We investigate the role of heavy quarks in the production of light flavored hadrons and in the determination of the corresponding nonperturbative hadronization probabilities. We define a general mass variable flavor number scheme for fragmentation functions that accounts for heavy quark mass effects, and perform a global QCD analysis to an up-to-date data set including very precise Belle and BABAR results. We show that the mass dependent picture provides a much more accurate and consistent description of the data.Instituto de Física La PlataConsejo Nacional de Investigaciones Científicas y Técnica

Epele, Manuel

García Canal, Carlos Alberto

Sassot, Rodolfo

Physical Review D

English

Servicio de Difusión de la Creación Intelectual

arXiv:1604.08427v1  [hep-ph]  28 Apr 2016The role of heavy quarks in light hadron fragmentation.Manuel Epele∗ and Carlos Garc´ıa Canal†Instituto de F´ısica La Plata, UNLP, CONICET Departamento de F´ısica,Facultad de Ciencias Exactas, Universidad de La Plata, C.C. 69, La Plata, ArgentinaR. Sassot‡Departamento de F´ısica and IFIBA, Facultad de Ciencias Exactas y Naturales,Universidad de Buenos Aires, Ciudad Universitaria, Pabello´n 1 (1428) Buenos Aires, ArgentinaWe investigate the role of heavy quarks in the production of light flavored hadrons and in thedetermination of the corresponding non perturbative hadronization probabilities. We define a gen-eral mass variable flavor number scheme for fragmentation functions that accounts for heavy quarkmass effects, and perform a global QCD analysis to an up-to-date data set including very preciseBelle and BaBar results. We show that the mass dependent picture provides a much more accurateand consistent description of data.PACS numbers: 13.87.Fh, 13.85.Ni, 12.38.BxIntroduction.— The effects of heavy quark masses inhard processes and the appropriate definition of partonprobability densities for such species of quarks have beenvery actively studied in recent years [1]. When a heavyquark participates in a hard process, and the character-istic energy of the process under consideration is not farfrom the heavy quark mass scale, the most natural choiceis to treat these particles as massive throughout the cal-culation, rather than appeal to the more conventionalmassless parton approximation. However, when the scaleof the process exceeds by large the mass scale of the heavyquarks, mass corrections not only become negligible, butthe all-order resummations implicit in massless partonapproaches become crucial. This situation clearly rep-resents a challenge for a precise and consistent compari-son of processes occurring at very different energy scales,as it necessarily happens in a global QCD analysis de-signed to extract non-perturbative parton distributionfunctions (PDFs) [2] or fragmentation functions (FFs)[3, 4] from the data. To overcome this problem, in mostmodern global QCD analysis of data, the so called generalmass variable flavor number (GMVFN) schemes [5–8] forparton densities are introduced, as they allow to retainthe advantages of massive schemes near the mass thresh-olds and those of the massless approach at high energies,smoothly interpolating between both regimes.In fact, different GMVFN approaches have been ap-plied to the analysis of fragmentation probabilities ofheavy quarks into heavy flavored hadrons [9–11], but un-til now little attention has been paid in this respect tolight hadron fragmentation processes. Since the charmand bottom content in the proton is rather limited, theproduction of light mesons via heavy quarks of course isstrongly suppressed relative to that via light quarks both∗Electronic address: manuepele@fisica.unlp.edu.ar†Electronic address: garcia@fisica.unlp.edu.ar‡Electronic address: sassot@df.uba.arin proton-proton collisions and in semi-inclusive deep in-elastic scattering (SIDIS). Heavy quark corrections arethus expected to be negligible for these processes. Butthat is not the case for single inclusive electron-positronannihilation (SIA) into light mesons, where the charmand bottom contribution to the cross section is estimatedto be comparable in size to that of the light flavors [12].Whereas charm and bottom mass corrections may still benegligible in SIA experiments tuned at the mass of theZ boson, they are certainly relevant at the energy scalesof the more recent BaBar and Belle experiments [13, 14],which are just above the bottom mass threshold.In this paper we compute the single inclusive electron-positron annihilation cross section with non zero massesfor the charm and bottom quarks at first order in thestrong coupling constant αs, and we estimate the effectsof retaining these mass corrections in pion production.Since we find the mass dependence to be large but wewant to keep the advantages of the NLO massless ap-proximation at high energies, we define a general massvariable flavor number scheme for fragmentation func-tions in the lines of the FONLL scheme [15, 16], com-monly used for PDFs. We implement numerically thisscheme in Mellin moment space for the fast computationof the cross sections as required by QCD global analyses,and we perform them including recent Belle and BaBardata. The mass dependent picture introduced by theGMVFN scheme is found to be relevant in the extractionof fragmentation functions both in terms of the quality ofthe fit to data and in the reduction of the normalizationshifts applied to data that are customarily included inglobal fits. The shape of the charm into pion fragmenta-tion function, that contributes significantly to the crosssection at the energies of the Belle and BaBar experi-ments, is noticeably modified relative to the results ob-tained within a massless scheme. The bottom fragmenta-tion, constrained mainly by higher energy data, remainssimilar to the one found in the massless approximation.Factorization Schemes.— Most analyses of quark frag-mentation into light flavored hadrons [3, 4] rely on2the massless perturbative QCD approximation, supple-mented with heavy quark mass thresholds, where thecorresponding heavy quarks become active, contribute tocross sections and enter the scale or evolution equations[17]. This zero mass variable flavor number (ZMVFN)scheme is the simplest framework to compute the SIAcross section [18]:dσdzZMVFN=Xi=q,g,hσˆZMi (z,Q)⊗DZMi (z,Q) (1)where σˆZMi is the massless partonic SIA cross section intoa parton of flavor i, and DZMi are the corresponding FFs,for which we omit in what follows the dependence on thescaled hadron energy fraction z. ⊗ represent the appro-priate convolution over z, and Q is the center of massenergy. DZMi evolves in the scale Q through masslessQCD evolution equations for any parton flavor i, thatinclude light (q) and heavy (h) quarks and antiquarksand gluons (g). Eq.(1) gives a remarkably good approx-imation at NLO and NNLO well above mass thresholdmh [12, 19], i.e. Q ≫ mh, but fails to account for masseffects. Alternatively, in a massive scheme (M), heavyquark masses are kept at the partonic cross section level,dσdzM=Xi=q,gσˆMi (Q,mh)⊗DMi (Q) + σˆMh (Q,mh)⊗DMh (2)the light flavored FFs DMq,g(Q) still evolve through mass-less QCD evolution equations in NLO, although heavyquark loops contribute above O(α2S), and heavy quarkFFs DMh decouple from the QCD evolution [20]. Themassive scheme gives a good description near the massthresholds, but fail to converge to the massless limitat high energies because of potentially large logarith-mic contributions (αkS logk(mh/Q)) present in the par-tonic cross sections σˆMi that spoil the convergence of theperturbation expansion. These logarithmic contributionsare effectively resummed in the renormalization groupimproved ZM approximation. In fact, it has been shown[21] that in the massless limit, i.e. Q≫ mh,σˆMi (Q,mh) −−−−→mh→0Xj=q,g,hσˆZMj (Q)⊗Aji(Q/mh) (3)where all logarithmic contributions can be factorized inan operator matrix Aij that is independent of the hardprocess under consideration [10, 11, 22].The advantages of both the zero mass and the mas-sive schemes can be exploited in a general mass (GM)scheme, which defines the corresponding fragmentationprobabilities throughdσdzGMVFN=Xi=q,g,hσˆGMj (Q,mh)⊗DGMj (Q) (4)where a subtracted massive partonic cross sectionσˆGMj (Q,mh) =Xi=q,g,hσˆMi (Q,mh)⊗A−1ij (Q/mh) (5)46810120.20.40.60.8134567810 2BELLE TPC LEP1 dσpicσ dz z= 0.2σZMσMσGMσGM*∧∧∧∧BELLE TPC LEP1 dσpicσ dz z= 0.51 dσpibσ dz z= 0.2Q1 dσpibσ dz z= 0.5Q0.10.150.20.250.30.3510 2FIG. 1: Charm and bottom contribution to the SIA cross sec-tion computed with different approximations for the partoniccross sections and the same set of FFs.guarantees the correct massless behavior at high ener-gies. The fragmentation functions DGMj (Q) obey stan-dard evolution equations as in the ZM scheme and thecontinuity across the thresholds can be ensured by im-posing the following matching conditionDGMj (mh) =Xi=q,g,hAji(1)⊗DMi (mh) (6)analogously as in the FONLL scheme for PDFs [15, 16].In order to illustrate the effects arising from the use ofthe different approximations for the partonic cross sec-tions, in Fig.1 we show the charm and bottom contribu-tion to the SIA pion cross section (upper and lower pan-els, respectively) as a function of the center of mass en-ergy Q at two reference values of z. For the comparison,the same set of NLO FFs that will be described in detailin the next section, is convoluted with the NLO mass-less (ZM) partonic cross section (red solid lines), the fullO(αS) massive (M) result (black dashed lines), and thesubtracted (GM) approximation (blue dot-dashed lines).Even though the scheme defined by Eqs.(4-6) interpo-lates between the massive behavior near thresholds andthe desired zero mass limit at Q ≫ mh, this solution isclearly not unique. The same limits can be satisfied inalternative factorization schemes where the convergenceto the massless limit happens at a lower or higher energyscale. For instance, substitutingσˆGMj −→ σˆGM*j = (1− f(Q)) σˆMj + f(Q) σˆGMj (7)in eq.(4), with an f(Q) that vanishes in the thresholdf(mh) = 0, and saturates to 1 for Q ≫ mh such as3f(Q) = 1−2mh/Q, would delay the onset of the massless-like behaviour, as shown for the charm contribution bythe grey dotted lines in the upper panels of Fig.1, whileσˆGMj −→ σˆGM*j = (1− f(Q)) σˆGMj + f(Q) σˆZMj (8)would suppress mass effects, as shown for the bottomcontribution in the lower panels. In any case, it is clearthat while the GM schemes introduce negligible correc-tions at the energy scale of the LEP experiments, masseffects are significant at the scales of Belle and BaBar.The remarkably precise measurements of the SIA crosssection as a function of z performed by these experimentsallows to check if a particular scheme is favored by thedata.GMVFN scheme global analysis for FFs.— In this sec-tion, we discuss the actual relevance of heavy quark masscorrections in charged pion production, implementing dif-ferent factorization schemes in a NLO QCD global anal-ysis for the extraction of FFs, performed along the linesof that of ref. [12]. The method for the global analysishas been described in detail in [3, 12]. It is based onan efficient Mellin moment technique that allows one totabulate and store the computationally most demandingparts of the NLO calculation of SIA, SIDIS and proton-proton hadroproduction cross sections prior to the ac-tual analysis. In this way, the evaluation of the relevantcross sections becomes so fast that can be easily per-formed inside a standard χ2 minimization. At variancewith [12], where SIA cross sections were evaluated onlyin the ZM approximation, here they are extended to theGM framework what requires to define different contoursin the complex moment space to perform the Mellin in-version. Hadroproduction and SIDIS cross sections arestill computed in the ZM framework, to asses in this firststep the impact introduced by SIA corrections. Never-theless, the heavy quark contributions to these processesare negligibly small. Also a minor difference with [12] isthat we remove from the data sets included in the analy-sis TASSO and OPAL light flavor tagged data, that havecomparatively large errors.In Tab. I we compare the quality of a fit performedwithin the standard ZMVFN factorization scheme, and aGMVFN variant where the prescriptions of Eqs.(7) and(8) have been adopted for the charm and bottom coeffi-cients respectively, as in the example of Fig.1. Among thedifferent prescriptions we have explored, the above men-tioned one reproduces best the data, significantly betterthat in the ZMVFN scheme, with much lower χ2i valuesand smaller normalization shifts Ni. No significant im-provement is found with more sophisticated weight func-tions f(Q). On the other hand, the most simple subtrac-tion of Eq.(5), produces fits of much poor quality, sug-gesting that such prescription oversubtracts for charm,and converges much slower to the massless limit thanthe data require for bottom.As expected, heavy quark mass effects are most no-ticeable for Belle and Babar experiments, since at theirrelatively low center of mass energies heavy quarks areTABLE I: Individual χ2 values and normalization shifts Ni forthe data sets included in global analyses where the ZMVFNand GMVFN schemes have been implemented.experiment data # data ZMVFN GMVFNtype in fit Ni χ2 Ni χ2Aleph [23] incl. 22 0.968 21.6 0.994 23.3BaBar [13] incl. 39 1.019 76.7 1.002 58.2Belle [14] incl. 78 1.044 19.5 1.019 11.0Delphi [24] incl. 17 0.978 6.7 1.003 9.3uds tag 17 0.978 20.8 1.003 9.5b tag 17 0.978 10.5 1.003 7.8Opal [25] incl. 21 0.946 27.9 0.970 15.9Sld [26] incl. 28 0.938 28.0 0.963 9.5uds tag 17 0.938 21.3 0.963 11.3c tag 17 0.938 34.0 0.963 19.8b tag 17 0.938 11.1 0.963 9.9Tpc [27] incl. 17 0.997 31.7 1.006 27.9uds tag 9 0.997 2.0 1.006 2.0c tag 9 0.997 5.9 1.006 4.3b tag 9 0.997 9.6 1.006 10.9Compass [28] pi± (d) 398 1.003 378.7 1.008 382.9Hermes [29] pi± (p) 64 0.981 74.0 0.986 69.9pi± (d) 64 0.980 107.3 0.985 103.7Phenix [30] pi0 15 1.174 14.3 1.167 14.4Star [31] pi±, pi0 38 1.205 31.2 1.202 33.8Alice [32] pi0 11 0.696 33.3 0.700 31.2TOTAL: 924 966.4 875.8far from behaving as massless. On the other hand, eventhough these experiments are above the bottom thresh-old 2mb, where the bottom should be considered as anactive flavor in the ZMVFN scheme, the bottom is actu-ally strongly suppressed, feature that is well accountedfor in the GMVFN scheme. In Fig.2 we show the dif-ferences between the fit estimates and Belle and BaBardata, against the relative experimental error.Notice that there is also a considerable improvementin the description of data from experiments at a higherenergy scale. The difference between the massless andthe mass dependent picture comes in this case from the-0.100.1z0.2 0.4 0.8Belle: (data - theory)/theoryGMVFN schemeZMVFN schemerelative exp. error0.2 0.4 0.8BaBar: (data - theory)/theoryzGMVFN schemeZMVFN schemerelative exp. errorFIG. 2: Comparison between data from Belle and BaBar andestimates from the ZMVFN and GMVFN schemes40.20.40.60.811.21.41.60.20.40.60.811.21.41.61u + us + su = dzDpi+(z)iZMVFNGMVFNQ = 4 GeVc + cgluonzDpi+(z)iZMVFNGMVFNQ = 4 GeVu + us + su = dzDpi+(z)iZMVFNGMVFNQ = MZzc + cgluonb + bzDpi+(z)iZMVFNGMVFNzQ = MZ1FIG. 3: FFs at Q = 4 GeV and Q = MZ coming from theZMVFN and GMVFN schemesfact that the QCD scale dependence preserves the differ-ence between the charm fragmentation probabilities con-strained by lower energy data, as shown in Fig.3. Thebottom fragmentation probability is mainly constrainedby high energy flavor tagged SIA data, for which massdependent corrections become negligible, and thereforethe results for this flavor in both pictures agree. TheGMVFN however guarantees that the bottom contribu-tion at lower energies is conveniently suppressed, improv-ing the overall agreement and consistency. No significantdifferences are found for the light flavors, that are con-strained mainly by light flavor tagged SIA and SIDISdata.Conclusions and outlook.— We have shown that an ac-curate determination of the fragmentation probabilitiesof quarks and gluons into pions, matching the precision ofthe present generation of hadroproduction experiments,requires a picture sensitive to heavy quarks dynamics.Such a picture was presented here, together with the re-sults of a NLO QCD global analysis where it was im-plemented. Heavy quark mass dependence is speciallyrelevant in single inclusive electron-positron annihilationinto pions, where the detailed energy scale dependence ofthe charm contribution and the suppression of the bot-tom above their respective thresholds is non negligible.These effects are expected to be even more apparent inthe production of heavy flavored mesons, where the massdependent fragmentation probabilities are dominant.We warmly acknowledge Daniel de Florian and MarcoStratmann for help and encouragment. This work wassupported by CONICET, ANPCyT and UBACyT.[1] See, e.g. R. D. Ball, M. Bonvini and L. Rottoli, JHEP1511, 122 (2015), and references therein.[2] For a review see, e.g. J. Rojo et al., J. Phys. G 42, 103103(2015)[3] D. de Florian, R. Sassot and M. Stratmann, Phys. Rev.D 75, 114010 (2007).[4] S. Albino, B. A. Kniehl, and G. Kramer, Nucl. Phys. B725, 181 (2005); ibid. 803, 42 (2008); M. Hirai, S. Ku-mano, T. -H. Nagai, and K. Sudoh, Phys. Rev. D 75,094009 (2007).[5] M. A. G. Aivazis, J. C. Collins, F. I. Olness andW. K. Tung, Phys. Rev. D 50, 3102 (1994)[6] R. S. Thorne and R. G. Roberts, Eur. Phys. J. C 19, 339(2001)[7] R. S. Thorne, Phys. Rev. D 73, 054019 (2006).[8] M. Guzzi, P. M. Nadolsky, H. L. Lai and C.-P. Yuan,Phys. Rev. D 86, 053005 (2012).[9] M. Cacciari, P. Nason and C. Oleari, JHEP 0510, 034(2005)[10] T. Kneesch, B. A. Kniehl, G. Kramer and I. Schienbein,Nucl. Phys. B 799, 34 (2008)[11] B. A. Kniehl, G. Kramer, I. Schienbein and H. Spies-berger, Phys. Rev. D 77, 014011 (2008).[12] D. de Florian, R. Sassot, M. Epele, R. J. Hernndez-Pinto and M. Stratmann, Phys. Rev. D 91, no. 1, 014035(2015).[13] J. P. Lees et al. [BaBar Collaboration], Phys. Rev. D88, 032011 (2013).[14] M. Leitgab et al. [Belle Collaboration], Phys. Rev. Lett.111, 062002 (2013).[15] S. Forte, E. Laenen, P. Nason and J. Rojo, Nucl. Phys.B 834, 116 (2010).[16] R. D. Ball et al. Phys. Lett. B 754, 49 (2016)[17] G. Altarelli and G. Parisi, Nucl. Phys. B 126, 298 (1977);Y. L. Dokshitzer, Sov. Phys. JETP 46, 641 (1977) [Zh.Eksp. Teor. Fiz. 73, 1216 (1977)]; V. N. Gribov andL. N. Lipatov, Sov. J. Nucl. Phys. 15, 438 (1972).[18] G. Altarelli, R.K. Ellis, G. Martinelli, and S.Y. Pi, Nucl.Phys. B160, 301 (1979).[19] D. P. Anderle, F. Ringer and M. Stratmann, Phys. Rev.D 92, no. 11, 114017 (2015).[20] J. C. Collins, Phys. Rev. D 58, 094002 (1998)[21] M. Buza, Y. Matiounine, J. Smith and W. L. van Neer-ven, Eur. Phys. J. C 1, 301 (1998).[22] B. A. Kniehl, G. Kramer, I. Schienbein and H. Spies-berger, Phys. Rev. D 71, 014018 (2005); Eur. Phys. J. C41, 199 (2005).[23] D. Buskulic et al. [Aleph Collaboration], Z. Phys. C 66,355 (1995).[24] P. Abreu et al. [Delphi Collaboration], Eur. Phys. J. C5, 585 (1998).[25] R. Akers et al. [Opal Collaboration], Z. Phys. C 63, 181(1994).[26] K. Abe et al. [Sld Collaboration], Phys. Rev. D 59,052001 (1999).[27] H. Aihara et al. [Tpc/Two Gamma Collaboration],Phys. Lett. B 184, 299 (1987); Phys. Rev. Lett. 61, 1263(1988).[28] N. Makke [Compass Collaboration], PoS DIS 2013, 202(2013).5[29] A. Airapetian et al. [Hermes Collaboration], Phys. Rev.D 87, 074029 (2013).[30] A. Adare et al. [Phenix Collaboration], Phys. Rev. D 76,051106 (2007).[31] L. Adamczyk et al. [Star Collaboration], Phys. Rev. D89, 012001 (2014).[32] B. Abelev et al. [Alice Collaboration], Phys. Lett. B717, 162 (2012).

Role of heavy quarks in light hadron fragmentation

SEDICI - Repositorio de la UNLP

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The role of heavy quarks in light hadron fragmentation.
Manuel Epele∗ and Carlos Garc´ıa Canal†
Instituto de F´ısica La Plata, UNLP, CONICET Departamento de F´ısica,
Facultad de Ciencias Exactas, Universidad de La Plata, C.C. 69, La Plata, Argentina
R. Sassot‡
Departamento de F´ısica and IFIBA, Facultad de Ciencias Exactas y Naturales,
Universidad de Buenos Aires, Ciudad Universitaria, Pabello´n 1 (1428) Buenos Aires, Argentina
We investigate the role of heavy quarks in the production of light flavored hadrons and in the
determination of the corresponding non perturbative hadronization probabilities. We define a gen-
eral mass variable flavor number scheme for fragmentation functions that accounts for heavy quark
mass effects, and perform a global QCD analysis to an up-to-date data set including very precise
Belle and BaBar results. We show that the mass dependent picture provides a much more accurate
and consistent description of data.
PACS numbers: 13.87.Fh, 13.85.Ni, 12.38.Bx
Introduction.— The effects of heavy quark masses in
hard processes and the appropriate definition of parton
probability densities for such species of quarks have been
very actively studied in recent years [1]. When a heavy
quark participates in a hard process, and the character-
istic energy of the process under consideration is not far
from the heavy quark mass scale, the most natural choice
is to treat these particles as massive throughout the cal-
culation, rather than appeal to the more conventional
massless parton approximation. However, when the scale
of the process exceeds by large the mass scale of the heavy
quarks, mass corrections not only become negligible, but
the all-order resummations implicit in massless parton
approaches become crucial. This situation clearly rep-
resents a challenge for a precise and consistent compari-
son of processes occurring at very different energy scales,
as it necessarily happens in a global QCD analysis de-
signed to extract non-perturbative parton distribution
functions (PDFs) [2] or fragmentation functions (FFs)
[3, 4] from the data. To overcome this problem, in most
modern global QCD analysis of data, the so called general
mass variable flavor number (GMVFN) schemes [5–8] for
parton densities are introduced, as they allow to retain
the advantages of massive schemes near the mass thresh-
olds and those of the massless approach at high energies,
smoothly interpolating between both regimes.
In fact, different GMVFN approaches have been ap-
plied to the analysis of fragmentation probabilities of
heavy quarks into heavy flavored hadrons [9–11], but un-
til now little attention has been paid in this respect to
light hadron fragmentation processes. Since the charm
and bottom content in the proton is rather limited, the
production of light mesons via heavy quarks of course is
strongly suppressed relative to that via light quarks both
∗Electronic address: manuepele@fisica.unlp.edu.ar
†Electronic address: garcia@fisica.unlp.edu.ar
‡Electronic address: sassot@df.uba.ar
in proton-proton collisions and in semi-inclusive deep in-
elastic scattering (SIDIS). Heavy quark corrections are
thus expected to be negligible for these processes. But
that is not the case for single inclusive electron-positron
annihilation (SIA) into light mesons, where the charm
and bottom contribution to the cross section is estimated
to be comparable in size to that of the light flavors [12].
Whereas charm and bottom mass corrections may still be
negligible in SIA experiments tuned at the mass of the
Z boson, they are certainly relevant at the energy scales
of the more recent BaBar and Belle experiments [13, 14],
which are just above the bottom mass threshold.
In this paper we compute the single inclusive electron-
positron annihilation cross section with non zero masses
for the charm and bottom quarks at first order in the
strong coupling constant αs, and we estimate the effects
of retaining these mass corrections in pion production.
Since we find the mass dependence to be large but we
want to keep the advantages of the NLO massless ap-
proximation at high energies, we define a general mass
variable flavor number scheme for fragmentation func-
tions in the lines of the FONLL scheme [15, 16], com-
monly used for PDFs. We implement numerically this
scheme in Mellin moment space for the fast computation
of the cross sections as required by QCD global analyses,
and we perform them including recent Belle and BaBar
data. The mass dependent picture introduced by the
GMVFN scheme is found to be relevant in the extraction
of fragmentation functions both in terms of the quality of
the fit to data and in the reduction of the normalization
shifts applied to data that are customarily included in
global fits. The shape of the charm into pion fragmenta-
tion function, that contributes significantly to the cross
section at the energies of the Belle and BaBar experi-
ments, is noticeably modified relative to the results ob-
tained within a massless scheme. The bottom fragmenta-
tion, constrained mainly by higher energy data, remains
similar to the one found in the massless approximation.
Factorization Schemes.— Most analyses of quark frag-
mentation into light flavored hadrons [3, 4] rely on
2the massless perturbative QCD approximation, supple-
mented with heavy quark mass thresholds, where the
corresponding heavy quarks become active, contribute to
cross sections and enter the scale or evolution equations
[17]. This zero mass variable flavor number (ZMVFN)
scheme is the simplest framework to compute the SIA
cross section [18]:
dσ
dz
ZMVFN
=
X
i=q,g,h
σˆZMi (z,Q)⊗D
ZM
i (z,Q) (1)
where σˆZMi is the massless partonic SIA cross section into
a parton of flavor i, and DZMi are the corresponding FFs,
for which we omit in what follows the dependence on the
scaled hadron energy fraction z. ⊗ represent the appro-
priate convolution over z, and Q is the center of mass
energy. DZMi evolves in the scale Q through massless
QCD evolution equations for any parton flavor i, that
include light (q) and heavy (h) quarks and antiquarks
and gluons (g). Eq.(1) gives a remarkably good approx-
imation at NLO and NNLO well above mass threshold
mh [12, 19], i.e. Q ≫ mh, but fails to account for mass
effects. Alternatively, in a massive scheme (M), heavy
quark masses are kept at the partonic cross section level,
dσ
dz
M
=
X
i=q,g
σˆMi (Q,mh)⊗D
M
i (Q) + σˆ
M
h (Q,mh)⊗D
M
h (2)
the light flavored FFs DMq,g(Q) still evolve through mass-
less QCD evolution equations in NLO, although heavy
quark loops contribute above O(α2S), and heavy quark
FFs DMh decouple from the QCD evolution [20]. The
massive scheme gives a good description near the mass
thresholds, but fail to converge to the massless limit
at high energies because of potentially large logarith-
mic contributions (αkS log
k(mh/Q)) present in the par-
tonic cross sections σˆMi that spoil the convergence of the
perturbation expansion. These logarithmic contributions
are effectively resummed in the renormalization group
improved ZM approximation. In fact, it has been shown
[21] that in the massless limit, i.e. Q≫ mh,
σˆMi (Q,mh) −−−−→
mh→0
X
j=q,g,h
σˆZMj (Q)⊗Aji(Q/mh) (3)
where all logarithmic contributions can be factorized in
an operator matrix Aij that is independent of the hard
process under consideration [10, 11, 22].
The advantages of both the zero mass and the mas-
sive schemes can be exploited in a general mass (GM)
scheme, which defines the corresponding fragmentation
probabilities through
dσ
dz
GMVFN
=
X
i=q,g,h
σˆGMj (Q,mh)⊗D
GM
j (Q) (4)
where a subtracted massive partonic cross section
σˆGMj (Q,mh) =
X
i=q,g,h
σˆMi (Q,mh)⊗A
−1
ij (Q/mh) (5)
4
6
8
10
12
0.2
0.4
0.6
0.8
1
3
4
5
6
7
8
10 2
BELLE TPC LEP
1 dσpic
σ dz z= 0.2
σZM
σM
σGM
σGM*
∧
∧
∧
∧
BELLE TPC LEP
1 dσpic
σ dz z= 0.5
1 dσpib
σ dz z= 0.2
Q
1 dσpib
σ dz z= 0.5
Q
0.1
0.15
0.2
0.25
0.3
0.35
10 2
FIG. 1: Charm and bottom contribution to the SIA cross sec-
tion computed with different approximations for the partonic
cross sections and the same set of FFs.
guarantees the correct massless behavior at high ener-
gies. The fragmentation functions DGMj (Q) obey stan-
dard evolution equations as in the ZM scheme and the
continuity across the thresholds can be ensured by im-
posing the following matching condition
DGMj (mh) =
X
i=q,g,h
Aji(1)⊗D
M
i (mh) (6)
analogously as in the FONLL scheme for PDFs [15, 16].
In order to illustrate the effects arising from the use of
the different approximations for the partonic cross sec-
tions, in Fig.1 we show the charm and bottom contribu-
tion to the SIA pion cross section (upper and lower pan-
els, respectively) as a function of the center of mass en-
ergy Q at two reference values of z. For the comparison,
the same set of NLO FFs that will be described in detail
in the next section, is convoluted with the NLO mass-
less (ZM) partonic cross section (red solid lines), the full
O(αS) massive (M) result (black dashed lines), and the
subtracted (GM) approximation (blue dot-dashed lines).
Even though the scheme defined by Eqs.(4-6) interpo-
lates between the massive behavior near thresholds and
the desired zero mass limit at Q ≫ mh, this solution is
clearly not unique. The same limits can be satisfied in
alternative factorization schemes where the convergence
to the massless limit happens at a lower or higher energy
scale. For instance, substituting
σˆGMj −→ σˆ
GM*
j = (1− f(Q)) σˆ
M
j + f(Q) σˆ
GM
j (7)
in eq.(4), with an f(Q) that vanishes in the threshold
f(mh) = 0, and saturates to 1 for Q ≫ mh such as
3f(Q) = 1−2mh/Q, would delay the onset of the massless-
like behaviour, as shown for the charm contribution by
the grey dotted lines in the upper panels of Fig.1, while
σˆGMj −→ σˆ
GM*
j = (1− f(Q)) σˆ
GM
j + f(Q) σˆ
ZM
j (8)
would suppress mass effects, as shown for the bottom
contribution in the lower panels. In any case, it is clear
that while the GM schemes introduce negligible correc-
tions at the energy scale of the LEP experiments, mass
effects are significant at the scales of Belle and BaBar.
The remarkably precise measurements of the SIA cross
section as a function of z performed by these experiments
allows to check if a particular scheme is favored by the
data.
GMVFN scheme global analysis for FFs.— In this sec-
tion, we discuss the actual relevance of heavy quark mass
corrections in charged pion production, implementing dif-
ferent factorization schemes in a NLO QCD global anal-
ysis for the extraction of FFs, performed along the lines
of that of ref. [12]. The method for the global analysis
has been described in detail in [3, 12]. It is based on
an efficient Mellin moment technique that allows one to
tabulate and store the computationally most demanding
parts of the NLO calculation of SIA, SIDIS and proton-
proton hadroproduction cross sections prior to the ac-
tual analysis. In this way, the evaluation of the relevant
cross sections becomes so fast that can be easily per-
formed inside a standard χ2 minimization. At variance
with [12], where SIA cross sections were evaluated only
in the ZM approximation, here they are extended to the
GM framework what requires to define different contours
in the complex moment space to perform the Mellin in-
version. Hadroproduction and SIDIS cross sections are
still computed in the ZM framework, to asses in this first
step the impact introduced by SIA corrections. Never-
theless, the heavy quark contributions to these processes
are negligibly small. Also a minor difference with [12] is
that we remove from the data sets included in the analy-
sis TASSO and OPAL light flavor tagged data, that have
comparatively large errors.
In Tab. I we compare the quality of a fit performed
within the standard ZMVFN factorization scheme, and a
GMVFN variant where the prescriptions of Eqs.(7) and
(8) have been adopted for the charm and bottom coeffi-
cients respectively, as in the example of Fig.1. Among the
different prescriptions we have explored, the above men-
tioned one reproduces best the data, significantly better
that in the ZMVFN scheme, with much lower χ2i values
and smaller normalization shifts Ni. No significant im-
provement is found with more sophisticated weight func-
tions f(Q). On the other hand, the most simple subtrac-
tion of Eq.(5), produces fits of much poor quality, sug-
gesting that such prescription oversubtracts for charm,
and converges much slower to the massless limit than
the data require for bottom.
As expected, heavy quark mass effects are most no-
ticeable for Belle and Babar experiments, since at their
relatively low center of mass energies heavy quarks are
TABLE I: Individual χ2 values and normalization shifts Ni for
the data sets included in global analyses where the ZMVFN
and GMVFN schemes have been implemented.
experiment data # data ZMVFN GMVFN
type in fit Ni χ
2 Ni χ
2
Aleph [23] incl. 22 0.968 21.6 0.994 23.3
BaBar [13] incl. 39 1.019 76.7 1.002 58.2
Belle [14] incl. 78 1.044 19.5 1.019 11.0
Delphi [24] incl. 17 0.978 6.7 1.003 9.3
uds tag 17 0.978 20.8 1.003 9.5
b tag 17 0.978 10.5 1.003 7.8
Opal [25] incl. 21 0.946 27.9 0.970 15.9
Sld [26] incl. 28 0.938 28.0 0.963 9.5
uds tag 17 0.938 21.3 0.963 11.3
c tag 17 0.938 34.0 0.963 19.8
b tag 17 0.938 11.1 0.963 9.9
Tpc [27] incl. 17 0.997 31.7 1.006 27.9
uds tag 9 0.997 2.0 1.006 2.0
c tag 9 0.997 5.9 1.006 4.3
b tag 9 0.997 9.6 1.006 10.9
Compass [28] pi± (d) 398 1.003 378.7 1.008 382.9
Hermes [29] pi± (p) 64 0.981 74.0 0.986 69.9
pi± (d) 64 0.980 107.3 0.985 103.7
Phenix [30] pi0 15 1.174 14.3 1.167 14.4
Star [31] pi±, pi0 38 1.205 31.2 1.202 33.8
Alice [32] pi0 11 0.696 33.3 0.700 31.2
TOTAL: 924 966.4 875.8
far from behaving as massless. On the other hand, even
though these experiments are above the bottom thresh-
old 2mb, where the bottom should be considered as an
active flavor in the ZMVFN scheme, the bottom is actu-
ally strongly suppressed, feature that is well accounted
for in the GMVFN scheme. In Fig.2 we show the dif-
ferences between the fit estimates and Belle and BaBar
data, against the relative experimental error.
Notice that there is also a considerable improvement
in the description of data from experiments at a higher
energy scale. The difference between the massless and
the mass dependent picture comes in this case from the
-0.1
0
0.1
z
0.2 0.4 0.8
Belle: (data - theory)/theory
GMVFN scheme
ZMVFN scheme
relative exp. error
0.2 0.4 0.8
BaBar: (data - theory)/theory
z
GMVFN scheme
ZMVFN scheme
relative exp. error
FIG. 2: Comparison between data from Belle and BaBar and
estimates from the ZMVFN and GMVFN schemes
40.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1
u + u
s + s
u = d
zDpi+(z)i
ZMVFN
GMVFN
Q = 4 GeV
c + c
gluon
zDpi+(z)i
ZMVFN
GMVFN
Q = 4 GeV
u + u
s + s
u = d
zDpi+(z)i
ZMVFN
GMVFN
Q = MZ
z
c + c
gluon
b + b
zDpi+(z)i
ZMVFN
GMVFN
z
Q = MZ
1
FIG. 3: FFs at Q = 4 GeV and Q = MZ coming from the
ZMVFN and GMVFN schemes
fact that the QCD scale dependence preserves the differ-
ence between the charm fragmentation probabilities con-
strained by lower energy data, as shown in Fig.3. The
bottom fragmentation probability is mainly constrained
by high energy flavor tagged SIA data, for which mass
dependent corrections become negligible, and therefore
the results for this flavor in both pictures agree. The
GMVFN however guarantees that the bottom contribu-
tion at lower energies is conveniently suppressed, improv-
ing the overall agreement and consistency. No significant
differences are found for the light flavors, that are con-
strained mainly by light flavor tagged SIA and SIDIS
data.
Conclusions and outlook.— We have shown that an ac-
curate determination of the fragmentation probabilities
of quarks and gluons into pions, matching the precision of
the present generation of hadroproduction experiments,
requires a picture sensitive to heavy quarks dynamics.
Such a picture was presented here, together with the re-
sults of a NLO QCD global analysis where it was im-
plemented. Heavy quark mass dependence is specially
relevant in single inclusive electron-positron annihilation
into pions, where the detailed energy scale dependence of
the charm contribution and the suppression of the bot-
tom above their respective thresholds is non negligible.
These effects are expected to be even more apparent in
the production of heavy flavored mesons, where the mass
dependent fragmentation probabilities are dominant.
We warmly acknowledge Daniel de Florian and Marco
Stratmann for help and encouragment. This work was
supported by CONICET, ANPCyT and UBACyT.
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Role of heavy quarks in light hadron fragmentation

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