![Page 1: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/1.jpg)
QCD analysis of the nucleon spin structure function data in next to
leading order in αS
The polarized gluon distribution
in the nucleon
Jechiel LichtenstadtTel Aviv University
![Page 2: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/2.jpg)
World data of nucleon spin structure functionsQ2 evolution with DGLAP equationsResultsContribution of the polarized gluon distributionDirect measurements of the polarized gluon distribution Prospective future measurements of the spin structure functionsConclusions
Outline
![Page 3: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/3.jpg)
History
SLAC-YALE E80/E130
G. Baum et al. 1983
0.2 < x < 0.7EMC 1988
J. Ashman et al; 198
0.01 < x < 0.7
Determine the first moment :
xxg d)(1
0 1p
1
![Page 4: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/4.jpg)
World data of theSpin Structure Functions
Proton:EMC, SMC, SLAC-E143, E154, HERMES, JLAB-CLAS
Deuteron: SMC, SLAC-E143, E155, HERMES, JLAB-CLAS
Neutron: SLAC-E142, HERMES,JLAB-MIT
![Page 5: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/5.jpg)
Spin structure function data
![Page 6: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/6.jpg)
QCD Evolution - DGLAP equations
• In the parton model, the spin structure function g1 is related to the polarized quark and gluon distributions in the nucleon by an integral equation involving quark and gluon coefficient functions Ci
q and Cg
(Dokshitzer, Gribov, Lipatov, Altarelli and Parisi)
![Page 7: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/7.jpg)
QCD Evolution - DGLAP equations (Cont.)
)],())(,( ),())(,(2
),())(,([d
2
1),(
1
1
2
1
tyqty
xtyGt
y
xCn
tyty
x
y
ytx
NS
s
NS
qsgf
s
S
qx
n
k f
k
C
Cnef
g
)],(),([),(1
txqtxqtx i
n
ii
f
Where: t = ln(Q2/Λ2), and Λ is a scale parameter
ΔG(x) - the polarized gluon distribution
(singlet)
(non-singlet)
),(),(),(1 1
22
1
1
2
txqtxqn
ee
n
etxq ii
n
i
n
k f
ki
n
k f
kNSf ff
![Page 8: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/8.jpg)
QCD analysis- DGLAP equations (Cont.)
• Parameterization at initial scale (1 (GeV/c)2 )
∆f(x)= η N(αf,βf,af,bf) xα (1-x)β(1+ax+bx1/2)
for ΔΣ(x), ΔG(x), ∆qpNS(x), ∆qn
NS(x)
• Normalization ∫Nxα(1-x)β(1+ax+bx1/2)dx =1 η the first moment of the parton distribution
• Fit procedure: Set parameters at initial scale Evolve parton distributions –using the DGLAP equations and calculate g1(x,Q2) for each data Modify parameters and obtain best fit
![Page 9: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/9.jpg)
QCD analysis- DGLAP equations (Cont.)
QCD analysis in the Adler-Bardeen scheme:First moment of Cg -
(G. Altarelli, R. Ball, DS. Forte and G. Ridolfi Nucl. Phys. B 496 (1997) 337.)
Compare also to the scheme – C1g = 0.
The first moments of the singlet distributions differ by:
(a0 – the singlet axial current matrix element)
MS
4
1 s
gC
)(2
)()( 2
22
0 QGQ
nQa sfABMS
![Page 10: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/10.jpg)
QCD analysis in NLO- results
• Fit to data (shown at measured Q2)
![Page 11: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/11.jpg)
QCD analysis in NLO - results
• 12 parameters• 286 data 2=251
(C.L.=84%)
at Q2=1 (GeV/c)2
stat.)(62.0d)( 13.012.0
1
0
xxGG
![Page 12: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/12.jpg)
1) Open charm production:
q = c
charm fragmentation:- D0, D* (60%)
- D+ (20%)
- DS+, ΛC
+ (10% each)
2) High-Pt hadron pair production:
q=u,d,s
ΔG/G from photon gluon fusion
![Page 13: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/13.jpg)
Measurements of ΔG/G
• Determination of ΔG/G from high PT
hadron pairs
-
PGF QCD Compton Leading order DIS
![Page 14: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/14.jpg)
Measurements of the gluon polarization ∆G/G(x)
Photoproduction of High PT hadron pairs
SMC: Phys. Rev C D70 (2004) 012002 ∆G/G(xg=0.07; μ2=3 GeV2) =-0.20 ± 0.28 ± 0.10
HERMES: Phys. Rev. Lett. 84 (2000) 2584
∆G/G(xg=0.17; μ2=2 GeV2) =0.41 ± 0.18 ± 0.03
![Page 15: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/15.jpg)
Determination of ΔG/G from open charm production
D0 K-+BR = 4%
D*+ Ds+ ~ 30%(*)
K-+s+
c
c
COMPASS at CERN
![Page 16: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/16.jpg)
Measurements of the gluon polarization ∆G/G(x) (Cont.)
• COMPASS: High PT (low Q2) ∆G/G(xg=0.085; μ2=3 GeV2) =0.016 ± 0.058 ±
0.055
High PT (high Q2) ∆G/G(xg=0.13; μ2=3 GeV2) =0.06 ± 0.31 ± 0.06
Open Charm ∆G/G(xg=0.15; μ2=13 GeV2) =-0.57 ± 0.41 ± 0.17 Use CTEQ parameterization of G(x) to obtain ∆G(x)
![Page 17: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/17.jpg)
Comparison of best-fit polarized gluon distribution
with G measurements
Best fit compatible with presently measured G values
Including G data in fit has a very small effect.
G = 0.57 0.11 (stat.)
![Page 18: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/18.jpg)
x dependence of the polarized gluon distribution
• Fit data with 11 parameters: no mid-range node
• 286 data 2=261 (C.L.=72%) G = 0.480.14
• Not quite compatible with measured G• (C.L. -> 67%)• Additional measurements of G are essential
![Page 19: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/19.jpg)
PHENIX at RHICAsymmetry in 0 production
from p+p collisions
GeVsatpp 200
Calculations: NLO with
pQCD
Glück, Reya, Stratmann and
Vogelsang
0.02 < Xg< 0.3 for each PT bin.
![Page 20: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/20.jpg)
Sensitivity to G x=(0.02-0.3)
• Compare data with LL(pT) calculated
with different G(x) and
3.0
02.0
)3.002.0( d)( xxGG xGRSV
![Page 21: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/21.jpg)
ALL( 0) with G from best-fit to world g1 data
Caculation by W. Vogelsang
![Page 22: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/22.jpg)
Systematic errors
Preliminary (rough) estimates
• Systematic errors in g1 data ~ 0.10
• Factorization and renormalization scales ~ 0.05 • Functional form ~ 0.15
• Total estimate ~ 0.20
![Page 23: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/23.jpg)
Measurements with a future Electron Ion Collider
• Extend kinematic range:
• High luminosity
• Ee = 10 GeV
• Ep= 250 GeV
• ∫L = 2fb-1
GeVs 100
![Page 24: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/24.jpg)
Projected EIC data in QCD analysis
• Use best-fit distributions to calculate asymmetries over EIC
kinematic region• Randomize data • Repeat QCD analysis with projected data
G = 0.64 0.05 (stat.)
![Page 25: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/25.jpg)
CONCLUSIONS (I)
• New measurements of the polarized gluon distribution are compatible with g1 data.
• Need more g1 data at low x (proton, deuteron).
• Need more precise G data to determine the x dependence of the gluon distribution
(Expected in near future)
![Page 26: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/26.jpg)
CONCLUSIONS (II)• Determine the axial vector coupling
constant – The Bjorken sum-rule.
with
Note: The non-singlet distribution evolves independently! (Measurements at same (x,Q2)
),())(,(
d
2
1),(),( 1
122
12
1 tyqty
xC
y
yeQxgQxg NSs
NS
x
np
)1()1(2
3),( 2/12 bxaxxNx
g
gQxq
V
ANS
![Page 27: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/27.jpg)
CONCLUSIONS (III)
Remaining tasks:• Evaluate systematic error. Future tasks• Introduce model-independent SF’s
(Fourier-Bessel, Sum of Gaussians) to get a realistic estimate on the uncertainty (G is not determined at low x)
• Incorporate RHIC measurement (G(x:0.02-
0.3)) in the analysis (iterative procedure)• Fit strong coupling constant s
![Page 28: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/28.jpg)
The last slide
• Thanks to many people who were involved and contributed to this work:
• A. Deshpande • W. Vogelsang• R. Windmolders
![Page 29: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/29.jpg)
History – EMC 1988• Measurement of the proton spin structure
function and determination of its first moment: Γ= ∫g1
P(x)dx (Ellis and Jaffe Sum
rule)
0.01 ≤ X ≤ 0.7
![Page 30: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/30.jpg)
The t dependence of the distributions follows the DGLAP equations:Gluon and singlet:
Note: The non-singlet evolves independently of the singlet and the gluon distributions:
The coefficient functions Ciq, Cg, and splitting functions Pi,j can be
expanded in powers of αs .
Use next to leading order (NLO) coefficient functions.
QCD Evolution - DGLAP equations (Cont.)
![Page 31: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/31.jpg)
QCD Evolution - DGLAP equations (Cont.)
The t dependence of the distributions follows the DGLAP equations:Singlet and gluon distributions:
t)G(y,))(,(),())(,(d
2),(G
d
d
t)G(y,))(,(2),())(,(d
2),(
d
d
1
1
ty
xPtyt
y
xP
y
ytx
t
ty
xPntyt
y
xP
y
ytx
t
sggsSgq
x
s
sqgfsSqq
x
s
),())(,(
d
2),(
d
d 1
tyqty
xP
y
ytxq
tNS
sNSqq
x
sNS
Note: The non-singlet evolves independently of the singlet and the gluon distributions:
The coefficient functions Ciq, Cg, and splitting functions Pi,j can be expanded in
powers of αs .
Use next to leading order (NLO) coefficient functions.
![Page 32: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/32.jpg)
Spin structure function data- deuteron
![Page 33: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/33.jpg)
Spin structure function data - proton
![Page 34: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/34.jpg)
Spin structure function data- deuteron (301)
![Page 35: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/35.jpg)
Spin structure function data - neutron
![Page 36: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/36.jpg)
1) Open charm production:
q = c
charm fragmentation:- D0, D* (60%)
- D+ (20%)
- DS+, ΛC
+ (10% each)
2) High-Pt hadron pair production:
q=u,d,s
ΔG/G from photon gluon fusion
![Page 37: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/37.jpg)
![Page 38: QCD analysis of the nucleon spin structure function data in next to leading order in α S The polarized gluon distribution in the nucleon Jechiel Lichtenstadt](https://reader035.vdocuments.mx/reader035/viewer/2022062720/56649f145503460f94c298aa/html5/thumbnails/38.jpg)