First global next-to-leading order determination of diffractive parton distribution functions and their uncertainties within the xFitter framework

This is an elec onic ep in o he o iginal a icle. This ep in may di e om he o iginal in pagina ion and ypog aphic de ail. Au ho (s): Ti le: Yea : Ve sion: Please ci e he o iginal e sion: All ma e ial supplied ia JYX is p o ec ed by copy igh and o he in ellec ual p ope y igh s, and duplica ion o sale o all o pa o any o he eposi o y collec ions is no pe mi ed, excep ha ma e ial may be duplica ed by you o you esea ch use o educa ional pu poses in elec onic o p in o m. You mus ob ain pe mission o any o he use. Elec onic o p in copies may no be o e ed, whe he o sale o o he wise o anyone who is no an au ho ised use . 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- 123 309 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 123 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 123 309 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 . 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