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Fi s global nex - o-leading o de de e mina ion o di ac i e pa on dis ibu ion
unc ions and hei unce ain ies wi hin he xFi e amewo k
Goha ipou , Muhammad; Khanpou , Hamzeh; Guzey, Vadim
Goha ipou , M., Khanpou , H., & Guzey, V. (2018). Fi s global nex - o-leading o de
de e mina ion o di ac i e pa on dis ibu ion unc ions and hei unce ain ies
wi hin he xFi e amewo k. Eu opean Physical Jou nal C, 78(4), A icle 309.
h ps://doi.o g/10.1140/epjc/s10052-018-5787-z
2018
Eu . Phys. J. C (2018) 78:309
h ps://doi.o g/10.1140/epjc/s10052-018-5787-z
Regula A icle - Theo e ical Physics
Fi s global nex - o-leading o de de e mina ion o di ac i e
pa on dis ibu ion unc ions and hei unce ain ies wi hin he
xFi e amewo k
Muhammad Goha ipou 1,a, Hamzeh Khanpou 1,2,b, Vadim Guzey3,4,5,c
1School o Pa icles and Accele a o s, Ins i u e o Resea ch in Fundamen al Sciences (IPM), P.O.Box 19395-5531, Teh an, I an
2Depa men o Physics, Uni e si y o Science and Technology o Mazanda an, P.O.Box 48518-78195, Behshah , I an
3Depa men o Physics, Uni e si y o Jy äskylä, P.O. Box 35, 40014 Jy äskylä, Finland
4Helsinki Ins i u e o Physics, P.O. Box 64, 00014 Helsinki, Finland
5Na ional Resea ch Cen e “Ku cha o Ins i u e”, Pe e sbu g Nuclea Physics Ins i u e (PNPI), Ga china 188300, Russia
Recei ed: 6 Feb ua y 2018 / Accep ed: 8 Ap il 2018
© The Au ho (s) 2018
Abs ac We p esen GKG18-DPDFs, a nex - o-leading
o de (NLO) QCD analysis o di ac i e pa on dis ibu ion
unc ions (di ac i e PDFs) and hei unce ain ies. This is
he i s global se o di ac i e PDFs de e mined wi hin
he xFi e amewo k. This analysis is mo i a ed by
all a ailable and mos up- o-da e da a on inclusi e di ac-
i e deep inelas ic sca e ing (di ac i e DIS). Hea y qua k
con ibu ions a e conside ed wi hin he amewo k o he
Tho ne–Robe s (TR) gene al mass a iable la o numbe
scheme (GM-VFNS). We o m a mu ually consis en se o
di ac i e PDFs due o he inclusion o high-p ecision da a
om H1/ZEUS combined inclusi e di ac i e c oss sec ions
measu emen s. We s udy he impac o he H1/ZEUS com-
bined da a by p oducing a a ie y o de e mina ions based
on educed da a se s. We ind ha hese da a se s ha e a
signi ican impac on he di ac i e PDFs wi h some sub-
s an ial educ ions in unce ain ies. The p edic ions based on
he ex ac ed di ac i e PDFs a e compa ed o he analyzed
di ac i e DIS da a and wi h o he de e mina ions o he
di ac i e PDFs.
Con en s
1 In oduc ion ......................
2 Theo e ical amewo k and assump ions ........
2.1 C oss sec ion o di ac i e DIS .........
2.2 QCD ac o iza ion heo em ............
ae-mail: [email p o ec ed]
be-mail: [email p o ec ed]
ce-mail: Guzey_ [email p o ec ed]
2.3 Hea y la ou con ibu ions o he di ac i e
DIS s uc u e unc ion ...............
3 The me hod o di ac i e PDFs global QCD analysis
3.1 GKG18-DPDFs pa ame iza ions o he di ac-
i e PDFs .....................
3.2 Di ac i e DIS da a se s used in GKG18-DPDFs
i s ........................
3.3 The me hod o minimiza ion and di ac i e
PDF unce ain ies .................
4 Resul s and discussions ................
4.1 Q2e olu ion and compa ison o o he di ac-
i e PDFs .....................
4.2 Compa ison o he di ac i e DIS da a ......
5 Summa y and conclusions ...............
Re e ences .........................
1 In oduc ion
High p ecision calcula ions o ha d sca e ing c oss sec-
ions in lep on-had on deep inelas ic sca e ing (DIS) and
had on-had on collide expe imen s can be done wi hin
he amewo k o pe u ba i e quan um ch omodynamics
(pQCD). The compu a ions o c oss sec ions can be pe -
o med using he so-called ac o iza ion heo em ha allows
o a sys ema ic sepa a ion o pe u ba i e and nonpe u ba-
i e physics [1,2]. Some examples o desc ibing he la e
in a ious p ocesses a e he well-known pa on dis ibu ion
unc ions (PDFs) [3–7], nuclea PDFs [8–11], and pola ized
PDFs [12–18], which a e a he igh ly cons ained by global
QCD i s o DIS and had on collide da a. In ac , hey a e
c ucial asse s in all sca e ing p ocesses in ol ing had ons
(nucleons and nuclei) in he ini ial s a e. In his espec , phe-
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309 Page 2 o 20 Eu . Phys. J. C (2018) 78:309
nomenological and expe imen al s udies o e he pas h ee
decades ha e p o ided impo an in o ma ion on he s uc u e
o had ons. A signi ican amoun o PDF se s has been de e -
mined conside ing he mos p ecise da a om LHC Run I and
II [3,5,7,19–24]. In he li e a u e, he ela i e impo ance o
LHC da a has been subjec o conside able discussion. These
new and up- o-da e se s o PDFs ha e played an impo an
ole in he sea ch o new physics, o example in he op
qua k and Higgs boson sec o s [3,25].
Di ac i e p ocesses, ep →epX, whe e X ep esen s
had onic inal s a e sepa a ed om he ecoiled p o on by a
apidi y gap and he p o on in he inal s a e ca ies mos o
he beam momen um (see Fig. 1), ha e been s udied ex en-
si ely in he H1 and ZEUS expe imen s a he elec on-p o on
(ep) collide HERA [2,26–31]. A HERA, a subs an ial ac-
ion o up o 10% o all ep DIS in e ac ions p oceeds ia
he di ac i e sca e ing p ocess ini ia ed by a highly i ual
pho on. In he amewo k o he collinea ac o iza ion heo-
em, he heo e ical calcula ion o di ac i e c oss sec ions
equi es a special ype o nonpe u ba i e unc ions as inpu ,
so ha he uni e sal di ac i e PDFs may be de ined. To be
mo e p ecise, he ac o iza ion heo em p edic s ha he c oss
sec ion can be exp essed as he con olu ion o nonpe u ba-
i e di ac i e PDFs and pa onic c oss sec ions o he ha d
subp ocess calculable wi hin he amewo k o pQCD. Con-
sequen ly, he dynamics o he di ac i e p ocesses can be
o mula ed in e ms o qua k and gluon densi ies. The di ac-
i e PDFs ha e p ope ies simila o he PDFs o he ee
nucleon, bu wi h he cons ain o a leading p o on o i s low
mass exci a ions being p esen in he inal s a e. Like PDFs,
i is well es ablished ha he di ac i e PDFs a e uni e sal
quan i ies, which can be ex ac ed om di ac i e DIS da a
h ough global QCD analyses. The knowledge o di ac i e
PDFs o di e en had on species as well as he es ima ion
o hei unce ain ies is he e o e i al o p ecise heo e ical
Fig. 1 Rep esen a i e Feynman diag am o he neu al cu en di ac-
i e DIS p ocess ep →epX
and expe imen al calcula ions and, hence, has ecei ed qui e
some in e es s in he pas (see, o example, Re . [32] o a
ecen e iew).
The main sou ces o cons ain he di ac i e PDFs a e he
inclusi e di ac i e DIS da a measu ed a HERA. Gi en he
di ac i e PDFs, pe u ba i e QCD calcula ions a e expec ed
o be applicable o o he p ocesses such as he je and hea y
qua k p oduc ion in di ac i e DIS a HERA [29–31,33–35].
A ull discussion o di ac i e dije p oduc ion a HERA will
be he main subjec o ou u u e wo k. Indeed, he nex - o-
leading o de (NLO) QCD p edic ions using di ac i e PDFs
desc ibe hese measu emen s a he well. The e a e se e al
s udies in which he di ac i e PDFs ha e been de e mined
om he QCD analyses o di ac i e DIS da a [27,28,36–
41]. In his pape , we p esen a new se o di ac i e PDFs,
e e ed o as GKG18-DPDFs, h ough a comp ehensi e
NLO QCD analysis. The GKG18-DPDFs di ac i e PDFs
a e de e mined using all a ailable and up- o-da e da a om
di ac i e DIS c oss sec ion [42–44], including, o he i s
ime, he H1 and ZEUS combined inclusi e di ac i e c oss
sec ion measu emen s [45].
The ou line o his pape is as ollows: In Sec . 2.1,we
b ie ly p esen he heo e ical o malism adop ed o desc ib-
ing he di ac i e DIS a HERA. A e e iewing he QCD
ac o iza ion heo em in Sec . 2.2, we explain he hea y
la o con ibu ions o he di ac i e DIS s uc u e unc-
ion in Sec . 2.3. The phenomenological amewo k used in
GKG18-DPDFs global QCD analysis is p esen ed in Sec . 3.
This sec ion includes ou pa ame iza ions o he di ac i e
PDFs (Sec . 3.1), a de ailed discussion o he desc ip ion
o di e en da a se s included in GKG18-DPDFs global i
(Sec . 3.2), and he me hod o minimiza ion and di ac-
i e PDF unce ain ies (Sec . 3.3). In Sec . 4, we p esen
GKG18-DPDFs esul s o di ac i e PDFs ob ained om
global i s o H1 di ac i e DIS c oss sec ions [42–44], and
H1 and ZEUS combined inclusi e di ac i e da a [45]. In
Sec . 4.1, we compa e he di ac i e PDFs ob ained in his
wo k o he p e iously de e mined by o he g oups. Sec-
ion 4.2 is also de o ed o compa ing he heo e ical p e-
dic ions based on he ex ac ed di ac i e PDFs wi h he
analyzed di ac i e DIS da a. Finally, in Sec . 5, we p esen
ou summa y and conclusions.
2 Theo e ical amewo k and assump ions
In he ollowing we desc ibe he s anda d heo e ical ame-
wo k adop ed o he di ac i e DIS. Al hough, he e a e
di e en heo e ical app oaches o desc ibe he di ac i e
p ocesses in li e a u e [46], i is well known now ha he
app oach, whe e he di ac i e DIS is media ed by he
exchange o he ha d Pome on and a seconda y Reggeon
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Eu . Phys. J. C (2018) 78:309 Page 3 o 20 309
can be ema kably success ul o he desc ip ion o mos o
di ac i e DIS da a.
2.1 C oss sec ion o di ac i e DIS
In o de o discuss he c oss sec ion o di ac i e DIS, one
needs o in oduce he kinema ic a iables i s . The common
a iables in any DIS p ocess a e as ollows: he pho on i -
uali y Q2=−q2, whe e q=k−kis he di e ence o
he ou -momen a o he incoming (k) and ou going (k) lep-
ons; he longi udinal momen um ac ion x=−q2
2P.q, whe e
Pis he ou -momen um o he incoming p o on; and he
inelas ici y y=P.q
P.k.
The ep esen a i e Feynman diag am o he neu al cu -
en di ac i e DIS p ocess ep →epX, p oceeding ia
a i ual pho on exchange, is depic ed in Fig. 1.In he
case o di ac i e DIS, as illus a ed in Fig. 1, he addi-
ional a iables a e he squa ed ou -momen um ans e ed
=(P−P)2, whe e Pis he ou -momen um o he
ou going p o on, and he mass MXo he di ac i e inal
s a e, which is p oduced by di ac i e dissocia ion o he
exchanged i ual pho on. This mass is much smalle han
he in a ian pho on-p o on ene gy and should be conside ed
as a u he deg ee o eedom. I is usually eplaced by he
ligh -cone momen um ac ion o he di ac i e exchange β,
β=Q2
2(P−P).q=Q2
M2
X+Q2− .(1)
The -in eg a ed di e en ial c oss sec ion o he di ac-
i e p ocess, ep →epX, is p esen ed in he o m o a di ac-
i e educed c oss sec ion σD(3)
(β, Q2;xIP )as
dσep→epX
dβdQ2dxIP =2πα2
βQ41+(1−y)2σD(3)
(β, Q2;xIP ),
(2)
whe e xIP =(P−P).q
P.q e e s o he longi udinal momen um
ac ion los by he incoming p o on, which is ca ied away by
he di ac i e exchange; and is he ou -momen um ans-
e squa ed a he p o on e ex. No e ha he longi udinal
momen um ac ion βo he s uck pa on wi h espec o
he colou less exchange can be also exp essed as β=x/xIP .
The di ac i e educed c oss sec ion is gi en by
σD(3)
(β, Q2;xIP )=FD(3)
2(β, Q2;xIP )
−y2
1+(1−y)2FD(3)
L(β, Q2;xIP ),
(3)
whe e FD(3)
2and FD(3)
La e he di ac i e s uc u e unc-
ions. I should be emphasized he e ha o he yno o
close o uni y, one can neglec he con ibu ion om FD(3)
L
and σD(3)
(β, Q2;xIP )≈FD(3)
2(β, Q2;xIP )holds o e y
good app oxima ion. Since ou analysis is based on ecen
measu emen s o inclusi e di ac i e DIS a HERA o he
educed c oss sec ions, we conside he con ibu ions o
bo h FD(3)
2and he longi udinal di ac i e s uc u e unc-
ion FD(3)
L.
2.2 QCD ac o iza ion heo em
I has been shown ha he di ac i e DIS c oss sec ions a
HERA [27,28,30] a e well in e p e ed assuming he “p o-
on e ex ac o iza ion” app oach which p o ides a good
desc ip ion o di ac i e DIS da a in e ms o a esol ed
Pome on (IP )[47,48]. Wi hin he Regge phenomenol-
ogy [49], he c oss sec ions o di ac i e p ocesses a high
ene gies a e desc ibed by he exchange o so-called Regge
ajec o ies. The di ac i e c oss sec ion is domina ed by a
ajec o y usually called he Pome on, while he subleading
Reggeon (IR ) con ibu ion is signi ican only o xIP >0.01.
I has been shown ha he QCD ac o iza ion heo em and
he well-known DGLAP pa on e olu ion equa ions can be
applied o desc ibe he dependence o he c oss sec ion on β
and Q2, while a Regge inspi ed app oach is used o exp ess
he dependence on xIP and .
In he QCD ac o iza ion app oach, he di ac i e s uc-
u e unc ions can be w i en as a con olu ion o ha d sca -
e ing coe icien unc ions wi h he di ac i e PDFs,
FD(4)
2/L(β, Q2;xIP , )
=
i1
β
dz
zC2/L,iβ
z D
i(z,Q2;xIP , ), (4)
whe e he sum uns o e qua ks and gluons.
Conside ing QCD ac o iza ion heo em, a ious ha d
sca e ing di ac i e p ocesses a e calculable by means o
di ac i e PDFs, such as he di ac i e je p oduc ion in
DIS. The concep o QCD ha d ac o iza ion o he di ac-
i e PDFs as well as he alidi y o he assump ion o QCD
ha d ac o iza ion ha e been heo e ically p edic ed o hold
in di ac i e DIS p ocesses [1]. We should men ioned he e
ha he ha d QCD ac o iza ion has been es ed a HERA in
a ious di ac i e p ocesses. In ecen H1 analyses he alid-
i y o he ha d ac o iza ion has been success ully examined
o open cha m p oduc ion in pho op oduc ion and DIS wi h
Dmesons [29,50] and in di ac i e p oduc ion o dije s
in DIS [30,34,35,51]. These s udies suppo he alidi y o
QCD ha d sca e ing ac o iza ion in di ac i e DIS.
We should no ice he e ha in DGLAP NLO QCD global
i s, NLO con ibu ions o he spli ing unc ions go e ning
he e olu ion o unpola ized nonsingle and single combi-
na ions o qua k densi ies a e he same as in ully inclusi e
DIS. Hence, he di ac i e pa on densi ies sa is y he same
(DGLAP) e olu ion equa ions as he usual pa on dis ibu-
ions in inclusi e DIS [52–54]. The Wilson coe icien unc-
ions C2and CLin Eq. (4) a e also he same as in inclusi e
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309 Page 4 o 20 Eu . Phys. J. C (2018) 78:309
DIS and calculable in pe u ba i e QCD [55]. The di ac i e
PDFs D
i(β, Q2;xIP , )a e uni e sal and non-pe u ba i e
quan i ies, which can be ob ained om he QCD i o he
inclusi e di ac i e da a. No e ha di ac i e PDFs can be
de ined in e ms o ma ix elemen s o qua k and gluon ope -
a o s; he eno maliza ion o di e gencies a nex - o-leading
o de is ca ied ou simila ly o he inclusi e case and leads
o he DGLAP e olu ion equa ions.
In GKG18-DPDFs analysis, he p o on e ex ac o iza-
ion [47] is assumed, whe e he xIP and dependencies o he
di ac i e PDFs ac o ize om he dependencies on βand
Q2. In his amewo k, he di ac i e PDFs can be w i en
as,
D
i/p(β, Q2;xIP , )= IP /p(xIP , ) i/IP (β, Q2)
+ IR /p(xIP , ) IR
i/IR (β, Q2), (5)
whe e i/IP (β, Q2)and IR
i/IR (β, Q2)a e he pa onic s uc-
u es o Pome on and Reggeon, espec i ely. The emission o
Pome on and Reggeon om he p o on can be desc ibed by
he lux- ac o s o IP /p(xIP , )and IR /p(xIP , ). The de ail
discussion on he pa ame iza ion o he di ac i e PDFs in
Eq. (5) will be p esen ed in a sepa a e sec ion.
2.3 Hea y la ou con ibu ions o he di ac i e DIS
s uc u e unc ion
In his sec ion, we discuss a gene al amewo k o he inclu-
sion o hea y qua k con ibu ions o di ac i e DIS s uc-
u e unc ions. The co ec ea men o hea y qua k la ou s
in an analysis o di ac i e PDFs is essen ial o p ecision
measu emen s a DIS collide s as well as o he LHC phe-
nomenology. As an example, he c oss sec ion o he W-
boson p oduc ion a he LHC depends c ucially on p ecise
knowledge o he cha m qua k dis ibu ion. A de ailed dis-
cussion on he impac o he hea y qua k mass ea men s
in he pa on dis ibu ions as well as he de e mina ion o
he hei unce ain y due o unce ain y in he hea y qua k
masses can be ound in Re . [56].
Like o he case o inclusi e DIS, he ea men o hea y
la ou s has an impo an impac on he di ac i e PDFs
ex ac ed om he global analysis o di ac i e DIS, due o
he hea y la ou con ibu ion o he o al s uc u e unc ion a
small alues o z. Recall ha he e a e a ious choices ha can
be used o conside he hea y qua k con ibu ions. These a e
he so-called a iable la ou numbe scheme (VFNS), ixed
la ou numbe scheme (FFNS) and gene al-mass a iable-
la o -numbe scheme (GM-VFNS).
In he case o FFNS, Q2≃m2
c,m2
b, he massi e qua k
may be ega ded as being only p oduced in he inal s a e
and no as pa ons wi hin he nucleon. Hence, he ligh up-,
down- and s ange-qua ks a e ac i e pa ons and he numbe
o la ou s is ixed o n =3. Howe e one can also con-
side cha m o bo om qua k as ligh qua k a high scales.
I has been shown ha he accu acy o he FFNS becomes
inc easingly unce ain as Q2inc eases abo e he hea y qua k
mass h eshold m2
H[57]. In he ze o-mass VFNS, he mas-
si e qua ks beha e like massless pa ons o Q2m2
c,m2
b.
The ZM-VFNS misses ou O(m2
H/Q2)con ibu ions com-
ple ely in he pe u ba i e expansion, and hence, his scheme
is no accu a e enough o be used in a QCD analysis. One can
also see a discon inui y in he pa on dis ibu ions and o al
s uc u e unc ion a Q2=m2
Hin ZM-VFNS [57].
The GM-VFNS is he app op ia e scheme o in e pola e
be ween hese wo egions and could co ec FFNS a low Q2
and ZM-VFNS a high Q2→∞, and hence, could imp o e
he smoo hness o he ansi ion egion whe e he numbe
o ac i e la ou s is changed by one [57]. The e o e, o a
p ecise analysis o s uc u e unc ions and o he inclusi e
DIS o had on collide s da a, one can use he GM-VFNS,
which smoo hly connec s he wo well-de ined scheme o
VFNS and FFNS [57]. This scheme is ha mos commonly
app oach in a ie y o global i s. In H1-DPDFs-2006 [27]
and ZEUS-DPDFs-2010 [28] di ac i e PDFs analyses, he
hea y qua k s uc u e unc ions ha e been compu ed using
he FFNS and gene al-mass a iable- la o -numbe scheme
o Tho ne and Robe s (TR GM-VFNS), espec i ely. Ou
app oach is based on he TR GM-VFNS [5,58,59] which
ex apola es smoo hly om he FFNS a low Q2 o he ZM-
VFNS a high Q2and p oduces a good desc ip ion o he
e ec o hea y qua ks on s uc u e unc ions o e he whole
ange o Q2.
In ou analysis, we ollow he MMHT14 PDFs analysis
and adop hei de aul alues o he hea y qua k masses as
mc=1.40 and mb=4.75 GeV [60]. In Re . [60], he a i-
a ion in he MMHT14 PDFs when he hea y qua k masses
mcand mbwe e a ied away om hei de aul alues o
mc=1.40 and mb=4.75 GeV has been in es iga ed. The
dependence o he MMHT14 PDFs and he quali y o he
compa ison o analyzed da a, unde a ia ions o he hea y
qua k masses away om hei de aul alues has been s ud-
ied. I has been shown ha he e ec s o a ying mcand mb
in he p edic ions o c oss sec ions o s anda d p ocesses a
he LHC a e small and he unce ain ies on PDFs due o he
a ia ion o qua k masses a e no hugely impo an [60].
3 The me hod o di ac i e PDFs global QCD analysis
In he ollowing, we p esen he me hod o GKG18-DPDFs
global QCD analysis. This sec ion also includes ou
pa ame iza ions o he di ac i e PDFs, he de ailed discus-
sion o he desc ip ion o di e en da a se s included in ou
global i , and he me hod o minimiza ion and unce ain ies
o ou esul ing di ac i e PDFs.
123

Eu . Phys. J. C (2018) 78:309 Page 5 o 20 309
3.1 GKG18-DPDFs pa ame iza ions o he di ac i e
PDFs
As we al eady men ioned, he scale dependence o he dis-
ibu ions i=q,g(β, Q2)o he qua ks and gluons can be
ob ained by he DGLAP e olu ion equa ions, p o ided he
di ac i e PDFs a e pa ame ized as unc ions o βa some
s a ing scale Q2
0. In ou analysis, he di ac i e PDFs a e
modelled a he s a ing scale Q2
0=1.8GeV
2(below he
cha m h eshold) in e ms o qua k z q(z,Q2
0), and gluon
z g(z,Q2
0)dis ibu ions. He e, zis he longi udinal momen-
um ac ion o he s uck pa on, which en e s he ha d sub-
p ocess, wi h espec o he di ac i e exchange. Conside -
ing he lowes -o de qua k-pa on model p ocess, we ha e
z=β, while he inclusion o highe -o de p ocesses leads
o 0 <β<z. Fo he qua k dis ibu ions we assume ha
all ligh -qua ks and hei an iqua ks dis ibu ions a e equal,
u= d= s= ¯u= ¯
d= ¯s. The hea y qua k dis i-
bu ions q(=c,b)a e gene a ed dynamically a he scale Q2
>m2
c,babo e he co esponding mass h eshold in he TR
GM-VFN scheme.
Due o he signi ican ly smalle amoun o da a o inclu-
si e di ac i e DIS da a han o he o al DIS c oss sec ion,
we adop a sligh ly less lexible, mo e economical unc ional
o m o pa ame ize he nonpe u ba i e di ac i e PDFs a
he ini ial scale Q2
0=1.8GeV
2. Ou s anda d pa ame iza-
ions o he qua ks and gluon di ac i e PDFs a e as
ollows:
z q(z,Q2
0)=αqzβq(1−z)γq1+ηq√z,(6)
z g(z,Q2
0)=αgzβg(1−z)γg1+ηg√z.(7)
An addi ional ac o o e−0.001
1−zis included o ensu e ha
he dis ibu ions anish o z→1. The e o e, he pa ame-
e s γqand γgha e he eedom o ake nega i e as well as
posi i e alues in he i . We ha e es ed ha Eqs. (6) and (7)
ne e heless yield a e y sa is ac o y desc ip ion o he ana-
lyzed di ac i e DIS da a. We ound ha he wo pa ame e s
ηqand ηghad o be ixed o ze o since he da a do no con-
s ain hem well enough. These simple unc ional o ms wi h
signi ican ly ewe pa ame e s ha e he addi ional bene i o
g ea ly acili a ing he i ing p ocedu e.
The xIP dependence o di ac i e PDFs D
i/p(z,Q2;xIP , )
in Eq. (5) is pa ame ized by he Pome on and Reggeon lux
ac o s
IP ,IR (xIP , )=AIP ,IR
eBIP ,IR
x2αIP ,IR ( )−1
IP
,(8)
whe e he ajec o ies a e assumed o be linea , αIP ,IR ( )=
αIP ,IR (0)+α
IP ,IR . The Pome on and Reggeon in e cep s,
αIP (0)and αIR (0), and he no maliza ion o he Reggeon
e m, AIR , a e ee pa ame e s and should be ex ac ed om
he i o da a. No e ha he alue o he no maliza ion pa am-
e e AIP is abso bed in αqand αg.
The Reggeon pa on densi ies IR
i/IR (z,Q2)p esen ed in
Eq. (5) a e ob ained om he GRV pa ame iza ion de i ed
om a i o pion s uc u e unc ion da a [61]. The alues o
he pa ame e s, which a e ixed in GKG18-DPDFs i , a e
he ollowing:
α
IP =0.0,
α
IR =0.90 GeV−2,
BIP =7.0GeV
−2,
BIR =2.0GeV
−2.
These alues a e aken om he ollowing expe imen al mea-
su emen s [26,62],
α
IP =−0.01 ±0.06 (s a .)+0.04
−0.08 (sys .)±0.04 (model)GeV−2,
α
IR =0.90 ±0.10 GeV−2,
BIP =7.1±0.7(s a .)+1.4
−0.7(sys .)GeV−2,
BIR =2.0±2.0GeV
−2.
In o al, 9 ee pa ame e s a e le in GKG18-DPDFs QCD
analysis, which a e αq,βq,γq,αg,βg,γg,αIP (0),αIR (0),
and AIR .
3.2 Di ac i e DIS da a se s used in GKG18-DPDFs i s
In his sec ion, we p esen he new expe imen al da a and hei
ea men in GKG18-DPDFs di ac i e PDFs analysis. A e
e iewing he analyzed da a se s, which include he ecen H1
and ZEUS combined da a, we discuss each o he new da a
se s in u n. We inally e iew he way in which he o al
di ac i e DIS da a se s a e cons uc ed and, in pa icula ,
which da a and which cu s a e included.
A lis o all di ac i e DIS da a poin s used in
GKG18-DPDFs global analysis is p esen ed in Tables 1and
2. These ables co espond o ou wo di e en scena ios o
including inclusi e di ac i e DIS da a in GKG18-DPDFs
global analyses, namely Fi A and Fi B.
Fo each da a se p esen ed in hese ables, we ha e p o-
ided he co esponding e e ences, he kinema ical co e age
o β,xIP , and Q2and he numbe o da a poin s. We s i e o
include as much o he a ailable di ac i e DIS expe imen-
al da a as possible in ou di ac i e PDF analysis. Howe e ,
some cu s ha e o be applied in o de o ensu e ha only
p ope da a a e included in he analysis.
The i s da a se we ha e used in ou QCD analysis is he
inclusi e di ac i e DIS da a om H1-LRG-11, which we e
aken wi h he H1 de ec o in he yea s 2006 and 2007. These
da a co espond o h ee di e en cen e -o -mass ene gies o
√s=225, 252 and 319 GeV [42,43]. In his measu emen ,
he educed c oss sec ions ha e been measu ed in he ange
123
309 Page 6 o 20 Eu . Phys. J. C (2018) 78:309
Table 1 Lis o all di ac i e DIS da a poin s used in Fi A global analysis. Fo each da ase we ha e p o ided he e e ences, he kinema ical
co e age o β,xIP ,andQ2and he numbe o da a poin s. The de ails o kinema ic cu s explained in he ex
Expe imen Obse able [βmin,βmax][xmin
IP ,xmax
IP ]Q2[GeV2]# o poin s
H1-LRG-11 √s=225 [42,43]σD(3)
[0.089–0.699] [5.0×10−4−3.0×10−3] 11.5–44 13
H1-LRG-11 √s=252 [42,43]σD(3)
[0.089–0.699] [5.0×10−4−3.0×10−3] 11.5–44 12
H1-LRG-11 √s=319 [42,43]σD(3)
[0.089–0.699] [5.0×10−4−3.0×10−3] 11.5–44 12
H1-LRG-12 [44]σD(3)
[0.0067–0.80] [3.0×10−4−3.0×10−2] 12–1600 165
H1/ZEUS combined [45]σD(3)
[0.0056–0.562] [9.0×10−4−9.0×10−2] 15.3–200 96
To al da a 298
Table 2 Lis o all di ac i e DIS da a poin s used in Fi B global analysis. See he cap ion o Table 1 o mo e de ails
Expe imen Obse able [βmin,βmax][xmin
IP ,xmax
IP ]Q2[GeV2]# o poin s
H1-LRG-11 √s=225 [42,43]σD(3)
[0.089–0.699] [5.0×10−4−3.0×10−3] 11.5–44 13
H1-LRG-11 √s=252 [42,43]σD(3)
[0.089–0.699] [5.0×10−4−3.0×10−3] 11.5–44 12
H1-LRG-11 √s=319 [42,43]σD(3)
[0.089–0.699] [5.0×10−4−3.0×10−3] 11.5–44 12
H1-LRG-12 [44]σD(3)
[0.0067–0.80] [3.0×10−4−3.0×10−2] 12–1600 165
H1/ZEUS combined [45]σD(3)
[0.0056–0.562] [9.0×10−4−9.0×10−2] 26.5–200 70
To al da a 272
o pho on i uali ies 4.0≤Q2≤44.0GeV
2and o he
longi udinal momen um ac ion o he di ac i e exchange
5×10−4≤xIP ≤3×10−3.
In addi ion o he H1-LRG-11 da a se , we ha e used o
he i s ime he H1-LRG-12 da a, whe e he di ac i e p o-
cess ep →eXY wi h MY<1.6 GeV and | |<1GeV
2has
been s udied wi h he H1 expe imen a HERA [44]. This
high s a is ics measu emen co e ing he da a aking pe i-
ods 1999–2000 and 2004–2007, has been combined wi h
p e iously published esul s [27] and co e s he ange o
3.5<Q2<1600 GeV2,0.0017 ≤β≤0.8, and 0.0003 ≤
xIP ≤0.03.
Finally, o he i s ime, we ha e used he ecen and up-
o-da e H1/ZEUS combined da a se o he educed di ac-
i e c oss sec ions, σD(3)
(ep →epX)[45]. This measu e-
men used samples o di ac i e DIS ep sca e ing da a a
a cen e-o -mass ene gy o √s=318 GeV and combined
he p e ious he H1 FPS HERA I [63], H1 FPS HERA
II [64],ZEUSLPS1[65] and ZEUS LPS 2 [26] da a se s.
This combined da a co e he pho on i uali y ange o
2.5<Q2<200 GeV2,3.5×10−4<xIP <0.09 in p o-
on ac ional momen um loss, 0.09 <| |<0.55 GeV2in
squa ed ou -momen um ans e a he p o on e ex, and
1.8×10−3<β<0.816.
While all H1-LRG da a a e gi en o he ange | |<
1GeV
2, he combined H1/ZEUS di ac i e DIS, which is
based upon p o on- agged samples, a e es ic ed o he ange
0.09 <| |<0.55 GeV2, so one needs o use a global no -
maliza ion ac o be ween hose wo measu emen egions.
Assuming an exponen ial dependence o he inclusi e
di ac i e c oss sec ion, he ex apola ion om 0.09 <| |<
0.55 GeV2 o | |<1GeV
2has been done using he H1 alue
o exponen ial slope pa ame e b≃6GeV
−2[45,64]. The
slope pa ame e can be ex ac ed om i s o he educed
c oss sec ion xIP σD(4)
. Wi h he abo e choice o cons an
slope pa ame e , a good desc ip ion o he da a o e he ull
xIP ,Q2and β ange is ob ained [63,64].
In addi ion o he ex apola ion discussed abo e, dis inc
me hods ha e been employed by he H1 and ZEUS expe i-
men s, and hence, c oss sec ions a e no always gi en wi h
he co ec ions o p o on dissocia ion backg ound. The di -
e en con ibu ions om p o on dissocia ion in he di e -
en da a se s should be conside ed by applica ion o di e -
en global ac o s. P o on dissocia ion is simula ed using an
app oxima e dσ
dM2
Y∝1
M2
Y
dependence [27,41]. The combined
H1/ZEUS di ac i e DIS a e co ec ed by a global ac o o
1.21 o accoun o such con ibu ions.
I should be no ed ha he wo da a no maliza ion ac o s,
which we desc ibed abo e, b ing a small sys ema ic unce -
ain y o he i ed da a. Howe e , since he ex apola ion in
| |is a he modes and he slope pa ame e bis expe imen-
ally de e mined wi h be e han 10% accu acy [63] and he
ac o due o p o on dissocia ion is a he well-cons ained
phenomenologically and expe imen ally, his unce ain y is
a he le el o a ew pe cen . Hence, i can be sa ely neglec ed
compa ed o he o al expe imen al e o o he H1/ZEUS
combined da a [45].
123
Eu . Phys. J. C (2018) 78:309 Page 7 o 20 309
Fig. 2 Dependence o χ2/do on he minimum cu alue o Q2
min o
all da a se s used in he analysis
As in he case o H1-DPDFs-2006 [27] and ZEUS-
DPDFs-2010 [28] i s, we apply a cu on MX,βand Q2.
To de e mine ou di ac i e PDFs, we apply β≤0.80 o e
he da a se s. The da a wi h MX>2 GeV a e included in
he i and he da a wi h Q2<Q2
min a e excluded o a oid
egions, which a e mos likely o be in luenced by highe
wis (HT) co ec ions o o he p oblems wi h he chosen
heo e ical amewo k.
To ensu e he alidi y o he DGLAP e olu ion equa ions,
we ha e o impose ce ain cu s on he abo e men ioned da a
se s. In o de o inalize he cu on Q2, he sensi i i y o χ2
o a ia ions in Q2>Q2
min is in es iga ed o da a used in
he analysis. Conside ing hese χ2scans, ou ull di ac i e
PDFs i s a e epea ed o each di e en Q2>Q2
min cu .
In Fig. 2, he dependence o χ2pe numbe o deg ees o
eedom, χ2/do , on he minimum cu alue o Q2has been
p esen ed as a unc ion o Q2
min o all inclusi e di ac i e
DIS da a se s used in GKG18-DPDFs (see Table 1). The
Q2
min dependence is e lec ed om his plo and no u he
imp o emen on χ2/do can be expec ed o la ge alue o
Q2>Q2
min =9GeV
2. The e o e, he lowes Q2da a a e
omi ed om ou QCD i and Q2
min ≥9GeV
2is applied o
he di ac i e DIS da a se s. We e e his i o Fi A.
Howe e , his choice is somewha di e en om he cu
used in Re s. [27,28](Q2
min >8.5GeV
2). Since his issue
can be ela ed o he possible ension be ween he H1-LRG-
11 and H1-LRG-12 da a se s wi h he H1/ZEUS combined
da ainlow-Q2bins, some u he in es iga ions a e equi ed.
To esol e his issue, we also p esen simila plo s o he
H1/ZEUS combined da a as well as o all H1 LRG da a
se s. As one can see om he uppe panel o Fig. 3,an
imp o emen on χ2pe numbe o da a poin s, χ2/Np s, can
be expec ed o la ge alue o Q2>Q2
min =16 GeV2 o
he H1/ZEUS combined da a. In Fig. 3,weha ealsoshown
Fig. 3 Dependence o χ2/Np s on he minimum cu alue o Q2
min o
H1/ZEUS combined da a (up) and all H1 LRG da a se s (down)
he same plo o he H1 LRG da a se s. This plo clea ly
shows ha he app op ia e choice o he case o H1 LRG da a
se s is Q2
min >9GeV
2. This ac indica es ha he choice o
Q2
min >9GeV
2is s ill sui able o all da a se s excluding
he H1/ZEUS combined da a. Hence, we epea ed ou anal-
ysis by applying an addi ional cu s on Q2
min ≥16 GeV2 o
he H1/ZEUS combined and keeping Q2
min ≥9GeV
2 o
o he H1-LRG-11 and H1-LRG-12 da a se s. We e e his
i o Fi B. The numbe o da a poin s a e all cu s o
bo h Fi A and Fi B a e summa ized in Tables 1and 2,
espec i ely. No e ha since highe wis (HT) can be po en-
ially la ge is inclusi e di ac i e DIS [66], he choice o
la ge Q2
min also ends o educe he HT in luence.
3.3 The me hod o minimiza ion and di ac i e PDF
unce ain ies
As we al eady discussed, GKG18-DPDFs di ac i e PDFs
a e p o ided a NLO in pe u ba i e QCD and he da a used
in ou i s co e a wide ange o β,xIP and Q2kinema ics.
123
309 Page 8 o 20 Eu . Phys. J. C (2018) 78:309
In o de o achie e an accu a e heo e ical desc ip ions o
bo h he di ac i e PDFs e olu ion and he ha d sca e ing
c oss sec ions, a well- es ed so wa e package is necessa y. In
GKG18-DPDFs analysis, we ha e used he xFi e [67]
which is a s anda d package o pe o ming he global QCD
analysis o PDFs. Fo una ely, he necessa y ools o mak-
ing heo e ical p edic ions o he di ac i e DIS obse ables
ha e been implemen ed in he xFi e , allowing one o
pe o m also a global analysis o di ac i e PDFs. Fo he
minimiza ion, χ2de ini ion and ea men o expe imen-
al unce ain ies, we used he me hodology implemen ed in
xFi e o de e mine he unknown pa ame e s o di ac-
i e PDFs.
The QCD i s a egy ollows closely he one adop ed o
he de e mina ion o he PDFs in he HERAPDF me hodol-
ogy [68,69]. The QCD p edic ions o he inclusi e di ac-
i e c oss sec ion a e ob ained by sol ing he DGLAP e o-
lu ion equa ions a NLO. As we men ioned, he hea y
qua k coe icien unc ions a e calcula ed in he TR GM-
VFNS [5,58] and he hea y qua k masses o cha m and
beau y a e chosen as mc=1.40 GeV and mb=4.75
GeV [60]. The s ong coupling cons an is ixed o he
αs(M2
Z)=0.1176 [70] which is close o he bes - i alue
o NNLO MMHT2014 global PDF analysis, αs(M2
Z)=
0.1172 +±0.0013 [71]. The χ2 unc ion is minimized using
he CERN MINUIT package [72]. The o m o he χ2mini-
mized du ing ou QCD i s is exp essed as ollows [69],
χ2({ξk})=
i
μi−Ti({ξk})(1−jγi
jbj)2
δ2
i,uncT2
i({ξk})+δ2
i,s a μiTi1−jγi
jbk
+
j
b2
j+
i
ln
δ2
i,uncT2
i({ξk})+δ2
i,s a μiTi({ξk})
δ2
i,uncμ2
i+δ2
i,s a μ2
i
,
(9)
whe e μiis he measu ed alue o inclusi e di ac i e c oss
sec ion a poin i, and Tiis he co esponding heo e ical p e-
dic ions. The pa ame e s δi,s a ,δi,unc, and γi
ja e he ela i e
s a is ical, unco ela ed sys ema ic, and co ela ed sys em-
a ic unce ain ies. The nuisance pa ame e s bja e associa ed
o he co ela ed sys ema ics which a e de e mined simul a-
neously wi h he unknown pa ame e s {ξk}o ou unc ional
o mso Eq.(6) and (7). We minimize he abo e χ2 alue
wi h he k=9 unknown i pa ame e s {ξk}o ou di ac i e
PDFs.
Table 3con ains he inal esul s o χ2/Np s o ou global
i s. Fo each da a se , he alue o χ2/Np s has been p e-
sen ed o bo h Fi A and Fi B. In he las ow o he
able, he alues o χ2/do ha e also been p esen ed as well.
These able illus a es he quali y o ou QCD i s o inclu-
si e di ac i e c oss sec ion a NLO accu acy in e ms o he
indi idual χ2 alues ob ained o each expe imen . Fo Fi
Table 3 The alues o χ2/Np s o he da a se s included in he global
i s
Expe imen Fi A Fi B
χ2/Np s χ2/Np s
H1-LRG-11 √s=
225 GeV [42,43]
11/13 12/13
H1-LRG-11 √s=
252 GeV [42,43]
20/12 21/12
H1-LRG-11 √s=
319 GeV [42,43]
6.5/12 6.2/12
H1-LRG-12 [44] 135/165 138/165
H1/ZEUS
combined [45]
128/96 85/70
Co ela ed χ210 11
Log penal y χ2+11 +6.9
χ2/do 322/289 =1.11 280/263 =1.06
Aand Fi B, we ob ain χ2o 322 and 280 wi h he o al
289 and 263 da a poin s, espec i ely. As one can see om
his Table, a Q2
min ≥16 GeV2cu on he H1/ZEUS com-
bined da a se signi ican ly educes he χ2/Np s om 128/96
o 85/70. No e also ha he alues o χ2/Np s o H1-LRG-11
da a se s a √s=225 and 252 GeV do no change om Fi
A o Fi B and jus a e y small educ ion is obse ed o he
H1-LRG-11 (√s=319 GeV)and H1-LRG-12 da a se s. In
conclusion, he quali y o Fi B is sligh ly be e han ha
o Fi A, indica ing a be e desc ip ion o he inclusi e
di ac i e DIS da a. A subs an ial pa o he imp o emen
in he desc ip ion is d i en by he H1/ZEUS combined da a.
In o de o ob ain he unce ain ies on he di ac i e PDFs,
we use he xFi e amewo k, which includes bo h he
expe imen al s a is ical and sys ema ic e o s on he da a
poin s and hei co ela ions in he de ini ion o he χ2 unc-
ion. The unce ain ies on he di ac i e PDFs as well as
he co esponding obse ables h oughou ou analysis a e
compu ed using he s anda d “Hessian” e o p opaga ion
[57,73,74].
4 Resul s and discussions
Key esul s o he cu en NLO di ac i e PDFs i com-
pa ed o all p e ious analyses a e he inclusion o all new
and up- o-da e expe imen al di ac i e DIS da a, in pa ic-
ula , he H1/ZEUS combined da a se [45], and he e o
analysis o he ex ac ed di ac i e PDFs. Since hese new
da a se s may ha e he po en ial o p o ide mo e in o ma ion
on he ex ac ed di ac i e PDFs, i is impo an o p ecisely
s udy hei impac on he di ac i e PDFs as well as on hei
unce ain y bands. The second signi ican addi ion is he i s
123
Eu . Phys. J. C (2018) 78:309 Page 15 o 20 309
0.02 0.03 0.1 0.2 0.3
educed
σ
IP
x
0.01
0.015
0.02
0.025
0.03
0.035
0.04 LRG 2012
=0.01
p
= 12, x
2
H1 Da a Q
unco ela edδ
o alδ
Theo y + shi s
Fi A
Fi B
β
0.02 0.03 0.1 0.2 0.3 0.4
Theo y/Da a
0.9
0.95
1
1.05
1.1 0.02 0.03 0.1 0.2 0.3
educed
σ
IP
x
0.01
0.02
0.03
0.04
0.05
LRG 2012
=0.01
p
= 15, x
2
H1 Da a Q
unco ela edδ
o alδ
Theo y + shi s
Fi A
Fi B
β
0.02 0.03 0.1 0.2 0.3 0.4
Theo y/Da a
0.8
0.9
1
1.1
1.2 0.03 0.1 0.2 0.3 0.4
educed
σ
IP
x
0.015
0.02
0.025
0.03
0.035
0.04
0.045 LRG 2012
=0.01
p
= 20, x
2
H1 Da a Q
unco ela edδ
o alδ
Theo y + shi s
Fi A
Fi B
β
0.03 0.1 0.2 0.3 0.4
Theo y/Da a
0.95
1
1.05
1.1
0.03 0.1 0.2 0.3 1
educed
σ
IP
x
0.015
0.02
0.025
0.03
0.035
0.04
0.045
LRG 2012
=0.01
p
= 25, x
2
H1 Da a Q
unco ela edδ
o alδ
Theo y + shi s
Fi A
Fi B
β
0.03 0.1 0.2 0.3 1
Theo y/Da a
0.8
1
1.2 0.1 0.2 0.3 0.4 1
educed
σ
IP
x
0.015
0.02
0.025
0.03
0.035
0.04
0.045 LRG 2012
=0.01
p
= 35, x
2
H1 Da a Q
unco ela edδ
o alδ
Theo y + shi s
Fi A
Fi B
β
0.04 0.1 0.2 0.3 0.4 1
Theo y/Da a
0.95
1
1.05
1.1 0.1 0.2 0.3 0.4 0.5 1
educed
σ
IP
x
0.015
0.02
0.025
0.03
0.035
0.04
LRG 2012
=0.01
p
= 45, x
2
H1 Da a Q
unco ela edδ
o alδ
Theo y + shi s
Fi A
Fi B
β
0.07 0.1 0.2 0.3 0.4 0.5 1
Theo y/Da a
0.95
1
1.05
0.2 0.3 0.4 0.5 0.6
educed
σ
IP
x
0.015
0.02
0.025
0.03
0.035
0.04 LRG 2012
=0.01
p
= 60, x
2
H1 Da a Q
unco ela edδ
o alδ
Theo y + shi s
Fi A
Fi B
β
0.2 0.3 0.4 0.5 0.6
Theo y/Da a
0.95
1
1.05 0.2 0.3 0.4 0.5 0.6 0.7
educed
σ
IP
x
0.015
0.02
0.025
0.03
0.035
LRG 2012
=0.01
p
= 90, x
2
H1 Da a Q
unco ela edδ
o alδ
Theo y + shi s
Fi A
Fi B
β
0.2 0.3 0.4 0.5 0.6 0.7
Theo y/Da a
0.9
0.95
1
1.05
1.1 0.3 0.4 0.5 0.6 0.7 0.8
educed
σ
IP
x
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045 LRG 2012
=0.01
p
= 200, x
2
H1 Da a Q
unco ela edδ
o alδ
Theo y + shi s
Fi A
Fi B
β
0.3 0.4 0.5 0.6 0.7 0.8
Theo y/Da a
0.9
1
1.1
0.8
educed
σ
IP
x
0.01
0.012
0.014
0.016
0.018
0.02
0.022
0.024
0.026 LRG 2012
=0.01
p
= 400, x
2
H1 Da a Q
unco ela edδ
o alδ
Theo y + shi s
Fi A
Fi B
β
0.8
Theo y/Da a
0.8
0.9
1
1.1
1.2
Fig. 10 The esul s o ou NLO pQCD i based on Fi B o he educed di ac i e c oss sec ion xIP σD(3)
as a unc ion o β o xIP =0.01 in
compa ison wi h H1-LRG-2012 da a [44]. See he cap ion o Fig. 8 o u he de ails
2012 da a used in he analysis as well as he p edic ions based
on he H1-2006 Fi B. Fo he case o xIP =0.003, as
he alues o βand Q2a e inc eased, a be e ag eemen
be ween ou esul s and he H1-2006 Fi B is obse ed.
In he case o H1-LRG-2011 da a [42,43], we p esen in
Fig. 14 he NLO pQCD i s o bo h ou Fi A and Fi
B. This igu e shows, o ins ance, he NLO heo y p edic-
ions o he educed di ac i e c oss sec ion xIP σD(3)
as a
123

309 Page 16 o 20 Eu . Phys. J. C (2018) 78:309
0.01 0.02 0.03
educed
σ
IP
x
0
0.01
0.02
0.03
0.04
0.05
LRG 2012
=0.03
p
= 12, x
2
H1 Da a Q
unco ela edδ
o alδ
Theo y + shi s
Fi A
Fi B
β
0.006 0.01 0.02 0.03
Theo y/Da a
0.8
1
1.2
0.01 0.02 0.03 0.1
educed
σ
IP
x
0.02−
0
0.02
0.04
0.06
0.08 LRG 2012
=0.03
p
= 15, x
2
H1 Da a Q
unco ela edδ
o alδ
Theo y + shi s
Fi A
Fi B
β
0.005 0.01 0.02 0.03 0.1
Theo y/Da a
0.8
1
1.2 0.01 0.02 0.03 0.1
educed
σ
IP
x
0
0.01
0.02
0.03
0.04
0.05
0.06 LRG 2012
=0.03
p
= 20, x
2
H1 Da a Q
unco ela edδ
o alδ
Theo y + shi s
Fi A
Fi B
β
0.009
0.01 0.02 0.03 0.1
Theo y/Da a
0.8
1
1.2
1.4
1.6
0.01 0.02 0.03 0.1 0.2
educed
σ
IP
x
0
0.01
0.02
0.03
0.04
0.05
0.06
LRG 2012
=0.03
p
= 25, x
2
H1 Da a Q
unco ela edδ
o alδ
Theo y + shi s
Fi A
Fi B
β
0.009
0.01 0.02 0.03 0.1 0.2
Theo y/Da a
0.6
0.8
1
1.2
1.4 0.02 0.03 0.1 0.2 0.3
educed
σ
IP
x
0.02−
0
0.02
0.04
0.06
0.08
LRG 2012
=0.03
p
= 35, x
2
H1 Da a Q
unco ela edδ
o alδ
Theo y + shi s
Fi A
Fi B
β
0.02 0.03 0.1 0.2 0.3
Theo y/Da a
0.8
1
1.2 0.03 0.1 0.2 0.3
educed
σ
IP
x
0.02−
0
0.02
0.04
0.06
0.08 LRG 2012
=0.03
p
= 45, x
2
H1 Da a Q
unco ela edδ
o alδ
Theo y + shi s
Fi A
Fi B
β
0.03 0.1 0.2 0.3
Theo y/Da a
0.8
1
1.2
1.4
0.04 0.1 0.2 0.3 0.4 0.5
educed
σ
IP
x
0.01−
0
0.01
0.02
0.03
0.04
0.05
0.06 LRG 2012
=0.03
p
= 60, x
2
H1 Da a Q
unco ela edδ
o alδ
Theo y + shi s
Fi A
Fi B
β
0.04 0.1 0.2 0.3 0.4 0.5
Theo y/Da a
0.8
1
1.2
1.4 0.06 0.1 0.2 0.3 0.4 0.5
educed
σ
IP
x
0
0.01
0.02
0.03
0.04
0.05
0.06 LRG 2012
=0.03
p
= 90, x
2
H1 Da a Q
unco ela edδ
o alδ
Theo y + shi s
Fi A
Fi B
β
0.06 0.1 0.2 0.3 0.4 0.5
Theo y/Da a
0.5
1
1.5 0.1 0.2 0.3 0.4 0.5 0.6
educed
σ
IP
x
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05 LRG 2012
=0.03
p
= 200, x
2
H1 Da a Q
unco ela edδ
o alδ
Theo y + shi s
Fi A
Fi B
β
0.1 0.2 0.3 0.4 0.5 0.6
Theo y/Da a
0.8
1
1.2
0.3 0.4 0.5 0.6 0.7
educed
σ
IP
x
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045 LRG 2012
=0.03
p
= 400, x
2
H1 Da a Q
unco ela edδ
o alδ
Theo y + shi s
Fi A
Fi B
β
0.3 0.4 0.5 0.6 0.7
Theo y/Da a
0.8
1
1.2 0.5 0.6
educed
σ
IP
x
0.01−
0
0.01
0.02
0.03
0.04
0.05
0.06 LRG 2012
=0.03
p
= 800, x
2
H1 Da a Q
unco ela edδ
o alδ
Theo y + shi s
Fi A
Fi B
β
0.5 0.6 0.7
Theo y/Da a
0.8
1
1.2
educed
σ
IP
x
0.005
0.01
0.015
0.02
0.025
0.03
0.035 LRG 2012
=0.03
p
= 1600, x
2
H1 Da a Q
unco ela edδ
o alδ
Theo y + shi s
Fi A
Fi B
β
Theo y/Da a
0.6
0.8
1
1.2
1.4
Fig. 11 The esul s o ou NLO pQCD i based on Fi B o he educed di ac i e c oss sec ion xIP σD(3)
as a unc ion o β o xIP =0.03 in
compa ison wi h H1-LRG-2012 da a [44]. See he cap ion o Fig. 8 o u he de ails
unc ion o β o xIP =0.003 and Q2=11.5GeV
2in com-
pa ison wi h H1-LRG-2011 da a a √s=225 GeV (le )
and 319 GeV ( igh ). The e o ba s on he da a poin s and he
yellow bands ep esen he unco ela ed unce ain ies and he
o al unco ela ed and co ela ed unce ain ies, espec i ely.
As can be seen, in he kinema ics conside ed, he heo y is
again in good ag eemen wi h he expe imen .
123
Eu . Phys. J. C (2018) 78:309 Page 17 o 20 309
10 100
Q2
0.1
1
10
3i. xIPσ
D(3)
H1 LRG 2012
H1 LRG 1997
Fi A
Fi B (H1)
xIP= 0.003
β=0.067 (i=5)
β=0.11 (i=4)
β=0.17 (i=3)
β=0.27 (i=2)
β=0.43 (i=1)
β=0.67 (i=0)
Fig. 12 The esul s o ou NLO pQCD i based on Fi A o he
educed di ac i e c oss sec ion xIP σD(3)
as a unc ion o Q2 o di -
e en alues o βand xIP =0.003. The da a a e co espond o he
H1-LRG-2012 [44] and H1-LRG-1997 [27] measu emen s. The da a
a e mul iplied by a u he ac o o 3i o isibili y, wi h ias indica ed
in pa en heses
F om he esul s p esen ed in his sec ion, one can con-
clude ha ou NLO QCD p edic ions based on he DGLAP
app oach and using di ac i e PDFs ex ac ed om ou QCD
analysis o inclusi e di ac ion DIS da a desc ibe all ana-
lyzed da a well.
5 Summa y and conclusions
In his pape , we ha e p esen ed GKG18-DPDFs, he i s
global QCD analysis o di ac i e PDFs ha makes use o
he H1/ZEUS combined and he mos ecen H1 da a se s on
he educed c oss sec ion o inclusi e di ac i e DIS. P e-
ious de e mina ions o non-pe u ba i e di ac i e PDFs in
he pa on model o QCD [27,28,41] we e based on he olde
di ac i e inclusi e DIS da a om H1 and ZEUS collabo-
a ion. The ad en o p ecise da a om he H1 [42–44] and
H1/ZEUS combined [45] da a se s as well as he widely used
xFi e package o e us he oppo uni y o ob ain a new
se o di ac i e PDFs. The TR GM-VFNS p o ides a igo -
ous heo e ical amewo k o conside ing he hea y-qua ks
10 100 1000
Q2
0.01
0.1
1
10
100
1000
3i. xIPσ
D(3)
H1 LRG 2012
H1 LRG 1997
Fi A
Fi B (H1)
xIP= 0.01
β=0.02 (i=8)
β=0.03 (i=7)
β=0.05 (i=6)
β=0.08 (i=5)
β=0.13 (i=4)
β=0.2 (i=3)
β=0.32 (i=2)
β=0.5 (i=1)
β=0.8 (i=0)
Fig. 13 The esul s o ou NLO pQCD i based on Fi A o he
educed di ac i e c oss sec ion xIP σD(3)
as a unc ion o Q2 o di -
e en alues o βand xIP =0.01. The da a a e co espond o he H1
LRG 2012 [44] and H1-LRG-1997 [27] measu emen s. See he cap ion
o Fig. 12 o u he de ails
con ibu ions and is employed he e o de e mine di ac i e
PDFs o hea y qua ks. The GKG18-DPDFs deli e s o he
i s ime he op imized Hessian e o analysis.
We s udy he impac o he new inclusi e di ac i e DIS
da a se s by p oducing wo di ac i e PDFs using wo di -
e en scena ios. Fi s ly, by conside ing simul aneously he
Q2
min =9GeV
2cu on all analyzed di ac i e DIS da a
se s, and secondly by emo ing H1/ZEUS combined da a
wi h Q2
min <16 GeV2in o de o in es iga e possible en-
sion be ween hese da a se s a small alues o Q2. In o de o
alida e he e iciency and emphasize he phenomenological
impac o his selec ion, he di e ences be ween hese wo
di ac i e PDFs se s a e p esen ed and discussed. We ind
ha bo h o ou di ac i e PDFs de e mina ions a e in e y
good ag eemen wi h he esul s in he li e a u e o he o al
qua k single densi ies.
We also ind di e ences be ween ou esul s and he H1-
2006 DPDFs i o he gluon densi y. The e is much be e
ag eemen be ween GKG18 and ZUES-2010 o he gluon
densi y. Fo he cha m and bo om qua k densi ies, he e a e
insigni ican disc epancies be ween GKG18-DPDFs esul s
and ZEUS-2010 o he small alues o z;z<0.01. Ou he-
o y p edic ions based on he de e mined di ac i e PDFs o
123
309 Page 18 o 20 Eu . Phys. J. C (2018) 78:309
0.08 0.1 0.2 0.3 0.4 0.5
educed
σ
IP
x
0.01−
0
0.01
0.02
0.03
0.04
0.05
0.06 LRG-S225 2011
=0.003
p
= 11.5, x
2
H1 Da a Q
unco ela edδ
o alδ
Theo y + shi s
Fi A
Fi B
β
0.08 0.1 0.2 0.3 0.4 0.5
Theo y/Da a
0.8
1
1.2 0.1 0.2 0.3
educed
σ
IP
x
0.015
0.02
0.025
0.03
0.035
0.04 LRG-S319 2011
=0.003
p
= 11.5, x
2
H1 Da a Q
unco ela edδ
o alδ
Theo y + shi s
Fi A
Fi B
β
0.08 0.1 0.2 0.3 0.4
Theo y/Da a
0.9
0.95
1
1.05
1.1
Fig. 14 The esul s o ou NLO pQCD i based on Fi A and Fi
B o he educed di ac i e c oss sec ion xIP σD(3)
as a unc ion o β
o xIP =0.003 and Q2=11.5GeV
2in compa ison wi h H1-LRG-
2011 da a [42,43]a √s=225 (le ) and 319 ( igh ). The e o ba s on
he da a poin s ep esen he unco ela ed unce ain ies and he yellow
bands ep esen he o al unco ela ed and co ela ed unce ain ies
he educed di ac i e c oss sec ion a e also in sa is ac o y
ag eemen s wi h he da a se s analyzed as well as wi h he
p e ious se o H1 da a se s. The mos signi ican changes a e
seen o he hea y qua k densi ies a small alues o zand
in he inc eased p ecision in he de e mina ion o he gluon
di ac i e PDF due o he inclusion o new p ecise da a. Fo
he u u e, ou main aim is o include he e y ecen di ac-
i e dije p oduc ion da a, which could p o ide an addi ional
cons ain on he de e mina ion o he di ac i e gluon den-
si y.
AFORTRAN sub ou ine, which e alua es he leading
o de (LO) and NLO di ac i e PDFs p esen ed he e o
gi en alues o β,xIP and Q2, can be ob ained om he
au ho s upon eques ia elec onic mail.
Acknowledgemen s We a e g a e ul o Se gey Le onian om H1 col-
labo a ion, Ma hew Wing and Woj ek Slominski om ZEUS collabo a-
ion o many help ul discussions and commen s. We also hank To bj n
Sjs and, Ch is ine O. Rasmussen and Fede ico Albe o Ceccopie i o
de ailed discussions on he GKG18 di ac i e PDFs. Au ho s hank
School o Pa icles and Accele a o s, Ins i u e o Resea ch in Funda-
men al Sciences (IPM) o inancial suppo o his p ojec . HK also
acknowledges he Uni e si y o Science and Technology o Mazan-
da an o inancial suppo p o ided o his esea ch.
Open Access This a icle is dis ibu ed unde he e ms o he C ea i e
Commons A ibu ion 4.0 In e na ional License (h p://c ea i ecomm
ons.o g/licenses/by/4.0/), which pe mi s un es ic ed use, dis ibu ion,
and ep oduc ion in any medium, p o ided you gi e app op ia e c edi
o he o iginal au ho (s) and he sou ce, p o ide a link o he C ea i e
Commons license, and indica e i changes we e made.
Funded by SCOAP3.
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