Introduction to Parton Distribution Functions in the Nucleon

Introduction to Parton Distribution Functions in the Nucleon and Nuclei Shunzo Kumano High Energy Accelerator Research Organization (KEK) Graduate University for Advanced Studies (GUAS) shunzo. kumano@kek. jp http: //research. kek. jp/people/kumanos/ Meeting on New Interaction Code For Cosmic Air Shower ICRR, Kashiwa, Japan http: //cosmos. n. kanagawa-u. ac. jp/new. Code/ December 26, 2008

Contents 1. Introduction • Parton Distribution Functions (PDFs) in air-shower model 2. • Deep inelastic scattering and Parton picture 2. Determination of PDFs 3. 4. • PDFs in the nucleon Global analysis Comments on higher-twist effects • Nuclear PDFs Mechanisms for nuclear modifications Our global analysis, Comparison with other analyses 3. Summary

Introduction

Typical Air Shower Model (SENECA) Soft interactions My talk is on this “hard” part. http: //th. physik. uni-frankfurt. de/~drescher/SENECA/

In an air-shower model (e. g. SIBYLL) R. S. Fletcher, T. K. Gaisser, P. Lipari, and T. Stanev, Phys. Rev. D 50 (1994) 5710. High-energy part is described by the following cross sections SIBYLL (1994): PDFs by Eichten-Hinchliffe-Lane-Quigg (EHLQ) in 1984 The PDFs at large x 1 and small x 2 should affect simulation results of the air shower.

hard processes. Soft and Hard processes Soft Hard scale Resonances ~1 Ge. V (e. g. transverse momentum p. T ) Partons p. QCD + Parton Distribution Functions (PDFs) (+ Fragmentation Functions) p, …, Fe (R. Engel) • • • N, O Most energetic particles (namely large x. F ) contribute mainly to subsequent shower development. Nuclear PDFs at small x (N, O) Nucleonic and Nuclear PDFs at large x (p, …, Fe) Fragmentation functions --- LHC --- JLab, J-PARC --- Belle, Babar

Momentum fraction x in the forward region p, …, Fe N, O

Deep Inelastic Lepton-Nucleon Scattering and Parton Picture

Kinematics of e+p e’+X z

Meaning of x consider the frame where the proton is moving fast proton momentum: p x = momentum fraction carried by the struck parton For example, x=0. 5 means that the struck parton carries 50% momentum of the proton.

Kinematical range of x : 0 ≤ x ≤ 1 for the nucleon

Meaning of Q 2 Laboratory frame Breit frame is defined as the frame in which exchanged boson is completely spacelike: q=(0, 0, 0, q). Breit frame q 0=0: photon does not transfer any energy Spatial resolution = reduced wavelength Q 2 corresponds to the “spatial resolution” in the Breit frame.

Bjorken scaling Example of electronproton scattering data at SLAC (1972). W 2=F 2 x = 0. 25 Structure function F 2 is independent of Q 2. (Bjorken scaling) It means that F 2 does not change even if the spatial resolution is increased. existence of small particles which cannot be resolved: This point-like particle was named “parton”. Q 2 (Ge. V 2/c 2) parton quark, gluon

Scaling violation (Q 2 dependence) DGLAP (Dokshitzer, Gribov, Lipatov, Altarelli, Parisi) small Q 2 large Q 2 ZEUS, Eur. Phys. J. C 21 (2001) 443. Q 2 corresponds to “spatial resolution”. As Q 2 becomes large, the virtual starts to probe the gluon, quark, and antiquark “clouds”.

Parton Distribution Functions in the Nucleon

Related web sites Parton distribution functions (PDFs), Experimental data: ・http: //durpdg. dur. ac. uk/HEPDATA/ CTEQ: ・http: //www. phys. psu. edu/~cteq/ Our activities: ・ Nuclear PDFs, Q 2 evolution codes, Fragmentation functions http: //research. kek. jp/people/kumanos/

Recent papers on unpolarized PDFs It is likely that I miss some papers! CTEQ (uncertainties) D. Stump (J. Pumplin) et al. , Phys. Rev. D 65 (2001) 14012 & 14013. (CTEQ 6) D. Pumplin et al. , JHEP, 0207 (2002) 012; 0506 (2005) 080; 0602 (2006) 032; 0702 (2007) 053; PRD 78 (2008) 013004. (charm) PR D 75 (2007) 054029; (strange) PRL 93 (2004) 041802; Eur. Phys. J. C 40 (2005) 145; JHEP 0704 (2007) 089. GRV (GRV 98) M. Glück, E. Reya, and A. Vogt, Eur. Phys. J. C 5 (1998) 461. (GJR 08) M. Glück, P. Jimenez-Delgado, and E. Reya, Eur. Phys. J. C 53 (2008) 355. MRST A. D. Martin, R. G. Roberts, W. J. Stirling, and R. S. Thorne, (MRST 2001, 2002, 20033) Eur. Phys. J. C 23 (2002) 73; Eur. Phys. J. C 28 (2003) 455; (theoretical errors) Eur. Phys. J. C 35 (2004) 325; (2004) PL B 604 (2004) 61; (QED) Eur. Phys. J. C 39 (2005) 155; PL B 636 (2006) 259; (2006) PRD 73 (2006) 054019; PL B 652 (2007) 292. (2008) ar. Xiv: 0806. 4890 Alekhin S. I. Alekhin, PRD 68 (2003) 014002; D 74 (2006) 054033. BB J. Blümlein and H. Böttcher, Nucl. Phys. B 774 (2007) 182 -207. NNPDF S. Forte et al. , JHEP 0205 (2002) 062; 0503 (2005) 080; 0703 (2007) 039. H 1 C. Adloff et al. , Eur. Phys. J. C 21 (2001) 33; C 30 (2003) 1. ZEUS S. Chekanov et al. , PRD 67 (2003) 012007; Eur. Phys. J. C 42 (2005) 1. Recent activities uncertainties NNLO QED s – s charm

Parton distribution functions are determined by fitting various experimental data.

Available data for determining PDFs (Ref. MRST, hep/ph-9803445)

Determination of each distribution Valence quark m– X q W nm p N

Sea quark e/ scattering Drell-Yan (lepton-pair production) projectile target

Gluon scaling violation of F 2 K. Prytz, Phys. Lett. B 311 (1993) 286. jet production

Global Analysis for PDFs

Outline of analysis 1. Express x-dependent PDFs with parameters at a fixed Q 2 ( Q 02) * Choice of Q 02 2. Evolve the PDFs to experimental Q 2 data points * Q 2 evolution methods 3. Convolute with coefficient functions to calculate observables 4. Determine 2 in comparison with data 5. Repeat 2, 3, and 4 processes until minimal 2 is obtained

Unpolarized Parton Distribution Functions (PDFs) in the nucleon The PDFs could be obtained from http: //durpdg. dur. ac. uk/hepdata/pdf. html Gluon distribution / 5 Valence-quark distributions

PDF uncertainty u q(x) at large x CTEQ 6 (J. Pumplin et al. ), JHEP 0207 (2002) 012 CTEQ 5 M 1 MRS 2001 CTEQ 5 HJ g (unknown)2 for cosmic-ray studies There also large nuclear corrections in these regions. g(x) at small x “gluon saturation” d

Comments on Higher-Twist Effects

Higher-twist effects by CTEQ in 2002 error estimate ? CTEQ (J. Pumplin et al. ), JHEP 07 (2002) 012.

Higher-twist effects by BB in 2008 proton J. Blüemlein, H. Böttcher, Phys. Lett. B 662 (2008) 336. deuteron

Higher-twist effects by BB in 2008

For extracting reliable higher-twist effects Full analysis is needed! “Full” means at least Parametrization and fit also for the HT terms Uncertainty range of a determined HT function together with target-mass corrections and higher-order effects.

Nuclear Parton Distribution Functions

Brief Introduction to Nuclear Modifications of Parton Distribution Functions

Nuclear modifications of structure function F 2 1. 2 Fermi motion of the nucleon EMC NMC E 139 E 665 1. 1 1 Could affect cosmic-ray studies 0. 9 0. 8 0. 7 0. 001 0. 01 x 0. 1 Shadowing (q q fluctuation of photon) 1 D. F. Geesaman, K. Saito, A. W. Thomas, Ann. Rev. Nucl. Part. Sci. 45 (1995)337. Nuclear binding (+ Nucleon modification)

EMC (European Muon Collaboration) effect Theoretical Description q k p P Quark (a) Hadron (T) Nucleus (A)

Binding Model

0. 98 Because the peak shifts slightly (1 0. 98), nuclear modification of F 2 is created. 0. 20 Fermi motion binding

Shadowing Models: Vector-Meson-Dominance (VMD) type

Determination of Nuclear Parton Distribution Functions

Nuclear modification of F 2 A / F 2 D is well known in electron/muon scattering. 1. 2 Fermi motion EMC NMC E 139 E 665 1. 1 1 0. 9 0. 8 original EMC finding shadowing 0. 7 0. 001 0. 01 sea quark x 0. 1 1 valence quark

Drell-Yan and Antiquark Distributions Drell-Yan cross-section ratio is roughly equal to antiquark ratio. The Fermilab E 772 Drell-Yan data suggested that nuclear modification of antiquark distributions should be small in the region, x≈0. 1.

References There are only a few papers on the parametrization of nuclear PDFs! Need much more works. (EKRS) K. J. Eskola, V. J. Kolhinen, and P. V. Ruuskanen, Nucl. Phys. B 535 (1998) 351; K. J. Eskola, V. J. Kolhinen, and C. A. Salgado, Eur. Phys. J. C 9 (1999) 61. K. J. Eskola et al. , JHEP 0705 (2007) 002; 0807 (2008) 102. (HKM, HKN) M. Hirai, SK, M. Miyama, Phys. Rev. D 64 (2001) 034003; M. Hirai, SK, T. -H. Nagai, Phys. Rev. C 70 (2004) 044905; 2 analysis M. Hirai, SK, T. -H. Nagai, Phys. Rev. C 76 (2007) 065207. (DS) D. de Florian and R. Sassot, Phys. Rev. D 69 (2004) 074028. See also S. A. Kulagin and R. Petti, Nucl. Phys. A 765 (2006) 126 (2006); L. Frankfurt, V. Guzey, and M. Strikman, Phys. Rev. D 71 (2005) 054001. The recent HKN 07 analysis is explained in this talk.

NLO Determination of Nuclear Parton Distribution Functions by M. Hirai, SK, T. -H. Nagai Phys. Rev. C 76 (2007) 065207 NPDF codes can be obtained from http: //research. kek. jp/people/kumanos/nuclp. html Related papers. M. Hirai, SK, M. Miyama, Phys. Rev. D 64 (2001) 034003; M. Hirai, SK, T. -H. Nagai, Phys. Rev. C 70 (2004) 044905.

Experimental data: total number = 1241 (1) F 2 A / F 2 D 896 data NMC: p, He, Li, C, Ca SLAC: He, Be, C, Al, Ca, Fe, Ag, Au EMC: C, Ca, Cu, Sn E 665: C, Ca, Xe, Pb BCDMS: N, Fe HERMES: N, Kr (2) F 2 A / F 2 A’ 293 data NMC: Be / C, Al / C, Ca / C, Fe / C, Sn / C, Pb / C, C / Li, Ca / Li (3) DYA / DYA’ 52 data E 772: E 866: C / D, Ca / D, Fe / D, W / D Fe / Be, W / Be

Functional form Nuclear PDFs “per nucleon” If there were no nuclear modification Isospin symmetry： Take account of nuclear effects by wi (x, A) at Q 2=1 Ge. V 2 ( Q 02 )

Functional form of wi (x, A) Note: The region x > 1 cannot be described by this parametrization. x A simple function = cubic polynomial Three constraints

Comparison with F 2 Ca/F 2 D & DYp. Ca/ DYp. D data LO analysis NLO analysis (Rexp-Rtheo)/Rtheo at the same Q 2 points R= F 2 Ca/F 2 D, DYp. Ca/ DYp. D

Nuclear PDFs

Nuclear PDFs and uncertainties Future experiments • Some NLO improvements, but not significant ones. • Impossible to determine gluon modifications. • Antiquark distributions are not determined at large x. Fermilab J-PARC GSI RHIC LHC e. LIC LHe. C e. RHIC JLab J-PARC? GSI? n. Factory MINAR A RHIC LHe. C e. LIC e. RHIC

Comparison with Other Global Analyses

Comparison (EKRS) Eskola, Kolhinen, Ruuskanen, Salgado (HKN) Hirai, Kumano, Nagai (DS) de Florian, Sassot

Comparison of used data set HKN data table data in EKRS data in DS data not included

Other works: EKRS (Eskola, Kolhinen, Ruuskanen, Salgado)

Results for nuclear PDFs gluon Ca/D valence antiquark Pb/D

EPS 08 Analysis for Nuclear PDFs An Improved global analysis of nuclear parton distribution functions including RHIC data K. J. Eskola, H. Paukkunen, C. A. Salgado JHEP 07 (2008) 102 New point: RHIC-BRAHMS hadron production data in a forward rapidity region very large gluon shadowing Note: still LO analysis

Analysis conditions 15 parameters KKP for Dih

Results: RHIC BRAHMS forward rapidity

Data weight Note: very large weight for BRAHMS

Comparison of EPS, HKN, DS parametrizations • Valence- and sea-quark distributions agree well. • Large variations in gluon distributions: EPS 08 (with BRAHMS forward data) Huge gluon shadowing!

Uncertainties of fragmentation functions “in including hadron -production data in the global analysis. ” • Gluon and light-quark fragmentation functions have large uncertainties. • Large differences between the functions of various analysis groups. • Gluon function at large-z is important for hadron-productions at RHIC. (Torii) Global analysis results for RHIC M. Hirai et al. , PRD 75 (2007) 094009.

Why is it so difficult to determine nuclear gluon distributions?

Current nuclear data are kinematically limited. (from H 1 and ZEUS, hep-ex/0502008) F 2 data for the proton 500 SLAC 100 EMC E 665 F 2 & Drell-Yan data for nuclei BCDMS 2 Q (Ge. V 2) NMC (F 2 A /F 2 D ) HERMES )2 10 NMC (F 2 A /F 2 A') E 772/E 886 DY 1 0. 001 0. 01 x 0. 1 1 region of nuclear data

Scaling Violation and Gluon Distributions dominant term at small x K. Prytz, PLB 311 (1993) 286. Q 2 dependence of F 2 is proportional to the gluon distribution. No experimental consensus of Q 2 dependence! GA(x) determination is difficult.

Summary Hard interactions are discussed in my talk. In order to understand the shower profile, it is important to study Nucleonic and Nuclear PDFs at small x (LHC) Nucleonic and Nuclear PDFs at large x (JLab, J-PARC, …) Higher-twist effects --- numerical studies are in progress at large x. Nuclear gluon distributions --- not obvious, studied at LHC

The End