Introduction to Introduction to Parton Distribution FunctionsParton Distribution Functions

in the Nucleon and Nuclei in the Nucleon and NucleiShunzo KumanoShunzo Kumano

High Energy Accelerator Research Organization (KEK) High Energy Accelerator Research Organization (KEK)

Graduate University for Advanced Studies (GUAS)Graduate University for Advanced Studies (GUAS)

December 26, 2008December 26, 2008

shunzo.kumano@kek.jpshunzo.kumano@kek.jphttp://research.kek.jp/people/kumanhttp://research.kek.jp/people/kumanos/os/Meeting on New Interaction Code

For Cosmic Air ShowerICRR, Kashiwa, Japan

http://cosmos.n.kanagawa-u.ac.jp/newCode/

ContentsContents

1.1. Introduction Introduction

Parton Distribution Functions (PDFs) in air-shower model • Parton Distribution Functions (PDFs) in air-shower model • • • Deep inelastic scattering and Parton pictureDeep inelastic scattering and Parton picture

2.2. Determination of PDFsDetermination of PDFs

• • PDFs in the nucleonPDFs in the nucleon Global analysisGlobal analysis Comments on higher-twist effectsComments on higher-twist effects

• • Nuclear PDFsNuclear PDFs Mechanisms for nuclear modificationsMechanisms for nuclear modifications Our global analysis, Comparison with other analysesOur global analysis, Comparison with other analyses

3. Summary3. Summary

IntroductionIntroduction

http://th.physik.uni-frankfurt.de/~drescher/SENECA/

Typical Air Shower Model (SENECA)

My talk is on this “hard” part.

Soft interactionsSoft interactions

In an air-shower modelIn an air-shower model ((e.g.e.g. SIBYLL) SIBYLL)R. S. Fletcher, T. K. Gaisser, P. Lipari, R. S. Fletcher, T. K. Gaisser, P. Lipari, and T. Stanev, Phys. Rev. D 50 (1994) 5710.and T. Stanev, Phys. Rev. D 50 (1994) 5710.

High-energy part is described by the following cross sectionsHigh-energy part is described by the following cross sections

SIBYLL (1994): PDFs by Eichten-Hinchliffe-Lane-Quigg (EHLQ) in 1984SIBYLL (1994): PDFs by Eichten-Hinchliffe-Lane-Quigg (EHLQ) in 1984

The PDFs at large The PDFs at large xx11 and small and small xx22 should affect should affectsimulation results of the air shower. simulation results of the air shower.

Soft and Hard processesSoft and Hard processes My talk is on hard processes.My talk is on hard processes.

• • Nuclear PDFs at small Nuclear PDFs at small xx (N, O) --- LHC (N, O) --- LHC • • Nucleonic and Nuclear PDFs at large Nucleonic and Nuclear PDFs at large xx (p, …, Fe) --- JLab, J-PARC (p, …, Fe) --- JLab, J-PARC • • Fragmentation functions --- Belle, BabFragmentation functions --- Belle, Bab

arar

Soft Hard

~1 GeV

Hard scale (e.g. transverse momentum pT )Resonances Partons

pQCD + Parton Distribution Functions (PDFs)(+ Fragmentation Functions)

(R. Engel)

p, …, Fe N, O

Most energetic particles (namely large Most energetic particles (namely large xxF F ) contribute) contributemainly to subsequent shower development.mainly to subsequent shower development.

x1x2 =Q2

s =

Q2

2ME for fixed targets

For the forward region of x1 ~1 (large x), Q2 ~10 GeV

x2 ~10

s(GeV2 ) =10−7 for s=(14 TeV)2

x2 ~10

2E(GeV) =10−10 = 1 for cosmic rays with E =1020 eV

(extremely small x)

Momentum fraction Momentum fraction x x in the forward regionin the forward region

x1 x2

N, Op, …, Fe

Deep Inelastic Deep Inelastic

Lepton-Nucleon ScatteringLepton-Nucleon Scattering

andand

Parton PictureParton Picture

Kinematics of e+pKinematics of e+pee’’+X+X

k

k '

p

pX

q Bjorken scaling variable: x =Q2

2p⋅q

q2 =(k−k')2 =−Q2 note: q2 =q0

2 −rq2

s =(p+ k)2 s =c.m. energy

invariant mass: W 2 =pX2 =(p+q)2

y =p⋅qp⋅k

ν =p ⋅qM

: energy transfer in the rest frame of N

a⋅b=a+b−+ a−b+ −raT ⋅

rbT

light-cone variables: a± =a0 ±a3

2,

convenient choice: q =(ν, 0, 0, − v2 +Q2 )

q+ =−

M x2, q− =

2ν + M x2

= 2ν ? 1 e.g. p⋅q ; p+q−

γ z

Meaning ofMeaning of x x

x = momentum fraction carried by the struck parton

For example, x=0.5 means that the struck parton carries 50% momentum of the proton.

consider the frame wherethe proton is moving fast

proton momentum: p

k

′k

q (k +q)2 = ′k 2

(k +q)2 =mq2 + 2k⋅q−Q2

′k 2 =mq2

2k ⋅q=Q2

if k =ξp, 2ξp⋅q=Q2

ξ =Q2

2 p ⋅q= x

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

0 < x%< A for a nucleus

x =Q2

2p⋅q=

Q2

2MNν

Q2 =rq2 −ν 2 =(

rk−

rk')2 −(E −E')2

; 2 |rk ||

rk'|(1−cosθ) ≥0 for m= E

ν =E − E ' in the rest frame of N ≥ 0

x ≥0

Q2 ≥0 : spacelike(e+e- annihilation, Q2 ≤0 : timelike)

W 2 =(p+q)2 =MN 2 + 2MNν +q2 ≥MN

2

2M Nν +q2 ≥0 x = − q2

2MNν≤1

In the same way by MN → MA, we have xA = Q2

2MAν≤1.

xA = Q2

2MAν=

MN

MA

Q2

2MNν;

1A

x≤1 ⇒ x%< A for a nucleus

Meaning of QMeaning of Q22

Breit frameLaboratory frame

pxp−xp2xp

Spatial resolution = reduced wavelength

q0=0: photon does not transfer any energy

D=

1rq

=1

ν 2 + Q2≈

1

ν D=

1rq

=1

Q2 in the Breit frame

Q2 corresponds to the “spatial resolution”in the Breit frame.

Breit frame is defined as the frame in whichexchanged boson is completely spacelike:q=(0, 0, 0, q).

Q2 (GeV2/c2)

Structure function F2 isindependent of Q2.(Bjorken scaling)

It means that F2 does not change even if

the spatial resolution is increased. existence of small particles which cannot be resolved:

This point-like particle wasnamed “parton”. parton quark, gluon

x = 0.25

Bjorken scalingBjorken scaling

Example of electron-proton scattering data at SLAC (1972).

νW2=F2

size <1 / Q2

F2 (x,Q2 ) ≈F2 (x)

Scaling violation (QScaling violation (Q2 2 dependence)dependence)

ZEUS, Eur. Phys. J. C21 (2001) 443.

small Qsmall Q22

large Qlarge Q22

1

Q2

1

Q2

gluon, qq clouds

As QAs Q2 2 becomes large, the virtualbecomes large, the virtual γγstarts to probe the gluon, quark,starts to probe the gluon, quark,and antiquark “clouds”.and antiquark “clouds”.

∂∂logQ2

q x,Q2( )

g x,Q2( )

⎛

⎝⎜

⎞

⎠⎟ =

α s

2π

dy

yx

1∫

Pqq x / y( ) Pqg x / y( )

Pgq x / y( ) Pgg x / y( )

⎛

⎝⎜⎞

⎠⎟q y,Q2( )

g y,Q2( )

⎛

⎝⎜

⎞

⎠⎟

DGLAP (Dokshitzer, Gribov, Lipatov, Altarelli, Parisi)

Q2 corresponds to“spatial resolution”.

Parton Distribution FunctionsParton Distribution Functions

in the Nucleonin the Nucleon

Related web sitesRelated 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, Q2 evolution codes, Fragmentation functions http://research.kek.jp/people/kumanos/

Recent papers on unpolarized PDFsCTEQ (uncertainties) D. Stump (J. Pumplin) et al., Phys. Rev. D65 (2001) 14012 & 14013. (CTEQ6) D. Pumplin et al., JHEP, 0207 (2002) 012; 0506 (2005) 080; 0602 (2006) 032; 0702 (2007) 053; PRD78 (2008) 013004. (charm) PR D75 (2007) 054029; (strange) PRL 93 (2004) 041802; Eur. Phys. J. C40 (2005) 145; JHEP 0704 (2007) 089.

GRV (GRV98) M. Glück, E. Reya, and A. Vogt, Eur. Phys. J. C5 (1998) 461. (GJR08) M. Glück, P. Jimenez-Delgado, and E. Reya, Eur. Phys. J. C53 (2008) 355.

MRST A. D. Martin, R. G. Roberts, W. J. Stirling, and R. S. Thorne, (MRST2001, 2002, 20033) Eur. Phys. J. C23 (2002) 73; Eur. Phys. J. C28 (2003) 455; (theoretical errors) Eur. Phys. J. C35 (2004) 325; (2004) PL B604 (2004) 61; (QED) Eur. Phys. J. C39 (2005) 155; PL B636 (2006) 259; (2006) PRD73 (2006) 054019; PL B652 (2007) 292. (2008) arXiv:0806.4890

Alekhin S. I. Alekhin, PRD68 (2003) 014002; D74 (2006) 054033.

BB J. Blümlein and H. Böttcher, Nucl. Phys. B774 (2007) 182-207.

NNPDF S. Forte et al., JHEP 0205 (2002) 062; 0503 (2005) 080; 0703 (2007) 039.

H1 C. Adloff et al., Eur. Phys. J. C 21 (2001) 33; C30 (2003) 1.

ZEUS S. Chekanov et al., PRD67 (2003) 012007; Eur. Phys. J. C42 (2005) 1.

It is likely that I miss some papers!

Recent activities uncertainties NNLO QED s – s charm

Parton distribution functions are determined by fitting various experimental data.

g electron/muon: μ + p→ μ + Xg neutrino: νμ + p→ μ + X

g Drell-Yan: p+ p→ μ+μ−+ Xg ⋅⋅⋅

(1) assume functional form of PDFs at fixed Q2 (≡Q02 ) :

e.g. fi (x,Q02 ) =Aix

αi (1−x)βi (1+γix),

where i =uv, dv, u, d, s, g

(2) calculate observables at their experimental Q2 points.(3) then, the parameters Ai , α i , βi , γiare determined so as

to minimize χ 2 in comparison with data.

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

€

MW1= F1 , νW2 = F2 , νW3 = F3 , x = Q2

2p⋅q , y = p⋅qp⋅k

€

dσ ν ,ν CC

dx dy =GF

2 (s − M 2)2π (1+Q2 / MW

2 )2 x y2F1CC + 1− y − M x y

2 E ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ F2

CC ± x y 1− y2

⎛

⎝ ⎜

⎞

⎠ ⎟ F3

CC ⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

N

X

q

p W

νμ

μ –

Determination of each distribution Valence quark

€

12 [F3

νp +F3ν p]CC = uv +dv + s−s +c −c

M = 11+Q2 / MW

2 GF

2 u( ′k , ′λ ) γμ (1−γ5) u(k,λ) <X|J μ

CC |p,λp>

dσd ′E dΩ =

GF2

(1+Q2 /MW2 )2

′k32π 2E

Lμν Wμν

νμ + p → μ − + X

Lμν =8 kμ ′k ν + kν ′k μ −gμνk⋅ ′k + iε μνρσkρ ′kσ⎡⎣ ⎤⎦where ε0123 =+1

Wμν =−W1 gμν −qμqν

q2

⎛

⎝⎜⎞

⎠⎟+W2

1MN

2 pμ −p⋅qq2 qμ⎛

⎝⎜⎞⎠⎟

pν −p⋅qq2 qν⎛

⎝⎜⎞⎠⎟+

i2MN

2 W3εμνρσ pρqσ

Note: Issue of nuclear correctionsin CCFR/NuTeV (ν+Fe)unless we will have a ν factory.

Sea quark

e/μ scattering

Drell-Yan (lepton-pair production)

(q1)

q1

q2

μ–

μ+

(q2)

projectile target

F2N =

F2p + F2

n

2=

518

x u+u+d+d( ) +218

x u+u+d+d( )

=518

xV +418

xS if q distributions are flavor symmetric

p1 + p2 → μ+μ−+ X

dσ ∝ q(x1) q(x2 ) +q(x1) q(x2 )

dσ ∝ qV (x1) q(x2 )at large xF =x1 −x2

q(x2 ) can be obtained if qV (x1) is known.

Gluon

scaling violation of F2

∂∂ lnQ2( )

qs x,t( )

g x,t( )

⎛

⎝⎜⎞

⎠⎟=

α s

2π

dy

yx

1∫

Pqq x / y( ) Pqg x / y( )

Pgq x / y( ) Pgg x / y( )

⎛

⎝⎜⎞

⎠⎟qs y,t( )

g y,t( )

⎛

⎝⎜⎞

⎠⎟

at small x ∂F2

∂ lnQ2( )≈10 αs

27πG

jet production

K. Prytz, Phys. Lett. B311 (1993) 286.

Global Analysis for PDFsGlobal Analysis for PDFs

Outline of analysis

1. Express x-dependent PDFs with parameters at a fixed Q2 ( Q02)

2. Evolve the PDFs to experimental Q2 data points

5. Repeat 2, 3, and 4 processes until minimal 2 is obtained

3. Convolute with coefficient functions to calculate observables

4. Determine 2 in comparison with data

* Choice of Q02

* Q2 evolution methods

Unpolarized Parton Distribution Functions (PDFs) in the nucleonUnpolarized Parton Distribution Functions (PDFs) in the nucleon

The PDFs could be obtained from http://durpdg.dur.ac.uk/hepdata/pdf.html

0

0.2

0.4

0.6

0.8

1

0.00001 0.0001 0.001 0.01 0.1 1

x

Q2

= 2 Ge V2

xg/5

xd

xu

xs

xuv

xdv

Valence-quarkdistributions

Gluon distribution / 5

PDF uncertaintyPDF uncertainty

CTEQ5M1

MRS2001

CTEQ5HJ

CTEQ6 (J. Pumplin et al.),

JHEP 0207 (2002) 012

u d

g

other PDF

CTEQ6q(x)q(x) at large at large xx

g(x)g(x) at small at small xx

(unknown)(unknown)22

for cosmic-ray studiesfor cosmic-ray studies

““gluon saturation”gluon saturation”

There are also large nuclearThere are also large nuclearcorrections in these regions.corrections in these regions.

Comments onComments onHigher-Twist EffectsHigher-Twist Effects

Higher-twist effects by CTEQ in 2002Higher-twist effects by CTEQ in 2002

F2 (x,Q2 ) =F2NLO(x,Q2 ) 1+

H(x)Q2

⎡

⎣⎢

⎤

⎦⎥

H(x) =h0 +h1x+h2x2 +h3x

3 +h4x4

error estimate ?

CTEQ (J. Pumplin CTEQ (J. Pumplin et al.)et al.),,JHEP 07 (2002) 012.JHEP 07 (2002) 012.

Initial PDFs defined at Q02 =(1.3)2 GeV2

Kinematical cut for data

Q2 ≥4 GeV2 , W2 ≥(3.5)2=12.25 GeV2

Higher-twist corrections are not needed.

Higher-twist effects by BB in 2008Higher-twist effects by BB in 2008

F2 (x,Q2 ) =F2LT (x,Q2 )

× (target-mass corrections)+CHT (x)

Q2

⎡

⎣⎢

⎤

⎦⎥

J. Blüemlein, H. Böttcher, Phys. Lett. B 662 (2008) 336.

How to extract higheer-twist (HT) effects? • Systematic studies of lnQ2 slopes from large Q2

Actual analysis (1) Leading-twist (LT) fit at large W2 (W2 > 12.5 GeV 2 ) (2) Taget-mass corrections (3) Extraction of HT effects by extrapolating the results (1) with (2) to the region 4 GeV 2 < W2 < 12.5 GeV 2 .

QuickTime˛ Ç∆ êLí£ÉvÉçÉOÉâÉÄ

ǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB

proton deuteron

Higher-twist effects by BB in 2008Higher-twist effects by BB in 2008

F2 (x,Q2 ) =F2LT (x,Q2 ) (target-mass corrections)+

CHT (x)Q2

⎡

⎣⎢

⎤

⎦⎥

QuickTime˛ Ç∆ êLí£ÉvÉçÉOÉâÉÄ

ǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB

Depletion of HT effectswith increasing α s order at large x

For extracting reliable higher-twist effectsFor extracting reliable higher-twist effects

Full analysis is needed!Full analysis is needed!

““Full” means at leastFull” means at least

Parametrization and fit also for the HT termsParametrization and fit also for the HT terms

Uncertainty range of a determined HT functionUncertainty range of a determined HT function

together with target-mass corrections and higher-order effects.together with target-mass corrections and higher-order effects.

Nuclear Nuclear

Parton Distribution FunctionsParton Distribution Functions

Brief Introduction toBrief Introduction toNuclear Modifications ofNuclear Modifications of

Parton Distribution FunctionsParton Distribution Functions

0.7

0.8

0.9

1

1.1

1.2

0.001 0.01 0.1 1

EMC

NMC

E139

E665

Nuclear binding

(+ Nucleon modification)

Fermi motionof the nucleon

x

Could affectCould affectcosmic-ray studiescosmic-ray studies

Nuclear modifications of structure function Nuclear modifications of structure function FF22

D. F. Geesaman, K. Saito, A. W. Thomas, Ann. Rev. Nucl. Part. Sci. 45 (1995)337.

Shadowing (q q fluctuation of photon)

EMC (European Muon Collaboration) effectEMC (European Muon Collaboration) effect

Theoretical DescriptionTheoretical Description

fa/A(q2,P⋅q) =ΣT

d4p(2π)4 fa/T(p,q) fT/A(P,p)

Q2 rescaling model, ⋅⋅⋅

fa/A(x,Q2) = Σ

TdyA

x A

1

fa/TxAyA

fT/A(yA)

nuclear binding, nuclear pion, ⋅⋅⋅

(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(xA ).

(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.

PP

pp

Nucleus (A)Nucleus (A)

Hadron (T)Hadron (T)

kk

qq

Quark (a) Quark (a)

Binding ModelBinding Model

Convolution: WμνA (pA,q) = d4∫ p S(p) Wμν

N (pN ,q)

S(p) = Spectral function = nucleon momentum distribution in a nucleus

In a simple shell model: S(p) = φi (

rp) 2

i∑ δ(p0 −MN −ε i )

Single-particle energy: ε i

Projecting out F2 : F2A (x,Q2 ) = d∫ z fi (z)

i∑ F2

N (x / z,Q2 )

fi (z) = d3∫ p z δ z−

p⋅qMNν

⎛

⎝⎜⎞

⎠⎟ φi (

rp) 2 lightcone momentum distribution for a nucleon i

z =

p⋅qMNν

;p⋅q

pA ⋅q / A;

p+

pA+ / A

lightcone momentum fractiona± =

a0 ±a3

2

p⋅q=p+q−+ p−q+ −rpT ⋅

rqT ; p+q−

F2A (x,Q2 ) = d∫ z fi (z)

i∑ F2

N (x / z,Q2 ) fi (z) = d3∫ p z δ z−

p⋅qMNν

⎛

⎝⎜⎞

⎠⎟ φi (

rp) 2

z =

p⋅qMNν

=p0ν −

rp⋅

rq

MNν=1−

|ε i |MN

−rp⋅

rq

MNν≈1.00 −0.02 ±0.20 for a medium-size nucleus

f (z)

z

0.980.98

0.200.20

If fi (z) were fi (z) =δ(z−1), there is no nuclear

modification: F2A(x,Q2 ) =F2

N (x,Q2 ).

Because the peak shifts slightly (1Because the peak shifts slightly (1 0.98), 0.98),nuclear modification of Fnuclear modification of F22 is created. is created.

F2A (x,Q2 ) ; F2

N (x / 0.98,Q2 )

For x =0.60, x / 0.98 =0.61F2

N (x =0.61)F2

N (x=0.60)=0.0210.024

=0.88

x

F2A / F2

N

binding

Fermi motion

Shadowing Models: Vector-Meson-Dominance (VMD) typeShadowing Models: Vector-Meson-Dominance (VMD) typeA

q

q

V Virtual photon splits into a qq pair and

it becomes a vector meson, which interacts

with a nucleus, especially in the surface region.

propagation length of V: λ =1

EV −Eγ

=2ν

MV2 +Q2 =

0.2 fmx

> 2 fm at x< 0.1

At small x, the virtual photon interacts with

the target nucleus as if it were a vector meson.

F2A (x,Q2 ) =

Q2

πdM 2∫

M 2 (M 2 )∏(M 2 +Q2 )2

σVA

(M 2 )∏ =σ(e+e−→ hadrons)σ(e+e−→ μ+μ−)

=vector mesons+qq continuum

Determination of Nuclear Determination of Nuclear Parton Distribution FunctionsParton Distribution Functions

Nuclear modificationNuclear modification

Nuclear modification of F2A / F2

D iswell known in electron/muon scattering.

0.7

0.8

0.9

1

1.1

1.2

0.001 0.01 0.1 1

EMC

NMC

E139

E665

shadowingoriginal

EMC finding

Fermi motion

x sea quark valence quark

F2A (LO) = ei

2

i∑ x qi (x) +qi (x)[ ]A

Drell-Yan and Antiquark DistributionsDrell-Yan and Antiquark Distributions

σ DYpCa

σ DYpD

≈q Ca

q D

The Fermilab E772 Drell-Yan data suggested that nuclearmodification of antiquark distributions should be smallin the region, x≈0.1.

0.7

0.8

0.9

1

1.1

1.2

0.001 0.01 0.1 1x

772E

Drell-Yan cross-section ratiois roughly equal to antiquark ratio.

p + A→ μ+μ−+ X

ReferencesReferences

(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; 0807 (2008) 102.

(HKM, HKN) M. Hirai, SK, M. Miyama, Phys. Rev. D64 (2001) 034003;

M. Hirai, SK, T.-H. Nagai, Phys. Rev. C70 (2004) 044905;

M. Hirai, SK, T.-H. Nagai, Phys. Rev. C76 (2007) 065207.

(DS) D. de Florian and R. Sassot, Phys. Rev. D69 (2004) 074028.

There are only a few papers onthe parametrization of nuclear PDFs! Need much more works.

χ2 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) 054001.

The recent HKN07 analysis is explained in this talk.The recent HKN07 analysis is explained in this talk.

NLO Determination ofNLO Determination of

Nuclear Parton Distribution FunctionsNuclear Parton Distribution Functions

by M. Hirai, SK, T.-H. Nagaiby M. Hirai, SK, T.-H. Nagai

Phys. Rev. C 76 (2007) 065207

Related papers. M. Hirai, SK, M. Miyama, Phys. Rev. D64 (2001) 034003;

M. Hirai, SK, T.-H. Nagai, Phys. Rev. C70 (2004) 044905.

NPDF codes can be obtained from http://research.kek.jp/people/kumanos/nuclp.html

Experimental data: Experimental data: total number = 1241total number = 1241

(1) F2A / F2

D 896 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 / F2

A’ 293 data NMC: Be / C, Al / C, Ca / C, Fe / C, Sn / C, Pb / C, C / Li, Ca / Li

(3) σ DYA / σ DY

A’ 52 data E772: C / D, Ca / D, Fe / D, W / D E866: Fe / Be, W / Be

1

10

100

500

0.001 0.01 0.1 1

x

NMC (F

2

A

/F

2

D

)

SLAC

EMC

E665

BCDMS

HERMES

NMC (F

2

A

/F

2

A'

)

E772/E886 DY

NMC (F

2

D

/F

2

p

)

Functional formFunctional form

If there were no nuclear modificationIf there were no nuclear modification

Isospin symmetryIsospin symmetry ：：

Take account of nuclear effects by Take account of nuclear effects by wwi i (x, A)(x, A)

uvA x( ) =wuv

x,A( )Zuv x( ) + Ndv x( )

A, dv

A x( ) =wdvx,A( )

Zdv x( ) + Nuv x( )A

uA x( ) =wq x,A( )Zu x( ) + Nd x( )

A, dA x( ) =wq x,A( )

Zd x( ) + Nu x( )A

sA x( ) =wq x,A( )s x( )

gA x( ) =wg x,A( )g x( )

→ uA x( ) =Zu x( ) + Nd x( )

A, d A x( ) =

Zd x( ) + Nu x( )

A

un =dp ≡d, dn =up ≡u

Nuclear PDFs “per nucleon”Nuclear PDFs “per nucleon”

AuA x( ) =Zup x( ) + Nun x( ), AdA x( ) =Zdp x( ) + Ndn x( ) p = proton, n = neutron

at at QQ22==1 GeV1 GeV2 2 (( QQ002 2 ))

Functional form of Functional form of wwi i (x, A)(x, A)

fiA (x,Q0

2 ) =wi (x,A) fi (x,Q02 ) i =uv, dv, u, d, s, g

wi (x, A) =1+ 1−1Aα

⎛⎝⎜

⎞⎠⎟

ai +bix+ cix2 +dix

3

(1−x)β

Nuclear charge: Z =A dx23

uA −uA( )−13

dA −dA( )−13

sA −sA( )⎡⎣⎢

⎤⎦⎥∫ =A dx

23

uvA −

13

dvA⎡

⎣⎢⎤⎦⎥∫

Baryon number: A=A dx13

uA −uA( ) +13

dA −dA( ) +13

sA −sA( )⎡⎣⎢

⎤⎦⎥∫ =A dx

13uv

A +13

dvA⎡

⎣⎢⎤⎦⎥∫

Momentum: A=A dx uA +uA +dA +dA + sA + sA + g⎡⎣ ⎤⎦∫ =A dx uv

A +dvA + 2 uA +dA + sA( ) + g⎡⎣ ⎤⎦∫

Three constraintsThree constraints

xx

A simple function = cubic polynomialA simple function = cubic polynomial

Note: The regionNote: The region x x > 1 cannot be > 1 cannot be described by this parametrization.described by this parametrization.

0.7

0.8

0.9

1

1.1

1.2

0.03 0.1 1

x

772E

Q

2

= 50 GeV

2

LO

NLO

H

H

H

H

H

H

H

0.7

0.8

0.9

1

1.1

1.2

0.001 0.01 0.1 1

x

EMC

NMC

HE136

E665

Q

2

= 10 GeV

2

Comparison with FComparison with F22CaCa/F/F22

DD & & σσDYDYpCapCa/ / σσDYDY

pDpD data data

(R(Rexpexp-R-Rtheotheo)/R)/Rtheo theo at the same Qat the same Q22 points points R= FR= F22CaCa/F/F22

DD, , σσDYDYpCapCa/ / σσDYDY

pDpD

H

H

H HHH H

F F

F

F

F

-0.2

0

0.2

0.001 0.01 0.1 1

x

EMC

NMC

H E139

F E665

-0.2

0

0.2

x

E772

NLO analysisNLO analysisLO analysisLO analysis

Nuclear PDFsNuclear PDFs

0.6

0.7

0.8

0.9

1

1.1

1.2

0.001 0.01 0.1 1

x

0.6

0.7

0.8

0.9

1

1.1

1.2

0.001 0.01 0.1 1

x

D

4He

Li

Be

C

N

Al

Ca

Fe

Cu

Kr

Ag

Sn

Xe

W

Au

Pb

0.6

0.7

0.8

0.9

1

1.1

1.2

0.001 0.01 0.1 1

x

0.6

0.7

0.8

0.9

1

1.1

1.2

0.001 0.01 0.1 1

x

Wd v

Q2

= 1 GeV2

Wu v

Q2

= 1 GeV2

Q2

= 1 GeV2

Q2

= 1 GeV2

WgW q

Future experimentsFuture experiments

0.4

0.6

0.8

1

1.2

0.001 0.01 0.1 1

x

LO

NLO

uv

Q 2 = 1 GeV

2

JLab

νFactoryMINARνA

0.4

0.6

0.8

1

1.2

0.4

0.6

0.8

1

1.2

0.001 0.01 0.1 1

x

q

gluon

FermilabJ-PARCGSI

RHICLHCLHeC

RHICLHCLHeC

J-PARC?GSI?

eLICeRHIC

eLICeRHIC

Nuclear PDFs and uncertaintiesNuclear PDFs and uncertainties

• • Some NLO improvements, Some NLO improvements, but not significant ones.but not significant ones. • • Impossible to determine Impossible to determine

gluon modifications.gluon modifications. • • Antiquark distributions are Antiquark distributions are

not determined at large not determined at large xx..

Comparison withComparison withOther Global AnalysesOther Global Analyses

ComparisonComparison

EKRS HKN DS

É‘2 analysis Å~Å®ÅZ ÅZ ÅZ

NPDF error Å~ ÅZ Å~

LO ÅZ ÅZ ÅZ

NLO Å~ ÅZ ÅZ

nucleonic PDFs GRV92, CTEQ4L MRST01,98 GRV98

initial Q2 (GeV2) 2.25 1 0.4 (0.26)

data sets F2, Drell-Yan F2, Drell-Yan F2, Drell-Yan

number of data 250~300? 951, 1241 420

Q2 cut of data (GeV2) 2.25 1 1

deuteron modification Å~ ÅZ ÅZ

number of parameters 27 9, 12 27

É‘2/d.o.f. — 1.58, 1.21 0.76

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

Comparison of used data setComparison of used data setHKN data table

data in EKRS data in DS

data not included

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

EKR (1998), EKS (1999) analysis

g "eye fit" (not χ 2 analysis)→ χ 2 analysis after 2007g F2 data, Drell-Yan data (see the table for a detailed comparison) g LO (Leading Order of αs)g Q0

2 =2.25 GeV2 =mc2 [Data with Q2 ≥2.25 GeV2 are used.]

g deuteron (=nucleon): no nuclear modification

g uA /u =dA / d =sA / s ≡RSA, uv

A /uv =dvA / dv ≡Rv

A

g Divide x into 3 regions (xp ≈0.1, xeq ≈0.4)

g RSA =Rv

A =RF2A at x> xeq, RS

A(xp < x< xeq,Q02 ) =const=RS

A(xp,Q02 )

g RGA(x,Q0

2 ) ≈RF2A (x,Q0

2 ) at small x→ not in the 2008 version

g 27 parameters (a few of them are fixed)→ no error analysisg baryon-number and momentum conservationsg nucleonic PDFs: GRV92-LO, CTEQ4L

xxp xeq

p =plateau: RSA =const

Results for nuclear PDFs

Pb/D

€

Q2 = 2.25 GeV2

valence

antiquark

gluon

0.4

0.6

0.8

1

1.2

0.001 0.01 0.1 1

x

Ca/D

Q2 =1 GeV2

EPS08 Analysis for Nuclear PDFsEPS08 Analysis for Nuclear PDFs

An Improved global analysis of nuclear parton distribution functionsincluding RHIC data

K. J. Eskola, H. Paukkunen, C. A. Salgado

JHEP 07 (2008) 102

New point: RHIC-BRAHMS hadron production dataNew point: RHIC-BRAHMS hadron production data in a forward rapidity regionin a forward rapidity region very large gluon shadowingvery large gluon shadowing

Note: still LO analysisNote: still LO analysis

Analysis conditionsAnalysis conditions

KKP forKKP for D Diihh

15 parameters15 parameters

Results: RHICResults: RHIC

BRAHMS forward rapidityBRAHMS forward rapidity

η =−ln tanθh

2

⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥

QuickTime˛ Ç∆ êLí£ÉvÉçÉOÉâÉÄ

ǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB

QuickTime˛ Ç∆ êLí£ÉvÉçÉOÉâÉÄ

ǙDZÇÃÉsÉNÉ`ÉÉÇ å©ÇÈÇΩÇflÇ…ÇÕïKóvÇ≈Ç∑ÅB

DataData

Note: very large weightNote: very large weight for BRAHMS for BRAHMS

weightweight

Comparison of EPS, HKN, DS parametrizationsComparison of EPS, HKN, DS parametrizations

• • Valence- and sea-quark distributions agree well.Valence- and sea-quark distributions agree well. Large variations in gluon distributions:• Large variations in gluon distributions:•

EPS08 (with BRAHMS forward data) EPS08 (with BRAHMS forward data) Huge gluon shadowing! Huge gluon shadowing!

-0.5

0

0.5

1

1.5

-0.5

0

0.5

1

1.5

-0.5

0

0.5

1

1.5

-0.5

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1z

-0.5

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1z

gluon

u quark

c quark b quark

Q2 = 2 GeV2

Q2 = 2 GeV2 Q2 = 2 GeV2

Q2 = 10 GeV2 Q2 = 100 GeV2

KKPAKK Kretzer

HKNS

s quark

DSS

Uncertainties of fragmentationUncertainties of fragmentationfunctions functions “in including hadron“in including hadron-production data in the global analysis.”-production data in the global analysis.”

RHICRHIC • • Gluon and light-quark fragmentationGluon and light-quark fragmentation

functions have large uncertainties.functions have large uncertainties.

• • Large differences between the functionLarge differences between the functionss of various analysis groups.of various analysis groups.

• • Gluon function at large-z is importantGluon function at large-z is important for hadron-productions at RHIC.for hadron-productions at RHIC.

Global analysisresults for π

M. Hirai M. Hirai et al., et al., PRD 75 (2007) 094009.PRD 75 (2007) 094009.

(Torii)

Why is it so difficultWhy is it so difficultto determine to determine

nuclear gluon distributions?nuclear gluon distributions?

Current nuclear data areCurrent nuclear data arekinematically limited.kinematically limited.

x =

Q2

2p⋅q;

Q2

ys

fixed target: min(x) =Q2

2MNElepton

≤1

2Elepton(GeV)

if Q2 ≥1 GeV2

for Elepton (NMC) =200 GeV, min(x) =1

2 ⋅200=0.003

(from H1 and ZEUS, hep-ex/0502008)

F2 datafor the proton

x

1

10

100

500

0.001 0.01 0.1 1

Q2 (

GeV

2 )

2 )

NMC (F2A /F2

D)

SLAC

EMC

E665

BCDMS

HERMES

NMC (F2A /F2

A')

E772/E886 DY

F2 & Drell-Yan datafor nuclei

region of nuclear data

x =0.65

x =0.013

x =0.0005

Scaling Violation and Gluon DistributionsScaling Violation and Gluon Distributions

at small x

∂F2

∂ lnQ2( )≈20 αs

27πxg

0 .811 .20 .811 .20 .811 .211 011 0H E R M E S11 0x=0 .0 35x=0 .0 45x=0 .0 55x=0 .0 7x=0 .0 9x=0 .1 25x=0 .1 75x=0 .25x=0 .35Q2 ( G e V2 )

0.8

1

1.2

1 10 1001 10 100

x=0.035 x=0.045

Q2 ( GeV2 )

HERMES

x=0.055

0.8

1

1.2

0.8

1

1.2

NMC

x=0.0125 x=0.0175 x=0.025

x=0.035 x=0.045 x=0.055

No experimental consensus ofQ2 dependence! GA(x) determination is difficult.

∂∂logQ2

qi+ (x,Q2 ) =

α s

2π

dy

y

x

1

∫ Pqi q j(x / y) q j

+ (y,Q2 )j

∑ + Pqg (x / y) g(y,Q2 )⎡

⎣⎢

⎤

⎦⎥

dominant term at small xqi+ =qi +qi

Q2 dependence of F2 is proportionalto the gluon distribution.

K. Prytz, PLB 311 (1993) 286.

SummarySummary

Hard interactions are discussed in my talk.Hard interactions are discussed in my talk.

In order to understand the shower profile, it is importantIn order to understand the shower profile, it is importantto studyto study

Nucleonic and Nuclear PDFs at small Nucleonic and Nuclear PDFs at small xx (LHC) (LHC)

Nucleonic and Nuclear PDFs at large Nucleonic and Nuclear PDFs at large xx (JLab, J-PA (JLab, J-PARC, …)RC, …)

Higher-twist effects --- numerical studies are in pHigher-twist effects --- numerical studies are in progressrogress at large at large xx. .

Nuclear gluon distributions --- not obvious, studied Nuclear gluon distributions --- not obvious, studied at LHCat LHC

The End

The End