Quark Orbital Angular Momentum

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1 Quark Orbital Angular Momentum Matthias Burkardt NMSU September 21, 2015

2 Outline 3D imaging of the nucleon, Single-Spin Asymmetries (SSAs) quark-gluon correlations color force angular momentum decompositions (Jaffe v. Ji), change in OAM due to FSI Summary 2

3 Deeply Virtual Compton Scattering (DVCS) 3 form factor electron hits nucleon & nucleon remains intact Compton scattering electron hits nucleon, nucleon remains intact & photon gets emitted study amplitude that nucleon remains intact as function of momentum transfer F (q 2 ) F (q 2 ) = dxgp D(x, q 2 ) GPDs provide momentum disected form factors study both energy & q 2 dependence additional information about momentum fraction x of active quark generalized parton distributions GP D(x, q 2 )

4 Physics of GPDs: 3D Imaging of the Nucleon 4 MB, PRD62, (2000) form factors: F T ρ( r) GP Ds(x,0, 2 ): form factor for quarks with momentum fraction x suitable FT of GP Ds should provide spatial distribution of quarks with momentum fraction x Impact Parameter Dependent Quark Distributions d 2 q(x, b ) = (2π) 2 GP D(x, 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 probabilistic interpretation! no relativistic corrections: Galilean subgroup! (MB,2000) corollary: interpretation of 2d-FT of F 1 (Q 2 ) as charge density in transverse plane also free from relativistic corrections (MB,2003;G.A.Miller, 2007)

$Impact parameter dependent quark distributions 5 unpolarized proton q(x, b ) = d 2 (2π) 2 H(x, 0, 2 )e ib F 1 ( 2 ) = dxh(x, 0, 2 ) x = momentum fraction of the quark$

5 Impact parameter dependent quark distributions 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

6 Impact parameter dependent quark distributions 6 proton polarized in +ˆx direction no axial symmetry! 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 Physics: relevant density in DIS is j + j 0 + j 3 and left-right asymmetry from j 3 intuitive explanation moving Dirac particle with anomalous magnetic moment has electric dipole moment to p and magnetic moment γ sees flavor dipole moment of oncoming nucleon

7 Impact parameter dependent quark distributions 6 proton polarized in +ˆx direction no axial symmetry! 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 Physics: relevant density in DIS is j + j 0 + j 3 and left-right asymmetry from j 3

8 Impact parameter dependent quark distributions 7 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 sign & magnitude of the average shift model-independently related to p/n anomalous magnetic moments: b q y dx d 2 b q(x, b )b y = 1 2M dxeq (x, 0, 0) = κq 2M

9 Impact parameter dependent quark distributions 7 sign & magnitude of the average shift model-independently related to p/n anomalous magnetic moments: b q y dx d 2 b q(x, b )b y = 1 2M dxeq (x, 0, 0) = κq 2M κ p = = 2 3 κp u 1 3 κp d +... u-quarks: κ p u = 2κ p + κ n = shift in +ŷ direction d-quarks: κ p u = 2κ n + κ p = shift in ŷ direction b q y = O (±0.2fm)!!!!

10 GPD Single Spin Asymmetries (SSA) 8 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)

11 Sign of Boer-Mulders Function 9 q with polarization unpolarized target all q polns. equally likely q in ground state orbit counterclockwise current from lower component q distribution shifted to top

12 Sign of Boer-Mulders Function 9 q with polarization unpolarized target q with pol. shifted to left q in ground state orbit counterclockwise current from lower component q distribution shifted to top

13 Sign of Boer-Mulders Function 9 q with polarization unpolarized target q with pol. shifted to right q in ground state orbit counterclockwise current from lower component q distribution shifted to top

14 Sign of Boer-Mulders Function 9 q with polarization unpolarized target q with pol. shifted to top q in ground state orbit counterclockwise current from lower component q distribution shifted to top

15 Sign of Boer-Mulders Function 9 q with polarization unpolarized target q with pol. shifted to bottom q in ground state orbit counterclockwise current from lower component q distribution shifted to top

16 Sign of Boer-Mulders Function 9 q with polarization unpolarized target lattice calculations (QCDSF) transversity distribution in unpol. target described by chirally odd GPD ĒT Ē T > 0 for u & d (QCDSF) connection h 1 (x, k ) ĒT similar to f 1T (x, k ) E. h 1 (x, k ) < 0 for u/p, d/p, u/π, d/π,..(mb+bh, 2008) different valence quarks add coherently h 1 > f 1 (MB+BH; Musch)

17 Calculating TMDs in Lattice QCD 10 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

18 Quasi Light-Like Wilson Lines from Lattice QCD 11 f 1T,SIDIS = f 1T,DY (Collins) f 1T (x, k ) is k -odd term in quark-spin averaged momentum distribution in polarized target

19 Quasi Light-Like Wilson Lines from Lattice QCD 11 f 1T,SIDIS = f 1T,DY (Collins) f 1T (x, k ) is k -odd term in quark-spin averaged momentum distribution in polarized target

20 Average force on quarks in DIS 12 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)

21 Average force on quarks in DIS 12 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)

22 Angular Momentum Carried by Quarks 13 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 p z & b use quark GPDs to determine angular momentum carried by quarks Jq x = 1 2 dx x [H(x, 0, 0) + E(x, 0, 0)] (X.Ji, 1996) interpretation in terms of 3D distribution (MB,2001,2005)

23 Angular Momentum Carried by Quarks 14 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

24 Photon Angular Momentum in QED 15 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!

framework & gauge A + = 0 1 2 = q 1 2 q + L q + G + L g 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

25 The Nucleon Spin Pizzas 16 Ji decomposition Jaffe-Manohar decomposition 1 2 = q 1 2 q + L q + J g 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 = [ )] 3 d 3 x P, S x ( E B P, S i D = i g A light-cone framework & gauge A + = = q 1 2 q + L q + G + L g 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 term exists (Lorcé, Pasquini; Hatta)

26 The Nucleon Spin Pizzas 17 Ji decomposition 1 2 = q 1 2 q + L q + J g 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 = [ )] 3 d 3 x P, S x ( E B P, S i D = i g A GPDs L q p p G L i q,g Li Jaffe-Manohar decomposition light-cone framework & gauge A + = = q 1 2 q + L q + G + L g 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 invariant definitions for each term exist ( Hatta) QED: L e L e [M.B. + Hikmat BC, PRD 79, (2009)] L q L q =? can we calculate/predict/measure the difference? what does it represent?

27 OAM from Wigner Distributions 18 Wigner Functions (Belitsky, Ji, Yuan; Lorcé, Pasquini; Metz et al.) 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 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 for U ±LC 0ξ light-cone staple yields L Jaffe Manohar

28 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 JM Lq Ji = 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

29 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 JM Lq Ji = 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)

30 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 JM Lq Ji = 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

31 Quark OAM 20 difference L q L q L q JM Lq Ji = Lq F SI = change in OAM as quark leaves nucleon example: torque in magnetic dipole field QED - effect from e in its own dipole field L e F SI = α 3π > 0 (MB+H.BC, PRD 2009)

32 Quark OAM 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

33 Quark OAM - sign of L q L q 22 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

34 Quark OAM - sign of L q L q 22 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

35 Quark OAM - G 2 Sum Rule 23 Leader & Lorcé - Phys. Repts. Twist-3 GPD Angular momentum Sum Rule - Polyakov Scalar diquark model sum rule not satisfied issue not Ji vs. JM as L Ji = L JM in that model origin of violation still under investigation...

36 Digression: Aharonov Bohm Effect in SSAs? 24 Aharonov Bohm effect double slit experiment with solenoid between beams no magnetic field at beam vector( potential A modifies QM phase ) ( ) exp i e A d r = exp i e B ds B 0 shifts interference pattern

37 Digression: Aharonov Bohm Effect in SSAs? 25 Aharonov Bohm effect double slit experiment with solenoid between beams no magnetic field at beam vector( potential A modifies QM phase ) ( ) exp i e A d r = exp i e B ds B 0 shifts interference pattern transverse SSA Wilson( line phase factor exp i g ) A d r crucial for SSAs in SIDIS et al. However, contrary to SSAs in SIDIS, AB-effect does NOT generate a net (average), k example for Ehrenfest s theorem: k = dt F No net momentum f(k ) f dbl f sgl f sgl sin2 ( ) k a 2 (k a) 2 ) f dbl cos ( 2 k d+φ 2 a width of each slit d center-to-center distance of slits Φ = e A( r) d r dk I(k )k = 0

38 Summary 26 GPDs F T q(x, b ) 3d imaging polarization deformation nucleon anomalous magnetic moment deformation sign of SSAs d 2 : force on quarks in DIS sign and magnitude of d 2 L q JM Lq Ji = change in OAM as quark leaves nucleon (due to torque from FSI) SSAs not a pure Aharonov-Bohm effect

39 Calculating Jaffe-Monohar OAM in Lattice QCD 27 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

40 Quasi Light-Like Wilson Lines from Lattice QCD 28 f 1T,SIDIS = f 1T,DY (Collins) f 1T (x, k ) is k -odd term in quark-spin averaged momentum distribution in polarized target

41 Quasi Light-Like Wilson Lines from Lattice QCD 28 f 1T,SIDIS = f 1T,DY (Collins) f 1T (x, k ) is k -odd term in quark-spin averaged momentum distribution in polarized target

42 Calculating Jaffe-Monohar OAM in Lattice QCD 29 TMDs in lattice QCD B. Musch, P. Hägler, M. Engelhardt 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 next: 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 renormalization same as f 1T study ratios...

43 Angular Momentum Carried by Quarks 30 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

44 Connection with Jaffe-Manohar-Bashinsky 31 PT (Hatta) PT L + = L (different from SSAs due to factor x in OAM) alternative: Bashinsky-Jaffe A + = 0 x i x A ( x ) = antisymm. boundary condition A + = 0 fix residual gauge inv. A µ A µ + µ φ( x ) by imposing A (, x ) = A (, x ) A (, x ) A (, x ) = dx F + gauge inv. L + involves i D + = i g A(, x ) L involves i D = i g A(, x ) L + = L no contribution from A(, x ) naive JM OAM L JM = L + = L [ i ga( x ] ) dx A (x, x ) dx

45 Connection with Jaffe-Manohar-Bashinsky 31 PT (Hatta) PT L + = L (different from SSAs due to factor x in OAM) alternative: Bashinsky-Jaffe A + = 0 x i x A ( x ) = antisymm. boundary condition A + = 0 fix residual gauge inv. A µ A µ + µ φ( x ) by imposing A (, x ) = A (, x ) A (, x ) A (, x ) = dx F + gauge inv. L + involves i D + = i g A(, x ) L involves i D = i g A(, x ) L + = L no contribution from A(, x ) naive JM OAM L JM = L + = L [ i ga( x ] ) dx A (x, x ) dx = 1 2 ( A (, x ) + A (, x ) L JB = 1 2 (L + + L ) = L + = L )

Quark Orbital Angular Momentum

Quark Orbital Angular Momentum Matthias Burkardt NMSU May 29, 2015 Outline 3D imaging of the nucleon, Single-Spin Asymmetries (SSAs) quark-gluon correlations color force angular momentum decompositions