Quark Orbital Angular Momentum
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1 Quark Orbital Angular Momentum Matthias Burkardt New Mexico State University August 31, 2017
2 Outline 2 GPDs F T q(x, b ) 3d imaging polarization deformation L q JM Lq Ji = change in OAM as quark leaves nucleon (due to torque from FSI) force on quarks in DIS twist-3 GPD G 2 Summary
3 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
4 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
5 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
6 Physics of GPDs - Transverse Imaging 5 form factors: F T ρ( r) (nonrelativistic) reference point is center of mass GP Ds(x, ): form factor for quarks with momentum fraction x suitable FT of GP Ds should provide spatial distribution of quarks with momentum fraction x careful: cannot measure longitudinal momentum (x) and longitudinal position simultaneously (Heisenberg) consider purely transverse momentum transfer Impact Parameter Dependent Quark Distributions d 2 q(x, b ) = (2π) 2 H(x, ξ = 0, 2 )e ib q(x, b ) = parton distribution as a function of the separation b from the transverse center of momentum R i q,g r,ix i MB, Phys. Rev. D62, (2000) No relativistic corrections (Galilean subgroup!) corollary: interpretation of 2d-FT of F 1 (Q 2 ) as charge density in transverse plane also free of relativistic corrections ( G.Miller) probabilistic interpretation
7 Physics of GPDs: 3D Imaging 6 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
8 From 2015 Long Range Plan for Nuclear Science 7
9 From 2015 Long Range Plan for Nuclear Science 8
10 Physics of GPDs: 3D Imaging MB, IJMPA 18, 173 (2003) 9 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
11 GPD Single Spin Asymmetries (SSA) 10 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)
12 DIgression: GPDs for x = ξ 11 imaging: ξ = 0 probabilistic interpretation variable conjugate to is b (1 x)r distance to COM of hadron t-slope same as 2 slope both 0 as x 1 imaging (x = ξ) no probabilistic interpretation variable conjugate to is r distance to COM of spectators no a priori reason for 2 -slope B 2 (ξ) to vanish as ξ 1 (size of system) imaging: ξ 0 center of momentum of hadron not conserved when ξ 0, distance of active quark to COM not conserved position of each parton is conserved, and so is distance r of active quark to spectators (any ξ) variable conjugate to is 1 x 1 ξ r
13 DIgression: GPDs for x = ξ 11 imaging: ξ = 0 probabilistic interpretation variable conjugate to is b (1 x)r distance to COM of hadron t-slope same as 2 slope both 0 as x 1 imaging (x = ξ) no probabilistic interpretation variable conjugate to is r distance to COM of spectators no a priori reason for 2 -slope B 2 (ξ) to vanish as ξ 1 (size of system) imaging: x = ξ 0 t = t ξ slope: B 2 = 1 1 ξ 2 B t B 2 (ξ) does not have to vanish for ξ 1 B t (ξ) does have to vanish for ξ 1
14 Angular Momentum Carried by Quarks 12 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 only for component (MB,2001,2005)
15 Angular Momentum Carried by Quarks 13 lattice: (lattice hadron physics collaboration - LHPC) no disconnected quark loops chiral extrapolation J q = 1 2 dx x [H(x, 0, 0) + E(x, 0, 0)] L q = J q 1 2 Σq
16 Photon Angular Momentum in QED 14 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!
17 The Nucleon Spin Pizzas 15 Ji decomposition Jaffe-Manohar decomposition 1 2 = ( 1 q 2 q + L ) q + Jg 1 2 q = 2 1 d 3 x P, S q ( x)σ 3 q( x) P, S L q = ( 3q( x) P,S d 3 x P,S q ( x) x id) J g = [ )] z d 3 x P, S x ( E B P, S i D = i g A 1 2 = ( 1 q 2 q + L ) q + G + Lg light-cone gauge A + = 0 L q = d 3 r P,S q( r)γ + ( r i ) zq( r) P,S G=ε + ij d 3 r P, S TrF +i A j P, S L g =2 d 3 r P,S TrF +j ( x i ) za j P,S manifestly gauge inv. def. for each
18 The Nucleon Spin Pizzas 16 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 How large is difference L q L q in QCD and what does it represent?
19 Quark OAM from Wigner Functions 17 5-D Wigner Functions (Lorcé, Pasquini) W (x, b, d 2 q d 2 ξ dξ k ) (2π) 2 (2π) 3 e ik ξ e i q 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 crucial for SSAs in SIDIS et al. 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
20 Light-Cone Staple Jaffe-Manohar-Bashinsky 18 L /L L z with light-cone staple at x = ± PT (Hatta) Bashinsky-Jaffe A + = 0 not completely gauge fixed: residual gauge inv. A µ A µ + µ φ( x ) x i [ L JB x i ga( x ] ) PT L = L (different from SSAs due to factor x in OAM) A ( x ) = dx A (x, x ) dx L JB inv. under A µ A µ + µ φ( x ) Bashinsky-Jaffe light-cone staple A + = 0 [ ] L / = x i ga (±, x ) [ ] L JB = x i ga( x ) A ( x ) = dx A (x, x ) dx = 1 2 ( A (, x ) + A (, x ) L JB = 1 2 (L + L ) = L = L )
21 Quark OAM from Wigner Distributions 19 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 i D = i g A(x =, x ) 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 2F +y = F 0y + F zy = E y + B x
22 Quark OAM from Wigner Distributions 19 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) ) ( ) y 2F +y = F 0y + F zy = E y + B x = E + v B for v = (0, 0, 1)
23 Quark OAM from Wigner Distributions 19 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) ) ( ) y 2F +y = F 0y + F zy = E y + B x = E + v B for v = (0, 0, 1) Torque along the trajectory of q Change in OAM [ ( )] z T z = x E ˆ z B L z = ( )] z dr [ x E x ˆ z B
24 Torque in Scalar Diquark Model (with C.Lorcé) 20 (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)
25 Quark OAM - sign of L q L q 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 e + moving through dipole field of e consider e polarized in +ẑ direction µ in ẑ direction (Figure) e + moves in ẑ direction net torque negative sign of L q L q in QCD color electric force between two q in nucleon attractive same as in positronium spectator spins positively correlated with nucleon spin expect L q L q < 0 in nucleon
26 Quark OAM - sign of L q L q 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 e + moving through dipole field of e consider e polarized in +ẑ direction µ in ẑ direction (Figure) e + moves in ẑ direction net torque negative sign of L q L q in QCD color electric force between two q in nucleon attractive same as in positronium spectator spins positively correlated with nucleon spin expect L q L q < 0 in nucleon 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
27 Calculating Jaffe-Monohar OAM in Lattice QCD 22 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
28 Quasi Light-Like Wilson Lines from Lattice QCD 23 f 1T,SIDIS = f 1T,DY (Collins) f 1T (x, k ) is k -odd term in quark-spin averaged momentum distribution in polarized target
29 Quasi Light-Like Wilson Lines from Lattice QCD 23 f 1T,SIDIS = f 1T,DY (Collins) f 1T (x, k ) is k -odd term in quark-spin averaged momentum distribution in polarized target
30 Digression: Average Force on Quarks in DIS 24 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) ) ( ) 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
31 Digression: Average Force on Quarks in DIS 25 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) ) ( ) 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)
32 Digression: Average Force on Quarks in DIS 25 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) ) ( ) 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)
33 OAM from twist 3 GPDs (with F.Aslan & C.Lorcé) 26 twist-3 GPDs (Polyakov & Kitpily) dz e ixz p + p q(z /2)γ x q( z /2) p = 1 [ x 2 p + ū(p ) 2M G 1 + γ x (H +E+G 2 ) + x γ + p + G 3 + i y γ + ] γ 5 p + G 4 u(p) Lorentz invariance relations dxg q 1 (x, ξ, t) = 0 dxg q 2 (x, ξ, t) = 0 dxg q 3 (x, ξ, t) = 0 dxg q 4 (x, ξ, t) = 0 Tests test above relations in scalar diquark model & QED compare L(x) with xg 2 (x) QCD Eqs. of motion dx xg q 2 (x, 0, 0) = Lq results xg 2 (x, 0, 0) 2x(1 x) ln Λ 2 L JM (x) (1 x) 2 ln Λ 2 [ L Ji (x) 1 2 H +E H ] [ ] 1 x 2 ln Λ integrals equal x-dependence not same how about dxg 2 (x)? = 0???
34 OAM from twist 3 GPDs (with F.Aslan & C.Lorcé) 27 Lorentz invariance relations dxg q 2 (x, ξ, t) = 0 xg 2 (x, 0, 0) 2x(1 x) ln Λ 2 δ(x) in G 2 (x, 0, t) (QED/QCD) G 2 (x, ξ, t) has term Θ( ξ<x<ξ) ξ rep. of δ(x) for ξ 0 without δ(x): dxg 2 (x, 0, t) 0 with δ(x): dxg 2 (x, 0, t) = 0 δ(x) and twist 3 δ(x) terms possible in twist 3 PDFs twist 3 GPDs allow studying those contributions in detail cannot use Lorentz invariance relation to constrain G 2 (x, 0, 0) twist-3 PDFs (MB&Y.Koike 02) Lorentz invariance relations: dxg1 (x) = dxg T (x) (probably satisfied in QCD) dxh1 (x) = dxh L (x) (violated in QCD) h L contains term δ(x) missed in lim ε 0 1 ε dx h L(x) similar for e(x) (scalar twist 3)
35 Twist-3 Factorization (F.Aslan,M.B.,C.Lorcé,A.Metz & B.Pasquini) 28 potential issue DVCS amplitude [ dx 1 x ξ 1 x+ξ ] GP D(x, ξ, t) integral undefined if GP D(x, ξ, t) discontinuous at x = ξ potential issue with twist-3 factorization relevant DVCS amplitude involves G 2 (x, ξ, t) ± 1 ξ G 2(x, ξ, t) verified that discontinuities in 1-loop QCD at x = ξ cancel for G 2 (x, ξ, t) ± 1 ξ G 2 (x, ξ, t) good news made world safer for twist-3 factorization bad news may not be possible to extract G 2 from DVCS but only G 2 ± 1 G ξ 2 dx xg q 2 (x) = Lq may not be practically useful
36 Summary G 2 only 29 verified explicitly dx xg q 2 (x) = Lq in QED/QCD and scalar diquark model to one loop discontinuities for twist-3 GPDs at x = ±ξ from terms only contributing in ERBL region for G 2 cancel in relevant DVCS amplitudes discontinuities give rise to δ(x) contributions for ξ 0 careful using Lorentz invariance relations in fits: δ(x) may have to be uncluded in parameterizations
37 Summary 30 GPDs F T q(x, b ) 3d imaging polarization deformation 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) d 2 : force on quarks in DIS sign and magnitude of d 2 verfied OAM sum rule for twist-3 GPD G 2 twist-3 GPDs contain discontinuities/δ(x)
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