Impact of unidentified light charged hadron data on the determination of pion fragmentation functions
IImpact of unidentified light charged hadron data on the determination of pionfragmentation functions
Maryam Soleymaninia , , ∗ Muhammad Goharipour , † and Hamzeh Khanpour , ‡ Institute of Advanced Technologies, Shahid RajaeeTeacher Training University, Lavizan, Tehran, 16788, 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
In this paper a new comprehensive analysis of parton-to-pion fragmentation functions(FFs) is performed for the first time by including all experimental data sets on single in-clusive pion as well as unidentified light charged hadron production in electron-positron( e + e − ) annihilation. We determine the pion FFs along with their uncertainties using thestandard “Hessian” technique at next-to-leading order (NLO) and next-to-next-to leadingorder (NNLO) in perturabative QCD. It is shown that the determination of pion FFs usingsimultaneously the data sets from pion and unidentified light charged hadron productionsleads to the reduction of all pion FFs uncertainties especially for the case of strange quarkand gluon FFs by significant factors. In this study, we have quantified the constraints thatthese data sets could impose on the extracted pion FFs. Our results also illustrate the sig-nificant improvement in the precision of FFs fits achievable by inclusion of higher ordercorrections. The improvements on both FFs uncertainties as well as fit quality have beenclearly discussed. PACS numbers: 11.30.Hv, 14.65.Bt, 12.38.Lg ∗ Maryam [email protected] † [email protected] ‡ [email protected] a r X i v : . [ h e p - ph ] F e b ONTENTS
I. Introduction 2II. Experimental data selection 5III. Theoretical methodology for calculations and fitting 7IV. Analysis results 12A. Comparison of χ values 13B. Comparison of the relative uncertainties 15C. Comparison of the “ pion+hadron fit ” at NLO and NNLO accuracy 25D. Comparison of the data and theory predictions 25V. Summary and Conclusions 27Acknowledgments 29References 29 I. INTRODUCTION
Essential ingredients of theoretical predictions for the present or future hadron colliderssuch as the large hadron collider (LHC) and large hadron-electron collider (LHeC), are thedetailed understanding of the quark and gluon structure of the nucleon [1–4]. These arequantified by the parton distribution functions (PDFs) [5–7] as well as the fragmentationfunctions (FFs) [8–20]. In recent years, precise determination of PDFs as well as FFsincluding their experimental uncertainties had become an active topic for many LHCprocesses, including top-quark and Higgs boson sector, searches for new heavy beyondthe Standard Model (BSM) particles, searches for new physics (NP) as well as in themeasurement of fundamental SM parameters such as the strong coupling constant. Forore details, we refer the readers to the literature [21–23] and a recent study on the PDFsat the High-Luminosity LHC (HL-LHC) [1].In a hard-scattering collision, PDFs determine how the proton’s momentum is sharedamong its constituents. Likewise, the FFs describe the probability density for the fragmen-tation of the final-state parton with a certain momentum into the hadron with a fractionof the parton’s momentum. PDFs and the FFs depend on the factorization scale. Thisdependence is described by the DGLAP evolution equations [24–27], which allow thecalculation of the PDFs and FFs, if they are known at a given initial scale, i.e. µ = µ . Itis well known that the PDFs and the FFs can not be calculable in perturbation theory, andhence, these distributions need to be extracted from experimental information through aQCD fit. In addition, these non-perturbative functions are also universal. The universalityof PDFs and FFs commonly refers that, since the hadronization processes are not sensitiveto the particular choices of hard scattering process in short range, these non-perturbativefunctions can be extracted from certain kind of scattering experimental observables. Thenthe extracted distributions can be used for the theory predictions of scattering observablein high energy collisions.The new and precise data sets are vital for the precise determination of FFs. Thesedata sets have been and currently been collected from di ff erent high energy processes atvariety of lepton and hadron colliders. These processes include the hadron productiondata in single-inclusive electron-positron ( e + e − ) annihilation (SIA), semi-inclusive deepinelastic scattering (SIDIS), and proton-proton and proton-antiproton collisions measuredby TEVATRON, RHIC and LHC. For a list of all available data sets, we refer the readers tothe recent analysis by NNPDF collaboration and references therein [8, 9]. Several analyseshave been done so far to extract FFs using the observables mentioned above. Amongthem are the recent determination of charged hadron FFs from collider data by NNPDFcollaboration, NNFF1.1h [8]. This collaboration also have determined the pions, kaons, andproton FFs using the SIA data sets at NNLO in perturbative QCD based on the NNPDFethodology,
NNFF1.0 [9]. The recent analyses by
HKKS16 [13] and
JAM16 [28] also havebeen performed using the SIA data only. Other analyses in literature can be found forexample in Refs. [29–36]Recently, we also have performed the First determination of D ∗± -meson FFs and theiruncertainties at NNLO, SKM18 [10]. In Ref. [11] we presented our QCD analysis of chargedhadron FFs and their uncertainties at NLO and NNLO (
SGK18 ) which is the first determi-nation of light charged hadron FFs at NNLO accuracy. Finally in Ref. [12] the contributionsfrom residual light charged hadrons in the inclusive charged hadrons have been extractedusing the e + e − annihilation data sets. Since the QCD framework for FFs at NNLO arenot accessible for SIDIS, and hadron-hadron collisions, both of our analyses are restrictedto the single-inclusive charged hadron production in electron-positron annihilation. Theuncertainties in our recent analyses on FFs as well as the corresponding observables areestimated using the “Hessian” technique.In this work, an extraction of pion FFs from QCD analysis of electron-positron anni-hilation experimental data in zero-mass variable flavor number scheme (ZM-VFNS) hasbeen presented. The main aim of this paper is to examine, for the first time, the impactof unidentified light charged hadron experimental data on the determination of pion FFsand their uncertainties at NLO and NNLO accuracy. In this respect, we have attempted adetermination of pion FFs considering two di ff erent scenarios. First, we present a determi-nation of pion FFs through a QCD analysis of pion data sets. In this first study of FFs, whichis performed within ZM-VFNS at both NLO and NNLO approximations and referred toas “ pion fit ”, we simplify the analysis by considering the pion data sets only. Secondly,we determine pion FFs through a QCD analysis by including both pion and unidentifiedlight charged hadron data sets. We show that the fitting simultaneously the pion FFs usingboth data sets leads to a well-constrained determination of pion FFs including significante ff ect on the extracted uncertainties. Our second fit entitled as “ pion+hadron fit ”.The outline of this paper is as follows. In section II, we present in details all availableIA data sets for pion production as well as the SIA data sets for the unidentified lightcharged hadrons. In Section III, we discuss the theoretical formalism of single-hadroninclusive production in electron-positron ( e + e − ) annihilation. This section also includesthe detailed discussions of our fitting process and parameterization for the pion FFs.Section IV is then dedicated to our results. The obtained results are clearly discussed forvariety of aspect in this section, and comparison with other analyses in literature alsopresented. This section also includes our theory predictions based on the extracted pionFFs including a comparison with all data analyzed. Finally, Section V includes a summaryand our conclusions. II. EXPERIMENTAL DATA SELECTION
In this section, we present the experimental data sets that are included in our “ pionfit ” and “ pion+hadron fit ” analyses. As we mentioned in the Introduction, our QCDfits are performed by inducing the electron-positron annihilation data in two scenarios:In the first analysis, we use the available SIA data for pion from Refs. [37–46] to extractthe pion FFs. In the second analysis, the SIA data sets for the unidentified chargedhadrons [41, 44, 46–51] along with the pion data sets are included in our fits to calculatethe FFs of pion. All the data sets for pion and unidentified hadrons are listed in Tables. I andII for inclusive and flavor-tagged SIA data which are reported by di ff erent experiments.Note that, the measured observables for these data sets, specially for pion, are di ff erentand a complete explanation about SIA pion data and the relations between the scalingvariables are available in related analysis done by NNPDF collaboration in NNFF1.0 [9].In addition, we have used the unidentified light charged hadron experimental data in ourrecent study of (
SGK18 ) [11]. The details of corrections to these data sets and the kinematiccuts applied are presented in Ref. [11].According to the data sets presented in second column of Tables. I and II, the observ-ables are di ff erent and provide limited sensitivity to the separation between light andeavy quark FFs due to the flavor-tagged data. Since the gluon receives its leading order(LO) accuracy at O ( α ∫ ), the total SIA cross sections are poor to constrain this density.However, the longitudinal cross sections can impose a comparable sensitivity to the gluonFF because the longitudinal coe ffi cient functions start at O ( α ∫ ). Hence, the longitudinalobservables that are available for the unidentified hadrons could constrain the gluon FFwell enough. It should be noted that the NNLO QCD corrections for longitudinal struc-ture functions are not available in the literature, and hence, such corrections can not beconsidered in our analyses.In this paper, we plan to study the e ff ects arising from the unidentified light chargedhadron experimental data on the calculation of pion FFs by including both pion andunidentified hadron data sets, and then, compare the extracted pion FFs with the resultscalculated from the QCD analysis using pion data sets alone. Since the most contributionof FFs into the unidentified light charged hadron cross sections mainly comes from theidentified pion FFs, it motivates us to investigate the e ff ect of unidentified light chargedhadron data sets on the reduction of pion FFs uncertainties. In Tables. I and II, our resultsare reported at NLO and NNLO accuracies of perturbative QCD. In both tables, the forthcolumn presents our fit results for the value of χ per number of data points ( χ / N pts . )considering pion data sets in the fit, while in the fifth column the same quantity arereported considering the pion and light hadron experimental data sets in the analysis.One of the most important findings from these tables are the significant reduction of χ / do f by going from NLO to the NNLO corrections. We will return to this issue in thenext section.In order to avoid the sensitivity of behaviors of FF parametrization in the low andhigh regions of z , we apply cuts on the momentum fraction z . We exactly follow the cutsapplied in our recent study on light charged hadron FFs, SGK18 [11]. These selections arealso imposed for the pion experimental data. For data sets at √ s = M Z , we include thedata points with the scaling variable of z ≥ .
02 and for √ s < M Z , the data points with ≥ .
075 are included in our QCD fits. The data points with z > . III. THEORETICAL METHODOLOGY FOR CALCULATIONS AND FITTING
In this section, a brief review of the theoretical framework and our methodology hasbeen presented. According to the factorization theorem, the SIA di ff erential cross sectionnormalized to the total cross section σ tot d σ H ± dz at a given center-of-mass energy of √ S = Q is written by, 1 σ tot d σ H ± dz = σ tot (cid:104) F H ± T ( z , Q ) + F H ± L ( z , Q ) (cid:105) . (1)This equation is used for identified charged hadrons such as π ± , K ± and p / ¯ p and, unidenti-fied hadrons h ± . In Eq. (1), H ± is defined as sum of di ff erent charge of hadrons H = H + + H − and z = E H √ s is the scaling variable. The total cross section σ tot depends to the perturbativeorder of QCD corrections and detail explanations can be found, for example, in Ref. [11].According to the Eq. (1), in the case of multiplicities, the di ff erential cross section for SIAprocesses can be decomposed into time-like structure functions F T and F L which are thetransverse (T) and longitudinal (L) perturbative parts, respectively. The time-like struc-ture functions can be written as convolutions of a perturbative part, coe ffi cient functions C i ( z , α s ), and a nonperturbative part, FFs D H ± ( z , Q ), F H ± ( z , Q ) = (cid:88) i C i ( z , α s ) ⊗ D H ± ( z , Q ) . (2)The coe ffi cient functions have been calculated in Refs. [52–54] and they are available upto NNLO accuracy for electron positron annihilations. It should be mentioned here that,in this analysis, the renormalization scale µ R and the factorization scale µ F considered tobe equal to the center-of-mass energy of collision, µ R = µ F = √ s . ataset observable √ s [GeV] χ / pts . “pion” χ / pts . “pion + hadron”BELLE [37] inclusive 10.52 38.37 /
70 42.28 / /
40 81.64 / / / / / / / /
13 8.25 / / / / / /
23 43.07 / /
21 22.86 / uds tag 91.2 21.47 /
21 22.70 / b tag 91.2 21.12 /
21 11.11 / /
24 37.41 / /
34 76.20 / uds tag 91.2 90.98 /
34 92.04 / c tag 91.2 38.83 /
34 40.13 / b tag 91.2 19.81 /
34 38.28 / / / / / / / / uds tag 91.2 —- 10.52 / b tag 91.2 —- 51.76 / / b tag 91.2 —- 20.12 / / uds tag 91.2 —- 7.34 / c tag 91.2 —- 14.18 / b tag 91.2 —- 26.85 / / / uds tag 91.2 —- 15.86 / c tag 91.2 —- 29.26 / b tag 91.2 —– 81.21 / χ / dof 1.42 1.44 TABLE I. The data sets included in the analyses of π ± FFs at NLO. For each experiment, we indicatethe corresponding reference, the measured observables, the center-of-mass energy √ s , the χ / pts . values for every data set, as well as the total χ / do f . ataset observable √ s [GeV] χ / pts . “pion” χ / pts . “pion + hadron”BELLE [37] inclusive 10.52 27.39 /
70 29.96 / /
40 57.80 / / / / / / / /
13 9.26 / / / / / /
23 35.50 / /
21 24.78 / uds tag 91.2 22.22 /
21 23.57 / b tag 91.2 19.96 /
21 10.57 / /
24 35.74 / /
34 47.80 / uds tag 91.2 68.97 /
34 66.70 / c tag 91.2 31.73 /
34 35.18 / b tag 91.2 19.36 /
34 40.38 / / / / / / / / uds tag 91.2 —- 11.66 / b tag 91.2 —- 50.99 / / b tag 91.2 —- 9.37 / / uds tag 91.2 —- 8.53 / c tag 91.2 —- 14.56 / b tag 91.2 —- 26.41 / / / uds tag 91.2 —- 10.97 / c tag 91.2 —- 29.74 / b tag 91.2 —– 80.62 / χ / dof 1.17 1.06 TABLE II. Same as Table. I but at NNLO accuracy. ince the universal FFs are nonperturbative functions, in order to determine the FFs,one needs to parametrize the functions of partons i = q , ¯ q , g at a given initial scale. The z parameter represents the fraction of the parton momentum which carried by hadron.Theoretically, the renormalization equations govern the scale dependence of the FFs andthey can be evaluate to a given higher energy scale using the DGLAP evolution equations.In our analysis, we use the publicly APFEL package [55] in order to calculate of the SIAcross sections as well as the evolution of FFs by DGLAP equations up to NNLO accuracy.In addition, the ZM-VFNS is considered to account the heavy quarks contributions, andhence, the e ff ects of heavy quark mass are not taken into account in our analysis.Our main aim in this analysis is to study the e ff ect of adding all the unidentified lightcharged hadrons experimental data to the pion ones from SIA processes in the procedureof determination of pion FFs. Hence, we need the theoretical definition of unidentifiedcharge hadron FFs in our calculations. Experimentally, the unidentified light chargedhadrons contain all identified light hadrons such as pion, kaon, proton and a small residual light hadrons. Then unidentified charged hadron cross sections of SIA can be calculatedby summing of individual cross sections of the identified light ones ( π ± , K ± and p / ¯ p ) andthe residual contribution. The SIA coe ffi cient functions for all final states are the same, andhence, the FFs of unidentified light charged hadrons ( D h ± ) can be defined as the sum of thepion, kaon and proton FFs ( D π ± , D K ± , D p / ¯ p ) including the residual light hadron FFs D res ± D h ± = D π ± + D K ± + D p / ¯ p + D res ± . (3)Since our aim in this analysis is a new determination of pion FFs D π ± , we use the kaon andproton FFs from NNFF1.0 parton set [9] both at NLO and NNLO accuracies. Recently, wehave calculated the residual light hadron FFs D res ± in Ref. [12] up to NNLO QCD correction.In Ref. [12], we have shown that the contribution of the residual light hadrons are small,and hence, one can ignore this small contribution in Eq. (3). The contribution from thissmall distribution are not significant for the case of total or light charged cross sections,however, for the case of c - and b -tagged cross sections they are sizable.or the uncertainty from NNFF1.0 , we follow the analysis by
DSS07 in Ref. [36] andestimate an average uncertainty of 5% in all theoretical calculations of the inclusive chargedhadron cross sections stemming from the large uncertainties of kaon and proton FFs from
NNFF1.0 set. In addition, our recent study shows that an additional uncertainty due tothe contributions of residual charged hadrons FFs [12] also need to be taken into account.Overall, we believe that a 8% of the cross section value seems to be reasonable. Theseadditional uncertainties are included in the χ minimization procedure for determiningthe pion FFs. In order to add these uncertainties, we apply such a simplest way to include a“theory” error which we add it in quadrature to the statistical and systematic experimentalerror in the χ expression. This is the standard approach that one can use to add thisadditional uncertainty to the QCD analysis. The method of the present study are alsoconsistent with those of DSS07 [36] who used the same approach, and hence, our resultsshare a number of similarities with
DSS07 findings. This method was chosen because it isone of the most practical and economic ways to include such uncertainty and in agreementfrom previous results reported in the literature. However this method may su ff ers froma number of pitfalls. One need to use a rigorous approach and include the full NNFF1.0 uncertainties in the kaon and proton FFs in Eq. (3). In order to ensure the a ff ect of thisalternative method on our conclusions, we also examined this approach. Our study showsthat one can reaches the same conclusions, finding no increase in the size of uncertainty. Forthe physical parameters, we exactly follow the analysis by NNFF collaboration, NNFF1.0 .We use the heavy flavor masses for charm and bottom as m c = .
51 GeV and m b = .
92 GeV [8, 9], respectively. Also the Z-boson mass is chosen to be M Z = .
187 GeV andthe QCD coupling constant is fixed to the world average α s ( M Z ) = . Q which we also used in ourery recent analysis of unidentified light charged hadrons [11], D π ± i ( z , Q ) = N i z α i (1 − z ) β i [1 + γ i (1 − z ) δ i ] B [2 + α i , β i + + γ i B [2 + α i , β i + δ i + , (4)where i = u + , d + , s + , c + , b + and g , q + = q + ¯ q . In order to normalize the parameter N i weuse the Euler Beta function B [ a , b ]. Since we include the FF sets of NNFF1.0 for kaon andproton, we choose the initial scale of energy Q = n f =
5. In addition, the charge conjugation andisospin symmetry D π ± u + = D π ± d + are assumed. More specifically, the γ and δ parameters for s + , c + and g could not well constrain by the SIA data and we are forced to fix them as γ s + , c + , g = δ s + , c + , g =
0. Then the best fit is only achieved with all five parameters ofEq. (4) for u + and b + . We determine 19 free parameters by a standard χ minimizationstrategy in which the details can be found in Refs. [11, 57].The free parameters are determined from the best fit, and we list them in Table. III. Inthe second and third columns of this table, we report our best fit parameters for only piondata analysis at NLO and NNLO accuracy, respectively. The parameters reported by theforth and fifth columns are for the analyses with both pion and unidentified hadron datasets at both perturbative orders. IV. ANALYSIS RESULTS
After the detailed presenting of the experimental data sets included in the presentwork and the theoretical and phenomenological framework of the analysis in the previoussections, in the following we present the numerical results obtained for the pion FFs fromdi ff erent analyses and compare them with each other. As we mentioned before, the maingoal of the present work is to investigate, for the first time, the impact of unidentified lightcharged hadron experimental data on the pion FFs at both NLO and NNLO accuracy. Inthis respect, the pion FFs should be determined by performing two di ff erent analyses: 1)etermination of pion FFs through a QCD analysis of only pion data sets as usual ( pionfit ), and 2) determination of pion FFs through a simultaneous analysis of both pion andunidentified light charged hadron data sets ( pion+hadron fit ).The important point that should be noted is the presence of the kaon, proton and residual FFs in the theoretical calculation of the unidentified light charged hadron cross sectionswhich is required for the second analysis. As discussed in Sec. III, we use the kaon andproton FFs from the
NNFF1.0 analysis [9] and ignore the small residual contribution. Hence,some theoretical uncertainties should be taken into account in the analysis containing theunidentified light charged hadron data. One of the most common methods is adding apoint-to-point uncertainty to the experimental data as a systematic error source, 8% in ouranalyses.
A. Comparison of χ values The list of experimental data sets including their references as well as the results of ouranalyses introduced above have been summarized in Tables. I and II at NLO and NNLO,respectively. In each table, the second column indicates the kind of observable measuredby each experiment and the third column specifies its related value of center-of-massenergy. Note also that the columns labeled by “ pion ” and “ pion+hadron ” are containingthe results of the first and second analyses, respectively. The values of χ per number ofdata points ( χ / N pts. ) have been presented in these columns for each data set. Moreover,the value of total χ divide by the number of degrees of freedom ( χ / dof) for each analysisis presented in the last raw of the table. The total number of data points included in the“ pion fit ” analysis is 405, while it is 879 for the “ pion+hadron fit ” analysis. Accordingto the results obtained, the following conclusions can be drawn. For the case of NLOanalyses, although the values of χ / N pts. have increased almost for each pion data set afterthe inclusion of the unidentified light charged hadron data, but the values of χ / dof forthe “ pion fit ” and “ pion+hadron fit ” analyses are almost equal. Such behavior is seen arameter “pion” NLO “pion” NNLO “pion + hadron” NLO “pion + hadron” NNLO N u + .
123 1 .
062 1 .
133 1 . α u + − . − . − . − . β u + .
737 1 .
854 1 .
757 1 . γ u + .
324 6 .
550 9 .
705 7 . δ u + .
175 5 .
843 5 .
314 6 . N s + .
239 0 .
456 0 .
124 0 . α s + .
634 0 .
598 3 .
376 0 . β s + .
714 8 .
468 12 .
658 8 . N c + .
739 0 .
777 0 .
724 0 . α c + − . − . − . − . β c + .
662 5 .
055 4 .
520 4 . N b + .
694 0 .
735 0 .
673 0 . α b + − . − . − . − . β b + .
346 5 .
057 4 .
728 4 . γ b + .
014 7 .
356 9 .
098 8 . δ b + .
102 8 .
567 10 .
573 9 . N g .
616 0 .
571 0 .
705 0 . α g .
406 0 . − . − . β g .
210 16 .
174 8 .
658 13 . TABLE III. The best fit parameters for the fragmentation of partons into the π ± for both pionfit and pion+hadron fit analyses at NLO and NNLO accuracy. The starting scale is taken to be Q = for some of the data sets in the case of NNLO analyses, but with the di ff erence that thevalue of χ / dof has decreased by including the unidentified light charged hadron data inthe analysis. Another point should be noted here is the significant reduction in the valueof χ / dof when we move from NLO to NNLO. The optimum values of fit parametershave been presented in Table. III, where the first and second columns are related to thepion data analyses at NLO and NNLO, respectively, while the third and fourth columnscontain the results of the simultaneous analyses of the pion and hadron data at NLO andNNLO accuracy. . Comparison of the relative uncertainties In order to investigate the impact arising from the inclusion of unidentified lightcharged hadron experimental data on pion FFs both in behavior and uncertainty, theresults obtained from “ pion fit ” and “ pion+hadron fit ” can be compared in variousways. One of the best approaches to check the validity and excellency of the new resultsobtained, specifically in view of the uncertainties, is comparing the relative uncertaintiesof the extracted distributions which are obtained, for each analysis separately, by dividingthe upper and lower bands to the central values. Fig. 1 shows a comparison between therelative uncertainties of pion FFs obtained from the “ pion fit ” and “ pion+hadron fit ”analyses at NLO accuracy. We have presented the results for all flavors parameterizedin the analysis at the initial scale of Q = s + ¯ s FF, the relative uncertainties of pion FFs obtained from the simultaneous analysis ofthe pion and hadron data are smaller than those obtained by fitting the pion data alone,especially for the case of gluon FF. In fact, the amount of the uncertainty of s + ¯ s FF from“ pion+hadron fit ” analysis is also less than “ pion fit ” analysis (as will be shown later),but since its central value is smaller by a factor of two, it has overall a relative uncertaintywhich is somewhat larger.Fig. 2 shows the same results as Fig. 1, but this time for our NNLO analysis. One canclearly conclude that the inclusion of the unidentified light charged hadron data in thepion FFs analysis at NNLO accuracy can also lead to a smaller relative uncertainty for allflavors. Note that, compared with the NLO results, the relative uncertainty of s + ¯ s FF from“ pion+hadron fit ” analysis has now remarkably decreased at lower z values rather thanits distribution from “ pion fit ” analysis. Overall, the results obtained indicate that byperforming a simultaneous analysis of pion and unidentified light charged hadron data, apion FFs set with more acceptable uncertainties can be obtained at both NLO and NNLOaccuracies.To study the e ff ects of the evolution and also evaluate the results at a given higher .01 0.1 10.51.01.5 NLO zD u+u (z, =5 GeV) Z pion fit pion+hadron fit R e l a ti v e e rr o r NLO zD s+s (z, =5 GeV) Z pion fit pion+hadron fit R e l a ti v e e rr o r NLO zD c+c (z, =5 GeV) Z pion fit pion+hadron fit R e l a ti v e e rr o r NLO zD b+b (z, =5 GeV) Z pion fit pion+hadron fit R e l a ti v e e rr o r NLO zD g (z, =5 GeV) Z pion fit pion+hadron fit R e l a ti v e e rr o r FIG. 1. Comparison between the relative uncertainties of pion FFs at Q = pion fit ” and “ pion+hadron fit ” analyses at NLO. .01 0.1 10.51.01.5 zD u+u (z, =5 GeV) Z pion fit pion+hadron fit R e l a ti v e e rr o r NNLO zD s+s (z, =5 GeV) Z pion fit pion+hadron fit R e l a ti v e e rr o r NNLO zD c+c (z, =5 GeV) Z pion fit pion+hadron fit R e l a ti v e e rr o r NNLO zD b+b (z, =5 GeV) Z pion fit pion+hadron fit R e l a ti v e e rr o r NNLO zD g (z, =5 GeV) Z pion fit pion+hadron fit R e l a ti v e e rr o r NNLO
FIG. 2. Same as Fig. 1 but at NNLO. .01 0.1 10.51.01.5
NLO zD u+u (z, =M Z ) Z pion fit pion+hadron fit R e l a ti v e e rr o r NLO zD s+s (z, =M Z ) Z pion fit pion+hadron fit R e l a ti v e e rr o r NLO zD c+c (z, =M Z ) Z pion fit pion+hadron fit R e l a ti v e e rr o r NLO zD b+b (z, =M Z ) Z pion fit pion+hadron fit R e l a ti v e e rr o r NLO zD g (z, =M Z ) Z pion fit pion+hadron fit R e l a ti v e e rr o r FIG. 3. Same as Fig. 1 but at Q = M Z . .01 0.1 10.51.01.5 zD u+u (z, =M Z ) Z pion fit pion+hadron fit R e l a ti v e e rr o r NNLO zD s+s (z, =M Z ) Z pion fit pion+hadron fit R e l a ti v e e rr o r NNLO zD c+c (z, =M Z ) Z pion fit pion+hadron fit R e l a ti v e e rr o r NNLO zD b+b (z, =M Z ) Z pion fit pion+hadron fit R e l a ti v e e rr o r NNLO zD g (z, =M Z ) Z pion fit pion+hadron fit R e l a ti v e e rr o r NNLO
FIG. 4. Same as Fig. 1 but for Q = M Z at NNLO. .01 0.1 10.00.51.01.52.0 zD u+u (z, =M ) Z pion fit pion+hadron fit NNFF NLO- R a t i o t o p i on f i t zD s+s (z, =M ) Z pion fit pion+hadron fit NNFF NLO- R a t i o t o p i on f i t zD c+c (z, =M ) Z pion fit pion+hadron fit NNFF NLO- R a t i o t o p i on f i t zD b+b (z, =M ) Z pion fit pion+hadron fit NNFF NLO- R a t i o t o p i on f i t zD g (z, =M ) Z pion fit pion+hadron fit NNFF NLO R a t i o t o p i on f i t FIG. 5. Comparison between the pion FFs ratios from the “ pion fit ”, “ pion+hadron fit ” and
NNFF1.0 analyses to the pion FFs from “pion” analysis at NLO for Q = M Z . .01 0.1 10.00.51.01.52.0 zD u+u (z, =M ) Z pion fit pion+hadron fit NNFF NNLO- R a t i o t o p i on f i t zD s+s (z, =M ) Z pion fit pion+hadron fit NNFF NNLO- R a t i o t o p i on f i t zD c+c (z, =M ) Z pion fit pion+hadron fit NNFF NNLO- R a t i o t o p i on f i t zD b+b (z, =M ) Z pion fit pion+hadron fit NNFF NNLO- R a t i o t o p i on f i t zD g (z, =M ) Z pion fit pion+hadron fit NNFF NNLO R a t i o t o p i on f i t FIG. 6. Same as Fig. 5 but at NNLO. nergy, we recalculate the predictions of Figs. 1 and 2, but this time for Q = M z . Theresults obtained have been shown in Figs. 3 and 4 at NLO and NNLO, respectively. Thereduction in the relative uncertainty of all flavors after the inclusion of the unidentifiedlight charged hadron data in the analysis is clearly seen from these figures. Note thatthe shift observed in the relative uncertainty of s + ¯ s and gluon FFs from “ pion+hadronfit ” analysis compared with the “ pion fit ” analysis at NLO (see Fig. 3) is due to theconsiderable change in the central values of these distributions after the inclusion of thehadron data.Another way for comparing the results of two aforementioned analyses is using theratio plots in which any change in the central values of the distribution can be alsoinvestigated, in addition to their uncertainties. Fig. 5 shows a comparison between theratios of pion FFs obtained from the “ pion+hadron fit ” analysis (yellow band) and also NNFF1.0 [9] (green band) to those obtained from the “ pion fit ” analysis (blue band) at Q = M z and NLO. According to the results obtained, one can sees that the uncertaintiesof all flavor distributions have been decreased by inclusion the unidentified light chargedhadron data in the analysis compared with the “ pion fit ” analysis. Overall, our FFswhether from the “ pion fit ” analysis or “ pion+hadron fit ” one, have less uncertaintiesthan the NNFF1.0 results, especially for the case of up, strange and gluon distributions.Let us focus on each flavor separately to discuss about the changes in more details.For the case of u + ¯ u FF, no significant change can be seen between the “ pion fit ” and“ pion+hadron fit ” analyses. However, both of these analyses have di ff erent results thanthe u + ¯ u FF of
NNFF1.0 , almost for all values of z . Actually, the di ff erence is more significantat lower values of z and reaches even to 30%. The second panel of Fig. 5 shows that theinclusion of hadron data in the analysis of pion FFs at NLO can put further constraintson s + ¯ s FF, especially at medium to small z regions, so that the uncertainty is remarkablyreduced. Moreover, it decreases the s + ¯ s distribution in magnitude at medium and largevalues of z . It should be noted that our results for the s + ¯ s FF are very di ff erent to NNFF1.0 esult and have smaller magnitude up to 100% at smaller z values. For the case of c + ¯ c and b + ¯ b FFs, all three analyses have almost same results both in magnitude and uncertaintiesat medium to small values of z , but di ff er at larger values. To be more precise, the c + ¯ c FFof “ pion fit ” and “ pion+hadron fit ” analyses are similar even at large values of z , butthe NNFF1.0 result is grows rapidly in this region. In contrast, the b + ¯ b FF of “ pion+hadronfit ” analysis behaves more similar to the
NNFF1.0 and grows rapidly at large z valuescompared with the “ pion fit ” analysis. Overall, one can conclude that the inclusion ofthe hadron data in the analysis does not a ff ect the c + ¯ c FF, but can change the b + ¯ b FF at largevalues of z . The last panel of Fig. 5 shown again the immense impact of the unidentifiedlight charged hadron data on the gluon FF of pion, especially at medium values of z . Ascan be seen, in addition to the significant reduction of the gluon FF uncertainty, its centralvalue has changed considerably at around z = . NNFF1.0 result at this region. However, there are still some di ff erences at 0 . (cid:46) z (cid:46) . z values. Another importantpoint should be noted is the very less uncertainty of our results compared with the NNFF1.0 one, in particular at large z regions which can be attributed to the low flexibility of ourparameterization for the gluon FF.Fig. 6 shows the same results as Fig. 5, but at NNLO accuracy. Overall, the inter-pretation of results obtained for each flavor distribution is similar to NLO case, with thedi ff erence that now the discrepancy observed between the s + ¯ s and also gluon FFs from“ pion fit ” and “ pion+hadron fit ” analyses at medium z regions is more moderate thanbefore. For example, the di ff erence between the gluon FFs obtained from these two anal-yses at z (cid:39) . u + ¯ u and c + ¯ c FFsremain still unchanged after the inclusion of the unidentified light charged hadron datain the analysis, and the b + ¯ b FF is rapidly grown at large z values just similar to NLO case. .01 0.1 1012345 = 5 GeV z D h u + u ( z , ) z Model (NLO) Model (NNLO)pion+hadron fit - = 5 GeV z D h s + s ( z , ) z Model (NLO) Model (NNLO)pion+hadron fit - = 5 GeV z D h c + c ( z , ) z Model (NLO) Model (NNLO)pion+hadron fit - = 5 GeV z D h b + b ( z , ) z Model (NLO) Model (NNLO)pion+hadron fit - = 5 GeV z D h g ( z , ) z Model (NLO) Model (NNLO)pion+hadron fit
FIG. 7. Comparison between the NLO and NNLO pion FFs determined from a simultaneousanalysis of pion and unidentified light charged hadron data , “ pion+hadron fit ”, for all flavordistributions at Q = . Comparison of the “ pion+hadron fit ” at NLO and NNLO accuracy Considering the “ pion+hadron fit ” analysis as a final and more excellent analysis todetermine the pion FFs from SIA data, it is also of interest to compare the distributionsobtained at NLO and NNLO accuracy. A comparison between the NLO and NNLOpion FFs determined from a simultaneous analysis of pion and unidentified light chargedhadron data for all flavor distributions at Q = u + ¯ u and gluon FFs follow similar manner. Tobe more precise, although the size of the changes is not too large, but both of them areincreased at smaller values of z and decreased at larger values since the NNLO correctionsare included. The c + ¯ c and b + ¯ b FFs are partially changed just at smaller values of z . Butthe situation is completely di ff erent for the case of s + ¯ s FF. Actually, the magnitude ofits distribution grows to a great extent by considering the NNLO corrections. Note that,although the uncertainty band of s + ¯ s FF at NNLO is bigger than NLO one, but the relativeuncertainties of two distributions (similar to Fig. 1) are of the same order.
D. Comparison of the data and theory predictions
Now we are in a position to complete our study of the fit quality as well as the data vs.theory comparisons.Here we will focus on the theory prediction based on the extracted pion FFs from our“ pion+hadron fit ” analysis. We turn to consider only the NNLO results to calculate thenormalized cross section for the total, light, c -tagged and b -tagged. To begin with, in Fig. 8,we show the detailed comparisons of σ tot d σ π ± dz with the SIA data sets analyzed in this study.These data sets include the charged pion productions at ALEPH , DELPHI , SLD and
OPAL experiments. As we can see from this comparison, the agreement between the analyzed .1 110 -6 -5 -4 -3 -2 -1 / t o t d t o t a l / d z Z Model (NNLO Theory) hadron+pion fit ALEPH ( ) DELPHI ( ) SLD ( ) OPAL ( ) -6 -5 -4 -3 -2 -1 / t o t d li gh t / d z z Model (NNLO Theory) hadron+pion fit SLD ( ) DELPHI ( ) -6 -5 -4 -3 -2 -1 / t o t d c - t ag / d z z Model (NNLO Theory) hadron+pion fit SLD ( ) -6 -5 -4 -3 -2 -1 / t o t d b - t ag / d z z Model (NNLO Theory) hadron+pion fit SLD ( ) DELPHI ( )
FIG. 8. Detailed comparisons of σ tot d σ π ± dz with the SIA data sets for the charged pion productions at ALEPH , DELPHI , SLD and
OPAL experiments. data sets and theoretical predictions for wide range of z are excellent, which show boththe validity and the quality of the QCD fits. In Fig. 9, we show the comparison betweenthe NNLO theory based on our “ pion+hadron fit ” with the charged pion productions at BABAR and
BELLE experiments. From the comparisons in this figure, we can see again thatthe data vs. theory comparisons are excellent.As a short summary, considering the impact of these two types of data on the pionFFs, shown in plots presented in this section, one sees that in the case of “ pion+hadron .1 110 -6 -5 -4 -3 -2 -1 / t o t d t o t a l / d z z Model (NNLO Theory) hadron+pion fit BABAR ( )=10.54 GeV -6 -5 -4 -3 -2 -1 / t o t d t o t a l / d z z Model (NNLO Theory) hadron+pion fit BELLE ( )=10.52 GeV
FIG. 9. Detailed comparisons of σ tot d σ π ± dz with the SIA data sets for the charged pion productions at BABAR and
BELLE experiments. fit ” analysis there is a visible reduction on the pion FFs uncertainties at wide range of z ,showing that the inclusion of two data sets simultaneously is somewhat more constraining. V. SUMMARY AND CONCLUSIONS
In this study, we have quantified the constraints that the unidentified light chargedhadron data sets could impose on the determination of pion FFs. To achieve this goal, newdeterminations of pion FFs at NLO and NNLO QCD corrections have been carried outbased on a comprehensive data sets of SIA processes. In this respect, we calculate the pionFFs from QCD analyses of two di ff erent data sets. Firstly, the pion FFs are determinedthrough QCD analyses of pion experimental data sets alone, which is referred to as “ pionfit ”. In addition to the determination of pion FFs using pion experimental data sets, onemay certainly expects further constraints to become available for pion FFs studies and animproved knowledge of the FFs will become possible from other source of experimentalinformation. Although the data sets of pion production in electron-positron annihilationnclude inclusive, uds -tagged, c -tagged and b -tagged observables, some of the parametersof pion FFs at initial scale can not be constrained well enough. Since the most contributionof unidentified light charged hadrons cross sections in SIA measurements is related to theidentified pion production, one can expects further constraints by adding these data setsinto the QCD fits. Hence, to achieve the first and new determination of pion FFs, we haveexplicitly chosen our input dataset and calculated pion FFs adding simultaneously thepion and unidentified light charged hadron data sets in our analysis, which is entitled as“ pion+hadron fit ”. Our main finding is that using the pion experimental data along withthe unidentified light charged hadron data sets has the potential to significantly reducethe pion FFs uncertainties in a wide kinematic range of momentum fraction z .According to the plots presented in this study, one can clearly sees the reduction ofpion FFs uncertainties in almost all range of z . The most e ff ects of adding unidentifiedlight charged hadron data sets in “ pion+hadron fit ” analysis are seen for the s + ¯ s andgluon FFs. Not only the uncertainties of s + ¯ s and gluon decrease, but also the behaviorof their central values have changed considerably. Consequently, our study shows thatapplying unidentified light charged hadron observables together with pion productiondata sets in a calculation of pion FFs leads to somewhat a better fit quality. Since thehigher-order corrections are significant, we plan to study the e ff ect arising from higherorder correction in the determination of pion FFs. Since we include the SIA data sets inour analyses, the perturbative QCD corrections up to NNLO accuracy can be considered.We found that our results at NNLO corrections improved the fit quality in comparison tothe NLO accuracy and it leads to reduction of the χ for all data sets separately as well asfor the total χ . By considering the NNLO corrections, similar slight improvements in theFFs uncertainty are also found in some region of z .The two analyses presented in this study share, however, a common limitation. In bothcases, it has indeed been necessary to include other source of experimental informationsuch as the data from semi-inclusive deep inelastic scattering (SIDIS), and proton-protonnd proton-antiproton collisions measured by TEVATRON, RHIC and LHC. However, theNNLO calculations for such processes are not yet available, which would require a relent-less e ff ort for the QCD calculations. It is worth mentioning here that our investigations inthis study could be extended to the new determination of kaon and proton FFs consideringthe unidentified light charged hadron data sets as well as the identified charged hadronproduction observables. More detailed discussions of these new determination of kaonand proton FFs will be presented in our upcoming study. ACKNOWLEDGMENTS
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