1. Strangeness and glue in the nucleon from lattice QCD
Takumi Doi
(Univ. of Kentucky)
In collaboration with
Univ. of Kentucky:
M. Deka, S.-J. Dong, T. Draper, K.-F. Liu, D. Mankame
Tata Inst. of Fundamental Research:
N. Mathur
Univ. of Regensburg:
T. Streuer
cQCD Collaboration
07/17/2008
Lattice 2008

2. Introduction Nucleon structure
Fundamental particle, but a whole understanding of its structure has not been obtained yet
Spin “crisis”
The EMC experiments (1989)  quark spin is only 30%
Orbital angular momentum and/or gluon must carry the rest
Exciting results are coming from experiments
RHIC, JLAB, DESY, …
Inputs from theoretical prediction are necessary for some quantities: e.g., strangeness <x2>
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Lattice 2008

3. Introduction The ingredients: valence/sea quark and gluon
Quark “connected” diagrams
Quark“disconnected insertion” diagrams
Glue  what is suitable “glue” operator ?
Disconnected Insertion (D.I.) terms
Now is the full QCD Era: dynamical sea quark !
Strangeness in <x>, <x2>, electric/magnetic form factors
Glue terms
Glue in <x>
Glue contribution to nucleon spin
 necessary to complete (angular) momentum sum rules
Tough calculation in lattice
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Lattice 2008

4. Outline Energy-momentum tensor <x> from disconnected insertion
<x> and spin
<x> from disconnected insertion
<x> from glue
Glue operator from overlap operator
Outlook
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Lattice 2008

5. Methodology
The energy momentum tensor can be decomposed into quark part and gluon part gauge invariantly
Nucleon matrix elements can be decomposed as
(angular) momentum sum rules (reduce renormalization consts.)
Orbital part
X.Ji (1997)
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Lattice 2008

6. Methodology <x> can be obtained by q p p’=p-q
To improve S/N, we take a sum over t1=[t0+1, t2-1] t1 t0 t2
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Lattice 2008

7. Methodology Spin components can be obtained by q
p’=p-q
N.B. we use one more equation to extract T1 and T2 separately
(q^2 dependence could be different)
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Lattice 2008

8. Analysis for <x> (D.I.)
c.f. Analysis for <x> (connected)  talk by D. Mankame (Mon.)
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Lattice 2008

9. Analysis (1)
Nf=2+1 dynamical clover fermion + RG improved gauge configs (CP-PACS/JLQCD)
About 800 configs
Beta=1.83, (a^-1=1.62GeV, a=0.12fm)
16^3 X 32 lattice, L=2fm
Kappa(ud)= , ,
M(pi)= 610 – 840 MeV
Kappa(s)=
(Figures are for kappa(ud)= )
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Lattice 2008

10. Analysis (2) Wilson Fermion + Wilson gauge Action
500 configs with quenched approximation
Beta=6.0, (a^-1=1.74GeV, a=0.11fm)
16^3 X 24 lattice, L=1.76fm
kappa=0.154, 0.155,
M(pi)= MeV
Kappa(s)= , kappa(critical)=0.1568
(Figures are for kappa=0.154)
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Lattice 2008

11. D.I. calculation
Disconnected diagrams are estimated Z(4) noise (color, spin, space-time) method
#noise = 300 (full), 500 (quenched) (To reduce the possible autocorrelation, we take different noise for different configurations)
We also take many nucleon sources
(full: #src=64/32 (lightest mass/others), quenched: #src=16 )
We found that this is very effective
(autocorrelation between different sources is small)
CH, H and parity symmetry:
(3pt)=(2pt) X (loop)(3pt) = Im(2pt) X Re(loop) + Re(2pt) X Im(loop)
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Lattice 2008

12. Results for <x>(s)
Nf=2+1
Linear slope corresponds to signal
By increasing the nucleon sources #src = 1  32, the signal becomes prominent
Error bar reduced more than factor 5 !
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Lattice 2008

13. Chiral Extrapolation <x>(s) <x>(ud) [D.I.]
Nf=2+1
<x>(s)
<x>(ud) [D.I.]
We expect we can furhter reduce the error by subtraction technique using hopping parameter expansion
Note: The values are not renormalized
07/17/2008
Lattice 2008

14. Ratio of <x>(s) and <x>(ud)[D.I.]
Nf=2+1
<x>(s) / <x>(ud)[D.I.] =0.857(40)
Preliminary
c.f. Quenched
<x>(s) / <x>(ud)[D.I.] =0.88(7)
M. Deka
Note: The values are not renormalized
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Lattice 2008

15. Glue calculation Gluon Operator
Glue operator constructed from link variables are known to be very noise
Smearing ? (Meyer-Negele. PRD77(2008)037501, glue in pion)
Field tensor constructed from overlap operator
Ultraviolet fluctuation is expected to be suppressed
In order to estimate D_ov(x,x), Z(4) noise method is used, where color/spin are exactly diluted, space-time are factor 2 dilution + even/odd dilution, #noise=2
K.-F.Liu, A.Alexandru, I.Horvath PLB659(2008)773
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Lattice 2008

16. Results for <x>(g) (quenched)
Linear slope corresponds to signal
First time to obtain the signal of glue in nucleon !
c.f. M.Gockeler et al., Nucl.Phys.Proc.supp..53(1997)324
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Lattice 2008

17. Summary/Outlook
We have studied the <x> from strangeness, u, d (disconnected insertion[D.I.]) and glue
Nf=2+1 clover fermion and quenched for <x>(q)
<x>(s) is as large as <x>(ud) [D.I.]
Renormalization is necessary for quantitative results
Glue <x> has been studied using overlap operator
We have obtained a promising signal !
Outlook
Angular momentum is being studied  origin of nuc spin
Various quantities of D.I., strangeness electric/magnetic form factor, pi-N-sigma term, etc.
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Lattice 2008

18. Supplement
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Lattice 2008

19. Renormalization We have two operators: T4i(q), T4i(G)
It is known that the RG can be parametrized as
Two unknown parameters can be determined by two sum rules
Momentum sum rule:
Spin sum rule:
X.Ji, PRD52 (1995) 271
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Lattice 2008