Nucleon form factors and moments of GPDs in twisted mass lattice QCD

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1 Nucleon form factors and moments of GPDs in twisted mass lattice QCD European Collab ora tion M. Constantinou, C. Alexandrou, M. Brinet, J. Carbonell P. Harraud, P. Guichon, K. Jansen, C. Kallidonis, T. Korzec, M. Papinutto HEP211, Grenoble July 22, 211

2 A Motivation OUTLINE B Nucleon Generalized Parton Distributions C Evaluation on the Lattice 2pt and 3pt functions Renormalization Cut-off and Volume effects D Nucleon form factors E Nucleon Moments

3 Motivation Characterization of nucleon structure is considered a milestone in hadronic physics many experiments have been carried out to measure form factors and structure functions. New generation experiments using polarized beams and target are yielding high precision data spanning larger Q 2 ranges. They provide ideal probes of the charge and magnetization, determination of shape in analogy to e.g. deuteron and other nuclei Non-relativistically form factors are related to density distribution: F( q 2 ) = d 3 xe i q. x < ψ ρ( x) ψ >

4 Nucleon Generalized Parton Distributions (GPDs) High energy scattering: Formulate in terms of light-cone correlation functions F Γ(x,ξ,q 2 ) = 1 2 ig dλ 2π eixλ p ψ( λn/2)γpe λ/2 λ/2 where q = p p, P = (p + p)/2, n: light-cone vector ( P.n = 1). Choices of operators: Γ = /n 1 ) [/nh(x,ξ,q 2 ) + i nµqνσµν 2ū(p Γ = /nγ 5 1 [ ) 2ū(p Γ = n µσ µν tensor GPDs dαn A(nα) ψ(λn/2) p ] 2m E(x,ξ,q2 ) u(p) ] u(p) /nγ 5 H(x,ξ,q 2 ) + n.qγ5 Ẽ(x,ξ,q 2 ) 2m 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) twist-2 operators O µ 1...µn = qγ {µ1 id µ2...id µn} q Õ µ 1...µn = qγ 5γ {µ1 id µ2...id µn} q unpolarized x n q = helicity x n q = 1 dxx n [ q(x) ( 1) n q(x) ] 1 dxx n [ q(x) + ( 1) n q(x) ] q = q + q, q = q q,δq = q T + q

5 Nucleon generalized form factors Decomposition of matrix elements into GFFs: contain form factors, parton distributions N(p ) O / n µ 1...µn N(p) = [ ū(p ( n 1 ) A i= ni (q 2 )γ {µ 1 + B ni (q 2 ) iσµ 1 α ) qα q 2m µ 2...q µ i+1 µ P i+2...p µn} even Similarly for O /nγ5 in terms of à ni(q 2 ), B ni(q 2 ) Special cases: n = 1: ordinary nucleon form factors +δ n even C n (q2 ) 1 m q{µ 1...q µn} ]u(p) 1 vector operator = γ µf 1(q 2 )+ iσµνqν 2m F2(q2 ) 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 ) 2 axial operator = i [ γ µγ 5G A(q 2 )+ qµγ 5 2m Gp(q2 ) ] τ a 2 n = 2: moments of parton distributions 1 vector operator = A 2(q 2 ), B 2(q 2 ), C 2(q 2 ) x q = A 2(): spin independend moment 2 axial operator = Ã2(q2 ), B2(q 2 ) x q = Ã2(): helicity moment

6 Evaluation on the Lattice Issues to be addressed: Evaluation of three-point correlators and renormalization Choice of operators - avoid mixing, consider iso-vector operators no disconnected contributions Cut-off effects Finite volume effects Chiral expansions - not as developed as in the light meson case Volume and cut-off effects more difficult to assess Extrapolation to physical point more demanding This work focuses on: N F = 2 twisted mass gauge configurations (produced by ETMC) Nucleon form factors Nucleon lower moments of GPDs Dynamical simulations, pion mass m π < 5 MeV, L > 2 fm

7 Fermion and Gluon Action N F = 2 Twisted mass fermions (twisted basis) S F = a 4 x χ(x) (1 2 γµ( µ + µ ) ar µ µ 2 + m + iµγ5τ3) χ(x) physical basis at maximal twist ψ= 1 2 [1 + iτ 3 γ 5]χ ψ= χ 1 2 [1 + iτ 3 γ 5] Tree-level Symanzik improved gluons S g = β 3 ( 5 x 3 4 µ,ν=1 1 µ<ν { } 1 Re Tr(U 1 1 x,µ,ν ) µ,ν=1 µ ν { } ) 1 Re Tr(U 1 2 x,µ,ν )

8 Ensembles β a (fm) aµ m π (GeV) L 3 T C. Alexandrou et al. (ETM Collaboration), Phys. Rev. D83 (211) 451 C. Alexandrou et al. (ETM Collaboration), Phys. Rev. D83 (211) 9452 C. Alexandrou et al. (ETM Collaboration), Phys. Rev. D83 (211) C. Alexandrou et al. (ETM Collaboration), Phys. Rev. D83 (211) 1453

9 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 () OΓ q = p p 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 () ( xf, tf) ( x, t) ( xi, ti) Sequential inversion through the sink fix sink-source separation t f t i, final momentum p f =, Γ Smearing techniques improvement of ground state dominance in three-point correlators Ratios: Leading time dependence cancels G(Γ, q,t) R(Γ, q,t) = G( q,t f t)g(,t)g(,t f ) G(,t f ) G(,t f t)g( q,t)g( q,t f ) lim t f t t t lim R(Γ, q,t) Π(Γ, q) i R(Γ, q, t) depends on the indices of current insertion Variational approach, possible improvement on plateaux extend to Q GeV 2 t/a R 1 (Γ,p=(1,1,) R 22 (Γ,p=(,1,) R (Γ,p=(1,,) R (Γ,p=(,,)

10 Renormalization Constants RI -MOM renormalization scheme Fix configurations to Landau gauge. 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 m π (GeV) Z q Z V Z A Z T Z V Z A (a p) 2

11 Nucleon EM and Axial form factors (m π :3-35MeV) Can we get results at physical point?

12 g A = G A() Nucleon axial charge N F = 2 twisted mass fermions, ETMC N F = Domain wall fermions, RBC-UKQCD N F = hybrid action, LHPC 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?

13 g A = G A() Nucleon axial charge - - TMF at 3 β-values Continuum limit of our volume corrections results 1-loop chiral perturbation theory in the small scale expansion (SSE) Fitting volume corrected and extrapolated to the continuum results g A = 1.12(7) Fitting lattice results directly g A = 1.8(8) (black dashed line)

14 Chiral extrapolation of EM form factors Baryon χpt to 1-loop, with d.o.f.(sse) and isovector N coupling included in LO 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 (g N or c A) couplings and two counterterms

15 Nucleon Moments GFFs: ūγ {µ D ν} u dγ {µ D ν} d and ūγ {µ γ 5 D ν} u dγ {µ γ 5 D ν} d Results given in the MS scheme at µ = 2 GeV.4 TMF: 3 MeV QCDSF: 35MeV Hybrid: 355MeV.4 TMF: 3MeV Hybrid: 355MeV.3.3 A 2.2 ~ A Q 2 (GeV 2 ) Q 2 (GeV 2 ) C 2 B TMF: 3MeV Hybrid: 355MeV QCDSF: 35MeV ~ B TMF: 3MeV Hybrid: 355MeV Q 2 (GeV 2 ) Q 2 (GeV 2 )

16 spin-independent moment: A 2() helicity moment: Ã 2()

17 Chiral extrapolation of A 2, B 2, Ã 2, B2 at Q 2 = HBχPT: A combined fit to raw data A 2 and B 2 is carried out (agreement with a results). The mass of the nucleon at the chiral limit is used as input Isovector unpolarized and polarized first moments of quark distribution A 2.2 A ~ B m π (GeV ) B ~ m π (GeV )

18 Proton Spin One also needs the isoscalar moments A u+d 2 and B u+d 2 since the total spin of a quark is J q = 1 ( q A ()) L q : orbital angular momentum Σ q : spin components J q = 1 2 Σq + L q Σ u+d = g u+d A no disconnected contributions.4 CBχPT.4 HBχPT Contributions to nucleon spin J u J d contributions to nucleon spin J u J d m π 2 (GeV 2 ) m π 2 (GeV 2 )

19 Chiral extrapolation using HBχPT left: total angular momentum and total spin component right: Angular momentum and spin carried by u- d- quarks Σ u+d / 2 L u+d contributions to nucleon spin Σ u /2 L d L u Σ d / m π 2 (GeV 2 ) m π (GeV 2 ) Physical points from HERMES 27 analysis

20 Conclusions Nucleon form factors provide a benchmark for lattice QCD beyond hadron masses. Need results at both lower Q 2 extract radii and magnetic moments and higher Q 2 Cut-off effects small for a <.1 fm Finite volume corrections difficult to assess Within current statistical errors of 3% results on G E, G M, G A, < x > q and < x > q are consistent for Lm π > 3.5 Lmπ = 4 Finite volume corrections significant for G p Need to include the disconnected contributions Make a lattice determination of a number of couplings used as input in chiral extrapolations explore GPDs that yield more detailed information on both longitudinal and transverse distributions

21 THANK YOU