Nuclear Modifications ofNuclear Modifications ofParton Distribution FunctionsParton Distribution Functions

Shunzo 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 22, December 22, 20072007

shunzo.kumano@kek.jpshunzo.kumano@kek.jp

http://research.kek.jp/people/http://research.kek.jp/people/kumanos/kumanos/

Workshop on Workshop on

Description of Lepton-Nucleus Description of Lepton-Nucleus ReactionsReactions

KEK, Tsukuba, December 22, 2007KEK, Tsukuba, December 22, 2007

Collaborators: Masanori Hirai (Juntendo) Collaborators: Masanori Hirai (Juntendo) Takahiro Nagai (GUAS)Takahiro Nagai (GUAS)

Ref. Phys. Rev. C 76 (2007) 065207.

ContentsContents

(1) IntroductionMotivation Comments on parton distribution functions (PDFs) in the nucleon

(2) Determination of PDFs in Nuclei Analysis methodLO and NLO results and their comparisonsSummary

Motivations for studyingMotivations for studying structure functionsstructure functions and and parton parton distribution functionsdistribution functions

(1)(1)To establish QCDTo establish QCD

Perturbative QCDPerturbative QCD • In principle, theoretically

established in many processes. (There are still issues on small-x

physics.)

• Experimentally confirmed (unpolarized, polarised ?)

Non-perturbative QCD (PDFs)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 Determination of the PDFs from experimental data. experimental data.

(2) For discussing any high-energy reactions, accur(2) For discussing any high-energy reactions, accurate PDFsate PDFs are needed.are needed.

origin of nucleon spin:origin of nucleon spin: quark- and gluon-spin quark- and gluon-spin contributionscontributions

exotic events at large Qexotic events at large Q22:: physics of beyond physics of beyond current frameworkcurrent framework

heavy-ion reactions:heavy-ion reactions: quark-hadron matter quark-hadron matter

neutrino oscillations: neutrino oscillations: nuclear effects in nuclear effects in + + 1616O O

cosmology: cosmology: ultra-high-energy cosmic raysultra-high-energy cosmic rays

New structure functions and new investigations at New structure functions and new investigations at factory! factory!

Nuclear PDFs in neutrino reactionsNuclear PDFs in neutrino reactions

(1)(1) CCFR and NuTeV: CCFR and NuTeV: 5656Fe target Fe target

Nuclear effects are important in Nuclear effects are important in extracting nucleonic PDFs. extracting nucleonic PDFs.

(2) Oscillation experiments (2) Oscillation experiments

Nuclear corrections in Nuclear corrections in 1616OO

Low QLow Q22 data: High Q data: High Q22 (PDFs) (PDFs) Low Q Low Q22 is needed.is needed.

(Quark-hadron duality)(Quark-hadron duality)

(3) Neutrino Factory(3) Neutrino Factory

New investigations with proton and New investigations with proton and deuteron targets,deuteron targets,

so that nuclear modifications could so that nuclear modifications could be studied bybe studied by

measuring measuring A/DA/D ratios. ratios.

Parton Distribution Functions (PDFs) in the NucleonParton Distribution Functions (PDFs) in the Nucleon

PDFs fromPDFs from http://durpdg.dur.ac.uk/hepdata/pdf.htmlhttp://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

factoryfactory

Suppose E =50GeV

x=Q2

2MN

xmin =min(Q2 )

2MN max()

:1

2 ⋅1⋅50=0.01

wheremin(Q2 ) : 1GeV2

max() =E =50GeV

Valence-quark distributions are Valence-quark distributions are determined determined from data includingfrom data includingCCFR and NuTeV ones withCCFR and NuTeV ones withthe the ironiron target. target. It should be worth investigating them It should be worth investigating them for the real nucleonfor the real nucleon at a neutrino factory. at a neutrino factory.

PDF uncertaintyPDF uncertainty

CTEQ5M1

MRS2001

CTEQ5HJMRS2001

CTEQ5M1

CTEQ6 (J. Pumplin et al.), JHEP 0207 (2002) 012

Parton Distribution Parton Distribution

FunctionsFunctions

in “Nuclei”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, , …)

“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…

s−s

J-PARC = Japan Proton Accelerator Research Complex

Situation of data for nuclear PDFs

Neutrino factory: ~10 years later ?(CCFR, NuTeV)Small-x, high-energy electron facility?

RHIC, LHC

RHIC, LHC

RHIC, LHC, J-PARC

Available data for nuclear PDFs

Jlab at large x

Table from MRST, hep/ph-9803445

Current nuclear data arekinematically limited.

x =

Q2

2p⋅q;

Q2

ys

fixed target: min(x) =Q2

2MNElepton

≤1

2Elepton(GeV)

ifQ2 ≥1GeV2

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

2 ⋅200=0.003

1

10

100

500

0.001 0.01 0.1 1

x

NMC (F2A/F2

D)

SLAC

EMC

E665

BCDMS

HERMES

NMC (F2A/F2

A')

E772/E886 DY

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

F2 datafor the proton

F2 & Drell-Yan datafor nuclei

region of nuclear data

x =0.65

x =0.013

x =0.0005

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

EMCNMCE139E665

shadowingoriginal EMC finding

Fermi motion

x sea quark valence quark

F2A (LO) = ei

2

i∑ x qi (x) +qi (x)[ ]A

Binding ModelBinding Model

Convolution: WμA (pA,q) = d4∫ pS(p)Wμ

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 Recoil energy

rp2

2M A−i

isneglected.

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

i∑ F2

N (x / z,Q2 )

fi (z) = d3∫ pzδ z−

p⋅qMN

⎛

⎝⎜⎞

⎠⎟φi (

rp) 2 lightconemomentumdistributionforanucleoni

z =

p⋅qMN

;p⋅q

pA ⋅q / A;

p+

pA+ / A

lightconemomentumfractiona± =

a0 ±a3

2

p⋅q=p+q−+ p−q+ −rpT ⋅

rqT ; p+q−

F2A (x,Q2 ) = d∫ zfi (z)

i∑ F2

N (x / z,Q2 ) fi (z) = d3∫ pzδ z−

p⋅qMN

⎛

⎝⎜⎞

⎠⎟φi (

rp) 2

z =

p⋅qMN

=p0 −

rp⋅

rq

MN=1−

|ε i |MN

−rp⋅

rq

MN≈1.00 −0.02 ±0.20foramedium-sizenucleus

f (z)

z

0.980.98

0.200.20

If fi (z) were fi (z) =δ(z−1),thereisnonuclear

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

=2

MV2 +Q2 =

0.2fmx

> 2fmatx< 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−)

=vectormesons+qqcontinuum

EMC (European Muon Collaboration) EMC (European Muon Collaboration) effecteffectTheoretical DescriptionTheoretical Description

q q

a k

T p

AP

fa/T(k,p)

fT/A(p,P)

fa/A(q2,P⋅q) =Σ

T

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.

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.

(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.

2 analysis

The recent HKN report (KEK-TH-1013) is explained in this talk.The recent HKN report (KEK-TH-1013) is explained in this talk.

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.

NLO Determination ofNLO Determination ofNuclear Parton Distribution FunctionNuclear Parton Distribution Function

ss

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

arXiv:0709.3038 [hep-ph]Phys. Rev. C 76 (2007) 065207

Related refs. 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

New pointsNew points

(1) Both LO and NLO global analyses(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 LOEstimation 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(2) Discussions on deuteron modifications Comparison with FComparison with F22

DD/F/F22pp data 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: 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) σDYA / σDYA’ 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 If there were no nuclear modificationmodification

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+ 1−1A

⎛⎝⎜

⎞⎠⎟

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⎡

⎣⎢⎤⎦⎥∫

Baryonnumber: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

Analysis conditionsAnalysis conditions

· Nucleonic PDFs:MRST98 [ QCD = 174 MeV (LO), 300 MeV (NLO) ]

· Total number of data： 1241 ( Q2≧1 GeV2 )

896 (F2A/F2

D) + 293 (F2A/F2

A´) + 52 (Drell-Yan)

· Subroutine for 2 analysis： CERN-Minuit

2min ( /d.o.f.) = 1653.3 (1.35) ….. LO = 1485.9 (1.21) ….. NLO

2 =Ri

data − Ritheo( )

2

σ idata( )

2i

∑σ i

data = σ isys

( )2

+ σ istat

( )2

R =F2

A

F2D ,

F2A

F2′A ,

σ pA

σ p ′A

· Total number of parameter：12

· Error estimate： Hessian method

δF(x)[ ]2

= Δχ 2 ∂F(x)

∂ξ ii, j∑ H ij

−1 ∂F(x)

∂ξ j

H ij =Hessian

ξi =parameter

22 values in LO and NLO values in LO and NLONLO improvementNLO improvement

NLO disimprovementNLO disimprovement

NLO improvements mainly in light NLO improvements mainly in light nuclei;nuclei;however, disimprovements for however, disimprovements for Drell-Yan data.Drell-Yan data.

Total Total 22 improvements improvements in NLO.in NLO.

0.7

0.8

0.9

1

1.1

1.2

0.03 0.1 1

x

772E

Q

2

=50GeV

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 datadata

(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

Comparison with FComparison with F22AA/F/F22

DD data: data: Light Light nucleinuclei

J JJ

J

JJ

J

JJJ

J

JJJJJJJ J

JJ

JJJJJJ

J

JJ

JJ

JJ

J

J

JJJJJ

J

JJJ

J

JJ

J

J

JJJJJJJJJJ

JJ

J

JJ

JJJJJJJJJJ

J

JJJJ

JJJJ

JJJJJ

J

J J

J

JJ

JJJ

JJJJJ

J

J

JJJ

J

J

J

J

J

JJJ

JJJJ

J

J

JJ

J

J

J

J

J

JJ

J

JJJJ

J

J

JJJ

J

J

JJJJJJ

JJ

J

JJJ

JJ

J

J

JJJ

J

JJ

J

J

J

J

JJ

JJ

J

J

J

J

J

J

JJ

JJ

JJ

J

J

JJ

J

J

J

J

JJJ

JJ

J

J

J

J

JJ

J

J

J

J

JJ

J

J

J

J

J

JJ

J

J

J

JJ

J

J

J

J

J

JJJJ

J

J

J

J

J

JJ

J

J

J

J

JJ

J

JJ

J

J

JJ

J

J

J

J

J

J

J

J

J

J

J

J

J

J

J

J

J

J

JJ

J

J

J

J

J

JJ J J J

JJ J

J

J

J

-0.2

0

0.2

J NMC

H

HH

HH

H

HH

HH

H

H

H

H

H

H

H

-0.2

0

0.2

0.001 0.01 0.1 1

x

H E139

J

J

J

J J JJ J

J

J

J

J

J J

J

J

J

H

H

H

H

H HH

H HHHH

HHHH

H

H

-0.2

0

0.2

J NMC

H E139

JJ

J JJ

J

J

JJ J J J J

J

J

J

J

-0.2

0

0.2

J NMC

He/D

Be/D

Li/D

D/p

H

HH H H

HH

FF

F

F

F

-0.2

0

0.2

EMC NMC H E139 F E665

-0.2

0

0.2

BCDMS

HERMES

-0.2

0

0.2

E139 E49

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

Ca/D

Al/D

N/D

C/D

Comparison with FComparison with F22AA/F/F22

DD data: data: Heavy Heavy nucleinuclei

Ç

Ç

ÇÇ Ç

Ç ÇÇ Ç

Ç

Ç

Ç

Ç

Ç

H

HH

HH HH

H

HH

H

HH

H

H

H

H

H

HHHH

H

ÑÑÑÑ

ÑÑÑ Ñ

ÑÑ

-0.2

0

0.2

BCDMS Ç E87 H E139 Ñ E140

-0.2

0

0.2

EMC

-0.2

0

0.2

HERMES

-0.2

0

0.2

0.001 0.01 0.1 1

x

E139

Cu/D

Kr/D

Ag/D

Fe/D

-0.2

0

0.2

EMC

-0.2

0

0.2

E665

Ñ

-0.2

0

0.2

E139 Ñ E140

-0.2

0

0.2

0.001 0.01 0.1 1

x

E665

Xe/D

Au/D

Pb/D

Sn/D

QQ22 dependence dependence

0.75

0.8

0.85

0.9

0.95

0.8

0.85

0.9

0.95

1

x = 0.01

0.95

1

1.05

1.1

1.15

1 10 100

0.8

0.85

0.9

0.95

1

1 10 100

x = 0.1 x = 0.7

Q

2

(GeV

2

)Q

2

(GeV

2

)

LO

x = 0.001

NLO

Only NLO uncertainty bands are Only NLO uncertainty bands are shown.shown.

0.8

1

1.2

x = 0.03 5 x = 0.04 5

HERMES

x = 0.05 5

0.8

1

1.2

x = 0.07

x = 0.09x = 0.12 5

0.8

1

1.2

1 10

x = 0.17 5

1 10

x = 0.2 5

1 10

x = 0.3 5

Q

2

( GeV

2

)

The differences between LO and NLO The differences between LO and NLO become obvious only at small become obvious only at small xx..

• 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.

Scaling Violation and Gluon DistributionsScaling Violation and Gluon Distributions

at small x

∂F2

∂ lnQ2( )≈20s

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.

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

PDFs in PDFs in 4040Ca and Ca and uncertaintiesuncertainties• • Some NLO improvements, but not Some NLO improvements, but not significant ones.significant ones.

• • Impossible to determine gluon Impossible to determine gluon modifications.modifications.

• • Antiquark distributions are not Antiquark distributions are not determined at large determined at large xx..

• • Flavor separation is needed for Flavor separation is needed for antiquarks antiquarks factoryfactory

• • Confirmation of valence Confirmation of valence modifications at small modifications at small xx factoryfactory

0.4

0.6

0.8

1

1.2

x

LO

NLO

uv

Q 2

= 1 GeV 2

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

Summary on nuclear-PDF Summary on nuclear-PDF determination in NLOdetermination in NLOLO 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.

ExtraExtra

Nuclear corrections in iron (A=56, Z=26)Nuclear corrections in iron (A=56, Z=26)KP (Kulagin, Petti)KP (Kulagin, Petti)

SumamrySumamryNuclear 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 −sAsymmetry

• Nucleon does not have net strangeness: dx s(x) − s (x)[ ]0

1

∫ = 0. However, it does not mean s(x) = s (x). → could be s(x) ≠ s (x)

• If s and s are created perturbatively, they should be equal s(x) = s (x).

• Hadron models predict the asymmetry: s(x) ≠ s (x).

Motivations for s(x)−s(x)

p(uud)→ KY[K +(us)(uds), K +(us)Σ0 (uds), K 0 (ds)Σ+(uus),⋅⋅⋅]

1 / mK+ =1 / 494MeV

=0.40fm1 / m =1 /1116MeV=0.18fm

• The asymmetry could be important for NuTeV anomaly.

0.18 fm0.40 fm

ss

x

s

s

s+ =r(u+d)

this analysis

CTEQ, (F. Olness et al., Eur. Phys. J. C40 (2005) 145) H.-L. Lai et al., JHEP 04 (2007) 089.

Global analysis for s(x) and s (x)

The 2007 paper includes the final NuTeV data for dimuons.

First, s+ =s+ s

s+(x,Q02 ) =A0x

A1 (1−x)A2 P+(x), P+(x) =eA3 x+A4x+A5x2

2 reduction with respect to CTEQ6.5 with s+ = r(u + d )

g Strange shape, s+(x), issignificantlydifferentfromnonstrangeu(x) +d(x).gNosignificantimprovementfromtheparametersA3, A4 , A5 .

gImprovementismainlyfrom-induceddimuondata(−Δμ+μ−2 =46).

Theotherdataarenotmuchsensitivetothestrangedistributions(−Δglobal2 =65).

Analysis for s−(x) =s(x)−s(x)

s−(x,Q02 ) =s+(x,Q0

2 )2π

tan−1 cxa 1−xb

⎛⎝⎜

⎞⎠⎟edx+ex2⎡

⎣⎢

⎤

⎦⎥

g 2reductionisinsignificant.gNosignificantimprovementfromtheparametersd, e..

− 0.001 < x s−< 0.005

0.018 < x s+< 0.040( )

best fit

x s−=0.005

x s−=−0.001

s(x)−s(x)cannotbedeterminedatthisstage.

NuTeV analysiss+(x,Q0

2 ) =κ +xγ+(1−x)

+u(x,Q0

2 ) +d(x,Q02 )⎡⎣ ⎤⎦

s−(x,Q02 ) =s+(x,Q0

2 )tan−1 κ −xγ−(1−x)

−1−

xx0

⎛

⎝⎜⎞

⎠⎟⎡

⎣⎢

⎤

⎦⎥

Consistent with CTEQ 2007

−0.001 < x s− < 0.005

x s−=0.00196 ±0.0046(stat)

±0.0045(syst) ±0.00119(external)Q2 =16GeV2

C. Bourrely et al., PLB 648 (2007) 39

x s−=−0.00194

The NuTeV result is not much different from CTEQ onealthough it used to differ in hep-ex/0405037.

Global analysis for determiningGlobal analysis for determiningfragmentation functionsfragmentation functionsand their uncertaintiesand their uncertainties

Shunzo KumanoShunzo KumanoHigh Energy Accelerator Research Organization High Energy Accelerator Research Organization

(KEK)(KEK)Graduate University for Advanced Studies (GUAS)Graduate University for Advanced Studies (GUAS)shunzo.kumano@kek.jpshunzo.kumano@kek.jp

http://research.kek.jp/people/kumahttp://research.kek.jp/people/kumanos/nos/

with M. Hirai (TokyoTech), T.-H. Nagai (GUAS), K. Sudoh (KEK)with M. Hirai (TokyoTech), T.-H. Nagai (GUAS), K. Sudoh (KEK)

Reference: Reference: Phys. Rev. D75 (2007) 094009.

ContentsContents

(1) Introduction to fragmentation functions (FFs)

Definition of FFsMotivation for determining FFs

(2) Determination of FFs Analysis methodResultsComparison with other parameterizations

(3) Summary

IntroductionIntroduction

Fragmentation FunctionFragmentation Function

Fragmentation function is defined by

e+

e–

γ, Z

q

q

h

Fragmentation: hadron production from a quark, antiquark, or gluon

Fh (z,Q2 ) =1

σ tot

dσ(e+e−→ hX)dz

σ tot =totalhadroniccrosssection

z ≡Eh

s / 2=2Eh

Q=

Eh

Eq, s=Q2

Variable Variable zz• • Hadron energy / Beam energyHadron energy / Beam energy• • Hadron energy / Primary quark energyHadron energy / Primary quark energy

A fragmentation process occurs from quarks, antiquarks, and gluons,A fragmentation process occurs from quarks, antiquarks, and gluons,so that so that FFhh is expressed by their individual contributions: is expressed by their individual contributions:

F h(z,Q2 ) =

dyyz

1∫

i∑ Ci

zy,Q2⎛

⎝⎜⎞

⎠⎟Di

h(y,Q2 )

Ci (z,Q2 ) =coefficientfunction

Dih(z,Q2 ) =fragmentationfunctionofhadronhfromapartoni

Calculated in perturbative QCDCalculated in perturbative QCD

Non-perturbative (determined from experiments)

Momentum (energy) sum Momentum (energy) sum rulerule

Dih z,Q2( ) =probabilitytofindthehadronhfromapartoni

withtheenergyfractionz

Energy conservation: dz z

0

1

∫

h∑ Di

h z,Q2( ) =1

h =π + ,π 0 ,π −,K + ,K 0 ,K 0 ,K −,p,p,n,n,⋅⋅⋅

Simple quark model: π +(ud),K +(us),p(uud),⋅⋅⋅

Favored fragmentation: Duπ+

,Ddπ+

,...

(fromaquarkwhichexistsinanaivequarkmodel)

Disfavoredfragmentation:Ddπ+

,Duπ+

,Dsπ+

,...

(fromaquarkwhichdoesnotexistinanaivequarkmodel)

Favored and disfavored fragmentation Favored and disfavored fragmentation functionsfunctions

Nulceonic PDFs Polarized PDFs Nuclear PDFs FFs

Determination **** ** ** **Uncertainty ÅZ ÅZ ÅZ Å~

Comments

Accuratedeterminationfrom small x tolarge x

Gluon &antiquarkpolarization?Flavorseparation?

Gluon?Antiquark atmedium x?Flavorseparation?

LargedifferencesbetweenKretzer and KKP(AKK)

Status of determining Status of determining fragmentation functionsfragmentation functions

Uncertainty ranges of determined fragmentation funUncertainty ranges of determined fragmentation functionsctionswere not estimated, although there are such studiewere not estimated, although there are such studies in s in nucleonic and nuclear PDFs.nucleonic and nuclear PDFs.The large differences indicate thatThe large differences indicate that

the determined FFs have much ambiguitthe determined FFs have much ambiguities.ies.

Parton Distribution Functions (PDFs), Fragmentation FunctionParton Distribution Functions (PDFs), Fragmentation Functions (FFs)s (FFs)

Situation of fragmentation functionsSituation of fragmentation functionsThere 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.

-0.5

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1

z

gluon

Q2 = 2 GeV2

-0.5

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1

z

u quark

Q2 = 2 GeV2

KKPAKK Kretzer

-0.5

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1

z

Q2 = 2 GeV2

s quark

zDu(π+ +π−)/2 (z)

zDs(π+ +π−)/2 (z) zDg

(π+ +π−)/2 (z)

Purposes of investigating fragmentation functionsPurposes of investigating fragmentation functions

Semi-inclusive reactions have been used for investigating

・ origin of proton spin

re +

rp→ ′e +h+ X(e.g.HERMES),

rp+

rp→ h+ X(RHIC-Spin)

A + ′A → h+ X(RHIC,LHC)・ properties of quark-hadron matters

Quark, antiquark, and gluon contributions to proton spin

(flavor separation, gluon polarization)

Nuclear modification

(recombination, energy loss, …)

σ = fa(xa,Q2 )⊗ fb(xb,Q

2 )a,b,c∑

⊗ σ (ab→ cX)⊗ Dcπ (z,Q2 )

A code for calculating the FFs is available atA code for calculating the FFs is available athttp://research.kek.jp/people/kumanos/ffs.htmlhttp://research.kek.jp/people/kumanos/ffs.html

Determination of fragmentation functionDetermination of fragmentation functionand their uncertaintiesand their uncertainties

M. Hirai, SK, T.-H. Nagai, K. SudohM. Hirai, SK, T.-H. Nagai, K. SudohPhys. Rev. D75 (2007) 094009.

Determination ofDetermination of

Fragmentation Fragmentation FunctionsFunctions

New aspectsNew aspects in our in our analysisanalysis • • Determination of fragmentation functions (FFs) andDetermination of fragmentation functions (FFs) and their uncertainties their uncertainties in LO and NLO.in LO and NLO.

• • Discuss NLO improvement in comparison with LODiscuss NLO improvement in comparison with LO by considering the uncertainties.by considering the uncertainties. (Namely, roles of NLO terms in the determination (Namely, roles of NLO terms in the determination of FFs)of FFs)

• • Comparison with other parametrizationsComparison with other parametrizations

• • Avoid assumptions on parameters as much as we can,Avoid assumptions on parameters as much as we can, Avoid contradiction to the momentum sum ruleAvoid contradiction to the momentum sum rule

• • SLD (2004) data are included.SLD (2004) data are included.

HKNS (Ours) Kretzer KKP (AKK)

Function form

# 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 Q0

2

(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

N iπ +

z iπ+

(1−z)βiπ+

N iπ +

z iπ+

(1−z)βiπ+

N iπ±

z iπ±

(1−z)βiπ±

Duπ +

=(1−z)Duπ +

Mg =Mu + Mu

2

M i

h ≡ zDih(z,Q2 )

0.05

1∫ dz

Comparison with other Comparison with other analysesanalyses

Initial functions for pionInitial functions for pion

Duπ+

(z,Q02 ) =Nu

π+zu

π+(1−z)βu

π+=Dd

π+(z,Q0

2 )

Duπ+

(z,Q02 ) =Nu

π+zu

π+(1−z)βu

π+=Dd

π+(z,Q0

2 ) =Dsπ+

(z,Q02 ) =Ds

π+(z,Q0

2 )

Dcπ+

(z,mc2 ) =Nc

π+zc

π+(1−z)βc

π+=Dc

π+(z,mc

2 )

Dbπ+

(z,mb2 ) =Nb

π+zb

π+

(1−z)βbπ+

=Dbπ+

(z,mb2 )

Dgπ+

(z,Q02 )=Ng

π+zgπ+

(1−z)βgπ+

Dq

π−=Dq

π+

nf=

3, μ02 <Q2 < mc

2

4, mc2 <Q2 < mb

2

5, mb2 <Q2 < mt

2

6, mt2 <Q2

⎧

⎨

⎪⎪

⎩

⎪⎪

N =

MB( + 2,β +1)

, M ≡ zD(z)dz(2ndmoment)0

1

∫ ,B( + 2,β +1) =betafunction

Constraint: 2nd moment should be finite and less than 1

0 < Mi

h <1becauseofthesumrule

h∑Mi

h =1

Note: constituent-quark composition π + =ud, π - =ud

Experimental data for pionExperimental data for pion

# of data

TASSOTCPHRSTOPAZSLDSLD [light quark]SLD [ c quark]SLD [ b quark]ALEPHOPALDELPHIDELPHI [light quark]DELPHI [ b quark]

12,14,22,30,34,44292958

91.2

91.291.291.2

291824

292929292222171717

s (GeV)

Total number of data ： 264

0

20

40

60

80

100

0 0.2 0.4 0.6 0.8 1

z

TASSO

TPC

HRS

TOPAZ

SLD

ALEPH

OPAL

DELPHI

AnalysisAnalysis

Initial scale: Q02 =1GeV 2

Scaleparameter:QCD

nf =4 =0.220(LO), 0.323(NLO)

svarieswithnf

Heavy-quarkmasses:mc =1.43GeV,mb =4.3GeV

2 /d.o.f. =1.81(LO), 1.73 (NLO)

Δ 2 ≡ 2(a+δa) − 2(a) = Hijδaii, j∑ δaj , Hij =

∂2 2(a)∂ai∂aj

δD(z)⎡⎣ ⎤⎦2=Δ 2 ∂D(z, a)

∂ai

Hij−1 ∂D(z, a)

∂aji, j∑

Uncertainty estimation: Hessian method

Results for the pion

Comparison with pion dataComparison with pion data

1E-3

1E-2

1E-1

1E+0

1E+1

1E+2

1E+3

0 0.2 0.4 0.6 0.8 1

z

SLD

ALEPH

OPAL

DELPHI

Q = MZ

-1-0.5

00.5

1

0 0.2 0.4 0.6 0.8 1

z

Fπ±(z,Q2 ) =

1σ tot

dσ(e+e−→ π ±X)dz

Our NLO fitwith uncertainties

Fπ±(z,Q2 )data−Fπ±

(z,Q2 )theory

Fπ±(z,Q2 )theory

Rational difference between data and theory

Our fit is successful to reproduce the pion data.

The DELPHI data deviate from our fit at large z.

Comparison with pion data: (data-theory)/theoryComparison with pion data: (data-theory)/theory

-1

-0.5

0

0.5

1SLD

ALEPH

OPAL DELPHI

-1

-0.5

0

0.5

1

TPC HRS

-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1

z

-1

-0.5

0

0.5

1TASSO Q=12 GeV 14 GeV 22 GeV

Q = 29 GeV

Q = Mz

Q = Mz

Charm quark

-1

-0.5

0

0.5

1TASSO Q=34 GeV 44 GeV

-1

-0.5

0

0.5

1

TOPAZ

-1

-0.5

0

0.5

1SLD DELPHI

-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1

z

Q = Mz

Q = Mz

Q = 58 GeV

Light quark (u, d, s)

Bottom quark

Determined fragmentation functions for pionDetermined fragmentation functions for pion

-1.5-1

-0.50

0.51

1.52

2.5

-1-0.5

00.51

1.52

2.53

-1-0.5

00.51

1.52

2.53

-1-0.5

00.51

1.52

2.53

0 0.2 0.4 0.6 0.8 1z

-1-0.5

00.51

1.52

2.53

0 0.2 0.4 0.6 0.8 1z

gluon

u quark

c quark b quark

=1Q GeV

=1.43Q GeV =4.3Q GeV

=1Q GeV =1Q GeV

LO

NLO

u quark

• 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 dataComparison with kaon data

-1

-0.5

0

0.5

1SLD

ALEPH

OPAL DELPHI

-1

-0.5

0

0.5

1TPC HRS

-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1

z

HH

-1

0

1

2

=29Q GeV

=Q Mz

=Q Mz

Charm quark

-1012

=12TASSO Q GeV

14GeV

22GeV

-1

0

1

2TASSO Q=34 GeV

-1

-0.5

0

0.5

1

TOPAZ

-1

-0.5

0

0.5

1SLD DELPHI

[[

[[

-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1

z

Q = Mz

Q = Mz

Q = 58 GeV

Light quark (u, d, s)

Bottom quark

Determined functions for kaonDetermined functions for kaon

-0.3

0

0.3

0.6

-1

-0.5

0

0.5

1

1.5

-1

-0.5

0

0.5

1

1.5

-0.3

0

0.3

0.6

0 0.2 0.4 0.6 0.8 1

z

-0.3

0

0.3

0.6

0 0.2 0.4 0.6 0.8 1

z

gluon

u quark

c quark b quark

Q = 1 GeV

Q = 1.43 GeV Q = 4.3 GeV

-0.3

0

0.3

0.6

Q = 1 GeV

Q = 1 GeV Q = 1 GeV

LO

NLO

u quark

s quark

• Gluon and light-quark fragmentation functions have large uncertainties. • Uncertainty bands become smaller in NLO in comparison with LO.

The situation is similar to the pion functions.

• Heavy-quark functions are relatively well determined.

Comparison with other parametrizations in pionComparison with other parametrizations in pion

-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=2GeV2

Q2=2GeV2 Q2=2GeV2

Q2=10GeV2 Q2=100GeV2

KKPAKK Kretzer

HKNS

s quark

• 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 agreein charm and bottom functions.

(KKP) Kniehl, Kramer, Pötter(AKK) Albino, Kniehl, Kramer(HKNS) Hirai, Kumano, Nagai, Sudoh

Comparison with other parametrizations in kaon and protonComparison with other parametrizations in kaon and proton

-0.2

0

0.2

0.4

-0.2

0

0.2

0.4

-0.2

0

0.2

0.4

0 0.2 0.4 0.6 0.8 1

z

-0.2

0

0.2

0.4

0 0.2 0.4 0.6 0.8 1

z

-0.2

0

0.2

0.4-0.2

0

0.2

0.4gluon

u quark

c quark b quark

Q2 = 2 GeV2 Q2 = 2 GeV2

Q2 = 2 GeV2 Q2 = 2 GeV2

Q2 = 10 GeV2 Q2 = 100 GeV2

KKP

AKK Kretzer

HKNS

d quark

s quark

-0.1

0

0.1

0.2-0.1

0

0.1

0.2

-0.1

0

0.1

0.2-0.1

0

0.1

0.2

-0.1

0

0.1

0.2

0 0.2 0.4 0.6 0.8 1

z

-0.1

0

0.1

0.2

0 0.2 0.4 0.6 0.8 1

z

gluon

u quark

c quark b quark

Q2 = 2 GeV2

Q2 = 2 GeV2

Q2 = 2 GeV2

Q2 = 10 GeV2

d quark

Q2 = 2 GeV2

Q2 = 100 GeV2

s quark

KKP

AKK

HKNS

kaonkaon protonproton

Comments on “low-energy” experiments, Belle & BaBarComments on “low-energy” experiments, Belle & BaBar

Gluon fragmentation function is very important for Gluon fragmentation function is very important for hadron production at small phadron production at small pTT at RHIC (heavy ion, sp at RHIC (heavy ion, spin) and LHC, in) and LHC, (see the next transparency)(see the next transparency)and it is “not determined” as shown in this analysiand it is “not determined” as shown in this analysis.s. Need to determine it accurately.Need to determine it accurately. Gluon function is a NLO effect with the coefficieGluon function is a NLO effect with the coefficientnt function and in Qfunction and in Q22 evolution. evolution.

We have precise data such as the SLD ones at Q=Mz,We have precise data such as the SLD ones at Q=Mz,so that so that accurate small-Qaccurate small-Q22 data are needed for probin data are needed for probinggthe Qthe Q22 evolution, namely the gluon fragmentation fun evolution, namely the gluon fragmentation functions.ctions.(Belle, BaBar ?)(Belle, BaBar ?)

Pion production at RHIC: p + pPion production at RHIC: p + p ππ 00 + X + X

S. S. Adler et al. (PHENIX), PRL 91 (2003) 241803

• 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

s =200GeV pTp p

π

Blue band indicates the scale uncertaintyby taking Q=2pT and pT/2.

SummarySummaryDetermination of the optimum fragmentation functions for π, 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 http://research.kek.jp/people/kumanos/ffs.html .

rp +

rp→ π 0 + X, A+ ′A → h+ X(RHIC,LHC),HERMES,JLab,...

The End

The End