GPDs and Quark Orbital Angular Momentum

Transcription

1 GPDs and Quark Orbital Angular Momentum Matthias Burkardt NMSU May 14, 2014

2 Outline background proton spin puzzle 3D imaging of the nucleon, Single-Spin Asymmetries (SSAs), Quark orbital angular momentum angular momentum decompositions (Jaffe v. Ji) quark-gluon correlations 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, ): 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, 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)

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, 2 )e ib 1 d 2 2M b y (2π) 2 E q(x, 2 )e ib Physics: relevant density in DIS is j + j 0 + j z and left-right asymmetry from j z 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, 2 )e ib 1 d 2 2M b y (2π) 2 E q(x, 2 )e ib Physics: relevant density in DIS is j + j 0 + j z and left-right asymmetry from j z

8 Impact parameter dependent quark distributions 7 proton polarized in +ˆx direction d 2 q(x, b ) = (2π) 2 H q(x, 2 )e ib 1 d 2 2M b y (2π) 2 E q(x, 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) = 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) = 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 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 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)

12 Angular Momentum Carried by Quarks 10 lattice results for J q (LHPC) no disconnected quark loops ( K.-F. Liu) chiral extrapolation J q = 1 2 dx x [H(x, 0, 0) + E(x, 0, 0)] L q = J q 1 2 Σq

13 The Nucleon Spin Pizzas 11 Ji decomposition Jaffe-Manohar decomposition light-cone framework & gauge A + = = 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 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 +i A j P, S L g =2 d 3 r P,S TrF +j ( x i ) za j P,S manifestly gauge invariant definition for each term exists ( Hatta)

14 Photon Angular Momentum in QED 12 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 r ) 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! can also be done for only part of A Chen/Goldman, Wakamatsu

15 The Nucleon Spin Pizzas 13 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?

16 OAM from Wigner Functions 14 Wigner Functions (Belitsky, Ji, Yuan; 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. (quasi) probabilty distribution for b and k f(x, k ) = d 2 b W (x, b, k ) q(x, b ) = d 2 k W (x, b, k ) OAM from Wigner (Lorcé, Pasquini,...) L z = dx d 2 b d 2 k W (x, b, k )(b x k y b y k x ) = d 3 r P,S q( r)γ +( r i ) zq( r) P,S = L q Gauge Invariance? need to include Wilson-line gauge link to connect 0 and ξ (Ji, Yuan; Hatta; Lorcé;...)

17 Quark momentum from Wigner Distributions 15 Wigner Functions with gauge link U 0ξ (Ji, Yuan; Hatta) 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 + U 0ξ q(ξ) P S. k dx d 2 b d 2 k W (x, b, k ) k depends on choice of path! light-cone staple straight-line gauge link k = P,S q( x)γ + i Dq( x) P,S i D = i g A( x) k = 0 (T-odd!) correct choice for k distributions relevant for SIDIS K = P,S q( x)γ + i Dq( x) P,S i D = i g A(x =, x ) A + = 0 K 0 (FSI! Brodsky,Hwang,Schmidt)

18 Quark momentum from Wigner Distributions 15 difference K q k q K q k q = g d 3 x P,S q( x)γ + x dr F + (r, x )q( x) P,S 2F +y = F 0y + F zy = E y + B x light-cone staple straight-line gauge link k = P,S q( x)γ + i Dq( x) P,S i D = i g A( x) k = 0 (T-odd!) correct choice for k distributions relevant for SIDIS K = P,S q( x)γ + i Dq( x) P,S i D = i g A(x =, x )+g x dr A + id j = i j ga j (x, x ) g x dr F +j

19 Quark momentum from Wigner Distributions 15 difference K q k q K q k q = g d 3 x P,S q( x)γ + x dr F + (r, x )q( x) P,S color Lorentz Force acting on ejected quark (Qiu, Sterman) ( ) y 2F +y = F 0y + F zy = E y + B x = E + v B for v = (0, 0, 1) Impulse due to FSI k q K q k q = (average) change in momentum due to FSI! straight-line gauge link k = P,S q( x)γ + i Dq( x) P,S i D = i g A( x) k = 0 (T-odd!) light-cone staple correct choice for k distributions relevant for SIDIS K = P,S q( x)γ + i Dq( x) P,S i D = i g A(x =, x )+g x dr A + id j = i j ga j (x, x ) g x dr F +j

20 Quark momentum from Wigner Distributions 15 difference K q k q K q k q = g d 3 x P,S q( x)γ + x dr F + (r, x )q( x) P,S color Lorentz Force acting on ejected quark (Qiu, Sterman) ( ) y 2F +y = F 0y + F zy = E y + B x = E + v B for v = (0, 0, 1) Corollary: d 2 average force on quark in DIS from pol target polarized DIS: MB, PRD 88 (2013) 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 matrix element defining d 2 1 st integration point in QS-integral

21 Quark momentum from Wigner Distributions 15 difference K q k q K q k q = g d 3 x P,S q( x)γ + x dr F + (r, x )q( x) P,S color Lorentz Force acting on ejected quark (Qiu, Sterman) ( ) y 2F +y = F 0y + F zy = E y + B x = E + v B for v = (0, 0, 1) Corollary: d 2 average force on quark in DIS from pol target polarized DIS: MB, PRD 88 (2013) 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 sign of d q 2 opposite Sivers f q 1T deformation of quark distributions

22 OAM from Wigner Functions 16 Wigner Functions with gauge link U 0ξ (Ji, Yuan; Hatta) 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 + U 0ξ q(ξ) P S. W and thus L z = dx d 2 b d 2 k W (x, b, k )(b x k y b y k x ) may depend on choice of path! straight line (Ji et al.) straigth Wilson line from 0 to ξ yields L q = ( zq( x) P,S d 3 x P,S q ( x) x id) i D = i g A( x) same as Ji-OAM f(x, k ) = d 2 b W (x, b, k ) not the TMDs relevant for SIDIS (missing FSI!)

23 OAM from Wigner Functions 16 Wigner Functions with gauge link U 0ξ (Ji, Yuan; Hatta) 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 + U 0ξ q(ξ) P S. W and thus L z = dx d 2 b d 2 k W (x, b, k )(b x k y b y k x ) may depend on choice of path! Light-Cone Staple for U ±LC 0ξ (Hatta) want Wigner function that yields TMDs relevant for SIDIS when integrated d 2 b path for gauge link light-cone staple U +LC 0ξ L q + = d 3 x P,S q( x)γ + ( x i D) zq( x) P,S i D = i g A(, x ) (A + = 0)

24 OAM from Wigner Functions 16 Wigner Functions with gauge link U 0ξ (Ji, Yuan; Hatta) 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 + U 0ξ q(ξ) P S. W and thus L z = dx d 2 b d 2 k W (x, b, k )(b x k y b y k x ) may depend on choice of path! Light-Cone Staple for U ±LC 0ξ (Hatta) want Wigner function that yields TMDs relevant for SIDIS when integrated d 2 b path for gauge link light-cone staple U +LC 0ξ L q + = d 3 x P,S q( x)γ + ( x i D) zq( x) P,S i D = i g A(, x ) + g x dr A + id j = i j ga j (x, x ) g x dr F +j

25 Quark OAM from Wigner Distributions 17 straight line ( Ji) L q = d 3 x P,S q( x)γ + ( x i D) zq( x) P,S light-cone staple ( Jaffe-Manohar) 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

26 Quark OAM from Wigner Distributions 17 straight line ( Ji) L q = d 3 x P,S q( x)γ + ( x i D) zq( x) P,S light-cone staple ( Jaffe-Manohar) 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 acting on ejected quark (MB: arxiv: ) ( ) y 2F +y = F 0y + F zy = E y + B x = E + v B for v = (0, 0, 1)

27 Quark OAM from Wigner Distributions 17 straight line ( Ji) L q = d 3 x P,S q( x)γ + ( x i D) zq( x) P,S light-cone staple ( Jaffe-Manohar) 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 acting on ejected quark (MB: arxiv: ) ( ) 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

28 Quark OAM from Wigner Distributions 18 straight line ( Ji) L q = d 3 x P,S q( x)γ + ( x i D) zq( x) P,S light-cone staple ( Jaffe-Manohar) L q = d 3 x P,S q( x)γ + ( x i D) zq( x) P,S i D = i g A( x) i D = i g A(x =, x ) difference L q L q ( Wakamatsu: L q pot) L q L q = L q F SI = change in OAM as quark leaves nucleon example: torque in magnetic dipole field

29 Connection with Jaffe-Manohar-Bashinsky 19 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

30 Connection with Jaffe-Manohar-Bashinsky 19 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 )

31 Nucleon Spin Decompositions 20 The Difference L q L q [MB, PRD88, (2013)] L q L q = d 3 x P,S q( x)γ +[ x x dr gf + (r, x ) ] z q( 0) P,S /N in QCD, additional Wilson lines (along r ) compare k dx d 2 k f(x, k )k f(x, k ) d 2 ξ dξ (2π) 3 e ik ξ P S q(0)γ + U 0ξ q(ξ) P S Wilson line U 0ξ along x to properly account for FSI acting on ejected quark, i.e. f(x, k ) momentum distribution incl. FSI relevant for SIDIS (JLab, EIC) and DY (Rhic) k = P,S q(0)γ + dr gf + (r, 0 0 )q( 0) P,S k = (average) change in momentum due to FSI as quark leaves target (Qiu, Sterman) Light-Cone Staple for U ±LC 0ξ Color Lorentz Force 2F +y = F 0y + F zy for v = (0, 0, 1) = E y + B ( x ) y = E + v B

32 Nucleon Spin Decompositions 20 The Difference L q L q [MB, PRD88, (2013)] L q L q = d 3 x P,S q( x)γ +[ x x dr gf + (r, x ) ] z q( 0) P,S /N L q L q = (average) change in OAM due to FSI as quark leaves target compare k dx d 2 k f(x, k )k f(x, k ) d 2 ξ dξ (2π) 3 e ik ξ P S q(0)γ + U 0ξ q(ξ) P S Wilson line U 0ξ along x to properly account for FSI acting on ejected quark, i.e. f(x, k ) momentum distribution incl. FSI relevant for SIDIS (JLab, EIC) and DY (Rhic) k = P,S q(0)γ + dr gf + (r, 0 0 )q( 0) P,S k = (average) change in momentum due to FSI as quark leaves target (Qiu, Sterman) Light-Cone Staple for U ±LC 0ξ Color Lorentz Force 2F +y = F 0y + F zy for v = (0, 0, 1) = E y + B ( x ) y = E + v B

33 Calculating Jaffe-Monohar OAM in Lattice QCD 21 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 talk by M. Engelhardt tomorrow...

34 Calculating Jaffe-Monohar OAM in Lattice QCD 22 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 talk by M. Engelhardt tomorrow... 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...

35 Summary 23 generalized parton distributions and 3d imaging final state interactions in deep-inelastic scattering GPDs and OAM TMDs and OAM from Wigner distributions straight-(wilson)line gauge link L q ( Ji-OAM ) light-cone staple- gauge link L q + ( JM-OAM ) L q + L q = change in OAM as quark leaves nucleon (torque due to FSI) A + = 0 gauge (with anti-symmetric boundary condition) L q + canonical OAM (Jaffe-Manohar-OAM) JM-OAM from lattice QCD