Hadron Structure in Lattice QCD

Transcription

1 Hadron Structure in Lattice QCD C. Alexandrou University of Cyprus and Cyprus Institute From Quarks and Gluons to Hadrons and Nuclei Erice, September 2011 C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

2 Outline 1 Introduction QCD on the lattice Computational cost Recent results 2 Nucleon Generalized form factors Definitions Results on nucleon form factors Nucleon spin 3 N to transition form factors Nγ N to axial-vector and pseudoscalar form factors 4 form factors electromagnetic form factors axial-vector form factors Pseudo-scalar form factors 5 Conclusions C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

3 Introduction Numerical simulation of QCD already provides essential input for a wide class of strong interaction phenomena QCD phase diagram relevant for Quark-Gluon Plasma: t s and T 10 27, studied in heavy ion collisions at RHIC and LHC Hadron structure: t 10 6 s, experimental program at JLab, Mainz. Momentum distribution of quarks and gluons in the nucleon Hadron form factors e.g. the nucleon axial charge g A Nuclear forces: t 10 9 years, affect the large scale structure of the Universe Exa-scale machines are required to go beyond hadrons to nuclei C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

4 QCD on the lattice a µ ψ(n) U µ (n)=e igaaµ(n) Discretization of space-time in 4 Euclidean dimensions Rotation into imaginary time is the most drastic modification Lattice acts as a non-perturbative regularization scheme with the lattice spacing a providing an ultraviolet cutoff at π/a no infinities Gauge fields are links and fermions are anticommuting Grassmann variables defined at each site of the lattice. They belong to the fundamental representation of SU(3) Construction of an appropriate action such that when a 0 (and Volume ) it gives the continuum theory Construction of the appropriate operators with their renormalization to extract physical quantities Can be simulated on the computer using methods analogous to those used for Statistical Mechanics systems Allows calculations of correlation functions of hadronic operators and matrix elements of any operator between hadronic states in terms of the fundamental quark and gluon degrees of freedom with only input parameters the coupling constant and the quarks masses. = Lattice QCD provides a well-defined approach to calculate observables non-perturbative starting directly from the QCD Langragian. Consider simplest isotropic hypercubic grid: a = a S = a T and size N S N S N S N T, N T > N S. Finite Volume: 1. Finite volume effects need to be studied Take box sizes such that L S m π > Only discrete values of momentum in units of 2π/L S are allowed. Finite lattice spacing: Need at least three values of the lattice spacing in order to extrapolate to the continuum limit. q 2 -values: Fourier transform of lattice results in coordinate space taken numerically for large values of momentum transfer results are too noisy = Limited to Q 2 = q 2 2 GeV 2. C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

5 QCD on the lattice a µ ψ(n) U µ (n)=e igaaµ(n) Discretization of space-time in 4 Euclidean dimensions Rotation into imaginary time is the most drastic modification Lattice acts as a non-perturbative regularization scheme with the lattice spacing a providing an ultraviolet cutoff at π/a no infinities Gauge fields are links and fermions are anticommuting Grassmann variables defined at each site of the lattice. They belong to the fundamental representation of SU(3) Construction of an appropriate action such that when a 0 (and Volume ) it gives the continuum theory Construction of the appropriate operators with their renormalization to extract physical quantities Can be simulated on the computer using methods analogous to those used for Statistical Mechanics systems Allows calculations of correlation functions of hadronic operators and matrix elements of any operator between hadronic states in terms of the fundamental quark and gluon degrees of freedom with only input parameters the coupling constant and the quarks masses. = Lattice QCD provides a well-defined approach to calculate observables non-perturbative starting directly from the QCD Langragian. Consider simplest isotropic hypercubic grid: a = a S = a T and size N S N S N S N T, N T > N S. Finite Volume: 1. Finite volume effects need to be studied Take box sizes such that L S m π > Only discrete values of momentum in units of 2π/L S are allowed. Finite lattice spacing: Need at least three values of the lattice spacing in order to extrapolate to the continuum limit. q 2 -values: Fourier transform of lattice results in coordinate space taken numerically for large values of momentum transfer results are too noisy = Limited to Q 2 = q 2 2 GeV 2. C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

6 Computational cost ( Simulation cost: C sim ) cm ( 300MeV L mπ 2fm ) cl ( 0.1fm ) ca a Coefficients c m, c L and c a depend on the discretized action used for the fermions. State-of-the-art simulations use improved algorithms: Mass preconditioner, M. Hasenbusch, Phys. Lett. B519 (2001) 177 Multiple time scales in the molecular dynamics updates = for twisted mass fermions: c m 4, c L 5 and c a 6. Results at physical quark masses require O(1) Pflop.Years. After post-diction of well measured quantities the goal is to predict quantities that are difficult or impossible to measure experimentally. L=2.1 fm, a=0.089 fm, K. Jansen and C. Urbach, arxiv: C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

7 Mass of low-lying hadrons N F = smeared Clover fermions, BMW Collaboration, S. Dürr et al. Science 322 (2008) N F = 2 twisted mass fermions, ETM Collaboration, C. Alexandrou et al. PRD (2008) BMW with N F = 2 + 1: 3 lattice spacings: a 0.125, 0.085, fm set by m Ξ Pion masses: m π > 190 MeV Volumes:m min π L > 4 ETMC with N F = 2: 3 lattice spacings: a = 0.089, 0.070, a = fm, set by m N m π > 260 MeV Volumes:m min π L > 3.3 Good agreement between different discretization schemes = Significant progress in understanding the masses of low-lying mesons and baryons C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

8 Mass of low-lying hadrons N F = smeared Clover fermions, BMW Collaboration, S. Dürr et al. Science 322 (2008) N F = 2 twisted mass fermions, ETM Collaboration, C. Alexandrou et al. PRD (2008) Baryon Spectrum M(GeV) ! N # * # " K $ * $ % Exp. results Width Input ETMC results BMW results BMW with N F = 2 + 1: 3 lattice spacings: a 0.125, 0.085, fm set by m Ξ Pion masses: m π > 190 MeV Volumes:m min π L > 4 ETMC with N F = 2: 3 lattice spacings: a = 0.089, 0.070, a = fm, set by m N m π > 260 MeV Volumes:m min π L > Strangeness Good agreement between different discretization schemes = Significant progress in understanding the masses of low-lying mesons and baryons C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

9 Pion form factor Several Collaborations using dynamical quarks with pion masses down to about 300 MeV ETMC, N F = 2, R. Frezzotti, V. Lubicz and S. Simula, PRD 79, (2009) Examine volume and cut-off effects estimate continuum and infinite volume values Twisted boundary conditions to probe small Q 2 = q 2, Talk by A. Juttner All-to-all propagators and one-end trick to obtain accurate results ( ) 1 Chiral extrapolation using NNLO r 2 and F π(q 2 ) = 1 + r 2 Q 2 /6 f π π PACS-CS (N f =2+1) m π =411 MeV PACS-CS (N f =2+1) m π =296 MeV ETMC (N f =2) physical point UKQCD (N f =2+1) m π =330 MeV N5 F6 A (q*r 0 ) 2 B.B. Brandt, A. Juttner, H. Wittig (CLS) arxiv: C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

10 Pion form factor Several Collaborations using dynamical quarks with pion masses down to about 300 MeV ETMC, N F = 2, R. Frezzotti, V. Lubicz and S. Simula, PRD 79, (2009) Examine volume and cut-off effects estimate continuum and infinite volume values Twisted boundary conditions to probe small Q 2 = q 2, Talk by A. Juttner All-to-all propagators and one-end trick to obtain accurate results ( ) 1 Chiral extrapolation using NNLO r 2 and F π(q 2 ) = 1 + r 2 Q 2 /6 C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

11 Pion form factor Several Collaborations using dynamical quarks with pion masses down to about 300 MeV ETMC, N F = 2, R. Frezzotti, V. Lubicz and S. Simula, PRD 79, (2009) Examine volume and cut-off effects estimate continuum and infinite volume values Twisted boundary conditions to probe small Q 2 = q 2, Talk by A. Juttner All-to-all propagators and one-end trick to obtain accurate results ( ) 1 Chiral extrapolation using NNLO r 2 and F π(q 2 ) = 1 + r 2 Q 2 /6 1.0 CERN 0.8 DESY F! (Q 2 ) JLAB experimental radius ETMC radius Q 2 (GeV 2 ) C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

12 ρ-meson width Consider π + π in the I = 1-channel Estimate P-wave scattering phase shift δ 11 (k) using finite size methods Use Lüscher s relation between energy in a finite box and the phase in infinite volume Use Center of Mass frame and Moving frame Use effective range formula: tanδ 11 (k) = g2 ρππ 6π g ρππ and then extract Γ ρ = g2 ρππ 6π m π = 309 MeV, L = 2.8 fm k R 3 m R 2, k R = k 3 E (m R 2, k = E 2 /4 m 2 E2) π determine m R and mr 2 /4 m2 π 1 sin 2 (δ) 0.5 CMF MF1 MF2 sin 2 (δ)=1=>am R ae CM N F = 2 twisted mass fermions, Xu Feng, K. Jansen and D. Renner, Phys. Rev. D83 (2011) C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

13 ρ-meson width Consider π + π in the I = 1-channel Estimate P-wave scattering phase shift δ 11 (k) using finite size methods Use Lüscher s relation between energy in a finite box and the phase in infinite volume Use Center of Mass frame and Moving frame Use effective range formula: tanδ 11 (k) = g2 ρππ 6π g ρππ and then extract Γ ρ = g2 ρππ 6π m π = 309 MeV, L = 2.8 fm k R 3 m R 2, k R = k 3 E (m R 2, k = E 2 /4 m 2 E2) π determine m R and mr 2 /4 m2 π 1 sin 2 (δ) 0.5 CMF MF1 MF2 sin 2 (δ)=1=>am R Γ R (GeV) a=0.086fm a=0.067fm PDG data ae CM m π (GeV ) N F = 2 twisted mass fermions, Xu Feng, K. Jansen and D. Renner, Phys. Rev. D83 (2011) C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

14 Definition of Generalized Form Factors High energy scattering: Formulate in terms of light-cone correlation functions, M. Diehl, Phys. Rep. 388 (2003) Consider one-particle states p and p Generalized Parton Distributions (GPDs), X. Ji, J. Phys. G24(1998)1181 F Γ (x, ξ, q 2 ) = 1 2 ig dλ 2π eixλ p ψ( λn/2)γpe λ/2 λ/2 dαn A(nα) ψ(λn/2) p where q = p p, P = (p + p)/2, n is a light-cone vector and P.n = 1 Γ = /n 1 ] 2 ū(p ) [/nh(x, ξ, q 2 ) + i nµqνσµν 2m E(x, ξ, q2 ) u(p) Γ = /nγ 5 1 [ ] 2 ū(p ) /nγ 5 H(x, ξ, q 2 ) + n.qγ5 Ẽ(x, ξ, q 2 ) u(p) 2m Γ = n µσ µν tensor GPDs Handbag diagram Expansion of the light cone operator leads to a tower of local twist-2 operators O µ 1...µn : Diagonal matrix element P O(x) P (DIS) parton distributions: q(x), q(x), δq(x) O µµ 1...µn q Õ µµ 1...µn q O ρµµ 1...µn δq = qγ {µ id µ 1... id µn} q = qγ 5γ {µ id µ 1... id µn} q = qσ ρ{µ id µ 1... id µn} q unpolarized x n q = helicity x n q = transversity x n δq = 1 dx x n [ q(x) ( 1) n q(x) ] 0 1 dx x n [ q(x) + ( 1) n q(x) ] 0 1 dx x n [ δq(x) ( 1) n δ q(x) ] 0 where q = q + q, q = q q, δq = q T + q Off-diagonal matrix elements (DVCS) generalized form factors C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

15 Definition of Generalized Form Factors High energy scattering: Formulate in terms of light-cone correlation functions, M. Diehl, Phys. Rep. 388 (2003) Consider one-particle states p and p Generalized Parton Distributions (GPDs), X. Ji, J. Phys. G24(1998)1181 F Γ (x, ξ, q 2 ) = 1 2 ig dλ 2π eixλ p ψ( λn/2)γpe λ/2 λ/2 dαn A(nα) ψ(λn/2) p where q = p p, P = (p + p)/2, n is a light-cone vector and P.n = 1 Γ = /n 1 ] 2 ū(p ) [/nh(x, ξ, q 2 ) + i nµqνσµν 2m E(x, ξ, q2 ) u(p) Γ = /nγ 5 1 [ ] 2 ū(p ) /nγ 5 H(x, ξ, q 2 ) + n.qγ5 Ẽ(x, ξ, q 2 ) u(p) 2m Γ = n µσ µν tensor GPDs Handbag diagram Expansion of the light cone operator leads to a tower of local twist-2 operators O µ 1...µn : Diagonal matrix element P O(x) P (DIS) parton distributions: q(x), q(x), δq(x) O µµ 1...µn q Õ µµ 1...µn q O ρµµ 1...µn δq = qγ {µ id µ 1... id µn} q = qγ 5γ {µ id µ 1... id µn} q = qσ ρ{µ id µ 1... id µn} q unpolarized x n q = helicity x n q = transversity x n δq = 1 dx x n [ q(x) ( 1) n q(x) ] 0 1 dx x n [ q(x) + ( 1) n q(x) ] 0 1 dx x n [ δq(x) ( 1) n δ q(x) ] 0 where q = q + q, q = q q, δq = q T + q Off-diagonal matrix elements (DVCS) generalized form factors C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

16 Nucleon generalized form factors Decomposition of matrix elements into generalized form factors: N(p, s ) O µµ 1...µn [ ( q N(p, s) = ū(p, s n ) i=0 even A n+1,i (q 2 )γ {µ + B n+1,i (q 2 ) iσ{µα qα 2m +mod(n, 2)C n+1,0 (q 2 ) 1 m q{µ q µ 1... q µn} ]u(p, s) And similarly for O q in terms of Ãni (q 2 ), B ni (q 2 ) and O δq in terms of A T ni, BT ni, CT ni and D T ni ) q µ 1... q µ i P µ i+1... P µn} Special cases: n = 1: ordinary nucleon form factors A 10 (q 2 ) = F 1 (q 2 ) = 1 1 dxh(x, ξ, q2 ), B 10 (q 2 ) = F 2 (q 2 ) = 1 1 dxe(x, ξ, q2 ) Ã 10 (q 2 ) = G A (q 2 ) = 1 1 dx H(x, ξ, q 2 ), B10 (q 2 ) = G p(q 2 ) = 1 1 dxẽ(x, ξ, q2 ) where j µ = ψγ µψ = γ µf 1 (q 2 iσµν qν ) + 2m F 2 (q 2 ) The Dirac F 1 and Pauli F 2 are related to the electric and magnetic Sachs form factors: G E (q 2 ) = F 1 (q 2 ) q2 (2m) 2 F 2(q 2 ), G M (q 2 ) = F 1 (q 2 ) + F 2 (q 2 ) j µ = ψγ [ ] τ µγ a 5 2 ψ(x) = i γ µγ 5G A (q 2 ) + qµ γ 5 2m Gp(q2 τ ) a 2 A n0 (0), Ãn0(0), A T n0 (0) are moments of parton distributions, e.g. x q = A 20(0) and x q = Ã20(0) are the spin independent and helicity distributions can evaluate quark spin, J q = 1 2 [A 20(0) + B 20 (0)] = 1 2 Σq + Lq nucleon spin sum rule: 1 2 = 1 2 Σq + Lq + Jg, momentum sum rule: x g = 1 A 20(0) C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

17 Nucleon generalized form factors Decomposition of matrix elements into generalized form factors: N(p, s ) O µµ 1...µn [ ( q N(p, s) = ū(p, s n ) i=0 even A n+1,i (q 2 )γ {µ + B n+1,i (q 2 ) iσ{µα qα 2m +mod(n, 2)C n+1,0 (q 2 ) 1 m q{µ q µ 1... q µn} ]u(p, s) And similarly for O q in terms of Ãni (q 2 ), B ni (q 2 ) and O δq in terms of A T ni, BT ni, CT ni and D T ni ) q µ 1... q µ i P µ i+1... P µn} Special cases: n = 1: ordinary nucleon form factors A 10 (q 2 ) = F 1 (q 2 ) = 1 1 dxh(x, ξ, q2 ), B 10 (q 2 ) = F 2 (q 2 ) = 1 1 dxe(x, ξ, q2 ) Ã 10 (q 2 ) = G A (q 2 ) = 1 1 dx H(x, ξ, q 2 ), B10 (q 2 ) = G p(q 2 ) = 1 1 dxẽ(x, ξ, q2 ) where j µ = ψγ µψ = γ µf 1 (q 2 iσµν qν ) + 2m F 2 (q 2 ) The Dirac F 1 and Pauli F 2 are related to the electric and magnetic Sachs form factors: G E (q 2 ) = F 1 (q 2 ) q2 (2m) 2 F 2(q 2 ), G M (q 2 ) = F 1 (q 2 ) + F 2 (q 2 ) j µ = ψγ [ ] τ µγ a 5 2 ψ(x) = i γ µγ 5G A (q 2 ) + qµ γ 5 2m Gp(q2 τ ) a 2 A n0 (0), Ãn0(0), A T n0 (0) are moments of parton distributions, e.g. x q = A 20(0) and x q = Ã20(0) are the spin independent and helicity distributions can evaluate quark spin, J q = 1 2 [A 20(0) + B 20 (0)] = 1 2 Σq + Lq nucleon spin sum rule: 1 2 = 1 2 Σq + Lq + Jg, momentum sum rule: x g = 1 A 20(0) C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

18 Nucleon generalized form factors Decomposition of matrix elements into generalized form factors: N(p, s ) O µµ 1...µn [ ( q N(p, s) = ū(p, s n ) i=0 even A n+1,i (q 2 )γ {µ + B n+1,i (q 2 ) iσ{µα qα 2m +mod(n, 2)C n+1,0 (q 2 ) 1 m q{µ q µ 1... q µn} ]u(p, s) And similarly for O q in terms of Ãni (q 2 ), B ni (q 2 ) and O δq in terms of A T ni, BT ni, CT ni and D T ni ) q µ 1... q µ i P µ i+1... P µn} Special cases: n = 1: ordinary nucleon form factors A 10 (q 2 ) = F 1 (q 2 ) = 1 1 dxh(x, ξ, q2 ), B 10 (q 2 ) = F 2 (q 2 ) = 1 1 dxe(x, ξ, q2 ) Ã 10 (q 2 ) = G A (q 2 ) = 1 1 dx H(x, ξ, q 2 ), B10 (q 2 ) = G p(q 2 ) = 1 1 dxẽ(x, ξ, q2 ) where j µ = ψγ µψ = γ µf 1 (q 2 iσµν qν ) + 2m F 2 (q 2 ) The Dirac F 1 and Pauli F 2 are related to the electric and magnetic Sachs form factors: G E (q 2 ) = F 1 (q 2 ) q2 (2m) 2 F 2(q 2 ), G M (q 2 ) = F 1 (q 2 ) + F 2 (q 2 ) j µ = ψγ [ ] τ µγ a 5 2 ψ(x) = i γ µγ 5G A (q 2 ) + qµ γ 5 2m Gp(q2 τ ) a 2 A n0 (0), Ãn0(0), A T n0 (0) are moments of parton distributions, e.g. x q = A 20(0) and x q = Ã20(0) are the spin independent and helicity distributions can evaluate quark spin, J q = 1 2 [A 20(0) + B 20 (0)] = 1 2 Σq + Lq nucleon spin sum rule: 1 2 = 1 2 Σq + Lq + Jg, momentum sum rule: x g = 1 A 20(0) C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

19 Lattice evaluation Evaluation of two-point and three-point functions G( q, t) = xf e i x f q Γ 4 βα Jα( x f, t f )J β (0) G µν (Γ, q, t) = xf, x e i x q Γ βα J α( x f, t f )O µν ( x, t)j β (0) Sequential inversion through the sink fix sink-source separation t f t i, final momentum p f = 0, Γ Apply smearing techniques to improve ground state dominance in three-point correlators Ratios: Leading time dependence cancels ae eff( q, t) = ln [ G( q, t)/g( q, t + a) ] ae( q) R µν (Γ, q, t) = Π µν ( q, Γ) Gµν (Γ, q,t) G( p i,t f t)g( 0,t)G( 0,t f ) G( 0,t f ) G( 0,t f t)g( p i,t)g( p i,t f ) C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

20 Lattice evaluation Evaluation of two-point and three-point functions G( q, t) = xf e i x f q Γ 4 βα Jα( x f, t f )J β (0) G µν (Γ, q, t) = xf, x e i x q Γ βα J α( x f, t f )O µν ( x, t)j β (0) Sequential inversion through the sink fix sink-source separation t f t i, final momentum p f = 0, Γ Apply smearing techniques to improve ground state dominance in three-point correlators Ratios: Leading time dependence cancels ae eff( q, t) = ln [ G( q, t)/g( q, t + a) ] ae( q) R µν (Γ, q, t) = Π µν ( q, Γ) Gµν (Γ, q,t) G( p i,t f t)g( 0,t)G( 0,t f ) G( 0,t f ) G( 0,t f t)g( p i,t)g( p i,t f ) Variational approach in the case of mesons leads to improvement on plateaux: B. Blossier et at., (Alpha Collaboration), JHEP 0904 (2009) C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

21 Lattice evaluation Evaluation of two-point and three-point functions G( q, t) = xf e i x f q Γ 4 βα Jα( x f, t f )J β (0) G µν (Γ, q, t) = xf, x e i x q Γ βα J α( x f, t f )O µν ( x, t)j β (0) Sequential inversion through the sink fix sink-source separation t f t i, final momentum p f = 0, Γ Apply smearing techniques to improve ground state dominance in three-point correlators Ratios: Leading time dependence cancels ae eff( q, t) = ln [ G( q, t)/g( q, t + a) ] ae( q) G E ( t sink t src ) / a = 12 ( t sink t src ) / a = 8 R µν (Γ, q, t) = Π µν ( q, Γ) Gµν (Γ, q,t) G( p i,t f t)g( 0,t)G( 0,t f ) G( 0,t f ) G( 0,t f t)g( p i,t)g( p i,t f ) Q 2 in GeV 2 Electric form factor: t f t i > 1 fm However, this might be operator dependent C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

22 Study of excited state contributions N F = with m π 380 MeV and a = 0.08 fm Vary source- sink separation: ga fixed sink method, t sink = 12a PDG open sink method t sink/a S. Dinter, C.A. M. Constantinou, V. Drach, K. Jansen and D. Renner, arxiv: C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

23 Study of excited state contributions N F = with m π 380 MeV and a = 0.08 fm Vary source- sink separation: < x >u d fixed sink method, t sink = 12a 0.05 ABMK fit open sink method t sink/a = Excited contributions are operator dependent g A unaffected, x u d 10% lower S. Dinter, C.A. M. Constantinou, V. Drach, K. Jansen and D. Renner, arxiv: C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

24 Non-perturbative renormalization Most collaborations use non-perturbative renormalization. ETMC: RI -MOM renormalization scheme as in e.g. M. Göckeler et al., Nucl. Phys. B544,699 Fix to Landau gauge and compute: S u (p) = a8 V x,y e ip(x y) u(x)ū(y) G(p) = a12 V x,y,z,z e ip(x y) u(x)ū(z)j (z, z )d(z ) d(y) Amputated vertex functions: Γ(p) = (S u (p)) 1 G(p) (S d (p)) 1 Renormalization functions: Z q and Z O Mass independent renormalization scheme need chiral extrapolations Subtract O(a 2 ) perturbatively. Z V Z A Z sub DV1 Z sub DA m π (GeV) (a p) 2 C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

25 Non-perturbative renormalization Most collaborations use non-perturbative renormalization. RBC: Also uses a RI -MOM renormalization scheme but with momentum independent source, Y. Aoki al. arxiv: Z sub DV1 Z sub DA m π (GeV) Similarly for < x > u d non-perturbative renormalization may explain the lower values observed by LHPC Z V Z A (a p) 2 C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

26 Disconnected contributions G M (Q 2 ) C G M (Q 2 ) D Approximate using stochastic techniques A study was carried out comparing stochastic methods to the exact evaluation enabled using GPUs; N f = 2 Wilson fermions (SESAM Collaboration) Disconnected loops contributing to nucleon form factors show slow convergence The truncated solver method is best suited, G. Bali, S. Collins, A. Schafer Comput.Phys.Commun. 181 (2010) 1570 Loops with a scalar inversion are much easier to compute Q 2 (GeV) Exact 3000 Noise Truncated Q 2 (GeV) C.A., K. Hadjiyiannakou, G. Koutsou, A. O Cais, A. Strelchenko, arxiv: G s (Q 2 ) C G s (Q 2 ) D Q 2 (GeV) exact 500 noise Truncated Q 2 (GeV) C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

27 Nucleon form factors Many lattice studies down to lowest pion mass of m π 300 MeV = Lattice data in general agreement Axial-vector FFs: A a µ = τ ψγ µγ a [ 5 2 ψ(x) ] = 1 2 γ µγ 5G A (q 2 ) + qµ γ 5 2m Gp(q2 ) Axial charge is well known experimentally, straight forward to compute in lattice QCD Agreement among recent lattice results - all use non-perturbative Z A Weak light quark mass dependence What can we say about the physical value of g A? Similar discrepancy also for the momentum fraction C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

28 Physical results on g A What can we say about the physical value of g A? Use results obtained with twisted mass fermions C. A. et al. (ETMC), Phys. Rev. D83 (2011) Take continuum limit and estimate volume corrections, A. Ali Khan, et al., PRD 74, (2006) Use one-loop chiral perturbation theory in the small scale expansion (SSE), T. R. Hemmert, M. Procura and W. Weise, PRD 68, (2003). 3 fit parameters, g 0 A = 1.10(8), g =2.1(1.3), C SSE (1 GeV ) = 0.7(1.7), axial N coupling fixed to 1.5: g A = 1.14(6) Fitting lattice results directly leads to g A = 1.12(7) Lattice determination of the axial charges of other baryons can provide input for χpt, H.- W. Lin and K. Orginos, PRD 79, (2009); M. Gockeler et l., arxiv: Volume and discretization errors small compared to the uncertainty in the chiral expansion C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

29 Physical results on g A What can we say about the physical value of g A? Use results obtained with twisted mass fermions C. A. et al. (ETMC), Phys. Rev. D83 (2011) Take continuum limit and estimate volume corrections, A. Ali Khan, et al., PRD 74, (2006) Use one-loop chiral perturbation theory in the small scale expansion (SSE), T. R. Hemmert, M. Procura and W. Weise, PRD 68, (2003). 3 fit parameters, g 0 A = 1.10(8), g =2.1(1.3), C SSE (1 GeV ) = 0.7(1.7), axial N coupling fixed to 1.5: g A = 1.14(6) Fitting lattice results directly leads to g A = 1.12(7) Lattice determination of the axial charges of other baryons can provide input for χpt, H.- W. Lin and K. Orginos, PRD 79, (2009); M. Gockeler et l., arxiv: Volume and discretization errors small compared to the uncertainty in the chiral expansion C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

30 Nucleon form factors G E (q 2 ) = F 1 (q 2 ) q2 (2m) 2 F 2(q 2 ), G M (q 2 ) = F 1 (q 2 ) + F 2 (q 2 ) Results from ETMC (arxiv: ), LHPC using DWF (S. N. Syritsyn, PRD 81, (2010)) and a hybrid action (J. D. Bratt et al., arxiv:1001:3620), and from CLS using Clover, (H. Wittig) Can we get results at physical point? C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

31 Chiral extrapolation of electromagnetic form factors Baryon chiral perturbation theory to one-loop, with d.o.f.(sse) and iso-vector N coupling included in LO, T. R. Hemmert and W. Weise, Eur. Phys. J. A 15,487 (2002); M. Gockeler et al., PRD 71, (2005). Fit F 1 (m π, Q 2 ) and F 2 (m π, Q 2 ) with 5 parameters: (κ v ), the isovector (c v ) and axial N to (c A ) couplings and two counterterms C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

32 Nucleon momentum fraction Momentum fraction x u d = A isovector 20 is calculated similarly to g A ( xf, tf) OΓ ( x, t) q = p p ( xi, ti) Physical point: x u d from S. Alekhin et al. arxiv: HBχPT for x u d and x u d, D. Arndt, M. Savage, NPA 697, 429 (2002); W. Detmold, W Melnitchouk, A. Thomas, PRD 66, (2002) Fit ETMC results with λ 2 = 1 GeV 2 x u d = C [ 1 3g2 A + 1 (4πf π) 2 m2 π ln m2 π λ 2 ] + c 8(λ 2 )m 2 π (4πf π) 2 x u d = C [ 1 2g2 A + 1 (4πf π) 2 m2 π ln m2 π λ 2 ] + c 8(λ 2 )m 2 π (4πf π) 2 C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

33 Nucleon spin Results using N F = 2 TMF for 270 MeV < m π < 500 MeV, C. Alexandrou et al. (ETMC), arxiv: In qualitative agreement with J. D. Bratt et al. (LHPC), PRD82 (2010) contributions to nucleon spin J u J d m π 2 (GeV 2 ) C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

34 Nucleon spin Results using N F = 2 TMF for 270 MeV < m π < 500 MeV, C. Alexandrou et al. (ETMC), arxiv: In qualitative agreement with J. D. Bratt et al. (LHPC), PRD82 (2010) contributions to nucleon spin 0.4 J u J d m π (GeV 2 ) contributions to nucleon spin J u J d Π J u d FIG. 42: Chiral extrapolations of J u,d using = Total spin for u-quarks J u 0.25 and for d-quark J d 0 BChPT. Note that the displayed lattice data points were not directly employed in the chiral fits. Details are given in the text. FIG. 43: Chir HBChPT includ The fit and error Since the GFF B 20(t) cannot be extracted directly at t = 0, we have first pe of B u and B d to t = 0, and combined this with our values for x u+d = 20(t) 20(t) B20 u+d (0))/2. The resulting lattice data points, including the full jackknife B 20(t) to the forward limit, are displayed in Fig. 43. Chiral fits based on Eq b qn (A + B) 0,u+d 20, b q and J mπ,u+d, to the data with m π 600MeV and Disconnected contributions neglected shaded error band and the curves (representing the upper and lower bounds cases, we have fixed = 0.3GeV and used the large-n c relation g πn = 3/(2 C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) The fit to our lattice results with m π 600 MeV gives (A + B) 0,u+d 20 = J u+d HBChPT+ (mphys π ) = 0.245(30) at the physical pion mass. Including the d Hadron Structure we find in consistent Lattice QCD values with somewhat smaller Erice, errors, Sept. (A B) 0,u+d 20 = / 32 b qn

35 Nγ form factors A dominant magnetic dipole, M1 An electric quadrupole, E2 and a Coulomb, C2 signal a deformation in the nucleon/ 1/2-spin particles have vanishing quadrupole moment in the lab-frame Probe nucleon shape by studying transitions to its excited -state Difficult to measure/calculate since quadrupole amplitudes are sub-dominant R EM (EMR) = G E2 (Q2 ) G M1 (Q 2 ), R SM (CMR) = q 2m G C2 (Q 2 ) G M1 (Q 2 ), in lab frame of the. Precise data strongly suggesting deformation in the Nucleon/ At Q 2 = GeV 2 : EMR=( 2.00 ± 0.40 stat+sys ± 0.27 mod)%, CMR=( 6.27 ± 0.32 stat+sys ± 0.10 mod)% C. N. Papanicolas, Eur. Phys. J. A18 (2003); N. Sparveris et al., PRL 94, (2005) C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

36 Lattice evaluation (p, s ) j µ N(p, s) = i 2 Evaluation of two-point and three-point functions 3 ( ) ] m m 1/2 N ū σ(p, s )[ G E (p M1 ) E N (p) (q2 )K M1 σµ + G E2 (q2 )K E2 σµ + G C2 K C2 σµ u(p, s) OΓ G( q, t) = xf e i x f q Γ 4 βα Jα( x f, t f )J β (0) (xf,tf) (x,t) q = p p (xi,ti) G µν (Γ, q, t) = xf, x e i x q Γ βα J α( x f, t f )O µ ( x, t)j β (0) R J σ (t 2, t 1 ; p, p ; Γτ ; µ) = G JµN σ (t 2, t 1 ; p, p; Γτ ) [ G (t ii 2, p ; Γ 4 ) G NN (t 2, p; Γ 4 ) G NN (t 2 t 1, p; Γ 4 ) G ii (t 1, p ; Γ 4 ) ] 1/2 G ii (t 2, p ; Γ 4 ) G ii (t 2 t 1, p ; Γ 4 ) G NN (t 1, p; Γ 4 ) Construct optimized sources to isolate quadrupoles three-sequential inversions needed 3 3 S J 1 (q; J) = Π J σ (0, q ; Γ 4; J), S J 2 (q; J) = Π J σ (0, q ; Γ k ; J) σ=1 σ k=1 S J 3 (q; J) = ΠJ 3 (0, q ; Γ 3; J) 1 [ ] Π J 1 2 (0, q ; Γ 1; J) + Π J 2 (0, q ; Γ 2; J) Use the coherent sink technique: create four sets of forward propagators for each configuration at source positions separated in time by one-quarter of the total temporal size, Syritsyn et al. (LHPC), Phys. Rev. D81 (2009) C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

37 Results on magnetic dipole C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

38 Results on magnetic dipole Slope smaller than experiment, underestimate G M1 at low Q2 pion cloud effects? New results using N f = dynamical Domain Wall Fermions, simulated by RBC-UKQCD Collaborations = No visible improvement. C. A., G.Koutsou, J.W. Negele, Y. Proestos, A. Tsapalis, Phys. Rev. D83 (2011) Situation like for nucleon form factors, independent of lattice discretization C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

39 Results on EMR and CMR Systematic errors may cancel in rations: G E2 and G C2 are suppressed at low Q 2 like G M1 = look at EMR and CMR New results using N f = dynamical domain wall fermions by RBC-UKQCD Collaborations Need large statistics to reduce the errors = as m π 140 MeV O(10 3 ) need to be analyzed. C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

40 N to axial-vector and pseudoscalar form factors (p, s ) A 3 µ N(p, s) = Aūλ (p, s ) ( ) A = i 23 m m 1/2 N E (p )E N (p) [ CA 3 (q2 ) m N γ ν + C A C A 5 (q2 ) analogous to the nucleon G A (q 2 ) 4 (q2 ) m N 2 p ν ( g λµ gρν g λρ gµν C A 6 (q2 ), analogous to the nucleon G p(q 2 ) pion pole behaviour C A 3 (q2 ) and C A 4 (q2 ) are suppressed (transverse part of the axial-vector) Study also the pseudo-scalar transition form factor G πn (q 2 ) = Non-diagonal Goldberger-Treiman relation: Pion pole dominance relates C A 6 to G πn through: Goldberger-Treiman relation becomes C A 5 (q2 ) + q2 C A mn 2 6 (q2 ) = 1 G πn (q 2 )f πm 2 π 2m N mπ 2. q2 1 m N C A 6 (q2 ) 1 2 G πn (q 2 )f π m 2 π q2 G πn (q 2 ) f π = 2m N C A 5 (q2 ) ) q ρ +C 5 A (q2 )g λµ + CA 6 (q2 ] ) m N 2 q λ qµ u(p, s) C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

41 N to axial-vector and pseudoscalar form factors (p, s ) A 3 µ N(p, s) = Aūλ (p, s ) ( ) A = i 23 m m 1/2 N E (p )E N (p) [ CA 3 (q2 ) m N γ ν + C A C A 5 (q2 ) analogous to the nucleon G A (q 2 ) 4 (q2 ) m N 2 p ν ( g λµ gρν g λρ gµν C A 6 (q2 ), analogous to the nucleon G p(q 2 ) pion pole behaviour C A 3 (q2 ) and C A 4 (q2 ) are suppressed (transverse part of the axial-vector) Study also the pseudo-scalar transition form factor G πn (q 2 ) = Non-diagonal Goldberger-Treiman relation: Pion pole dominance relates C A 6 to G πn through: Goldberger-Treiman relation becomes C A 5 (q2 ) + q2 C A mn 2 6 (q2 ) = 1 G πn (q 2 )f πm 2 π 2m N mπ 2. q2 1 m N C A 6 (q2 ) 1 2 G πn (q 2 )f π m 2 π q2 G πn (q 2 ) f π = 2m N C A 5 (q2 ) ) q ρ +C 5 A (q2 )g λµ + CA 6 (q2 ] ) m N 2 q λ qµ u(p, s) C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

42 N to Goldberger-Treiman relation 2 HYB, mπ = 353 MeV DWF, mπ = 330 MeV DWF, mπ = 297 MeV 2 HYB, mπ = 353 MeV DWF, mπ = 330 MeV DWF, mπ = 297 MeV mnfπgπn 2(m 2 π+q 2 )C 6 A 1 fπgπn 2mNC 5 A Q 2 (GeV 2 ) 1 Pion-pole dominance: m C A N 6 (Q2 ) 1 G πn (Q 2 )fπ 2 mπ 2 Goldberger-Treiman rel.: G +Q2 πn (Q 2 )f π = 2m N C A 5 (Q2 ) Similar behavior as in the nucleon system, i.e. between G πnn and G p, and G πnn and G A. Ratio: G πn /G πnn Q 2 (GeV 2 ) C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

43 electromagnetic form factors (p, s ) j µ (0) (p, s) = ūα(p, s ) F 1 (Q2 )g αβ + F 3 (Q2 ) qα q β (2M ) 2 γ µ + F 2 (Q2 )g αβ + F 4 (Q2 ) qα q β (2M ) 2 ( with e.g. the quadrupole form factor given by: G E2 = F 1 ) τf 2 1 ( 2 (1 + τ) F 3 ) τf, where τ Q 4 2 /(4M 2 ) Construct an optimized source to isolate G E2 additional sequential propagators needed. Neglect disconnected contributions in this evaluation. iσµν qν 2M u β (p, s) Transverse charge density of a polarized along the x-axis can be defined in the infinite momentum frame ρ T 2 3 ( b) and ρ T 1 ( b). 2 Using G E2 we can predict shape of G E2 2 b y [fm] 0.0 b y [fm] quenched Wilson, m π = 410 MeV hybrid, m π = 353 MeV dynamical Wilson, m π = 384 MeV Q 2 in GeV b x [fm] b x [fm] with spin 3/2 projection elongated along spin axis compared to the Ω C. A., T. Korzec, G. Koutsou, C. Lorcé, J. W. Negele, V. Pascalutsa, A. Tsapalis, M. Vanderhaeghen, NPA825,115 (2009). C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

44 axial-vector form factors Axial-vector current: A a τa µ (x) = ψ(x)γµγ5 2 ψ(x) (p, s ) A 3 µ (0) (p, s) = ūα(p, s ) 1 ( g αβ 2 g 1 (q 2 )γ µ γ 5 + g 3 (q 2 ) qµ ) γ 5 + qα q β 2M 4M 2 (h1 (q 2 )γ µ γ 5 + h3 (q 2 ) qµ ) γ 5 uβ (p, s) 2M i.e. 4 axial form-factors, g 1, g 3, h 1 and h 3 at q 2 = 0 we can extract the axial charge C. A., E. Gregory, T. Korzec, G. Koutsou, J. W. Negele, T. Sato, A. Tsapalis, arxiv: C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

45 pseudoscalar couplings Pseudoscalar current: P a τ (x) = ψ(x)γ a 5 2 ψ(x) matrix element: [ ] (p, s ) P 3 (0) (p, s) = ū α(p, s ) 1 2 g αβ g(q 2 )γ 5 + qα q β h(q 2 4M 2 )γ 5 u β (p, s) i.e. two π couplings = two Goldberger-Treiman relations. G π is given by: m q g(q 2 ) fπ m2 π G π (Q2 ) (m 2 π +Q2 ) and H π is given by: m q h(q 2 ) fπ m2 π H π (Q2 ) (m 2 π +Q2 ) Goldberger-Treiman relations: f πg π (Q 2 ) = m g 1 (Q 2 ), f πh π (Q 2 ) = m h 1 (Q 2 ) C. A., E. Gregory, T. Korzec, G. Koutsou, J. W. Negele, T. Sato, A. Tsapalis, arxiv: C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

46 Global chiral fit to the axial couplings Use heavy baryon χpt to describe the pion mass dependence of the nucleon and axial charge g A and g as well as the N to axial transition C A 5. T. R. Hemmert, M. Procura, W. Weise, PRD68, (2003); F. J. Jiang and B. C. Tiburzi, PRD 78, (2008); M. Procura, PRD 78, (2008). C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

47 Conclusions Large scale simulations using the underlying theory of the Strong Interactions have made spectacular progress = we now have simulations of the full theory at near physical parameters The low-lying hadron spectrum is reproduced Nucleon form factors are being computed by a number of collaborations aiming at reproducing the experimental values N to transition form factors can be extracted in a similar way to the nucleon ones EMR and CMR allow comparison to experiment form factors are predicted Resonance width can be computed within Euclidean Lattice QCD as illustrated for the ρ-meson similar techniques can be applied to = We expect many physical results using dynamical simulations at physical pion mass in the next few years. C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32

48 Thank you for your attention C. Alexandrou (Univ. of Cyprus & Cyprus Inst.) Hadron Structure in Lattice QCD Erice, Sept / 32