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

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

3. C. Alexandrou,, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 LATTICE FERMIONS a μ U μ (n)= e igaA μ (n) ψ(n) Det[D]: Difficult to simulate  90’s Det[D]=1 : quenched Integrate out It is well known that naïve discretization of Dirac action leads to 16 species 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… valence

4. C. Alexandrou, University of Cyprus CHIRAL FERMIONS Instead of a D such that {γ 5, D}=0 of the no-go theorem, find a D such that {γ 5,D}=2a D γ 5 D Ginsparg - Wilson relation Luescher 1998: realization of chiral symmetry but NO no-go theorem!  Kaplan: Construction of D in 5-dimensions  Domain Wall fermions Left-handed fermion right-handed fermion L5L5  Neuberger: Construction of D in 4-dimensions but with use of sign function  Overlap fermions Equivalent formulations of chiral fermions on the lattice 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

5. C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 REACHING THE CHIRAL REGIME  Impressive progress in unquenched simulations using various fermions 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 N F =2+1 fermions reported simulations with ~160 MeV pions! But volume effects might be a problem  RBC-UKQCD: Dynamical N F =2+1 domain wall fermions reaching m π ~300 MeV on ~3 fm volume  JLQCD: Dynamical overlap N F =2 and N F =2+1, m l ~m s /6, small volume (~1.8 fm)  Progress in algorithms for calculating D -1 as m π physical value Deflation methods: project out lowest eigenvalues Lüscher, Wilcox, Orginos/Stathopoulos  Chiral extrapolations more reliable as simulations use m π in chiral regime e.g. using lattice results on f π /m π yield LECs to unprecedented accuracy  Begin to address more complex observables e.g. excited states and resonances, decays, finite density density

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

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

8. C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 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 from χPT Chiral perturbation theory in finite volume: Correct for volume effects depending on pion mass and lattice volume Like in infinite volume W. Detmold

9. C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 NUCLEON MASS Good agreement for the nucleon mass r0mNr0mN No common scaling r 0 f π

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

11. INCLUDING THE STRANGE C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 (r 0 m π ) 2 r0mNr0mN

12. C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 LIGHT HADRON MASSES S. Durr et al. Science 322 (2008) 1224 N F =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

13. C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 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: g A, g πNN, g πΝΔ, …  Form factors: G A (Q 2 ), G p (Q 2 ), G E (Q 2 ), G M (Q 2 ), …., where Q 2 =-q 2 =(p’-p) 2  Parton distribution functions: q(x), Δq(x),…  Generalized parton distribution functions: H(x,ξ,Q 2 ), E(x,ξ,Q 2 ),… Form factors, GPDS: three-point functions: Four-point functions: yield detailed info on quark distribution only simple operators used up to now - need all-to-all propagators Masses: two-point functions: t2t2 t0t0 t1t1 t2t2 t0t0 t1t1 t2t2 t0t0 t’1t’1 e.g. O for EM is

14. C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 NUCLEON ELECTROMAGNETIC FORM FACTORS connected disconnected g valence quarks sea quarks Disconnected diagram omitted so far. Better methods are being developed to allow their computation e.g. dilution, lower eigenvalue projection, one-end trick,… From connected diagram we calculate the isovector Sachs form factors: with G E and G M given in terms of F 1 and F 2 PQCD: G v E,M = G p E,M – G n E,M Nucleon 3-point functions

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

16. C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 Nucleon EM form factors Results using domain wall valence on staggered sea are from LHPC Cyprus/MIT Collaboration, C. A., G. Koutsou, J. W. Negele, A. Tsapalis q 2 =- Q 2 QCDSF/UKQCD Dirac FF

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

18. C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 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 valenceseaNFNF a (fm)L (fm) lowest m π (GeV) C. AlexandrouWilson - 00.09230.411 C. AlexandrouWilson 20.0771.80.380 C. Alexandrou (ETMC) Twisted mass 20.089 0.067 2.1, 2.70.268 Ph. Hagler (LHPC)DWFstaggered2+10.1242.5, 3.50.293 M. Gockeler (QCDSF) Clover-00.107 0.077 0.058 1.7 1.8 1.9 0.553 0.617 0.505 M. Gockeler (QCDSF) Clover 20.08 0.065 20.350 H. -W. LinDWF 20.1131.90.490 Sh. Ohta (RBC)DWF 2+10.1131.8, 2.70.330 S. SasakiDWF-00.151.8, 2.4, 3.60.390

19. RECENT RESULTS ON EM NUCLEON FORM FACTORS C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 C.A. et al. ETMC M. Lin et al. LHPC JLab group is developing methods to go to higher Q 2. Results obtained in quenched approximation for lowest m π ~ 500 MeV

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

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

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

23. C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 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 G πNN (q 2 ) PCAC relates G A and G p to G πNN : Generalized Goldberger- Treiman relation Pion pole dominance: Goldberger-Treiman relation

24. C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 G A AND G P dipole fit pion pole dominance: dipole fit to experimental results Results in the hybrid approach by the LPHC in agreement with dynamical Wilson results Unquenching effects large at small Q 2 in line with theoretical expectations that pion cloud effects are dominant at low Q 2 Cyprus/MIT collaboration

25. DISTRIBUTION AMPLITUDES C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 Exclusive processes at large Q 2 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 * [S 11 (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 Helicity amplitudes A 1/2 and S 1/2 can be expressed in terms of G 1 (q 2 ) and G 2 (q 2 ) V. M. Braun et al., arXiv:0902.3087 experiment lattice Small volumes and pion mass higher than ~ 400 MeV

26. Δ ELECTROMAGNETIC FORM FACTORS C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 P μ total momentum A linear combination of a 1, a 2, c 1, c 2 gives G E0, G M1, G E2 and G M3 subdominant C. A. et al., PRD79:014507(2009), arXiv:0901.3457 Exponential fitsDipole fits

27. Δ ELECTRIC QUADRUPOLE FORM FACTOR C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 Exponential fits Also P. Moran et al. [Adelaide], quenched results Quark transverse charge densities with definite light-cone helicity Quark transverse charge densities in Δ + polarized along the x-axis extracted from lattice data ρ Δ Τ3/2 ρ Δ Τ1/2 Cyprus-MIT/Mainz

28. C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 N TO Δ TRANSITION FORM FACTORS Electromagnetic N to Δ: Write in term of Sachs form factors: Magnetic dipole: dominant Electric quadrupole Coloumb quadrupole Axial vector N to Δ: four additional form factors, C 3 A (Q 2 ), C 4 A (Q 2 ), C 5 A (Q 2 ), C 6 A (Q 2 ) Dominant: C 5 A G A C 6 A G p PCAC relates C 5 A and C 6 A to G πΝΔ : Generalized non-diagonal Goldberger-Treiman relation Pion pole dominance: Non-diagonal Goldberger-Treiman relation

29. C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2008 COMPARISON WITH EXPERIMENT G M1 Thanks L. Tiatorπ-cloud?

30. C. Alexandrou University of Cyprus Hadron Physics on the Lattice, Milos, Sept. 2007 QUADRUPOLE FORM FACTORS AND DEFORMATION Q 2 =0.127 GeV 2 Large pion effects  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)

31. CHIRAL EXTRAPOLATION C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 NLO chiral extrapolation on the ratios using m π /Μ~δ 2, Δ/Μ~δ. G M1 itself not given. V. Pascalutsa and M. Vanderhaeghen, hep-ph/0508060 Just beginning to enter chiral regime

32. C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 AXIAL N TO Δ M. Procura, arXiv:0803.4291 SSE to O (ε 3 ): C 5 A (Q 2,m π )=a 1 +a 2 m π 2 +a 3 Q 2 +loop Again unquenching effects at small Q 2 clearly visible Pion-pole dominance: heavier pion mass required C 6 (Q 2 )= C 5 (Q 2 ) c 0 /(m 2 +Q 2 ) dipole exponential

33. G ΠΝΝ AND G ΠΝΔ C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 G πΝN =a(1-bQ 2 /m π 2 ) GTR: G πNΝ =G A m N /f π G πΝΔ =1.6G πNΝ  Q 2 -dependence for G πΝΝ (G πΝΔ ) extracted from G A (C 5 A ) using the Goldberger- Treiman relation deviates at low Q 2  Approximately linear Q 2 -dependence (small deviations seen at very low Q 2 in the case of G πΝΔ need to be checked)  G πΝΔ /G πΝN = 2C 5 A /G A as predicted by taking ratios of GTRs  G πΝΔ /G πΝN =1.60(1) Curves refer to the quenched data

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

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

36. TRANSVERSE QUARK DENSITIES C. Alexandrou, University of Cyprus Hadron Electromagnetic form factors, ECT*, May 12-23 2008 Fourier transform of form factors/GPDs gives probability Transverse quark distributions: M. Burkardt, PRD62 (2000 ) as n increases slope of A n0 decreases

37. GENERALIZED FORM FACTORS A 20, B 20, C 20 C. Alexandrou,, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 LHPC : PRD 77, 094502(2008), 0705.4295

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

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

40. SPIN STRUCTURE OF THE NUCLEON C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 QCDSF/UKQCD Dynamical Clover For m π ~340 MeV: J u =0.279(5) J d =-0.006(5) L u+d =-0.007(13) ΔΣ u+d =0.558(22) In agreement with LPHC, Ph. Hägler et al., hep-lat/07054295 Total spin of quark: Spin of the nucleon: A 20 (0)

41. EXCITED STATES C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 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. χ 2 -method: C. Alexandrou, E. Stiliaris, C.N. Papanicolas, PoS LAT2008, arXiv:0810.3982 Histogram method: V. Bernard et al., arXiv:0806.4495 Several approaches: N-Roper transition matrix element : H.-W. Lin et al. So far quenched, heavy pions

42. MATRIX OF CORRELATORS C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 For a given NxN correlator matrix one defines the N principal correlators λ k (t,t 0 ) as the eigenvalues of where t 0 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)

43. η c spectrum C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 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 χ 2 -analysis: 1.3171(13) 1.608(9) 2.010(11) χ 2 - method

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

45. C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 CONCLUSIONS  Accurate results on coupling constants, form factors, moments of generalized parton distributions are becoming available close to the chiral regime (m π ~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 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 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

46. STRANGENESS C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 GsEGsE 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 G s M =-0.082(8)(25) G s E = 0.00044(1)(130) J. Zanotti, Lattice 2008

47. C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 PION SECTOR Applications to pion and nucleon mass and decay constant e.g. G. Colangelo, St. Dürr and Haefeli, 2005 r 0 =0.45(1)fm NLO continuum χPT Pion decay constant Applied by QCDSF to correct lattice data

48. C. Alexandrou, University of Cyprus ΠΠ-SCATTERING LENGTHS m π a 0 2 =-0.04330(42) A. Walker-Loud, arXiv:0706.3026  More statistics  mixed action χPT formula ππ s-wave length a 0 0 and a 0 2 Leutwyler (hep-ph/0612112) ETMC

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

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

51. C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 Dirac and Pauli rms radii C.A. Sh. Ohta H.-W. Lin S. Sasaki Sh. Ohta and T. Yamazaki, PoS LAT2008

52. C. Alexandrou, University of Cyprus TRANSVERSE SPIN DENSITY Unpolarized u d N N u d q unpolarizedN unpolarized QCDSF : Ph. Hägler et al. PRL 98, 222001 (2007)

53. C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 MAGNETIC DIPOLE FORM FACTOR

54. C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 TWISTED MASS DYNAMICAL FERMIONS Consider two degenerate flavors of quarks: Action is Wilson Dirac operator untwisted mass Twisted mass μ τ 3 acts in flavor space Advantages: - Automatic O (a) improvement (at maximal twist) - Only one parameter to tune (zero PCAC mass at smallest μ) - No additional operator improvement Disadvantages: - explicit chiral symmetry breaking (like in all Wilson) - explicit breaking of flavour symmetry appears only at O (a 2 ) in practice only affects π 0 Physics results reported for ~270 MeV pions on spatial size > 2.1 fm with a < 0.1fm

55. C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 TWISTED MASS FORMULATION The mass term can be written as: with α=tan -1 (μ/m 0 ), M 2 =m 0 2 +μ 2 Perform axial transformation to the physical basis The mass term transforms as and if α=ω then two degenerate flavors we recover the standard action At maximal twist ω=π/2 and the mass is due to the twisted mass term One parameter to tune just like with Wilson fermions. We tune m 0 to its critical value by requiring the PCAC mass to vanish. This is done at each β value at the lowest μ-value. This procedure is shown to preserve O(a) improvement Continuum fermionic action:

56. C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 DELTA MASS SPLITTING Symmetry: u d  Δ ++ is degenerate with Δ - and Δ + with Δ 0 Check for flavor breaking by computing mass splitting between the two degenerate pairs Splitting consistent with zero in agreement with theoretical expectations that isospin breaking only large for neutral pions (~16% on finest lattice) * β=3.8 L=2.4 fm

57. AXIAL NUCLEON FORM FACTORS C. Alexandrou, University of Cyprus Baryon Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009

58. C. Alexandrou, University of Cyprus Hadron Electromagnetic form factors, ECT*, May 12-23 2008 PION FORM FACTOR Comparison with experiment S. Simula, Lattice07

59. C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 BARYON SECTOR - Volume effects: small - cutoff effects: small - Agreement with staggered fermion results - Effect of dynamical strange quark small Use continuum chiral PT to extrapolate to physical point

60. C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 NUCLEON MASS β=3.9 β=3.9, 4.05 HBχPT to lowest order: Two fit parameters  m 0 =0.865(10), c 1 =-1.224(17) GeV -1  σ-term =66(3) MeV Higher order yield results in agreement with the lowest order statistical

61. C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 FIX LATTICE SPACING Value of lattice spacing using the nucleon mass at the physical point in agreement with that extracted from the pion sector non-trivial check of our lattice systematics

62. C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 CONTINUUM LIMIT Use the data at β=3.9 and 4.05 to estimate the continuum limit and check consistency with β=3.8 Impressive agreement - cutoff effects non-visible

63. C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 FITS TO CONTINUUM RESULTS Lattice prediction of nucleon mass in agreement with experiment

64. C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 FITS TO Δ MASS AT THE CONTINUUM LIMIT Using r 0 from the nucleon mass

65. C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 GOLDBERGER-TREIMAN RELATIONS Deviations from GTR Improvement when using chiral fermions

66. C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 RHO FORM FACTORS decomposed into three form factors G c (Q 2 ), G M (Q 2 ), G Q (Q 2 ) Related to the ρ quadrupole moment G Q (0)=m ρ 2 Q ρ Adelaide group: Q ρ non-zero --> rho-meson deformed in agreement with wave function results using 4pt functions Non-relativistic approx. |Ψ(y)| 2 One-end trick improves signal, application to baryons is underway Decreasing quark mass G. Koutsou, Lattice07

67. C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 HADRON-SHAPE For mesons we applied the one-end trick  improves statistical noise Rho clearly prolate Density-density correlators evaluated exactly using all-to-all propagators on dynamical two degenerate flavors of Wilson fermions m π =380 MeV m π = 680 MeV For Δ we have results with dynamical quarks for the first time. More noisy, small deformation in agreement with an oblate shape X 0.1

68. C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 PION FORM FACTOR Set initial and final pion momentum to zero --> no disadvantage in applying the one-end trick G. Koutsou, Lattice07 q -q 4-point function: first application of the one-end trick (C. Michael): S. Simula, ETM Collaboration one-trick for 3pt function Wilson

69. COMPARISON WITH EXPERIMENT C. Alexandrou, University of Cyprus Hadron Electromagnetic form factors, ? Thanks V. Burkert Thanks L. Tiator Data, assuming only M1 Analysis by Cole Smith to the CLAS π 0 data, B. Julia-Diaz, et al. PRC75 (2007 ) π-cloud

70. C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 QUADRUPOLE FORM FACTOR G C2