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BARYON STRUCTURE FROM LATTICE QCD

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  1. C. Alexandrou University of Cyprus and Cyprus Institute Κ. Αλεξάνδρου BARYON STRUCTURE FROM LATTICE QCD Πανεπιστήμιο Κύπρου First European CLAS12 Workshop, Feb.25-28, 2009, Genoa, Italy

  2. CLAS12 PROJECT • Science Program: • Baryon form factors • Generalized Parton Distributions • Electroproduction at very small Q2 • Inclusive nucleon structure • Semi-Inclusive DIS • Properties of QCD from nuclear medium C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  3. a ψ(n) μ Integrate out Uμ(n)=eigaAμ(n) It is well known that naïve discretization of Dirac action leads to 16 species LATTICE FERMIONS Det[D]: Difficult to simulate 90’s Det[D]=1 : quenched valence • A number of different discretization schemes have been developed to avoid doubling: • Wilson : add a second derivative-type term  breaks chiral symmetry for finite a • Staggered: “distribute” 4-component spinor on 4 lattice sites  still 4 times more species, take 4th root, non-locality Nielsen-Ninomiya no go theorem: impossible to have doubler-free, chirally symmetric, local, translational invariant fermion lattice action BUT… C. Alexandrou,, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  4. L5 Left-handed fermion right-handed fermion But still expensive to simulate despite progress in algorithms Compromise: - Use staggered sea (MILC) and domain wall valence (hybrid) - LHPC - Twisted mass Wilson - ETMC - Clover improved Wilson - QCDSF, UKQCD, PACS-CS, BMW CHIRAL FERMIONS Instead of a D such that {γ5, D}=0 of the no-go theorem, find a D such that {γ5,D}=2a D γ5D Ginsparg - Wilson relation • Kaplan: Construction of D in 5-dimensions  Domain Wall fermions Luescher 1998: realization of chiral symmetry but NO no-go theorem! • Neuberger: Construction of D in 4-dimensions but with use of sign function  Overlap fermions Equivalent formulations of chiral fermions on the lattice C. Alexandrou, University of Cyprus

  5.  Progress in algorithms for calculating D-1 as mπ physical value Deflation methods: project out lowest eigenvalues Lüscher, Wilcox, Orginos/Stathopoulos REACHING THE CHIRAL REGIME  Impressive progress in unquenched simulations using variousfermions discretization schemes • With staggered, Clover improved and Twisted mass dynamical fermions results are emerging with pion mass of ~300 MeV and in some cases lower • PACS-CS using Clover NF=2+1 fermions reported simulations with ~160 MeV pions! But volume effects might be a problem • RBC-UKQCD: Dynamical NF=2+1 domain wall fermions reaching mπ~300 MeV on ~3 fm volume • JLQCD: Dynamical overlap NF=2 and NF=2+1, ml ~ms/6, small volume (~1.8 fm)  Chiral extrapolations more reliable as simulations use mπin chiral regime e.g. using lattice results on fπ/mπ yield LECs to unprecedented accuracy  Begin to address more complex observables e.g. excited states and resonances, decays, finite density density C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  6. DYNAMICAL SIMULATIONS • Reaching the chiral regime: • Twisted mass: ETMC (Cyprus, France, Germany, Italy, Netherlands, UK, Spain, Switzerland) automaticO(a2) but breaks isospin • Clover: QCDSF, UKQCD, PACS-CS, BMW (requires improvement of operators) • Hybrid approach: LHPC/Cyprus • Domain wall: RBC and UKQCD • Overlap: JLQCD O(a2) chiral fermions (expensive) Most simulations are done using volumes larger than 2 fm Chiral extrapolation of lattice results are becoming more reliable C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  7. SUMMARY OF DYNAMICAL SIMULATIONS K. Jansen, Lattice 2008 C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  8. CHIRAL EXTRAPOLATIONS • Controlled extrapolations in volume, lattice spacing and quark masses: still needed for reaching the physical world - this may change in the next 2-3 years • Quark mass dependence of observables teaches about QCD - can not be obtained from experiment • Lattice systematics are checked using predictions fromχPT Chiral perturbation theory in finite volume: Correct for volume effects depending on pion mass and lattice volume W. Detmold Like in infinite volume C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  9. NUCLEON MASS r0 fπ r0mN Good agreement for the nucleon mass No common scaling C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  10. LIGHT HADRON MASSES C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  11. INCLUDING THE STRANGE r0mN preliminary (r0mπ)2 C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  12. LIGHT HADRON MASSES • NF=2+1 Clover fermions + stout smearing • 3 lattice spacings: 0.125 fm, 0.085 fm and 0.065 fm • volumes > 2 fm • lightest pion mass ~ 190 MeV S. Durr et al. Science 322 (2008) 1224 C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  13. t2 t2 t2 t0 t0 t0 BARYON STRUCTURE • Hybrid (mixed) action (LHPC/Cyprus), • Twisted mass fermions (ETMC), • Clover fermions (QCDSF, UKQCD, CPACS-CS, BMW, CERN), • Domain wall fermions (RBC-UKQCD), • Overlap fermions (JLQCD) • Coupling constants: gA, gπNN, gπΝΔ, … • Form factors: GA(Q2), Gp(Q2), GE(Q2), GM(Q2), …., where Q2=-q2=(p’-p)2 • Parton distribution functions: q(x), Δq(x),… • Generalized parton distribution functions: H(x,ξ,Q2), E(x,ξ,Q2),… • Masses: two-point functions: • Form factors, GPDS: three-point functions: t1 t1 t’1 e.g. O for EM is • Four-point functions: yield detailed info on quark distribution • only simple operators used up to now - need all-to-all propagators C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  14. with GE and GM given in terms of F1 and F2 NUCLEON ELECTROMAGNETIC FORM FACTORS Nucleon 3-point functions Disconnected diagram omitted so far. Better methods are being developed to allow their computation e.g. dilution, lower eigenvalue projection, one-end trick,… valence quarks g From connected diagram we calculate the isovector Sachs form factors: disconnected GvE,M= GpE,M– GnE,M sea quarks PQCD: connected C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  15. EXPERIMENT C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  16. Cyprus/MIT Collaboration, C. A., G. Koutsou, J. W. Negele, A. Tsapalis Nucleon EM form factors QCDSF/UKQCD Dirac FF q2 =- Q2 Results using domain wall valence on staggered sea are from LHPC C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  17. CHIRAL EXTRAPOLATION C. Alexandrou et al., PRD 71 Heavy baryon effective theory with explicit Δ degrees of freedom C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  18. Recent lattice results • C. Alexandrou et al., PRD 74 (2006) 034508; (ETMC) PoS LAT2008, arXiv:0811.0724 • Ph. Hagler et al.,(LHPC) PRD 77 (2008) 094502 • M. Gockeler et al., (QCDSF) PRD 71 (2005) 034508; PoS LAT2007,161, Pos LAT 2006 • H.-W. Lin et al., PRD 78 (2008) 014505 • Sh. Ohta and T. Yamazaki et al., (RBC-UKQCD) PoS LAT2008, arXiv:0810.0045 • S. Sasaki and T. Yamazaki, PRD 78 (2008) 014510 C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  19. RECENT RESULTS ON EM NUCLEON FORM FACTORS C.A. et al. ETMC M. Lin et al. LHPC JLab group is developing methods to go to higher Q2. Results obtained in quenched approximation for lowest mπ~ 500 MeV C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  20. Dirac and Pauli r.m.s. radii Twisted mass and domain wall fermions C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  21. Nucleon axial charge Obtain axial nucleon form factors without computational cost Forward matrix element yields gA=GA(0) • accurately measured: 1.2695(29) • no-disconnected diagrams • chiral PT  benchmark for lattice calculations M. Gockeler et al. PRD74 (2006) 094508 R. G. Edwards et al. PRL 96, 052001, 2006 Improved Wilson (QCDSF) Mixed action, LHPC HBχPT with Δ Hemmert et al. Savage & Beane, hep-ph/0404131 Pleiter, Lat07 C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  22. Finite volume dependence RECENT RESULTS ON THE NUCLEON AXIAL CHARGE QCDSF Lattice volume (2.7 fm)3 RBC/UKQCD, T. Yamazaki et al. PRL 100, 171602, 2008 C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  23. PCAC relates GA and Gp to GπNN: Generalized Goldberger-Treiman relation Pion pole dominance: Goldberger-Treiman relation NUCLEON AXIAL FORM FACTORS Within the fixed sink method for 3pt function we can evaluate the nucleon matrix element of any operator with no additional computational cost • use axial vector current to obtain the axial form factors • pseudoscalar density to obtain GπNN(q2) C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  24. GA AND GP Results in the hybrid approach by the LPHC in agreement with dynamical Wilson results Unquenching effects large at small Q2 in line with theoretical expectations that pion cloud effects are dominant at low Q2 dipole fit to experimental results pion pole dominance: Cyprus/MIT collaboration dipole fit C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  25. DISTRIBUTION AMPLITUDES Exclusive processes at large Q2 can be factorized into: • perturbative hard scattering amplitude (process dependent) • nonperturbative wave functions describing the hadron's overlap with lowest Fock state (process independent) First results on nucleon and N* [S11(1535)] distribution amplitudes are obtained by the QCDSF-UKQCD collaboration Light cone sum rules derived for and making use of dispersion relations one connects distribution amplitudes for N* to the transition form factors for N* to N Operator with the quantum numbers of the nucleon experiment lattice Helicity amplitudes A1/2 and S1/2can be expressed in terms of G1(q2) and G2(q2) Small volumes and pion mass higher than ~ 400 MeV V. M. Braun et al., arXiv:0902.3087 C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  26. Δ ELECTROMAGNETIC FORM FACTORS Pμ total momentum A linear combination of a1, a2, c1, c2 gives GE0, GM1, GE2 and GM3 subdominant C. A. et al., PRD79:014507(2009), arXiv:0901.3457 Dipole fits Exponential fits C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  27. Δ ELECTRIC QUADRUPOLE FORM FACTOR Also P. Moran et al. [Adelaide], quenched results Exponential fits Quark transverse charge densities with definite light-cone helicity Quark transverse charge densities in Δ+polarized along the x-axis extracted from lattice data ρΔΤ3/2 Cyprus-MIT/Mainz ρΔΤ1/2 C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  28. Magnetic dipole: dominant Electric quadrupole Coloumb quadrupole Axial vector N to Δ: four additional form factors, C3A(Q2), C4A(Q2), C5A(Q2), C6A(Q2) Dominant: C5A GA C6A Gp PCAC relates C5A and C6A to GπΝΔ : Generalized non-diagonal Goldberger-Treiman relation Pion pole dominance: Non-diagonal Goldberger-Treiman relation N TO Δ TRANSITION FORM FACTORS Electromagnetic N to Δ: Write in term of Sachs form factors: C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  29. Thanks L. Tiator GM1 π-cloud? COMPARISON WITH EXPERIMENT C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2008

  30. QUADRUPOLE FORM FACTORS AND DEFORMATION Q2=0.127 GeV2 • Quantities measured in the lab frame of the Δ are the ratios: • Precise experimental data strongly suggest deformation of Nucleon/Δ • First conformation of non-zero EMR and CMR in full QCD C. N. Papanicolas, Eur. Phys. J. A18 (2003) Large pion effects C. AlexandrouUniversity of Cyprus Hadron Physics on the Lattice,Milos, Sept. 2007

  31. CHIRAL EXTRAPOLATION Just beginning to enter chiral regime NLO chiral extrapolation on the ratios using mπ/Μ~δ2 , Δ/Μ~δ. GM1 itself not given. V. Pascalutsa and M. Vanderhaeghen, hep-ph/0508060 C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  32. AXIAL N TO Δ dipole M. Procura, arXiv:0803.4291 SSE to O(ε3): C5A(Q2,mπ)=a1+a2mπ2+a3Q2+loop exponential Again unquenching effects at small Q2 clearly visible Pion-pole dominance: heavier pion mass required C6(Q2)= C5(Q2) c0/(m2+Q2) C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  33. GΠΝΝ AND GΠΝΔ Curves refer to the quenched data • Q2-dependence for GπΝΝ (GπΝΔ) extracted from GA (C5A) using the Goldberger-Treiman relation deviates at low Q2 • Approximately linear Q2-dependence (small deviations seen at very low Q2 in the case of GπΝΔ need to be checked) • GπΝΔ/GπΝN = 2C5A/GA as predicted by taking ratios of GTRs • GπΝΔ/GπΝN=1.60(1) GTR: GπNΝ=GAmN/fπ GπΝN=a(1-bQ2/mπ2) GπΝΔ=1.6GπNΝ C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  34. MOMENTS OF PARTON DISTRIBUTIONS Parton distributions measured in deep inelastic scattering Forward matrix elements: Moments of parton distributions 3 types of operators: unpolarized moment polarized moment transversity C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  35. MOMENTS OF GENERALIZED PARTON DISTRIBUTIONS Off diagonal matrix element Genelarized parton distributions GPD measured in Deep Virtual Compton Scattering Generalized FF GPDs • Momentum sum rule: t=0 (forward) yield moments of parton distributions • Spin sum rule C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  36. Fourier transform of form factors/GPDs gives probability Transverse quark distributions: M. Burkardt, PRD62 (2000) TRANSVERSE QUARK DENSITIES as n increases slope of An0 decreases C. Alexandrou, University of Cyprus Hadron Electromagnetic form factors, ECT*, May 12-23 2008

  37. GENERALIZED FORM FACTORS A20, B20, C20 LHPC: PRD 77, 094502(2008), 0705.4295 C. Alexandrou,, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  38. CHIRAL EXTRAPOLATION LHPC <x>u-d NLO: gA,0--> gA,lat, same with fπ Self consistent HBχPT QCDSF, Lattice 2007: 0710.1534 C. Alexandrou,, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  39. <X>u-d AND <X>Δu-Δd Finite size effect? C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  40. SPIN STRUCTURE OF THE NUCLEON Total spin of quark: A20(0) Spin of the nucleon: QCDSF/UKQCD Dynamical Clover For mπ~340 MeV: Ju=0.279(5) Jd=-0.006(5) Lu+d=-0.007(13) ΔΣu+d=0.558(22) In agreement with LPHC, Ph. Hägler et al., hep-lat/07054295 C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  41. EXCITED STATES Several approaches: • Phase shift method: M. Luscher • Correlation matrix: C. Michael; M. Luscher; S. Barak et al., PDR76, 074504 (2007); C. Gattringer, arXiv:0802.2020; B. G. Lasscock et al., PRD 76 (2007) 054510; C. Morningstar, Lattice 2008. • Maximum entropy methods: P. Lepage et al., Nucl. Phys. Proc. Suppl. 106 (2002) 12; C. Morningstar, Nucl. Phys. Proc. Suppl. 109A (2002) 185. • χ2-method: C. Alexandrou, E. Stiliaris, C.N. Papanicolas, PoS LAT2008,arXiv:0810.3982 • Histogram method: V. Bernard et al., arXiv:0806.4495 N-Roper transition matrix element: H.-W. Lin et al. So far quenched, heavy pions C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  42. MATRIX OF CORRELATORS • For a given NxN correlator matrix one defines the N principal correlators λk(t,t0) as the eigenvalues of • where t0 is small • The N principal effective masses tend (plateau) to the N lowest-lying stationary-state energies C. Morningstar, Lattice 2008 • Crucial to use very good operators so noise does not swamp signal • construct operators using smeared fields • - link variable smearing • quark field smearing • spatially extended operators • use large set of operators (variational coefficients) C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  43. ηcspectrum χ2 - method Analysis by C. Davies et al. using priors (P. Lepage et al. hep-lat/0110175): 1.3169(1) 1.62(2) 1.98(22) Compare to χ2-analysis: 1.3171(13) 1.608(9) 2.010(11) C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  44. NUCLEON SPECTRUM C. Alexandrou, E. Stiliaris, C.N. Papanicolas, PoS LAT2008,arXiv:0810.39 C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  45. CONCLUSIONS • Lattice QCD is entering an era where it can make significant contributions in the interpretation of current experimental results. • A valuable method for understanding hadronic phenomena • Accurate results on coupling constants, form factors, moments of generalized parton distributions are becoming available close to the chiral regime (mπ~300 MeV)  expected to below 200 MeV in the next couple of years • More complex observables are evaluated e.g excited states, polarizabilities, scattering lengths, resonances, finite density • First attempts into Nuclear Physics e.g. nuclear force • Computer technology and new algorithms will deliver 100´s of Teraflop/s in the next five years • Provide dynamical gauge configurations in the chiral regime • Enable the accurate evaluation of more involved matrix elements We expect a lot of interesting Physics C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  46. STRANGENESS • Indirect: charge symmetry + chiral extrapolation: D.Leinweber et al., PRL 94, 212001 (2005); 97, 022001(2006); H.-W Lin and K. Orginos (mixed action) • Direct: disconnected contributions, still noisy: R. Babich et al. , LAT2008 GsE GsM=-0.082(8)(25) GsE= 0.00044(1)(130) J. Zanotti, Lattice 2008 C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  47. PION SECTOR Applications to pion and nucleon mass and decay constant e.g. G. Colangelo, St. Dürr and Haefeli, 2005 Pion decay constant NLO continuum χPT Applied by QCDSF to correct lattice data r0=0.45(1)fm C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  48. More statistics • mixed action χPT formula ΠΠ-SCATTERING LENGTHS mπa02 =-0.04330(42) A. Walker-Loud, arXiv:0706.3026 ππ s-wave length a00 and a02 ETMC Leutwyler (hep-ph/0612112) C. Alexandrou,University of Cyprus

  49. RATIOS C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

  50. RATIOS C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009