Determination of contributions from residual light charged hadrons to inclusive charged hadrons from e + e − annihilation data
Alireza Mohamaditabar, F. Taghavi-Shahri, Hamzeh Khanpour, Maryam Soleymaninia
aa r X i v : . [ h e p - ph ] O c t Determination of contributions from residual light charged hadrons to inclusivecharged hadrons from e + e − annihilation data Alireza Mohamaditabar , ∗ F. Taghavi-Shahri , † Hamzeh Khanpour , , ‡ and Maryam Soleymaninia § Department of Physics, Ferdowsi University of Mashhad, P.O.Box 1436, Mashhad, Iran Department of Physics, University of Science and Technologyof Mazandaran, P.O.Box 48518-78195, Behshahr, Iran School of Particles and Accelerators, Institute for Research inFundamental Sciences (IPM), P.O.Box 19395-5531, Tehran, Iran (Dated: October 8, 2019)In this paper, we present an extraction of the contribution from the “ residual ” light chargedhadrons to the inclusive unidentified light charged hadron fragmentation functions (FFs) atnext-to-leading (NLO) and, for the first time, at next-to-next-to-leading order (NNLO) ac-curacy in perturbative QCD. Considering the contributions from charged pion, kaon and(anti)proton FFs from recent
NNFF1.0 analysis of charged hadron FFs, we determine thesmall but efficient residual charged hadron FFs from QCD analysis of all available singleinclusive unidentified charged hadron data sets in electron-positron ( e + e − ) annihilations.The zero-mass variable flavor number scheme (ZM-VFNS) has been applied to account theheavy flavor contributions. The obtained optimum set of residual charged hadron FFs is ac-companied by the well-known Hessian technique to assess the uncertainties in the extractionof these new sets of FFs. It is shown that the residual contributions of charged hadron FFshave very important impact on the inclusive charged hadron FFs and substantially on thequality and the reliability of the QCD fit. Furthermore, this study shows that the residual contributions become also sizable for the case of heavy quark FFs as well as for the c - and b -tagged cross sections. Contents
I. Introduction ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: Maryam [email protected]
II. QCD analysis framework up to NNLO accuracy III. Phenomenological parametrization up to NNLO IV. Description of experimental observables residual charged hadrons FFs 12B. Uncertainties of residual charged hadrons FFs 13
V. Discussion of QCD fit results and residual charged hadrons FFs
VI. Summary and Conclusions Acknowledgments References I. INTRODUCTION
Quantum chromodynamics (QCD) is known as a fundamental theory of the strong interaction,and hence, it has been a topic of active research in the last decades [1, 2]. Using the asymptoticfreedom in QCD as well as in the perturbative theory, the high energy scattering processes can beanalyzed. The factorization theorem separates the perturbative calculation part of the partoniccross section from the non-perturbative parts of both parton distribution functions (PDFs) andfragmentation functions (FFs). In hadronization processes, when specific hadrons are identified inthe final state, FFs can explain how color-carrying quarks and gluons turn into the color-neutralhadrons. The common method to determine the FFs is to use the experimental data sets which aresensitive to the FFs. The well-known FFs are non-perturbative quantities in factorization theorem,and hence, they need to be extracted from a global QCD analysis [3–20].In the last decade, collinear or integrated FFs have been determined from neutral and chargedhadrons in different high energy processes such as single inclusive electron-positron annihilation(SIA), semi-inclusive deep inelastic lepton-nucleon scattering (SIDIS) and single-inclusive hadronproduction in proton-proton collisions [3, 4, 13–18, 21]. Collinear or integrated FFs are denotedby D hi ( z ) and describe the fragmentation of an unpolarized partons f i into unpolarized hadron h ,where the fraction z of the parton momentum is carried by the hadron. Since the FFs for differenthadrons are universal quantities, they are calculated from different high energy processes at variouscenter-of-mass energies. The recent hadron production data sets incorporate: The SIA measuredby BELLE [22, 23] and
BaBar [24], semi-inclusive deep-inelastic scattering (SIDIS) measured by
HERMES [25] and
COMPASS [26, 27] and, the (anti)proton collisions measured by
CMS [28, 29] and
ALICE [30] at the LHC,
STAR [31] and
PHENIX [32] at RHIC and
CDF [33, 34] at the Tevatron. Thesedata sets cover a wide range of µ and z , and hence, they are sensitive to different parton species.Among them, the measurements of the p T charged-hadron spectra in proton-proton collisionsare sensitive to the gluon FF, and therefore, provide the most stringent constraint on the gluondensity [3].Some of recent analyses determined different identified light charged hadrons, i.e. π ± , K ± and p/ ¯ p [4, 14, 15] and unidentified light charge hadrons h ± [3, 21]. Recently, QCD analyses of heavierhadrons such as D ∗ also have been done up to next-to-next-to leading order (NNLO) [13, 18]. Sincethe calculations for the hadronization processes in SIDIS and pp collisions at NNLO are not yetaccessible, these NNLO analyses are not global and only the SIA experimental data can be usedin such analyses. In recent years, there have been many studies to determine the unpolarized FFsfor light or heavy hadrons. We refer the readers to the Refs. [3, 16, 19, 35, 36] for more details.Although SIA experimental data provide the cleanest access to the FFs, and in comparison to SIDISand pp collision, the FFs are the only non-perturbative objects in the SIA cross sections, SIDIS dataprovides studying the flavor structure of FFs separately and pp collision data is indispensable forconstraining the gluon FFs. Recently, new analyses have been done to determine the unidentifiedlight charged hadron FFs, and these FFs apply to the measurements of the charged-particle spectrain proton-ion and ion-ion collisions by RHIC [37] and LHC [38].In this paper, we determine the FFs of residual light charged hadrons at NLO and NNLO accura-cies using the e + e − annihilation experimental observables. The unidentified light charged hadronsare considered as sum of the identified light charged hadrons such as pion, kaon, (anti)proton andthe residual heavier charged hadrons. Hence, the residual light charged hadrons refers to the FFsfor the fraction of hadrons that are not attributable to charged pions, charged kaons, protons orantiprotons. Although the contribution of the residual charged hadrons is small, but it is non-negligible. Most recently, the NNPDF Collaboration extracted the FFs for unidentified chargedhadrons and they have used the hadron production in proton-(anti)proton collisions data as wellas electron-positron annihilation data in their analysis entitled NNFF1.1h [3]. In addition, the
NNFF1.0 have recently presented the FFs for charged pion, charged kaon and (anti)proton froman analysis of SIA hadron production data [4]. Considering the pion, kaon and (anti)proton asthe most important contributions in unidentified charged hadron production, we apply the FFsof
NNFF1.0 analysis in order to determine the FFs of residual charged hadrons. Moreover, sincethe
NNFF1.0 present the FFs of π ± , K ± and p (¯ p ) up to NNLO accuracy, it enable us to calculatethe residual hadrons FFs up to NNLO approximation. We will show that the consideration of thissmall and important contributions of residual charged hadrons improves the agreements betweentheoretical predictions and the experimental data sets of unidentified charged hadron productionsin SIA. Furthermore, this study shows that the residual contributions become also sizable for thecase of heavy quark FFs as well as for the c - and b -tagged cross sections.The paper is organized as follows. In Sec. II, we discuss the perturbative QCD analysis ofsingle-inclusive hadron production in electron-positron annihilation up to NNLO accuracy. Ourmethodology for the input parametrization at initial scale for the residual charged hadrons ispresented in Sec. III. In Sec. IV, we present all the experimental data sets analyzed in this study aswell as the χ values calculated from our analyses for every data set. The minimization strategy todetermine the FFs at initial scale and the Hessian uncertainty approach to calculate the errors ofFFs are presented in Sec. IV A. In Sec. V, we discuss the behavior of our FFs and compare them toother available FF sets in the literature. We also present a detailed comparison of our theoreticalpredictions with the experimental data in this section. Finally, our summary and conclusion aregiven in Sec. VI. II. QCD ANALYSIS FRAMEWORK UP TO NNLO ACCURACY
In this section, we discuss in details the QCD analysis framework of FFs which is a well estab-lished perturbative QCD (pQCD) framework for analyzing the single-inclusive hadron productionprocesses in e − e + annihilation. The cross section observables are defined based on the structurefunctions F T,L,A ( z, µ ) for the single inclusive e + e − annihilation process of e + e − → γ/Z → h + X at a given center-of-mass energy √ s . The general form for unpolarized inclusive single-particleproduction is given by1 σ d σ h dzd cos θ = 38 (1 + cos θ ) F hT ( z, µ ) + 34 sin θ F hL ( z, µ )+ 34 cos θ F hA ( z, µ ) , (1)where z = 2 E h / √ s is the scaled energy of the hadron h , and θ is the hadron angle relative tothe electron beam. In above equation, the F T and F L are the transverse and longitudinal time-likestructure functions, respectively. The normalization factors σ is equal to σ = 4 πα N c / s . Alsothe asymmetric structure function F A will be omitted by integration of Eq. (1) over θ , and hence,the total cross section can be written as1 σ tot dσ h dz = F hT ( z, µ ) + F hL ( z, µ )= X i X a Z z dxx C i,a ( x, α s ( µ ) , sµ ) D hi ( z/x, µ ) + O ( 1 √ s ) , (2)with i = u, ¯ u, d, ¯ d, ..., g and a = T and L . The differential cross section has been normalized tothe total cross section for e + e − annihilation into hadrons ( σ tot ) which reads σ tot = P q e q σ (1 + α s ( µ ) /π ).The function D hi ( z, µ ) is the fragmentation densities in which describe the probability thatthe parton i fragments to a hadron h . In above equation, C i are the process dependent coefficientfunctions which are given by C a,i ( x, α s ) = (1 − δ aL ) δ iq + a s c (1) a,i ( x ) + a s c (2) a,i ( x ) + h.c.. (3)The coefficient functions are known up to NNLO approximation that have been reported inRefs. [39–41]. According to Eq. (3), the coefficient functions for F L are vanished at leading orderand the C L leading contribution is of order α s . The NNLO QCD corrections to the F L coefficientfunctions, which are O ( a s ), are not known in the literature. Since the perturbative corrections tothe coefficient functions of the longitudinal cross section are only known up to O ( α s ), one cannotanalyses the longitudinal structure function F L data at NNLO accuracy. In addition, we use thepublicly available APFEL code [42] to perform our analysis and the NNLO QCD corrections to the F L are not included in this code. Since NNLO QCD corrections to the corresponding coefficientfunctions, which are O ( α s ), are not known in the literature nor in the APFEL code, we do notinclude the longitudinal experimental data sets in our analysis.Perturbative QCD corrections lead to logarithmic scaling violations via the DGLAP evolutionequations [43–46] which evaluate the FFs with the energy scale Q as ∂∂ ln µ D i ( z, µ ) = X j Z z dxx P ji ( x, α s ( µ )) D j ( zx , µ ) (4)where P ji ( x, α s ( µ )) are purturbative splitting functions P j,i ( x, α s ) = α s π P (0) ji ( x ) + ( α s π ) P (1) ji ( x ) + ( α s π ) P (2) ji ( x ) + h.c.. (5)Commonly, the DGLAP equation is decomposed into a 2 × D i ( z, µ ) in Eq. (4)as well as the numerical computation of the cross section in Eq. (2) are performed using the publiclyavailable APFEL package [42, 47] at NLO and NNLO accuracy in pQCD. This package has beenused in our pioneering works in Refs. [13, 21] as well as many other analyses in literature such as
NNFF [3, 4, 20].
III. PHENOMENOLOGICAL PARAMETRIZATION UP TO NNLO
In this section, we will describe all techniques including the phenomenological parametrizationas well as the assumptions we use for the global analysis of residual charged hadrons FFs. Theunidentified charged hadrons are sum of the identified light charged hadrons that are produced inthe fragmentation of a parton. The light charged hadrons include pion ( π ± ), kaon ( K ± ), proton( p/ ¯ p ) and residual light hadrons. Then, the unidentified charged hadron cross sections can bewritten as a sum of the individual cross sections of π ± , K ± , p/ ¯ p and residual hadrons. Followingthat, the unidentified charged hadron FFs are sum of the FFs of π ± , K ± , p/ ¯ p and residual lighthadrons, D h ± i ( z, µ ) = D π ± i ( z, µ ) + D K ± i ( z, µ ) + D p/ ¯ pi ( z, µ ) + D res ± i ( z, µ ) . (6)Consequently, in order to calculate the FFs of residual hadrons we use the following relation, D res ± i ( z, µ ) = D h ± i ( z, µ ) − X l D li ( z, µ ) , l = π ± , K ± , p/ ¯ p. (7)Our main aim in this analysis is the determination of D res ± ( z, µ ) by including SIA experimen-tal data of the unidentified light charged hadrons and also using the FFs of charged pions, chargedkaons, and (anti)protons from the recent NNFF1.0 sets [4]. The FFs of
NNFF1.0 have been deter-mined from an analysis of single inclusive hadron production data in electron-positron annihilationat leading order (LO), NLO and NNLO accuracy.In the following, we introduce the methodology and the assumptions of our analysis to determinethe residual charged hadrons FFs. In comparison to the other light hadrons, pion productions aremuch more copiously and after pions the production of kaons and protons are more than the others.Then we expect that the D h ± is strongly dominated by these three light hadrons and then thecontribution of residual light hadrons in Eq. (7) seems to be small but rather important. Hence,we choose the most simple functional form for all the parton FFs as follows, D res ± i ( z, µ ) = N i z α i (1 − z ) β i B [2 + α i , β i + 1] , i = u + ¯ u, d + ¯ d, s + ¯ s, c + ¯ c, b + ¯ b, g. (8)The N i in above equation represents the normalizations of FFs and along with the free pa-rameters { ξ i = α i , β i } , they need to be determined from QCD fit to the data. The variation ofthe residual light hadrons distributions at small and large values of momentum fraction z will becontrolled by the α i and β i , respectively.The extraction of charged hadrons FFs in a global QCD analysis of a large body of data atNLO as well as NNLO accuracy requires an extensive number of time-consuming computations ofthe FFs evolution and the corresponding observables in each step of the usual χ minimizationprocedure. The large number of parameters specifying the functional form of the charged hadronsFFs in the QCD fit and the need for a proper assessment of their uncertainties, add to this. Hence,we prefer to choose a simple standard form for our residual charged hadrons FFs as presented inEq. (8). In addition, the available SIA data are not accurate enough to determine all the shapeparameters with sufficient accuracy, and hence, it encourages us to assume a very simple form forthe residual charged hadrons FFs.It should be noted that in our analysis, the initial scale for the above parametrization formis µ = 5 GeV for all partons. Since we use very recent NNFF1.0 sets for π ± , K ± and p/ ¯ p , wechoose the NNFF1.0 initial scale in our analysis. Also the value of charm and bottom masses in ouranalysis are same as the
NNFF1.0 , and hence, we fixed them to m c = 1 .
51 and m b = 4 .
92. We shouldemphasize here that we use the fragmentation functions of pions, kaons, and protons/antiprotonsfrom
NNFF1.0 set [4] at the input parametrization scale µ = 5 GeV, and then we evolved theseFFs with our residual charged hadrons FFs using the APFEL kernel [42].Let us now discuss our final definitions of the residual charged hadrons FFs considered inthis analysis. As a first assumption, the SU (3) flavor symmetry is considered for the light quarks( u, d, s ) since the data are not sensitive to the kind of light quarks, such that D res ± u +¯ u = D res ± d + ¯ d = D res ± s +¯ s . (9)We should mentioned that the charge conjugation symmetry is another assumption that we con-sidered in our analysis, i.e., D res + q = D res − ¯ q . As we previously discussed, based on the SIA taggeddata sets we included, they are only sensitive to the flavor combinations of u + ¯ u + d + ¯ d + s + ¯ s , c + ¯ c and b + ¯ b . Then we can choose separate parametrization form for the heavy quark FFs. Hence, intotal, we have 12 free parameters in our parametrization for the residual charged hadrons FFs. Weshould highlight here that, during the fit procedure and constraining the fit parameters, we foundthat the data used in this analysis can not really put enough constrain for all the parameters andthen some of the parameters should be fixed in the best values of the first minimization. Hence, wefix two of parameters, namely α NLO u +¯ u = 154 . α NLO g = 27 . α NNLO u +¯ u = 153 .
47 and α NNLO g = 24 . χ function and the various methods for the analysis of residual charged hadrons FFsuncertainties. Most of the discussions presented here will follow the pioneering work in Ref. [21]. IV. DESCRIPTION OF EXPERIMENTAL OBSERVABLES
In this section, we will review the experimental data sets used in this analysis to determinethe residual charged hadrons FFs. As we mentioned earlier, we restrict this analysis to SIA andconsider all available tagged and flavor-untagged resentments performed by different experiments,including
ALEPH , OPAL and
DELPHI experiments at CERN,
TASSO experiment at DESY, and
TPC and
SLD experiments at SLAC. The analyzed SIA data sets are summarized in Tables. I, II, III,IV, V and VI. For each data sets we specify the name of the experiment, the correspondingreference, the observable, the center-of-mass energy √ s and the number of analyzed data pointsfor each experiment. These tables also include the χ values for both NLO and NNLO analyses.As we mentioned, SIA data sets are fundamental quantities providing information about quarkfragmentation and are also sensitive to the flavor of q + ¯ q fragmentation functions.In order to avoid the resummation effects at small and large z regions, we exclude the data intheses regions. According to the reasonable result in our analysis, we choose the value z min = 0 . µ = M Z and z min = 0 .
075 for µ < M Z . The kinematic cut for large z is taken tobe z max = 0 . NNFF collaboration [3, 20]. Considering the kinematic cuts, we include the total 474data points at both NLO and NNLO QCD fits. It should be mentioned here that since we includepion, kaon, and proton FFs from NNFF1.0 analysis, their uncertainties should be considered inthe theoretical calculations of the unidentified charged hadron cross sections. For the uncertaintyfrom
NNFF1.0 , we follow the analysis by
DSS07 in Ref. [19] and estimate an average uncertaintyof 5% in all theoretical calculations of the inclusive charged hadron cross sections stemming fromthe uncertainties of pion, kaon, and proton FFs from
NNFF1.0 set. This additional uncertaintyis included in the χ minimization procedure for determining the residual charged hadrons FFs.We apply the simplest way to include a “theory” error which is to add it in quadrature to thestatistical and systematic experimental error in the χ expression. We should mentioned herethat the uncertainties from NNFF1.0 parameterizations are not flat over z and also depend onthis variable, hence one need to properly propagate these uncertainties through the QCD analysis.However, like for the case of DSS07 analysis, we believe that a 5% of the cross section value seemsto be reasonable.In the following, we begin with discussing the measurements of single-inclusive charged hadronproduction in electron-positron annihilation, collected by different experiments. The first source ofinformation on the unidentified charged hadrons is provided by
TASSO experiment at DESY for thetotal inclusive cross section measurements for annihilation into hadron according to the reaction e + e − → hadrons [48]. As indicated in Table. I, these data sets correspond to the four differentcenter-of-mass energies of √ s = 14, 22, 35 and 44 GeV. This measurement covers the range of14 ≤ Q ≤
44 GeV and 0 . ≤ z ≤ .
9. After applying kinematical cuts on the analyzed data sets,we use 60 data points from
TASSO experiment.
Experiment Reference Observable √ s [ GeV ] Number of data points N NLO χ , NLO n N NNLO χ , NNLO n TASSO-14 [48] σ total dσ h ± dz TASSO-22 [48] σ total dσ h ± dz TASSO-35 [48] σ total dσ h ± dz TASSO-44 [48] σ total dσ h ± dz TABLE I: The data sets by
TASSO experiment at DESY used in the present analysis of FFs for residual charged hadrons. For each experiment, we present the observables and corresponding reference, the center-of-mass energy √ s , the number of analyzed data points after kinematical cuts, and the χ values for eachdata set. The details of corrections and the kinematical cuts applied are contained in the text. TPC experiment at SLAC [49] for unidentifiedcharged hadrons is presented in Table. II. This data correspond to the center-of-mass energy of √ s = 29 GeV for the momentum interval 0 . ≤ z ≤ . Experiment Reference Observable √ s [ GeV ] Number of data points N NLO χ , NLO n N NNLO χ , NNLO n TPC [49] σ total dσ h ± dz TABLE II: The data sets by
TPC experiment at SLAC used in the present analysis of FFs for residual charged hadrons. See the caption of Table. I for further details.
Another source of information for unidentified charged hadrons comes from the data collectedby
ALEPH experiment at CERN [50]. As one can see from Table. III, these data sets correspond tothe totalinclusive cross section measurements of charged particles for the center-of-mass energy of √ s = M Z . Experiment Reference Observable √ s [ GeV ] Number of data points N NLO χ , NLO n N NNLO χ , NNLO n ALEPH [50] σ total dσ h ± dz TABLE III: The data sets by
ALEPH experiment at CERN used in the present analysis of FFs for residual charged hadrons. See the caption of Table. I for further details.
In Tables. IV and V, we indicate another key ingredient in our residual charged hadrons FFsanalysis which are the single inclusive hadron production data sets from electron-positron collisionsat
DELPHI and
OPAL experiments at CERN [51–53].
Experiment Reference Observable √ s [ GeV ] Number of data points N NLO χ , NLO n N NNLO χ , NNLO n DELPHI [51] σ total dσ h ± dp h σ total dσ h ± dp h (cid:12)(cid:12)(cid:12)(cid:12) uds σ total dσ h ± dp h (cid:12)(cid:12)(cid:12)(cid:12) b TABLE IV: The data sets by
DELPHI experiment at CERN used in the present analysis of FFs for residual charged hadrons. See the caption of Table. I for further details. Experiment Reference Observable √ s [ GeV ] Number of data points N NLO χ , NLO n N NNLO χ , NNLO n OPAL [53] σ total dσ h ± dz σ total dσ h ± dz (cid:12)(cid:12)(cid:12)(cid:12) uds σ total dσ h ± dz (cid:12)(cid:12)(cid:12)(cid:12) c σ total dσ h ± dz (cid:12)(cid:12)(cid:12)(cid:12) b TABLE V: The data sets by
OPAL experiment at CERN used in the present analysis of FFs for residual charged hadrons. See the caption of Table. I for further details.
Finally, the last source of information on the unidentified charged hadrons is provided by the
SLD experiments at SLAC, (see Table. VI).
SLD data sets correspond to the center-of-mass energyof √ s = 91 .
28 GeV [54]. In total, after kinematic cuts, we use 136 data points provided by thisexperiment.
Experiment Reference Observable √ s [ GeV ] Number of data points N NLO χ , NLO n N NNLO χ , NNLO n SLD [54] σ total dσ h ± dp h σ total dσ h ± dz (cid:12)(cid:12)(cid:12)(cid:12) uds σ total dσ h ± dz (cid:12)(cid:12)(cid:12)(cid:12) c σ total dσ h ± dz (cid:12)(cid:12)(cid:12)(cid:12) b TABLE VI: The data sets by
SLD experiment at CERN used in the present analysis of FFs for residual charged hadrons. See the caption of Table. I for further details.
As one can see from the experiments outlined in this section, variety of SIA data sets have beenused in our analysis to extract the residual charged hadrons FFs. The flavor tagged cross sections,could help to distinguish between the sum of light u , d and s -quarks, as well as the charm andbottom FFs.Stringent constraint for bottom FF comes mainly from the DELPHI , OPAL and
SLD flavor taggeddata sets. For charm FF,
OPAL and
SLD data sets with slightly larger errors are available. As onecan expect, the singlet combination q + ¯ q is constrained well enough with the electron-positronannihilation data, while for gluon FF these data sets could not provide enough information. Gluon2FF is constrained in our fit by the scale dependence of the data. The measurements of the lon-gitudinal inclusive cross sections can be used to extract the fragmentation function for the gluon.Due to the lack of precise data that cover a wide range of energies, this data could also helps toconstrain the gluon FF. The motivation for using the c -tagged and b -tagged data in our analysiscomes mainly from the ability to separate the heavy flavor FFs for charm and bottom. We showthat all the analyzed data sets are reasonably well described by our QCD fits.In the next section, we present the calculation method of uncertainties for the resulting new setof residual charged hadrons FFs. A. The minimization of residual charged hadrons FFs
To determine the best values of the known parameters at NLO and NNLO accuracies, one needto minimize the χ with respect to four free input residual charged hadrons FFs parameters ofEqs. (8). In a global QCD analyses of PDFs as well as FFs, the global goodness-of-fit procedureusually follows the usual method with χ ( { ξ } ) defined as χ ( { ξ } ) = n data X i =1 ( D data i − T theory i ( { ξ } ) ) ( σ data i ) , (10)where { ξ } denotes the set of independent free parameters in the fit, and n data is the number ofdata points included in this analysis, which is n data = 474. In Eq. (10), the quantity D data is themeasured value of a given observable and T theory is the corresponding theoretical estimate for agiven set of parameters { ξ } at the same experimental z and Q points. The widely-used CERNprogram library MINUIT [55] is applied to obtain the best parametrization of the residual chargedhadrons FFs. The experimental errors are calculated from systematic and statistical errors addedin quadrature, ( σ data i ) = ( σ sys i ) + ( σ stat i ) .For all analyzed data sets, we obtained χ / dof = 0 .
699 for the NLO analysis and χ / dof = 0 . χ / dof . The χ values corresponding to each individual data set for each of the NLO andNNLO fits are presented in Tables. I, II, III, IV, V and VI for the TASSO , TPC , ALEPH , DELPHI , OPAL and
SLD , respectively. As one can see, almost for all data set, the NNLO QCD correction lead tothe reduction of individual χ .As one can see, for some certain experiments such as TASSO-35 we obtained relatively largevalue of χ showing a lower agreement in comparison with the other datasets between our theory3predictions and this particular set of data. For other experiment such as ALEPH , the χ is slightlytoo small. These treatments may deserve some detailed discussions. By refereeing to the analysisby DSS07 [19] in which residual unidentified light charged hadron is determined at NLO, one cansee the same conclusion in their analysis. They obtained a relatively small value of χ for the TPC and large value for the DELPHI. It should be emphasize now that the kinematical cuts for the z in DSS07 analysis are different with the cuts we applied in our analysis. They excluded the datasets in the z ≤ . z min = 0 .
02 for data sets at Q = M Z and z min = 0 .
075 for
Q < M Z .We should mention here that most single-inclusive charged hadron production data in electron-positron annihilation come with an additional information on the fully correlated normalizationuncertainty. Since, the simple χ ( { ξ } ) definition needs to be modified in order to account for suchnormalization uncertainties. Therefore, the modified χ function is given by: χ global ( { ξ i } ) = n exp X n =1 (cid:18) − N n ∆ N n (cid:19) + N data n X j =1 ( N n D data j − T theory j ( { ξ i } ) N n δ D data j ! , (11)∆ N n in above equation are the experimental normalization uncertainties quoted by the exper-iments. The relative normalization factors N can be fitted along with the fitted parameters { ξ } of residual charged hadrons FFs and then kept fixed. The relative normalization factors for ourNLO ( N NLO ) and NNLO ( N NNLO ) analyses extracted from fit to the data, are presented in Tables. I,II, III, IV, V and VI.
B. Uncertainties of residual charged hadrons FFs
An important objective in a global QCD analysis of FFs is to estimate uncertainties of thecharged hadrons FFs obtained from the χ optimization. To this end, in the following section,we present our method for the calculation of the residual charged hadrons FFs uncertainties anderror propagation from experimental data points. To obtain the uncertainties in any global FFsanalyses, there are well-defined procedures for propagating experimental uncertainties on the fitteddata points through to the FFs uncertainties. In this paper, “Hessian method” will mainly be themethod of our choice for estimating uncertainties of the residual charged hadrons FFs. Hence, in4our analysis, we apply the “Hessian method” (or error matrix approach), which is based on linearerror propagation and involves the production of eigenvector FFs sets suitable for convenient useby the end user.Originally, the “Hessian method” was widely used in MRST [56] and MSTW08 [57] global QCDanalyses and we also applied this approach in our previous works [58–60]. Therefore, in the presentanalysis, we again follow this method and extract the uncertainties of residual charged hadronsFFs. Following that, an error analysis can be obtained by using the “Hessian matrix”, which isdetermined by running the CERN program library MINUIT [55].The most commonly applied Hessian approach, which is based on the covariance matrix diag-onalization, provides us a simple and efficient method for calculating the uncertainties of residual charged hadrons FFs. The basic assumption of the Hessian approach is a quadratic expansion ofthe global goodness-of-fit quantity, χ , in the fit parameters ξ i near the global minimum,∆ χ global ≡ χ global − χ min = n X i,j =1 ( ξ i − ξ i ) H ij ( ξ j − ξ j ) , (12)where H ij are the elements of the Hessian matrix and n stands for the number of parameters inthe global fit.The uncertainty on a residual charged hadrons FFs D res ( z, ξ i ) is then given by δD res ( z, ξ i ) = (cid:20) ∆ χ n X i,j (cid:18) ∂D res ( z, ξ ) ∂ξ i (cid:19) ξ =ˆ ξ H − ij (cid:18) ∂D res ( z, ξ ) ∂ξ j (cid:19) ξ =ˆ ξ (cid:21) / , (13)where ξ i stand for the fit parameters in the input residual charged hadrons FFs, and ˆ ξ indicatesthe number of parameters which make an extreme value for the related derivative. Running theCERN program library MINUIT , the Hessian or covariance matrix elements for free parameters inour NLO and NNLO residual charged hadrons FFs analyses can be obtained. The uncertaintiesof residual charged hadrons FFs as well as the related observable are estimated using the “Hessianmatrix” explained above and their values at higher µ ( µ > µ ) are calculated using the DGLAPevolution equations.5 V. DISCUSSION OF QCD FIT RESULTS AND
RESIDUAL
CHARGED HADRONS FFS
Now we turn to the numerical results for the residual charged hadrons FFs extracted fromthe following analyses at NLO and NNLO accuracy. In Tables. VII and VIII we present thebest fit parameters for the fragmentation of quarks and gluon into the D res ± at NLO and NNLOaccuracy in pQCD. As we mentioned before, the starting scale is taken to be Q = 5 GeV for allparton species. The values labeled by (*) have been fixed after the first minimization, since theanalyzed SIA data dose not constrain all unknown fit parameters well enough. Regarding the simpleparameterization that we considered in this analysis, one can see from Tables. VII and VIII thatwe fixed the parameter α for the u + ¯ u and gluon FFs. These parameters not being well constrainedby the analyzed datasets. For other densities such as c + ¯ c , this parameter also determined withslightly large uncertainties showing that the heavy flavor tagged cross sections can not constrainthese parameters well enough. However, we prefer to let these parameters to be free in the fit andin the FFs uncertainty determination to give more flexibility to the parameterizations.In addition to the much more flexible input parametrization for residual charged hadrons FFsproposed in Sec. III (see Eq. (8)), we have repeated our QCD analysis with variety of alternativeparameterizations, even more flexible than the one we finally used in our analysis. For example,we have chosen the (1 + γ i z + η i √ z ), even allowing the fit to vary these new parameters. We alsoexamine another parametrization form such as D i ( z, µ ) = N i z α i (1 − z ) β i [1 − e − γ i z ] [36]. None ofthese modifications resulted in any significant improvement in χ optimizations, in the quality ofthe fit to the analyzed SIA data sets, or decreasing of the residual charged hadrons FFs uncertaintybands. This clearly indicates that the present SIA data sets are not really able to discriminatebetween various forms of the input distributions for the small residual charged hadrons FFs andthe stability of the corresponding FFs is not affected, as long as a sufficiently flexible choice ismade. Therefore, we mainly focused on a very simple standard parameterizations for the residual charged hadrons FFs as presented in Eq. (8).In Fig. 1 we present the resulting residual charged hadrons FFs entitled “ Model ” along withestimates of their uncertainty bands at NLO and NNLO accuracy. The resulting residual chargedhadrons FFs zD res ± i ( z, µ ) are shown at the scale of µ = M Z for the singlet Σ = u + ¯ u + d + ¯ d + s + ¯ s , c + ¯ c , b + ¯ b , and gluon FFs at NLO and NNLO accuracy. The shaded bands provide uncertaintyestimates using a criterion of ∆ χ = 1 as allowed tolerance on the χ value of our QCD fit.We have mentioned earlier that in our fit we consider the symmetric total up, down and strangedistributions: zD res ± u +¯ u ( z, µ ) = zD res ± d + ¯ d ( z, µ ) = zD res ± s +¯ s ( z, µ ) = zD res ± Σ ( z, µ ). In this figure,6 TABLE VII: Fit parameters for the fragmentation of quarks and gluon into the D res ± at NLO accuracy.The starting scale is taken to be Q = 5 GeV for all parton species. The values labeled by (*) have beenfixed after the first minimization, since the available SIA data dose not constrain all unknown fit parameterswell enough.flavor i N i α i β i u + u . ± . . ∗ . ± . g . ± . . ∗ . ± . c + c . ± . . ± .
540 20 . ± . b + b . ± . . ± .
369 5 . ± . χ / dof . α s ( M Z ) 0 . ∗ [3, 4] m c . ∗ [3, 4] m b . ∗ [3, 4]TABLE VIII: Same as Table VII but for the NNLO analysis.flavor i N i α i β i u + u . ± . . ∗ . ± . g . ± . . ∗ . ± . c + c . ± . . ± .
503 25 . ± . b + b . ± . . ± .
384 5 . ± . χ / dof . α s ( M Z ) 0 . ∗ [3, 4] m c . ∗ [3, 4] m b . ∗ [3, 4] the yellow bands represent the uncertainty for the NNLO accuracy and green bands indicate theuncertainty of NLO analysis. As one can see, considering the NNLO accuracy leads to a smallerFFs uncertainties. From Tables. VII and VIII one also can conclude that the inclusion of higherorder corrections leads to a smaller values of χ . A. Charged hadrons FFs and comparison with other FF sets
In this section, we give a detailed discussions of the first QCD analysis of residual chargedhadrons FFs at NLO and NNLO which in the following will be referred to as “
Model ”. We now turn7 = M z D r e s ( z , ) z Model (NNLO) Model (NLO) = M z D r e s b + b ( z , ) z Model (NNLO) Model (NLO) - = M z D r e s c + c ( z , ) z Model (NNLO) Model (NLO) - = M z D r e s g ( z , ) z Model (NNLO) Model (NLO)
FIG. 1:
Residual charged hadrons FFs determined from this analysis are shown for zD res ± i ( z, µ ) at thescale of µ = M Z for singlet, c + ¯ c , b + ¯ b , and gluon FFs at NLO and NNLO. The shaded bands provideuncertainty estimates using a criterion of ∆ χ = 1 as an allowed tolerance on the χ value of our QCD fit,as described in the text. to present the charged hadrons FFs determined by using our residual charged hadrons FFs in thisanalysis and compare it with other results in literature. Firstly, in Fig. 2 we compare our results forthe unidentified charged hadron FFs at NLO (sum of our residual FFs with the π ± , K ± and p/ ¯ p FFs from
NNFF1.0 ) with those of the previous charged hadrons FFs,
DSS07 [19], as well as themost recent results from
NNFF1.0 by NNPDF collaboration [4]. It should be mentioned here thatthe
NNFF1.0
FFs are determined for the light identified charged hadrons of π ± , K ± and p/ ¯ p FFs.However the
DSS07
FFs for unidentified light charge hadrons are calculated by sum of the FFs from residual and π ± , K ± and p/ ¯ p . In order to present the impact of our residual FFs in calculation ofunidentified charged hadron FFs, we calculate the total
NNFF1.0
FFs for charged pion, kaon and(anti) proton entitled as “ π ± + K ± + p/ ¯ p NNFF1.0 ” and compare with other FF sets.Since in our analysis we parameterize the q + ¯ q combinations for FFs, and hence, the u + ¯ u, d +¯ d, s + ¯ s, c + ¯ c, b + ¯ b and g FFs can be compared directly to those of other analyses in literature. The8comparison in Fig. 2 is shown at Q = M Z for the NLO analysis. We should mentioned here that,in order to quantitatively assess the impact of the contribution from light quark and antiquarkFFs for the residual charged hadrons, in Fig. 2 we plot the total light quarks and antiquarkscontributions D h ± Σ .The main differences between our charged hadrons FFs results and DSS07 are found for thegluon FF in the region z < .
1. In this region, the gluon FFs from
DSS07 is smaller than ourgluon FFs. At small to large values of z , in the kinematical coverage of the SIA data sets, asexpected, our charged hadrons FFs and π ± + k ± + p/ ¯ p NNFF1.0 for D h ± Σ and gluon are statisticallyequivalent.On the other hand, our total heavy quark-antiquark combinations c + ¯ c and b + ¯ b FFs areonly moderately affected by the residual contributions, which leads to a minor enhancement incomparison with π ± + k ± + p/ ¯ p NNFF1.0 mostly for the whole z region. For these distributions,the inclusion of residual contributions visibly affects the shape of this distribution and the smallcontributions from residual charged hadrons FFs are evident.It should be mentioned here that the NNFF1.1h
FFs for unidentified charged hadrons have beendetermined independently from residual and other light ( π ± , k ± and p/ ¯ p ) FFs [3]. The comparisonof our results with NNFF1.1h , Fig. 2, shows that there are no big difference for the b + ¯ b and D h ± Σ FFs, except for very small values of z . For the c + ¯ c FF, we see a small enhancement for the
NNFF1.1h for all range of momentum fraction z . A relatively big difference has been found for thegluon FF, specially for the region of z < . π ± + k ± + p/ ¯ p NNFF1.0
SIA QCD fit as well as the most recent analysis of
SGK18 [21] at Q = M Z . The main conclusions from the comparison in Fig. 3 are the following: The D h ± Σ andgluon FFs of our analysis at NNLO accuracy and π ± + k ± + p/ ¯ p NNFF1.0 are qualitatively similar,which indicate a very small contributions from the residual charged hadrons FFs. Note howeverthat for the case of heavy flavors, c + ¯ c and b + ¯ b FFs the difference are a little bigger, especiallyfor the case of bottom quark FF over the whole range of z . As already noticed in the discussionof our NLO results, inclusion of the residual charged hadrons FFs in a FFs analysis visibly affectsthe shape of the c + ¯ c and b + ¯ b distributions. The comparison of our results with those of SGK18 are also shown in Fig. 3. As one can see, all the distributions obtained by
SGK18 are larger thanour results for all range of z . Big difference for the gluon FF between these two analyses are alsoevident from Fig. 3. In view of the comparison with other charged hadrons FF sets and clearevidence of different shape of the heavy flavor distributions, it is interesting to consider the small9contributions of residual charged hadrons FFs in any global QCD analysis of FFs.As a final point, we should mentioned here that the uncertainty bands presented in Figs. 2and 3 are only correspond to the uncertainty calculations of our residual charged hadrons FFs asindicated in Fig. 1. m = M z D h S ( z , m ) z Model (NLO) p +k +p/p NNFF1.0 (NLO) DSS07 NNFF1.1h (NLO)- - m = M z D h b + b ( z , m ) z Model (NLO) p +k +p/p NNFF1.0 (NLO) DSS07 NNFF1.1h (NLO) - - m = M z D h c + c ( z , m ) z Model (NLO) p +k +p/p NNFF1.0 (NLO) DSS07 NNFF1.1h (NLO) - - m = M z D h g ( z , m ) z Model (NLO) p +k +p/p NNFF1.0 (NLO) DSS07 NNFF1.1h (NLO)
FIG. 2: Charged hadrons FFs determined from this analysis (solid line) are shown for zD h ± i ( z, µ ) at thescale of µ = M Z for Σ, b + ¯ b , c + ¯ c , and gluon FFs at NLO. The corresponding result from DSS07 [19](dot-dashed lines) as well as the recent identified light FFs from
NNFF1.0 [4] (dashed lines) have been shownfor comparison. Also our results are compared with the most recent unidentified charged hadron FFs from
NNFF1.1h [3] (short-dashed lines). The shaded bands provide the uncertainty calculations of our residual charged hadrons FFs using a criterion of ∆ χ = 1 as an allowed tolerance on the χ value of our QCD fit. In the next section, we compute the theory predictions for the SIA processes based on ourresults for the charged hadrons FFs, and compare results to the analyzed data sets. In addition,in order to discuss the size of contributions from residual charged hadrons FFs, we also presentthe data/theory ratio based on the extracted residual charged hadrons FFs.0 - m = M z D h S ( z , m ) z Model (NNLO) p +k +p/p NNFF1.0 (NNLO) SGK18 (NNLO) m = M z D h b + b ( z , m ) z Model (NNLO) p +k +p/p NNFF1.0 (NNLO) SGK18 (NNLO) - - - m = M z D h c + c ( z , m ) z Model (NNLO) p +k +p/p NNFF1.0 (NNLO) SGK18 (NNLO) - m = M z D h g ( z , m ) z Model (NNLO) p +k +p/p NNFF1.0 (NNLO) SGK18 (NNLO)-
FIG. 3: Charged hadrons FFs determined from this analysis (solid line) are shown for zD h ± i ( z, µ ) at thescale of µ = M Z for Σ, b + ¯ b , c + ¯ c , and gluon FFs at NNLO accuracy. The corresponding result fromthe most recent SGK18 [21] (dot-dashed lines) as well as the recent identified light FFs from
NNFF1.0 [4](dashed lines) have been shown for comparison. The shaded bands provide the uncertainty calculations ofour residual charged hadrons FFs using a criterion of ∆ χ = 1 as an allowed tolerance on the χ value ofour QCD fit. B. Discussion of fit quality and data/theory comparison
After our detailed discussion on the determined residual charged hadrons FFs and details pre-sentation as well as comparison with other results in literature, we are now in position to presentour theory prediction using the extracted charged hadrons FFs. In order to discuss the size ofcontributions from residual charged hadrons FFs, we present in Fig. 4 the data/theory ratio basedon the extracted residual charged hadrons FFs at NNLO accuracy. These ratios are presentedfor the total inclusive, light, heavy quark c - and b -tagged normalized cross sections at √ s = 91 . TASSO data sets which correspond to the smaller values of center-of-mass energy, √ s = 14 , ,
35 and 441 D a t a / T h e o r y ALEPH inclusive Model NNFF Relative uncertainty OPAL inclusive SLD inclusive OPAL uds-tagged DELPHI uds-tagged SLD uds-tagged DELPHI inclusive OPAL c-tagged SLD c-tagged Z DELPHI b-tagged ZZ OPAL b-tagged SLD b-tagged
FIG. 4: The data/theory for our NNLO results for each of the data sets used in this analysis. The ratiosare presented for the total inclusive, light, heavy quark c - and b -tagged normalized cross sections. Theuncertainty bands originate from the uncertainty calculations of our residual charged hadrons FFs. GeV. The uncertainty bands originating from the uncertainty calculations of our residual chargedhadrons FFs also have been presented in these figures. As one can conclude from these figures, themost important effects of the inclusion of residual charged hadrons FFs are for the case of heavyquark c - and b -tagged normalized cross sections.Figure 6 shows the comparison of our QCD fit to the fitted SIA data sets at NLO accuracy,while the comparison for the NNLO analysis is shown in Fig. 7. Notice that in both figures, ourresults are labeled as “ Model ”. It should be noted that in these figures the theoretical predictionsfor cross sections of unidentified light charged hadrons h ± have four contributions: the residual charged hadron contribution determined in our analyses at both NLO and NNLO accuracies andthe charged pion, charged kaon and (anti) proton contributions that are determined by the NNPDFcollaboration in their NNFF1.0 analysis [4]. In order to present the efficient of the residual chargedhadron FFs in theoretical prediction of unidentified charged hadrons, we compare our results2 D a t a / T h e o r y TASSO-14 Model NNFF Relative uncertainty TASSO-22 D a t a / T h e o r y Z TASSO-35 Z TASSO-44
FIG. 5: Same as Fig. 4 but this time for the
TASSO data sets with the smaller values of center-of-massenergy, √ s = 14 , ,
35 and 44 GeV. “ Model ” with the sum of the identified light charge hadrons pion, kaon and (anti) proton from
NNFF1.0 analysis in the following labeled as “
NNFF1.0 ” [4].As one can see from the results presented in these figures, the overall agreement of the SIAexperimental data sets in our global QCD analysis of residual charged hadrons FFs is excellent.All data can be very satisfactorily described by the universal set of residual charged hadrons FFsdetermined from this analysis. It is clear that in Figs. 6 and 7, considering the residual chargedhadron contribution to the unidentified charged hadrons has important role and the theoreticalpredictions for unidentified charged hadrons by adding the residual
FFs (
Model ) have been improvedin comparison with the only pion, kaon and proton FFs (
NNFF1.0 ) at both NLO and NNLOaccuracies. According to these figures, the most improvements are related to the c − and b -taggednormalized cross sections. This finding also is in good agreement with our discussions for the heavyquark FFs and also for the data/theory plots presented in Fig. 4.3 -2 -1 E x c l ud e d fr o m t h e f it / t o t d h / d z | b t a g Z DELPHI OPAL SLD Model (NLO) NNFF1.0 (NLO) -2 -1 E x c l ud e d fr o m t h e f it / t o t d h / d z | c t a g OPAL SLD Model (NLO) NNFF1.0 (NLO) Z -2 -1 E x c l ud e d fr o m t h e f it / t o t d h / d z | L i gh t Z DELPHI OPAL SLD Model (NLO) NNFF1.0 (NLO) -2 -1 E x c l ud e d fr o m t h e f it / t o t d h / d z | T o t a l i n c l u s i v e Z ALEPH DELPHI OPAL Model (NLO) NNFF1.0 (NLO)
FIG. 6: SIA data sets compared to the best-fit results of our NLO QCD analysis of residual chargedhadrons FFs (
Model ; solid lines) for variety of SIA observables including total inclusive, c -tagged, b -taggednormalized cross sections. The theory predictions ( NNFF1.0 ; dashed lines) based on very recent
NNPDF collaboration [4] have also been shown as well.
VI. SUMMARY AND CONCLUSIONS
Let us now come to our summary and conclusions. In this paper, we have presented details ofa new study of the residual charged hadrons contributions in unidentified light charged hadronsat NLO and NNLO approximations, which used experimental information available from single-inclusive unidentified charged hadron production in electron-positron annihilation. The data setsincluded in our analysis are the
ALEPH , OPAL and
DELPHI experiments at CERN; the
TPC and
SLD experiments at SLAC, and
TASSO experiment at DESY. These data sets were used jointly in bothof our NLO and NNLO QCD analyses, allow us to extract the set of residual charged hadrons FFsthat provides the optimal overall description of the SIA data, along with the estimates of theiruncertainties. Since we do not access to the calculations for the hadronization processes in SIDIS4 -2 -1 E x c l ud e d fr o m t h e f it / t o t d h / d z | b t a g Z DELPHI OPAL SLD Model (NNLO) NNFF1.0 (NNLO) -2 -1 E x c l ud e d fr o m t h e f it / t o t d h / d z | c t a g OPAL SLD Model (NNLO) NNFF1.0 (NNLO) Z -2 -1 E x c l ud e d fr o m t h e f it / t o t d h / d z | L i gh t Z DELPHI OPAL SLD Model (NNLO) NNFF1.0 (NNLO) -2 -1 E x c l ud e d fr o m t h e f it / t o t d h / d z | T o t a l i n c l u s i v e Z ALEPH DELPHI OPAL Model (NNLO) NNFF1.0 (NNLO)
FIG. 7: Same as Fig. 6 but for NNLO accuracy. and proton-proton collisions at NNLO, we can not include SIDIS and proton-proton collisionsexperimental data sets in this analysis.The unidentified light charged hadron cross sections are total of the identified light chargedhadron cross sections, i.e. pion, kaon and (anti) proton, and also the residual light charged hadroncross sections. Consequently, in order to determine the residual charged hadron FFs by using theunidentified light charged hadron observables, we need to include the charged pion, charged kaonand (anti) proton FFs. In our analyses we use the light charged hadron FFs of
NNFF1.0
FFs fromthe NNPDF collaboration.We have presented the techniques, the analyzed data sets, the parameterization and our com-putational methods of our residual charged hadrons FFs analysis. Our technique is formulated in z -space using the publicly available APFEL code. We have performed uncertainty estimates for our residual charged hadrons FFs, using the “Hessian method”. We found that the “Hessian approach”yielded consistent results for moderate departures from the best fit, typically for the tolerance of5∆ χ = 1.With the information from SIA data sets alone, one can only obtain the q + ¯ q and gluon FFs.This clearly demonstrates the need for improvements on the FFs from other experiments. For thefuture, one can use any observable from hadron productions as well as SIDIS processes. These datasets would give information on the gluon FFs for a wide range of z , and also would provide a cleannew probe of the light and heavy residual charged hadrons FFs. Our results in this study indicatethat there is significant potential for the small residual charged hadrons contributions in inclusivecharged hadrons and considering the small but efficient of the residual charged hadrons improvethe agreement between the theoretical predictions and experimental observables. Furthermore, thestudy presented in this paper has also shown that the residual contributions become also sizablefor the heavy quark FFs as well as the c - and b -tagged cross sections. To provide further importantinsights into charged hadrons FFs, it will be straightforward to include all the forthcoming datasets in a certain global QCD analysis. Acknowledgments
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