Nucleon Structure & GPDs

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1 Nucleon Structure & GPDs Matthias Burkardt New Mexico State University August 15, 2014

2 Outline 2 GPDs: Motivation impact parameter dependent PDFs angular momentum sum rule physics of form factors DVCS Q2 evol. GPDs for x = ξ torque in DIS Summary GPDs

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, ξ, 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, ξ, q 2 )

4 What can we learn from GPDs? 4 Transverse Imaging Quark (Orbital) Angular Momentum d 2 X.Ji, 1996: q(x, b ) = (2π) J q = 1 2 H q(x, 0, 2 )e ib dx x [H q (x, ξ, 0) + E q (x, ξ, 0)] 1 d 2 2 2M b y (2π) 2 E q(x, 0, 2 )e ib Physics of large Q 2 Form Factors Feynman: form factor at large Q 2 dominated by high-x quarks pqcd mechanism (SJB+...): dominated by typical valence quarks accompanied by hard gluon exchange GPDs F 1 (Q 2 ) = F 2 (Q 2 ) = dxh(x, ξ, Q 2 ) dxe(x, ξ, Q 2 )

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

7 Impact parameter dependent quark distributions 7 Chromodynamic Lensing nucleon size at small x J.Qiu nucleon size at large x MB & G.A.Miller (2006): mathematical model H(x, 0, Q 2 ) = (1 x) 3 exp( µ 2 (1 x) 2 Q 2 ) F 1 (Q 2 ) = dxh(x, 0, Q 2 ) 1 Q 4 large Q 2 behaviour dominated by large x regime Feynman qualitative connection between deformation and SSA one gluon exchange: SSA density-dentity correlations LHC physics Frankfurt, Strikman, Weiss: spatial distributions hard parton-parton processes at LHC

8 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)

9 Angular Momentum Carried by Quarks 9 L x = yp z zp y J x = d 3 r [ yt 0z zt 0y] if state invariant under rotations about ˆx axis then yp z = zp y L x = 2 yp z J z = 2 d 3 r yt 0z 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) partonic interpretation in terms of 3D distribution (MB,2001,2005)

10 Angular Momentum Carried by Quarks 10 Total (Spin+Orbital) Quark Angular Momentum Jq x = L x q + Sq x = d 3 r [ ytq 0z ( r) ztq 0y ( r) ] T q µν ( r) energy momentum tensor (T q µν ( r) = Tq νµ ( r)) Tq 0i ( r) momentum density [Pq i = d 3 rtq 0i ( r) ] think: ( r p) x = yp z zp y relate to impact parameter dependent quark distributions q ψ (x, r ): Consider spherically symmetric wave packet with nucleon polarized in +ˆx direction eigenstate under rotations about x-axis both terms in Jq x equal: Jq x = 2 d 3 r ytq 0z d 3 r ytq 00 J x q = ( r) = d 3 r y [ Tq 0z ( r) ( r) = 0 = d 3 r ytq zz ( r) + T z0 q ( r) ] d 3 r yt ++ q ( r) with T ++ T 00 + T 0z + T z0 + T zz

11 Angular Momentum Carried by Quarks 10 relate to impact parameter dependent quark distributions q ψ (x, r ): Consider spherically symmetric wave packet with nucleon polarized in +ˆx direction eigenstate under rotations about x-axis both terms in Jq x equal: Jq x = 2 d 3 r ytq 0z d 3 r ytq 00 J x q = ( r) = d 3 r y [ Tq 0z ( r) ( r) = 0 = d 3 r ytq zz ( r) + T z0 q ( r) ] d 3 r yt ++ q ( r) with T ++ T 00 + T 0z + T z0 + T zz dx xq(x, r ) = 1 2m N dr z T ++ ( r) (note: here x is momentum fraction and not r x ) ψ J x q ψ = m N dx d 2 b xb y q ψ (x, b )

12 Angular Momentum Carried by Quarks 11 distribution in delocalized wave packet (pol. in +ˆx direction) q ψ (x, b ) = ( ) d 2 r q(x, b r ) ψ(r ) 2 1 2M r y ψ(r ) 2 with d 2 q(x,b ) = (2π) 2 H q(x, 0, 2 )e ib 1 2M two contributions to shift b y intrinsic shift relative to center of momentum R overall shift of R for polarized nucleon d 2 (2π) 2 E q(x, 0, 2 )e ib insert into ψ Jq x ψ = dx d 2 b q ψ (x, b ) MB, PRD72, (2005) ψ Jq x ψ = 1 2 dx x [Hq (x, 0, 0) + E q (x, 0, 0)] (here: derived for p = 0 only!) X.Ji (1996): rotational invariance apply to all components of J result for J z q also applies to p z 0 partonic interpretation ( shift) exists only for components of J q!

13 Angular Momentum Carried by Quarks 11 insert into ψ Jq x ψ = dx d 2 b q ψ (x, b ) MB, PRD72, (2005) ψ Jq x ψ = 1 2 dx x [Hq (x, 0, 0) + E q (x, 0, 0)] (here: derived for p = 0 only!) X.Ji (1996): rotational invariance apply to all components of J result for J z q also applies to p z 0 partonic interpretation ( shift) exists only for components of J q! gauge invariance matrix element of T q ++ = qγ + i + q in A + = 0 gauge same as that of qγ + (i + ga + ) q in any gauge identify 1 2 dx x [H(x, 0, 0) + E(x, 0, 0)] with Jq in decomposition where L q = ( d 3 x P,S q ( x) x id ) q( x) P,S

14 Angular Momentum Carried by Quarks 11 insert into ψ Jq x ψ = dx d 2 b q ψ (x, b ) MB, PRD72, (2005) ψ Jq x ψ = 1 2 dx x [Hq (x, 0, 0) + E q (x, 0, 0)] (here: derived for p = 0 only!) caution! X.Ji (1996): rotational invariance apply to all components of J result for J z q also applies to p z 0 partonic interpretation ( shift) exists only for components of J q! made heavily use of rotational invariance itentification ψ Jq x ψ = 1 2 dx x [H(x, 0, 0) + E(x, 0, 0)] does not apply to unintegrated quantities [ d 2 e ib x 2 H(x, 0, 2 ) + E(x, 0, 2 ) ] [ not equal to J z (b) J q(x) x 2 Hq(x, 0, 0) + E q(x, 0, 2 ) ] not x-distribution of angular momentum Jq z (x) in long. pol. target (only x-distribution of half of J q ) regardless whether one takes gauge covariant definition or not

15 Angular Momentum Carried by Quarks 12 lattice: 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 L u + L d 0 signs of L q counter-intuitive

16 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 imaging requires 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 )

17 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 imaging requires 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

18 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

19 GPDs for x = ξ (arxiv: ) 15 imaging: ξ 0 imaging: ξ = 0 probabilistic interpretation variable conjugate to is b (1 x)r distance to COM of hadron 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 (any ξ) distance r of active quark to spectators (any ξ) variable conjugate to is 1 x 1 ξ r imaging (x = ξ) no probabilistic interpretation variable conjugate to is r distance to COM of spectators expect no dramatic ξ-dependence of 2 -slope (size of system)

20 The Nucleon Spin Pizzas 16 Ji decomposition Jaffe 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

21 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 Jaffe 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 q evolution and/or p p G L i q,g Li L q L q L q L q =? can we calculate/predict the difference? what does it represent?

22 The Nucleon Spin Pizzas - What s New? 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 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 (A(, x ) A( x))] z q( x) P,S A (, x ) A ( x) = x dr F + (r, x ) 2F +y = F 0y + F zy = E y + B x 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

23 The Nucleon Spin Pizzas - What s New? 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 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 (A(, x ) A( x))] z q( x) P,S A (, x ) A ( x) = x dr F + (r, x ) color Lorentz Force acting on ejected quark (MB: arxiv: ) 2F +y = F 0y + F zy = E y + B x 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 Difference Jaffe/Manohar vs. Ji 19 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 L q L q = change in OAM as the quark leaves the nucleon example: torque in magnetic dipole field

25 Calculating Jaffe-Monohar OAM in Lattice QCD 20 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

26 Quasi Light-Like Wilson Lines from Lattice QCD 21 f 1T,SIDIS = f 1T,DY (Collins) f 1T (x, k ) is k -odd term in quark-spin averaged momentum distribution in polarized target

27 Quasi Light-Like Wilson Lines from Lattice QCD 21 f 1T,SIDIS = f 1T,DY (Collins) f 1T (x, k ) is k -odd term in quark-spin averaged momentum distribution in polarized target

28 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 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...

29 quark-photon corellations 23 inclusive SSA Hall A: inclusively scattered (summed over all final states) electrons off polarized target show left-right asymmetry relatve to target spin for n target (i.e. 3 He), the e is preferentially deflected to beam-right

30 quark-photon corellations 23 inclusive SSA Hall A: inclusively scattered (summed over all final states) electrons off polarized target show left-right asymmetry relatve to target spin Metz et al. arxiv: for n target (i.e. 3 He), the e is preferentially deflected to beam-right beam e approx. at same position as struck quark k e q ( e)f q with F q dr P, S q(0)γ + ef + QED (0, r )q(0) P, S

31 quark-photon corellations 23 Metz et al. arxiv: beam e approx. at same position as struck quark k e q ( e)f q with F q dr P, S q(0)γ + ef + QED (0, r )q(0) P, S model (Metz et al. arxiv: ) if FSI in QCD is caused by one-gluon exchange between valence quarks then F q = 2 e qα 4 3 α F q QCD s e s charge of spectators, e.g. e q = 1 3 for u quarks in proton

32 Metz et al. vs. MB 24 Metz et al. F u/p F d/p multiply by charge of spectator flavor, e.g. e q = 1 3 for u in p divide by 4 3 α s em (x, x) = α 6πC F α s M em (x, x) = 2α 3πC F α s M F u/n α (x, x) = F d/n em 3πC F α s M em (x, x) = α 6πC F α s M violates 2 3 F u/n assume T q/p F 1 3 F d/n u/p gtf (x, x) d/p gtf (x, x) d/p gtf (x, x) u,p gtf (x, x).! = 0 MB saturate sum rule q=u,d multiply by charge of spectator flavor, e.g. e q = 1 3 for u in p divide by 2 3 α s for u in p F u/p α (x, x) = F d/p em 3πC F α s M em (x, x) = 2α 3πC F α s M F u/n α em (x, x) = 3πC F α s M F d/n em (x, x) = 2α 3πC F α s M satisfies 2 3 F u/n 1 3 F d/n q/p dxt F (x, x) = 0. u/p gtf (x, x) d/p gtf (x, x) d/p gtf (x, x) u,p gtf (x, x).! = 0

33 Metz et al. vs. MB 25 Metz et al. MB F u/p F d/p em (x, x) = α 6πC F α s M em (x, x) = 2α 3πC F α s M F u/n α (x, x) = F d/n em 3πC F α s M em (x, x) = α 6πC F α s M violates 2 3 F u/n 1 3 F d/n u/p gtf (x, x) d/p gtf (x, x) d/p gtf (x, x) u,p gtf (x, x).! = 0 ( σ p UT 2αemg 3πC F α s T u/p F ( σut n 2αemg 3πC F α s 2T d/p F σ p UT σn UT ) T d/p F ) 1 4 T u/p F F u/p α (x, x) = F d/p em 3πC F α s M em (x, x) = 2α 3πC F α s M F u/n α em (x, x) = 3πC F α s M F d/n (x, x) = 2α 3πC F α s M satisfies 2 3 F u/n em 1 3 F d/n ( σ p UT 2αemg 3πC F α s 2T u/p F ( σut n 2αemg 3πC F α s 2T d/p F σ p UT σn UT u/p gtf (x, x) d/p gtf (x, x) d/p gtf (x, x) u,p gtf (x, x).! = 0 ) T d/p F T u/p F ) inclusive SSA σ UT 4F u + F d assume T d F T u F

34 Inclusive SSA - Lensing Picture 26 neutron target proton target

35 Summary: Hadron Structure and GPDs 27 GPDs impact parameter dependent PDFs parton interpretation for Ji-relation (in principle) GP D(x, ξ, Q 2 ) from QCD evolution of GP D(ξ, ξ, Q 2 ) EIC (need wide Q 2 range) Ji vs. Jaffe-Manohan quark OAM: interpretation of L q L q as change in OAM of ejected quark

36 The Nucleon Spin Pizzas 28 L q matrix element of [ ( q r i ga)] [ z ( q = qγ 0 r i g A)] z q L z q matrix element of (γ + = γ 0 + γ z ) [ qγ + r i ] z q A+=0 ( (for p = 0) matrix element of qγ [ r z i g A)] z q vanishes (parity!) L q identical to matrix element of qγ + [ r at rest) ( i g A)] z q (nucleon even in light-cone gauge, L z q and L z q still differ by matrix element of q ( r ga ) z q = q A+=0 (r x ga y r y ga x ) q A+ =0 how significant is that difference?

37 What is Orbital Angular Momentum 29 first: QED without electrons apply a ( b c) = b( a c) b( a c) to E ( ) A J = = ( ) d 3 r x E B = d 3 r [ ( )] d 3 r x E A [ ( E j x ) A j x ( E ) A ] integrate by parts (drop surface term) [ J ( = d 3 r E j x ) ( A j + x A ) ] E + E A drop 2 nd term (eq. of motion E = 0), yielding J = L + S with L ( = d 3 r E j x ) A j S = d 3 r E A note: L and S not separately gauge invariant as written, but can be made so ( nonlocal)

38 Example: Photon Angular Momentum in QED 30 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! can also be done for only part of A Chen/Goldman, Wakamatsu