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NNLO nuclea pa on dis ibu ion unc ions wi h elec oweak-boson p oduc ion da a
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Helenius, Ilkka; Wal , Ma ina; Vogelsang, We ne
Helenius, I., Wal , M., & Vogelsang, W. (2022). NNLO nuclea pa on dis ibu ion unc ions wi h
elec oweak-boson p oduc ion da a om he LHC. Physical Re iew D, 105(9), A icle 094031.
h ps://doi.o g/10.1103/PhysRe D.105.094031
2022
NNLO nuclea pa on dis ibu ion unc ions wi h elec oweak-boson
p oduc ion da a om he LHC
Ilkka Helenius ,1,2,* Ma ina Wal ,3,‡and We ne Vogelsang3,†
1Uni e si y o Jy askyla, Depa men o Physics, P.O. Box 35, FI-40014 Uni e si y o Jy askyla, Finland
2Helsinki Ins i u e o Physics, P.O. Box 64, FI-00014 Uni e si y o Helsinki, Finland
3Ins i u e o Theo e ical Physics, Uni e si y o Tübingen,
Au de Mo gens elle 14, 72076 Tübingen, Ge many
(Recei ed 14 Janua y 2022; accep ed 11 Ap il 2022; published 24 May 2022)
We p esen new se s o nuclea pa on dis ibu ion unc ions (nPDFs) a nex - o-leading o de and
nex - o-nex - o-leading o de in pe u ba i e QCD. Ou analyses a e based on deeply inelas ic sca e ing
da a wi h cha ged-lep on and neu ino beams on nuclea a ge s, and expe imen al da a om
measu emen s o W;Zboson p oduc ion in p þPb collisions a he LHC. In addi ion, a se o p o on
baseline PDFs is i ed wi hin he same amewo k and wi h he same heo e ical assump ions.
The esul s o ou global QCD analysis a e compa ed o exis ing nPDF se s and o he p e ious nPDF
se TUJU19, which was based on deep-inelas ic sca e ing da a only. Ou wo k is pe o med using an
open-sou ce ool,
X
F
ITTER
, and he equi ed ex ensions o he code a e discussed as well. We ind
good ag eemen wi h he da a included in he i and a lowe alue o χ2=Ndp when pe o ming he
i a nex - o-nex - o-leading o de . We apply he esul ing nuclea PDFs o elec oweak boson
p oduc ion in Pb þPb collisions a he LHC and compa e he esul s o he mos ecen da a om
ATLAS and CMS.
DOI: 10.1103/PhysRe D.105.094031
I. INTRODUCTION
In he amewo k o collinea ac o iza ion [1], he
di e en ial c oss sec ion o a gi en p ocess in had onic
collisions can be calcula ed in e ms o con olu ions o
p ocess-independen pa on dis ibu ion unc ions (PDFs)
and p ocess-speci ic pe u ba i ely calculable coe icien
unc ions. By combining da a o inclusi e deep-inelas ic
sca e ing (DIS) om HERA, ixed- a ge expe imen s, o
neu ino sca e ing, wi h a ious LHC da a, including, e.g.,
(di)je , op-pai and elec oweak-boson p oduc ion, p o on
PDFs ha e by now eached accu acies a he le el o a ew
pe cen o be e o e wide kinema ic egions. This is i al
o p ecision s udies o p ocesses in he S anda d Model
and beyond (see [2] and e e ences he ein).
One may also apply he same ac o iza ion o p ocesses
in ol ing high-ene gy nuclei, such as e þAo pþA. This
p o ides in o ma ion on nuclea pa on dis ibu ion unc-
ions (nPDFs). Being undamen al p ope ies o a omic
nuclei, nPDFs a e o much impo ance o ou gene al
unde s anding o he s ong in e ac ions. A he same ime,
ex ac ing “cold-nuclea ma e ”e ec s om collisions
wi h a small p ojec ile and a a ge nucleus also p o ides
a baseline o qua k-gluon plasma s udies.
The backbone o analyses o nPDFs ha e been ixed-
a ge DIS e þA da a, o en accompanied by D ell-Yan
(DY) dilep on p oduc ion da a. In addi ion o DIS wi h
cha ged lep on beams, also a good amoun o neu ino DIS
da a a e nowadays a ailable ha p o ide some addi ional
sensi i i y o he la o dependence o nuclea e ec s. In
ecen yea s, da a om p þPb collisions om he LHC
ha e ul illed he p omise o ex ending he kinema ic each
and u he cons aining he la o dependence o he
nPDFs. P ime examples a e he D-meson p oduc ion a
o wa d apidi ies measu ed by LHCb [3] which is pa -
icula ly sensi i e o small-xgluon nPDFs [4], and Wand
Zboson p oduc ion da a om ATLAS [5] and CMS [6,7],
which a e p ima ily sensi i e o di e en qua k- la o
combina ions a mode a e alues o xand p o ide po en ial
cons ain s o la o dependence [8] and also o s ange
qua ks and gluons [9]. Fu he mo e, dije p oduc ion [10]
can o e u he cons ain s on gluon nPDFs in he
in e media e x egion [11]. O he possible obse ables
include inclusi e di ec -pho on and pion p oduc ion in
*[email p o ec ed]
†we ne [email p o ec ed]
‡P esen add ess: HQS Quan um Simula ions GmbH, Haid-
und-Neu-S aße 7, 76131 Ka ls uhe, Ge many.
Published by he Ame ican Physical Socie y unde he e ms o
he C ea i e Commons A ibu ion 4.0 In e na ional license.
Fu he dis ibu ion o his wo k mus main ain a ibu ion o
he au ho (s) and he published a icle’s i le, jou nal ci a ion,
and DOI. Funded by SCOAP3.
PHYSICAL REVIEW D 105, 094031 (2022)
2470-0010=2022=105(9)=094031(19) 094031-1 Published by he Ame ican Physical Socie y
pþPb collisions a he LHC. The o me has been ecen ly
measu ed by ATLAS [12] and o he la e he e a e
al eady se e al da ase s a ailable om ALICE [13–15].
Nume ous nPDF analyses a e a ailable by now which
can be classi ied in e ms o he da a used in he analysis
and o hei pe u ba i e p ecision. Ea ly “global”analyses
include he EPS09 [16] and DSSZ [17] se s, which we e
based on cha ged-lep on and neu ino DIS and DY da a
om ixed- a ge expe imen s, and inclusi e pion p oduc-
ion da a om d þAu collisions a RHIC. The EPPS16
analysis [18] was he i s o include also da a om p þPb
collisions a he LHC by analyzing un I da a o dije s and
Wand Zboson p oduc ion. The o iginal nCTEQ15
analysis [19] did no include any LHC da a, bu i has
ecen ly been ex ended o include, e.g., un II Wand Z
da a om p þPb collisions (nCTEQ15wz analysis [9]).
Simila da ase s ha e been conside ed also by he NNPDF
collabo a ion in hei mos ecen analysis nNNPDF2.0
[20]. The nCTEQ15 analysis has been ecen ly upda ed o
con ain also single-inclusi e had on p oduc ion [21] and
also o he ac i e g oups ha e ecen ly p epa ed new
analyses. The EPPS21 [22] and nNNPDF3.0 [23] analyses
bo h include a signi ican amoun o new LHC da a
including, e.g., dije , Wand Zboson and inclusi e D-
meson p oduc ion om he LHCb. All analyses men ioned
so a ha e been pe o med a he nex - o-leading-o de
(NLO) o pe u ba i e QCD. The e a e a ew ecen
analyses ha ha e been pe o med a nex - o-nex - o-
leading o de (NNLO): nNNPDF1.0 [24], TUJU19 [25],
and KSASG20 [26]. So a , hese ha e only conside ed
cha ged-lep on and neu ino DIS, and ixed- a ge DY da a.
In his wo k we p esen new nuclea PDF analyses a NLO
and NNLO, using he same amewo k as in ou p e ious
analysis, TUJU19, bu now including in addi ion o neu al-
cu en DIS wi h a lep on beam and cha ged-cu en
neu ino DIS da a also new elec oweak (EW)-boson p o-
duc ion da a om he LHC. This is he i s ime whe e LHC
da a a e employed in a ull NNLO analysis o nuclea PDFs.
We include he LHC da a also o a p o on “baseline” i . In
his way, we ob ain a ully consis en way o compu ing
c oss sec ions o p þAcollisions a NLO o NNLO. The
esul ing nPDFs may also be eadily used o u he c oss
sec ion calcula ions a NNLO in nuclea collisions. As an
applica ion we s udy EW-boson p oduc ion in Pb þPb
collisions and compa e ou esul s o ecen ATLAS and
CMS da a [27–29] which, con a y o expec a ions, ha e
p e iously been ound o be di icul o desc ibe wi hin a
ac o iza ion-based NNLO calcula ion [30].
Ou pape is o ganized as ollows. We desc ibe he
heo e ical amewo k o ou analysis in Sec. II, hen
discuss he analysis p ocedu e in Sec. III and he selec ion
o expe imen al da a in Sec. III C. The esul s o he
analysis a e p esen ed in Sec. IV. We discuss he applica ion
o Pb þPb collisions a he LHC in Sec. V. Finally, we
summa ize ou wo k in Sec. VI, whe e also an ou look
owa d u u e de elopmen s is gi en.
II. THEORETICAL FRAMEWORK
The heo e ical amewo k is e y simila o ha adop ed
in ou ea lie analysis, TUJU19 [25], which includes also a
de ailed desc ip ion o neu al-cu en (NC) and cha ged-
cu en (CC) DIS p ocesses. He e we gi e an o e iew o
he calcula ional amewo k ha we ha e se up o use he
new LHC da a.
A. D ell-Yan and W,Zboson p oduc ion p ocesses
As new cons ain s we include LHC da a o inclusi e
had op oduc ion o elec oweak (EW) bosons, Wand
Z=γ. O en such p ocesses a e e e ed o as DY p ocesses,
o iginally ela ing o eac ions whe e wo had ons collide
and o m a highly i ual pho on ha decays o a lep on pai
[31]. The signa u e o such an e en is a lep on pai (l¯
lwi h
l¼e,μ,τ) o he same la o wi h a la ge in a ian mass. In
leading o de (LO) such a p ocess can ake place only by
annihila ion o a qua k-an iqua k pai o he same la o ,
q¯
q→γ→l¯
l: ð1Þ
A high enough ene gies, simila sca e ing may o m
also o he EW bosons, namely Zand W. As he o me
has he same quan um numbe s as a pho on, he p oduc ion
and decay channels a e he same as in adi ional DY. In he
la e case, howe e , he LO p ocess equi es, e.g., a u¯
d
ini ial s a e and he lep onic decay channel o include
p oduc ion o a cha ged lep on and a co esponding
neu ino,
q¯
q→Z→l¯
l; and qi¯
qj→W→lνl:ð2Þ
As he (an i)neu inos om he W−ðþÞ decays canno be
di ec ly measu ed in he de ec o s and only he momen um
o he cha ged lep on is known, some u he kinema ical
cu s a e equi ed o inc ease he sensi i i y o he signal
p ocess.
A ac o iza ion heo em o DY has been explici ly
p o ed o all o de s in pe u ba ion heo y [32–34].
In o de o la e be able o assess he impac o DY (o
EW-boson p oduc ion) da a on ou analysis, i is use ul o
w i e down he double di e en ial DY c oss sec ion a
LO [35,36]:
d2σ
dM2dy¼4πα2τ
3NCM4X
q
e2
q½qðx1Þ¯
qðx2Þþ¯
qðx1Þqðx2Þ;ð3Þ
whe e NCis he numbe o colo s, α he ine s uc u e
cons an , τ¼M2=s (wi h ffiffiffi
s
p he cen e -o -mass ene gy),
and we ha e in oduced he a iables
HELENIUS, WALT, and VOGELSANG PHYS. REV. D 105, 094031 (2022)
094031-2
M2¼ðp1þp2Þ2≡ˆ
s¼x1x2sðin a ian massÞ;
y¼1
2logx1
x2ð apidi yÞ:ð4Þ
He e p1;2a e momen a o he incoming pa ons ca ying
momen um ac ion x1;2o he o al had on momen um. In
e ms o M2and ywe ha e (again, o LO):
x1¼M
ffiffiffi
s
pexpðyÞand x2¼M
ffiffiffi
s
pexpð−yÞ:ð5Þ
These ela ions can be used o es ima e he sensi i i y o
di e en egions o momen um ac ion when s udying EW
boson p oduc ion a speci ic kinema ics. O en in he
expe imen al analysis, howe e , pseudo apidi y ηis used
ins ead o yas he o me does no equi e in o ma ion o
he mass o he gi en sys em.
In Eq. (3),qðx1Þand ¯
qðx2Þa e he qua k and an iqua k
PDFs, whose scale dependence we ha e omi ed.
Compa ing o he LO DIS c oss sec ion o mula, we no ice
ha in DY we always ha e a combina ion o qua k and
an iqua k PDFs. Thus we expec DY p ocesses o p o ide
mo e in o ma ion on he sea qua k densi ies.
The well-known ac o ized all-o de exp ession o he
D ell-Yan c oss sec ion is based on con olu ions o he
PDFs wi h pe u ba i e ha d-sca e ing unc ions. Beyond
LO, he e a e also con ibu ions om p ocesses o he han
q¯
qð0Þ annihila ion. Fo example, s a ing om NLO, also
con ibu ions om gluon-(an i)qua k ini ial s a es a ise, and
a NNLO and beyond gluon-gluon sca e ing pa icipa es.
Deno ing he pa onic ha d-sca e ing unc ion o a gi en
pa onic channel ab →EW boson þXby ωab, we ha e he
pe u ba i e expansion
ωab ¼ωð0Þ
ab þαs
πωð1Þ
ab þαs
π2
ωð2Þ
ab þOðα3
sÞ;ð6Þ
whe e αsis he s ong coupling e alua ed a some eno m-
aliza ion scale μR. The NLO coe icien unc ions ωð1Þ
ab ha e
been known o a long ime [37]. The NNLO co ec ions
ωð2Þ
ab a e also ully a ailable [38–43], and hei inclusion is a
new ea u e o ou TUJU21 analysis.
B. Inpu pa ame iza ion
Apa om he pe u ba i e ha d-sca e ing unc ions,
ano he key ing edien in a global analysis o collinea
PDFs is hei Dokshi ze –G ibo –Lipa o –Al a elli–Pa isi
(DGLAP) e olu ion [44–47], which desc ibes how he
PDFs depend on he ac o iza ion scale. The pe u ba i e
o de o he spli ing ke nels o be used in he e olu ion
equa ions needs o be chosen in acco dance wi h he o de
in he ha d-sca e ing unc ions. As only he pe u ba i e
scale e olu ion o he PDFs is gi en by he e olu ion
equa ions, a nonpe u ba i e inpu a some ini ial scale is
equi ed o ob ain a PDF se . In his analysis he baseline
pa on dis ibu ions o a p o on a e pa ame ized simila ly
as in [25]
x p
iðx; Q2
0Þ¼c0xc1ð1−xÞc2ð1þc3xþc4x2Þ;ð7Þ
whe e he index i uns o e pa on la o i¼
g; d ;u
;¯
u; ¯
d; ¯
s, wi h he subsc ip “ ” e e ing o he
up o down alence-qua k con ibu ion. We ha e kep he
la o dependence in he alence sec o , bu o he sea
qua ks we had o assume ¯
u¼¯
d¼¯
s¼sas i u ned
ou ha wi hin he applied da a and he adop ed amewo k
i is di icul o ha e a con e ged i wi hou such a
condi ion. We de ine he inpu pa ame iza ion a he
cha m mass h eshold, Q2
0¼m2
c¼1.69 GeV2.Wedo
no include any in insic cha m con en a his scale and
apply he FONLL-A(C) gene al-mass a iable- la o -
numbe -scheme [48,49] o calcula e he hea y-qua k c oss
sec ions a Q>Q
0in ou NLO (NNLO) analysis.
To ob ain he PDFs o p o ons bound in a nucleus, we
add dependence on he nuclea mass numbe A o he
pa ame e s ciby epa ame izing hem as
ck→ckðAÞ¼ck;0þck;1ð1−A−ck;2Þ;ð8Þ
whe e k¼0;…;4. A simila o m has been used also in he
nCTEQ15 analysis [9,19]. I is wo h poin ing ou ha wi h
A¼1 he A-dependen igh -hand pa o Eq. (8) anishes
and he ee p o on PDFs a e eco e ed when inden i ying
ck;0¼ck. In o de o ob ain he ull nuclea PDF, we need
o add also he con ibu ion om neu ons in he nucleus.
The PDFs o neu ons a e no i ed sepa a ely bu a e
de e mined om he p o on PDFs based on isospin sym-
me y, gi ing un=A ¼dp=A and dn=A ¼up=A, and likewise
o he ligh an iqua ks. Then, an a e age PDF o a nucleon
bound in a nucleus wi h Zp o ons and A−Zneu ons is
ob ained as
N=A
iðx; Q2Þ¼Z· p=A
iþðA−ZÞ· n=A
i
A:ð9Þ
III. ANALYSIS PROCEDURE
A. Minimiza ion p ocedu e
A he hea o PDF i ing is he minimiza ion o he
di e ence be ween he expe imen al da a and he calcula ed
c oss sec ions, p o iding he op imal PDF se wi hin he
adop ed amewo k. In his wo k, as well as in ou p e ious
analysis [25], his is done by minimizing χ2de ined as
χ2ðm;bÞ¼X
i
½mi−Σαγi
αμibα−μi2
ðδi;s a ffiffiffiffiffiffiffiffiffi
μimi
pÞ2þðδi;unco miÞ2
þX
α
b2
α:ð10Þ
NNLO NUCLEAR PARTON DISTRIBUTION FUNCTIONS WITH …PHYS. REV. D 105, 094031 (2022)
094031-3
He e, μiis he alue o he measu ed da a poin o a gi en
obse able, whe eas miis he ac ual heo e ical alue
calcula ed using DGLAP-e ol ed PDFs wi h gi en pa am-
e e s ckg. The unce ain ies a e ep esen ed by he δi;s a and
δi;unco , which a e he ela i e s a is ical and unco ela ed
sys ema ic unce ain ies, espec i ely, as well as by he γi
αμi,
which a e he co ela ed e o s. Fu he mo e, he bαa e he
so-called nuisance pa ame e s ha a e de e mined du ing
he i ing. To accoun o he co ela ed sys ema ic unce -
ain ies o each da ase , we allow o shi s o he calcula ed
c oss sec ion wi hin he quo ed unce ain y, penalizing such
shi s by he addi ional b2
αcon ibu ions o χ2.
The minimiza ion o χ2, Eq. (10), p o ides a cen al se
o PDFs wi h he pa ame e alues allowing he bes
desc ip ion o he used da a. Since he expe imen al da a
always con ain se e al unce ain ies, as lis ed abo e, a
sepa a e e o analysis needs o be pe o med o s udy how
hese unce ain ies p opaga e in o he i ed PDFs. In his
QCD analysis he Hessian me hod [50,51] is used o he
analysis o he unce ain ies. The Hessian e o analysis is
pe o med assuming a quad a ic expansion o he unc ion
χ2¼χ2
0þΔχ2a ound i s global minimum. He e, χ2
0is he
alue o he unc ion a he global minimum (wi h he bes -
i pa ame e s k0g) and Δχ2is he displacemen om he
minimum [50,51]. Du ing he pe o med e o analysis Δχ2
de ines he ole ance c i e ion de e mining he allowed
g ow h o χ2. As a gued in ou p e ious wo k [25], o
he p o on baseline wi h 13 ee i pa ame e s we selec
Δχ2¼20 and o he nuclea PDF e o analysis we choose
Δχ2¼50 o ou 16 ee pa ame e s.
FIG. 1. Schema ic iew o he high-le el
X
F
ITTER
unc ionali ies as ele an o D ell-Yan and W,Zboson p oduc ion p ocesses,
wi h he newly implemen ed con olu ion ou ine in
X
F
ITTER
as used o he nPDF i TUJU21.
X
F
ITTER
logo c edi ed om [60].
HELENIUS, WALT, and VOGELSANG PHYS. REV. D 105, 094031 (2022)
094031-4

B. The i ing amewo k
Ou global analyses o he baseline p o on and nuclea
PDFs a e pe o med wi h he
X
F
ITTER
[52,53] ool. The
main goal o he
X
F
ITTER
p ojec is o p o ide an open-
sou ce ool o i p o on PDFs wi h a ied heo e ical
assump ions. In o de o pe o m a nuclea PDF analysis
se e al modi ica ions o he code we e pe o med in he
con ex o he wo k p esen ed in Re . [25].
When going om DIS o collisions be ween wo had ons
whe e c oss sec ion calcula ions in ol e con olu ions o
mo e han one PDF, he equi ed calcula ions become
compu a ionally expensi e, and e en mo e so when inc eas-
ing he pe u ba i e p ecision o NNLO. The e o e, o
implemen da a o DY and EW-boson p oduc ion beyond
LO, i is necessa y o p epa e as in e pola ion g ids allowing
o e icien compa isons wi h he da a when he PDF
pa ame e s a e i e a ed. S anda d ools o handle such
con olu ions wi h in e pola ion g ids, such as APPL
GRID
[54], can be linked wi h
X
F
ITTER
, acili a ing as c oss
sec ion calcula ions using he p ecompu ed g ids. To p epa e
he ac ual g ids used in he i ing, a code capable o
calcula ing he gi en p oduc ion c oss sec ion needs o be
linked wi h an in e pola ion code. Depending on he a ail-
able da a iles, u he p ocessing in
X
F
ITTER
equi es g ids in
a ROOT o ma [55]. I is also possible o use K ac o s when
going om NLO o NNLO c oss sec ions o educe he
compu ing e o . We ha e used such K ac o s in ou p o on
baseline i in cases whe e hey we e a ailable in
X
F
ITTER
o a gi en da ase . The se ings and PDFs used o calcula e
he K ac o s can be ound om he e e ences p o ided in
Table II.
The schema ic o e iew o he i ing ou ine and he
equi ed ool se (
X
F
ITTER
and all o he modules) as applied
o he nuclea PDF i is shown in Fig. 1. Fo his, a new
con olu ion ou ine had o be implemen ed in
X
F
ITTER
whe e one i s needs o p epa e as in e pola ion g ids by
calcula ing he obse ables in he ele an kinema ic egion,
using an exis ing PDF se , e.g., in he LHAPDF6 o ma
[56]. In his wo k, MCFM8.3 [57–59] has been used o
calcula e he in e pola ion g ids o EW boson p oduc ion
a e sui able modi ica ion o handle asymme ic collisions
as needed o p þPb. The ob ained g ids we e hen used as
he inpu o he op imiza ion ou ine. This se up p o ides
he possibili y o use as in e pola ion g ids in a plain ex
o ma gene a ed wi h MCFM up o o de NNLO, wi hou
elying on app oxima i e K ac o s. Du ing he i ing
p ocedu e he ac ual heo e ical p edic ions we e ob ained
by con olu ing he i ed nPDFs based on upda ed
TABLE I. Summa y o expe imen al DIS da a used o de e -
mine ou p o on PDF baseline. In he las wo columns he χ2
alues a NLO and NNLO ob ained in ou analysis a e p o ided.
Exp. Da ase Yea Re . Ndp χ2NLO χ2NNLO
BCDMS F2p 100 GeV 1996 [65] 83 96.30 93.69
F2p 120 GeV 90 70.54 68.70
F2p 200 GeV 79 91.81 86.32
F2p 280 GeV 75 67.52 69.71
HERA 1þ2NCep 920 2015 [66] 377 459.71 482.23
NCep 820 70 72.91 73.47
NCep 575 254 222.64 231.35
NCep 460 204 218.84 225.68
NCem 159 227.80 232.79
CCep 39 46.59 43.17
CCem 42 60.49 63.60
NMC-97 NCep 1997 [67] 100 117.72 111.31
In o al: 1559
TABLE II. Summa y o expe imen al DYand W;Zboson p oduc ion da a used o de e mine ou p o on PDFs. In
he las wo columns he χ2 alues a NLO and NNLO ob ained in ou analysis a e p o ided.
Exp. Da ase Yea Re . Ndp χ2NLO χ2NNLO
ATLAS DY high mass DY 2013 [61] 13 10.91 11.54
low mass DY 2014 [62] 8 22.86 8.68
ATLAS W;Z Wþlep on η2012 [63] 11 15.77 14.00
W−lep on η11 7.98 8.54
Zy 8 4.10 2.71
Wþlep on y2016 [64] 11 19.57 10.71
W−lep on y11 11.11 11.82
high mass CC Zy 6 7.61 6.05
high mass CF Zy 6 3.92 3.95
low mass Zy 6 41.32 23.84
peak CC Zy 12 45.76 14.40
peak CF Zy 9 21.85 7.55
CMS WWþσ8TeV 2016 [68] 11 10.64 4.81
W−σ8 TeV 11 8.14 9.27
In o al: 134
NNLO NUCLEAR PARTON DISTRIBUTION FUNCTIONS WITH …PHYS. REV. D 105, 094031 (2022)
094031-5
pa ame e s wi h he p ecalcula ed di e en ial c oss sec ions
p o ided in o m o g ids. Fo he con olu ion s ep one
needs o speci y he p o on PDF baseline ha was used o
he gene a ion o he as in e pola ion g ids in MCFM. In
ou case, we a e adop ing ou own p o on PDF baseline
p epa ed in he o m o an LHAPDF6 lib a y.
C. Expe imen al da a
We build upon ou p e ious analysis, TUJU19, including
all he same cha ged-lep on and neu ino DIS da a as
be o e. On op o hese we include da a o DY and EW
boson p oduc ion o bo h ou p o on baseline and he
nuclea PDFs.
The DIS da a used o he p o on baseline i a e
summa ized in Table I, also showing he esul ing χ2
alues a NLO and NNLO ob ained in he analysis. The
newly included expe imen al da a om DY and Wand Z-
boson p oduc ion o he p o on baseline a e lis ed in
Table II. Fo he expe imen al p o on da a used, he as
in e pola ion g ids we e publicly a ailable and he de ails o
g id gene a ion o each p o on PDF da ase can be ound
om he e e ences p o ided in Table I. These da a include
high- and low-mass DY da a om ATLAS a ffiffiffi
s
p¼7TeV
[61,62], EW boson p oduc ion da a om ATLAS a
ffiffiffi
s
p¼7TeV [63], and inc eased-luminosi y da a o hese
om ATLAS a ffiffiffi
s
p¼7TeV [64]. In addi ion, also W
p oduc ion c oss sec ions om CMS a ffiffiffi
s
p¼8TeV ha e
been included. In o al he newly included da a se s consis
o 134 da a poin s.
Table III p o ides a lis o nuclea -DIS da a as also used
in he TUJU19 analysis, bu wi h he χ2 alues ob ained in
his analysis. The inpu da a iles and he as in e pola ion
g ids used o he i ing p ocedu e laid ou in Fig. 1 o
nPDFs ha e been collec ed and p epa ed as pa o his
analysis o he newly included da a poin s summa ized in
Table IV. The added da a include un I measu emen s o Z
boson p oduc ion in p þPb collisions a he LHC by
ATLAS [5] and CMS [7] a ffiffiffiffiffiffiffiffi
sNN
p¼5.02 TeV and he
mo e ecen un II measu emen o Wboson p oduc ion in
pþPb collisions a ffiffiffiffiffiffiffiffi
sNN
p¼8.16 TeV by CMS [69].In
o al hese add 74 da a poin s o he nPDF analysis. In
addi ion o hese, he e a e mo e EW-boson da a a ailable
om he LHC expe imen s. In pa icula , he e a e da a
om ALICE [70] and LHCb [71] ha ex end he kinema ic
each bu ha e only wo da a poin s pe se and su e om
la ge s a is ical unce ain ies. The e a e also un I da a o
Wp oduc ion om CMS [72] a he same kinema ics han
he mo e ecen da a bu wi h signi ican ly la ge unce -
ain ies. The e a e also ixed- a ge DY da a a ailable, e.g.,
om E772 [73], E866 [74], and CLAS [75] expe imen s
ha ha e been used p e iously in simila analyses. As he
g id compu a ions a NNLO a e compu a ionally e y
hea y, we included only he da ase s ha we expec o
p o ide he s onges cons ain s o he nPDFs in he
selec ed se up. In Sec. VB we conside he ecen un II
CMS measu emen o Z=γp oduc ion o which i has
p o en di icul o ob ain χ2=Ndp alues close o uni y in a
NLO QCD analysis [22,23]. As we ha e discussed in he
con ex o he TUJU19 nPDF analysis, some au ho s ha e
ound ension be ween he neu ino- and cha ged-lep on
DIS da a. To check o po en ial ension wi h he new LHC
da a we ha e also pe o med i s wi hou any neu ino-DIS
TABLE III. Summa y o expe imen al DIS da a used o
de e mine he nuclea PDFs. In he las wo columns he χ2
alues a NLO and NNLO ob ained in ou analysis a e p o ided.
Nucleus Exp. Yea Re . Ndp χ2NLO χ2NNLO
D NMC 97 1996 [67] 120 151.61 121.52
EMC 90 1989 [76] 21 24.31 22.89
He=D HERMES 2002 [77] 7 6.79 8.92
NMC 95, e. 1995 [78] 13 10.67 10.42
SLAC E139 1994 [79] 11 6.47 4.42
Li=D NMC 95 1995 [80] 12 9.10 9.00
Be=D SLAC E139 1994 [79] 10 11.58 11.51
Be=C NMC 96 1996 [81] 14 13.56 16.06
C EMC 90 1989 [76] 17 13.41 13.44
C=D FNAL E665 1995 [82] 3 2.00 1.83
SLAC E139 1994 [79] 6 20.69 13.86
EMC 88 1988 [83] 9 3.70 4.22
NMC 95, e. 1995 [78] 13 34.96 19.49
C=Li NMC 95, e. 1995 [78] 10 7.77 10.18
N=D HERMES 2002 [77] 1 0.95 2.08
Al=D SLAC E139 1994 [79] 10 18.49 9.49
Al=C NMC 96 1996 [81] 14 7.29 6.23
Ca EMC 90 1989 [76] 19 13.41 13.44
Ca=D NMC 95, e. 1995 [78] 12 34.75 21.82
FNAL E665 1995 [82] 3 1.84 2.41
SLAC E139 1994 [79] 6 16.74 9.01
Ca=Li NMC 95, e. 1995 [78] 10 1.45 1.33
Ca=C NMC 95, e. 1995 [78] 10 9.35 8.00
NMC 96 1996 [81] 14 8.45 6.42
Fe SLAC E140 1993 [84] 2 0.14 0.04
Fe=D SLAC E139 1994 [79] 14 44.53 32.07
Fe=C NMC 96 1996 [81] 14 11.17 9.93
νFe CDHSW 1991 [85] 464 404.26 358.19
¯
νFe CDHSW 1991 [85] 462 439.53 395.99
Cu=D EMC 88 1988 [83] 9 8.38 5.71
EMC 93 1993 [86] 19 26.38 12.58
K =D HERMES 2002 [77] 1 2.02 2.02
Ag=D SLAC E139 1994 [79] 6 21.37 18.80
Sn=D EMC 88 1988 [83] 8 14.37 13.98
Sn=C NMC 96 1996 [81] 14 6.48 8.52
NMC 96,
Q2dep.
1996 [87] 134 76.13 75.03
Xe=D FNAL E665 1992 [88] 3 1.64 1.34
Au=D SLAC E139 1994 [79] 11 16.89 18.66
Pb=D FNAL E665 1995 [82] 2 8.28 7.72
Pb=C NMC 96 1996 [81] 14 8.32 5.42
νPb CHORUS 2005 [89] 405 259.48 237.85
¯
νPb CHORUS 2005 [89] 405 356.01 352.09
In o al: 2336
HELENIUS, WALT, and VOGELSANG PHYS. REV. D 105, 094031 (2022)
094031-6
da a and ound ha he new da a was equally well desc ibed
as when he neu ino da a we e included. Thus we do no
ind any ension be ween hese da ase s.
IV. RESULTS
A. P o on baseline
Analyses o nuclea PDFs ha e o en been pe o med by
using an exis ing p o on PDF se as a baseline o he
nuclea modi ica ions. In his wo k, howe e , we ha e
i ed he p o on PDFs using he same se up as o he
nuclea PDFs. This ensu es ha all assump ions like sum
ules, pa on la o decomposi ion, e c., as well as all
pa ame e s like coupling cons an s and qua k masses, and
also u he se ings like, e.g., he hea y la o mass
scheme, a e applied in a consis en way. Fu he mo e, his
pa es he way o a u u e combined analysis o p o on and
nuclea PDFs.
In his sec ion he upda ed ee p o on PDF se s in
TUJU21 a e compa ed o hose o ou ea lie TUJU19
analysis, which we e de e mined using DIS da a only. The
ee p o on PDFs used as a baseline o he nuclea pa o
he QCD analysis we e upda ed by including expe imen al
da a o DY, W;Z boson p oduc ion p ocesses aken by
he ATLAS and CMS collabo a ions a he LHC; see
Sec. III C o de ails. The compa isons a e p esen ed in
Fig. 2 o NLO and in Fig. 3 o NNLO.
The impac o he newly added LHC da a is a he mild a
NLO. We obse e ha he unce ain ies o he PDFs ha e
become sligh ly smalle in some cases, especially o he
alence qua ks. A NNLO, he esul ing dis ibu ions o
he alence qua ks, and especially o gluons, a e sligh ly
dec eased wi h espec o ou p e ious analysis. The esul s
ob ained o he upda ed ee p o on PDF baseline con i m
ha DY, and W;Z boson p oduc ion da a can be accom-
moda ed oge he wi h he DIS da a and p o ide u he
cons ain s in a global analysis. Using hese da a o pin
down he p o on PDFs in he same amewo k as he
nuclea ones will ensu e ha he baseline is well con-
s ained in he egion whe e new da a a e included o
he nPDFs.
The pa ame e s o he inpu dis ibu ions o ou bes i
o he p o on baseline a e collec ed in he Appendix.
B. Nuclea PDFs
The esul ing nuclea PDFs, e e ed o as TUJU21, a e
p esen ed in Figs. 4a NLO and 5a NNLO o he bound-
p o on-in-lead-nucleus PDFs, including also a ios o ou
baseline ee p o on PDFs. We also compa e o he nPDFs
o ou p e ious DIS-only analysis TUJU19. A NLO, he
la ges di e ences be ween he wo analyses occu o
gluons and sea qua ks. Fo gluons he small-xsupp ession
is signi ican ly milde han in TUJU19, along wi h a
TABLE IV. Summa y o expe imen al W;Zboson p oduc ion da a om LHC p þPb collisions in un I and un
II used o de e mine ou nuclea PDFs. In he las wo columns he χ2 alues a NLO and NNLO ob ained in ou
analysis a e p o ided.
Nucleus P oc. Exp. Yea Re . Ndp χ2NLO χ2NNLO
pPb ZLHC un I ATLAS 2015 [5] 14 16.40 12.82
ZLHC un I CMS 2015 [7] 12 8.76 7.30
W−LHC un II CMS 2019 [69] 24 39.58 42.83
WþLHC un II CMS 24 41.08 39.07
In o al: 74
FIG. 2. P o on baseline PDFs in TUJU21 a NLO compa ed o he p e ious TUJU19 esul s, shown a he ini ial scale Q2
0¼
1.69 GeV2(uppe panels) and a Q2¼100 GeV2(lowe panels) a e DGLAP e olu ion.
NNLO NUCLEAR PARTON DISTRIBUTION FUNCTIONS WITH …PHYS. REV. D 105, 094031 (2022)
094031-7
sligh ly educed unce ain y. Also, he s ong an ishadow-
ing enhancemen a in e media e alues o xwe ound
p e iously is now amed o a mo e mode a e ∼10% e ec .
A he ini ial scale o he i he sea qua k nPDFs a e now
sligh ly lowe in he small-x egion, wi h a somewha
smalle unce ain y band, bu ha e emained e y simila a
la ge x. Because o he la ge gluon nPDF in he upda ed
i , he sea qua k dis ibu ions become la ge a highe
scales h ough scale e olu ion. O e all, he gluon unce -
ain ies a e likely s ill unde es ima ed due o he a he igid
o m o he inpu pa ame iza ion. Fo he alence qua ks
he esul ing nPDFs a e e y simila as in ou p e ious
analysis, sugges ing ha he added EW-boson da a do no
p o ide signi ican cons ain s o he alence sec o .
A NNLO he changes wi h espec o ou p e ious
analysis a e clea ly milde . The unce ain ies ha e now
become sligh ly la ge o gluons and smalle o sea qua ks,
bu o he wise hese a e well consis en wi h ou p e ious
analysis. Also o he alence qua ks he di e ences a e
small, wi h unce ain ies sligh ly educed. The p e iously
obse ed opposi e beha io o he nuclea modi ica ions o
u and d is now less p onounced. E en hough he e is no
signi ican educ ion in he esul ing unce ain y bands, he
mu ual ag eemen be ween he nuclea e ec s ound in ou
NLO and NNLO i s sugges s ha such e ec s a e now be e
cap u ed han in he DIS-only i .
The pa ame e s o he inpu dis ibu ions o ou bes i
o nPDFs a e also collec ed in he Appendix. The e o se s,
FIG. 3. Same as o Fig. 2, bu a NNLO.
FIG. 4. NLO nuclea pa on dis ibu ion unc ions in TUJU21 o a lead nucleus, compa ed o he p e ious TUJU19 esul s, shown a
he ini ial scale Q2
0¼1.69 GeV2(uppe panels) and a Q2¼100 GeV2a e DGLAP e olu ion (cen e panels). The lowe panels show
he co esponding a ios o PDFs o a p o on bound in lead o e he ee p o on PDFs.
HELENIUS, WALT, and VOGELSANG PHYS. REV. D 105, 094031 (2022)
094031-8
FIG. 14. Compa ison o Zboson p oduc ion in Pb þPb collisions a ffiffiffiffiffiffiffiffi
sNN
p¼5.02 TeV a NLO (le ) and NNLO (cen e ) wi h (solid
wi h unce ain y band) and wi hou (dashed) nuclea PDF modi ica ions o ATLAS [28] (uppe panels) and CMS [29] (lowe panels)
da a. In he igh pa we plo he a ios o he NNLO ( ed wi h unce ain y) and NLO (do -dashed b own wi h ha ched unce ain y)
oge he wi h he da a.
FIG. 15. Compa ison o DY p oduc ion in p þPb collisions a ffiffiffiffiffiffiffiffi
sNN
p¼8.16 TeV a NLO (le ) and NNLO (cen e ) esul s wi h (solid
wi h unce ain y band) and wi hou (dashed) nuclea PDF modi ica ions in wo in a ian mass bins, 15 <M<60 GeV (uppe panels)
and 60 <M<120 GeV (lowe panels) o CMS da a [98]. In he igh pa we plo he a ios o he NNLO ( ed wi h unce ain y) and
NLO (do -dashed b own wi h ha ched unce ain y) oge he wi h he da a.
NNLO NUCLEAR PARTON DISTRIBUTION FUNCTIONS WITH …PHYS. REV. D 105, 094031 (2022)
094031-15

la ges apidi ies. The ATLAS da a seem o ag ee wi h he
NNLO esul , whe eas he CMS esul s seem o all a bi
below he NNLO calcula ion a la ge apidi ies and a e
be e in line wi h he NLO esul . The e o e ou esul s
poin o possible ensions be ween he wo da ase s. Tha
said, one should keep in mind ha he e a e some
di e ences in he expe imen al analyses: in case o
ATLAS, he Glaube model was used o calcula e he
no maliza ion, whe eas CMS applied he measu ed lumi-
nosi y. Also, ATLAS p o ides he esul only in he iducial
phase-space egion, while he CMS da a has been co ec ed
o include also he phase space emo ed by cu s on he
inal-s a e lep ons. These ea u es make di ec compa isons
o he wo da ase s di icul .
B. DY p oduc ion in p + Pb
A ecen da ase ha has p o ed di icul o include in an
nPDF analysis a he NLO is he CMS DY p oduc ion in
pþPb collisions [98]. I has been an icipa ed ha o he
lowe -mass bin (15 <M<60 GeV) he NNLO co ec-
ions could be signi ican [23] and o he highe -mass bin
(60 <M<120 GeV) i has been no ed ha due o la ge
luc ua ions a he mid apidi y i is di icul o ha e
accep able χ2 alues wi h any PDF-based calcula ion
[22]. He e we quan i y he impac o he NNLO co ec ions
on hese da a o s udy whe he hese could explain he
obse ed di e ences in he low-mass bin.
Compa ison wi h his CMS da a is p esen ed in Fig. 15
o bo h mass windows as a unc ion o he apidi y y o
he dilep on pai including also he a ios be ween he
NNLO and NLO esul s. The compa isons a e made o
he iducial c oss sec ion ha has no been co ec ed o he
limi ed accep ance. He e we no ice ha he NNLO co -
ec ions a e a he mild, a ound 5% o he high-mass bin
bu become signi ican o he low-mass bin, eaching 20%
a he la ges (absolu e) apidi ies. To u he quan i y his
e ec we ha e calcula ed he χ2=Ndp alues o hese da a a
NLO and NNLO, shown in Table Vsepa a ely o he low-
and high-mass bins and o he combina ion o hese
wo. The common luminosi y unce ain y is no included
in he da a unce ain ies o χ2calcula ion bu has been
accoun ed o by inding a common no maliza ion ac o
ha minimizes he combined χ2=Ndp. In bo h cases he
scaling ac o is consis en wi h he quo ed luminosi y
unce ain y o 3.5%. Fo he high-mass bin he da a ac ually
seem o be be e desc ibed by he NLO calcula ion a
nega i e apidi ies whe eas o posi i e apidi ies i is in
good ag eemen wi h he NNLO esul . Fo he lowe mass
bin i seems clea ha he NNLO co ec ions a e needed o
ha e a good ag eemen wi h his da a and also he
combined χ2=Ndp is signi ican ly smalle a NNLO (1.554)
han a NLO (2.261).
VI. SUMMARY AND OUTLOOK
We ha e p esen ed new analyses o nuclea PDFs a
NLO and NNLO, TUJU21. We ha e adop ed he same
amewo k as in ou p e ious TUJU19 analysis, bu in
addi ion o neu al-cu en DIS and cha ged-cu en neu-
ino DIS da a we ha e now also included new elec oweak-
boson p oduc ion da a om he LHC, bo h o ou p o on
baseline i and o he nuclea modi ica ions. The esul ing
nPDFs p o ide a ully consis en se up o c oss sec ion
calcula ions a NNLO in nuclea collisions, o he i s ime
inco po a ing he LHC da a in a ull NNLO analysis o
nuclea PDFs. The compa isons o he exis ing nPDF se s
demons a e a easonable ag eemen wi hin he e o bands,
al hough some disc epancies in la o dependence we e
obse ed. We do poin ou , howe e , ha he adop ed
pa ame iza ion is a he es ic i e, likely esul ing in
unce ain ies ha a e unde es ima ed in he egion x<
0.001 whe e no da a ha e been included. The esul ing
c oss sec ions show e y good ag eemen wi h he included
expe imen al da a, as con i med by he o al χ2=Ndp <1.0
o he nuclea pa o he analysis. In he p esen ed
amewo k, he i pe o med a NNLO was ound o ha e
a signi ican ly lowe χ2=Ndp alue han he NLO one, 0.84
ins ead o 0.94. The esul ing PDFs will become a ailable
in LHAPDF6 o ma om he LHAPDF home page1o by
eques om he au ho s.
As an applica ion, we ha e s udied EW-boson p oduc ion
in Pb þPb collisions a he LHC and ha e compa ed he
esul s o ecen ATLAS and CMS da a. We ound ha o
ATLAS bo h he NLO and he NNLO compu a ion wi h he
i ed nuclea PDFs ends o be below he da a, e en hough
he c oss sec ions we e well ep oduced o p þPb colli-
sions. We ind be e ag eemen when compa ing o he e y
ecen CMS da a o Zboson p oduc ion [29], which hin s a
a possible ension be ween he wo expe imen al da ase s.
We compa e ou esul s also o he ecen CMS da a o DY
dilep on p oduc ion in p þPb collisions [98] ha we e no
included in he p esen ed analysis. He e we ind ha he
NNLO co ec ions a e signi ican , especially o he lowe
mass bin, and necessa y o ha e a good desc ip ion o he
da a. This demons a es ha o some obse ables he
NNLO co ec ions can be la ge han he unce ain ies in
TABLE V. The alues o χ2=Ndp o he CMS DY da a in
Fig. 15. The da a poin s ha e been scaled by a ac o ha
minimizes χ2 o accoun o he co ela ed luminosi y unce ain y
(3.5%).
χ2=NdpðNLOÞχ2=NdpðNNLOÞ
15 <M<60 GeV 3.002 0.735
60 <M<120 GeV 1.894 2.009
Combined 2.261 1.554
Scaling ac o 0.989 1.037 1h ps://lhapd .hep o ge.o g/.
HELENIUS, WALT, and VOGELSANG PHYS. REV. D 105, 094031 (2022)
094031-16
nuclea PDF analyses and ha hese co ec ions should be
aken in o accoun when conside ing such da a.
A possible u u e imp o emen would be o analyze
he W-boson p oduc ion da a emo ing he cons ain
¯
u¼¯
d¼¯
s. Howe e , a elease o ha cons ain will
inc ease he numbe o ee pa ame e s and ha e an impac
on he con e gence o he i , likely equi ing an ex ension
o he analysis. Ano he a enue o a s ep o wa d will be a
combined analysis o p o on and nuclea PDFs. The
ensuing doubling o he numbe o i pa ame e s will
demand a ull e hinking o he minimiza ion and PDF
de e mina ion p ocedu e.
Clea ly, ou esul s—especially he compa isons o o he
se s o nPDFs—show ha we s ill ha e a long way o go
un il we can be con iden o ha e a good unde s anding o
nuclea modi ica ions o pa on dis ibu ions. Despi e he
s ill la ge unce ain ies, we a e encou aged by he imp o e-
men ha he inclusion o NNLO co ec ions appea s o
p o ide. Ul ima ely, we hope ha he u u e elec on ion
collide will p o ide p ecision cons ain s on he nuclea
PDFs. On he ime scale o he elec on ion collide , we also
expec new analysis echnologies o become a ailable ha
o e new me hods o ex ending he possibili ies o
heo e ical in es iga ions. Among hem could be physics
simula ions on a quan um compu e . As an example, a
ecen s udy [99] p esen s an algo i hm o compu ing
p edic ions o pa on dis ibu ion unc ions and he had-
onic enso , whe e i is also specula ed ha “quan um
sup emacy”could possibly be demons a ed in his a ea o
esea ch as such a ask has p o en e y di icul wi h
classical compu e s. As ano he example, a ecen s udy has
p esen ed a p oo -o -p inciple demons a ion o he de e -
mina ion o p o on PDFs wi h a quan um compu e [100],
pa ing he way o u he applica ions such as, o
example, p o ons bound in a nucleus. These new me hods
and new echnologies will be mos ele an when aiming a
he highes possible pe u ba i e p ecision, and we a e
con iden ha ou NNLO analysis will p o ide a solid
e e ence o such u u e s udies.
ACKNOWLEDGMENTS
The au ho s acknowledge suppo by he s a e o
Baden-Wü embe g h ough bwHPC p o iding he pos-
sibili y o un he compu a ional calcula ions on he high-
pe o mance clus e and also wish o acknowledge CSC-IT
Cen e o Science, Finland, o addi ional compu a ional
esou ces h ough P ojec No. jyy2580. Also he suppo
om he Academy o Finland, P ojec s No. 308301 and
No. 331545, and h ough he Cen e o Excellence in Qua k
Ma e , is acknowledged (I. H.). Fu he mo e, he au ho s
hank HQS Quan um Simula ions GmbH o he suppo i e
wo king en i onmen allowing he inaliza ion o he
publica ion o he p esen ed esul s. This wo k was also
suppo ed in pa by he Bundesminis e ium ü Bildung
und Fo schung (BMBF), G an No. 05P18VTCA1. We
hank he
X
F
ITTER
eam o he suppo and I. No iko o
he help wi h he APPL
GRID
in e acing. Thanks also o H.
Paukkunen o many use ul discussions.
APPENDIX: PDF PARAMETERS
He e we collec he inpu pa ame e s ob ained o he
p o on and nuclea pa on dis ibu ion unc ions p esen ed
in Sec. IV. The naming con en ion co esponds o he PDF
pa ame iza ion gi en in Eqs. (7) and (8). Table VI p o ides
he NLO pa ame e s, while Table VII p esen s he NNLO
ones. Some o he pa ame e s we e manually se o ze o i
he da a ha we e used did no p o ide enough sensi i i y
o cons ain hem wi hou la ge unce ain y. The Adepend-
ence was implemen ed o a subse o pa ame e s, again
selec ed such ha he da a p o ided enough sensi i i y o
esul in a con e ged i .
TABLE VI. Values o he NLO i pa ame e s a he ini ial scale, Q2
0¼1.69 GeV2. (SR) means ha he
no maliza ion o ha pa icula pa on is ixed by he momen um and alence numbe sum ules. A dash indica es
ha his pa ame e was excluded om he i . Pa ame e alues o he sea qua ks, apa om ¯
u, we e de i ed om
he applied cons ain s ¯
s¼s¼¯
d¼¯
u.
gValue u Value d Value ¯
uValue
cg
0;08.9596 cu
0;0(SR) cd
0;0(SR) c¯
u
0;0(SR)
cg
1;00.3270 cu
1;00.7121 cd
1;00.7629 c¯
u
1;0−0.1815
cg
2;013.438 cu
2;03.4290 cd
2;02.0996 c¯
u
2;05.2593
cg
3;06.4371 cu
3;01.4506 cd
3;0−1.4391 c¯
u
3;02.4151
cg
4;0 cu
4;0 cd
4;0 c¯
u
4;0
cg
1;1−5.4728 cu
1;1−0.0462 cd
1;1−19.16 c¯
u
1;1251.91
cg
1;2−0.0013 cu
1;20.3411 cd
1;2−0.0026 c¯
u
1;20.0002
cg
2;1−2.000 cu
2;14.2325 cd
2;11.2264 c¯
u
2;1−276.53
cg
2;20.3695 cu
2;20.0025 cd
2;20.4273 c¯
u
2;2−0.0017
NNLO NUCLEAR PARTON DISTRIBUTION FUNCTIONS WITH …PHYS. REV. D 105, 094031 (2022)
094031-17
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2;115.614 cd
2;11.2905 c¯
u
2;1−0.1323
cg
2;20.4518 cu
2;2−0.0011 cd
2;20.3396 c¯
u
2;2−0.4051
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094031-19