Simultaneous extraction of fragmentation functions of light charged hadrons with mass corrections

Achieving the highest possible precision for theoretical predictions at the present and future high-energy lepton and hadron colliders requires a precise determination of fragmentation functions (FFs) of light and heavy charged hadrons from a global QCD analysis with great accuracy. We describe a simultaneous determination of unpolarized FFs of charged pions, charged kaons and protons/antiprotons from single-inclusive hadron production in electron-positron annihilation (SIA) data at next-to-leading order and next-to-next-to-leading order accuracy in perturbative QCD. A new set of FFs, called {\tt SGKS20}, is presented. We include data for identified light charged hadrons ($\pi^\pm, K^\pm$ and $p/\bar{p}$) as well as for unidentified light charged hadrons, $h^\pm$. We examine the inclusion of higher-order perturbative QCD corrections and finite-mass effects. We compare the new {\tt SGKS20} FFs with other recent FFs available in the literature and find in general reasonable agreement, but also important differences for some parton species. We show that theoretical predictions obtained from our new FFs are in very good agreement with the analyzed SIA data, especially at small values of $z$. The {\tt SGKS20} FF sets presented in this work are available via the {\tt LHAPDF} interface.

The two non-perturbative elements in theoretical high energy cross sections of hard scattering processes are the parton distribution functions (PDFs) and the collinear unpolarized fragmentation functions (FFs) [1][2][3][4][5][6][7][8][9][10][11][12]. The factorization theorem of Quantum Chromodynamics * Maryam_Soleymaninia@ipm.ir † Muhammad.Goharipour@ipm.ir ‡ Hamzeh.Khanpour@cern.ch § spiesber@uni-mainz.de (QCD) tells us that these are universal and their evolution can be calculated from perturbative QCD. A precise determination of FFs is crucial for studies of the strong interaction in high energy scattering processes. FFs describe how high energy colored partons produced in the hard interactions are turned into the hadrons measured and identified in an experiment. As is the case for PDFs, FFs need to be determined through a QCD analysis of high-energy experimental data due to their non-perturbative nature. Currently, several experimental measurements from different processes are available which can be used for the determination of FFs. Hadron production in single-inclusive e + e − annihilation (SIA) provides the main information on FFs, but measurements from semi-inclusive deep inelastic scattering (SIDIS) and from proton-(anti)proton collisions at hadron colliders can also be used to determine well-constrained FFs. SIDIS and proton-proton collisions are particularly important for a complete flavour decomposition of FFs into quark and anti-quark components. However, among these high-energy processes, SIA is the cleanest process and the interpretation of it does not require a simultaneous knowledge of PDFs.
There have been several analyses aiming to extract FFs of the lightest charged hadrons π ± , K ± and p/p [2][3][4][13][14][15][16][17][18][19]. The most important experimental information for determining the FFs comes from SIA data and most of the recent analyses have considered only these data to determine FFs up to next-to-next-to-leading order (NNLO) in perturbative QCD. For the case of charged pion, kaon and proton/antiproton analyses which include SIDIS and pp data, we refer to Refs. [11,13,14].
The analyses performed so far for extracting π ± , K ± arXiv:2008.05342v1 [hep-ph] 12 Aug 2020 and p/p FFs differ in various aspects, such as the experimental data included, the QCD perturbative order, the phenomenological framework, the error calculation procedure, and so on. In particular we note that up to now, it was customary to analyze the light charged hadron data independently from each other, i.e. the extraction of FFs for one type of hadron was solely performed through the analysis of production data for that type of hadron. In contrast, in our most recent study [2], we have shown, for the first time, that a simultaneous analysis of pion and unidentified light charged hadron data for extracting pion FFs is also possible and leads to a reduction in the uncertainties of the extracted pion FFs.
The main goal of the following study, referred to as SGKS20 FFs, is to revisit our previous analysis [2] and extract π ± , K ± and p/p FFs simultaneously by including all available SIA data for pion, kaon, proton production along with data for unidentified light charged hadrons h ± . We perform a QCD analysis at next-to-leading order (NLO) as well as at next-to-next-to-leading order (NNLO). Moreover, in the present analysis, we also study hadron mass corrections. We find that these corrections are important at small z, the ratio of momentum transferred from the parton to the observed hadron, and at low values of center-of-mass energy √ s. Since the contribution of unidentified light charged hadrons h ± is mostly related to the pion, kaon and proton, we show that the extraction of π ± , K ± and p/p FFs in a simultaneous analysis of identified and unidentified light charged hadron production data and including the hadron mass corrections significantly improves the fit quality and leads to well-constrained FFs.
This article is organized in the following manner. In Sec. II we present the SIA data used in our NLO and NNLO FFs analyses, along with their corresponding observables and the kinematic cuts we impose on the data. Then, in Sec. III we discuss the theoretical details of the SGKS20 FFs determination of π ± , K ± and p/p FFs, including the parameterizations and the evolution of FFs. Our assumptions and the hadron mass corrections are discussed in this section as well. Sec. IV deals with the method of χ 2 minimization and estimation of the SGKS20 FFs uncertainties. Considering the best fit parameters, the main results of this study are presented in Sec. V. We first turn to discuss the SGKS20 FFs sets. Then, we compare our best fit obtained for pion, kaon and proton/antiproton FFs at NNLO with other results in the literature. We also present a detailed comparison between all analyzed SIA data and the corresponding theoretical predictions obtained using the SGKS20 FFs. Finally, in Sec. VI we present our summary and conclusions. We also outline in this section some possible future developments.

II. EXPERIMENTAL OBSERVABLES
The SIA processes have provided us with a wealth of high-precision experimental data carrying information about how partons fragment into a low-mass charged hadron. In this section, we provide details of the experimental measurements used as input for the determination of the SGKS20 FFs along with the corresponding observables and kinematic cuts applied. The simultaneous determination of light charged hadron FFs presented in this work is based on comprehensive data sets from electron-positron annihilation into a single identified and unidentified hadron. In addition to the inclusive measurements, the data set entering the SGKS20 analysis also includes flavor-tagged measurements.
We note that SIA data are particularly clean, however, they provide only a limited sensitivity to the flavor separation of different light quark FFs. In addition, it is known that the gluon FF is poorly constrained by the total SIA cross section measurements. Hence, in order to discriminate between different quark and antiquark flavors, one would have to include SIDIS and hadron collider observables. This is, however, beyond the scope of the present work.
For the case of unidentified light charged hadron h ± data, we use the SIA measurements by the TASSO, TPC, ALEPH, DELPHI, OPAL and SLD Collaborations [23,24,27,28,30,31]. The SIA data included in our analysis are listed in Tables I and II. The second column of these tables contains the value of the center-of-mass energy for each experiment. The data cover center-of-mass energies between 10.52 GeV and 91.2 GeV. The total number of data points included is 1492. This combines 392 data points for unidentified light charged hadrons h ± , 412 for pions, 369 for kaons and 319 for protons.
The details of our fitting procedure will be discussed below, but we present already here, in the last four columns of Tables I and II, the values of χ 2 per number of data points, χ 2 /(N pts. ), for each data set. The value of the total χ 2 per number of degrees of freedom, , is shown in the last line of these tables. It should be noted that the number of data points of each data set shown in the tables is subject to kinematic cuts. Actually, we remove data points at small-and large-z in order to avoid regions where re-summation effects are sizeable.
We have examined a variety of kinematic cuts for dif- (p/p)  The list of input data sets for π ± , K ± , p/p, and h ± production included in the present analysis. For each data set, we have indicated the corresponding reference and the center-of-mass energy √ s. In the last four columns we show the value of χ 2 /Npts. resulting from the FF fit at NLO order. The total value of χ 2 /d.o.f. is shown at the bottom of the table.
ferent hadrons at small values of z. Since we include hadron mass effects in our analysis which could affect the small-z region, we include more small-z data points in our QCD fits than has been done in previous studies. Here we provide some details about the choice of the interval [z min , z max ] in which data points are included in our fit. In general, our choice for z min and z max varies with the center-of-mass energy. Choosing the same values of z min = 0.02 for all experiments and for all center-of-mass energies leads to χ 2 /d.o.f. =1.45 and 1.25 for our NLO and NNLO analyses, respectively. Choosing the values of z min = 0.075 instead of 0.02 leads to χ 2 /d.o.f.=1.19 and 1.14 for the NLO and NNLO analyses, respectively. We found that it is indeed much better to include the data points with z ≥ 0.02 for the center-of-mass energy of √ s = M Z , and z ≥ 0.075 for √ s < M Z , where M Z is the mass of Z boson, for all different hadrons considered in the analysis. After imposing these kinematical cuts, we end up with a total of N pts. = 1492. As shown in Tables I and II, with these choices of kinematic cuts we find χ 2 /d.o.f. = 1.20 for NLO and χ 2 /d.o.f. = 1.10 for the NNLO fit, i.e. the NNLO fit shows in general a much better fit quality.
Compared with the most recent analysis by the NNFF1.0 collaboration [3], we use the same data sets for the identified light charged hadron production. However, our analysis is enriched with the additional unidentified light charged hadron production data sets. We agree with NNFF1.0 in the choice of z min : z min = 0.02 for experiments at √ s = M Z , and z min = 0.075 for all other experiments. However, NNFF1.0 use only data up to z max = 0.9 for all experiments.
(p/p)   Table. I but for the SGKS20 FFs fit at NNLO.

III. THE QCD FRAMEWORK FOR THE SGKS20 FFS
In this section, we turn to present our theoretical framework to perform a simultaneous determination of charged pion, charged kaon and proton/antiproton FFs using the available SIA experimental data, together with data for unidentified light charged hadron production.
In the present analysis, following Ref. [2], we parameterize all the charged pion, charged kaon and proton/antiproton FFs at the input scale µ 0 = 5 GeV, using the following functional form: where B[a, b] is the Euler Beta function, H refers to the type of hadron, H = π ± , K ± or p/p, and N i is the normalization constant for each flavor that is considered to be a fit parameter.
For the case of π ± FFs, the index i denotes the flavor combinations of u + , s + , c + , b + , and gluon g, where q + = q +q. For the case of K ± and p/p FFs, we consider i = u + , d + , s + , c + , b + , g. As we mentioned before, the reason for combining quark and anti-quark FFs in specific flavor combinations is that we use SIA data in our analysis which provide information on certain hadron species summed over the two charge states.
The number of free parameters is reduced by assuming relations between the FFs following from isospin symmetry. In our analysis, we assume SU(2) isospin symmetry for the pion FFs and use D π ± u + = D π ± d + . In the case of kaon FFs we cannot assume a similar relation, i.e. D K ± u + = D K ± d + due to the fact that the d quark is unfavored for kaon production. For the proton/antiproton FFs, we parameterize d + and s + FFs, as described above, but assume that the u + FF has the same shape as the d + FF, i.e. these two FFs are related by a z-independent normalization factor N , (2) The currently available SIA data do not fully constrain the entire z dependence of quark and gluon FFs presented in Eq. (1). Consequently, we are forced to make some further restrictions on the parameter space of the FFs. In particular, we found that the parameters γ and δ are not well constrained by the SIA data. Therefore we consider them equal to zero for each flavor i of the K ± and p/p FFs, and also for the s + , c + and g FFs of π ± . To be more precise, just the u + and b + FFs of pions are considered to include five free parameters. In addition we found that the parameters α π ± s + , α K ± s + , β p/p c + and β p/p b + are not well constrained by SIA data and we have fixed them at their best values which were found in pre-fits.
We find that these restrictions of the shape parameters of FFs only marginally limit the freedom of the input functional form for the kaon and proton/antiproton FFs. In total, we have 49 free parameters in our fit which we include later in our FFs uncertainty estimation.
Unidentified light charged hadron data contain additional information which can provide further constraints on the determination of FFs. In our recent analysis of pion FFs [2], the inclusion of data for unidentified light charged hadrons in the analysis of pion FFs has led to a reduction in their uncertainties and also changed their central values in some kinematic regions. Therefore, it is of interest to extend that analysis also for the present case including kaon and proton FFs.
By definition, the unidentified light charged hadrons include all identified light charged hadrons π ± , K ± and p/p, in addition to a small residual contribution from other light hadrons. Hence, the FFs of unidentified light charged hadrons is given by with i = u + , d + , s + , c + , b + , g. The residual light hadrons contribution is expected to be rather small. However, the most recent study in Ref. [33] shows that the contributions from residual hadrons are significant for the case of c-and b-tagged cross section. Hence, we decide to take into account such contribution in the present analysis. We include the residual light hadron FFs D res ± as described in Ref. [33] using their sets of residual light hadrons FFs available at NLO and NNLO accuracy in the standard LHAPDF format.
Our results show that taking into account these residual contributions decreases the χ 2 /d.o.f. from 1.35 to 1.20 for our NLO analysis and from 1.30 to 1.10 for our NNLO analysis which in general indicates a better agreement of data and theory.
As indicated, mass effects in pion, kaon and proton production are included in our QCD analysis. According to the definition of unidentified light charged hadrons in Eq. (3) and considering the fact that most of the contributions of light hadrons in unidentified light hadrons is relevant to the pion, kaon and proton, respectively, including their mass corrections is expected to improve the results, especially in the region of small z and small √ s. Hadron mass effects have been studied in Ref. [11,34] for e + e − annihilation processes. We follow the strategy described in these references and incorporate hadron mass effects in single inclusive hadron production in SIA. For zero hadron mass, the scaling variable is expressed as z = 2E H / √ s. A finite value of the hadron masses can be incorporated by a specific choice of the scaling variable. We define the light-cone scaling variable η as where m H is the hadron mass. Accordingly, the differential cross section in the presence of hadron mass effects reads According to Eqs. (4) and (5), the hadron mass corrections are most relevant in the small-z and low-√ s regions. These formulas are applied for all three types of hadrons, i.e. pions, kaons and protons. The values of the hadron masses used in Eqs. (4) and (5) are considered to be m π = 0.140 GeV, m K = 0.494 GeV, and m p = 0.938 GeV. We omit the hadron mass corrections for unidentified hadrons.
We note that the effects of accounting for non-zero hadron masses in extracting the light hadron FFs have been explored recently also by NNFF1.0 for the case of pions, kaons, and protons FFs [3]. It was observed that hadron-mass corrections can become significant in the kinematic region covered by the SIA data. Indeed, our present analysis confirms that hadron-mass corrections do improve the fit quality. Our detailed investigations show that ignoring these correction in our QCD fits would lead to larger values of χ 2 . At NLO we find χ 2 /d.o.f. = 1.34 and at NNLO χ 2 /d.o.f. = 1.31 if mass effects are omitted, while with mass effects included the corresponding values decrease to 1.20 and 1.10 for NLO and NNLO, respectively.
In the present study, we use the publicly available APFEL package [35] for both evolving FFs and performing the numerical calculation of the SIA cross sections. Note that, using APFEL, the related calculations can be performed up to NNLO accuracy in QCD. We should stress here that measurements of the longitudinal SIA crosssection (dσ h ± L /dz) are only available for the production of unidentified hadrons, h ± . However, one cannot analyze these data at NNLO as perturbative corrections to the coefficient functions are only available up the NLO accuracy in this case [1]. Hence, we omit the data from the measurements of the longitudinal SIA cross-section. The effect of heavy quark masses are not taken into account in the present analysis and we use the zero mass variable flavor number scheme (ZM-VFNS) with five active  Best-fit parameters for the fragmentation of partons into π ± , K ± and p/p obtained through a simultaneous analysis at NLO accuracy within the framework described in Sec III. The starting scale has been taken to be µ0 = 5 GeV for all parton species. Parameters marked with an asterisk are fixed input quantities.
flavors, including charm and bottom FFs. Moreover, the value of the strong coupling constant at the scale of the Z boson mass is considered to be α s (M 2 Z ) = 0.118 [36]. For performing minimization and determination of fit parameters, we use the CERN program MINUIT [37]. The definition of χ 2 is the same as the one we used in our previous works [2,6], including the overall normalization errors of the experimental data sets. For calculating the uncertainties of the extracted FFs, we use the standard "Hessian" approach [38,39] with ∆χ 2 = 1 (for further details, see Ref. [6]). We will briefly describe our method of minimization and uncertainty estimation in the next section.
The best values of the fit parameters for charged pion, charged kaon and proton/antiproton FFs determined at the initial scale µ 0 = 5 GeV are listed in Tables III and IV at NLO and NNLO accuracy, respectively. Note that the parameters labeled with an asterisk ( * ) are fixed input quantities.

IV. χ 2 MINIMIZATION AND UNCERTAINTY ESTIMATION
The best values of the independent fit parameters defined in Eq. (1) need to be determined from SIA data by  performing a minimization procedure using an effective χ 2 function. This function quantifies the goodness of fit to the SIA data for a given set of fit parameters, {p i }.
The simplest method to calculate the total χ 2 ({p i }) for a set of independent fit parameters {p i } is given by, where O data i refer to the experimental observables, and T theory i indicate the corresponding theoretical values at a given z and µ 2 . In this study, the experimental errors, σ data i , in the above equation are calculated from the statistical and systematical errors added in quadrature. However, the analyses available in the literature [38,[40][41][42] have shown that such a simple χ 2 ({p i }) definition needs to be modified to account for correlations in the experimental uncertainties. In particular, most of the SIA data come with an overall normalization uncertainty which is fully correlated within one data set, but uncorrelated between different data sets. Therefore we split the global χ 2 global ({p i }) into contributions from individual data sub-sets, where n exp is the number of individual experimental data sets and W n indicates a weight factor for the n th experiment. Then, χ 2 n ({p i }) defined in Eq. (6) needs to be corrected as in which i runs over all data points and N data n corresponds to the number of data points in each data set. In order to determine the best fit parameters of the SGKS20 light charged hadrons FFs, we minimize the χ 2 global ({p i }) function with respect to the fit parameters {p i } presented in Eq. (1). The normalization factors ∆N n need to be fitted along with the independent fit parameters ({p i }) and then can be kept fixed. The default value of the weight factors for each experimental data set is considered to be equal to 1 [43,44].
In the following, we briefly discuss our method to estimate the uncertainties of the SGKS20 light charged hadrons FFs. Three different approaches are available in the literature to estimate the uncertainty. They are based on Lagrange multipliers or Monte-Carlo sampling, but the most commonly used method is the so-called 'Hessian' approach [38]. Following the notation adopted in Refs. [39,45], our uncertainty estimation is done using the standard 'Hessian' approach. In this method, the uncertainty for a fragmentation function, ∆D(z), can be obtained from linear error propagation. It is given by where p i (with i = 1, 2, ..., n) denotes the independent free parameters for each FF, n refers to the total number of optimized parameters, andp i comprises the numerical values of the optimized parameters. C i,j ≡ H −1 i,j are the elements of the covariance matrix determined in this analysis at the input scale. In order to estimate the uncertainties of the SGKS20 light charged hadrons FFs, we follow the standard parameter-fitting criterion by considering contours of T = ∆χ 2 global = 1 defining the 68% (1-σ) confidence level (CL). For minimization and the determination of both fit parameters and elements of the covariance matrix we use the publicly available CERN program MINUIT [37].

V. RESULTS OF THE SGKS20 FF ANALYSIS
The following part of this article describes in greater details the results of the SGKS20 FFs analysis. We focus on the inclusion of higher-order QCD corrections in our NNLO results. We also compare our best fit pion, kaon and proton/antiproton FFs with their counterparts from the NNFF1.0 analysis [3].
In Tables III and IV we present the best fit parameters for the fragmentation functions of partons into π ± , K ± and p/p at NLO and NNLO accuracy, respectively.
The NNLO charged hadron FFs, zD H ± i (z, Q 2 ), for singlet (D H ± Σ = q (D H ± q + D H ± q ), q = u, d, s), heavyquark-and gluon-to-hadron fragmentation obtained from the combined fit are illustrated in Figs. 1, 2 and 3 together with their 1-σ uncertainty bands for charged pions, charged kaons and protons/antiprotons, respectively. The NNLO results from the most recent determination available in the literature, namely the NNFF1.0 FFs, are also shown for comparison. The results at Q 2 = 100 GeV 2 reveal the following findings. A noticeable feature of the distributions shown in Fig. 1 is the remarkable agreement between our zD π ± (z, Q 2 ) FFs for heavy and singlet quarks with the corresponding results from NNFF1.0. Fig. 1 shows a small difference for the gluon density, especially at small values of z. A further remarkable aspect of the comparison in this figure is related to the size of the uncertainties. For all cases, the SGKS20 1-σ error bands are smaller than those of the corresponding NNFF1.0 FFs.
Our charged kaon zD K ± (z, Q 2 ) FFs at NNLO accuracy are shown in Fig. 2 in comparison with the NNFF1.0 FFs. Concerning the shapes of the kaon FFs, a number of interesting differences between the SGKS20 and NNFF1.0 FFs can be seen. The differences in shape among the two FF sets are more marked than in the case of the charged pion FF. Moderate differences are observed for the central value of the singlet FF at smaller values of z, especially at z < 0.1, and for the uncertainty band of the bottom FF below z < 0.05. A more noticeable difference in shape is observed for the gluon and charm FFs for which the SGKS20 results are more suppressed and enhanced, respectively, at z < 0.4, than the gluon and charm FFs from NNFF1.0.
Let us now discuss the SGKS20 protons and antiprotons zD p/p (z, Q 2 ) FFs. A fair agreement is observed only in the case of the heavy-quark and singlet-quark FFs at large values of z. These FFs are more suppressed at medium to small z values, compared with the corresponding FFs from NNFF1.0. For zD p/p g , big differences can be seen both in the magnitude and the error band of the FFs in the whole range of z. Overall, the error bands for all heavy quark, singlet and gluon FFs for all light hadrons are dramatically reduced, except for the singlet FF of the kaon at medium to large z.
There are a number of similarities and differences between the SGKS20 and NNFF1.0 analyses. The QCD approach used in this study is similar to the one used by NNFF1.0. In both cases, NNLO QCD and hadron-mass corrections are taken into account. Also, the kinematic cuts imposed on data points in the small z region are the same in both analyses. The origin of the differences between the SGKS20 and NNFF1.0 FFs is likely to be due to the following reasons.
First, we have included data for unidentified light charged hadron production along with identified π ± , K ± and pp production simultaneously in one fit. We believe that this is the main reason that the SGKS20 FFs are much better constrained. In addition, the NNFF1.0 approach based on neural networks without fixing a priori a specific parametrization allows one to obtain much more flexibility in the description of FFs. It can, therefore, be expected that the uncertainties of the NNFF1.0 FFs are larger than those of SGKS20. This is indeed the case, as seen in the figures. The smaller error bands of SGKS20 FFs may also be due to the fact that we include more data in the analysis, especially the inclusion of unidentified light charged hadron data, and considering the residual light hadrons contributions.
Considering the fit quality upon inclusion of higherorder QCD corrections, one can conclude from Tables I and II that the NNLO corrections slightly improve the overall fit quality for almost all SIA data. As one can see from these tables, the χ 2 /(d.o.f.) values at NNLO accuracy are lower than at NLO. Moreover, the fit quality suggests that the inclusion of residual light-hadron contributions as well as unidentified light charged hadron data in our identified zD H ± (H ± = π ± , K ± , p/p) analysis leads to an improved agreement between theory and data.
Having at hand the SGKS20 NLO and NNLO light charged hadron FFs, we are now able to compare the analyzed data against the theory predictions for the normalized SIA cross sections. In Fig. 4, our theory predictions are compared to the total SIA cross section measurements for inclusive [26][27][28][29][30], light [28][29][30][31], c-tagged [29][30][31] and b-tagged [28][29][30][31] unidentified light charged hadron (h ± ) from ALEPH, DELPHI, OPAL and SLD experiments. In general, the agreement between data and theory is excellent. In addition, we observe that our NNLO results show a better agreement with the SIA data, especially for the total inclusive, c-tagged and light charged hadron cross sections at small values of z. One can also see that the error bands for the NLO and NNLO theory predictions are very similar, except for the case of c-tagged cross sections where the NNLO predictions show smaller uncertainties.
We also present a comparison of the charged pion, kaon and proton/antiproton data used in this analysis with the corresponding theoretical predictions obtained using our NNLO FFs. In Figs. 5, 6 and 7, data over theory ratios are displayed for the SLD [31], DELPHI [28] and BABAR [21] data for charged pion (π ± ), charged kaon (K ± ), and proton/antiproton (p/p) production in SIA.
For the case of pion production, Fig. 5, overall good agreement between measurements and the NNLO theory predictions is found for most of the data points, except for the large-z region of some experiments. The uncertainties of our theory predictions are getting large in this region for the case of SLD and DELPHI heavy quark production.
The comparison for charged kaons is presented in Fig. 6. We notice that for some data the agreement is good, in particular for the SLD and BABAR experiments, while for DELPHI we see some deviations in the small-z region. As one can see, the experimental data points for all data sets fluctuate inside the error bands of the theory predictions.
tagged data is reasonable. Deviations appear specifically for almost all experiments in the small-z region, except for the case of inclusive measurements from the BABAR experiment. For the inclusive measurements of SLD, DELPHI and BABAR, the agreement is acceptable in the mediumto-large range of z-values. The same conclusion can be made for the uds-tagged data from the SLD and DELPHI experiments.

VI. SUMMARY AND CONCLUSIONS
The main goal of the current study is to present a set of FFs, called SGKS20, for light charged hadron (π ± , K ± , p/p) production. These FFs are obtained in a simultaneous fit and we include both identified and unidentified light charged hadron data taken from electron-positron annihilation. We included finite-hadron mass corrections which are significant for small z and small √ s. For FFs which involve heavy quarks, we adopted the zero-mass variable-flavour-number scheme. As a third improvement, the residual light hadrons contributions have been included in our fit for unidentified light hadrons. We have shown that this approach improves the total χ 2 at both NLO and NNLO accuracy and also reduces the uncertainties for the FFs of light hadrons. Our results show that the inclusion of higher-order QCD corrections helped to obtain a much better agreement of data with theory. Finally, we compared our pion, kaon and proton FFs with the one recently extracted by the NNFF1.0 Collaboration.
The most important limitation of the present analysis is related to the fact that we have included data The data/theory ratio for the charged pion (π ± ) production data from SLD [31], DELPHI [28] and BABAR [21] experiments included in the SGKS20 fit. Our theoretical predictions are computed at NNLO accuracy with our best-fit NNLO FFs. from SIA measurements only. The precise data from proton-(anti)proton (pp) collisions, which cover a wide range in energy and momentum fractions, contain vital information about FFs, especially for the gluon FF, and also are sensitive to different partonic combinations [1].
These measurements include CDF [46,47] experiment at the Tevatron, STAR [48] and PHENIX [49] at RHIC, and CMS [50,51] and ALICE [52] experiments at the LHC. It is expected that the inclusion of these data will lead to much better constrained FFs. Hence, it will be inter- esting to repeat this analysis by considering the SIDIS data as well as hadron collider data which could provide a flavor separation between quark and antiquark FFs and also the gluon FF. In addition, a future study investigating the improvements of description of the data at low center-of-mass energy due to the effect arising from heavy quarks mass corrections would be very interesting.
The FF parametrizations at NLO and NNLO for identified light charged hadron, zD H ± (H ± = π ± , K ± , p/p), presented in this study are available in the standard LHAPDF format [53] from the authors upon request.