C. Alexandrou, University of Cyprus PSI, December 18th 2007 Hadron Physics on the Lattice Πανεπιστήμιο Κύπρου Κ. Αλεξάνδρου

C. Alexandrou, University of Cyprus PSI, December 18th 2007 Outline The Highlight of Lattice Reaching the chiral regime - Results using light dynamical fermions  Twisted mass - ETMC (Cyprus, France, Germany, Italy, Netherlands, UK, Spain, Switzerland)  Clover improved dynamical fermions - QCDSF/UKQCD  Overlap fermions - JLQCD  Hybrid approach - LHPC and Cyprus  Staggered fermions:Charm Physics - HPQCD - Chiral extrapolation of lattice results

C. Alexandrou, University of Cyprus PSI, December 18th 2007 Lattice 2007: Highlights Impressive progress in unquenched simulations using various fermions discretization schemes

C. Alexandrou, University of Cyprus PSI, December 18th 2007 a μ U μ (n)= e igaA μ (n) ψ(n) 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 Lattice fermions Basics: 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 PSI, December 18th 2007 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 Rely on new improved algorithms

C. Alexandrou, University of Cyprus PSI, December 18th 2007 Simulation of dynamical fermions Results from different discretization schemes must agree in the continuum limit Clark

C. Alexandrou, University of Cyprus PSI, December 18th 2007 Lattice 2007: Highlights  Impressive progress in unquenched simulations using various fermions discretization schemes With staggered, Clover improved and Twisted mass dynamical fermions results are reported with pion mass of ~300 MeV PACS-CS reported simulations with 210 MeV pions! But no results shown yet  RBC-UKQCD: Dynamical N F =2+1 domain wall simulations reaching ~300 MeV on ~3 fm volume are underway  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

C. Alexandrou, University of Cyprus PSI, December 18th 2007 Deflation methods in fermion inverters -Wilcox Twisted mass 20 3 x32 at κ cr Solve linear system

C. Alexandrou, University of Cyprus PSI, December 18th 2007 Lattice 2007: Highlights  Impressive progress in unquenched simulations using various fermions discretization schemes With staggered, Clover improved and Twisted mass dynamical fermions results are reported with pion mass of ~300 MeV PACS-CS reported simulations with 210 MeV pions! But no results shown yet  RBC-UKQCD: Dynamical N F =2+1 domain wall simulations reaching ~300 MeV on ~3 fm volume are underway  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 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

C. Alexandrou, University of Cyprus PSI, December 18th 2007 Chiral extrapolations  Controlled extrapolations in volume, lattice spacing and quark masses: still needed for reaching the physical world  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

C. Alexandrou, University of Cyprus PSI, December 18th 2007 Pion sector Applications to pion and nucleon mass and decay constants e.g. G. Colangelo, St. Dürr and Haefeli, 2005 Applied by QCDSF to correct lattice data r 0 =0.45(1)fm NLO continuum χPT Pion decay constant

C. Alexandrou, University of Cyprus PSI, December 18th 2007 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 ~300 MeV pions on spatial size of about 2.7 fm with a < 0.1fm

C. Alexandrou, University of Cyprus PSI, December 18th 2007 Pion decay constant with twisted mass fermions Simultaneous chiral fits to m π and f π Extract low energy constants r 0 =0.441(14)fm S. Necco, Lat07 N F =2

C. Alexandrou, University of Cyprus PSI, December 18th 2007 Leutwyler (hep-ph/ ) ETMC ππ s-wave length a 0 0 and a 0 2 ππ-scattering lengths

C. Alexandrou, University of Cyprus PSI, December 18th 2007 π-π scattering Lüscher’s finite volume approach: Calculate the energy levels of two particle states in a finite box with p n given by Expand ΔE 0 about p~0: where we use particle definition for the scattering length: Applied to I=2 two-pion system within the hybrid approach (staggered sea and domain wall fermions) m π a 0 2 = (27) S. Beans et al. PRD73 (2006)

C. Alexandrou, University of Cyprus PSI, December 18th 2007 Other applications: I=1 KK scattering N-N scattering - too noisy  no new LECs  form independent of sea quarks if valence are chiral π-π scattering - improved  More statistics  mixed action χPT formula m π a 0 2 = (42) A. Walker-Loud, arXiv:

C. Alexandrou, University of Cyprus PSI, December 18th 2007 Pion form factor Sh. Hashimoto Dynamical overlap, L~1.8 fm

C. Alexandrou, University of Cyprus PSI, December 18th 2007 Set initial and final pion momentum to zero --> no disadvantage in applying the one-end trick G. Koutsou, Lat07 q -q 4-point function: first application of the one-end trick (C. Michael): S. Simula, ETM Collaboration one-trick for 3pt function Wilson Pion form factor

C. Alexandrou, University of Cyprus PSI, December 18th 2007 Pion form factor Comparison with experiment S. Simula, Lat07

C. Alexandrou, University of Cyprus PSI, December 18th 2007 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, Lat07

C. Alexandrou, University of Cyprus PSI, December 18th 2007 Baryon sector Twisted mass and Staggered results Two fit parameters: m 0 =0.865(10), c 1 =-1.224(17) GeV -1 to be compared with ~-0.9 GeV -1 (π- N sigma term)

C. Alexandrou, University of Cyprus PSI, December 18th 2007 Delta mass splitting Symmetry: u d --> Δ ++ is degenerate with Δ - and Δ + with Δ 0 Check for flavour 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)

C. Alexandrou, University of Cyprus PSI, December 18th 2007 Baryon structure  Coupling constants: g A, g πNN, g πΝΔ, …  Form factors: G A (Q 2 ), G p (Q 2 ), F 1 (Q 2 ),….  Parton distribution functions: q(x), Δq(x),…  Generalized parton distribution functions: H(x,ξ,Q 2 ), E(x,ξ,Q 2 ),… Need to evaluate three-point functions: Four-point functions: yield detailed info on quark distribution only simple operators used up tp now - need all-to-all propagators Hybrid actions Wilson fermions In addition: Masses: two-point functions

C. Alexandrou, University of Cyprus PSI, December 18th 2007 Nucleon axial charge HBχPT with Δ Hemmert et al., Savage & Beane Finite volume dependence Forward nucleon axial vector matrix element: Improved Wilson (QCDSF) Pleiter, Lat07  accurately measured  no-disconnected diagrams  chiral PT

C. Alexandrou, University of Cyprus PSI, December 18th 2007 Nucleon form factors Isovector form factors obtained from connected diagram Isovector Sachs form factors: Disconnected diagram not evaluated with G E and G M given in terms of F 1 and F 2 In last couple of years better methods have become available, dilution, one-end trick,… Evaluation of 3pt function using the fixed sink approach --> any current to be inserted with no additional computational cost

C. Alexandrou, University of Cyprus PSI, December 18th 2007 Nucleon EM form factors Results using dynamical Wilson fermions and domain wall valence on staggered sea (from LHPC) are consistent QCDSF/UKQCD Cyprus/MIT Collaboration, C. A., G. Koutsou, J. W. Negele, A. Tsapalis q 2 =- Q 2 Dirac FF

C. Alexandrou, University of Cyprus PSI, December 18th 2007 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 πN (q 2 ) PCAC relates G A and G p to G πNN : Generalized Goldberger- Treiman relation Pion pole dominance: Goldberger-Treiman relation Th. Korzec

C. Alexandrou, University of Cyprus PSI, December 18th 2007 G A and G p monopole fit pion pole dominance monopole 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

C. Alexandrou, University of Cyprus PSI, December 18th 2007 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 PSI, December 18th 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. Pananicolas, Eur. Phys. J. A18 (2003)

C. Alexandrou, University of Cyprus PSI, December 18th 2007 Axial N to Δ Bending seen in HBχPT with explicit Δ - Procura et al. 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 large unquenching effects at small Q 2

C. Alexandrou, University of Cyprus PSI, December 18th 2007 G πΝN and G πΝΔ G πΝ =a(1-bQ 2 /m π 2 ) GTR: G πΝ =G A m N /f π G πΝΔ =1.6G πΝ  Q 2 -dependence for G πΝ (G πΝΔ ) extracted from G A (C 5 A ) using the Goldberger- Treiman relation deviates at low Q 2  Small deviations from linear Q 2 - dependence seen at very low Q 2 in the case of G πΝΔ  G πΝΔ /G πΝ = 2C 5 A /G A as predicted by taking ratios of GTRs  G πΝΔ /G πΝ =1.60(1) Curves refer to the quenched data

C. Alexandrou, University of Cyprus PSI, December 18th 2007 Goldberger-Treiman Relations Deviations from GTR Improvement when using chiral fermions

C. Alexandrou, University of Cyprus PSI, December 18th 2007 Δ electromagnetic form factors Four form factors: G E0, G E2, G M1, G M3 given in terms of a 1, a 2,c 1,c 2 Like for the nucleon form factors only the connected diagram is evaluated --> isovector part G E0 m π =0.56 GeV m π =0.49GeV m π =0.41 GeV Q 2 (GeV 2 ) G M1 m π =0.56 GeV m π =0.49 GeV m π =0.41 GeV Q 2 (GeV 2 ) Accurate results for the dominant form factors calculated in the quenched approximation. Unquenched evaluation is in progress - Cyprus/MIT Collaboration, C. A., Th. Korzec, Th. Leontiou J. W. Negele, A. Tsapalis

C. Alexandrou, University of Cyprus PSI, December 18th 2007 Electric quadrupole Δ form factor G E2 connected to deformation of the Δ: Q 2 (GeV 2 ) m π =0.56 GeV m π =0.49 GeV m π =0.41 GeV G E2 <0 Δ oblate in agreement with what is observed using density correlators to extract the wave function, C. A., Ph. de Forcrand and A. Tsapalis, PRD (2002) G M3 also evaluated, small and negative

C. Alexandrou, University of Cyprus PSI, December 18th 2007 Moments of parton distributions Forward matrix elements: unpolarized moment polarized moment transversity Parton distributions measured in deep inelastic scattering 3 types of operators: Moments of parton distributions

C. Alexandrou, University of Cyprus PSI, December 18th 2007 Moments of Generalized parton distributions 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

C. Alexandrou, University of Cyprus PSI, December 18th 2007 Transverse quark densities Fourier transform of form factors/GPDs gives probability Transverse quark distributions: M. Burkardt, PRD62 (2000 ) NLO: g A,0 --> g A,lat, same with f π Self consistent HBχPT u-d LHPC as n increases slope of A n0 decreases

C. Alexandrou, University of Cyprus PSI, December 18th 2007 Spin structure of the nucleon 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)

C. Alexandrou, University of Cyprus PSI, December 18th 2007 Transverse spin density Unpolarized u d N N u d q unpolarizedN unpolarized

C. Alexandrou, University of Cyprus PSI, December 18th 2007

Charm Physics HPQCD: C. Davies et al. Determination of CKM matrix, Test unitarity triangle Weak decays - calculate in lattice QCD N f =2+1 dynamical staggered fermions Quenched Full QCD results show impressive agreement with experiment across the board - from light to heavy quarks Lattice QCDFirst message: Use unquenched QCD Bench mark

C. Alexandrou, University of Cyprus PSI, December 18th 2007 Weak Decays and CKM matrix Leptonic D-meson decays: Lattice results versus experiment - CLEO-c cl ν For charm use highly improved staggered action remove further cut off effects at α(am c ) 2 and (am c ) 4 f Ds /f D =1.164(11) f Ds Semi-leptonic: in progress

C. Alexandrou, University of Cyprus PSI, December 18th 2007 B-decays Use non-relativistic QCD for b-quark with m b fixed from m Υ m q/ m s f Bs /f Bq Log- dependence on light quark mass to be compared with experiment ICHEP 06: f B =229(36)(34)MeV Semileptonic B-decays B-->πlν

C. Alexandrou, University of Cyprus PSI, December 18th 2007 Unitarity triangle Using lattice input Expectation: errors on lattice results should halve in the next two years

C. Alexandrou, University of Cyprus PSI, December 18th 2007 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).  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