Introduction to Parton Distribution Functions in the Nucleon and Nuclei
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Transcript of Introduction to Parton Distribution Functions in the Nucleon and Nuclei
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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
[email protected]@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/
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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
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IntroductionIntroduction
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http://th.physik.uni-frankfurt.de/~drescher/SENECA/
Typical Air Shower Model (SENECA)
My talk is on this “hard” part.
Soft interactionsSoft interactions
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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.
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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.
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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
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Deep Inelastic Deep Inelastic
Lepton-Nucleon ScatteringLepton-Nucleon Scattering
andand
Parton PictureParton Picture
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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
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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
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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
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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).
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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)
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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”.
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Parton Distribution FunctionsParton Distribution Functions
in the Nucleonin the Nucleon
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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/
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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
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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.
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Available data for determining PDFs (Ref. MRST, hep/ph-9803445)
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€
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.
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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.
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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.
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Global Analysis for PDFsGlobal Analysis for PDFs
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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
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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
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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.
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Comments onComments onHigher-Twist EffectsHigher-Twist Effects
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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.
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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
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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
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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.
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Nuclear Nuclear
Parton Distribution FunctionsParton Distribution Functions
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Brief Introduction toBrief Introduction toNuclear Modifications ofNuclear Modifications of
Parton Distribution FunctionsParton Distribution Functions
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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)
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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
Quark (a) Quark (a)
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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−
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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
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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
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Determination of Nuclear Determination of Nuclear Parton Distribution FunctionsParton Distribution Functions
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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
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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
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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.
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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
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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
)
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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 ))
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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.
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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
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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
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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..
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Comparison withComparison withOther Global AnalysesOther Global Analyses
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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
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Comparison of used data setComparison of used data setHKN data table
data in EKRS data in DS
data not included
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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
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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
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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
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Analysis conditionsAnalysis conditions
KKP forKKP for D Diihh
15 parameters15 parameters
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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
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DataData
Note: very large weightNote: very large weight for BRAHMS for BRAHMS
weightweight
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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!
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-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)
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Why is it so difficultWhy is it so difficultto determine to determine
nuclear gluon distributions?nuclear gluon distributions?
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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
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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.
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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
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The End
The End