Strangeness and glue in the nucleon from lattice QCD

1. Takumi Doi (Univ. of Kentucky) Strangeness and glue in the nucleon from lattice QCD 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 KY CCS seminar

2. electron Introduction • Nucleon structure • Fundamental particle as only stable baryons: the structure is crucial to understand not only nucleons themselves but also to understand the QCD • Electric/Magnetic structure • GE: electric form factor • GM: magnetic form factor 1950s KY CCS seminar

3. electron Introduction • Nucleon structure • Fundamental particle as only stable baryons: the structure is crucial to understand not only nucleons themselves but also to understand the QCD • Deep Inelastic Scattering (DIS) • W1 F1 structure function • W2 F2 structure function • Weak Q2 dependence & 2xF1=F2  Parton model 1960s KY CCS seminar

4. xP P Introduction • Nucleon structure • Parton model Parton carry (xP) momentum Q2 evolution  QCD interaction KY CCS seminar

5. Introduction • Nucleon structure • Yet, whole understanding of its structure has not been obtained • Q2-dependence  QCD pert. calc., but x-dependence ?? • 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> KY CCS seminar

6. Introduction • The ingredients: valence/sea quark and gluon • Quark “connected” insertion diagrams • Quark“disconnected insertion” diagrams • Glue  what is suitable “glue” operator ? • Glue terms • Glue in <x> • Glue contribution to nucleon spin •  necessary to complete (angular) momentum sum rules Tough calculation in lattice KY CCS seminar

7. Outline • Lattice QCD and our methodology • Energy-momentum tensor • <x> and spin • <x> from disconnected insertion • <x> from glue • Glue operator from overlap operator • Strangeness magnetic/electric form factor • Outlook KY CCS seminar

8. 1 time 2 3 space Lattice QCD Monte Carlo calculation (Weighted sum is taken) det=1: quenched det≠1: full QCD Non-perturbative fluctuations are included via Path-Integral KY CCS seminar

9. Lattice QCD • Various fermion formulations • Wilson fermion, clover fermion • Calculation cost is cheap • Chiral symmetry broken explicitly • Staggered fermion • Calculation cost is cheaper • Part of Chiral symmetry respected • Flavor cannot identified, Rooting problem • Domain Wall / Overlap fermion • Calculation cost is expensive • Good Chiral symmetry KY CCS seminar

10. Lattice QCD • Various fermion formulations • Wilson fermion, clover fermion • Calculation cost is cheap • Chiral symmetry broken explicitly • Staggered fermion • Calculation cost is cheap • Part of Chiral symmetry respected • Flavor cannot identified, Rooting problem • Domain Wall / Overlap fermion • Calculation cost is expensive • Good Chiral symmetry • Symmetry and (spontaneous) broken symmetry Nobel Prize 2008 Nambu Maskawa-Kobayashi KY CCS seminar

11. Methodology • Matrix elements • Requires 4pt function calc. • Operator Product Expansion (OPE)  3pt calc. KY CCS seminar

12. Disconnected Insertion (DI) • Why are DI diagrams important ? • Strangeness in nucleon • Strangeness electric/magnetic form factor • Structure function, <x>, <x2> • Quark spin and angular momentum in nuleon • Pion-Nucleon-Sigma term • Now is the full QCD Era: dynamical sea quark ! • Challenging subject • All-to-all propagator is necessary • Straightforward calculation impossible • O(105) inversion for O(106)XO(106) matrix Rich Physics ! KY CCS seminar

13. Disconnected Insertion (DI) • Stochastic Method for DI • Use Z(4) (or Z(2)) noises such that • DI loop can be calculated as • Introduce new source for noises (“off-diagonal” part) •  Unbiased subtraction using hopping parameter expansion (HPE) • Off-diagonal contaminations are estimated by HPE and subtracted (in unbiased way) S.-J.Dong, K.-F.Liu, PLB328(1994)130 KY CCS seminar

14. Analysis for <x> (D.I.) KY CCS seminar

15. Orbital part 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.) X.Ji (1997) KY CCS seminar

16. q p p’=p-q Methodology • <x> can be obtained by t1 t0 t2 KY CCS seminar

17. q p p’=p-q Methodology • We take the ratio of 3pt to 2pt To improve S/N, we take a sum over t1=[t0+1, t2-1] t1 Slope wrt. t2 (sink time) corresponds to the signal t0 t2 KY CCS seminar

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

19. 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)=0.13825, 0.13800, 0.13760 • M(pi)= 610 – 840 MeV • Kappa(s)=0.13760 • (Figures are for kappa(ud)=0.13760) KY CCS seminar

20. 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, 0.1555 • M(pi)=480-650 MeV • Kappa(s)=0.154 , kappa(critical)=0.1568 • (Figures are for kappa=0.154) KY CCS seminar

21. D.I. calculation • D.I. 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) KY CCS seminar

22. Results for <x>(s) Nf=2+1 Linear slope corresponds to signal #src = 1 Somewhat large errors KY CCS seminar

23. q p p’=p-q Many nucleon sources • Further improvement S/N improve by √Nnoise S/N improve by √Nsrc N.B. The calculations of loop part and 2pt part are independent ! KY CCS seminar

24. 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 ! KY CCS seminar

25. Chiral Extrapolation Nf=2+1 <x>(s) <x>(ud) [D.I.] We expect we can furhter reduce the error by subtraction technique using clover-fermion HPE Note: The values are not renormalized KY CCS seminar

26. 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 KY CCS seminar

27. 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 KY CCS seminar

28. 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 KY CCS seminar

29. Strangeness form factor • Very poor information available even today • Experiments • Even the sign of GM(Q2=0) unknown • Only few direct lattice QCD • Dong-Liu, PRD58(1998)074504 • Mathur-Dong, NP.Proc.Suppl.94(2001)311 • Lewis-Wilcox-Woloshyn, PRD67(2003)013003 KY CCS seminar arXiv:0805.2889 [hep-ex]

30. Strangeness form factor: Latt • Operator • Point-split operator  conserved, no RG factor • Electric/Magnetic form factor Electric Magnetic KY CCS seminar

31. Strangeness magnetic moment Linear slope corresponds to signal #src = 32 KY CCS seminar

32. Strangeness magnetic moment Q^2 dependence #src = 32 At each Q^2, s.m.m. is basically consistent with zero KY CCS seminar

33. Strangeness magnetic moment Q^2 dependence #src = 64 At each Q^2, s.m.m. is basically consistent with zero KY CCS seminar

34. Strangeness magnetic moment Chiral Extraplation s.m.m. is basically consistent with zero KY CCS seminar

35. Strangeness magnetic moment • Improvement ? • Dilution in color/spin in stochastic method •  did not work : probably because current is point-splitted • Dilution in even/odd • Does work, but unbiased subtraction w/ HPE compensate • Unbiased subtraction w/ HPE not w/ Wilson-type but w/ clover-type • Does work, but not so much • Deflation for the next overlap/domain-wall calc. KY CCS seminar

36. 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 ! • Strangeness magnetic/electric form factor • Outlook • Angular momentum is being studied  origin of nuc spin • Various quantities of D.I., pi-N-sigma term, etc. KY CCS seminar

37. 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 KY CCS seminar