1. Nuclear Modifications of Parton Distribution Functions
Shunzo Kumano
High Energy Accelerator Research Organization (KEK)
Graduate University for Advanced Studies (GUAS)
Collaborators: Masanori Hirai (Juntendo)
Takahiro Nagai (GUAS)
Ref. Phys. Rev. C 76 (2007)
Workshop on
Description of Lepton-Nucleus Reactions
KEK, Tsukuba, December 22, 2007
December 22, 2007

2. Contents (1) Introduction Motivation
Comments on parton distribution functions
(PDFs) in the nucleon
(2) Determination of PDFs in Nuclei
Analysis method
LO and NLO results and their comparisons
 Summary

3. Motivations for studying
structure functions and parton distribution functions
To establish QCD
Perturbative QCD
• In principle, theoretically established in many processes.
(There are still issues on small-x physics.)
• Experimentally confirmed (unpolarized, polarised ?)
Non-perturbative QCD (PDFs)
• Theoretical models: Bag, Soliton, …
(It is important that we have intuitive pictures of the nucleon.)
• Lattice QCD
Theoretical non-pQCD calculations are not accurate enough.
 Determination of the PDFs from experimental data.
Lattice QCD why only moments?

4.  New structure functions and new investigations at  factory!
(2) For discussing any high-energy reactions, accurate PDFs
are needed.
 origin of nucleon spin: quark- and gluon-spin contributions
 exotic events at large Q2: physics of beyond current framework
 heavy-ion reactions: quark-hadron matter
 neutrino oscillations: nuclear effects in n + 16O
 cosmology: ultra-high-energy cosmic rays
Actual effects on cosmic ray studies.
 New structure functions and new investigations at  factory!

5. Nuclear PDFs in neutrino reactions
CCFR and NuTeV: 56Fe target
Nuclear effects are important in extracting nucleonic PDFs.
(2) Oscillation experiments
Nuclear corrections in 16O
Low Q2 data: High Q2 (PDFs)  Low Q2 is needed.
(Quark-hadron duality)
(3) Neutrino Factory
New investigations with proton and deuteron targets,
so that nuclear modifications could be studied by
measuring A/D ratios.
Lattice QCD why only moments?

6. Parton Distribution Functions (PDFs) in the Nucleon
PDFs from
 factory
Valence-quark distributions are
determined from data including
CCFR and NuTeV ones with
the iron target.
It should be worth investigating them
for the real nucleon at a neutrino factory.

7. PDF uncertainty CTEQ5M1 MRS2001 CTEQ5M1 CTEQ5HJ MRS2001
CTEQ6 (J. Pumplin et al.),
JHEP 0207 (2002) 012

8. Parton Distribution Functions in “Nuclei”

9. J-PARC = Japan Proton Accelerator Research Complex
Status of PDF determinations
Unpolarized PDFs in the nucleon     Investigated by 3 major groups (CTEQ, GRV, MRST). Well studied from small x to large x in the wide range of Q The details are known. (Recent studies: NNLO, QED, error analysis, , …) “Polarized” PDFs in the nucleon   Investigated by several groups (GS, GRSV, LSS, AAC, BB, …). Available data are limited (DIS) at this stage (recent: HERMES, Jlab, COMPASS) New data from RHIC Future: J-PARC, eRHIC, eLIC, GSI… PDFs in “nuclei”   Investigated by only a few groups. Details are not so investigated! Available data are limited (inclusive DIS, Drell-Yan). New data from RHIC, LHC, Jlab, NuTeV Future: Fermilab, J-PARC, eRHIC, eLIC, GSI…
J-PARC = Japan Proton Accelerator Research Complex

10. Situation of data for nuclear PDFs
Available data for nuclear PDFs
Jlab at large x
Neutrino factory: ~10 years later ?
(CCFR, NuTeV)
Small-x,
high-energy electron facility?
RHIC, LHC, J-PARC
RHIC, LHC
Table from MRST,
hep/ph
RHIC, LHC

11. Current nuclear data are kinematically limited.
(from H1 and ZEUS, hep-ex/ )
F2 data
for the proton
F2 & Drell-Yan data
for nuclei
region of nuclear data

12. x Nuclear modification sea quark valence quark
Nuclear modification of F2A / F2D is
well known in electron/muon scattering.
Fermi motion
0.7
0.8
0.9 1
1.1
1.2
0.001
0.01
0.1
EMC
NMC
E139
E665
original
EMC finding
shadowing x
sea quark
valence quark

13. Binding Model
Separation (removal) energy <--> binding energy

14. Because the peak shifts slightly (1 0.98),
0.20
Because the peak shifts slightly (1 0.98),
nuclear modification of F2 is created.
Fermi motion
binding

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

16. EMC (European Muon Collaboration) effect
Theoretical Description

17. 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. B535 (1998) 351;
K. J. Eskola, V. J. Kolhinen, and C. A. Salgado, Eur. Phys. J. C9 (1999) 61.
K. J. Eskola et al., JHEP 0705 (2007) 002.
(HKM, HKN) M. Hirai, SK, M. Miyama, Phys. Rev. D64 (2001) ;
M. Hirai, SK, T.-H. Nagai, Phys. Rev. C70 (2004) ;
M. Hirai, SK, T.-H. Nagai, Phys. Rev. C76 (2007)
(DS) D. de Florian and R. Sassot, Phys. Rev. D69 (2004)
2 analysis
See also S. A. Kulagin and R. Petti, Nucl. Phys. A765 (2006) 126 (2006);
L. Frankfurt, V. Guzey, and M. Strikman, Phys. Rev. D71 (2005)
The recent HKN report (KEK-TH-1013) is explained in this talk.

18. Nuclear Parton Distribution Functions NPDF codes can be obtained from
NLO Determination of
Nuclear Parton Distribution Functions
by M. Hirai, SK, T.-H. Nagai
arXiv: [hep-ph]
Phys. Rev. C 76 (2007)
NPDF codes can be obtained from
Related refs. M. Hirai, SK, M. Miyama, Phys. Rev. D64 (2001) ;
M. Hirai, SK, T.-H. Nagai, Phys. Rev. C70 (2004)

19. New points (1) Both LO and NLO global analyses
(LO = Leading Order of s, NLO = Next to Leading Order)
Estimation of NPDF uncertainties both in NLO and LO
• Roles of NLO terms in the global analysis
• Better determination of gluon distributions (NLO terms)
(2) Discussions on deuteron modifications
Comparison with F2D/F2p data
• Deuteron modifications should be important in Gottfried sum,
RHIC d-Au collisions, …; however, they are not well studied.
• Note: Nuclear effects in the deuteron are partially contained
in the “nucleonic” PDFs.
Lattice QCD why only moments?

20. Experimental data: total number = 1241
(1) F2A / F2D data
NMC: p, He, Li, C, Ca
SLAC: He, Be, C, Al,
Ca, Fe, Ag, Au
EMC: C, Ca, Cu, Sn
E665: C, Ca, Xe, Pb
BCDMS: N, Fe
HERMES: N, Kr
(2) F2A / F2A’ data
NMC: Be / C, Al / C,
Ca / C, Fe / C,
Sn / C, Pb / C,
C / Li, Ca / Li
(3) DYA / DYA’ data
E772: C / D, Ca / D,
Fe / D, W / D
E866: Fe / Be, W / Be

21. Functional form Nuclear PDFs “per nucleon”
If there were no nuclear modification
Isospin symmetry：
Take account of nuclear effects by wi (x, A)
at Q2=1 GeV2 ( Q02 )

22. Functional form of wi (x, A)
A simple function = cubic polynomial
Three constraints

23. · Error estimate： Hessian method
Analysis conditions
· Nucleonic PDFs:
MRST98 [ QCD = 174 MeV (LO), 300 MeV (NLO) ]
· Total number of parameter：12
· Total number of data： 1241 ( Q2≧1 GeV2 )
896 (F2A/F2D) (F2A/F2A´) + 52 (Drell-Yan)
· Subroutine for 2 analysis： CERN-Minuit
2min ( /d.o.f.) = (1.35) ….. LO
= (1.21) ….. NLO
· Error estimate： Hessian method

24. 2 values in LO and NLO NLO improvement NLO disimprovement
Total 2 improvements in NLO.
NLO improvements mainly in light nuclei;
however, disimprovements for Drell-Yan data.

25. Comparison with F2Ca/F2D & DYpCa/ DYpD data
LO analysis
NLO analysis
(Rexp-Rtheo)/Rtheo at the same Q2 points
R= F2Ca/F2D, DYpCa/ DYpD

26. Comparison with F2A/F2D data: Light nuclei

27. Comparison with F2A/F2D data: Heavy nuclei

28. Q2 dependence The differences between LO and NLO
become obvious only at small x.
• Experimental data are not accurate enough
to find the differences.
 Determination of gluon distributions
(NLO terms) is not possible.
• The uncertainties become smaller in NLO
at small x.
Only NLO uncertainty bands are shown.

29. Scaling Violation and Gluon Distributions
dominant term at small x
Q2 dependence of F2 is proportional
to the gluon distribution.
No experimental consensus of
Q2 dependence!
 GA(x) determination is difficult.

30. Nuclear PDFs

31. PDFs in 40Ca and uncertainties
• Some NLO improvements, but not significant ones.
• Impossible to determine gluon modifications.
• Antiquark distributions are not determined at large x.
• Flavor separation is needed for antiquarks
  factory
• Confirmation of valence modifications at small x

32. Summary on nuclear-PDF determination in NLO
LO and NLO analysis for the nuclear PDFs and their uncertainties.
• Better determination of GA(x) is usually expected in NLO.
 However, the NLO improvement is not very clear due to
inaccurate measurement of Q2 dependence.
 The gluon modifications are also not determined well even in NLO.
Deuteron modifications
• At most 0.5%~2%; however, be careful that deuteron effects
could be contained in the PDFs of the nucleon.
NPDF codes at
 Comparison with (and analysis including) NuTeV nuclear corrections in future!
Small NuTeV nuclear corrections!? (J. F. Owens et al., PRD75, (2007); J. G.
Neutrino factory should be important for finding nuclear medium effects
in the valence-quark and (flavor-separated) antiquark distributions.

33. Extra

34. Nuclear corrections in iron (A=56, Z=26)
KP (Kulagin, Petti)
Nuclear PDFs from neutrino deep inelastic scattering,
I. Schienbein, J. Y. Yu, C. Keppel, J. G. Morfin, F. Olness, and
J. F. Owens (CTEQ Collaboration), arXiv: v1 [hep-ph].
Sumamry

36. 0.18 fm
0.40 fm x

37. CTEQ, (F. Olness et al. , Eur. Phys. J. C40 (2005) 145) H. -L
CTEQ, (F. Olness et al., Eur. Phys. J. C40 (2005) 145) H.-L. Lai et al., JHEP 04 (2007) 089.
The 2007 paper includes the final NuTeV data for dimuons.
this analysis

38. best fit

39. NuTeV analysis
The NuTeV result is not much different from CTEQ one although it used to differ in hep-ex/

40. Global analysis for determining fragmentation functions and their uncertainties
Shunzo Kumano
High Energy Accelerator Research Organization (KEK)
Graduate University for Advanced Studies (GUAS)
with M. Hirai (TokyoTech), T.-H. Nagai (GUAS), K. Sudoh (KEK)
Reference: Phys. Rev. D75 (2007)

41. Contents (1) Introduction to fragmentation functions (FFs)
 Definition of FFs
 Motivation for determining FFs
(2) Determination of FFs
 Analysis method
 Results
 Comparison with other parameterizations
(3) Summary

42. Introduction

43. Fragmentation Function
Fragmentation: hadron production
from a quark,
antiquark, or gluon e+ e–
, Z q h
Fragmentation function is defined by
Variable z
• Hadron energy / Beam energy
• Hadron energy / Primary quark energy
A fragmentation process occurs from quarks, antiquarks, and gluons,
so that Fh is expressed by their individual contributions:
Non-perturbative
(determined from experiments)
Calculated in perturbative QCD

44. Momentum (energy) sum rule
Favored and disfavored fragmentation functions

45. Status of determining fragmentation functions
Parton Distribution Functions (PDFs), Fragmentation Functions (FFs)
Uncertainty ranges of determined fragmentation functions
were not estimated, although there are such studies in
nucleonic and nuclear PDFs.
The large differences indicate that
the determined FFs have much ambiguities.

46. Situation of fragmentation functions
There are two widely used fragmentation functions by Kretzer and KKP.
An updated version of KKP is AKK.
(Kretzer) S. Kretzer, PRD 62 (2000)
(KKP) B. A. Kniehl, G. Kramer, B. Pötter, NPB 582 (2000) 514
(AKK) S. Albino, B.A. Kniehl, G. Kramer, NPB 725 (2005) 181
The functions of Kretzer and KKP (AKK) are very different.

47. Purposes of investigating fragmentation functions
Semi-inclusive reactions have been used for investigating
・origin of proton spin
Quark, antiquark, and gluon contributions to proton spin
(flavor separation, gluon polarization)
・properties of quark-hadron matters
Nuclear modification
(recombination, energy loss, …)

48. Determination of Fragmentation Functions
and their uncertainties
M. Hirai, SK, T.-H. Nagai, K. Sudoh
Phys. Rev. D75 (2007)
A code for calculating the FFs is available at

49. New aspects in our analysis
• Determination of fragmentation functions (FFs) and
their uncertainties in LO and NLO.
• Discuss NLO improvement in comparison with LO
by considering the uncertainties.
(Namely, roles of NLO terms in the determination of FFs)
• Comparison with other parametrizations
• Avoid assumptions on parameters as much as we can,
Avoid contradiction to the momentum sum rule
• SLD (2004) data are included.

50. Comparison with other analyses
HKNS (Ours)
Kretzer
KKP (AKK)
Function form
# of parameters 14 11
15 (18)
Mass threshold
mQ2
(mc,b=1.43, 4.3 GeV)
(mc,b=1.4, 4.5 GeV)
4mQ2
(2mc,b=2.98, 9.46 GeV)
Initial scale Q02 (NLO)
1.0 GeV2
0.4 GeV2
2.0 GeV2
Major ansatz
One constraint:
A gluon parameter is fixed.
Four constraints:
( issue of momentum sum)
No π+, π– separation

51. Initial functions for pion
Constraint: 2nd moment should be finite and less than 1

52. Experimental data for pion
Total number of data： 264
# of data
TASSO
TCP
HRS
TOPAZ
SLD
SLD [light quark]
SLD [ c quark]
SLD [ b quark]
ALEPH
OPAL
DELPHI
DELPHI [light quark]
DELPHI [ b quark]
12,14,22,30,34,44 29 58
91.2 18 2 4 22 17
Heavy-flavor separation

53. Analysis
Results for the pion
Uncertainty estimation: Hessian method

54. Comparison with pion data
Our fit is successful to reproduce the pion data.
The DELPHI data deviate from our fit at large z.
Our NLO fit
with uncertainties
Rational difference between data and theory

55. Comparison with pion data: (data-theory)/theory
Charm and bottom quark identification.
Because the charm and bottom data are separated and they are accurately determined, the charm and bottom FFs are accurately determined. On the other hand, light-quark FFs need flavor separation from non-flavor-separated data.--->large errors in light-quark FFs.

56. Determined fragmentation functions for pion
• Gluon and light-quark
fragmentation functions
have large uncertainties.
• Uncertainty bands
become smaller in NLO
in comparison with LO.
The data are sensitive to
NLO effects.
• The NLO improvement is
clear especially in gluon
and disfavored functions.
• Heavy-quark functions
are relatively well determined.

57. Comparison with kaon data

58. Determined functions for kaon
The situation is similar to the pion functions.
• Gluon and light-quark
fragmentation functions
have large uncertainties.
• Uncertainty bands
become smaller in NLO
in comparison with LO.
• Heavy-quark functions
are relatively well determined.

59. Comparison with other parametrizations in pion
(KKP) Kniehl, Kramer, Pötter
(AKK) Albino, Kniehl, Kramer
(HKNS) Hirai, Kumano, Nagai, Sudoh
• Gluon and light-quark
fragmentation functions have
large uncertainties, but they
are within the uncertainty bands.
The functions of KKP, Kretzer,
AKK, and HKNS are consistent
with each other.
All the parametrizations agree
in charm and bottom functions.

60. Comparison with other parametrizations in kaon and proton

61. Comments on “low-energy” experiments, Belle & BaBar
Gluon fragmentation function is very important for hadron production at small pT at RHIC (heavy ion, spin) and LHC,
(see the next transparency)
and it is “not determined” as shown in this analysis.
Need to determine it accurately.
Gluon function is a NLO effect with the coefficient
function and in Q2 evolution.
We have precise data such as the SLD ones at Q=Mz,
so that accurate small-Q2 data are needed for probing
the Q2 evolution, namely the gluon fragmentation functions.
(Belle, BaBar ?)

62. Pion production at RHIC: p + p  0 + X
S. S. Adler et al. (PHENIX), PRL 91 (2003)  pT p
• Consistent with NLO QCD calculation up to 10–8
• Data agree with NLO pQCD + KKP
• Large differences between Kretzer and KKP
calculations at small pT
 Importance of accurate fragmentation functions
Blue band indicates the scale uncertainty
by taking Q=2pT and pT/2.

63. Summary
Determination of the optimum fragmentation functions for , K, p
in LO and NLO by a global analysis of e++e– h+X data.
• This is the first time that uncertainties of the fragmentation functions
are estimated.
• Gluon and disfavored light-quark functions have large uncertainties.
 The uncertainties could be important for discussing physics in
 Need accurate data at low energies (Belle and BaBar).
• For the pion and kaon, the uncertainties are reduced in NLO
in comparison with LO.
For the proton, such improvement is not obvious.
• Heavy-quark functions are well determined.
• Code for calculating the fragmentation functions is available at .

64. The End
The End