Lattice QCD Calculations of Generalized Form Factors with Dynamical Fermions

Lattice QCD Calculations of Generalized Form Factors with Dynamical Fermions Sergey N. Syritsyn Lawrence Berkeley National Laboratory Nuclear Science Division INT Workshop Orbital angular momentum in QCD Seattle, Feb 7, 2012

QCD on a lattice 1 QCD on a lattice 2 Generalized Form Factors 3 Quark energy-momentum tensor in the nucleon 4 Excited States Contamination 5 Summary

QCD on a lattice QCD on a Lattice: Numerical Feynman Integration Monte Carlo sampling with Prob [ U, ψ, ψ ] S[U,ψ, ψ] e O = DUDψD ψ O [ U, ψ, ψ ] S[U,ψ, ψ] e = DU Õ Π [ ] f /D + mf e S g[u] 1 N N Õ[ U ]

QCD on a lattice QCD on a Lattice: Numerical Feynman Integration Monte Carlo sampling with Prob [ U, ψ, ψ ] S[U,ψ, ψ] e O = DUDψD ψ O [ U, ψ, ψ ] S[U,ψ, ψ] e = Euclidean QFT: DU Õ Π [ ] f /D + mf e S g[u] 1 N x 0 t ix 4 iτ p 0 E ip 4 N(t) N(0) e Eτ Fields on a discrete space-time grid: A a µ(x) U x,µ = Pe i x+ˆµ x dx (A a λa 2 ) ( ) Dµϕ 1 ( ) x Ux,µϕ x+ˆµ ϕ x a N Õ[ U ] S g [ Aµ ] (F a µν ) 2 A Tr ( ) + B Tr ( ) Fermions on a lattice: pick two from no doublers ; chiral symmetry; economy.

QCD on a lattice Lattice QCD is a Hard Problem Solving QCD numerically is hard because light quarks are expensive: cost 1 m π need large physical size of the box L 4 m π have to take continuum limit a 0, L lat = L a chiral symmetry is expensive to preserve in lattice regularization L [fm] 5 4 3 2 1 m π L=4 N f =2+1 DWF a=40 fm (LHP) N f =2+1 DWF a=0.084 fm (LHP) N f =2+1 Wilson-Clover (LHP) N f =2+1 Hyb a=24 fm (LHP) N f =2 Wilson-Clover (QCDSF) N f =2 Twisted Mass (ETM) N f =2+1 DWF (RBC) 0 0.6 m π [GeV]

QCD on a lattice Hadron Matrix Elements P t τ P q Extract P O P from 3-pt correlators x, y e i P x+i q y N( t, x) O(τ, y) N(0) where for the proton N α = ɛ abc u a α[ (u b ) T Cγ 5 d c ] All QCD states are present: N( t) O(τ) N(0) m,n Zm e Em( t τ) O mn e Enτ Z n Excited states can lead to systematic bias in m.e. : N lat Ω = N + C X, M = M X M N, N O N lat = N O N + C 2 X O X e M t + tails Signal / noise e (M N 3 2 mπ) t

QCD on a lattice Hadron Matrix Elements are Challenging 2008: The first calculation of hadron spectrum by the Budapest-Marseille-Wuppertal collaboration [Dürr et al, Science, 322:1224 (2008)] Hadron Structure Hadron Spectrum Quark-bilinear insertion P qγq P = N N + N N Disconnected contractions are noisy Gluon operators are noisy (especially with dynamical fermions) Only quenched calculations (no dynamical fermions) have been performed for gluon and disconnected quark EM tensors.

Generalized Form Factors 1 QCD on a lattice 2 Generalized Form Factors 3 Quark energy-momentum tensor in the nucleon 4 Excited States Contamination 5 Summary

Quark GPDs Generalized Form Factors Generalized Parton Distributions P O [γ5] (x) P {H, E, H, Ẽ}(x, ξ, q2 ), O [γ5] (x) = ] dλ 2π e2iλx q ( λn) [/n [γ 5 ] W( λn, λn) Moments O n = dx x n O(x) qγ + (i D + ) n q may be computed on a lattice using [ local operators O [γ5 ] n = q γ {µ1 [γ 5 ] id µ2 id ] µn} q and reduced to Generalized Form Factors P O n P {A ni, B ni, C n, Ãni, B ni }(Q 2 ) q (λn)

Generalized Form Factors Twist-2 Operators on a Hypercubic Lattice Mellin moments of GPDs symmetric, trace = 0 quark operators: In continuum: Lorentz symmetry preserves operators from mixing On a lattice: Hybercubic group has 20 irreducible representations n = 1 qγ µ q 4 1 n = 2 q [ γ {µ i D ν} Tr ] q 3 + 1 6+ 3 n = 3 q [ γ {µ i D ν i D ρ} Tr ] q 8 1 4 1 4 2 n = 4 q [ γ {µ i D ν i D ρ i D σ} Tr ] q 1 + 1 3+ 1 6+ 3 2+ 1 1+ 2 6+ 1 6+ 2... [Göckeler et al, Phys.Rev.D54,5705(1996)] Mixing coefficients Λ d1 d2 UV = ( ) 1 d1 d 2 a E.g. for n = 2 O lat = O phys + O(a 2 ) For higher n: subtraction with non-perturbative mixing coefficients QCD on a more symmetric (Celmaster) 4D lattice

u+d A 10, A Generalized Form Factors Generalizedform factors A n0 (Q 2 ) u+d A 10, u+d A20, u+d A30 0.0 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.0 Noise grows with n m! = 403 MeV Q 2 [GeV 2 ] A 10 A 20 A 30 0.0 0.6 0.8 1.0 (normalized A n,0 (Q 2 )/A n,0 (0)) Generalized radii r 2 n=1 > r 2 n=2 > r 2 n=3

! Generalized Form B 20 Factors C 20 n = 2 Gen. Form Factors A 20, B 20, C 2 u-d u-d, C20 u-d A 20, B20 u-d u-d u-d A 20, B20, C20 0.0 0.0 isovector m! = 403 MeV A 20 B 20 C 20 u+d u+d, C20 u+d A 20, B20 u+d u+d u+d A 20, B20, C20 0.0-0.6 0.0 - isoscalar! B 20 C 20 m! = 403 MeV A 20 B 20 C 20 0.0 0.6 0.8 1.0 Q 2 [GeV 2 ] 0.0 0.6 0.8 1.0 Q 2 [GeV 2 ] A u+d 20 Au d 20, Bu d 20 B u+d 20 0, C u+d 2 C u d 2 0. agree with large-n c scaling [Goeke et at, Prog. Part. Nucl. Phys. 47:401(2001)] fit with either dipole or linear form in 0 Q 2 GeV 2 extrapolate B 20 and C 2 to Q 2 0 extract forward values (Q 2 = 0) and slopes d {A 20, B 20, C 2 } / d Q 2, Q 2 0

Generalized Form Factors n = 2 Gen. Form Factors A 20, B 20, C 2 (cont.) isovector isoscalar u-d A 20, u-d B20, u-d C20 0.0 m π = 297 MeV A 20 B 20 C 20 u+d A 20, u+d B20, u+d C20 0.6 0.0 - m π = 297 MeV A 20 B 20 C 20 u-d A 20, u-d B20, u-d C20 0.0 m π = 355 MeV A 20 B 20 C 20 u+d A 20, u+d B20, u+d C20 0.6 0.0 - m π = 355 MeV A 20 B 20 C 20 u-d A 20, u-d B20, u-d C20 0.0 m π = 403 MeV A 20 B 20 C 20 u+d A 20, u+d B20, u+d C20 0.6 0.0 - m π = 403 MeV A 20 B 20 C 20 0.0 0.6 0.8 1.0 0.0 0.6 0.8 1.0 Q 2 [GeV 2 ] Q 2 [GeV 2 ]

Generalized Form Factors Chiral Extrapolations (isovector part) forward values Q 2 0 d {A 20, B 20, C 2 } / d Q 2, Q 2 0 A 20 u-d B 20 u-d C 20 u-d 0.8 0.6 0 0 0.6 m π [GeV] d C 20 u-d / dq 2 db 20 u-d / dq 2 da 20 u-d / dq 2 0 0 0.6 m π [GeV] simultaneously fit forward values and slopes at Q 2 0 Cov. Baryon χpt [Dorati et al Nucl. Phys. A798:96 (2008)] e.g., for (u d): 12 data points, 8 fit parameters, χ 2 /dof 1.5

Quark energy-momentum tensor in the nucleon 1 QCD on a lattice 2 Generalized Form Factors 3 Quark energy-momentum tensor in the nucleon 4 Excited States Contamination 5 Summary

Quark energy-momentum tensor in the nucleon Quark Momentum and Angular momentum Quark energy-momentum tensor T [ q µν = q γ {µ id ] ν} trace q T µν q N(P ) T µν q N(P ) {A 20, B 20, C 2 }(Q 2 ) quark momentum fraction x q = A q 20 (0) quark angular momentum [X. Ji 97]: J q = 1 2 [Aq 20 (0) + Bq 20 (0)] Separating contributions to nucleon spin: quark spin S q = 1 2 Σ q = 1 2 1 q quark orbital angular momentum L q = J q 1 2 Σ q gluons : the rest J glue = 1 2 1 2 Σ q L q

Quark energy-momentum tensor in the nucleon Quark Momentum Fraction x u d <x> u-d = A 20 u-d (0) 5 5 0.05 Phenomenology [CTEQ6] N f =2+1 DWF (LHP) N f =2+1 Hybrid (LHP) N f =2 Wilson-Clover (QCDSF) N f =2+1 DWF (RBC) 0 0 2 2 m π [GeV ] x q = A q 20 (0) = dx x ( q(x) + q(x) ) Sources of discrepancy: renormalization? finite volume? sea quarks? fermion action? excited states? perturbative vs. non-perturbative renormalization agreement between (2.5 fm) 3 and (3.5 fm) 3 at m π = 350 MeV results are consistently above the phenomenological value by 15 25%.

Quark energy-momentum tensor in the nucleon Quark Momentum Fraction x u+d <x> u+d = A 20 u+d (0) 0.8 0.7 0.6 Phenomenology [CTEQ6] N f =2+1 DWF (LHP) N f =2+1 Hybrid (LHP) N f =2 Wilson-Clover (QCDSF) 0 0 2 2 m π [GeV ] x q = A q 20 (0) = dx x ( q(x) + q(x) ) Sources of discrepancy: renormalization? finite volume? sea quarks? fermion action? excited states? perturbative vs. non-perturbative renormalization agreement between (2.5 fm) 3 and (3.5 fm) 3 at m π = 350 MeV qualitative agreement with phenomenology (no disc. contractions!)

Quark energy-momentum tensor in the nucleon Quarks Angular Momentum (1): J u+d J u+d 5 5 0.05 0 N f =2+1 Hybrid (LHP) N f =2+1 DWF (LHP) N f =2 Wilson-Clover (QCDSF) N f =2 Twisted mass (ETM) 0 m π 2 [GeV 2 ] Following [X. Ji PRL 97], J 3 q = N d 3 x M 012 N, M αµν = x µ Tq αν x ν Tq αµ and [ J q = 1 2 A q 20 (0) + Bq 20 (0)] Gluon contribution J g = 1 2 J q 52% of the nucleon spin result agrees with QCD sum rule estimations [Balitsky, Ji (1997)]

Quark energy-momentum tensor in the nucleon Quarks Angular Momentum (2): J u, J d J q 5 5 0.05 0 N f =2 Twisted Mass (ETM) N f =2 Wilson-Clover (QCDSF) N f =2+1 DWF (LHP) N f =2+1 Hybrid (LHP) -0.05 0 2 2 m π [GeV ] Following [X. Ji PRL 97], J 3 q = N d 3 x M 012 N, M αµν = x µ Tq αν x ν Tq αµ and [ J q = 1 2 A q 20 (0) + Bq 20 (0)] Most contribution to the nucleon spine comes from u-quarks: J d J u

Quark energy-momentum tensor in the nucleon Quark Spin and OAM 0 - L u Σ u / 2 L d Σ d / 2-0 0.6 m π [GeV] J d S d, L d L u+d L u, L d In agreement with [Hägler et al (2007)] L q = J q S q, S q = 1 2 Σ q = dx ( q(x) + q(x) ) L u+d 0 - N f =2+1 DWF (LHP) N f =2+1 Hybrid (LHP) N f =2 Wilson-Clover (QCDSF) N f =2 Twisted Mass (ETM) - 0 0.6 m π [GeV] L u+d 1 2

Excited States Contamination 1 QCD on a lattice 2 Generalized Form Factors 3 Quark energy-momentum tensor in the nucleon 4 Excited States Contamination 5 Summary

Excited States Contamination Mass Gap M = M exc M N Wilson-Clover 32 3 48, m π = 200 MeV M[MeV] am a M π 200 18 2π/L 334 96 N 1021 0.600 N ππ 1421 0.835 35 N π 1463 0.860 60 N(1440) (?)1520 0.894 94 N π 1637 0.962 62 N π 1786 1.050 50 N(1710) (?)1790 1.052 52... C 2pt best fit 2130 1.25(19) 0.65(19)

Excited States Contamination Momentum fraction x u d : vary t 0 5 T /a = 8, 10, 12 plateau values fit values 0 x u d (bare) 5 0 0.05 0.00 0.05 0 5 0 5 0 m π (GeV) Separate N and N with t = 0.93, 1.16, 1.39 fm m π 150, 200 and 250 MeV, m N 0.97... 1.06 GeV Fit to a 2-state model with fixed M Also reported in [Renner et al (ETMC Collab)]

Excited States Contamination Momentum fraction x u d : fix M 6 4 from fit from dt=8,10,12 plateaus <x> (bare) 2 syst.error 8 6 ΔM from fit 0 0.6 0.8 1 1.2 1.4 1.6 1.8 ΔM Separate N and N with t = 0.93, 1.16, 1.39 fm One point m π 200 MeV, m N 1.00 GeV Fit to a 2-state model with fixed M

Excited States Contamination Dirac radius r 2 1 u d : fix M 40 35 ΔM from fit from fit from dt=8,10,12 plateaus r 1 2 30 25 syst.error 20 15 0 0.6 0.8 1 1.2 1.4 1.6 1.8 ΔM m π 200 MeV, m N 1.00 GeV Separate N and N with t = 0.93, 1.16, 1.39 fm Fit to a 2-state model with fixed M

Summary 1 QCD on a lattice 2 Generalized Form Factors 3 Quark energy-momentum tensor in the nucleon 4 Excited States Contamination 5 Summary

Summary Summary Extrapolated results from different LQCD actions/volumes/discretizations agree near the physical point; disagreement with phenomenology for x u d qualitative agreement with phenomenology for quark angular momentum Calculations at m phys π are necessary Systematic bias due to excited states is likely to increase towards m phys π Need a systematic overhaul of hadron structure calculations: better control of excited states additional statistics disconnected contractions

BACKUP BACKUP SLIDES

BACKUP Generalized Radii 5 5 <r 2 perp > [fm2 ] 5 0.05 [DW] n = 1 [DW] n = 2 [DW] n = 3 [Hy] n = 1 [Hy] n = 2 [Hy] n = 3 <r 2 perp > [fm2 ] 5 0.05 [DW] n = 1 [DW] n = 2 [DW] n = 3 [Hy] n = 1 [Hy] n = 2 [Hy] n = 3 0 0 0.6 m π [GeV] (spin-independent) 0 0 0.6 m π [GeV] (spin-dependent)

B 20 BACKUP n = 2 Spin-dependent GFFs Ã20, B 20 ~ u-d A20, ~ u-d B20 ~ u-d A20, ~ u-d B20 0.8 0.7 0.6 0.0 0.9 0.8 0.7 0.6 0.0 isovector m! = 403 MeV A 20 B 20 ~ u+d A20, ~ u+d B20 ~ u+d A20, ~ u+d B20 0.8 0.7 0.6 0.0 0.9 0.8 0.7 0.6 0.0 isoscalar m! = 403 MeV B 20 A 20 B 20 0.0 0.6 0.8 1.0 Q 2 [GeV 2 ] 0.0 0.6 0.8 1.0 Q 2 [GeV 2 ]

BACKUP On a lattice, rotational symmetry is broken O(4) H(4) Tensors split into irred. reps. of H(4) For energy-momentum tensor there are two irred. of H(4): (4 1 ) 2 = 1 1 6 1 6 3 3 1 }{{} trace=0, symm. [Göckeler et al, hep-lat/9602029] (both 6 3 and 3 1 are needed to extract gen. form factors) Use non-perturbative renormalization (RI/MOM) for quark-bilinear T µν q : 1.79 1.78 1.77 1.76 1.75 1.74 1.73 1.72 1.71 1.7 1.69 0 2 4 6 8 10 12 14 µ 2 [GeV 2 ] Z n=2,τ (3) / Z RI n=2 Z n=2,τ (6) / Z RI n=2 Z scale-indep. O Z 31 O /Z63 O 0.98 P = Zlat O (aµ) Z pert O (µ) Normalize to MS at µ 2 = (2 GeV) 2 P

BACKUP Quark Angular Momentum: p,n-dvcs [JLab Hall A PRL`07; HERMES JHEP`08] LHPC arxiv:1001.3620 (this work) LHPC PRD `08 0705.4295 QCDSF (Ohtani et al.) 071534 Goloskokov&Kroll EPJC`09 0809 Wakamatsu 0908.0972 DiFeJaKr EPJC `05 hep-ph/040817 (Myhrer&)Thomas PRL`08 0803. Picture: Ph. Hägler, MENU 2010, W&M, Virginia, USA; J.Phys.Conf.Ser.295:012009,2011 MS at 2 GeV

BACKUP Quark Angular Momentum: p,n-dvcs [JLab Hall A PRL`07; HERMES JHEP`08] variation from disc. contribution to (u+d) LHPC arxiv:1001.3620 (this work) LHPC PRD `08 0705.4295 QCDSF (Ohtani et al.) 071534 Goloskokov&Kroll EPJC`09 0809 Wakamatsu 0908.0972 DiFeJaKr EPJC `05 hep-ph/040817 (Myhrer&)Thomas PRL`08 0803. Picture: Ph. Hägler, MENU 2010, W&M, Virginia, USA; J.Phys.Conf.Ser.295:012009,2011 MS at 2 GeV