Nuclear Modifications of Parton Distribution Functions
description
Transcript of Nuclear Modifications of Parton Distribution Functions
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
[email protected]@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)[email protected]@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