Nuclear Modifications of
Parton Distribution Functions
Shunzo Kumano
High Energy Accelerator Research Organization (KEK)
Graduate University for Advanced Studies (GUAS)
shunzo.kumano@kek.jp
http://research.kek.jp/people/kumanos/
Collaborators: Masanori Hirai (Juntendo)
Takahiro Nagai (GUAS)
Ref. Phys. Rev. C 76 (2007) 065207.
Workshop on
Description of Lepton-Nucleus Reactions
KEK, Tsukuba, December 22, 2007
December 22, 2007
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
Motivations for studying
structure functions and parton distribution functions
(1) 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.
(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
 New structure functions and new investigations at n factory!
Nuclear PDFs in neutrino reactions
(1) 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.
Parton Distribution Functions (PDFs) in the Nucleon
Suppose En  50 GeV
PDFs from
http://durpdg.dur.ac.uk/hepdata/pdf.html
Q2
x
2M N n
xmin
min(Q 2 )

2M N max(n )
1
:
2 1 50
 0.01
where min(Q 2 ) : 1 GeV2
max(n )  En  50 GeV
1
Q 2 = 2 GeV 2
0.8
0.6
n factory
xuv
xg/5
0.4
xs
0.2
Valence-quark distributions are
determined from data including
CCFR and NuTeV ones with
0
0.00001 0.0001 0.001
the iron target.
It should be worth investigating them
for the real nucleon at a neutrino factory.
xd v
xd
0.01
x
xu
0.1
1
PDF uncertainty
CTEQ5M1
MRS2001
CTEQ5M1
CTEQ5HJ
MRS2001
CTEQ6 (J. Pumplin et al.),
JHEP 0207 (2002) 012
Parton Distribution Functions
in “Nuclei”
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 Q2.
The details are known. (Recent studies: NNLO, QED, error analysis, s  s, …)
“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…
J-PARC = Japan Proton Accelerator Research Complex
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…
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
RHIC, LHC
Table from MRST,
hep/ph-9803445
Current nuclear data are
Q2
Q2
x
;
kinematically limited.
2 p  q ys
2
Q
1

2M N Elepton 2Elepton (GeV)
if Q 2  1 GeV2
fixed target: min(x) 
for Elepton (NMC)  200 GeV, min(x) 
1
 0.003
2  200
500
A
(from H1 and ZEUS, hep-ex/0502008)
x  0.0005
F2 data
for the proton
x  0.013
D
NMC (F2 /F2 )
SLAC
Q2 (GeV 2)
100
EMC
E665
F2 & Drell-Yan data
for nuclei
BCDMS
HERMES
10
NMC (F2 A/F2 A')
E772/E886 DY
1
0.001
0.01
x
0.1
1
region of nuclear data
x  0.65
Nuclear modification
F2A (LO)   ei2 x qi (x)  qi (x)A
i
Nuclear modification of F2A / F2D is
well known in electron/muon scattering.
1.2
Fermi motion
EMC
NMC
E139
E665
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
Binding Model
Convolution: WnA ( pA ,q)   d 4 p S( p) WnN ( pN ,q)
S( p) = Spectral function = nucleon momentum distribution in a nucleus
r 2
In a simple shell model: S( p)   i ( p)  ( p0  M N   i )
i
Single-particle energy:  i
Projecting out F2 : F2A (x,Q2 )    dz fi (z) F2N (x / z,Q2 )
i
pq
pq
p
z
;
;
lightcone momentum fraction
M Nn pA  q / A pA / A
r
p2
Recoil energy
is neglected.
2M Ai
a0  a3
a 
2

r r
p  q  p q  p q  pT  qT ; p q

pq 
r 2
fi (z)   d 3 p z   z 

(
p)
lightcone momentum distribution for a nucleon i
i

M Nn 

F2A (x,Q2 )    dz fi (z) F2N (x / z,Q2 )
i

pq 
r 2
fi (z)   d 3 p z   z 

(
p)
i
M Nn 

r r
r r
p  q p 0n  p  q
| i | p  q
z

 1

 1.00  0.02  0.20 for a medium-size nucleus
M Nn
M Nn
M N M Nn
0.98
If fi (z) were fi (z)   (z  1), there is no nuclear
f (z)
modification: F2A (x,Q 2 )  F2N (x,Q 2 ).
Because the peak shifts slightly (1 0.98),
nuclear modification of F2 is created.
0.20
F2A (x,Q2 ) ; F2N (x / 0.98,Q2 )
For x  0.60, x / 0.98  0.61
F2N (x  0.61) 0.021

 0.88
N
F2 (x  0.60) 0.024
F2A / F2N
Fermi motion
x
binding
z
Shadowing Models: Vector-Meson-Dominance (VMD) type
A
Virtual photon splits into a qq pair and
it becomes a vector meson, which interacts
with a nucleus, especially in the surface region.
q V
q
2n
0.2 fm
propagation length of V:  
 2

 2 fm at x  0.1
2
MV  Q
x
EV  E
1
At small x, the virtual photon interacts with
the target nucleus as if it were a vector meson.
M 2  (M 2 )
F (x,Q ) 
dM
 VA
2
2 2


(M  Q )
A
2
Q2
2
2
 (e e  hadrons)
 vector mesons  qq continuum
 (M ) 
 
 
 (e e    )
2
EMC (European Muon Collaboration) effect
Theoretical Description

fa/A(q ,Pq) = 
T
2

q
a k
d 4p
4 fa/T(p,q) f T/A (P,p)
(2)
f a/A(x,Q ) = 
T
1
2
dy A f a/T
xA
fa/T(k,p)
xA
y A f T/A(y A)
T
Q 2 rescaling model, 

nuclear
binding, nuclear pion, 
q
p
fT/A(p,P)
A
P
(1) A hadron T is distributed in a nucleus A with the momentum distribution fT/A (yA ).
(2) A quark a is distributed in the hadron T with the momentum distribution fa/T (x A ).
(3) The virtual photon interacts with the quark a.
(4) The quark momentum distribution in the nucleus A, fa/A (x), is given by
their convolution integral.
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) 034003;
M. Hirai, SK, T.-H. Nagai, Phys. Rev. C70 (2004) 044905;
2 analysis
M. Hirai, SK, T.-H. Nagai, Phys. Rev. C76 (2007) 065207.
(DS) D. de Florian and R. Sassot, Phys. Rev. D69 (2004) 074028.
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) 054001.
The recent HKN report (KEK-TH-1013) is explained in this talk.
NLO Determination of
Nuclear Parton Distribution Functions
by M. Hirai, SK, T.-H. Nagai
arXiv:0709.3038 [hep-ph]
Phys. Rev. C 76 (2007) 065207
NPDF codes can be obtained from
http://research.kek.jp/people/kumanos/nuclp.html
Related refs. M. Hirai, SK, M. Miyama, Phys. Rev. D64 (2001) 034003;
M. Hirai, SK, T.-H. Nagai, Phys. Rev. C70 (2004) 044905.
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.
Experimental data: total number = 1241
(1) F2A / F2D
896 data
p, He, Li, C, Ca
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’
Q2 (GeV2)
NMC:
SLAC:
293 data
NMC: Be / C, Al / C,
Ca / C, Fe / C,
Sn / C, Pb / C,
C / Li, Ca / Li
(3) DYA /  DYA’ 52 data
E772:
E866:
C / D, Ca / D,
Fe / D, W / D
Fe / Be, W / Be
500
100
NMC (F2A/F2D)
HERMES
SLAC
NMC (F2A/F2A' )
EMC
E772/E886 DY
E665
NMC (F2D/F2p)
BCDMS
10
1
0.001
0.01
0.1
x
1
Functional form
Nuclear PDFs “per nucleon”
If there were no nuclear modification
Au A x   Zu p x   Nu n x , Ad A x   Zd p x   Nd n x 
Isospin symmetry： u n  d p  d,
 u A x  
Zu x   Nd x 
,
A
p = proton, n = neutron
dn  u p  u
d A x  
Zd x   Nu x 
A
Take account of nuclear effects by wi (x, A)
Zuv x   Ndv x 
Zd x   Nuv x 
, dvA x   wdv x, A  v
A
A
Zu x   Nd x 
Zd x   Nu x 
u A x   wq x, A 
,
d A x   wq x, A 
A
A
s A x   wq x, A s x 
uvA x   wuv x, A 
g A x   wg x, A g x 
at Q2=1 GeV2 ( Q02 )
Functional form of wi (x, A)
fi A (x,Q02 )  wi (x, A) fi (x,Q02 )
i  uv , dv , u, d, s, g
1  ai  bi x  ci x 2  di x 3

wi (x, A)  1   1   

A 
(1  x)
x
A simple function = cubic polynomial
Three constraints
1 A
1 A
2 A
A
A
Nuclear charge: Z  A  dx  u  u  d  d  s  s A
3
3
3
1
1
1
Baryon number: A  A  dx  u A  u A  d A  d A  s A  s A
3
3
3
Momentum:

2 A 1 A
  A  dx  3 uv  3 dv 

1 A 1 A 

A
dx
  3 uv  3 dv 


 
 


 
 

A  A  dx u A  u A  d A  d A  s A  s A  g 


 A  dx uvA  dvA  2 u A  d A  s A  g 
Analysis conditions
MRST98 [ QCD = 174 MeV (LO), 300 MeV (NLO) ]
· Nucleonic PDFs:
· Total number of parameter：12
· Total number of data： 1241 ( Q2≧1 GeV2 )
896 (F2A/F2D) + 293 (F2A/F2A´) + 52 (Drell-Yan)
· Subroutine for 2 analysis： CERN-Minuit
2  
R
data
i
 Ritheo
 

2
data 2
i
i
F2A
R D,
F2
 idata 
F2A
 pA
,
A
F2
 pA 
 isys    istat 
2
2
2min ( /d.o.f.) = 1653.3 (1.35) ….. LO
= 1485.9 (1.21) ….. NLO
· Error estimate： Hessian method
F(x) 1 F(x)
H ij
 F(x)   


 j
i, j
i
2
2
H ij  Hessian
i  parameter
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.
Comparison with F2Ca/F2D & DYpCa/ DYpD data
LO analysis
NLO analysis
1.2
F2Ca/F2D
EMC
H
E136
1.1
NMC
E665
H H
1
H
H
0.9
H
HH
0.8
2
Q = 10 GeV
0.7
0.001
0.01
2
0.1
1
x
R= F2Ca/F2D, DYpCa/ DYpD
(Rexp-Rtheo)/Rtheo at the same Q2 points
0.2
0
-0.2
0.001
EMC
H
E139
NMC
F
E665
F
F
0.2
E772
F
H
F
0.01
0.1
x
F
H
HH
H HH
0
1
-0.2
x
Comparison with F2A/F2D data: Light nuclei
0.2
EMC
NMC
F
0
F
E139
H
F
F
E665
F
F
H H H HH
HH
F
H
C/D
-0.2
0.2
BCDMS
HERMES
0
N/D
-0.2
0.2
E139
E49
0
Al/D
-0.2
0.2
EMC
H
E139
NMC
F
E665
0
F
F
F
F
H
HH
H HH
Ca/D
-0.2
0.001
0.01
0.1
x
1
Comparison with F2A/F2D data: Heavy nuclei
0.2
EMC
0
Sn/D
-0.2
0.2
E665
0
Xe/D
-0.2
0.2
E139
Ñ
E140
0
Ñ
Au/D
-0.2
0.2
E665
0
Pb/D
-0.2
0.001
0.01
0.1
x
1
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.
Scaling Violation and Gluon Distributions

s

2
q (x,Q ) 
2 i
 logQ
2
qi  qi  qi

dy 

2
2
x y j Pqi q j (x / y) q j (y,Q )  Pqg (x / y) g(y,Q )
dominant term at small x
1
1.2
at small x
F2
20  s

xg
2
27
 lnQ


Q2 dependence of F2 is proportional
to the gluon distribution.
x=0.0175
x=0.025
1
NMC
0.8
1.2
1
0.8
No experimental consensus of
Q2 dependence!
 GA(x) determination is difficult.
x=0.0125
x=0.035
x=0.045
x=0.055
Nuclear PDFs
1.2
1.1
1.2
Q2 = 1 GeV 2
Q2 = 1 GeV 2
1.1
Wuv
1
Wdv
1
0.9
0.8
0.7
0.6
0.001
D
Al
Sn
4He
Ca
Xe
Li
Fe
W
Be
Cu
Au
C
Kr
Pb
N
Ag
0.01
0.1
0.9
0.8
0.7
1
0.6
0.001
0.01
x
1
0.1
1
x
1.2
1.2
Q2 = 1 GeV 2
Q2 = 1 GeV 2
1.1
1.1
Wg
Wq
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.001
0.1
0.01
0.1
x
1
0.6
0.001
0.01
x
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
 n factory
• Confirmation of valence modifications at small x
 n factory
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 http://research.kek.jp/people/kumanos/nuclp.html.
 Comparison with (and analysis including) NuTeV nuclear corrections in future!
Small NuTeV nuclear corrections!? (J. F. Owens et al., PRD75, 054030 (2007); J. G.
Morfin@WIN07)
Neutrino factory should be important for finding nuclear medium effects
in the valence-quark and (flavor-separated) antiquark distributions.
Extra
Nuclear corrections in iron (A=56, Z=26)
Sumamry
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:0710.4897v1 [hep-ph].
s  s Asymmetry
Motivations for s(x)  s (x)
 Nucleon does not have net strangeness:  dx s(x)  s (x)  0.
0
However, it does not mean s(x)  s (x).  could be s(x)  s (x)
1
 If s and s are created perturbatively, they should be equal s(x)  s (x).
 Hadron models predict the asymmetry: s(x)  s (x).
p(uud)  KY [K  (us )(uds), K  (us ) 0 (uds), K 0 (ds )  (uus),]
1 / mK   1 / 494 MeV
 0.40 fm
1 / m  1 / 1116 MeV
 0.18 fm
s
s
s
s
0.18 fm
0.40 fm
 The asymmetry could be important for NuTeV anomaly.
x
Global analysis for s(x) and s (x)
CTEQ, (F. Olness et al., Eur. Phys. J. C40 (2005) 145)
H.-L. Lai et al., JHEP 04 (2007) 089.
First, s  s  s
The 2007 paper includes the final NuTeV data for dimuons.
s (x,Q )  A0 x (1  x) P (x), P (x)  e
2
0
A1
A2
A3 x  A4 x A5 x 2
s  r(u  d )
 2 reduction with respect to CTEQ6.5 with s  r(u  d )
this analysis
Strange shape, s (x), is significantly different from nonstrange u (x)  d (x).
g No significant improvement from the parameters A3 , A4 , A5 .
g Improvement is mainly from n -induced dimuon data ( 2     46).
2
The other data are not much sensitive to the strange distributions ( global
 65).
Analysis for s (x)  s(x)  s (x)
x  dxex2 
 a
s (x,Q )  s (x,Q ) tan  cx  1   e



b


2
0
2
0
2
1
x s  0.005
 2 reduction is insignificant.
g No significant improvement
from the parameters d, e..
best fit
 0.001  x s  0.005
0.018  x
s
 0.040

s(x)  s (x) cannot be determined at this stage.
x s   0.001
NuTeV analysis
 

s (x,Q )   x (1  x)
2
0
u (x,Q02 )  d (x,Q02 ) 
  
x 
 
s (x,Q )  s (x,Q ) tan  x (1  x)  1   
x0  


2
0
2
0
Q 2  16 GeV2
1
x s  0.00196  0.0046(stat)
 0.0045(syst)  0.00119(external)
Consistent with CTEQ 2007
 0.001  x s  0.005
C. Bourrely et al., PLB 648 (2007) 39
x s   0.00194
The NuTeV result is not much different from CTEQ one
although it used to differ in hep-ex/0405037.
Global analysis for determining
fragmentation functions
and their uncertainties
Shunzo Kumano
High Energy Accelerator Research Organization (KEK)
Graduate University for Advanced Studies (GUAS)
shunzo.kumano@kek.jp
http://research.kek.jp/people/kumanos/
with M. Hirai (TokyoTech), T.-H. Nagai (GUAS), K. Sudoh (KEK)
Reference: Phys. Rev. D75 (2007) 094009.
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
Introduction
Fragmentation Function
e+
, Z
q
q
e–
Fragmentation: hadron production
from a quark,
antiquark, or gluon
h
Fragmentation function is defined by
1 d (e e  hX)
F (z,Q ) 
 tot
dz
 tot  total hadronic cross section
h
2
z
Eh
2Eh Eh


,
Q
Eq
s /2
s  Q2
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:
 z 2 h
F (z,Q )   
Ci  ,Q  Di ( y,Q 2 )
z y
y

i
Non-perturbative
h
2
1 dy
Calculated in perturbative QCD
(determined from experiments)
Ci (z,Q 2 )  coefficient function
Dih (z,Q 2 )  fragmentation function of hadron h from a parton i
Momentum (energy) sum rule


Dih z,Q 2  probability to find the hadron h from a parton i
with the energy fraction z
Energy conservation:
 0 dz z Dih z,Q
1
h
2
 1
h    ,  0 ,   , K  , K 0 , K 0 , K  , p, p, n, n,   
Favored and disfavored fragmentation functions
Simple quark model:   (ud ), K  (us ), p(uud ),   

Favored fragmentation: Du , D

d
, ...
(from a quark which exists in a naive quark model)



Disfavored fragmentation: Dd , Du , Ds , ...
(from a quark which does not exist in a naive quark model)
Status of determining fragmentation functions
Parton Distribution Functions (PDFs), Fragmentation Functions (FFs)
Nulceonic PDFs
Determination
Uncertainty
Comments
Polarized PDFs
Nuclear PDFs
FFs
****
**
**
**
〇
〇
〇
×
Accurate
determination
from small x to
large x
Gluon &
antiquark
polarization?
Flavor
separation?
Gluon?
Antiquark at
medium x?
Flavor
separation?
Large
differences
between
Kretzer and KKP
(AKK)
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.
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) 054001
(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.


1.5

1.5
1.5
s quark
u quark
1
Q2 = 2 GeV2
(     )/2
zDu
(z)
0.5
0
KKP
AKK
0.2
2
0.4
0.6
z
0.8
2
Q = 2 GeV
1
0.5
0.5
0
0


zDs(  )/2 (z)
Kretzer
-0.5
0
1
gluon
-0.5
1
0
0.2
0.4
0.6
z
0.8
-0.5
1 0
Q2 = 2 GeV2
(     )/2
zDg
(z)
0.2
0.4
0.6
z
0.8
1
Purposes of investigating fragmentation functions
Semi-inclusive reactions have been used for investigating
・origin of proton spin
r
r r
e  p  e  h  X (e.g. HERMES), p  p  h  X (RHIC-Spin)
Quark, antiquark, and gluon contributions to proton spin
(flavor separation, gluon polarization)
・properties of quark-hadron matters A  A  h  X (RHIC, LHC)
Nuclear modification
(recombination, energy loss, …)


a,b,c
f a (xa ,Q 2 )  fb (xb ,Q 2 )
 φ(ab  cX ) 
Dc (z,Q 2 )
Determination of
Fragmentation Functions
Determination of fragmentation function
and their uncertainties
M. Hirai, SK, T.-H. Nagai, K. Sudoh
Phys. Rev. D75 (2007) 094009.
A code for calculating the FFs is available at
http://research.kek.jp/people/kumanos/ffs.html
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.
Comparison with other analyses
HKNS (Ours)
Function form

  i
Ni z

(1 z)
i
Kretzer


  i
Ni z

(1 z)
KKP (AKK)
i


  i

Ni z
(1 z)
i

# of parameters
14
11
15 (18)
Mass threshold
mQ2
(mc,b=1.43, 4.3 GeV)
mQ2
(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:

Du  (1  z)Du
M  Mu
Mg  u
2

M ih 
1
0.05
zDih (z,Q 2 )dz
( issue of momentum sum)
No π+, π– separation
Initial functions for pion
Note: constituent-quark composition  +  ud,  -  ud


Du (z,Q02 )  N u z


u
  u

   c

Du (z,Q )  N u z

2
0
Dc (z,m )  N c z

2
c

  b
Db (z,m )  N b z

2
b

  g
Dg (z,Q )  N g z
2
0
(1 z)
(1 z)
(1 z)

(1 z)

(1 z)

u
u

c

b
 g



Dq  Dq

 Dd (z,Q02 )




 Dd (z,Q )  Ds (z,Q )  Ds (z,Q02 )
2
0
2
0

 Dc (z, mc2 )

 Db (z, mb2 )



nf  



3, 02  Q 2  mc2
4, mc2  Q 2  mb2
5, mb2  Q 2  mt2
6, mt2  Q 2
Constraint: 2nd moment should be finite and less than 1
M
N
,
B(  2,   1)
1
M   zD(z)dz (2nd moment), B(  2,   1)  beta function
0
0  M ih  1 because of the sum rule  M ih  1
h
Experimental data for pion
Total number of data： 264
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
29
58
91.2
91.2
91.2
91.2
# of data
29
18
2
4
29
29
29
29
22
22
17
17
17
100
80
Q (GeV)
s (GeV)
60
TASSO
TOPAZ
OPAL
TPC
SLD
DELPHI
HRS
ALEPH
40
20
0
0
0.2
0.4
0.6
z
0.8
1
Analysis
Q02  1 GeV 2
Initial scale:
n 4
f
Scale parameter:  QCD
 0.220 (LO), 0.323 (NLO)
 s varies with n f
Heavy-quark masses: mc  1.43 GeV, mb  4.3 GeV
 2 /d.o.f.  1.81 (LO), 1.73 (NLO)
Results for the pion
Uncertainty estimation: Hessian method
φ   H ij ai a j ,
   ( aφ  a)   ( a)
2
2
2
i, j
 D(z)    2 
2
i, j
φ 1 D(z, a)
φ
D(z, a)
H ij
ai
a j
φ
2  2 ( a)
H ij 
ai a j
Comparison with pion data
F

 

1
d

(e
e


X)
(z,Q 2 ) 
 tot
dz
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
F

(z,Q )data  F
2

(z,Q 2 )theory

F  (z,Q 2 )
theory
Comparison with pion data: (data-theory)/theory
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.
Comparison with kaon data
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.
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.
Comparison with other parametrizations in kaon and proton
kaon
proton
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 ?)
Pion production at RHIC: p + p  0 + X
S. S. Adler et al. (PHENIX), PRL 91 (2003) 241803
p

pT
s  200 GeV
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.
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
r
p  p   0  X, A  A  h  X (RHIC, LHC), HERMES, JLab, ...
 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
http://research.kek.jp/people/kumanos/ffs.html .
The End
The End