We present calculations of next-to-leading order and resummed QCD corrections for semi-inclusive deep-inelastic scattering and single-inclusive e+e− annihilation. The resummation is performed to next-to-leading logarithmic accuracy. Knowing these QCD corrections is important in order to extract parton distribution functions and fragmentation functions from present and future data for these reactions. We present phenomenological results relevant for the COMPASS, HERMES, and BELLE experiments

Anderle, D.

Ringer, F.

Vogelsang, W.

English

arXiv

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DOI 10.1393/ncc/i2013-11596-0Colloquia: QCDN12IL NUOVO CIMENTO Vol. 36 C, N. 5 Settembre-Ottobre 2013QCD resummation in hadron productionD. Anderle, F. Ringer and W. VogelsangInstitute for Theoretical Physics, Tu¨bingen UniversityAuf der Morgenstelle 14, 72076 Tu¨bingen, Germanyricevuto il 18 Aprile 2013Summary. — We present calculations of next-to-leading order and resummed QCDcorrections for semi-inclusive deep-inelastic scattering and single-inclusive e+e− an-nihilation. The resummation is performed to next-to-leading logarithmic accuracy.Knowing these QCD corrections is important in order to extract parton distributionfunctions and fragmentation functions from present and future data for these reac-tions. We present phenomenological results relevant for the COMPASS, HERMES,and BELLE experiments.PACS 13.85.Ni – Inclusive production with identified hadrons.PACS 12.38.Bx – Perturbative calculations.1. – IntroductionProcesses with identified final-state hadrons play important roles in QCD. In thepresent work, we address higher-order perturbative corrections to two of the key hadronproduction processes, single-inclusive annihilation (SIA) e+e− → hX and semi-inclusivedeep-inelastic scattering (SIDIS), p→ hX, where h denotes a final-state hadron. Mod-ern analyses [1-3] of fragmentation functions variously use data for these two processes.Our study is very much motivated by the recent advent of data for these reactions with un-precedented high precision. The BELLE collaboration at KEK has presented data [4] forpion and kaon multiplicities in SIA with a very fine binning and extremely high precisionat the sub-1% level. New preliminary high-statistics SIDIS data have been shown by theHERMES [5] and COMPASS [6] lepton scattering experiments over the past year or so.In the kinematic regimes accessed by these experiments, perturbative-QCD correc-tions are expected to be fairly significant. The infrared cancellations between virtual andreal-emission diagrams in higher orders leave behind logarithmic contributions. These“threshold logarithms” become large when the phase space for real-gluon radiation isshrinking and therefore have to be taken into account to all orders in perturbation the-ory. In this work, we will examine this resummation of the large corrections. We willrestrict ourselves to resummation at next-to-leading logarithmic (NLL) accuracy, whichshould capture the main effects. A more detailed description of the analysis presented inthis paper was published in ref. [7].c© Societa` Italiana di Fisica 153154 D. ANDERLE, F. RINGER and W. VOGELSANG2. – Resummation for SIDIS multiplicitiesWe consider semi-inclusive deep-inelastic scattering p→ hX. Using the usual kine-matic variables, the SIDIS cross section may be written as [8](1)d3σhdxdydz=4πα2Q2[1 + (1− y)22yFhT (x, z,Q2) +1− yyFhL(x, z,Q2)],where α is the fine structure constant and FhT,L are the transverse and the longitudinalstructure functions. SIDIS hadron multiplicities are defined by(2) RhSIDIS ≡d3σh/dxdydzd2σ/dxdy,where d2σ/dxdy is the cross section for inclusive DIS, p → X. In order to investigatehigher-order effects on SIDIS multiplicities, we have to consider QCD corrections to boththe SIDIS and the inclusive DIS cross section.Using factorization, the transverse and longitudinal structure functions in eq. (1) aregiven by (i = T,L)(3) Fhi (x, z,Q2) =∑f,f ′∫ 1xdxˆxˆ∫ 1zdzˆzˆf(xxˆ, μ2)Dhf ′(zzˆ, μ2)Cif ′f(xˆ, zˆ,Q2μ2, αs(μ2)),where f(ξ, μ2) denotes the distribution of parton f = q, q¯, g in the nucleon at momentumfraction ξ and scale μ, while Dhf ′(ζ, μ2) is the corresponding fragmentation function forparton f ′ going to the observed hadron h. The hard-scattering coefficient functions Cif ′fcan be computed in perturbation theory.Since threshold resummation can be derived in Mellin-moment space [9, 10], it isuseful to take Mellin moments of the structure functions FhT and FhL. Since x and z areindependent variables, we take moments separately in both. We find from eq. (3)(4) F˜hi (N,M,Q2) =∑f,f ′f˜N (μ2)D˜h,Mf ′ (μ2) C˜if ′f(N,M,Q2μ2, αs(μ2)).Thus, the Mellin moments of the structure functions are obtained from ordinary productsof the moments of the parton distribution functions and fragmentation functions, anddouble-Mellin moments of the partonic hard-scattering functions. To NLO, the lattermay be found in [11]. Using the techniques developed in ref. [12], we find that the finalNLL resummed coefficient function becomes in the MS scheme (see also [13])CT,resqq (N,M,αs(Q2)) = e2qHqq(αs(Q2),Q2μ2)exp[2∫ Q20dk2⊥k2⊥Aq(αs(k2⊥))(5)×{K0(√NM2k⊥Q)+ ln(k⊥Q√N¯M¯)}].Here Aq and Hqq are perturbative functions. For resummation at NLL, one needs toknow Hqq to first order in the strong coupling collecting the hard virtual corrections.QCD RESUMMATION IN HADRON PRODUCTION 155Fig. 1. – NLO and NLL resummed SIDIS multiplicities for π− compared to the preliminaryCOMPASS data [6].As the exponentiation of soft-gluon corrections is achieved in Mellin moment space,the hadronic structure function is obtained by taking the inverse Mellin transforms ofeq. (4), which are given by contour integrals in the complex plane.The NLL resummed results for DIS and SIA can be found in ref. [7] and refs. therein.3. – Phenomenological resultsWe now investigate the numerical size of the threshold resummation effects for SIDISand SIA multiplicities. For the parton distribution functions we use the NLO MSTW2008 set of [14], whereas we choose the NLO DSS [1] pion fragmentation functions.Starting with SIDIS, fig. 1 (fig. 2) shows our results for the NLO and the resummedmultiplicities with COMPASS (HERMES) kinematics. The center-of-mass energy ofCOMPASS is√s ≈ 17.4GeV. The kinematic cuts employed are 0.041 < x < 0.7,Fig. 2. – Same as fig. 1, but for HERMES kinematics. The preliminary data are from [5].156 D. ANDERLE, F. RINGER and W. VOGELSANGFig. 3. – π− multiplicity in electron-positron annihilation at√s = 10.52GeV. The data arefrom [4].0.1 < y < 0.9, Q2 > 1GeV2 and W 2 = Q2(1 − x)/x + m2p > 49GeV2, where mp is theproton mass. For HERMES the center-of-mass energy is√s ≈ 7.26GeV. The kinematiccuts are 0.023 < x < 0.6, 0.1 < y < 0.85, Q2 > 1GeV2 and W 2 > 10GeV2. We choosethe renormalization and factorization scales as Q and consider the SIDIS multiplicity forπ− as a function of z, but integrating numerator and denominator of eq. (2) over x andy. It turns out that resummation leads to a moderate, but significant, enhancement ofthe multiplicities.In fig. 3, we present our results for the SIA π− multiplicity at√s = 10.52 GeV and for−1 < cos θ < 1. The results can also be studied as ratios (Th′ −NLO)/NLO, where Th′denotes any of the higher-order SIDIS multiplicities generated by resummation. Theseratios are shown in fig. 4, making the large resummation effects even more apparent athigh xE .Fig. 4. – Ratios (Th′ −NLO)/NLO, where Th′ corresponds to the multiplicity at higher ordersas generated by resummation.QCD RESUMMATION IN HADRON PRODUCTION 1574. – ConclusionsWe have derived threshold-resummed expressions for the coefficient functions for theSIDIS and SIA processes. We have presented phenomenological results for pion multiplic-ities for these processes in the kinematic regimes presently accessed by the COMPASS,HERMES and BELLE experiments. We found that resummation leads to modest butsignificant enhancements of the multiplicities. Our main point is that, given the goodaccuracy of the new preliminary data sets, it will be crucial to include resummationeffects for both processes in the next generation of analyses of fragmentation functions.REFERENCES[1] de Florian D., Sassot R. and Stratmann M., Phys. Rev. D, 75 (2007) 114010.[2] Albino S., Kniehl B. A. and Kramer G., Nucl. Phys. B, 803 (2008) 42.[3] Hirai M., Kumano S., Nagai T. H. and Sudoh K., Phys. Rev. D, 75 (2007) 094009.[4] Leitgab M. et al. (BELLE Collaboration), Phys. Rev. Lett., 111 (2013) 062002.[5] Joosten S. (HERMES Collaboration), talk presented at DIS 2011, Newport News,VA USA; Airapetian A. et al. (HERMES Collaboration), Phys. Rev. D, 87 (2013)074029.[6] Makke N. (COMPASS Collaboration), PhD thesis, CERN-THESIS-2011-279.[7] Anderle D., Ringer F. and Vogelsang W., Phys. Rev. D, 87 (2013) 034014.[8] Altarelli G., Ellis R. K., Martinelli G. and Pi S. Y., Nucl. Phys. B, 160 (1979)301.[9] Sterman G. F., Nucl. Phys. B, 281 (1987) 310.[10] Catani S. and Trentadue L., Nucl. Phys. B, 327 (1989) 323.[11] Stratmann M. and Vogelsang W., Phys. Rev. D, 64 (2001) 114007.[12] Sterman G. F. and Vogelsang W., Phys. Rev. D, 74 (2006) 114002.[13] Cacciari M. and Catani S., Nucl. Phys. B, 617 (2001) 253.[14] Martin A. D., Stirling W., Throne R. and Watt G., Eur. Phys. J. C, 63 (2009) 189.

QCD resummation in hadron production

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