Table Of Content

The PDF4LHC Working Group Interim Report SergeyAlekhin1,2,SimoneAlioli1,Richard D. Ball3,ValerioBertone4,Johannes Blu¨mlein1,Michiel Botje5,JonButterworth6,FrancescoCerutti7,AmandaCooper-Sarkar8,AlbertdeRoeck9, LuigiDelDebbio3,JoelFeltesse10,Stefano Forte11,Alexander Glazov12,AlbertoGuffanti4,Claire Gwenlan8,JoeyHuston13,PedroJimenez-Delgado14,Hung-LiangLai15,Jose´ I.Latorre7,Ronan McNulty16,PavelNadolsky17,SvenOlafMoch1,JonPumplin13,VoicaRadescu18,JuanRojo11, Torbjo¨rnSjo¨strand19,W.J.Stirling20,DanielStump13,Robert S. Thorne6,MariaUbiali21,Alessandro 1 Vicini11,GraemeWatt22,C.-P.Yuan13 1 0 1 Deutsches Elektronen-Synchrotron, DESY,Platanenallee 6,D-15738Zeuthen,Germany 2 2 Institute forHighEnergyPhysics,IHEP,Pobeda1,142281 Protvino,Russia n 3 SchoolofPhysicsandAstronomy, UniversityofEdinburgh, JCMB,KB,MayfieldRd,Edinburgh a EH93JZ,Scotland J 4 Physikalisches Institut, Albert-Ludwigs-Universita¨t Freiburg, Hermann-Herder-Straße 3,D-79104 3 Freiburgi. B.,Germany ] 5 NIKHEF,SciencePark,Amsterdam,TheNetherlands h p 6 DepartmentofPhysicsandAstronomy,UniversityCollege,London,WC1E6BT,UK - 7 Departamentd’Estructura iConstituents delaMate`ria,UniversitatdeBarcelona, Diagonal647, p e E-08028Barcelona, Spain h 8 DepartmentofPhysics,OxfordUniversity,DenysWilkinson Bldg,KebleRd,Oxford,OX13RH,UK [ 9 CERN,CH–1211Gene`ve23,Switzerland;AntwerpUniversity, B–2610Wilrijk, Belgium;University 1 ofCalifornia Davis,CA,USA v 6 10 CEA,DSM/IRFU,CE-Saclay,Gif-sur-Yvetee, France 3 11 Dipartimento diFisica,Universita` diMilanoandINFN,SezionediMilano,ViaCeloria16,I-20133 5 Milano,Italy 0 . 12 Deutsches Elektronensynchrotron DESYNotkestraße 85D–22607Hamburg,Germany 1 0 13 PhysicsandAstronomyDepartment, MichiganStateUniversity, EastLansing, MI48824, USA 1 14 Institut fu¨rTheoretische Physik,Universita¨tZu¨rich,CH-8057Zu¨rich,Switzerland 1 15 TaipeiMunicipal UniversityofEducation, Taipei,Taiwan : v 16 SchoolofPhysics,UniversityCollegeDublinScienceCentreNorth,UCDBelfeld,Dublin4,Ireland i X 17 DepartmentofPhysics,SouthernMethodistUniversity, Dallas,TX75275-0175, USA r 18 Physikalisches Institut, Universita¨tHeidelberg Philosophenweg 12,D–69120Heidelberg, Germany a 19 DepartmentofAstronomyandTheoreticalPhysics,LundUniversity, So¨lvegatan14A,S-22362 Lund,Sweden 20 Cavendish Laboratory, UniversityofCambridge,CB3OHE,UK 21 Institut fu¨rTheoretische Teilchenhysik undKosmologie, RWTHAachenUniversity, D-52056 Aachen,Germany 22 TheoryGroup,PhysicsDepartment,CERN,CH-1211Geneva23,Switzerland Abstract Thisdocument isintended asastudy ofbenchmark crosssections attheLHC (at 7 TeV) at NLO using modern PDFs currently available from the 6 PDF fittinggroupsthathaveparticipated inthisexercise. Italsocontains asuccinct user guide to the computation of PDFs, uncertainties and correlations using available PDFsets. A companion note provides an interim summary of the current recommenda- tions of the PDF4LHCworking group for the use of parton distribution functions (PDFs)andofPDFuncertainties attheLHC,forcrosssection andcross section uncertainty calculations. 2 Contents 1. Introduction 4 2. PDFdeterminations-experimentaluncertainties 5 2.1 Features, tradeoffsandchoices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.11 DataSet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.12 Statistical treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.13 Partonparametrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 PDFdeliveryandusage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.21 Computation ofHessianPDFuncertainties . . . . . . . . . . . . . . . . . . . . 7 2.22 Computation ofMonteCarloPDFuncertainties . . . . . . . . . . . . . . . . . . 8 3. PDFdeterminations-Theoretical uncertainties 10 3.1 Thevalueofα anditsuncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 s 3.2 Computation ofPDF+α uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 s 3.21 CTEQ-CombinedPDFandα uncertainties . . . . . . . . . . . . . . . . . . . 12 s 3.22 MSTW-CombinedPDFandα uncertainties . . . . . . . . . . . . . . . . . . . 12 s 3.23 HERAPDF-α ,modelandparametrization uncertainties . . . . . . . . . . . . 13 s 3.24 NNPDF-CombinedPDFandα uncertainties . . . . . . . . . . . . . . . . . . 13 s 4. PDFcorrelations 15 4.1 PDFcorrelations intheHessianapproach . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.2 PDFcorrelations intheMonteCarloapproach . . . . . . . . . . . . . . . . . . . . . . . 17 5. ThePDF4LHCbenchmarks 18 5.1 Comparison betweenbenchmark predictions . . . . . . . . . . . . . . . . . . . . . . . . 19 5.2 TablesofresultsfromeachPDFset . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5.21 ABMK09NLO5Flavours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5.22 CTEQ6.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.23 GJR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5.24 HERAPDF1.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5.25 MSTW2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.26 NNPDF2.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5.3 Comparison ofW+,W−,Zo rapidity distributions . . . . . . . . . . . . . . . . . . . . 31 6. Summary 31 3 1. Introduction TheLHCexperimentsarecurrentlyproducingcrosssectionsfromthe7TeVdata,andthusneedaccurate predictions forthesecrosssectionsandtheiruncertainties atNLOandNNLO.Crucialtothepredictions andtheiruncertainties arethepartondistributionfunctions(PDFs)obtainedfromglobalfitstodatafrom deep-inelastic scattering, Drell-Yan and jet data. A number of groups have produced publicly available PDFsusingdifferentdatasetsandanalysisframeworks. ItisoneofthechargesofthePDF4LHCworking group toevaluate andunderstand differences amongthePDFsetstobeusedattheLHC,andtoprovide a protocol for both experimentalists and theorists to use the PDFsets to calculate central cross sections at the LHC, as well as to estimate their PDF uncertainty. This current note is intended to be an interim summary of our level ofunderstanding of NLOpredictions as the firstLHCcross sections at7 TeVare beingproduced 1. Theintention istomodifythisnoteasimprovementsindata/understanding warrant. For the purpose of increasing our quantitative understanding of the similarities and differences between available PDFdeterminations, a benchmarking exercise between the different groups was per- formed. This exercise was very instructive in understanding many differences in the PDF analyses: different input data, different methodologies and criteria for determining uncertainties, different ways of parametrizing PDFs, different number of parametrized PDFs, different treatments of heavy quarks, different perturbative orders, different ways of treating α (as an input or as a fit parameter), different s values of physical parameters such as α itself and heavy quark masses, and more. This exercise was s also very instructive in understanding where the PDFs agree and where they disagree: it established a broad agreement of PDFs(and uncertainties) obtained from data sets of comparable size and it singled outrelevantinstances ofdisagreementandofdependence oftheresultsonassumptionsormethodology. Theoutlineofthisinterimreportisasfollows. Thefirstthreesectionsaredevotedtoadescription of current PDF sets and their usage. In Sect. 2. we present several modern PDF determinations, with special regard to the way PDF uncertainties are determined. First we summarize the main features of various sets, then we provide an explicit users’ guide for the computation of PDF uncertainties. In Sect.3.wediscusstheoreticaluncertaintiesonPDFs. Wefirstintroducevarioustheoreticaluncertainties, then we focus on the uncertainty related to the strong coupling and also in this case we give both a presentation of choices made by different groups and a users’ guide for the computation of combined PDF+α uncertainties. FinallyinSect.4.wediscussPDFcorrelationsandthewaytheycanbecomputed. s In Sect. 5. weintroduce the settings forthe PDF4LHCbenchmarks on LHCobservables, present the results from the different groups and compare their predictions for important LHCobservables at 7 TeVatNLO.InSect.6.weconclude andbrieflydiscussprospects forfuturedevelopments. 1ComparisonsatNNLOforW,ZandHiggsproductioncanbefoundinref.[1] 4 2. PDFdeterminations-experimentaluncertainties Experimental uncertainties of PDFs determined in global fits (usually called “PDF uncertainties” for short) reflect three aspects ofthe analysis, and differ because of different choices made ineach of these aspects: (1) the choice of data set; (2) the type of uncertainty estimator used which is used to determine the uncertainties and which also determines the way in which PDFs are delivered to the user; (3) the form and size of parton parametrization. First, we briefly discuss the available options for each of these aspects (at least, those which have been explored by the various groups discussed here) and summarize the choices made by each group; then, we provide a concise user guide for the determination of PDF uncertainties for available fits. We will in particular discuss the following PDF sets (when several releases are available the most recent published ones are given in parenthesis in each case): ABKM/ABM[2, 3], CTEQ/CT(CTEQ6.6[4], CT10[5]), GJR [6, 7], HERAPDF(HERAPDF1.0[8]), MSTW(MSTW08[9]),NNPDF(NNPDF2.0[10]). Thereisasignificanttime-lagbetweenthedevelop- ment of a new PDF and the wide adoption of its use by experimental collaborations, so in some cases, wereportnotonthemostup-to-date PDFfromaparticular group,butinsteadonthemostwidely-used. 2.1 Features,tradeoffsandchoices 2.11 DataSet There is a clear tradeoff between the size and the consistency of a data set: a wider data set contains more information, but data coming from different experiment may be inconsistent to some extent. The choicesmadebythevarious groupsarethefollowing: The CTEQ, MSTW and NNPDF data sets considered here include both electroproduction and • hadroproduction data, in each case both from fixed-target and collider experiments. The electroproduction dataincludeelectron,muonandneutrinodeep–inelastic scattering data(bothinclusive and charm production). The hadroproduction data include Drell-Yan (fixed target virtual photon andcollider W andZ production) andjetproduction 2. The GJR data set includes electroproduction data from fixed-target and collider experiments, and • a smaller set of hadroproduction data. The electroproduction data include electron and muon inclusivedeep–inelasticscatteringdata,anddeep-inelasticcharmproductionfromchargedleptons andneutrinos. Thehadroproductiondataincludesfixed–targetvirtualphotonDrell-Yanproduction andTevatronjetproduction. TheABKM/ABMdata setsinclude electroproduction from fixed-target and collider experiments, • and fixed–target hadroproduction data. The electroproduction data include electron, muon and neutrino deep–inelastic scattering data (both inclusive and charm production). The hadroproduction data include fixed–target virtual photon Drell-Yan production. The most recent version, ABM10[11],includes Tevatronjetdata. TheHERAPDFdatasetincludes allHERAdeep-inelastic inclusive data. • 2.12 Statistical treatment AvailablePDFdeterminations fallintwobroadcategories: thosebasedonaHessianapproachandthose whichuseaMonteCarloapproach. ThedeliveryofPDFsisdifferentineachcaseandwillbediscussed inSect.2.2. WithintheHessianmethod,PDFsaredeterminedbyminimizingasuitablelog-likelihoodχ2func- tion. Different groups mayusesomewhatdifferent definitions ofχ2,forexample, byincluding entirely, 2AlthoughthecomparisonsincludedinthisnoteareonlyatNLO,wenotethat,todate,theinclusivejetcrosssection,unlike theother processes in thelistabove, has been calculated only toNLO,and not toNNLO.Thismay havean impact on the precisionofNNLOglobalPDFfitsthatincludeinclusivejetdata. 5 or only partially, correlated systematic uncertainties. While some groups account for correlated uncer- taintiesbymeansofacovariancematrix,othergroupstreatsomecorrelatedsystematics(specificallybut not exclusively normalization uncertanties) as a shift of data, with a penalty term proportional to some poweroftheshiftparameteraddedtotheχ2. Thereaderisreferredtotheoriginalpapersfortheprecise definition adopted by each group, but it should be born in mind that because of all these differences, valuesoftheχ2 quotedbydifferentgroupsareingeneralonlyroughly comparable. With the covariance matrix approach, we can define χ2 = 1 (d d¯)cov (d d¯), d¯ Ndat i,j i − i ij j − j i are data, d theoretical predictions, N is the number of data points (note the inclusion of the factor i dat P 1 inthe definition) andcov isthecovariance matrix. Different groups mayusesomewhat different Ndat ij definitions ofthecovariance matrix, byincluding entirely oronlypartially correlated uncertainties. The best fit is the point in parameter space at which χ2 is minimum, while PDF uncertainties are found by diagonalizing the (Hessian) matrix of second derivatives of the χ2 at the minimum (see Fig. 1) and then determining the range ofeach orthonormal Hessian eigenvector whichcorresponds to aprescribed increase oftheχ2 function withrespecttotheminimum. Inprinciple,thevariationoftheχ2whichcorresponds toa68%confidence(onesigma)is∆χ2 = 1. However, a larger variation ∆χ2 = T2, with T > 1 a suitable “tolerance” parameter [12, 13, 14] may turn out to be necessary for more realistic error estimates for fits containing a wide variety of input processes/data, and in particular in order for each individual experiment which enters the global fit to be consistent with the global best fit to one sigma (or some other desired confidence level such as 90%). Possible reasons why this is necessary could be related to data inconsistencies or incompatibili- ties,underestimatedexperimentalsystematics,insufficientlyflexiblepartonparametrizations, theoretical uncertainties orapproximation inthePDFextraction. Atpresent, HERAPDFandABKMuse∆χ2 = 1, GJR uses T 4.7 at one sigma (corresponding to T 7.5 at 90% c.l.), CTEQ6.6 uses T = 10 at ≈ ≈ 90% c.l. (corresponding to T 6.1 to one sigma) and MSTW08 uses a dynamical tolerance [9], i.e. a ≈ differentvalueofT foreacheigenvector, withvaluesforonesigmarangingfromT 1toT 6.5and ≈ ≈ mostvaluesbeing2 < T < 5. Within the NNPDF method, PDFs are determined by first producing a Monte Carlo sample of N pseudo-data replicas. Eachreplicacontainsanumberofpointsequaltothenumberoforiginaldata rep points. The sample is constructed in such a way that, in the limit N , the central value of the rep → ∞ i-th data point is equal to the mean over the N values that the i-th point takes in each replica, the rep uncertainty ofthesamepointisequaltothevarianceoverthereplicas, andthecorrelations betweenany two original data points is equal to their covariance over the replicas. From each data replica, a PDF replica is constructed by minimizing a χ2 function. PDF central values, uncertainties and correlations are then computed by taking means, variances and covariances over this replica sample. NNPDF uses a Monte Carlo method, with each PDF replica obtained as the minimum χ2 which satisfies a cross- validation criterion [15, 10], and is thus larger than the absolute minimum of the χ2. This method has beenusedinallNNPDFsetsfromNNPDF1.0onwards. 2.13 Partonparametrization Existing parton parametrizations differ in the number of PDFs which are independently parametrized andinthefunctionalformandnumberofindependent parametersused. Theyalsodifferinthechoiceof individual linear combinations of PDFswhich are parametrized. In what concerns the functional form, themostcommonchoiceisthateachPDFatsomereference scaleQ isparametrized as 0 f (x,Q ) = Nxαi(1 x)βig (x) (1) i 0 i − where g (x) is a function which tends to a constant both for x 1 and x 0, such as for instance i → → g (x) = 1+ǫ √x+D x+E x2 (HERAPDF).The fitparameters are α , β and the parameters in g . i i i i i i i 6 Someoftheseparametersmaybechosentotakeafixedvalue(includingzero). ThegeneralformEq.(1) isadopted inallPDFsetswhichwediscuss hereexceptNNPDF,whichinstead lets f (x,Q )= c (x)NN (x) (2) i 0 i i whereNN (x)isaneuralnetwork,andc (x)isisa“preprocessing” function. Thefitparametersarethe i i parameters whichdeterminetheshapeoftheneuralnetwork(a2-5-3-1feed-forward neuralnetworkfor NNPDF2.0). Thepreprocessing functionisnotfitted,butratherchosenrandomlyinaspaceoffunctions ofthegeneral formEq.(2)withinsomeacceptable rangeoftheparameters α andβ ,andwithg = 1. i i i Thebasisfunctions andnumberofparametersarethefollowing. ABKMparametrizes thetwolightestflavoursandantiflavours, thetotalstrangeness andthegluon • (fiveindependent PDFs)with21freeparameters. CTEQ6.6andCT10parametrizethetwolightestflavoursandantiflavoursthetotalstrangenessand • thegluon(sixindependent PDFs)withrespectively 22and26freeparameters. GJRparametrizes thetwolightest flavours andantiflavours andthegluon with20freeparameters • (five independent PDFs); the strange distribution is assumed to be either proportional to the light seaortovanishatalowscaleQ < 1GeVatwhichPDFsbecomevalence-like. 0 HERAPDFparametrizes thetwolightestflavours, u¯,thecombinationd¯+s¯andthegluonwith10 • freeparameters (sixindependent PDFs),strangeness isassumedtobeproportional tothed¯distribution; HERAPDF also studies the effect of varying the form of the parametrization and of and varyingtherelativesizeofthestrangecomponentandthusdetermineamodelandparametrization uncertainty (seeSect.3.23formoredetails). MSTWparametrizesthethreelightestflavoursandantiflavours andthegluonwith28freeparam- • eters(sevenindependent PDFs)tofindthebestfit,but8areheldfixedindetermining uncertainty eigenvectors. NNPDF parametrizes the three lightest flavours and antiflavours and the gluon with 259 free pa- • rameters(37foreachofthesevenindependent PDFs). 2.2 PDFdeliveryandusage Thewayuncertainties shouldbedeterminedforagivenPDFsetdependsonwhetheritisaMonteCarlo set (NNPDF) or a Hessian set (all other sets). We now describe the procedure to be followed in each case. 2.21 Computation ofHessianPDFuncertainties For Hessian PDF sets, both a central set and error sets are given. The number of eigenvectors is equal to the number of free parameters. Thus, the number of error PDFs is equal to twice that. Each error set corresponds to moving by the specified confidence level (one sigma or 90% c.l.) in the positive or negativedirection ofeachindependent orthonormal Hessianeigenvector. Consider avariable X; itsvalue using the central PDFfor an error set isgiven byX . X+ isthe 0 i − value of that variable using the PDF corresponding to the “+” direction for the eigenvector i, and X i thevalueforthevariableusingthePDFcorresponding tothe“ ”direction. − N ∆X+ = [max(X+ X ,X− X ,0)]2 max v i − 0 i − 0 ui=1 uX t N ∆X− = [max(X X+,X X−,0)]2 (3) max v 0 − i 0 − i ui=1 uX t 7 2-dim (i,j) rendition of d-dim (~16) PDF parameter space contours of constant 2global u: eigenvector in thec l-direction l aj p(i) ps(i:) :g ploobinat lo mf lianrigmesut mai with tolerance T zl p(i) u 0 l u l diagonalization and(cid:13) T s0 the irteesrcaatilvineg m beyt(cid:13)hod s0 zk a i H(cid:13) essian eigenvector basis sets (a)(cid:13) (b)(cid:13) Original parameter basis Orthonormal eigenvector basis Fig. 1: A schematic representation of the transformation from the PDF parameter basis to the orthonormal eigenvector basis[13]. ∆X+ addsinquadraturethePDFerrorcontributions thatleadtoanincreaseintheobservableX, − and ∆X thePDFerror contributions that lead toadecrease. Theaddition inquadrature isjustified by theeigenvectorsforminganorthonormalbasis. ThesumisoverallN eigenvectordirections. Ordinarily, oneofX+ X andX− X willbepositiveandonewillbenegative,andthusitistrivialastowhich i − 0 i − 0 termistobeincludedineachquadraticsum. Forthehighernumber(lesswell-determined)eigenvectors, however, the“+”and “ ”eigenvector contributions maybeinthesamedirection. Inthiscase, onlythe − morepositive term willbeincluded inthecalculation of∆X+ andthe morenegative inthe calculation − of ∆X [24]. Thus, there may be less than N non-zero terms for either the “+” or “ ” directions. A − symmetricversionofthisisalsousedbymanygroups, givenbytheequation below: N 1 ∆X = [X+ X−]2 2v i − i ui=1 uX t (4) Inmostcases,thesymmetricandasymmetricformsgiveverysimilarresults. Theextenttowhich thesymmetricandasymmetricerrorsdonotagreeisanindication ofthedeviationoftheχ2 distribution from a quadratic form. The lower number eigenvectors, corresponding to the best known directions in eigenvector space, tend to have very symmetric errors, while the higher number eigenvectors can have asymmetricerrors. Theuncertaintyforaparticularobservablethenwill(willnot)tendtohaveaquadratic form if it is most sensitive to lower number (higher number) eigenvectors. Deviations from a quadratic formareexpected tobegreaterforlarger excursions, i.e. for90%c.l. limitsthanfor68%c.l. limits. TheHERAPDFanalysisalsoworkswiththeHessianmatrix,definingexperimentalerrorPDFsin anorthonormalbasisasdescribedabove. ThesymmetricformulaEq.4ismostoftenusedtocalculatethe experimental errorbandsonanyvariable, butitispossible tousetheasymmetricformulaasforMSTW andCTEQ.(ForHERAPDF1.0theseerrorsareprovided at68%c.l. intheLHAPDFfile: HERAPDF10 EIG.LHgrid). Other methods of calculating the PDF uncertainties independent of the Hessian method, such as theLagrangeMultiplierapproach [12],arenotdiscussed here. 2.22 Computation ofMonteCarloPDFuncertainties FortheNNPDFMonteCarloset,aMonteCarlosampleofPDFsisgiven. Theexpectation valueofany observable [ q ](forexampleacross–section) whichdependsonthePDFsiscomputedasanaverage F { } 8 overtheensembleofPDFreplicas, usingthefollowingmasterformula: 1 Nrep [ q ] = [ q(k) ], (5) hF { }i N F { } rep k=1 X whereN isthenumberofreplicas ofPDFsintheMonteCarloensemble. Theassociated uncertainty rep isfoundasthestandard deviation ofthesample,according totheusualformula 1/2 N σF = rep [ q ]2 [ q ] 2 Nrep 1 F { } −hF { }i ! − (cid:16)D E (cid:17) 1/2 1 Nrep 2 = [ q(k) ] [ q ] . (6) N 1 F { } −hF { }i  rep − kX=1(cid:16) (cid:17)   Theseformulae mayalsobe usedfor thedetermination ofcentral values and uncertainties oftheparton distribution themselves, in which case the functional is identified with the parton distribution q : F [ q ] q. Indeed,thecentralvalueforPDFsthemselves isgivenby F { } ≡ 1 Nrep q(0) q = q(k) . (7) ≡ h i N rep k=1 X NNPDF provides both sets of N = 100 and N = 1000 replicas. The larger set ensures rep rep that statistical fluctuations are suppressed so that even oddly-shaped probability distributions such as non-gaussian orasymmetriconesarewellreproduced, andmoredetailed featuresoftheprobability distributions suchascorrelation coefficientsoruncertainties onuncertainties canbedetermined accurately. However, formostcommonapplications such asthedetermination oftheuncertainty onacross section the smaller replica set is adequate, and in fact central values can be determined accurately using a yet smaller number of PDFs (typically N 10), with the full set of N 100 only needed for the rep rep ≈ ≈ reliabledetermination ofuncertainties. NNPDFalsoprovides aset0intheNNPDF20 100.LHgridLHAPDFfile,asinprevious releases oftheNNPDFfamily, whilereplicas 1to100correspond toPDFsets1to100inthesamefile. Thisset 0contains the average of the PDFs,determined using Eq. (7): inother words, set 0contains the central NNPDF prediction for each PDF. This central prediction can be used to get a quick evaluation of a centralvalue. However,itshouldbenoticedthatforany [ q ]whichdependsnonlinearlyonthePDFs, F { } [ q ] = [ q(0) ]. Thismeansthatacrosssectionevaluatedfromthecentralsetisnotexactlyequal hF { }i 6 F { } to the central cross section (though it will be for example for deep-inelastic structure functions, which are linear in the PDFs). Hence, use of the 0 set is not recommended for precision applications, though in most cases it will provide a good approximation. Note that set q(0) should not be included when computing anaveragewithEq.(5),becauseitisitselfalready anaverage. Equation(6)providesthe1–sigmaPDFuncertaintyonageneralquantitywhichdependsonPDFs. However, an important advantage of the Monte Carlo method is that one does not have to rely on a Gaussian assumption or on linear error propagation. As a consequence, one may determine directly a confidence level: e.g. a 68% c.l. for [ q ] is simply found by computing the N values of and rep F { } F discarding the upper and lower 16% values. In a general non-gaussian case this 68% c.l. might be asymmetric and not equal to the variance (one–sigma uncertainty). For the observables of the present benchmark study the 1–sigma and 68% c.l. PDFuncertainties turn out to be very similar and thus only theformer are given, but this isnot necessarily thecase iningeneral. Forexample, the onesigma error bandontheNNPDF2.0largexgluonandthesmallxstrangeness ismuchlargerthanthecorresponding 68%CLband,suggestingnon-gaussianbehavioroftheprobabilitydistributionintheseregions,inwhich PDFsarebeingextrapolated beyondthedataregion. 9 3. PDFdeterminations-Theoretical uncertainties TheoreticaluncertaintiesofPDFsdeterminedinglobalfitsreflecttheapproximationsinthetheorywhich is used in order to relate PDFs to measurable quantities. The study of theoretical PDF uncertainties is currently less advanced that that of experimental uncertainties, and only some theoretical uncertainties have been explored. One might expect that the main theoretical uncertainties in PDF determination should be related to the treatment of the strong interaction: in particular to the values of the QCD parameters, specifically the value of the strong coupling α and of the quark masses m and m and s c b uncertainties related to the truncation of the perturbative expansion (commonly estimated through the variation of renormalization and factorization scales). Further uncertainties are related to the treatment ofheavy quarkthresholds, whicharehandled invarious waysbydifferent groups(fixedflavournumber vs. variable flavour number schemes, and in the latter case different implementations of the variable flavournumber scheme), andtofurther approximations such astheuseofK-factorapproximations. Fi- nally,moreuncertainties mayberelatedtoweakinteraction parameters (suchastheW mass)andtothe treatmentofelectroweakeffects(suchasQEDPDFevolution [16]). Of these uncertainties, the only one which has been explored systematically by the majority of the PDF groups is the α uncertainty. The way α uncertainty can be determined using CTEQ, HER- s s APDF, MSTW, and NNPDF will be discussed in detail below. HERAPDF also provides model and parametrization uncertainties which include the effect of varying m and m , as well as the effect of b c varyingthepartonparametrization, aswillalsobediscussed below. Setswithvaryingquarkmassesand their implications have recently been made available by MSTW [17], the effects of varying m and m c b have been included by ABKM [2] and preliminary studies of the effect of m and m have also been b c presentedbyNNPDF[18]. Uncertainties relatedtofactorization andrenormalization scalevariationand toelectroweakeffectsaresofarnotavailable. ForthebenchmarkingexerciseofSec.5.,resultsaregiven adoptingcommonvaluesofelectroweakparameters,andatleastonecommonvalueofα (thoughvalues s forother values ofα are alsogiven), butno attempt hasyet been madetobenchmark the other aspects s mentioned above. 3.1 Thevalueofα anditsuncertainty s We thus turn to the only theoretical uncertainty which has been studied systematically so far, namely theuncertainty onα . Thechoice ofvalue ofα isclearly important because itisstrongly correlated to s s PDFs, especially the gluon distribution (the correlation of α with the gluon distribution using CTEQ, s MSTW and NNPDF PDFs is studied in detail in Ref. [19]). See also Ref. [2] for a discussion of this correlationintheABKMPDFs. Therearetwoseparateissuesrelatedtothevalueofα inPDFfits: first, s the choice ofα (m )for which PDFsare madeavailable, and second the choice ofthe preferred value s Z of α to be used when giving PDFsand their uncertainties. Thetwo issues are related but independent, s andforeachofthetwoissuetwodifferent basicphilosophies maybeadopted. Concerning therangeofavailable valuesofα : s PDFsfitsareperformedforanumberofdifferentvaluesofα . ThoughaPDFsetcorrespondingto s • somereferencevalueofα isgiven,theuserisfreetochooseanyofthegivensets. Thisapproach s is adopted by CTEQ (0.118), HERAPDF (0.1176), MSTW (0.120) and NNPDF (0.119), where wehavedenoted inparenthesis thereference (NLO)valueofα foreachset. s α (m )istreatedasafitparametersandPDFsaregivenonlyforthebest–fitvalue. Thisapproach s Z • isadoptedbyABKM(0.1179)andGJR(0.1145),whereinparenthesisthebest-fit(NLO)valueof α isgiven. s Concerning thepreferred centralvalueandthetreatmentoftheα uncertainty: s Thevalueofα (m )istakenasanexternalparameter, alongwithotherparametersofthefitsuch s Z • as heavy quark masses or electroweak parameter. This approach is adopted by CTEQ, HERA- PDF1.0and NNPDF.Inthis case, there isno apriori central value ofα (m )and the uncertainty s Z 10