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

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

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

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

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

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

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

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

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

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

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

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: fm, fm and fm volumes > 2 fm lightest pion mass ~ 190 MeV

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

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

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

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

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

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) ; (ETMC) PoS LAT2008, arXiv: Ph. Hagler et al.,(LHPC) PRD 77 (2008) M. Gockeler et al., (QCDSF) PRD 71 (2005) ; PoS LAT2007,161, Pos LAT 2006 H.-W. Lin et al., PRD 78 (2008) Sh. Ohta and T. Yamazaki et al., (RBC-UKQCD) PoS LAT2008, arXiv: S. Sasaki and T. Yamazaki, PRD 78 (2008) valenceseaNFNF a (fm)L (fm) lowest m π (GeV) C. AlexandrouWilson C. AlexandrouWilson C. Alexandrou (ETMC) Twisted mass , Ph. Hagler (LHPC)DWFstaggered , M. Gockeler (QCDSF) Clover M. Gockeler (QCDSF) Clover H. -W. LinDWF Sh. Ohta (RBC)DWF , S. SasakiDWF , 2.4,

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

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

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)  accurately measured: (29)  no-disconnected diagrams  chiral PT R. G. Edwards et al. PRL 96, , 2006 Mixed action, LHPC Savage & Beane, hep-ph/

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, , 2008 Finite volume dependence QCDSF Lattice volume (2.7 fm) 3

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

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

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: experiment lattice Small volumes and pion mass higher than ~ 400 MeV

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: Exponential fitsDipole fits

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

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

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?

C. Alexandrou University of Cyprus Hadron Physics on the Lattice, Milos, Sept 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)

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/ Just beginning to enter chiral regime

C. Alexandrou, University of Cyprus Hadron Structure from Lattice QCD, CLAS12 Workshop, Feb.25-28, 2009 AXIAL N TO Δ M. Procura, arXiv: 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

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

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

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

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

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, (2008),

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:

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

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/ Total spin of quark: Spin of the nucleon: A 20 (0)

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, (2007); C. Gattringer, arXiv: ; B. G. Lasscock et al., PRD 76 (2007) ; C. Morningstar, Lattice 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: Histogram method: V. Bernard et al., arXiv: Several approaches: N-Roper transition matrix element : H.-W. Lin et al. So far quenched, heavy pions

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)

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/ ): (1) 1.62(2) 1.98(22) Compare to χ 2 -analysis: (13) 1.608(9) 2.010(11) χ 2 - method

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:

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

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, (2005); 97, (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 = (1)(130) J. Zanotti, Lattice 2008

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

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

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

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

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

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, (2007)

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

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

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:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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