1 . overview gpd and nucleon structure experiments – deeply virtual compton scattering –
DESCRIPTION
格子 QCD による核子の一般化パートン分布と クォーク角運動量の解析. 大谷 宗久 ( 杏林大学 ). 1 . overview GPD and nucleon structure Experiments – Deeply Virtual Compton Scattering – Theory – Chiral Quark Soliton Model – 2 . lattice QCD Axial FFs and quark spin Moments of GPD(vector) and total angular momentum - PowerPoint PPT PresentationTRANSCRIPT
1. overview GPD and nucleon structure Experiments – Deeply Virtual Compton Scattering –
Theory – Chiral Quark Soliton Model –
2. lattice QCD Axial FFs and quark spin Moments of GPD(vector) and total angular momentumSummary
格子格子 QCDQCD による核子の一般化パートン分布とによる核子の一般化パートン分布とクォーク角運動量の解析クォーク角運動量の解析
10 Jun. 2009@ Komaba
大谷 宗久 ( 杏林大学 )
IntroductionIntroduction
H, E, …P P'
x+ x -
momentum transfer squared:
t = (P' - P)2
longitudinal mt. transfer:
- n ·
Generalized Parton Distributions of Nucleon
GPDs encode important information on Nucleon structure !
q q'
Intrinsic quark momentum: k Impact parameter: b Momentum transfer: q
Wigner distribution in phase spaceWigner distribution in phase space
dkW(k,b)
GPD H(x,b), E(x,b),…F.T
.
GPD H(x,, t), E(x,, t),…
Wigner fn.
k dep. dist. (TMD)Sivers fn.: f1T
(x, k),Collins fn.: H1
(x, k), …
db W(k,b)
X.Ji, PRL91(2003) A.Belitsky et.al., PRD69(2004)
W(k,b) ∝ dr dq eik·r-iq·bP+q/2|†r/2r/2|Pq/2
Generalized Parton DistributionsGeneralized Parton Distributions
PDF q(x) q(x) q(x)
Fourier transf.
Quark density in b plane
q(x, b) = d2 e–i b H (x, , )
= H (x,0,0) = H (x,0,0) = HT(x,0,0) forward
limit
Form Factors
F1(t) = dx H (x, , t)
GA(t) = dx H (x, , t)
GT(t) = dx HT(x, , t)
local limit
as moments in the forward limit
J q = 1/2 dx x (H(x, , 0) + E(x, , 0)) u+d 1/2 ( A20 + B20 )
s q = 1/2 dx H(x, , 0)u+d 1/2 A10
Angular momentum
X.Ji, PRL78(1997)
GPDs
DVCS Bethe-Heitler
Deeply Virtual Compton ScatteringDeeply Virtual Compton Scattering
e -
P
FiH,E,..
F1×H, F2×E, …
Beam Charge AsymmetryBeam Charge Asymmetry HERMES coll. PRD 75 (2007) 011103
: angle btw e scat. plane & prod. plane
Longitudinal Target Spin AsymmetryLongitudinal Target Spin Asymmetry CLAS coll. PRL 97 (2006) 072002
: angle btw e - scat. plane & prod. plane
GPDs in Chiral quark soliton modelGPDs in Chiral quark soliton model M.Wakamatsu & H.Tsujimoto, PRD71(2005)
H(x, , 0) + E(x, , 0) 1/2 x (H(x, , 0) + E(x, , 0) )
K.Goeke et.al., PRC75(2007)
x dependence ca l culable no dynamical gluon
low energy scale
1. overview GPD and nucleon structure Experiments – Deeply Virtual Compton Scattering –
Theory – Chiral Quark Soliton Model –
2. lattice QCD Axial FFs and quark spin Moments of GPD and total angular momentumSummary
for UKQCD-QCDSF collab.
Moments of GPDMoments of GPD: : Generalized Form Factors Generalized Form Factors
An,2k , Bn,2k , Cn are related to P | DDn} |P'
X.Ji, J.Phys.G24(1998)1181 Polynomiality
LHPC, PRD68(2003)034505
Calculate ratio of 2pt & 3pt correlation functions on lattice
Extract GFF
Simulation parametersSimulation parameters
Nf =2 Wilson fermions w/ clover improvement
# of config: 400-2200 for each(,)
Physical unit translated by r0
c / a
O(a) improved operators
non-perturbative renormalization into MS @ = 2 GeV
1.3470.9560.670
0.0856163 32〃 〃〃〃
0.134200.135000.13550
5.20mGeVa [fm]volume
5.25
5.29
5.40
0.134600.135200.135750.136000.134000.135000.135500.135900.136200.136320.135000.135600.136100.136250.136400.13660
163 32〃
243 48〃
243 48〃〃〃
323 64
0.0794
0.0753
0.0672
1.2250.9490.6350.4571.5111.1020.8570.6290.4140.2791.1830.9170.6480.5590.451
243 48〃〃〃〃〃
323 64 0.255
V.Bernard et.al. J.Phys.G 28(2002)R1
cf. A10u-d
from expt: p + e e' + + n
t dependence of Axial Form Factort dependence of Axial Form Factor
t [GeV2]
Dipole form:
A10 22 )/1(
#
Amt
A10u+d (t) with = 5.29,= 1.3632
GFF and physical quantitiesGFF and physical quantities
A10 B10u-d
@ t = 0
Slope, mpole
Q
r 2 EM mV2
Vector Meson Dominance ?
PCAC &G-T relation
Tensor Meson Dominance ?
experiments · eN scattering
A10
u-d
gA gP
r 2 A m
A10u+d
2 s q
B10u-d
· N scattering· pion electroproduction· muon capture
A20 B20
x q 2 J q x q
mf2 , ma2 ?
· DVCS - spin asymmetry - charge asymmetry
0.402 0.024 (@ m= .14GeV)
Chiral extrapolation and quark spinChiral extrapolation and quark spin
A10u+d (t=0) 2 s u+d
= u+d
HERMES,
PRD75(2007)012007
Heavy Baryon Chiral Perturbation Theory
M.Diehl, A.Manashov, A.Schäfer, EPJ.A 29(2006)
m[GeV]
Strong m dependenceby “chiral log” term
t dependence of the 2t dependence of the 2ndnd Moments Moments
t [GeV2]
A20u-d (t), B20
u-d (t) and C20u-d (t) with = 5.29,= 1.3632
Chiral extrapolation of AChiral extrapolation of A2020u+du+d(t =0)(t =0)
A20u+d (t=0)
x u+d 0.563 0.014
(@ m= .14GeV)
M.Dorati et.al. nucl-th/0703073
m[GeV]
CTEQ6@2 = 4GeV2
Chiral extrapolation of AChiral extrapolation of A2020u-du-d(t =0)(t =0)
A20u-d (t=0)
x u-d 0.193 0.004 (@ m= .14GeV)
m[GeV]
CTEQ6 @2 = 4GeV2
Strong m dependenceby “chiral log” term
Generalized Form Factors in Chiral PerturbationGeneralized Form Factors in Chiral Perturbation M.Dorati, T.A.Gail and T.R.Hemmert, nucl-th/0703073
3 param. in each GFFs t dependence via
Dipole fit and forward limit of BDipole fit and forward limit of B2020u,du,d(t )(t )
B20q (t)
t [GeV2]
in CQSM
J u+d = 1/2
x u+d = 1 ) no dynamical gluon
2 J q x qt 0
K.Goeke et. al., PRC75(2007)in CQSM for comparison
BB2020u+du+d(( tt ) and covariantized Baryon ChPT) and covariantized Baryon ChPT
Chiral extrapolation of BChiral extrapolation of B2020u+du+d(0)(0)
B20u+d (0)
B20u+d (0) 0.120 0.023 (@ m= .14GeV)
m[GeV]
Strong m dependenceby “chiral log” term
BB2020u-du-d(t ) and covariantized Baryon ChPT(t ) and covariantized Baryon ChPT
Chiral extrapolation of BChiral extrapolation of B2020u-du-d(0)(0)
B20u-d (0)
B20u-d (0) 0.269 0.020 (@ m= .14GeV)
m[GeV]
m[GeV]
Chiral extrapolation of JChiral extrapolation of Ju u ,, JJdd
Ji’s sum rule : J q = 1/2 [ A20q (0) +B20
q(0) ]
J u 0.226 0.009 J d 0.005 0.009
(@ m= .14GeV)
decomposition of quark angular momentumdecomposition of quark angular momentum
m[GeV]
J u+d = 1/2 [ A20u+d (0) +B20
u+d(0) ]
s u+d = 1/2 A10u+d (0)
; Lu+d = J u+d s u+d
J u+d 0.222 0.014
s u+d 0.201 0.024
L u+d 0.021 0.028
(@ m= .14GeV)
Twisted boundary conditionTwisted boundary condition
q(xk +L) = e ik q(xk) t = (EpffLpiiL
2/L
Summary and outlookSummary and outlook
Generaized Parton Distribution spin content, transverse quark distribution, Form factors,…
- accessible experimentaly via DVCS - theoretical calculations in CQSM, Skyrme model,…
moments of GPD in lattice QCD - A20
u-d and B20u+d have strong “chiral log” corrections.
- Chiral extrapolation of A20 (0) & B20 (0) via BChPT nucl-th/0703073 leads J u 0.226 0.009 J d 0.005 0.009
- lighter m, larger volume (for t 0), Finite size corrections, Continuum limit, disconnected diagram, …
J u+d 0.222 0.014
s u+d 0.201 0.024 (@ m= .14GeV)
L u+d 0.021 0.028