Quark-Gluon Correlations in the Nucleon

Quark-Gluon Correlations in the Nucleon Matthias Burkardt New Mexico State University April 5, 2017

Outline 2 GPDs F T q(x, b ) 3d imaging polarization deformation force from twist 3 correlations L q JM Lq Ji = change in OAM as quark leaves nucleon (due to torque from FSI) lattice calcs. of ISI/FSI & L q JM Summary

Nucleon Spin Puzzle 3 spin sum rule 1 2 = 1 Σ + G + L 2 Longitudinally polarized DIS: Σ = q q 1 q 0 dx [q (x) q (x)] 30% only small fraction of proton spin due to quark spins Gluon spin G could possibly account for remainder of nucleon spin, but still large uncertainties EIC Quark Orbital Angular Momentum how can we measure L q,g need correlation between position & momentum how exactly is L q,g defined

Deeply Virtual Compton Scattering (DVCS) 4 form factor Compton scattering electron hits nucleon & nucleon remains intact form factor F (q 2 ) position information from Fourier trafo no sensitivity to quark momentum F (q 2 ) = dxgp D(x, q 2 ) GPDs provide momentum disected form factors electron hits nucleon, nucleon remains intact & photon gets emitted additional quark propagator additional information about momentum fraction x of active quark generalized parton distributions GP D(x, q 2 ) info about both position and momentum of active quark

Physics of GPDs: 3D Imaging MB,PRD 62, 071503 (2000) 5 unpolarized proton q(x, b ) = d 2 (2π) 2 H(x, 0, 2 )e ib F 1 ( 2 ) = dxh(x, 0, 2 ) x = momentum fraction of the quark b relative to center of momentum small x: large meson cloud larger x: compact valence core x 1: active quark becomes center of momentum b 0 (narrow distribution) for x 1

Physics of GPDs: 3D Imaging MB, IJMPA 18, 173 (2003) 6 proton polarized in +ˆx direction d 2 q(x, b ) = (2π) 2 H q(x,0, 2 )e ib 1 d 2 2M b y (2π) 2 E q(x,0, 2 )e ib relevant density in DIS is j + j 0 + j z and left-right asymmetry from j z av. shift model-independently related to anomalous magnetic moments: b q y dx d 2 b q(x, b )b y = 1 2M dxeq (x, 0, 0) = κq 2M

GPD Single Spin Asymmetries (SSA) 7 confirmed by Hermes & Compass data example: γp πx u, d distributions in polarized proton have left-right asymmetry in position space (T-even!); sign determined by κ u & κ d attractive final state interaction (FSI) deflects active quark towards the center of momentum FSI translates position space distortion (before the quark is knocked out) in +ŷ-direction into momentum asymmetry that favors ŷ direction chromodynamic lensing κ p, κ n sign of SSA!!!!!!!! (MB,2004)

Average Force on Quarks in DIS 8 d 2 average force on quark in DIS from pol target polarized DIS: σ LL g 1 2Mx ν g 2 σ LT g T g 1 + g 2 clean separation between g 2 and 1 Q 2 corrections to g 1 g 2 = g2 W W + ḡ 2 with g2 W W (x) g 1 (x) + 1 dy x y g 1(y) d 2 3 dx x 2 1 ḡ 2 (x) = P, S q(0)γ + gf +y (0)q(0) P, S 2MP +2 S x color Lorentz Force on ejected quark (MB, PRD 88 (2013) 114502) ( ) y 2F +y = F 0y + F zy = E y + B x = E + v B for v = (0, 0, 1) matrix element defining d 2 1 st integration point in QS-integral d 2 force QS-integral impulse sign of d 2 deformation of q(x, b ) sign of d q 2 : opposite Sivers magnitude of d 2 F y = 2M 2 d 2 = 10 GeV fm d 2 F y σ 1 GeV fm d 2 = O(0.01)

Average Force on Quarks in DIS 8 d 2 average force on quark in DIS from pol target polarized DIS: σ LL g 1 2Mx ν g 2 σ LT g T g 1 + g 2 clean separation between g 2 and 1 Q 2 corrections to g 1 g 2 = g2 W W + ḡ 2 with g2 W W (x) g 1 (x) + 1 dy x y g 1(y) d 2 3 dx x 2 1 ḡ 2 (x) = P, S q(0)γ + gf +y (0)q(0) P, S 2MP +2 S x color Lorentz Force on ejected quark (MB, PRD 88 (2013) 114502) ( ) y 2F +y = F 0y + F zy = E y + B x = E + v B for v = (0, 0, 1) sign of d 2 deformation of q(x, b ) sign of d q 2 : opposite Sivers magnitude of d 2 F y = 2M 2 d 2 = 10 GeV fm d 2 F y σ 1 GeV fm d 2 = O(0.01) consitent with experiment (JLab,SLAC), model calculations (Weiss), and lattice QCD calculations (Göckeler et al., 2005)

Angular Momentum Carried by Quarks 9 L x = yp z zp y if state invariant under rotations about ˆx axis then yp z = zp y L x = 2 yp z GPDs provide simultaneous information about longitudinal momentum and transverse position use quark GPDs to determine angular momentum carried by quarks Ji sum rule (1996) Jq x = 1 dx x [H(x, 0, 0) + E(x, 0, 0)] 2 parton interpretation in terms of 3D distributions (MB,2001,2005)

Photon Angular Momentum in QED 10 QED with electrons J γ = = = ( ) d 3 r r E B = d 3 r d 3 r [ ( )] d 3 r r E A [ ( E j r ) A j r ( E ) A ] [ ( E j r ) ( A j + r A ) ] E + E A replace 2 nd term (eq. of motion E = ej 0 = eψ ψ), yielding [ J γ = d 3 r ψ r eaψ ( + E j x ) A j + E A ] ψ r e Aψ cancels similar term in electron OAM ψ r ( p e A)ψ decomposing J γ into spin and orbital also shuffles angular momentum from photons to electrons!

The Nucleon Spin Pizzas 11 Ji decomposition Jaffe-Manohar decomposition 1 2 = ( 1 q 2 q + L ) q + Jg L q = ( 3q( x) P,S d 3 x P,S q ( x) x id) i D = i g A DVCS GPDs L q 1 2 = ( 1 q 2 q + L ) q + G + Lg L q = d 3 r P,S q( r)γ + ( r i ) zq( r) P,S light-cone gauge A + = 0 p p G L i q,g Li manifestly gauge inv. def. exists How large is difference L q L q in QCD and what does it represent?

Quark OAM from Wigner Functions 12 5-D Wigner Functions (Lorcé, Pasquini) W (x, b, k ) d 2 (2π) 2 e i b GT MD(x, k, )

Quark OAM from Wigner Functions 12 5-D Wigner Functions (Lorcé, Pasquini) W (x, b, d 2 d 2 ξ dξ k ) (2π) 2 (2π) 3 e ik ξ e i b P S q(0)γ + q(ξ) P S. TMDs: f(x, k ) = d 2 b W (x, b, k ) GPDs: q(x, b ) = d 2 k W (x, b, k ) L z = dx d 2 b d 2 k W (x, b, k )(b x k y b y k x ) ( need to include Wilson-line gauge link U 0ξ exp i g ξ A d r) 0 to connect 0 and ξ light-cone staple crucial for SSAs in SIDIS & DY straight line for U 0ξ Light-Cone Staple for U 0ξ straigth Wilson line from 0 to ξ yields Ji-OAM: L q = ( zq( x) P,S d 3 x P,S q ( x) x id) light-cone staple yields L Jaffe Manohar

Light-Cone Staple Jaffe-Manohar-Bashinsky 13 L /L L with light-cone staple at x = ± PT (Hatta) PT L = L (different from SSAs due to factor x in OAM) Bashinsky-Jaffe Bashinsky-Jaffe light-cone staple A + = 0 no complete gauge fixing residual gauge inv. A µ A µ + µ φ( x ) x i [ L JB x i ga( x ] ) A ( x ) = A + = 0 [ ] L / = x i ga (±, x ) [ ] L JB = x i ga( x ) A ( x ) = dx A (x, x ) dx dx A (x, x ) dx = 1 2 ( A (, x ) + A (, x ) L JB = 1 2 (L + L ) = L = L )

Quark OAM from Wigner Distributions 14 straight line ( Ji) 1 2 = q 1 2 q + L q + J g L q = d 3 x P,S q( x)γ + ( x i D) zq( x) P,S light-cone staple ( Jaffe-Manohar) 1 2 = q 1 2 q + L q + G + L g L q = d 3 x P,S q( x)γ + ( x i D) zq( x) P,S i D = i g A id j = i j ga j (x, x ) g x dr F +j difference L q L q L q L q = g d 3 x P,S q( x)γ +[ x x dr F + (r, x ) ] z q( x) P,S color Lorentz Force on ejected quark (MB, PRD 88 (2013) 014014) ( ) y 2F +y = F 0y + F zy = E y + B x = E + v B for v = (0, 0, 1)

Torque in General Nonzero! 15 difference L q L q L q JM Lq Ji = Lq F SI = change in OAM as quark leaves nucleon L q JM Lq Ji = g d 3 x P,S q( x)γ +[ x dr F + (r, x x ) ] z q( x) P,S e + moving through magnetic dipole field of e

Torque in Scalar Diquark Model (with C.Lorcé) 16 (Ji et al., 2016) for e : L JM L Ji = 0 to O(α)? L JM L Ji = 0 in general? how significant is L JM L Ji? L JM L Ji = qγ + ( r A) zq in scalar diquark model pert. evaluation of qγ + ( r A) zq L JM L Ji = O(α) same order as Sivers L JM L Ji as significant as SSAs why scalar diquark model? Lorentz invariant 1 st to illustrate: FSI SSAs (Brodsky,Hwang,Schmidt 2002) Sivers 0 calculation nonforward matrix elem. of qγ + A y q d =0 d x zq k q = 3mq+M 12 π qγ ( r + A)

Torque vs. Force 17 difference L q L q L q JM Lq Ji = g d 3 x P,S q( x)γ +[ x x dr F + (r, x ) ] z q( x) P,S change in OAM as quark leaves nucleon due to torque from FSI on active quark color Lorentz Force on ejected quark (MB, PRD 88 (2013) 014014) ( ) y 2F +y = F 0y + F zy = E y + B x = E + v B for v = (0, 0, 1) Single-Spin Asymmetries (Qiu-Sterman) single-spin asymmetry in semi-inclusive DIS governed by Qiu-Sterman integral k P,S q( x)γ + dr F + (r, x )q( x) P,S x semi-classical interpretation: F + (r, x ) color Lorentz Force acting on active quark on its way out integral yields impulse due to FSI

Quasi Light-Like Staples from Lattice QCD 18 challenge TMDs in lattice QCD B. Musch, P. Hägler, M. Engelhardt? v + ` 0 TMDs/Wigner functions relevant for SIDIS require (near) light-like Wilson lines on Euclidean lattice, all distances are space-like calculate space-like staple-shaped Wilson line pointing in ẑ direction; length L momentum projected nucleon sources/sinks remove IR divergences by considering appropriate ratios extrapolate/evolve to P z

Quasi Light-Like Staples from Lattice QCD 19 f 1T,SIDIS = f 1T,DY (Collins) f 1T (x, k ) is k -odd term in quark-spin averaged momentum distribution in polarized target

L q vs. L q in Lattice QCD 20 difference L q L q L q JM Lq Ji = Lq F SI = change in OAM as quark leaves nucleon L q JM Lq Ji = g d 3 x P,S q( x)γ +[ x dr F + (r, x x ) ] z q( x) P,S Orbital Angular Momentum same operator as for TMDs, only nonforward matrix elements: momentum transfer provides position space information ( r k ) staple with long side in ẑ direction (large) nucleon momentum in ẑ direction small momentum transfer in ŷ direction generalized TMD F 14 (Metz et al.) quark OAM lattice QCD (M.Engelhardt) L staple vs. staple length L q Ji for length = 0 L q JM for length shown L u staple Ld staple similar result for each L q F SI

Torque moments of twist-3 GPDs 21 difference L q L q L q JM Lq Ji = Lq F SI = change in OAM as quark leaves nucleon L q JM Lq Ji = g d 3 x P,S q( x)γ +[ x dr F + (r, x x ) ] z q( x) P,S local torque d 2 x P,S q( x)γ +[ x F + (x, x ) ] z q( x) P,S = 0 formal argument: PT intuitive: front vs. back cancellation in ensemble average lowest nontrivial moment d 2 x P,S q( x)γ +[ x F + (x, x ) ] z q( x) P,S = 0 off-forward matrix element of qγ 5 γ D 3 q (twist-3, x 3 moment)

Summary: May the Torque Always be With You 22 GPDs F T q(x, b ) 3d imaging polarization deformation sign of qg-correlations sign and magnitude of d 2 simultaneous info about position & long. momentum Ji sum rule for J q L q JM Lq Ji = change in OAM as quark leaves nucleon (due to torque from FSI) ISI/FSI and L q JM vs. Lq Ji in lattice QCD

Angular Momentum Carried by Quarks 23 lattice: (lattice hadron physics collaboration - LHPC) no disconnected quark loops chiral extrapolation QCD evolution J q = 1 2 dx x [H(x, 0, 0) + E(x, 0, 0)] L q = J q 1 2 Σq

A DV CS? GP Ds interesting GPD physics: J q = 1 dxx [H(x, ξ, 0) + E(x, ξ, 0)] 0 requires GP Ds(x, ξ, 0) for (common) fixed ξ for all x imagingrequires GP Ds(x, ξ = 0,t) ξ longitudinal momentum transfer on the target ξ = p+ p + p + +p + x (average) momentum fraction of the active quark x = k+ +p + IA DV CS (ξ, t) GP D(ξ, ξ, t) only sensitive to diagonal x = ξ limited ξ range p + +p + RA DV CS (ξ, t) 1 GP D(x,ξ,t) dx 1 x ξ limited ξ range Polynomiality/Dispersion Relations (GPV/AT DI) most sensitive to x ξ some sensitivity to x ξ, but RA(ξ, t, Q 2 ) = 1 1 dx H(x, ξ, t, Q2 ) x ξ = 1 1 dx H(x, x, t, Q2 ) x ξ + (t, Q 2 )

A DV CS? GP Ds interesting GPD physics: J q = 1 dxx [H(x, ξ, 0) + E(x, ξ, 0)] 0 requires GP Ds(x, ξ, 0) for (common) fixed ξ for all x imagingrequires GP Ds(x, ξ = 0,t) IA DV CS (ξ, t) GP D(ξ, ξ, t) RA DV CS (ξ, t) 1 GP D(x,ξ,t) dx 1 x ξ Polynomiality/Dispersion Relations (GPV/AT DI) RA(ξ, t, Q 2 ) = 1 1 dx H(x, ξ, t, Q2 ) x ξ = 1 1 dx H(x, x, t, Q2 ) x ξ + (t, Q 2 ) Can condense all information contained in contained in A DV CS (fixed Q 2 ) into GP D(x, x, t, Q 2 ) & (t, Q 2 ) if two models both satisfy polynomiality and are equal for x = ξ (but not for x ξ) and have same (t, Q 2 ) then DVCS at fixed Q 2 cannot distinguish between the two models

A DV CS? GP Ds Polynomiality/Dispersion Relations (GPV/AT DI) RA(ξ, t, Q 2 ) = 1 1 dx H(x, ξ, t, Q2 ) x ξ = 1 1 dx H(x, x, t, Q2 ) x ξ + (t, Q 2 ) Can condense all information contained in contained in A DV CS (fixed Q 2 ) into GP D(x, x, t, Q 2 ) & (t, Q 2 ) if two models both satisfy polynomiality and are equal for x = ξ (but not for x ξ) and have same (t, Q 2 ) then DVCS at fixed Q 2 cannot distinguish between the two models need Evolution! Q 2 evolution changes x distribution in a known way for fixed ξ measure Q 2 dependence to disentangle x vs. ξ dependence