Parton Distribution Functions of the Charged Pion Within The xFitter Framework
Ivan Novikov, Hamed Abdolmaleki, Daniel Britzger, Amanda Cooper-Sarkar, Francesco Giuli, Alexander Glazov, Aleksander Kusina, Agnieszka Luszczak, Fred Olness, Pavel Starovoitov, Mark Sutton, Oleksandr Zenaiev
PParton Distribution Functions of the Charged Pion Within The xFitter
Framework
Ivan Novikov,
1, 2, ∗ Hamed Abdolmaleki, Daniel Britzger, Amanda Cooper-Sarkar, Francesco Giuli, Alexander Glazov, † Aleksander Kusina, Agnieszka Luszczak, Fred Olness, Pavel Starovoitov, Mark Sutton, and Oleksandr Zenaiev (xFitter Developers’ team) Joint Institute for Nuclear Research, Joliot-Curie 6, Dubna, Moscow region, Russia, 141980 Deutsches Elektronen-Synchrotron (DESY), Notkestrasse 85, D-22607 Hamburg, Germany School of Particles and Accelerators, Institute for Research inFundamental Sciences (IPM), P. O. Box 19395-5531, Tehran, Iran. Max-Planck-Institut f¨ur Physik, F¨ohringer Ring 6, D-80805 M¨unchen, Germany Particle Physics, Denys Wilkinson Bdg, Keble Road, University of Oxford, OX1 3RH Oxford, UK University of Rome Tor Vergata and INFN, Sezione di Roma 2,Via della Ricerca Scientifica 1,00133 Roma, Italy Institute of Nuclear Physics Polish Academy of Sciences, PL-31342 Krakow, Poland T. Kosciuszko Cracow University of Technology, PL-30-084, Cracow, Poland Southern Methodist University, Department of Physics,Box 0175 Dallas, TX 75275-0175, United States of America Kirchhoff-Institut f¨ur Physik, Heidelberg University,Im Neuenheimer Feld 227, 69120 Heidelberg, Germany Department of Physics and Astronomy, The University of Sussex, Brighton, BN1 9RH, United Kingdom Hamburg University, II. Institute for Theoretical Physics,Luruper Chaussee 149, D-22761 Hamburg,Germany (Dated: August 14, 2020)We present the first open-source analysis of parton distribution functions (PDFs) of chargedpions using xFitter , an open-source QCD fit framework to facilitate PDF extraction and analyses.Our calculations are implemented at NLO using APPLgrids generated by MCFM. Using currentlyavailable Drell-Yan and photon production data, we find the valence distribution is well-constrained;however, the considered data are not sensitive enough to unambiguously determine sea and gluondistributions. Fractions of momentum carried by the valence, sea and gluon components arediscussed, and we compare with the JAM and GRVPI1 results.
INTRODUCTION
The pion plays an important role in our understand-ing of strong interactions. At the same time, it isa mediator of nucleon-nucleon interactions, a pseudo-Goldstone boson of dynamical chiral symmetry breakingand the simplest q ¯ q state in the quark-parton modelof hadrons. However, from the experimental point ofview, the pion structure is currently poorly understood,especially compared to the proton. Parton distributionfunctions (PDFs) are a primary theoretical constructused to describe hadron structure as it is probed in hardprocesses. Much progress has been made in mapping outthe parton distribution functions of the proton in the lastdecades [1].On the other hand, theoretically, the pion is a simplersystem than the proton. Consequently, the pion struc-ture has been investigated in several nonperturbativetheoretical models. Nambu-Jona-Lasinio model [2–4],Dyson-Schwinger equations [5–11] (DSE), meson cloudmodel [12], and nonlocal chiral-quark model [13–15] makepredictions about certain aspects of PDFs of the charged ∗ [email protected] † [email protected] pion, or even allow calculating PDFs themselves. In thelattice QCD approach first moments of the valence pionPDF have been calculated [16–18], and direct computa-tion of PDF has recently been achieved [19–22].Experimentally, the pion PDF is known mostly fromQCD analyses of Drell-Yan (DY) and prompt photonproduction data [23–26]. Within a dynamical approach,only the relatively well-known valence distribution isdetermined from DY data, with the sea and gluoncontent at a very low initial scale fixed by simplifyingassumptions [27] or constraints of the constituent quarkmodel [28, 29]. While all modern pion PDF extractionsare performed at next-to-leading order (NLO), additionalthreshold-resummation corrections and their impact onthe valence distribution at high x have been studied [30].In addition to DY data, a recent work by the JAMcollaboration [31] included leading neutron (LN) electro-production data obtained from the HERA collider (assuggested in [32]). The latest pion PDF fit by Bourellyand Soffer [33] uses a novel parameterisation at the initialscale Q .In this analysis we approach the pion PDFs from aphenomenological context and introduce a number ofunique features which provide a complementary perspec-tive relative to other determinations. In particular, thecombination of DY (E615 and NA10) and prompt photon a r X i v : . [ h e p - ph ] A ug FIG. 1. Leading order Feynman diagrams for the consideredprocesses: Drell-Yan dimuon production (left) and directphoton production (center and right). (WA70) data provide constraints on both the quarks andgluons in our kinematic range. We also explore the the-oretical uncertainties including variations of the strongcoupling, as well as the factorization and renormalizationscales; consequently, our PDF error bands reflect boththe experimental and theoretical uncertainties. Ouranalysis uses MCFM generated APPLgrids which allowfor efficient numerical computations; additionally, we im-plemented modifications to APPLgrid which allow bothmeson and hadron PDFs in the initial state. This workis implemented in the publicly available xFitter
PDFfitting framework [34]; as such, it is the first open-sourceanalysis of pion PDFs, and this will facilitate futurestudies of meson PDFs as new data become available.The paper is organised as follows: In Section I webriefly discuss the considered data. The adopted PDFparameterisation and decomposition are described in Sec-tion II. Calculation of theoretical predictions is discussedin Section III. Section IV is devoted to the statisticaltreatment used in this work and estimation of the uncer-tainty of the obtained PDFs. Finally, the results of theanalysis are presented and compared to results of otherstudies in Section V.
I. EXPERIMENTAL DATA
This analysis is based on Drell-Yan data from
NA10 [35]and
E615 [36] experiments, and on photon productiondata from the
WA70 [37] experiment. The
NA10 and
E615 experiments studied scattering of a π − beam off atungsten target, with E π = 194 and 286 GeV in the NA10 experiment and E π = 252 GeV in the E615 experiment.The
WA70 experiment used π ± beams and a protontarget. For the Drell-Yan data, the Υ-resonance range,which corresponds to bins with √ τ ∈ [0 . , . √ τ = m µµ / √ s , m µµ is the invariant mass of the muon pair, and √ s is thecenter-of-mass energy of pion-nucleon system.Leading order Feynman diagrams for the consideredprocesses are shown in Fig. 1. The Drell-Yan dataconstrain the valence distribution relatively well, but arenot sensitive to sea and gluon distributions. The promptphoton production data complement the DY data byproviding some sensitivity to the gluon distribution,but have smaller statistics and large uncertainties incomparison to the DY data. Additionally, the predictionsfor prompt photon production have significant theoretical uncertainty, as discussed in Section III. II. PDF PARAMETERISATION
The π − PDF xf ( x, Q ) is parameterized at an initialscale Q = 1 . , just below the charm mass thresh-old m c = 2 .
04 GeV . Neglecting electroweak correctionsand quark masses, charge symmetry is assumed: d = ¯ u ,and SU(3)-symmetric sea: u = ¯ d = s = ¯ s . Underthese assumptions, pion PDFs are reduced to threedistributions: total valence v , total sea S , and gluon g : v = d v − u v = ( d − ¯ d ) − ( u − ¯ u ) = 2( d − u ) = 2 d v ,S = 2 u + 2 ¯ d + s + ¯ s = 6 u,g = g, which we parameterise using a generic form: xv ( x ) = A v x B v (1 − x ) C v (1 + D v x α ) ,xS ( x ) = A S x B S (1 − x ) C S / B ( B S + 1 , C S + 1) , (1) xg ( x ) = A g ( C g + 1)(1 − x ) C g , where B is the Euler beta function, which ensures thatthe A S parameter represents the total momentum frac-tion carried by the sea quarks. The B -parameters deter-mine the low- x behavior, and C -parameters determinethe high- x behavior. Quark-counting and momentumsum rules have the following form for π − : (cid:90) v ( x )d x = 2 , (cid:90) x ( v ( x ) + S ( x ) + g ( x ))d x = 1 . (2)The sum rules determine the values of parameters A v and A g , respectively. The constant factors in the definitionsof v , S , g were chosen in such a way, that (cid:104) xv (cid:105) , (cid:104) xS (cid:105) , (cid:104) xg (cid:105) are momentum fractions of pion carried by the valencequarks, sea quarks, and gluons, respectively (here (cid:104) xf (cid:105) = (cid:82) xf ( x )d x ).The extension D v x α was introduced in xv ( x ) to mit-igate possible bias due to inflexibility of the chosen pa-rameterisation. This extension was omitted in the initialfits ( D v = 0). Afterwards, a parameterisation scan wasperformed by repeating the fit with free D v and differentfixed values of parameter α . The scan showed that only α = has noticeably improved the quality of the fit (seeTable I and Section V for discussion). The additionalfree parameter D v changes the shape of the valencedistribution only slightly (Fig. 2). Similar attempts toadd more parameters of the form (1 + D v x α + E v x β ) didnot result in significant improvement of χ . The finalpresented results use a free D v and α = . III. CROSS-SECTION CALCULATION
PDFs are evolved up from the starting scale Q by solv-ing the DGLAP equations numerically using QCDNUM [38].
TABLE I. Fitted parameter values and χ . The first columncorresponds to the fit with D v = 0. The second column showsresults of the fit with free D v and α = . The uncertaintiesof parameter values do not include scale variations. Thevalence and gluon normalization parameters A v and A g werenot fitted, but were determined based on sum rules (Eq.(2))and values of the fitted parameters. D v =0 free D v χ /N DoF A v .
60 1 . (cid:104) xv (cid:105) .
56 0 . B v . ± .
03 0 . ± . C v . ± .
03 0 . ± . D v − . ± . A S = (cid:104) xS (cid:105) . ± .
08 0 . ± . B S . ± . . ± . C S ± ± A g = (cid:104) xg (cid:105) .
23 0 . C g ± ± The evolution is performed using the variable flavor-number scheme with quark mass thresholds at m c =1 .
43 GeV, m b = 4 . APPLgrid [39] package was used for these calculations.The grids were generated using the
MCFM [40] generator.For Drell-Yan, the invariant mass of the lepton pairwas used for the renormalization and factorisation scales,namely µ R = µ F = m ll . For prompt photon production,the scale was chosen as the transverse momentum of theprompt photon, namely µ R = µ F = p T ( γ ).It was verified that the grid binning was sufficiently fineby comparing the convolution of the grid with the PDFsused for the grid generation and a reference cross-sectionproduced by MCFM . The deviation from the referencecross-section, as well as estimated statistical uncertaintyof the predictions, are an order of magnitude smaller thanthe uncertainty of the data. This check was performedfor each data bin.Both the evolution and cross-section calculations areperformed at next-to-leading order (NLO). For the tung-sten target, nuclear PDFs from nCTEQ15 [41] determina-tion were used. In the case of a proton target, the PDFsfrom ref. [42] were employed. These were also used asthe baseline in the nCTEQ15 study. The use of anotherpopular nuclear PDF set
EPPS16 [43] was omitted be-cause their fit had used the same pion-tungsten DY dataas the present analysis. Considering π − N data, EPPS16 fitted PDFs of tungsten using fixed pion PDFs from anold analysis by GRV [27]. Nevertheless, as the nCTEQ15 and
EPPS16
PDFs are comparable, within uncertainties,this choice should not be consequential.In the case of prompt photon production, the contri-bution of fragmentation photons cannot be accountedfor using the described techniques. The model used in
FIG. 2. The valence distribution when using minimal pa-rameterisation ( D v = 0) and the extended parameterisationwith free D v . The shown uncertainty bands do not includescale variations. the fit included only the direct photons. We estimatethe impact of the missing fragmentation contribution bycomparing the total integrated cross-sections computedusing MCFM for proton-proton collision at the
WA70 energywith and without fragmentation. The relative differenceof 32% is treated as the theoretical uncertainty in overallnormalization of the
WA70 data. In the run withoutfragmentation, Frixione isolation is used. In the otherrun the fragmentation function set
GdRG LO and coneisolation are used. The isolation cone size parameter is R = 0 . IV. STATISTICAL TREATMENT ANDESTIMATION OF UNCERTAINTIES
The PDF parameters are found by minimizing the χ function defined as χ = (cid:88) i ( d i − ˜ t i ) (cid:0) δ syst i (cid:1) + (cid:18)(cid:113) ˜ t i d i δ stat i (cid:19) + (cid:88) α b α , (3)where i is the index of the datapoint and α is theindex of the source of correlated error. The measuredcross-section is denoted by d i , with δ syst i and δ stat i beingrespectively the corresponding systematic and statisticaluncertainties. The t i ’s represent the calculated theorypredictions, and ˜ t i = t i (1 − (cid:80) α γ iα b α ) are theory predic-tions corrected for the correlated shifts. γ iα is the relativecoefficient of the influence of the correlated error source α on the datapoint i , and b α is the nuisance parameterfor the correlated error source α .The error rescaling ˜ δ stat = (cid:113) ˜ t i d i δ stat is used to correctfor Poisson fluctuations of the data. Since statisticaluncertainties are typically estimated as a square root ofthe number of events, a random statistical fluctuationdown in the number of observed events leads to asmaller estimated uncertainty, which gives such pointsa disproportionately large weight in the fit. The errorrescaling corrects for this effect. This correction was onlyused for the Drell-Yan data. TABLE II. The normalization and partial χ for the consid-ered datasets. The normalization uncertainty is presentedas estimated by corresponding experiments. In order toagree with theory predictions, the measurements must bemultiplied by the normalization factor. Deviations from 1 inthe normalization factor lead to a penalty in χ , as describedin Section IV.Experiment Normalizationuncertainty Normalizationfactor χ /N points E615
15 % 1 . ± .
020 206 / NA10 (194 GeV) 6.4% 0 . ± .
014 107 / NA10 (286 GeV) 6.4% 0 . ± .
013 95 / WA70
32% 0 . ± .
012 64 / The nuisance parameters b α are used to account forcorrelated uncertainties. In this analysis the correlateduncertainties consist of the overall normalization uncer-tainties of the datasets, the correlated shifts in predic-tions related to uncertainties from nuclear PDFs, and thestrong coupling constant α S ( M Z ) = 0 . ± . χ viathe penalty term (cid:80) α b α . For overall data normaliza-tion, the coefficients γ iα are relative uncertainties asreported by the corresponding experiments, and, in thecase of the WA70 data, the abovementioned additional32% theoretical uncertainty, (listed in Table II). For theuncertainties from nuclear PDFs and α S , the coefficients γ iα are estimated as derivatives of the theory predictionswith respect to α S and the uncertainty eigenvectors ofthe nuclear PDFs as provided by the nCTEQ15 set. Thislinear approximation is valid only when the minimisationparameters are close to their optimal values. It wasverified that this condition was satisfied for the performedfits.The uncertainty of the perturbative calculation isestimated by varying the renormalization scale µ R andfactorization scale µ F by a factor of two up and down,separately for µ R and µ F . The scales were variedusing APPLgrid , and the variations were coherent for alldata bins. Renormalization scale variation for DGLAPevolution was not performed. We observe a significantdependence of the predicted cross-sections on µ R and µ F :the change in predictions is ∼ Q = 1 . ± . . Thisvariation leads to only a small change in χ (∆ χ < ∼ Q = 2 . the mass threshold m c was shifted upby the same amount. The effect of such a change in thecharm mass threshold by itself was found to be negligible. TABLE III. Momentum fractions of the pion carried bythe valence, sea and gluon PDFs at different scales Q asdetermined in this work in comparison to other studies. (cid:104) xv (cid:105) (cid:104) xS (cid:105) (cid:104) xg (cid:105) Q (GeV )JAM [31] 0 . ± .
01 0 . ± .
02 0 . ± .
02 1.69JAM (DY) 0 . ± .
01 0 . ± .
05 0 . ± .
05 1.69this work 0 . ± .
06 0 . ± .
15 0 . ± .
16 1.69Lattice-3 [18] 0 . ± .
030 4SMRS [25] 0 .
47 4Han et al. [44] 0 . ± .
03 4GRVPI1 [27] 0 .
39 0 .
11 0 .
51 4Ding et al. [11] 0 . ± .
03 0 . ± .
02 0 . ± .
02 4this work 0 . ± .
05 0 . ± .
13 0 . ± .
13 4JAM 0 . ± .
01 0 . ± .
01 0 . ± .
02 5this work 0 . ± .
05 0 . ± .
12 0 . ± .
13 5Lattice-1 [16] 0 . ± .
166 5.76Lattice-2 [17] 0 . ± .
04 5.76this work 0 . ± .
05 0 . ± .
12 0 . ± .
13 5.76WRH [26] 0 . ± .
022 27ChQM-1 [13] 0 .
428 27ChQM-2 [15] 0 .
46 27this work 0 . ± .
04 0 . ± .
10 0 . ± .
10 27SMRS [25] 0 . ± .
02 49this work 0 . ± .
04 0 . ± .
09 0 . ± .
09 49
V. RESULTS
Figure 3 shows the obtained pion PDFs in comparisonto a recent analysis by JAM [31], and to
GRVPI1 [27].The new valence distribution presented here is in goodagreement with JAM, and both differ with the early GRVanalysis. The relatively difficult to determine sea andgluon distributions are different in all three PDF sets,however, this new PDF and the JAM determination agreewithin the larger uncertainties of our fit.In the case of valence distribution, the dominantcontribution to the uncertainty estimate is the variationof the scales µ R and µ F . For the sea and gluon distribu-tions, the missing fragmentation contribution to promptphoton production is the dominant uncertainty source,and the effect of scale variation is also significant. Recallthat JAM used the E615 and NA10 DY data (as we did),but used the HERA leading neutron electroproductiondata while we used a direct photon analysis with a largenormalization uncertainty ( c.f. , Table II).A comparison between experimental data and theorypredictions obtained with the fitted PDFs is presented inFig. 6. Reasonable agreement between data and theoryis observed, with no systematic trends for any of thekinematic regions.We now examine the high- x behavior of the valencePDF. The asymptotic limit of the valence PDF as x → v ( x ) ∼ (1 − x ) behavior while approaches based on theDyson-Schwinger equations (DSE) [8, 9] obtain a very FIG. 3. Comparison between the pion PDFs obtained in this work, a recent determination by the JAM collaboration [31],and the
GRVPI1 pion PDF set [27].FIG. 4. We display the scale variation of the cross sectionfor a sample E615 √ τ bin as a function of x F . Note, thenormalization factor of Table II (1 . ± . x ( x > ∼ . different v ( x ) ∼ (1 − x ) . The discrepancy betweenDSE predictions and fits to pion Drell-Yan data iswell-known [9, 26, 30], and it has been demonstratedthat soft-gluon threshold resummation (which was notincluded in this analysis) may be used to account for thisdisagreement [30]. Alternatively, DSE calculations usinginhomogeneous Bethe-Salpeter equations [9] can producePDFs consistent with the linear behavior of the v ( x ) inthe region covered by DY data, pushing the onset of the(1 − x ) regime to very high x .Although the asymptotic behavior of the valence PDFis a theoretically interesting measurement, we will ex- plain in the following why we are unable to determinethis with the current analysis; conversely, details of theasymptotic region therefore do not impact our extractedpion PDFs.First, the asymptotic DSE results only apply at asymp-totically large x values. While the precise boundaryis a subject of debate, Ref. [9] demonstrates that theperturbative QCD predictions may only set in very near x = 1; hence, the observed (1 − x ) behavior could bereal where the data exists. Consequently, it is entirelypossible to have (1 − x ) behavior at intermediate to large x , but then still find (1 − x ) asymptotically. Exceptfor the threshold-resummed calculation of Ref. [30], thefits to the E615 and NA10 data [28, 29, 31, 35, 36, 46]generally obtain high- x behaviors that are closer to(1 − x ) than the DSE result. This explains how thesemany fits can coexist with the asymptotic DSE limit.What would it take to be able to accurately explore the x → x → x measurements difficult. Fig. 6 displays the full set of datawe fit, and it is evident that the number of data at thelargest x F values is limited. The issues on the theoreticalside are also complex. In Fig. 4 we present the scaledependence for a sample subset of the E615 data. Wesee the relative scale dependence across the x F kinematicrange is generally under control, with the exception of thevery large x F limit; hence, the theoretical uncertaintiesof the NLO calculation increase precisely in the regionrequired to extract the asymptotic behavior. Therefore,we reiterate that this analysis does not possess sufficientprecision to infer definitive conclusions on the asymptotic x → x → { B v , C v , D v } , and FIG. 5. Momentum fractions of the pion as a function of Q . The error bands include all uncertainties described in SectionIV. Analogous momentum fractions in the proton PDF set NNPDF31 nlo as 0118 are shown for comparison. The labeled green,red, and blue points show respectively valence, sea, and gluon momentum fractions as reported by other studies. The referencesand numerical values for these points are listed in Table III. the large x behavior is dominantly controlled by the C v coefficient. In light of the results of Ref. [9], a moreflexible parameterization is required to accommodateseparate x -dependence at both intermediate to large x and then the asymptotic region.The threshold resummation calculation [47] has gen-erated significant interest, in part, because the result-ing pion PDFs had a valence structure closer to theDSE (1 − x ) form. [30] However it is important torecall that the PDFs themselves are not physical ob-servables, but depend fundamentally on the underlyingschemes and scales used for the calculation. If scheme-dependent PDFs are used with properly matched scheme-dependent hard cross sections, the result will yieldscheme-independent observables as in Fig. 6. Addition-ally, were we to perform our analysis with the thresholdresummed scheme, it would be most appropriate to dothis for all the processes including both the DY anddirect photon processes; however PDFs obtained withresummation corrections would also require resummedhard cross sections for the predictions. In contrast, ourNLO analysis effectively absorbs resummation correc-tions (approximately) into the PDFs; but, it can be usedto predict cross sections at NLO for future experimentsusing existing NLO open source tools.To study the restrictions of our parameterization, weintroduce an addition parameter D v for our valencePDF. This term has an impact on the intermediate tolarge x behavior as evidenced by the change on the C v parameter, c.f , Table I. However the improvementin the χ /N DoF is minimal (1.19 vs x -dependence as shown in Fig. 2. Fig. 5 shows the obtained momentum fractions in thepion as a function of Q . Recall that A S is a fit variableand { A v , A g } are determined by the sum rules of Eq.(2).Above the charm and bottom mass thresholds ( Q >m c , m b ), the c and b quarks and anti-quarks are includedin the sea distribution. For comparison, we have overlaidthe results from the other studies listed in Table III; theseresults are consistent within our uncertainties, exceptfor the lattice simulation of Ref. [18] (denoted by label”Lattice-3”) and the GRVPI1 set.Additionally, we have displayed the proton momentumfractions for the NNPDF31 nlo as 0118 [48] set. Relativeto the proton, we find the valence of the pion is larger,the gluon is smaller, and the sea component is similar,within uncertainties. We also note the Q -dependenceof the various components are similar for both the pionand the proton, as they are all determined by the sameDGLAP evolution equations. VI. SUMMARY AND OUTLOOK
We have presented the first open-source analysis ofpion PDFs. We have used Drell-Yan and prompt photonproduction data with
APPLgrids generated from
MCFM to extract the PDFs at NLO. Additionally, we haveperformed a complete analysis of both experimental andtheoretical uncertainties including renormalization andfactorization scale variation ( µ R , µ F ), strong couplingvariation ( α S ), and PDFs (both proton and nuclear).Comparing with other pion PDFs from the litera-ture, our results are similar to JAM, but differ fromthe GRVPI1 set. Although the valence distribution iscomparably well constrained, the considered data arenot sensitive enough to unambiguously determine thesea and gluon distributions. We note our uncertaintiesare larger than JAM due to i) the theoretical uncer-tainties discussed above, and ii) the large normalizationuncertainty on our direct photon analysis (JAM uses LNelectroproduction instead). This is an area where newdata, such as J/ Ψ production, could play an importantrole in constraining the gluon. [49]The data are reasonably well-described by NLO QCD,but the sensitivity to µ R and µ F indicates that next-to-next-to-leading order corrections could be significant,especially in the very large x region; this precludes usfrom extracting the asymptotic behavior of the valencedistribution.We will provide the extracted pion PDFs in the LHAPDF6
PDFs library, and the
APPLgrid grid files in the
Ploughshare [50] grid library. Since xFitter is an open-source program, it provides the community with a ver-satile tool to study meson PDFs which can be extendedto perform new analyses. In particular, when data fromfuture experiments, such as COMPASS++/AMBER,[50]becomes available, studying more flexible parameteriza-tion forms and including corrections beyond NLO will beof interest.
ACKNOWLEDGMENTS
The authors would like to thank the DESY IT de-partment for the provided computing resources and fortheir support of the xFitter developers. We thankTim Hobbs, Pavel Nadolsky, Nobuo Sato, Ingo Schien-bein, and Werner Vogelsang for useful discussions. A.K.acknowledges the support of Polish National Agencyfor Academic Exchange (NAWA) within The BekkerProgramme, and F.O. acknowledges US DOE grant DE-SC0010129.
FIG. 6. Considered experimental data and corresponding theory predictions. The displayed theory predictions includecorrelated shifts. Bands of different colors correspond to different datasets. Width of the bands shows uncertainty of thetheory predictions. The cross-sections are shown in the same format as adopted by corresponding experimental papers. The
E615 data is given as d σ/ (d √ τ d x F ) in nb/nucleon, averaged over each ( √ τ , x F ) bin. The DY data from the NA10 experimentis d σ/ (d √ τ d x F ) in nb/nucleus, integrated over each ( √ τ , x F ) bin. The WA70 data on direct photon production is given asinvariant cross-section E d σ/ d p in pb, averaged over each ( p T , x F ) bin. [1] J. Gao, L. Harland-Lang, and J. Rojo, Phys. Rept. ,1 (2018), arXiv:1709.04922 [hep-ph].[2] T. Hatsuda and T. Kunihiro, Phys. Lett. B185 , 304(1987).[3] S. P. Klevansky, Rev. Mod. Phys. , 649 (1992).[4] R. M. Davidson and E. Ruiz Arriola, Acta Phys. Polon. B33 , 1791 (2002), arXiv:hep-ph/0110291 [hep-ph].[5] T. Nguyen, A. Bashir, C. D. Roberts, and P. C. Tandy,Phys. Rev.
C83 , 062201 (R) (2011), arXiv:1102.2448[nucl-th].[6] L. Chang and A. W. Thomas, Phys. Lett.
B749 , 547(2015), arXiv:1410.8250 [nucl-th].[7] C. Chen, L. Chang, C. D. Roberts, S. Wan, and H.-S.Zong, Phys. Rev.
D93 , 074021 (2016), arXiv:1602.01502[nucl-th].[8] C. Shi, C. Mezrag, and H.-s. Zong, Phys. Rev.
D98 ,054029 (2018), arXiv:1806.10232 [nucl-th].[9] K. D. Bednar, I. C. Clot, and P. C. Tandy,arXiv:1811.12310 (2018), arXiv:1811.12310 [nucl-th].[10] M. Ding, K. Raya, D. Binosi, L. Chang, C. D. Roberts,and S. M. Schmidt, Chin. Phys. C , 031002 (2020),arXiv:1912.07529 [hep-ph].[11] M. Ding, K. Raya, D. Binosi, L. Chang, C. D. Roberts,and S. M. Schmidt, Phys. Rev. D , 054014 (2020),arXiv:1905.05208 [nucl-th].[12] A. Szczurek, H. Holtmann, and J. Speth, Nucl. Phys. A605 , 496 (1996).[13] S.-i. Nam, Phys. Rev.
D86 , 074005 (2012),arXiv:1205.4156 [hep-ph].[14] A. Watanabe, C. W. Kao, and K. Suzuki, Phys. Rev.
D94 , 114008 (2016), arXiv:1610.08817 [hep-ph].[15] A. Watanabe, T. Sawada, and C. W. Kao, Phys. Rev.
D97 , 074015 (2018), arXiv:1710.09529 [hep-ph].[16] C. Best, M. Gockeler, R. Horsley, E.-M. Ilgenfritz,H. Perlt, P. E. L. Rakow, A. Schafer, G. Schierholz,A. Schiller, and S. Schramm, Phys. Rev.
D56 , 2743(1997), arXiv:hep-lat/9703014 [hep-lat].[17] W. Detmold, W. Melnitchouk, and A. W. Thomas, Phys.Rev.
D68 , 034025 (2003), arXiv:hep-lat/0303015 [hep-lat].[18] A. Abdel-Rehim et al. , Phys. Rev.
D92 , 114513(2015), [Erratum: Phys. Rev.D93,no.3,039904(2016)],arXiv:1507.04936 [hep-lat].[19] J.-W. Chen, L. Jin, H.-W. Lin, Y.-S. Liu, A. Schfer, Y.-B. Yang, J.-H. Zhang, and Y. Zhao, arXiv:1804.01483(2018), arXiv:1804.01483 [hep-lat].[20] G. F. de Teramond, T. Liu, R. S. Sufian, H. G. Dosch,S. J. Brodsky, and A. Deur (HLFHS), Phys. Rev. Lett. , 182001 (2018), arXiv:1801.09154 [hep-ph].[21] R. S. Sufian, J. Karpie, C. Egerer, K. Orginos, J.-W. Qiu,and D. G. Richards, Phys. Rev.
D99 , 074507 (2019),arXiv:1901.03921 [hep-lat].[22] R. S. Sufian, C. Egerer, J. Karpie, R. G. Edwards, B. Jo,Y.-Q. Ma, K. Orginos, J.-W. Qiu, and D. G. Richards,(2020), arXiv:2001.04960 [hep-lat].[23] J. F. Owens, Phys. Rev.
D30 , 943 (1984).[24] P. Aurenche, R. Baier, M. Fontannaz, M. N. Kienzle-Focacci, and M. Werlen, Phys. Lett.
B233 , 517 (1989). [25] P. J. Sutton, A. D. Martin, R. G. Roberts, and W. J.Stirling, Phys. Rev.
D45 , 2349 (1992).[26] K. Wijesooriya, P. E. Reimer, and R. J. Holt, Phys. Rev.
C72 , 065203 (2005), arXiv:nucl-ex/0509012 [nucl-ex].[27] M. Gluck, E. Reya, and A. Vogt, Z. Phys.
C53 , 651(1992).[28] M. Gluck, E. Reya, and M. Stratmann, Eur. Phys. J. C2 , 159 (1998), arXiv:hep-ph/9711369 [hep-ph].[29] M. Gluck, E. Reya, and I. Schienbein, Eur. Phys. J. C10 ,313 (1999), arXiv:hep-ph/9903288 [hep-ph].[30] M. Aicher, A. Schafer, and W. Vogelsang, Phys. Rev.Lett. , 252003 (2010), arXiv:1009.2481 [hep-ph].[31] P. C. Barry, N. Sato, W. Melnitchouk, and C.-R. Ji,Phys. Rev. Lett. , 152001 (2018), arXiv:1804.01965[hep-ph].[32] H. Holtmann, G. Levman, N. N. Nikolaev, A. Szczurek,and J. Speth, Phys. Lett.
B338 , 363 (1994).[33] C. Bourrely and J. Soffer, Nuclear Physics A , 118(2019).[34] S. Alekhin et al. , Eur. Phys. J.
C75 , 304 (2015),arXiv:1410.4412 [hep-ph].[35] B. Betev et al. (NA10), Z. Phys.
C28 , 9 (1985), note thatthe original NA10 data have since been revised; updateddata are published in [46].[36] J. S. Conway et al. , Phys. Rev.
D39 , 92 (1989).[37] M. Bonesini et al. (WA70), Z. Phys.
C37 , 535 (1988).[38] M. Botje, Comput. Phys. Commun. , 490 (2011),arXiv:1005.1481 [hep-ph].[39] T. Carli, D. Clements, A. Cooper-Sarkar, C. Gwenlan,G. P. Salam, F. Siegert, P. Starovoitov, and M. Sutton,Eur. Phys. J.
C66 , 503 (2010), arXiv:0911.2985 [hep-ph].[40] J. M. Campbell and R. K. Ellis, Phys. Rev.
D60 , 113006(1999), arXiv:hep-ph/9905386 [hep-ph].[41] K. Kovarik et al. , Phys. Rev.
D93 , 085037 (2016),arXiv:1509.00792 [hep-ph].[42] J. F. Owens, J. Huston, C. E. Keppel, S. Kuhlmann,J. G. Morfin, F. Olness, J. Pumplin, and D. Stump,Phys. Rev.
D75 , 054030 (2007), arXiv:hep-ph/0702159[HEP-PH].[43] K. J. Eskola, P. Paakkinen, H. Paukkunen, andC. A. Salgado, Eur. Phys. J.
C77 , 163 (2017),arXiv:1612.05741 [hep-ph].[44] C. Han, H. Xing, X. Wang, Q. Fu, R. Wang, andX. Chen, “Pion Valence Quark Distributions from Maxi-mum Entropy Method,” (2018), arXiv:1809.01549 [hep-ph].[45] R. J. Holt and C. D. Roberts, Rev. Mod. Phys. , 2991(2010), arXiv:1002.4666 [nucl-th].[46] W. J. Stirling and M. R. Whalley, J. Phys. G19 , D1(1993).[47] E. Laenen, G. F. Sterman, and W. Vogelsang, Phys.Rev. D , 114018 (2001), arXiv:hep-ph/0010080.[48] R. D. Ball et al. (NNPDF), Eur. Phys. J. C77 , 663(2017), arXiv:1706.00428 [hep-ph].[49] W.-C. Chang, J.-C. Peng, S. Platchkov, and T. Sawada,(2020), arXiv:2006.06947 [hep-ph].[50] “Ploughshare,” https://ploughshare.web.cern.chhttps://ploughshare.web.cern.ch