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Hadron Physics on the Lattice

Πανεπιστήμιο Κύπρου. Κ. Αλεξάνδρου. Hadron Physics on the Lattice. Outline . The Highlight of Lattice 2007- Reaching the chiral regime - Results using light dynamical fermions Twisted mass - ETMC (Cyprus, France, Germany, Italy, Netherlands, UK, Spain, Switzerland)

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Hadron Physics on the Lattice

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

  2. Outline • The Highlight of Lattice 2007- 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 latticeresults C. Alexandrou, University of Cyprus PSI, December 18th 2007

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

  4. Lattice fermions a ψ(n) μ Integrate out Uμ(n)=eigaAμ(n) 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… Basics: Difficult to simulate --> 90’s Det[D]=1 : quenched C. Alexandrou, University of Cyprus PSI, December 18th 2007

  5. Chiral fermions 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 Rely on new improved algorithms 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 PSI, December 18th 2007

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

  7. Lattice 2007: Highlights  Progress in algorithms for calculating D-1 as mπ physical value Deflation methods: project out lowest eigenvalues Lüscher, Wilcox, Orginos/Stathopoulos  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 NF=2+1 domain wall simulations reaching ~300 MeV on ~3 fm volume are underway • JLQCD: Dynamical overlap NF=2 and NF=2+1, ml ~ms/6, small volume (~1.8 fm) C. Alexandrou, University of Cyprus PSI, December 18th 2007

  8. Deflation methods in fermion inverters -Wilcox Solve linear system Twisted mass 203x32 at κcr C. Alexandrou, University of Cyprus PSI, December 18th 2007

  9. Lattice 2007: Highlights  Progress in algorithms for calculating D-1 as mπ physical value Deflation methods: project out lowest eigenvalues Lüscher, Wilcox, Orginos/Stathopoulos  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 NF=2+1 domain wall simulations reaching ~300 MeV on ~3 fm volume are underway • JLQCD: Dynamical overlap NF=2 and NF=2+1, ml ~ms/6, small volume (~1.8 fm)  Chiral extrapolations Using lattice results on fπ/mπ yield LECs to unprecedented accuracy  Beginto address more complex observablese.g. excited states and resonances, decays, finitedensity C. Alexandrou, University of Cyprus PSI, December 18th 2007

  10. 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 fromχPT 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

  11. Pion sector Applications to pion and nucleon mass and decay constants 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 PSI, December 18th 2007

  12. Twisted mass dynamical fermions Consider two degenerate flavors of quarks: Action is Twisted mass μ τ3 acts in flavor space Wilson Dirac operator untwisted mass 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(a2) in practice only affects π0 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

  13. Pion decay constant with twisted mass fermions r0=0.441(14)fm S. Necco, Lat07 NF=2 Simultaneous chiral fits to mπ and fπ Extract low energy constants C. Alexandrou, University of Cyprus PSI, December 18th 2007

  14. ππ-scattering lengths ππ s-wave length a00 and a02 ETMC Leutwyler (hep-ph/0612112) C. Alexandrou, University of Cyprus PSI, December 18th 2007

  15. π-π scattering Lüscher’s finite volume approach: Calculate the energy levels of two particle states in a finite box with pn given by Expand ΔE0 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) mπa02 =-0.0426(27) S.Beans et al. PRD73 (2006) C. Alexandrou, University of Cyprus PSI, December 18th 2007

  16. More statistics • mixed action χPT formula mπa02 =-0.04330(42) A. Walker-Loud, arXiv:0706.3026 π-π scattering - improved • no new LECs • form independent of sea quarks if valence are chiral Other applications: I=1 KK scattering N-N scattering - too noisy C. Alexandrou, University of Cyprus PSI, December 18th 2007

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

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

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

  20. Rho form factors Decreasing quark mass <ρ(p’)|jμ|ρ(p)> decomposed into three form factors Gc(Q2), GM(Q2), GQ(Q2) Related to the ρ quadrupole moment GQ(0)=mρ2 Qρ Adelaide group: Qρ 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 G. Koutsou, Lat07 C. Alexandrou, University of Cyprus PSI, December 18th 2007

  21. Baryon sector Twisted mass and Staggered results Two fit parameters: m0=0.865(10), c1=-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

  22. Delta mass splitting Symmetry: u d --> Δ++ is degenerate with Δ- and Δ+ with Δ0 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

  23. Baryon structure Hybrid actions Wilson fermions • Coupling constants: gA, gπNN, gπΝΔ, … • Form factors: GA(Q2), Gp(Q2), F1(Q2),…. • Parton distribution functions: q(x), Δq(x),… • Generalized parton distribution functions: H(x,ξ,Q2), E(x,ξ,Q2),… Masses: two-point functions <J(tsink) J†(0)> Need to evaluate three-point functions: <J(p’,tsink) O (t,q) J†(p,0)> In addition: Four-point functions: yield detailed info on quark distribution only simple operators used up tp now - need all-to-all propagators C. Alexandrou, University of Cyprus PSI, December 18th 2007

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

  25. Nucleon form factors Isovector form factors obtained from connected diagram Isovector Sachs form factors: with GE and GM given in terms of F1 and F2 In last couple of years better methods have become available, dilution, one-end trick,… Disconnected diagram not evaluated 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

  26. Cyprus/MIT Collaboration, C. A., G. Koutsou, J. W. Negele, A. Tsapalis Nucleon EM form factors QCDSF/UKQCD Dirac FF q2 =- Q2 Results using dynamical Wilson fermions and domain wall valence on staggered sea (from LHPC) are consistent C. Alexandrou, University of Cyprus PSI, December 18th 2007

  27. Th. Korzec 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 GπN(q2) PCAC relates GA and Gp to GπNN: Generalized Goldberger-Treiman relation Pion pole dominance: Goldberger-Treiman relation C. Alexandrou, University of Cyprus PSI, December 18th 2007

  28. 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 monopole fit to experimental results pion pole dominance Cyprus/MIT collaboration monopole fit C. Alexandrou, University of Cyprus PSI, December 18th 2007

  29. Magnetic dipole: dominant Electric quadrupole Coloumb quadrupole N to Δ transition form factors Electromagnetic N to Δ: Write in term of Sachs form factors: 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 C. Alexandrou, University of Cyprus PSI, December 18th 2007

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

  31. Axial N to Δ Bending seen in HBχPT with explicit Δ - Procura et al. SSE to O(ε3): C5A(Q2,mπ)=a1+a2mπ2+a3Q2+loop Again large unquenching effects at small Q2 C. Alexandrou, University of Cyprus PSI, December 18th 2007

  32. GπΝN and GπΝΔ Curves refer to the quenched data • Q2-dependence for GπΝ (GπΝΔ) extracted from GA (C5A) using the Goldberger-Treiman relation deviates at low Q2 • Small deviations from linear Q2-dependence seen at very low Q2 in the case of GπΝΔ • GπΝΔ/GπΝ = 2C5A/GA as predicted by taking ratios of GTRs • GπΝΔ/GπΝ=1.60(1) GTR: GπΝ=GAmN/fπ GπΝ=a(1-bQ2/mπ2) GπΝΔ=1.6GπΝ C. Alexandrou, University of Cyprus PSI, December 18th 2007

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

  34. mπ=0.56 GeV GM1 mπ=0.56 GeV mπ=0.49 GeV GE0 mπ=0.49GeV mπ=0.41 GeV mπ=0.41 GeV Q2 (GeV2) Q2 (GeV2) Δ electromagnetic form factors Four form factors: GE0, GE2,GM1, GM3 given in terms of a1, a2,c1,c2 Like for the nucleon form factors only the connected diagram is evaluated --> isovector part 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

  35. Electric quadrupole Δ form factor GE2 connected to deformation of the Δ: mπ=0.56 GeV mπ=0.49 GeV mπ=0.41 GeV Q2 (GeV2) 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) GE2 <0 GM3 also evaluated, small and negative C. Alexandrou, University of Cyprus PSI, December 18th 2007

  36. 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 PSI, December 18th 2007

  37. 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

  38. Transverse quark densities LHPC <x>u-d Fourier transform of form factors/GPDs gives probability Transverse quark distributions: NLO: gA,0--> gA,lat, same with fπ Self consistent HBχPT M. Burkardt, PRD62 (2000) as n increases slope of An0 decreases C. Alexandrou, University of Cyprus PSI, December 18th 2007

  39. 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 PSI, December 18th 2007

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

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

  42. Charm Physics HPQCD: C. Davies et al. Determination of CKM matrix, Test unitarity triangle Weak decays - calculate in lattice QCD Nf=2+1 dynamical staggered fermions 0.9 1.0 1.1 0.9 1.0 1.1 Bench mark Quenched First message: Use unquenched QCD Lattice QCD Full QCD results show impressive agreement with experiment across the board - from light to heavy quarks C. Alexandrou, University of Cyprus PSI, December 18th 2007

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

  44. fBs/fBq mq/ms Log- dependence on light quark mass B-decays Use non-relativistic QCD for b-quark with mb fixed from mΥ to be compared with experiment ICHEP 06: fB=229(36)(34)MeV Semileptonic B-decays B-->πlν C. Alexandrou, University of Cyprus PSI, December 18th 2007

  45. 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

  46. 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). • 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 PSI, December 18th 2007

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