoretical calculation of the deuteron DIS cross section
without forcing the $d$-quark to deform to compensate for the missing nuclear
effect. A residual tension between the DIS deuteron (#4) and $W/Z$ Tevatron
data (#5) data is still visible at $x\\!\approx\\!0.5$. While the large-$x$
tensions are reduced, the small-$x$ pulls visibly change in shape for several
flavors.

For CT, the introduction of the fixed deuteron correction detailed in Sec. 2.1
instead preserves the qualitative $x$ dependence of the $L_{2}$ pulls (i.e.,
the shapes) but softens their magnitude in a few cases, for example, for the
Drell-Yan (#7) and the small-$x$ LHC $W/Z$ production (#6) data. Notably, the
DIS-deuterium sensitivity shifts to closely resemble that of the LHC $W/Z$
data (#6) in favoring a larger $d/u$ for $x\sim 0.2$. At the same time, the
size of the competing pulls at high $x\\!\gtrsim\\!0.3$ between the DIS-
deuterium and inclusive $\nu$-A data (#8) is enhanced, while the opposing
pulls of other experiments are modestly reduced over the same range in $x$.
This is especially clear for the CT Drell-Yan data (#7), which in the CT no
d.c. fit had preferred a softer $d/u$ ratio at low $x\\!<\\!0.01$ and an
enlarged value of $d/u$ over $0.01\\!\lesssim\\!x\\!\lesssim\\!0.2$. In CT
fixed d.c., these preferences mostly vanish. At the same time, outside this
interval of very high $x$, the opposing pulls of the inclusive $\nu$-A data
(Group #8) and both the DIS deuterium (#4) and LHC $W/Z$ (#6) experiments
increase and sharpen for $x\gtrsim 0.01$. In fact, this is the same collection
of experiments for which in Table 2 we observed increases (in the case of
Groups #6 and #8) in their respective values of $\chi^{2}/N_{\mathit{pt}}$
upon introducing deuteron corrections for $F^{d}_{2}$. Both the $\chi^{2}$
values in Table 2 and the $L_{2}$ analysis for CT therefore indicate a
noticeable rearrangement of the pulls of the inclusive neutrino-nucleus DIS
data and select other experiments introduced by the correction to the deuteron
DIS data. This rearrangement, being presently of similar order with respect to
other contributing effects, will require attention in the future.

Figure 10:  As in Fig. 9, we plot the PDF pulls on $d/u$ at 2 GeV with fixed
deuteron corrections present, but, in this case, removing select experiments
which have shown significant competing pulls with respect to the DIS deuterium
sets. For CT (left panel), we remove the inclusive $\nu$-A data (Group #8),
while for CJ, we remove the SLAC DIS experiments (part of Group #4) and
$W$-asymmetry information from the Tevatron (part of Group #5).

In the last row of Fig. 9 we present the $L_{2}$ pulls of the CJ free d.c. fit
in which the deuteron off-shellness degrees-of-freedom in Eq. (3) are
released. Comparing the vertical extents of the peaks with those of the CJ
fixed d.c. fit in the middle row, we see that freeing the offshell parameters
moderates the PDF pulls over the whole $x$ range, especially those at
$x\\!>\\!0.3$ between the DIS deuteron (#4) and the WZ Tevatron CJ Drell-Yan
(#5) data. This behavior can be generically understood as a consequence of
increasing the number of free parameters, but is not guaranteed, for example,
in the presence of incompatible data sets. The CJ free d.c. plot is thus an
indication of a good level of consistency between the considered data sets,
when the PDFs and deuteron corrections are fitted together.

Figure 11:  Analogous to Fig. 9, for the PDF pulls on $g(x,Q)$ at $Q=2$ GeV.

The results discussed so far suggest a nontrivial relationship between the
treatment of the DIS deuterium data and the description of other data sets in
each fitting framework: the impact of deuteron-structure corrections in a
global fit like CT and CJ cannot generally be expected to apply to deuterium
data alone, but have secondary effects on the patterns of pulls of other data
sets.

It is therefore interesting to study variations in the choices of experimental
data sets in both fits, in particular, removing from each analysis those data
sets that showed especially strong sensitivity to deuteron corrections or
otherwise played a major role in the foregoing discussion.

For CT, we remove the entire collection of inclusive $\nu$-A measurements
(Group #8), and refit with the fixed deuteron corrections of Sec. 2.1 in
place; the resulting $L_{2}$ sensitivity plot is displayed in the left panel
of Fig. 10. Overall, the magnitude of the PDF pulls is slightly reduced, with
the biggest change occurring for the DIS Deuterium Group (#4), which is now
more closely aligned with the pulls exerted by the DIS proton data (#3)
throughout the plotted domain in $x$. When considered in parallel with Fig. 9,
and in the light of the previous discussion of Fig. 9, the left panel of Fig.
10 suggests a connection between the pulls of the DIS deuteron and $\nu$-A
data in fits with and without deuteron corrections. For both groups of
experiments, the interplay between the theoretical description of deuterium
and heavy-nuclear data is relevant. To that end, investigating the systematic
treatment of nuclear effects for light and heavy nuclei is a critical subject
for future global analyses that aim to use such data for constraining the
nucleon PDFs to higher accuracy.

A similar consideration arises for CJ. As we have discussed, the combination
of $W$-boson charge asymmetries and SLAC DIS data is strongly constraining on
the $d/u$ ratio at large $x$, and the d.c. treatment influences also the PDF
pulls at smaller $x$ values (as seen in the right panels of Fig. 9). We
therefore remove these data sets from the fit to obtain the CJ no-W_slac fit
shown in the right panel of Fig. 10. In this instance, the removal of the
combined $W$ and SLAC DIS data relieves tensions seen in CJ fixed d.c., for
$x\\!<\\!0.1$. However, the large-$x$ tension between DIS deuteron and WZ
Tevatron data (that now only include the $W\to\ell$ decay lepton asymmetries)
remains largely intact and can in fact also be seen in the CT panel on the
left of Fig. 10. It remains to be seen whether this is of experimental origin,
or due to an as yet incomplete treatment of nuclear corrections in the
deuteron target.

Figure 12:  PDF pulls on $g$ at 2 GeV after removing the inclusive $\nu$-A
experiments for CT (left) and $W$ charge asymmetry and SLAC DIS data from CJ
(right), with deuteron corrections fixed.

###  4.6 PDF pulls in the gluon and light-quark sea sectors

At first sight, it might seem reasonable to suppose that deuteron-structure
corrections, being most sizable at high $x$ and more immediately connected to
extractions of the $d$-quark, would be relatively inconsequential for
determinations of the gluon PDFs. In actuality, constraining the gluon PDF
through DIS data requires an adequate prediction of the scale dependence of
both proton and deuteron DIS data sets, with the latter simultaneously
sensitive to the ($Q^{2}$ dependent) deuteron corrections discussed in Sec.
2.1. Moreover, the momentum sum rule requires that the changes in the total
momentum fraction from the large-$x$ and small-$x$ quark and gluon PDFs
compensate one another. The practical implementation of the deuteron
correction can therefore impact $g(x,Q)$ over a still broader range beyond
high $x$.

We therefore turn to an examination of the pulls on the gluon PDF in fits with
and without deuteron corrections, presented in Fig. 11, before examining CT
and CJ fits with the modified data sets in Fig. 12. Comparing the CJ no d.c.
fit in the upper-right panel of Fig. 11 to the CJ fixed d.c. fit in the
middle-right panel, one sees that adding a fixed deuteron correction clearly
aligns the pulls of the DIS proton group (#3) and DIS deuteron group (#4) on
the gluon. The $x$ dependence of these pulls was effectively uncorrelated
without the deuteron correction. In the presence of the fixed correction,
however, they are aligned and pronounced over the whole $x$ range and are
opposed mostly by the strong pull of the $WZ$ Tevatron data (Group #5).
Furthermore, after the off-shell parameters in the CJ deuteron correction are
freed (lowermost panel), the tension between Groups #3, #4, and #5 is
relieved, resulting in a more consistent data set, with weak pulls ($\lesssim
3$ units) everywhere. It is also interesting to note that a similar effect
arose in the $d/u$ sector discussed in the previous subsection, and therefore
seems to be a robust feature of fitting the deuteron corrections
simultaneously with the PDF parameters.

In the two CT fits in the upper left and middle left panels of Fig. 11, a
somewhat different pattern emerges. Inclusion of the deuteron correction in CT
fixed d.c. does have the effect of partially aligning the pulls of the DIS
Proton (Group #3) and DIS deuteron (Group #4) on the gluon, but this effect is
restricted to a narrower interval, $x\in[0.2,0.5]$. The pulls from the other
groups are relatively unaffected by the deuteron correction, although we see
some realignment of the pulls from LHC $WZ$ and fixed-target Drell-Yan
production experiments (Groups #6 and #7). The weaker dependence of the gluon
pulls in the CT fit on the deuteron correction compared to the CJ case is most
likely due to the absence of the SLAC DIS data from the former fit. One can
therefore investigate the effect of the removal of the SLAC data on the CJ
fit. Intriguingly, as shown in the right panel of Fig. 12, simultaneously
removing the SLAC DIS data and the Tevatron $W$-boson asymmetry data largely
alleviates the competing gluon pulls, which are now smaller than those
observed in the CT fit, especially from the LHC sets not included in CJ.

Clearly, the gluon pulls in the CT fit are due to the data other than the
large-$x$ SLAC and $W$ production data. In particular, in the absence of the
large-$x$ SLAC and $W$ production data in the CT fit, we notice a strong
preference for a harder gluon at $x\approx 0.3$ from the $\nu$-A DIS
experiments (Group #8) both with and without the nuclear correction. In fact,
the preference of the CDHSW $\nu$-A DIS deuteron data for a harder gluon at
large $x$ had already been identified in the CT18 analysis Hou:2019efy ,
although the net effect of including this data set in the CT18 fit does not
exceed the PDF uncertainty. However, as the left panel of Fig. 12 shows,
removing these data from the fit does not substantially alter the pulls of the
remaining experiments shown in the CT plots of Fig. 11, which are led by the
jet and $W/Z$ measurements from the LHC.

##  5 Conclusion

In this analysis, we have for the first time undertaken a comparative analysis
of two global fitting frameworks, CTEQ-JLab (CJ) and CTEQ-TEA (CT), using the
$L_{2}$ sensitivity statistical metric developed in Refs. Wang:2018heo ;
Hobbs:2019gob . This metric allowed our study to take advantage of the
complementary strengths of the two frameworks: the extended experimental
coverage and various theoretical developments implemented in the two
approaches, as well as the flexible PDF parametrizations available in CT and
the unique capabilities of CJ in describing low-energy and nuclear dynamics.
In doing so, we made a number of technical adjustments to each framework
(discussed in detail in the appendix) in order to reconcile the CT and CJ
treatment of PDF uncertainties and thereby render them sufficiently similar to
be meaningfully juxtaposed.

We have, in particular, concentrated on evaluating the impact on PDF
determinations of nuclear corrections which take into account the two-baryon
structure of the deuteron. In fact, as discussed in Sec. 1.1, DIS and Drell-
Yan measurements on deuterium are very informative in providing flavor
separation of $d$-type quarks from other parton flavors (under an assumption
of nucleon charge symmetry). At the same time, the introduction of deuterium
data into proton PDF fits brings along its own uncertainties associated with
nuclear and power-suppressed effects. Global analyses take diverse approaches
in handling the deuteron and heavy-nuclear effects, from selection of the
least affected experimental data Hou:2019efy ; Ball:2017nwa ; Bailey:2020ooq ,
to including some fixed Hou:2019efy  or free Accardi:2016qay ; Bailey:2020ooq
nuclear corrections and performing Bayesian marginalization Ball:2020xqw ;
Ball:2018twp  with respect to the nuclear parameters. It is therefore
important to understand the role of the deuteron corrections in a controlled
setting, by isolating them from other factors that affect the existing PDF
ensembles at comparable levels.111See examples in Hou:2019efy  for comparable
variations in $S_{f,L2}(E)$ caused by various assumptions.

By examining the fitted PDFs and resulting PDF pulls of experimental data
under several theoretical scenarios for the treatment of deuteron corrections,
in Sec. 4 we have gathered a substantial number of results that clarify these
questions. We reiterate here our overriding conclusions based on this
investigation:

  * •

While the compilation of $\chi^{2}$ values in Sec. 4.1 indicates good global
agreement of CJ and CT NLO theoretical predictions with deuteron data sets,
the $L_{2}$ sensitivity additionally provides insights about local
compatibility of fitted experiments in an $x$-dependent fashion. In the case
of CJ, the model of deuteron dynamic effects is crucial for the description of
the informative low-$Q$ DIS data from SLAC. The dependence on the deuteron
correction is reduced in the CT analysis with its more conservative cut on
$W^{2}$. Still, including the CJ deuteron correction reduces $\chi^{2}$ for
the aggregated DIS-deuteron experiments by about $\sim\\!\\!14$ units and also
reduces the cumulative $\chi^{2}$ for vector-boson production data sets by
several tens of units, with the modifications potentially comparable to the
NNLO scale dependence in an analysis like CT. Another effect of the deuteron
correction is to alleviate the competing pulls of deuteron and some other
experiments in the large-$x$ region.

  * •

The impact of a fixed deuteron correction is particularly evident in the
high-$x$ distribution for the $d$-quark, or the associated $d/u$ PDF ratio. A
number of commonalities exist between the CT and CJ analyses in the
qualitative effect of this correction on the extracted high-$x$ PDFs. The
fixed deuteron correction of Sec. 2.1 generally leads to the suppression of
the $d/u$ ratio at $x\\!>\\!0.5$ relative to the scenario without the deuteron
correction.

  * •

Due to the influence of sum rules and nontrivial correlations among fitted
PDFs of different parton flavors, deuteron corrections to DIS structure
functions at large $x$ can have important secondary effects on, e.g., the
gluon or sea-quark PDFs over a range of $x$, as well as the $d_{\mathit{val}}$
distribution at lower $x\\!\sim\\!0.03$ of relevance to precision studies in
the electroweak sector.

  * •

In both fitting frameworks, the modifications caused by the deuteron-structure
corrections are moderated by the inclusion of some non-deuteron data sets; for
CT, these are inclusive neutrino DIS data on heavy nuclear targets, while for
CJ, a combination of high-$x$ SLAC DIS data and reconstructed boson-level
Tevatron charge asymmetry requires special attention. Disentangling the
interplay among these fitted experiments will require a further study at NNLO
accuracy, including additional investigation of the implementation of
theoretical corrections for nuclear data sets (including both deuteron and
heavier targets) and the treatment of $W/Z$ data.

As the drive to realize next-generation accuracy in PDF analyses gains speed
with preparations for the High-Luminosity LHC ApollinariG.:2017ojx , Electron-
Ion Collider Accardi:2012qut , and Long-Baseline Neutrino Facility
Acciarri:2015uup , we recommend consideration of deuteron corrections and
broader nuclear effects in PDF analyses, as well as continued phenomenological
and model-based studies Melnitchouk:1994rv ; Bickerstaff:1985ax ;
Melnitchouk:1992eu ; Miller:2013hla  of deuteron structure in parallel.
Deuteron effects will become particularly unavoidable with increasing PDF
precision and in PDF-benchmarking studies, most obviously for the $d$-PDF at
$x\\!\gtrsim\\!0.2$ and beyond, but, ultimately, for consistency of the
extracted gluon density and over a widening range of $x$. Consideration of the
parton-level violation of the charge symmetry in the deuteron Hobbs:2011vy
may become relevant as precision goals advance still further. As emphasized in
Sec. 1.1 and 4.3, the achievement of ultimate precision in tests of the SM in
the electroweak sector will partly depend upon the successful treatment of the
issues described in this work.

###### Acknowledgements.

We appreciate insightful discussions with Aurore Courtoy, Lucian Harland-Lang,
Robert Thorne, and our colleagues within the CTEQ-JLab and CTEQ-TEA
collaborations, with special thanks to Jun Gao, Shujie Li, Wally Melnitchouk,
and C.-P. Yuan. Funding agency and grant acknowledgements appear on the first
page of this manuscript.

##  6 Appendix: Comparison procedure and technical details

To meaningfully compare two distinct global fits on a common footing as done
in this article, it has been necessary to conciliate their methodologies, cf.
Sec. 3.2. Part of this consisted in making a few technical adjustments to
ensure the mutual compatibility of the CJ and CT computations of the $L_{2}$
sensitivity introduced in Sec. 3.3 and employed in Sec. 4. In this Appendix,
we provide more detail about these adjustments.

The $L_{2}$ sensitivity can be interpreted as a fast approximation, based on
the Hessian error formalism Stump:2001gu , of the $\Delta\chi^{2}_{E}$ shifts
contained in the LM scans shown in the panels of Fig. 2. For that reason, we
expect the sum of $L_{2}$ sensitivities over all fitted experiments $E$ to
vanish for each parton flavor $f$; i.e.,

$L^{\mathrm{tot}}_{2,f}\,\equiv\,\sum_{E}S_{f,L2}(E)\,\approx\,0\ \quad\forall
f\ .$ (9)

This desired result — that the “total” $L_{2}^{\mathrm{tot}}$ sensitivities
approximately sum to zero for all flavors within a given fit — represents an
ideal scenario in which all data sets demonstrate full mutual compatibility,
and uncertainties are small enough to validate a quadratic approximation for
the $\chi^{2}$ function around the central PDF parameters, $\vec{a}_{0}$. In
practice, however, neither condition was perfectly realized when generating
the Hessian eigenvector sets, causing $L_{2}^{\mathrm{tot}}$ to deviate from
zero for both the CJ15 and CT18 eigenvector ensembles. We illustrate the
graphs of $L_{2}^{\mathrm{tot}}$ obtained with the standard eigenvector
computations, for tolerance $T^{2}=10$, in the left panels of Fig. 13. In the
upper figure, the total sensitivities in the standard CJ free d.c. show very
pronounced deviations from zero. In the lower figure, we see milder but not
entirely negligible deviations from zero in the counterpart default CT no d.c.
fit.

Figure 13:  Total sensitivities, $L^{\mathrm{tot}}_{2}$, summed over all experiments as in Eq. 9. The top left panel shows $L^{\mathrm{tot}}_{2}$ for the CJ free d.c. PDF ensemble with the standard CJ15 computation of the Hessian eigenvectors. The top right panel illustrates the same result after adjusting each CJ free d.c. eigenvector to have $\Delta\chi^{2}=T^{2}=10$ in both the positive and negative direction, thereby correcting the departures from Gaussianity observed for poorly constrained eigenvectors in the standard CJ Hessian analysis. In the bottom row, the left panel shows $L^{\mathrm{tot}}_{2}$ for the CT no d.c. PDF ensemble computed with the CT18 two-tier uncertainty constraints with $T^{2}=10$. On the right, the same but including only the Tier-1 uncertainty constraint that do not introduce a priori deviations from the Gaussianity condition.  | $\delta\tau^{E}$ | Expt ID | Experiment
---|---|---|---
1 | 37.2555 | 160 | HERA run I+II
2 | 32.6485 | 204 | E866 proton proton Drell-Yan process
3 | 28.1273 | 545 | CMS 8 TeV 19.7 $\mathrm{fb}^{-1}$, single incl. jet cross sec., R = 0.7
4 | 24.7211 | 250 | LHCb 8 TeV 2.0 $\mathrm{fb}^{-1}$ W/Z cross sec.
5 | 24.2715 | 101 | BCDMS $F_{2}^{p}$
6 | 20.5027 | 542 | CMS 7 TeV 5 $\mathrm{fb}^{-1}$, single incl. jet cross sec., R = 0.7
7 | 19.1007 | 245 | LHCb 7 TeV 1.0 $\mathrm{fb}^{-1}$ W/Z forward rapidity cross sec.
8 | 18.9541 | 102 | BCDMS $F_{2}^{d}$
9 | 17.935 | 203 | E866f Drell-Yan process
10 | 17.5018 | 544 | ATLAS 7 TeV 4.5 $\mathrm{fb}^{-1}$, single incl. jet cross sec., R = 0.6
… | … | … | …
39 | 0.446136 | 145 | H1 $\sigma_{r}^{b}$
Table 4: Impact of a given experiment on the difference between the Tier-1+2
and Tier-1 CT total sensitivities, quantified by the $\delta\tau_{E}$ distance
defined in Eq. (12). The first 3 experiments are well separated from the
remaining ones, which are more closely spaced.

We have identified the sources of this behavior and remedied this situation as
follows.

  * •

CJ: The standard CJ PDF error sets are obtained by the Hessian analysis around
the best-fit PDF parameters, $\vec{a}_{0}$, with each eigenvector scaled by a
given tolerance factor $T$ to nominally produce an increase of $T^{2}$ above
the minimum in the $\chi^{2}$ function ($T=1.646$ in the CJ15 analysis
Accardi:2016qay , $T=\sqrt{10}$ in this paper). For the computation of the
sensitivities, the $\chi^{2}$ function is instead scanned along every
eigenvector starting from the best-fit parameters, $\vec{a}_{0}$, until
parameters $\vec{a}_{i}$ are found in the plus- and minus-directions such that

$\displaystyle\Delta\chi^{2}(\vec{a}_{2i+1})$
$\displaystyle=\Delta\chi^{2}(\vec{a}_{2i})=T^{2}$ $\displaystyle\hskip
56.9055pt\forall\ i=1,\ldots,N_{par}\ .$ (10)

This way, the error PDFs correspond exactly to a given likelihood
$\mathcal{L}\propto e^{-T^{2}/2}$, and, by construction,
$L_{2}^{\mathrm{tot}}=0$ except for numerical uncertainties, see Eq. (6). The
total sensitivities after this adjustment of the eigenvector excursions are
shown in the upper right panel of Fig. 13. All $L_{2}^{\mathrm{tot}}$ values
are now well within $\pm 1$ unit from zero.

  * •

CT: The published general-purpose CT fits Hou:2019efy ; Dulat:2015mca ;
Lai:2010vv  apply a two-tier procedure to determine the published error bands.
Excursions along each eigenvector are constrained both by the Tier-1 penalty
imposed by the increase of the global $\chi^{2}$ above $T^{2}$ units, and by
the dynamical Tier-2 penalty based on effective Gaussian variables that ensure
that no single-experiment $\chi^{2}_{E}$ value increases above its best-fit
value by more than its uncertainty at the requested confidence level. The
Tier-2 procedure may lead to unequal excursions in the plus- and minus-
directions along a given eigenvector, therefore introducing a departure from
the condition for $L_{2}^{\mathrm{tot}}=0$. For this reason, we revert to the
Tier-1 PDF error determination, which nominally satisfies the desired
condition on $L_{2}^{\mathrm{tot}}$. The resulting total sensitivities are
compared to the standard CT18 calculation in the lower row of Fig. 13.222As a
technical observation, both Tier-1 and Tier-1+2 predictions in this case are
computed assuming $T^{2}=10$, without modifying the excursions of the
effective Gaussian parameters Lai:2010vv  in the Tier-2 penalty compared to
the default CT18 setup. All total sensitivities with the Tier-1 error sets are
within $\pm 1$ unit from zero, with a somewhat enhanced deviation observed for
the gluon PDF and for the charm PDF that follows it.

The very bad CJ $L_{2}^{\mathrm{tot}}$ obtained before the adjustment in the
upper left panel of Fig. 13 was traced primarily to a substantial deviation
from Gaussianity observed along a small number of eigenvector directions. This
can happen in a fit where certain linear combinations of parameters are poorly
constrained by the data, especially when exploiting an extended
parametrization such as in the CJ case, where additional parameters are
included to allow for a non-vanishing $d/u$ quark at large $x$, to describe
off-shell deformation of bound nucleons in a deuteron target, and to fit
higher-twist corrections to the standard twist-2 calculation of electron-
nucleon DIS. In fact, we have verified that the very large values of the total
sensitivity are primarily driven by the combined HERA data set, which accounts
for most of the calculated sensitivities across all parton flavors, with a
secondary large contribution provided by the DØ $W$-asymmetry measurements.

In the CT analysis, both the Tier-1+2 and Tier-1 total sensitivities in the
lower row of Fig. 13 are of a natural order, being generally $\approx\\!0$,
especially in the Tier-1 calculation at lower-right. This is aside from the
high-$x$ $L_{2}^{\mathrm{tot}}$ values for the $g$\- and $c$-PDFs, which are
somewhat larger than in the CJ15 adjusted analysis, but still $\lesssim\\!1$
in this case. The Tier-1 calculation is less affected by non-Gaussianities and
produces a smaller total sensitivity than the default Tier-1+2 analysis. As
with the CJ analysis, it is interesting to identify the key experiments that
affect the differences between the CT Tier-1+2 and Tier-1 total sensitivities.
Since these differences are comparatively small, we use a different procedure
than in the CJ case, where it was sufficient to analyze the size of the
experiment-by-experiment sensitivities before parameter adjustment. We notice
that the departure from $L_{2}^{\mathrm{tot}}\approx 0$ is determined by the
$\chi^{2}$ imbalance,

$\tau^{E}_{i}\equiv[\chi^{2}_{2i}]^{E}-[\chi^{2}_{2i-1}]^{E},\ \ \
i=1,\ldots,N_{\mathit{par}}\ .$ (11)

between the PDF error sets in each positive and negative eigenvector
direction, see Eq. (6). We can therefore quantify the impact of a given
experiment on the observed differences between the total sensitivities for the
Tier-1+2 and Tier-1 CT analysis by calculating

$\delta\tau^{E}=|\overrightarrow{\tau}^{E}_{\mathrm{Tier-1+2}}-\overrightarrow{\tau}^{E}_{\mathrm{Tier-1}}|\
.$ (12)

Note that we could also have considered other metrics, such as the quadratic
distance between the respective $\chi^{2}$ imbalances. However, our goal here
is to identify the experiments with the largest impact on the reduction of the
total sensitivity, rather than performing a detailed quantitative analysis and
ranking of each experiment.

We collect the largest values of the $\delta\tau^{E}$ metric in Table 4, where
experiments have been ordered from the highest to lowest impact. Intriguingly,
the most affected experiment in the CT case is also the HERA I+II combined
data set. We believe that the underlying reason(s) for this commonality with
the CJ analysis are the high precision and large kinematic coverage of the
HERA data, which are therefore sensitive to small variations in the PDFs and
subject to secondary constraints, such as those imposed on the gluon
distribution by PDF sum rules.

## References

  * (1) D. Stump, J. Pumplin, R. Brock, D. Casey, J. Huston, J. Kalk, H.L. Lai, W.K. Tung, Phys. Rev. D65, 014012 (2001).  DOI 10.1103/PhysRevD.65.014012
  * (2) P.M. Nadolsky, Z. Sullivan, eConf C010630, P510 (2001)
  * (3) P.M. Nadolsky, H.L. Lai, Q.H. Cao, J. Huston, J. Pumplin, D. Stump, W.K. Tung, C.P. Yuan, Phys. Rev. D78, 013004 (2008).  DOI 10.1103/PhysRevD.78.013004
  * (4) A. Armbruster, (2018).  Private communication
  * (5) P. Jimenez-Delgado, W. Melnitchouk, J.F. Owens, J. Phys. G 40, 093102 (2013).  DOI 10.1088/0954-3899/40/9/093102
  * (6) J. Gao, L. Harland-Lang, J. Rojo, Phys. Rept. 742, 1 (2018).  DOI 10.1016/j.physrep.2018.03.002
  * (7) K. Kovařík, P.M. Nadolsky, D.E. Soper, Rev. Mod. Phys. 92(4), 045003 (2020).  DOI 10.1103/RevModPhys.92.045003
  * (8) J.J. Ethier, E.R. Nocera, Ann. Rev. Nucl. Part. Sci. 70, 43 (2020).  DOI 10.1146/annurev-nucl-011720-042725
  * (9) A. Accardi, L.T. Brady, W. Melnitchouk, J.F. Owens, N. Sato, Phys. Rev. D93(11), 114017 (2016).  DOI 10.1103/PhysRevD.93.114017
  * (10) S. Alekhin, J. Blümlein, S. Moch, R. Placakyte, Phys. Rev. D96(1), 014011 (2017).  DOI 10.1103/PhysRevD.96.014011
  * (11) T.J. Hou, et al., Phys. Rev. D 103(1), 014013 (2021).  DOI 10.1103/PhysRevD.103.014013
  * (12) R.D. Ball, et al., Eur. Phys. J. C77(10), 663 (2017).  DOI 10.1140/epjc/s10052-017-5199-5
  * (13) S. Bailey, T. Cridge, L.A. Harland-Lang, A.D. Martin, R.S. Thorne, Eur. Phys. J. C 81(4), 341 (2021).  DOI 10.1140/epjc/s10052-021-09057-0
  * (14) A. Accardi, PoS DIS2015, 001 (2015).  DOI 10.22323/1.247.0001
  * (15) B.T. Wang, T.J. Hobbs, S. Doyle, J. Gao, T.J. Hou, P.M. Nadolsky, F.I. Olness, Phys. Rev. D98(9), 094030 (2018).  DOI 10.1103/PhysRevD.98.094030
  * (16) T. Hobbs, B.T. Wang, P.M. Nadolsky, F.I. Olness, Phys. Rev. D 100(9), 094040 (2019).  DOI 10.1103/PhysRevD.100.094040
  * (17) S. Dulat, T.J. Hou, J. Gao, M. Guzzi, J. Huston, P. Nadolsky, J. Pumplin, C. Schmidt, D. Stump, C.P. Yuan, Phys. Rev. D93(3), 033006 (2016).  DOI 10.1103/PhysRevD.93.033006
  * (18) Y. Fu, S. Yang, M. Liu, L. Han, T.J. Hou, C. Schmidt, C. Wang, C.P. Yuan, Chin. Phys. C 45(5), 053001 (2021).  DOI 10.1088/1674-1137/abe36d
  * (19) R. Abdul Khalek, S. Bailey, J. Gao, L. Harland-Lang, J. Rojo, Eur. Phys. J. C 78(11), 962 (2018).  DOI 10.1140/epjc/s10052-018-6448-y
  * (20) T.J. Hou, Z. Yu, S. Dulat, C. Schmidt, C.P. Yuan, Phys. Rev. D 100(11), 114024 (2019).  DOI 10.1103/PhysRevD.100.114024
  * (21) J. Pumplin, D. Stump, R. Brock, D. Casey, J. Huston, J. Kalk, H.L. Lai, W.K. Tung, Phys. Rev. D65, 014013 (2001).  DOI 10.1103/PhysRevD.65.014013
  * (22) M. Arneodo, et al., Nucl. Phys. B483, 3 (1997).  DOI 10.1016/S0550-3213(96)00538-X
  * (23) H. Abramowicz, et al., Eur. Phys. J. C 75(12), 580 (2015).  DOI 10.1140/epjc/s10052-015-3710-4
  * (24) M. Virchaux, A. Milsztajn, Phys. Lett. B 274, 221 (1992).  DOI 10.1016/0370-2693(92)90527-B
  * (25) M. Virchaux, A. Milsztajn, Physics Letters B 274(2), 221 (1992)
  * (26) A.C. Benvenuti, et al., Phys. Lett. B223, 485 (1989).  DOI 10.1016/0370-2693(89)91637-7
  * (27) A.C. Benvenuti, et al., Phys. Lett. B237, 592 (1990).  DOI 10.1016/0370-2693(90)91231-Y
  * (28) W. Seligman, et al., Phys. Rev. Lett. 79, 1213 (1997).  DOI 10.1103/PhysRevLett.79.1213
  * (29) R.J. Holt, C.D. Roberts, Rev. Mod. Phys. 82, 2991 (2010).  DOI 10.1103/RevModPhys.82.2991
  * (30) C.D. Roberts, R.J. Holt, S.M. Schmidt, Phys. Lett. B 727, 249 (2013).  DOI 10.1016/j.physletb.2013.09.038
  * (31) H.W. Lin, et al., Prog. Part. Nucl. Phys. 100, 107 (2018).  DOI 10.1016/j.ppnp.2018.01.007
  * (32) M. Constantinou, et al.  Parton distributions and lattice QCD calculations: toward 3D structure (2020)
  * (33) A. Courtoy, P.M. Nadolsky, Phys. Rev. D 103(5), 054029 (2021).  DOI 10.1103/PhysRevD.103.054029
  * (34) S.J. Brodsky, G.R. Farrar, Phys. Rev. Lett. 31, 1153 (1973).  DOI 10.1103/PhysRevLett.31.1153
  * (35) S.J. Brodsky, G.R. Farrar, Phys. Rev. D 11, 1309 (1975).  DOI 10.1103/PhysRevD.11.1309
  * (36) G.R. Farrar, D.R. Jackson, Phys. Rev. Lett. 35, 1416 (1975).  DOI 10.1103/PhysRevLett.35.1416
  * (37) R.D. Ball, E.R. Nocera, R.L. Pearson, Eur. Phys. J. C 81(1), 37 (2021).  DOI 10.1140/epjc/s10052-020-08826-7
  * (38) A.D. Martin, W.J. Stirling, R.S. Thorne, G. Watt, Eur. Phys. J. C63, 189 (2009).  DOI 10.1140/epjc/s10052-009-1072-5
  * (39) K.J. Eskola, P. Paakkinen, H. Paukkunen, C.A. Salgado, Eur. Phys. J. C77(3), 163 (2017).  DOI 10.1140/epjc/s10052-017-4725-9
  * (40) K. Kovařík, et al., Phys. Rev. D93(8), 085037 (2016).  DOI 10.1103/PhysRevD.93.085037
  * (41) D. de Florian, R. Sassot, P. Zurita, M. Stratmann, Phys. Rev. D85, 074028 (2012).  DOI 10.1103/PhysRevD.85.074028
  * (42) M. Hirai, S. Kumano, T.H. Nagai, Phys. Rev. C76, 065207 (2007).  DOI 10.1103/PhysRevC.76.065207
  * (43) H. Khanpour, S. Atashbar Tehrani, Phys. Rev. D93(1), 014026 (2016).  DOI 10.1103/PhysRevD.93.014026
  * (44) R. Abdul Khalek, J.J. Ethier, J. Rojo, Eur. Phys. J. C79(6), 471 (2019).  DOI 10.1140/epjc/s10052-019-6983-1
  * (45) S.I. Alekhin, S.A. Kulagin, R. Petti, Phys. Rev. D 96(5), 054005 (2017).  DOI 10.1103/PhysRevD.96.054005
  * (46) S.A. Kulagin, R. Petti, Nucl. Phys. A 765, 126 (2006).  DOI 10.1016/j.nuclphysa.2005.10.011
  * (47) D.A. Mason, (2006).  DOI 10.2172/879078.  URL https://www.osti.gov/biblio/879078
  * (48) M. Goncharov, et al., Phys. Rev. D 64, 112006 (2001).  DOI 10.1103/PhysRevD.64.112006
  * (49) A. Kayis-Topaksu, G. Önengüt, R. van Dantzig, M. de Jong, R. Oldeman, M. Güler, S. Kama, U. Köse, M. Serin-Zeyrek, P. Tolun, et al., Nuclear Physics B 798(1-2), 1–16 (2008).  DOI 10.1016/j.nuclphysb.2008.02.013.  URL http://dx.doi.org/10.1016/j.nuclphysb.2008.02.013
  * (50) L.L. Frankfurt, M.I. Strikman, Phys. Rept. 160, 235 (1988).  DOI 10.1016/0370-1573(88)90179-2
  * (51) M. Arneodo, Phys. Rept. 240, 301 (1994).  DOI 10.1016/0370-1573(94)90048-5
  * (52) B.Z. Kopeliovich, J.G. Morfin, I. Schmidt, Prog. Part. Nucl. Phys. 68, 314 (2013).  DOI 10.1016/j.ppnp.2012.09.004
  * (53) S. Malace, D. Gaskell, D.W. Higinbotham, I. Cloet, Int. J. Mod. Phys. E 23(08), 1430013 (2014).  DOI 10.1142/S0218301314300136
  * (54) J.P. Berge, et al., Z. Phys. C49, 187 (1991).  DOI 10.1007/BF01555493
  * (55) G. Moreno, et al., Phys. Rev. D 43, 2815 (1991).  DOI 10.1103/PhysRevD.43.2815
  * (56) E. Rondio, Nucl. Phys. A 553, 615C (1993).  DOI 10.1016/0375-9474(93)90668-N
  * (57) R.D. Ball, E.R. Nocera, R.L. Pearson, Eur. Phys. J. C 79(3), 282 (2019).  DOI 10.1140/epjc/s10052-019-6793-5
  * (58) S.A. Kulagin, Nucl. Phys. A 500, 653 (1989).  DOI 10.1016/0375-9474(89)90233-9
  * (59) J.F. Owens, A. Accardi, W. Melnitchouk, Phys. Rev. D 87(9), 094012 (2013).  DOI 10.1103/PhysRevD.87.094012
  * (60) W. Melnitchouk, A.W. Schreiber, A.W. Thomas, Phys. Lett. B 335, 11 (1994).  DOI 10.1016/0370-2693(94)91550-4
  * (61) R.K. Ellis, W. Furmanski, R. Petronzio, Nucl. Phys. B212, 29 (1983).  DOI 10.1016/0550-3213(83)90597-7
  * (62) J.w. Qiu, G.F. Sterman, Nucl. Phys. B353, 105 (1991).  DOI 10.1016/0550-3213(91)90503-P
  * (63) H. Georgi, H.D. Politzer, Phys. Rev. D 14, 1829 (1976).  DOI 10.1103/PhysRevD.14.1829
  * (64) A. De Rujula, H. Georgi, H.D. Politzer, Annals Phys. 103, 315 (1977).  DOI 10.1016/S0003-4916(97)90003-8
  * (65) I. Schienbein, et al., J. Phys. G 35, 053101 (2008).  DOI 10.1088/0954-3899/35/5/053101
  * (66) L.T. Brady, A. Accardi, T.J. Hobbs, W. Melnitchouk, Phys. Rev. D 84, 074008 (2011).  DOI 10.1103/PhysRevD.84.074008.  [Erratum: Phys.Rev.D 85, 039902 (2012)]
  * (67) V. Abazov, et al., Phys. Lett. B 666, 435 (2008).  DOI 10.1016/j.physletb.2008.06.076
  * (68) T. Aaltonen, et al., Phys. Rev. D 78, 052006 (2008).  DOI 10.1103/PhysRevD.78.052006.  [Erratum: Phys.Rev.D 79, 119902 (2009)]
  * (69) V. Abazov, et al., Phys. Rev. Lett. 101, 062001 (2008).  DOI 10.1103/PhysRevLett.101.062001
  * (70) F. Aaron, et al., Eur. Phys. J. C 71, 1579 (2011).  DOI 10.1140/epjc/s10052-011-1579-4
  * (71) A. Aktas, et al., Eur. Phys. J. C 40, 349 (2005).  DOI 10.1140/epjc/s2005-02154-8
  * (72) H. Abramowicz, et al., Eur. Phys. J. C 73(2), 2311 (2013).  DOI 10.1140/epjc/s10052-013-2311-3
  * (73) S. Malace, et al., Phys. Rev. C 80, 035207 (2009).  DOI 10.1103/PhysRevC.80.035207
  * (74) L. Whitlow, E. Riordan, S. Dasu, S. Rock, A. Bodek, Phys. Lett. B 282, 475 (1992).  DOI 10.1016/0370-2693(92)90672-Q
  * (75) A. Airapetian, N. Akopov, Z. Akopov, E. Aschenauer, W. Augustyniak, R. Avakian, A. Avetissian, E. Avetisyan, S. Belostotski, N. Bianchi, et al., Journal of High Energy Physics 2011(5), 126 (2011)
  * (76) E. Rondio, Nucl. Phys. B Proc. Suppl. 54A, 139 (1997).  DOI 10.1016/S0920-5632(97)00029-7
  * (77) S. Tkachenko, N. Baillie, S. Kuhn, J. Zhang, J. Arrington, P. Bosted, S. Bültmann, M. Christy, D. Dutta, R. Ent, et al., Physical Review C 89(4), 045206 (2014)
  * (78) F. Abe, et al., Phys. Rev. Lett. 77, 2616 (1996).  DOI 10.1103/PhysRevLett.77.2616
  * (79) F. Abe, et al., Phys. Rev. Lett. 81, 5754 (1998).  DOI 10.1103/PhysRevLett.81.5754
  * (80) D. Acosta, et al., Phys. Rev. D 71, 051104 (2005).  DOI 10.1103/PhysRevD.71.051104
  * (81) T.A. Aaltonen, et al., Phys. Lett. B 692, 232 (2010).  DOI 10.1016/j.physletb.2010.06.043
  * (82) V.M. Abazov, et al., Phys. Rev. D 91(3), 032007 (2015).  DOI 10.1103/PhysRevD.91.032007.  [Erratum: Phys.Rev.D 91, 079901 (2015)]
  * (83) V. Abazov, et al., Phys. Rev. D 77, 011106 (2008).  DOI 10.1103/PhysRevD.77.011106
  * (84) V.M. Abazov, et al., Phys. Rev. D 76, 012003 (2007).  DOI 10.1103/PhysRevD.76.012003
  * (85) T. Aaltonen, et al., Phys. Rev. Lett. 102, 181801 (2009).  DOI 10.1103/PhysRevLett.102.181801
  * (86) V.M. Abazov, et al., Phys. Rev. Lett. 112(15), 151803 (2014).  DOI 10.1103/PhysRevLett.112.151803.  [Erratum: Phys.Rev.Lett. 114, 049901 (2015)]
  * (87) V.M. Abazov, et al., Phys. Rev. D 88, 091102 (2013).  DOI 10.1103/PhysRevD.88.091102
  * (88) R. Aaij, et al., JHEP 08, 039 (2015).  DOI 10.1007/JHEP08(2015)039
  * (89) R. Aaij, et al., JHEP 05, 109 (2015).  DOI 10.1007/JHEP05(2015)109
  * (90) V. Khachatryan, et al., Eur. Phys. J. C 76(8), 469 (2016).  DOI 10.1140/epjc/s10052-016-4293-4
  * (91) R. Aaij, et al., JHEP 01, 155 (2016).  DOI 10.1007/JHEP01(2016)155
  * (92) G. Aad, et al., Eur. Phys. J. C 76(5), 291 (2016).  DOI 10.1140/epjc/s10052-016-4070-4
  * (93) S. Chatrchyan, et al., Phys. Rev. D 90(3), 032004 (2014).  DOI 10.1103/PhysRevD.90.032004
  * (94) S. Chatrchyan, et al., Phys. Rev. Lett. 109, 111806 (2012).  DOI 10.1103/PhysRevLett.109.111806
  * (95) G. Aad, et al., Phys. Rev. D85, 072004 (2012).  DOI 10.1103/PhysRevD.85.072004
  * (96) R. Towell, et al., Phys. Rev. D 64, 052002 (2001).  DOI 10.1103/PhysRevD.64.052002
  * (97) https://p25ext.lanl.gov/e866
  * (98) J.C. Webb, et al. (2003)
  * (99) U.K. Yang, et al., Phys. Rev. Lett. 86, 2742 (2001).  DOI 10.1103/PhysRevLett.86.2742
  * (100) A.M. Sirunyan, et al., Eur. Phys. J. C 77(7), 459 (2017).  DOI 10.1140/epjc/s10052-017-4984-5
  * (101) G. Aad, et al., Eur. Phys. J. C 76(10), 538 (2016).  DOI 10.1140/epjc/s10052-016-4366-4
  * (102) S. Chatrchyan, et al., Phys. Rev. D 90(7), 072006 (2014).  DOI 10.1103/PhysRevD.90.072006
  * (103) G. Aad, et al., JHEP 02, 153 (2015).  DOI 10.1007/JHEP02(2015)153.  [Erratum: JHEP 09, 141 (2015)]
  * (104) V. Khachatryan, et al., JHEP 03, 156 (2017).  DOI 10.1007/JHEP03(2017)156
  * (105) H.L. Lai, J. Huston, Z. Li, P. Nadolsky, J. Pumplin, D. Stump, C.P. Yuan, Phys. Rev. D82, 054021 (2010).  DOI 10.1103/PhysRevD.82.054021
  * (106) J. Pumplin, D.R. Stump, J. Huston, H.L. Lai, P.M. Nadolsky, W.K. Tung, JHEP 07, 012 (2002).  DOI 10.1088/1126-6708/2002/07/012
  * (107) S. Amoroso, et al., in _11th Les Houches Workshop on Physics at TeV Colliders: PhysTeV Les Houches_ (2020)
  * (108) CT18PDFs.  https://hep.pa.msu.edu/cteq/public/ct18.html and https://tinyurl.com/ct18pdfs-1
  * (109) K. Kovarik, I. Schienbein, F.I. Olness, J.Y. Yu, C. Keppel, J.G. Morfin, J.F. Owens, T. Stavreva, Phys. Rev. Lett. 106, 122301 (2011).  DOI 10.1103/PhysRevLett.106.122301
  * (110) H. Paukkunen, C.A. Salgado, Phys. Rev. Lett. 110(21), 212301 (2013).  DOI 10.1103/PhysRevLett.110.212301
  * (111) CTEQ-JLab, I. Fernando, et al., in preparation, 2021
  * (112) G. Apollinari, I. Béjar Alonso, O. Bruning, P. Fessia, M. Lamont, L. Rossi, L. Tavian, CERN Technical Design Report V. 0.1 4/2017 (2017).  DOI 10.23731/CYRM-2017-004
  * (113) A. Accardi, et al., Eur. Phys. J. A52(9), 268 (2016).  DOI 10.1140/epja/i2016-16268-9
  * (114) R. Acciarri, et al., (2015).  DOI arXiv:1512.06148[physics.ins-det]
  * (115) R.P. Bickerstaff, M.C. Birse, G.A. Miller, Phys. Rev. Lett. 53, 2532 (1984).  DOI 10.1103/PhysRevLett.53.2532
  * (116) W. Melnitchouk, A.W. Thomas, Phys. Rev. D 47, 3783 (1993).  DOI 10.1103/PhysRevD.47.3783
  * (117) G.A. Miller, Phys. Rev. C 89(4), 045203 (2014).  DOI 10.1103/PhysRevC.89.045203
  * (118) T.J. Hobbs, J.T. Londergan, D.P. Murdock, A.W. Thomas, Phys. Lett. B 698, 123 (2011).  DOI 10.1016/j.physletb.2011.02.040
  * (119) H.L. Lai, M. Guzzi, J. Huston, Z. Li, P.M. Nadolsky, J. Pumplin, C.P. Yuan, Phys. Rev. D82, 074024 (2010).  DOI 10.1103/PhysRevD.82.074024