Lattice Calculation of New Mesons in the Charm Region and Nucleon Structure Synopsis of Lattice QCD Study of X(2317) as a Tetraquark Mesonium Glue and Quark Momenta Renormalization with Sum Rules s, u+d (D.I.),, g Quark and glue angular momenta χQCD Collaboration: A. Alexandru, Y. Chen, T. Doi, S.J. Dong, T. Draper, M. Gong, I. Horvath, F. Lee, A. Li, K.F. Liu, N. Mathur, T. Streuer, J.B. Zhang 劉 克 非劉 克 非 刘克 菲克 菲

Taida, 2006, page 2 Introduction to Lattice Gauge Theory Path integral of the partition function of continuum QCD in Minkowski space Path integral of the partition function of continuum QCD in Minkowski space Imaginary time with Wick rotation Imaginary time with Wick rotation and and then then Integrating Grassmann numbers and gives Euclidean partition function Integrating Grassmann numbers and gives Euclidean partition function Note the Grassmann number integration Note the Grassmann number integration The Green’s function The Green’s function Path Integral Formulation in Discrete Euclidean Space-Time

GPW 2008, page 3 Lattice QCD – Path integral in Euclidean Space Regularization Regularization –Lattice spacing a –Hard cutoff, p ≤ π/a –Scale introduced (dimensional transmutation) Renormalization Renormalization –Perturbative –Non-perturbative Regularization independent Scheme Regularization independent Scheme Schroedinger functional Schroedinger functional Current algebra relations Current algebra relations Numerical Simulation Numerical Simulation –Quantum field theory classical statistical mechanics –Monte Carlo simulation (importance sampling) Why Lattice? a

Beijing, 2004, page 4 Hadron Mass and Decay Constant The two-point Green’s function decays exponentially at large separation of time Mass M= E p (p=0), decay constant ~ Φ

Beijing, 2004, page 5 Nucleon Form Factor The three-point Green’s function for the iso-vector axial current is proportional to asymptotically.

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C. Aubin, Lattice ‘09

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GPW 2008, page 25 Hadron Structure with Disconnected Insertion Calculation Pion-Nucleon Sigma Term, Strangeness Content in N Pion-Nucleon Sigma Term, Strangeness Content in N Quark Spin and Orbital Angular Momentum in Nucleon Quark Spin and Orbital Angular Momentum in Nucleon Sea Quark Contributions in, and Sea Quark Contributions in, and Strangeness Electric and Magnetic Form Factors Strangeness Electric and Magnetic Form Factors Muon Anomalous M. M. (g-2) (light-by-light) Muon Anomalous M. M. (g-2) (light-by-light) Neutron Electric Dipole Moment Neutron Electric Dipole Moment

Momenta and Angular Momenta of Quarks and Glue One can decompose the energy momentum tensor into quark part and gluon part gauge invariantly Nucleon matrix elements can be decomposed as where the angular momentum is X.Ji (1997) Orbital part

Methodology How to extract T1(q^2) and T2(q^2) ? q p p ’ =p-q N.B. we need one more equation to extract T1 and T2 separately (q^2 dependence could be different)

GPW 2008, page 28 Hadron Structure with Quarks and Glue Quark and Glue Momentum and Angular Momentum in the Nucleon Quark and Glue Momentum and Angular Momentum in the Nucleon

GPW 2008, page 29 Renormalization and Quark-Glue Mixing Momentum and Angular Momentum Sum Rules

GPW 2008, page 30 Strange Parton Moments Strange parton distribution is poorly known – H.L. Lai et al. (CTEQ), JHEP 0704:089 (2007) Strange parton distribution is poorly known – H.L. Lai et al. (CTEQ), JHEP 0704:089 (2007) s s < NuTeV measurement of sin 2 θ W is 3 σ above the standard model strangeness asymmetry, i.e. ? NuTeV measurement of sin 2 θ W is 3 σ above the standard model strangeness asymmetry, i.e. ? CTEQ: the sign of is uncertain. CTEQ: the sign of is uncertain. Lattice can calculate Lattice can calculate

CIPANP2009, page 31 s Full QCD with 2+1 Flavor Clover Fermions m π = 800, 700, 600 MeV s Full QCD with 2+1 Flavor Clover Fermions m π = 800, 700, 600 MeV

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GPW 2008, page 33 Quenched2+1 Flavor Full QCD

GPW 2008, page 34 Implication on Fitting of PDF Global PDF Fitting à la CTEQ Is there a discrepancy?

GPW 2008, page 35 Cat’s ears diagrams are suppressed by O(1/Q 2 ).

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GPW 2008 page 37 Large momentum frame Large momentum frame Parton degrees of freedom: valence, connected sea and disconnected sea Parton degrees of freedom: valence, connected sea and disconnected sea u d s u d s

GPW 2008, page 38 Physical Consequences Is it necessary to separate out the CS from the DS? Is it necessary to separate out the CS from the DS? 1)Small x behavior (Reggeon exchange, no pomeron exchange) (Reggeon exchange, no pomeron exchange) (Pomeron exchange) (Pomeron exchange)

GPW 2008, page 39 2) Gottfried Sum Rule Violation NMC: NMC: two flavor traces ( ) one flavor trace ( ) two flavor traces ( ) one flavor trace ( ) K.F. Liu and S.J. Dong, PRL 72, 1790 (1994) K.F. Liu and S.J. Dong, PRL 72, 1790 (1994)

GPW 2008, page 40 3) Fitting of experimental data CTEQ4, MRST CTEQ4, MRST But But A better fit A better fit where where

GPW 2008, page 41 HERMES – Kaon production in DIS, PL B666, 446 (2008)

Willam & Mary42 Results for Nf=2+1 Preliminary (s) (ud) [DI]

Results for (g) from overlap op Linear slope corresponds to signal Able to obtain a clear signal of glue in the nucleon. c.f. M.Gockeler et al., Nucl.Phys.Proc.supp..53(1997)324 Glue Momentum in Nucleon Glue operator from the overlap operator

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GPW 2008, page 45 Horst Fischer DIS2010See Talk 1193 by F. Kunne

GPW 2008, page 46 Nucleon Spin Quark spin ΔΣ ~ 30% of proton spin Quark spin ΔΣ ~ 30% of proton spin (DIS, Lattice) (DIS, Lattice) Quark orbital angular momentum? (recent lattice calculation  ~ 0) Quark orbital angular momentum? (recent lattice calculation  ~ 0) Glue spin ΔG/G small (COMPASS, STAR) ? Glue spin ΔG/G small (COMPASS, STAR) ? Glue orbital angular momentum is small (Brodsky and Gardner) ? Glue orbital angular momentum is small (Brodsky and Gardner) ? Dark Spin ?

GPW 2008, page 47 Flavor-singlet g A Quark spin puzzle (dubbed `proton spin crisis’) Quark spin puzzle (dubbed `proton spin crisis’) – NRQM – RQM –Experimentally (EMC, SMC, …

GPW 2008, page 48 S.J. Dong, J.-F. Lagae, and KFL, PRL 75, 2096 (1995) DI sea contribution independent of quark mass This suggests U(1) anomaly at work.

GPW 2008, page 49 Lattice Lattice Expt. (SMC) Expt. (SMC)NRQM RQM RQM 0.25(12 ) 1.20(10)0.61(13)0.79(11)-.42(11)-.12(1)0.45(6)0.75(11)0.60(2)0.22(10)1.2573(28)0.579(25)0.80(6)-0.46(6)-0.12(4)0.459(8)0.798(8)0.575(16) 1 5/3 5/ Lattice resolution: U(1) anomaly

GPW 2008, page 50 Summary Momentum fraction of quarks (both valence and sea) and gluons have been calculated for quenched and 2+1 flavor dynamical fermions. Momentum fraction of quarks (both valence and sea) and gluons have been calculated for quenched and 2+1 flavor dynamical fermions. Glue mometum fraction is ~ 50%. Glue mometum fraction is ~ 50%. g_A ~ 0.25 in agreement with expt. g_A ~ 0.25 in agreement with expt. Glue angular momentum (gauge invariant) is quite small. Glue angular momentum (gauge invariant) is quite small. Quark orbital angular momentum is small for the valence, but large for the sea quarks.Quark orbital angular momentum is small for the valence, but large for the sea quarks.

GPW 2008, page 51 Future Dynamical domain-wall fermion gauge Dynamical domain-wall fermion gauge (RBC + UKQCD configurations, lowest pion mass ~ 180 MeV on 4.5 fm box) (RBC + UKQCD configurations, lowest pion mass ~ 180 MeV on 4.5 fm box) + overlap fermion for the valence. + overlap fermion for the valence. The next set of lattices will have physical pion mass and 7 fm box. The next set of lattices will have physical pion mass and 7 fm box.

GPW 2008, page 52 Strangeness Magnetic Form Factors with 3 Quark Masses (m π = 0.6, 0.7, 0.8 GeV); T. Doi et al. ( QCD) arXiV: PRD (2009)

GPW 2008, page 53 The error band is due to chiral extrapolation. As a consequence, is only ~ 1 σ. Need larger lattices and smaller quark masses to be closer to the physical u/d mass. The error is about a factor of 10 smaller than previous direct lat calc and several times smaller than current experimental results.

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