Transverse (Spin) Structure of Hadrons

Transcription

1 Transverse (Spin) Structure of Hadrons Matthias Burkardt New Mexico State University Las Cruces, NM, 88003, U.S.A. Transverse (Spin) Structure of Hadrons p.1/138

2 Outline electromagnetic form factor charge distribution in position space deep-inelastic lepton-nucleon scattering (DIS) momentum distribution of quarks in nucleon deeply virtual Compton scattering (DVCS) generalized parton distributions (GPDs) simultaneous determination of (longitudinal) momentum and (transverse) position of quarks in the nucleon 3-d images of the nucleon where x y plane in position space and z axis in momentum space what is orbital angular momentum? single-spin asymmetries (SSA) transverse force on quarks in polarized DIS Transverse (Spin) Structure of Hadrons p.2/138

3 Physical Interpretation of Form Factors Nonrelativistic quantum mechanics (NRQM) Relativistic effects Breit frame interpretation Infinite momentum frame (IMF) Transverse (Spin) Structure of Hadrons p.3/138

4 Form Factors in nonrel. QM Potential scattering dσ dω = f( q) 2 Born approx. ( 1-photon exchange for Coulomb interaction) f( q) = d 3 re i q r V ( r) Electro-magnetic interactions: V ( r) = ea 0 ( r) with 2 A 0 ( r) = 4πρ( r), i.e. f( q) = 4πe q 2 d 3 re i q r ρ( r) dσ dω = f( q) 2 = dσ dω F( q) 2 with F( q) = point d 3 re i q r ρ( r) dσ dω F( q) ρ( r) Transverse (Spin) Structure of Hadrons p.4/138

5 Form Factor (Interpretation) small q 2 expansion F( q) = r2 q 2 + O( q 4 ) Hofstadter (NP 1961): first measurement of F( q) using electron nucleon scattering rp fm Transverse (Spin) Structure of Hadrons p.5/138

6 Issues Form factor of a moving target Relativistic effects Breit frame Infinite momentum frame (IMF) Transverse (Spin) Structure of Hadrons p.6/138

7 Moving Target (Meson, Nucleon,..) Fixed target: F( q) FT ρ( r) trivial Transverse (Spin) Structure of Hadrons p.7/138

8 Moving Target (Meson, Nucleon,..) Fixed target: F( q) FT ρ( r) trivial Moving target (nonrel.) H int = e d 3 r A 0 ( r)j 0 ( r) dσ dω = dσ dω dσ dω p H int p 2 F( q) 2 with F( q) point p j 0 ( 0) p ( q = p p ) Transverse (Spin) Structure of Hadrons p.7/138

9 Moving Target (Meson, Nucleon,..) Fixed target: F( q) FT ρ( r) trivial Moving target (nonrel.) H int = e d 3 r A 0 ( r)j 0 ( r) dσ dω = dσ dω dσ dω p H int p 2 F( q) 2 with F( q) point p j 0 ( 0) p ( q = p p ) Relation to ρ( r)? Transverse (Spin) Structure of Hadrons p.7/138

10 Moving Target (Meson, Nucleon,..) Fixed target: F( q) FT ρ( r) trivial Moving target (nonrel.) H int = e d 3 r A 0 ( r)j 0 ( r) dσ dω = dσ dω dσ dω p H int p 2 F( q) 2 with F( q) point p j 0 ( 0) p ( q = p p ) Relation to ρ( r)? What is ρ( r)? Transverse (Spin) Structure of Hadrons p.7/138

11 Form factor vs. charge distribution (nonrel.) plane wave states have uniform charge distribution meaningful definition of ρ( r) requires that state is localized in position space! define localized state (center of mass frame) R = 0 N d 3 p p define charge distribution (for this localized state) ρ( r) R = 0 j 0 ( r) R = 0 Transverse (Spin) Structure of Hadrons p.8/138

12 Form factor vs. charge distribution (non-rel.) use translational invariance to relate to same matrix element that appears in def. of form factor ρ( r) R = 0 j 0 ( r) R = 0 = N d 2 3 p d 3 p p j 0 ( r) p = N d 2 3 p d 3 p p j 0 ( 0) p e i r ( p p ), ( = N d 2 3 p d 3 p F ( p p) 2) e i r ( p p ) ρ( r) = d 3 q (2π) 3 F( q2 )e i q r back Transverse (Spin) Structure of Hadrons p.9/138

13 Form Factors (relativistic) Lorentz invariance, parity, current conservation p j µ (0) p = { ( p µ + p µ ) [ F(q 2 ) ] (spin 0) ū(p ) γ µ F 1 (q 2 ) + iσµν q ν 2M F 2(q 2 ) u(p) (spin 1 2 ) with q µ = p µ p µ. issues: energy factors spoil simple interpretation of form factors as FT of charge distributions q 2 depends on p + p Transverse (Spin) Structure of Hadrons p.10/138

14 Form Factor vs. Charge Distribution (rel.) wave packet Ψ = d 3 p ψ( p) 2E p (2π) 3 p, E p = M 2 + p 2 and covariant normalization p p = 2E p δ( p p) Fourier transform of charge distribution in the wave packet ρ( q) = = 1 2 d 3 xe i q x Ψ j 0 ( x) Ψ d 3 p 2E p 2E p Ψ ( p + q)ψ( p) p j 0 ( 0) p d 3 p E p + E p Ψ ( p + q)ψ( p)f(q 2 ). E p E p Transverse (Spin) Structure of Hadrons p.11/138

15 Form Factor vs. Charge Distribution (rel.) Nonrelativistic case: E p + E p 2 E p E p = 1 and q 2 = q 2 Fourier transform of charge distribution in the wave packet ρ( q) = d 3 pψ ( p + q)ψ( p)f( q 2 ) choose Ψ( p) very localized in position space Ψ ( p + q) Ψ ( p) ρ( q) = F( q 2 ) Transverse (Spin) Structure of Hadrons p.12/138

16 Form Factor vs. Charge Distribution (rel.) Relativistic corrections (example rms radius): ρ( q 2 ) = 1 R2 6 q2 R2 d 3 p Ψ( p) 2 ( q p) 2 6 E p 2 + d 3 p q Ψ( p) 2 1 d 3 p Ψ( p) 2 ( q p) 2 8 E 4 p, [R 2 defined as usual: F(q 2 ) = 1 + R2 6 q2 + O(q 4 )] If one completely localizes the wave packet, i.e. d 3 p q Ψ( p) 2 0, then relativistic corrections diverge ( x p 1) R 2 6 d 3 p Ψ( p) 2 ( q p) 2 E 2 p, 1 8 d 3 p Ψ( p) 2 ( q p) 2 E 4 p Transverse (Spin) Structure of Hadrons p.13/138

17 Sachs Form Factors in rest frame, rel. corrections contribute R 2 λ 2 C = 1 M 2 identification of charge distribution in rest frame with Fourier transformed form factor only unique down to scale λ C standard remedy: interpret F( q) as Fourier transform of charge distribution in Breit frame p = p G E (q 2 ) F 1 (q 2 ) + q2 4M 2 F 2 (q 2 ) G M (q 2 ) F 1 (q 2 ) + F 2 (q 2 ) Physics (Breit frame ): q/2 j 0 (0) q/2 = 2MG E (q 2 )δ s s q/2 j(0) q/2 = G M (q 2 )χ s i σ qχ s G E FT charge density; G M FT magnetization density; flaw of this interpretation: Breit frame is not one single frame but a different frame for each momentum transfer Transverse (Spin) Structure of Hadrons p.14/138

18 Infinite Momentum Frame Interpretation rel. corrections governed by p q E 2 p and q2 E 2 p consider wave packet Ψ( p ) in transverse direction, with sharp longitudinal momentum P z transverse size of wave packet r, with R r 1 P z take purely transverse momentum transfer ρ( q ) = F( q 2 ) form factor can be interpreted as Fourier transform of charge distribution w.r.t. impact parameter in momentum frame (without λ C uncertainties!) impact parameter measured w.r.t. center of momentum R = i q,g x ir i Transverse (Spin) Structure of Hadrons p.15/138

19 Derivation in LF-Coordinates light-front (LF) coordinates p + = 1 2 ( p 0 + p 3) p = 1 2 ( p 0 p 3) form factor for spin 1 2 conservation) p j µ (0) p = ū(p ) with q µ = p µ p µ. If q + = 0 (Drell-Yan-West frame) then target (Lorentz invariance, parity, charge [ ] γ µ F 1 (q 2 ) + iσµν q ν 2M F 2(q 2 ) u(p) p, j + (0) p, = 2p + F 1 ( q 2 ) Transverse (Spin) Structure of Hadrons p.16/138

20 F(q 2 ) ρ(r ) in LF-Coordinates define state that is localized in position: p +,R = 0, λ N d 2 p p +,p, λ Note: boosts in IMF form Galilean subgroup this state has R 1 P dx d 2 x + x T ++ (x) = 0 (cf.: working in CM frame in nonrel. physics) define charge distribution in impact parameter space ρ(b ) 1 2p + p +,R = 0 j + (0,b ) p +,R = 0 Transverse (Spin) Structure of Hadrons p.17/138

21 F(q 2 ) ρ(r ) in LF-Coordinates use translational invariance to relate to same matrix element that appears in def. of form factor ρ(b ) 1 p + 2p +,R = 0 j + (0,b ) p +,R = 0 N 2 = d 2 2p + p d 2 p p +,p j + (0,b ) p +,p N 2 = d 2 p 2p + d 2 p p +,p j + (0,0 ) p +,p e iq b = N 2 d 2 p d 2 p F 1 ( q 2 )e iq b ρ(b ) = d 2 q (2π) 2 F 1( q 2 )e iq b Transverse (Spin) Structure of Hadrons p.18/138

22 Boosts in NRQM x = x + vt t = t purely kinematical (quantization surface t = 0 inv.) 1. boosting wavefunctions very simple Ψ v ( p 1, p 2 ) = Ψ 0 ( p 1 m 1 v, p 2 m 2 v). 2. dynamics of center of mass R i x i r i with x i m i M decouples from the internal dynamics Transverse (Spin) Structure of Hadrons p.19/138

23 Relativistic Boosts t = γ (t + v ) c 2z, z = γ (z + vt) x = x generators satisfy Poincaré algebra: [P µ, P ν ] = 0 [M µν, P ρ ] = i(g νρ P µ g µρ P ν ) [ M µν, M ρλ] = i ( g µλ M νρ + g νρ M µλ g µρ M νλ g νλ M µρ) rotations: M ij = ε ijk J k, boosts: M i0 = K i. [K i, P j ] = iδ ij P 0 boost operator contains interactions! in general, no useful generalization of concept of center of mass to a relativistic theory Transverse (Spin) Structure of Hadrons p.20/138

24 Galilean subgroup of boosts introduce generator of boosts : B x M +x = K x + J y 2 B y M +y = K y J x 2 Poincaré algebra = commutation relations: [J 3, B k ] = iε kl B l [P k, B l ] = iδ kl P + [ P, B k ] = ip k [P +, B k ] = 0 with k, l {x, y}, ε xy = ε yx = 1, and ε xx = ε yy = 0. Transverse (Spin) Structure of Hadrons p.21/138

25 Galilean subgroup of boosts Together with [J z, P k ] = iε kl P l, as well as [ P, P k ] [ P +, P k ] = [ P, P +] = [ P, J z ] = 0 = [ P +, B k ] = [ P +, J z ] = 0. Same as commutation relations among generators of nonrel. boosts, translations, and rotations in x-y plane, provided one identifies P Hamiltonian P momentum in the plane P + mass L z rotations around z-axis B generator of boosts in the plane, Transverse (Spin) Structure of Hadrons p.22/138

26 Consequences of Galilean Subgroup many results from NRQM carry over to boosts in IMF, e.g. boosts kinematical Ψ (x,k ) = Ψ 0 (x,k x ) Ψ (x,k, y,l ) = Ψ 0 (x,k x, y,l y ) Transverse center of momentum R i x ir,i plays role similar to NR center of mass, e.g. d 2 p p +,p corresponds to state with R = 0. Transverse (Spin) Structure of Hadrons p.23/138

27 Summary: Form Factor vs. Charge Distribution fixed target: FT of form factor = charge distribution in position space moving target: nonrelativistically: FT of form factor = charge distribution in position space, where position is measured relative to center of mass relativistic corrections usually make idendification F(q 2 ) FT ρ( r) ambigous at scale R λ C = 1 M Sachs form factors have interpretation as charge and magnetization density in Breit frame Infinite momentum frame: form factors can be interpreted as transverse charge distribution in fast moving proton (without rel. corrections) Transverse (Spin) Structure of Hadrons p.24/138

28 Deep Inelastic Scattering (DIS) high-energy lepton (e ±, µ ±, ν, ) nucleon scattering usually inelastic for Q 2 = q 2 = (p p ) 2 M 2 p 1GeV 2, probe can resolve distance scales much smaller than the proton size deep inelastic scattering because of high Q 2, inclusive (i.e. sum over final states) cross section obtained from incoherent superposition of charged constituents DIS provided first direct evidence for the existence of quarks inside nucleons (Nobel Prize 1990: Freedman, Kendall, Taylor) Transverse (Spin) Structure of Hadrons p.25/138

29 e 0 Deep Inelastic Scattering (DIS) e ν = E E q Q 2 q 2 = 4EE sin 2 θ 2 X p d 2 σ dωde = 4πα2 MQ 4 { W 2 (Q 2, ν) cos 2 θ 2 + 2W 1(Q 2, ν) sin 2 θ } 2 exp. result: Bjorken scaling Q 2, ν x Bj = Q 2 /2Mν fixed 2M W 1(Q 2, ν)=f 1 (x Bj) ν W 2 (Q 2, ν)=f 2 (x Bj ) Transverse (Spin) Structure of Hadrons p.26/138

30 k 0 Physical Meaning of x Bj = Q2 2p q q k p Go to frame where q = 0, i.e. Q 2 = q 2 = 2q + q 2p q = 2q p + + 2q + p Bjorken limit: q, q + fixed q + q x Bj = q p + + q + p q+ p + Transverse (Spin) Structure of Hadrons p.27/138

31 Physical Meaning of x Bj = Q2 2p q x Bj = q+ p + LC energy-momentum dispersion relation k = m2 + k 2 2k + struck quark with k = k + q can only be on mass shell if k + = k + + q + 0 k + = q + x k+ p + = x Bj x Bj has physical meaning of light-cone momentum fraction carried by struck quark before it is hit by photon Transverse (Spin) Structure of Hadrons p.28/138

32 DIS light-cone correlations opt. theorem: inclusive cross section virtual, forward Compton amplitude X (ep e X) = k k k q -q P P struck quark carries large momentum: Q 2 Λ 2 QCD q -q q -q = + -q 2 crossed diagram suppressed (wavefunction!) asymptotic freedom neglect interactions of struck quark struck quark propagates along light-cone x 2 = 0 Transverse (Spin) Structure of Hadrons p.29/138

33 of the large momentum q through typical diagrams contributing Flow to the forward Compton amplitude. a) `handbag' suppression of crossed diagrams q q a) p p q q b) p p b) `cat's ears' diagram. Diagram b) is suppressed diagrams; large q due to the presence of additional propagators. at Transverse (Spin) Structure of Hadrons p.30/138

34 DIS light-cone correlations opt. theorem: inclusive cross section virtual, forward Compton amplitude X (ep e X) = k k k q -q P P struck quark carries large momentum: Q 2 Λ 2 QCD q -q q -q = + -q 2 crossed diagram suppressed (wavefunction!) asymptotic freedom neglect interactions of struck quark struck quark propagates along light-cone x 2 = 0 Transverse (Spin) Structure of Hadrons p.31/138

35 x x 0 x + x 3 DIS light-cone correlations light-cone coordinates: x + = ( x 0 + x 3) / 2 x = ( x 0 x 3) / 2 DIS related to correlations along light cone q(x Bj ) = dx 2π P q(0,0 )γ + q(x,0 ) P e ix x Bj P + Probability interpretation! No information about transverse position of partons! Transverse (Spin) Structure of Hadrons p.32/138

36 DIS light-cone correlations q(x Bj ) = dx 2π P q(0,0 )γ + q(x,0 ) P e ix x Bj P + Fourier transform along x filters out quarks with light-cone momentum k + = x Bj P + momentum distribution = FT of equal time correlation function boost to IMF tilts equal time plane arbitrarily close to x + = const. plane light-cone momentum distribution = momentum distribution in IMF q(x) = light-cone momentum distribution or momentum distribution in IMF Transverse (Spin) Structure of Hadrons p.33/138

37 unpolarized ep DIS σ elast eq Q 2 q e + p e + X sensitive to: [ u p (x) + ū p(x) + u p(x) + ū p(x) ] + [ d p (x) + d p(x) + d p(x) + d p(x) ] + [ s p (x) + s p(x) + s p(x) + s p(x) ] +... where e.g u p(x),u p(x), ū p(x),ū p(x),... are the distribution of u,ū,... in the proton with spin parallel/antiparallel to the proton s spin neutron target (charge symmetry) u n (x) = d p (x), d n (x) = u p (x), s n (x) = s p (x) sensitive to 4 9 d p(x) u p(x) s p(x) different linear combination of the same distribution functions! Transverse (Spin) Structure of Hadrons p.34/138

38 many important results from DIS: discovery of elementary, charged, spin 1 2 nucleon quarks constituents in the failure of momentum sum rule, i.e. quarks carry only about 50% of the nucleon s momentum gluons some recent puzzles: nuclear binding effect on structure functions (EMC collaboration): large and systematic modification of the nucleon s parton distribution in a bound nucleus failure of the "Ellis-Jaffe sum rule" for the spin dependent structure function g 1 (x) (EMC collaboration, also SMC, E142): spin fraction carried by quarks q=u,d,s 1 0 dx[ q (x) + q (x) q (x) q (x) ] 1 (nonrel quark model: 1) "spin crisis" Transverse (Spin) Structure of Hadrons p.35/138

39 Longitudinally Polarized DIS e /q helicity conserved in high-energy interactions e longitudinally polarized e preferentially scatter off q with spin opposite to that of the e e scattering long. pol. e off long. pol. nucleons quark/nucleon spin correlation q q dσ dσ g 1 (x) with g 1 (x) = q e 2 q [q (x) q (x)] q (x)/q (x) = probability that q has same/opposite spin as N q(x) = q (x) + q (x) g 1 (x) has been measured at CERN, SLAC, DESY, JLab,... future, more precise, measurements from JLab@12GeV, EIC Transverse (Spin) Structure of Hadrons p.36/138

40 Longitudinally Polarized DIS q (x)/q (x) = probability that q has same/opposite spin as N spin sum rule ( R.Jaffe) 1 2 = 1 2 Σ + G + L parton Σ = q q 1 q 0 dx[q (x) q (x)] = fraction of the nucleon spin due to quark spins G = fraction of the nucleon spin due to gluon spins L parton = angular momentum due to quarks & gluons EMC collaboration (1987): only small fraction of the proton spin due to quark spins incl. more recent data (CERN,SLAC,DESY): 30% was called spin crisis, because Σ much smaller than the quark model result Σ = 1 quest for the remaining 70% Transverse (Spin) Structure of Hadrons p.37/138

41 G gluons, like photons, descibed by a massless vector field and carry intrinsic angular momentum (spin) ±1 gluon contribution to nucleon momentum known to be large x g 1 0 dxxg(x) = 1 q 1 0 dxxq(x) 0.5 (physics of x g : think of E B) conceivable that G is of the same order of magnitude as x g (or larger) several explanations of the spin crisis even suggested G 4 6 G accessible e.g. through QCD-evolution of q(x) A LL in p p γ+ jet Transverse (Spin) Structure of Hadrons p.38/138

42 G from QCD-evolution sometimes a gluon is not just a gluon (quantum fluctuations): g qq g gg, ggg similar for quarks: q qg quantum fluctuations short distance effects become more visible as Q 2 of the probe increases resulting Q 2 dependence of PDFs descibed by QCD evolution equations (DGLAP evolution): coupled integro-differential equations with perturbatively calculable kernel qq pair inherits gluon spin in g qq infer G from Q 2 dependence of q need coverage down to small x (g(x) concentrated at very small x) and wide Q 2 range ( QCD evolution slow) planned EIC (electron ion collider) Transverse (Spin) Structure of Hadrons p.39/138

43 G from p + p (RHIC-spin) gg and gq scattering sensitive to (relative) helicity use double-spin asymmetry A LL = σ ++ σ + σ ++ +σ + in inelastic pp-scattering at RHIC to infer G directly Transverse (Spin) Structure of Hadrons p.40/138

44 G from p + p (RHIC-spin) global analysis (DIS & RHIC data): RHIC-spin substantially reduced error band (yellow) between x =.05 and x =.2 despite remaining uncertainties, now evident that G significantly less than 1 big deal: rules out a significant role of gluonic corrections to the quark spin as explanation for spin puzzle 500GeV at RHIC run with improved forward acceptance will reduce error band down to x.002 Transverse (Spin) Structure of Hadrons p.41/138

45 Theoretical Status perturbative effects in general well understood, e.g. Q 2 dependence (= evolution ) calculable in QCD (Altarelli, Parisi, Gribov, Lipatov eqs.): given q(x, Q 2 0) one can calculate q(x, Q 2 1) for Q 2 1 > Q 2 0 > a few GeV 2 important applications: compare two experiments at two different Q 2 compare low Q 2 models or sum rules with experiments at high Q 2 nonperturbative effects difficult! power law behavior for x 0 from Regge phenomenology typically, q(x, Q 2 0) from some QCD-inspired models lowest moments from lattice QCD Transverse (Spin) Structure of Hadrons p.42/138

46 Calculating PDFs in lattice QCD direct evaluation: NO: On a Euclidean lattice, all distances are spacelike (x 0 ix 0 E ). Therefore, a direct calculation of lightlike correlation functions on a Euclidean lattice is not possible! indirect evaluation: yes! Using analyticity, one can show that moments of parton distributions for a hadron h are related to expectation values of certain local operators in that hadron state 1 0 dxf(x)x n h ψd n ψ h r.h.s. of this equation can be calculated in Euclidean space (and then one could reconstruct f(x) from its moments)! In practice: replace n-th derivative on the r.h.s. by appropriate finite differences. Problem: statistical noise (Euclidean lattice calculations are done using Monte Carlo techniques) makes it very hard to calculate any moment n 1. Transverse (Spin) Structure of Hadrons p.43/138

47 Summary: DIS DIS Bj PDF q(x) q(x) is probability to find quark carrying fraction x of light-cone momentum (total momentum in IMF) no information about position of partons major results: 50% of nucleon momentum carried by glue significant [O(10%)] modification of quark distributions in nuclei ( EMC effect ) only 30% of nucleon spin carried by quark spin ( spin crisis ) more d than ū in proton ( violation of Gottfried sum rule ) G not very large theory: perturbative Q 2 evolution lattice calculations: lowest moments of PDFs and many QCD-inspired models... Transverse (Spin) Structure of Hadrons p.44/138

48 3D imaging of the nucleon b? (f m) 0 1 x Transverse (Spin) Structure of Hadrons p.45/138

49 Motivation (GPDs) X.Ji, PRL 78, 610 (1997): DVCS GPDs J q GPDs are interesting physical observable! But: do GPDs have a simple physical interpretation? what more can we learn from GPDs about the structure of the nucleon? Transverse (Spin) Structure of Hadrons p.46/138

50 Outline Probabilistic interpretation of GPDs as Fourier trafos of impact parameter dependent PDFs H(x, 0, 2 ) q(x,b ) H(x, 0, 2 ) q(x,b ) b? (fm) x E(x, 0, 2 ) distortion of PDFs when the target is polarized Chromodynamik lensing and SSAs } transverse distortion of PDFs SSA in γn π+x + final state interactions Summary p γ p N d u π + Transverse (Spin) Structure of Hadrons p.47/138

51 Generalized Parton Distributions (GPDs) GPDs: decomposition of form factors at a given value of t, w.r.t. the average momentum fraction x = 1 2 (x i + x f ) of the active quark dxh q (x, t) = F q 1 (t) dxe q (x, t) = F q 2 (t) dx H q (x, t) = G q A (t) dxẽ q (x, t) = G q P (t), x i and x f are the momentum fractions of the quark before and after the momentum transfer F q 1 (t), F q 2 (t), Gq A (t), and Gq P (t) are the Dirac, Pauli, axial, and pseudoscalar formfactors, respectively (t q 2 = (P P) 2 ) P, S j µ (0) P, S = ū(p, S ) [ ] γ µ F 1 (q 2 ) + i σµν q ν 2M F 2(q 2 ) u(p, S) GPDs can be probed in Deeply Virtual Compton Scattering (DVCS) Transverse (Spin) Structure of Hadrons p.48/138

52 Deeply Virtual Compton Scattering (DVCS) virtual Compton scattering: γ p γp (actually: e p e γp) deeply : q 2 γ M 2 p, t Compton amplitude dominated by (coherent superposition of) Compton scattering off single quarks only difference between form factor (a) and DVCS amplitude (b) is replacement of photon vertex by two photon vertices connected by quark propagator (depends on quark momentum fraction x) DVCS amplitude provides access to momentum-decomposition of form factor (GPDs). γ γ γ.. (a) (b) Transverse (Spin) Structure of Hadrons p.49/138

53 Deeply Virtual Compton Scattering (DVCS) need γ with several GeV 2 for Bjorken scaling DVCS X-section factor α = need high luminosity e beam with > 10 GeV smaller than elastic X-section facilities suitable for detailed GPD studies: 12 GeV upgrade at Jefferson Lab (higher x) e Ion Collider (EIC): lower x, higher Q 2 Transverse (Spin) Structure of Hadrons p.50/138

54 q q 0 p p 0 q q 0 p p 0 Deeply Virtual Compton Scattering (DVCS) Bj! T µν = i d 4 z e i q z p TJ µ ( z 2 ) ( J ν z ) p 2 Bj gµν ( ) 1 dx x ξ+iε + 1 H(x, ξ, 2 )ū(p )γ + u(p) +... x+ξ iε q = (q + q )/2 = p p x Bj q 2 /2p q = 2ξ(1 + ξ) Transverse (Spin) Structure of Hadrons p.51/138

55 Generalized Parton Distributions (GPDs) dx 2π eix p + x ( p q x 2 ) γ + q ( ) x p 2 = H(x, ξ, 2 )ū(p )γ + u(p) +E(x, ξ, 2 )ū(p ) iσ+ν ν 2M u(p) dx 2π eix p + x ( p q x 2 ) γ + γ 5 q ( ) x p 2 = H(x, ξ, 2 )ū(p )γ + γ 5 u(p)!+ẽ(x, ξ, 2 )ū(p ) γ 5 + 2M u(p) where = p p is the momentum transfer and ξ measures the longitudinal momentum transfer on the target + = ξ(p + + p + ). Transverse (Spin) Structure of Hadrons p.52/138

56 p Ô p = H(x, ξ, 2 )ū(p )γ + u(p) + E(x, ξ, 2 )ū(p ) iσ+ν ν What is Physics of GPDs? 2M with Ô dx 2π eix p +x q ( x 2 ) γ + q ( x 2 ) u(p) relation between PDFs and GPDs similar to relation between a charge and a form factor If form factors can be interpreted as Fourier transforms of charge distributions in position space, what is the analogous physical interpretation for GPDs? Transverse (Spin) Structure of Hadrons p.53/138

57 Form Factors vs. GPDs operator forward matrix elem. off-forward matrix elem. position space qγ + q Q F(t) ρ( r) dx e ixp+ x 4π ( q x 2 ) ( γ + x q 2 ) q(x) H(x, ξ, t)? Transverse (Spin) Structure of Hadrons p.54/138

58 Form Factors vs. GPDs operator forward matrix elem. off-forward matrix elem. position space qγ + q Q F(t) ρ( r) dx e ixp+ x 4π ( q x 2 ) ( γ + x q 2 ) q(x) H(x, 0, t) q(x,b ) q(x,b ) = impact parameter dependent PDF Transverse (Spin) Structure of Hadrons p.55/138

59 Impact parameter dependent PDFs define state that is localized in position: p +,R = 0, λ N d 2 p p +,p, λ Note: boosts in IMF form Galilean subgroup this state has R 1 P dx d 2 x + x T ++ (x) = 0 (cf.: working in CM frame in nonrel. physics) define impact parameter dependent PDF dx q(x,b ) p +,R = 0 x ψ( 4π 2,b )γ + ψ( x 2,b ) p +,R = 0 e ixp + x Transverse (Spin) Structure of Hadrons p.56/138

60 Impact parameter dependent PDFs q(x,b ) use translational invariance to relate to same matrix element that appears in def. of GPDs dx p +,R = 0 ψ( x = N 2 d 2 p d 2 p 2,b )γ + ψ( x 2,b ) p +,R = 0 e ixp + x dx p +,p x ψ( 2,b )γ + ψ( x 2,b ) p +,p e ixp + x Transverse (Spin) Structure of Hadrons p.57/138

61 Impact parameter dependent PDFs q(x,b ) use translational invariance to relate to same matrix element that appears in def. of GPDs dx p +,R = 0 ψ( x = N 2 d 2 p d 2 p = N 2 d 2 p d 2 p dx p +,p x ψ( 2,b )γ + ψ( x 2,b ) p +,R = 0 e ixp + x 2,b )γ + ψ( x 2,b ) p +,p e ixp + x dx p +,p x ψ( 2,0 )γ + ψ( x 2,0 ) p +,p e ixp + x e ib (p p ) Transverse (Spin) Structure of Hadrons p.58/138

62 Impact parameter dependent PDFs q(x,b ) use translational invariance to relate to same matrix element that appears in def. of GPDs dx p +,R = 0 ψ( x = N 2 d 2 p d 2 p = N 2 d 2 p d 2 p = N 2 d 2 p d 2 p H 2,b )γ + ψ( x 2,b ) p +,R = 0 e ixp + x dx p +,p x ψ( 2,b )γ + ψ( x 2,b ) p +,p e ixp + x dx p +,p x ψ( 2,0 )γ + ψ( x 2,0 ) p +,p e ixp + x e ib (p p ) ( x, 0, (p p ) 2) e ib (p p ) q(x,b ) = d 2 (2π) 2 H(x, 0, 2 )e ib Transverse (Spin) Structure of Hadrons p.59/138

63 Impact parameter dependent PDFs GPDs allow simultaneous determination of longitudinal momentum and transverse position of partons q(x,b ) = d 2 (2π) 2 H(x, 0, 2 )e ib q(x,b ) has interpretation as density (positivity constraints!) q(x,b ) p +,0 b (xp +,b )b(xp +,b ) p +,0 = b(xp +,b ) p +,0 2 0 q(x,b ) 0 for x > 0 q(x,b ) 0 for x < 0 Transverse (Spin) Structure of Hadrons p.60/138

64 Impact parameter dependent PDFs No relativistic corrections (Galilean subgroup!) corollary: interpretation of 2d-FT of F 1 (Q 2 ) as charge density in transverse plane also free from relativistic corrections q(x,b ) has probabilistic interpretation as number density ( q(x,b ) as difference of number densities) Reference point for IPDs is transverse center of (longitudinal) momentum R i x ir i, for x 1, active quark becomes COM, and q(x,b ) must become very narrow (δ-function like) H(x, 2 ) must become indep. as x 1 (MB, 2000) consistent with lattice results for first few moments ( J.Negele) Note that this does not necessarily imply that hadron size goes to zero as x 1, as separation r between active quark and COM of spectators is related to impact parameter b via r = 1 1 x b. Transverse (Spin) Structure of Hadrons p.61/138

65 q(x,b ) for unpol. p b y b x 6 4 x = b y b x b? (f m) 0 1 x x = 0.3 b y x = momentum fraction of the quark b x x = 0.5 b = position of the quark Transverse (Spin) Structure of Hadrons p.62/138

66 Summary form factor FT ρ( r) relativistic corrections! Can be avoided in infinite momentum frame interpretation of 2D FT of form factors DIS q(x) light-cone momentum distribution of quarks in nucleon DVCS GPDs FT q(x,b ) impact parameter dependent PDFs Transverse (Spin) Structure of Hadrons p.63/138

67 Transversely Deformed Distributions and E(x, 2 ) M.B., Int.J.Mod.Phys.A18, 173 (2003) distribution of unpol. quarks in unpol (or long. pol.) nucleon: q(x,b ) = d 2 (2π) 2 H(x, 2 )e ib H(x,b ) unpol. quark distribution for nucleon polarized in x direction: q(x,b ) = H(x,b ) 1 2M b y d 2 (2π) 2 E(x, 2 )e ib Physics: j + = j 0 + j 3, and left-right asymmetry from j 3 Transverse (Spin) Structure of Hadrons p.64/138

68 Intuitive connection with J q DIS probes quark momentum density in the infinite momentum frame (IMF). Quark density in IMF corresponds to j + = j 0 + j 3 component in rest frame ( p γ in ẑ direction) j + larger than j 0 when quark current towards the γ ; suppressed when away from γ For quarks with positive orbital angular momentum in ˆx-direction, j z is positive on the +ŷ side, and negative on the ŷ side p γ ẑ ŷ j z > 0 Details of deformation described by E q (x, 2 ) not surprising that E q (x, 2 ) enters Ji relation! J i q = S i dx[h q (x, 0) + E q (x, 0)] x. j z < 0 Transverse (Spin) Structure of Hadrons p.65/138

69 Transversely Deformed PDFs and E(x, 0, 2 ) q(x,b ) in polarized nucleon is deformed compared to longitudinally polarized nucleons! mean deformation of flavor q ( flavor dipole moment) d q y dx d 2 b q(x,b )b y = 1 2M dxe q (x, 0) = κ q/p 2M with κ q/p F u/d 2 (0) contribution from quark flavor q to the proton anomalous magnetic moment κ p = = 2 3 κ u/p 1 3 κ d/p κ u/p = 2κ p + κ n = κ n = = 2 3 κ d/p 1 3 κ u/p κ d/p = 2κ n + κ p = d q y = O(0.2fm) Transverse (Spin) Structure of Hadrons p.66/138

70 u(x,b ) d(x,b ) p polarized in +ˆx direction b y b y b x b x x = 0.1 ŷ j z > 0 b y b y p γ ẑ b x b x j z < 0 x = 0.3 x = 0.3 b y b y b x b x x = 0.5 x = 0.5 Transverse (Spin) Structure of Hadrons p.67/138

71 SSAs in SIDIS (γ + p π + + X) e q e π + D π+ q (z,p ) SIDIS = semi-inclusive DIS Single-Spin-Asymmetry (SSA) = left-right asymmery in the X-section when only one spin is measured (e.g. target spin) example: nucleon transversely (relative to e beam) polarized left-right asymmetry of produced π-mesons relative to target pol. p q(x,k ) infer transverse momentum distribution q(x,k ) of quarks in target from transverse momentum distribution of produced π (note: left-right asymmetry can also arise in fragmentation process (Collins effect), but resulting asymmetry has different angular dependence...) Transverse (Spin) Structure of Hadrons p.68/138

72 GPD SSA (Sivers) Sivers: distribution of unpol. quarks in pol. proton f q/p (x,k ) = f q 1 (x,k2 ) f q 1T (x,k2 ) (ˆP k ) S M without FSI, f(x,k ) = f(x, k ) f q 1T (x,k2 ) = 0 with FSI, f q 1T (x,k2 ) 0 (Brodsky, Hwang, Schmidt) Why interesting? (like κ), Sivers requires matrix elements between wave function components that differ by one unit of OAM (Brodsky, Diehl,..) probe for orbital angular momentum Sivers requires nontrivial final state interaction phases learn about FSI Transverse (Spin) Structure of Hadrons p.69/138

73 GPD SSA (Sivers) example: γp πx p γ p N d π + u, d distributions in polarized proton have left-right asymmetry in position space (T-even!); sign determined by κ u & κ d u attractive 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 correlation between sign of κ p q and sign of SSA: f q 1T κp q f q 1T κp q confirmed by HERMES data (also consistent with COMPASS deuteron data f1t u + f 1T d 0) Transverse (Spin) Structure of Hadrons p.70/138

74 Transversity Distribution in Unpolarized Target (sign) Consider quark in ground state hadron polarized out of the plane expect counterclockwise net current j associated with the magnetization density in this state virtual photon sees enhancement of quarks (polarized out of plane) at the top, i.e. virtual photon sees enhancement of quarks with polarization up (down) on the left (right) side of the hadron Transverse (Spin) Structure of Hadrons p.71/138

75 Transversity Distribution in Unpolarized Target Transverse (Spin) Structure of Hadrons p.72/138

76 IPDs on the lattice (Hägler et al.) lowest moment of distribution q(x,b ) for unpol. quarks in pol. proton (left) and of pol. quarks in unpol. proton (right): Transverse (Spin) Structure of Hadrons p.73/138

77 Boer-Mulders Function SIDIS: attractive FSI expected to convert position space asymmetry into momentum space asymmetry e.g. quarks at negative b x with spin in +ŷ get deflected (due to FSI) into +ˆx direction (qualitative) connection between Boer-Mulders function h 1 (x,k ) and the chirally odd GPD Ē T that is similar to (qualitative) connection between Sivers function f 1T (x,k ) and the GPD E. Boer-Mulders: distribution of pol. quarks in unpol. proton f q /p(x,k ) = 1 2 [ f q 1 (x,k2 ) h q 1 (x,k2 ) (ˆP k ) S q M h q 1 (x,k2 ) can be probed in Drell-Yan (RHIC, J-PARC, GSI) and tagged SIDIS (JLab, erhic), using Collins-fragmentation ] Transverse (Spin) Structure of Hadrons p.74/138

78 probing BM function in tagged SIDIS how do you measure the transversity distribution of quarks without measuring the transversity of a quark? consider semi-inclusive pion production off unpolarized target spin-orbit correlations in target wave function provide correlation between (primordial) quark transversity and impact parameter (attractive) FSI provides correlation between quark spin and quark momentum BM function Collins effect: left-right asymmetry of π distribution in fragmentation of polarized quark tag quark spin cos(2φ) modulation of π distribution relative to lepton scattering plane cos(2φ) asymmetry proportional to: Collins BM Transverse (Spin) Structure of Hadrons p.75/138

79 probing BM function in tagged SIDIS quark pol. Primordial Quark Transversity Distribution Transverse (Spin) Structure of Hadrons p.76/138

80 polarization and γ absorption QED: when the γ scatters off polarized quark, the polarization gets modified gets reduced in size gets tilted symmetrically w.r.t. normal of the scattering plane quark pol. before γ absorption quark pol. after γ absorption lepton scattering plane Transverse (Spin) Structure of Hadrons p.77/138

81 probing BM function in tagged SIDIS quark pol. Primordial Quark Transversity Distribution Transverse (Spin) Structure of Hadrons p.78/138

82 probing BM function in tagged SIDIS quark pol. Quark Transversity Distribution after γ absorption lepton scattering plane quark transversity component in lepton scattering plane flips Transverse (Spin) Structure of Hadrons p.79/138

83 probing BM function in tagged SIDIS quark pol. momentum due to FSI k q due to FSI lepton scattering plane on average, FSI deflects quarks towards the center Transverse (Spin) Structure of Hadrons p.80/138

84 Collins effect When a polarized struck quark fragments, the strucure of jet is sensitive to polarization of quark distribution of hadrons relative to polarization direction may be left-right asymmetric asymmetry parameterized by Collins fragmentation function Artru model: struck quark forms pion with q from q q pair with 3 P 0 vacuum quantum numbers pion inherits OAM in direction of spin of struck quark produced pion preferentially moves to left when looking into direction of motion of fragmenting quark with spin up Artru model confirmed by HERMES experiment more precise determination of Collins function under way (KEK) Transverse (Spin) Structure of Hadrons p.81/138

85 probing BM function in tagged SIDIS k due to Collins quark pol. momentum due to Collins k q due to FSI lepton scattering plane SSA of π in jet emanating from pol. q Transverse (Spin) Structure of Hadrons p.82/138

86 probing BM function in tagged SIDIS net momentum (FSI+Collins) k due to Collins k q due to FSI net k q lepton scattering plane in this example, enhancement of pions with momenta to lepton plane Transverse (Spin) Structure of Hadrons p.83/138

87 probing BM function in tagged SIDIS net k π (FSI + Collins) net k q lepton scattering plane expect enhancement of pions with momenta to lepton plane Transverse (Spin) Structure of Hadrons p.84/138

88 What is Orbital Angular Momentum Transverse (Spin) Structure of Hadrons p.85/138

89 Motivation polarized DIS: only 30% of the proton spin due to quark spins spin crisis spin puzzle, because Σ much smaller than the quark model result Σ = 1 quest for the remaining 70% quark orbital angular momentum (OAM) gluon spin gluon OAM How are the above quantities defined? How can the above quantities be measured Transverse (Spin) Structure of Hadrons p.86/138

90 example: angular momentum in QED consider, for simplicity, QED without electrons: J = ( ) d 3 r x E B = [ ( )] d 3 r x E A integrate by parts 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 Transverse (Spin) Structure of Hadrons p.87/138

91 example (cont.) total angular momentum of isolated system uniquely defined ambiguities arise when decomposing J into contributions from different constituents gauge theories: changing gauge may also shift angular momentum between various degrees of freedom decomposition of angular momentum in general depends on scheme (gauge & quantization scheme) does not mean that angular momentum decomposition is meaningless, but one needs to be aware of this scheme -dependence in the physical interpretation of exp/lattice/model results in terms of spin vs. OAM and, for example, not mix schemes, e.t.c. Transverse (Spin) Structure of Hadrons p.88/138

92 What is Orbital Angular Momentum? Ji decomposition Jaffe decomposition recent lattice results (Ji decomposition) model/qed illustrations for Ji v. Jaffe Transverse (Spin) Structure of Hadrons p.89/138

93 The nucleon spin pizza(s) Ji Jaffe & Manohar L q 1 2 Σ 1 L q 2 Σ J g L g G pizza tre stagioni pizza quattro stagioni only 1 2 Σ 1 2 q q common to both decompositions! Transverse (Spin) Structure of Hadrons p.90/138

94 T µν (x) Momentum Operator energy momentum tensor T µν = T νµ ; µ T µν = 0 T 00 energy density; T 0i momentum density P µ d 3 xt µ0 conserved d dt P µ = d 3 x x 0 T µ0 µt µν =0 = d 3 x x i T µi = 0 T µν contains interactions, e.g. T µν q = i 2 ψ (γ µ D ν + γ ν D µ )ψ T µ0 contains time derivative (don t want a Hamiltonian/momentum operator that contains time derivative!) replace by space derivative, using equation of motion, e.g. (iγ µ D µ m) ψ = 0 to replace id 0 ψ γ 0 ( iγ k D k m ) ψ some of the resulting space derivatives add up to total derivative terms which do not contribute to volume integral P µ d 3 x [ T µ0 + eq. of motion terms + surface terms ] Transverse (Spin) Structure of Hadrons p.91/138

95 Angular Momentum Operator angular momentum tensor M µνρ = x µ T νρ x ν T µρ ρ M µνρ = 0 J i = 1 2 εijk d 3 rm jk0 conserved d dt J i = 1 2 εijk d 3 x 0 M jk0 = 1 2 εijk d 3 x l M jkl = 0 M µνρ contains time derivatives (since T µν does) use eq. of motion to get rid of these (as in T 0i ) integrate total derivatives appearing in T 0i by parts yields terms where derivative acts on x i which then disappears J i usally contains both Extrinsic terms, which have the structure x Operator, and can be identified with OAM Intrinsic terms, where the factor x does not appear, and can be identified with spin Transverse (Spin) Structure of Hadrons p.92/138

96 Angular Momentum in QCD (Ji) following this general procedure, one finds in QCD J = d 3 x [ ψ Σψ + ψ x ( i ) ( )] g A ψ + x E B with Σ i = i 2 εijk γ j γ k Ji does not integrate gluon term by parts, nor identify gluon spin/oam separately Ji-decomposition valid for all three components of J, but usually only applied to ẑ component, where the quark spin term has a partonic interpretation (+) all three terms manifestly gauge invariant (+) DVCS can be used to probe J q = S q + L q (-) quark OAM contains interactions (-) only quark spin has partonic interpretation as a single particle density Transverse (Spin) Structure of Hadrons p.93/138

97 Ji-decomposition L q 1 2 Σ Ji (1997) 1 2 = q J q + J g = q ( ) 1 2 q + L q + J g J g with (P µ = (M, 0, 0, 1), S µ = (0, 0, 0, 1)) 1 2 q = 1 d 3 x P, S q ( x)σ 3 q( x) P, S Σ 3 = iγ 1 γ 2 2 ( L q = d 3 x P, S q ( x) x id) 3 q( x) P, S [ ( 3 J g = d 3 x P, S x E B)] P, S i D = i g A Transverse (Spin) Structure of Hadrons p.94/138

98 The Ji-relation (poor man s derivation) What distinguishes the Ji-decomposition from other decompositions is the fact that L q can be constrained by experiment: J q = S 1 1 dxx[h q (x, ξ, 0) + E q (x, ξ, 0)] (nucleon at rest; S is nucleon spin) L z q = J z q 1 2 q derivation (MB-version): consider nucleon state that is an eigenstate under rotation about the ˆx-axis (e.g. nucleon polarized in ˆx direction with p = 0 (wave packet if necessary) for such a state, T 00 q y = 0 = T zz q y and T 0y q z = T 0z q y T q ++ y = Tq 0y z Tq 0z y = Jq x relate 2 nd moment of flavor dipole moment to J x q Transverse (Spin) Structure of Hadrons p.95/138

99 The Ji-relation (poor man s derivation) derivation (MB-version): consider nucleon state that is an eigenstate under rotation about the ˆx-axis (e.g. nucleon polarized in ˆx direction with p = 0 (wave packet if necessary) for such a state, T 00 q y = 0 = T zz q y and T 0y q z = T 0z q y T q ++ y = Tq 0y z Tq 0z y = Jq x relate 2 nd moment of flavor dipole moment to Jq x effect sum of two effects: T ++ y for a point-like transversely polarized xnucleon T q ++ y for a quark relative to the center of momentum of a transversely polarized nucleon 2 nd moment of flavor dipole moment for point-like nucleon ψ = ( f(r) σ p E+m f(r) ) ( χ with χ = ) Transverse (Spin) Structure of Hadrons p.96/138

100 The Ji-relation (poor man s derivation) derivation (MB-version): T 0z q = i q ( γ 0 z + γ z 0) q since ψ z ψ is even under y y, i qγ 0 z q does not contribute to T 0z y using i 0 ψ = Eψ, one finds T 0z b y = E = 2E E + M d 3 rψ γ 0 γ z ψy = E d 3 rψ ( 0 σ z σ z 0 d 3 rχ σ z σ y χf(r)( i) y f(r)y = ) ψy E E + M consider nucleon state with p = 0, i.e. E = m & d 3 rf 2 (r) = 1 2 nd moment of flavor dipole moment is 1 2M overall shift of nucleon COM yields contribution 1 2 dxxhq (x, 0, 0) to T ++ q y d 3 Transverse (Spin) Structure of Hadrons p.97/138

101 The Ji-relation (poor man s derivation) derivation (MB-version): intrinsic distortion adds 1 2 dxxeq (x, 0, 0) to that Jq x = 1 2 dxx[hq (x, 0, 0) + E q (x, 0, 0)] rotational invariance: should apply to each vector component Ji relation Transverse (Spin) Structure of Hadrons p.98/138

102 Ji-decomposition L q J = q 1 2 q Σq + q ( r id ) q + r ( E B ) applies to each vector component of nucleon angular momentum, but Ji-decomposition usually applied only to ẑ component where at least quark spin has parton interpretation as difference between number densities q from polarized DIS J q = 1 2 q + L q from exp/lattice (GPDs) J g 1 2 Σ L q in principle independently defined as matrix elements of ) q ( r i D q, but in practice easier by subtraction L q = J q 1 2 q J g in principle accessible through gluon GPDs, but in practice easier by subtraction J g = 1 2 J q further decomposition of J g into intrinsic (spin) and extrinsic (OAM) that is local and manifestly gauge invariant has not been found Transverse (Spin) Structure of Hadrons p.99/138

103 L q for proton from Ji-relation (lattice) lattice QCD moments of GPDs (LHPC; QCDSF) insert in Ji-relation J i q = S i dx[h q (x, 0) + E q (x, 0)] x. L z q = J z q 1 2 q L u, L d both large! present calcs. show L u + L d 0, but disconnected diagrams..? m 2 π extrapolation parton interpret. of L q... Transverse (Spin) Structure of Hadrons p.100/138

104 Angular Momentum in QCD (Jaffe & Manohar) define OAM on a light-like hypesurface rather than a space-like hypersurface J 3 = d 2 x dx M 12+ where x = 1 2 ( x 0 x ) and M 12+ = 1 2 ( M M 123) Since µ M 12µ = 0 d 2 x dx M 12+ = d 2 x dx 3 M 120 (compare electrodynamics: B = 0 flux in = flux out) use eqs. of motion to get rid of time ( + derivatives) & integrate by parts whenever a total derivative appears in the T i+ part of M 12+ Transverse (Spin) Structure of Hadrons p.101/138

105 Jaffe/Manohar decomposition in light-cone framework & light-cone gauge A + = 0 one finds for J z = dx d 2 r M +xy q L q L g G 1 2 Σ 1 2 = 1 2 Σ + q L q + G + L g where (γ + = γ 0 + γ z ) ( L q = d 3 r P, S q( r)γ + r i ) z q( 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 ) z A j P, S Transverse (Spin) Structure of Hadrons p.102/138

106 Jaffe/Manohar decomposition Σ = q 1 2 = 1 2 Σ + q L q + G + L g q from polarized DIS (or lattice) q L q L g G 1 2 Σ G from p p or polarized DIS (evolution) G gauge invariant, but local operator only in light-cone gauge dxx n G(x) for n 1 can be described by manifestly gauge inv. local op. ( lattice) L q, L g independently defined, but no exp. identified to access them not accessible on lattice, since nonlocal except when A + = 0 parton net OAM L = L g + q L q by subtr. L = Σ G in general, L q L q L g + G J g makes no sense to mix Ji and JM decompositions, e.g. J g G has no fundamental connection to OAM Transverse (Spin) Structure of Hadrons p.103/138

107 L q L q L q matrix element of q [ r ( i g A)] 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 nucleon at rest, matrix element of L q same as that of ( qγ [ r + i g z A)] q even in light-cone gauge, L z q and L z q still differ by matrix element of q ( r ga ) z q A+=0 = q (xga y yga x ) q A +=0 Transverse (Spin) Structure of Hadrons p.104/138

108 Summary part 1: Ji: J z = 1 2 Σ + q L q + J g Jaffe: J z = 1 2 Σ + q L q + G + L g G can be defined without reference to gauge (and hence gauge invariantly) as the quantity that enters the evolution equations and/or p p represented by simple (i.e. local) operator only in LC gauge and corresponds to the operator that one would naturally identify with spin only in that gauge in general L q L q or J g G + L g, but how significant is the difference between L q and L q, etc.? Transverse (Spin) Structure of Hadrons p.105/138

109 OAM in scalar diquark model [M.B. + Hikmat Budhathoki Chhetri (BC), PRD 79, (2009)] toy model for nucleon where nucleon (mass M) splits into quark (mass m) and scalar diquark (mass λ) light-cone wave function for quark-diquark Fock component ψ (x,k ) = ( M + m x ) φ ψ 1 2 = k1 + ik 2 x φ with φ = c/ 1 x M 2 k2 +m2 x k2 +λ2 1 x. quark OAM according to JM: L q = 1 0 dx d 2 k 16π 3 (1 x) ψ 1 2 quark OAM according to Ji: L q = dxx[q(x) + E(x, 0, 0)] 1 2 q (using Lorentz inv. regularization, such as Pauli Villars subtraction) both give identical result, i.e. L q = L q not surprising since scalar diquark model is not a gauge theory 2 Transverse (Spin) Structure of Hadrons p.106/138

110 OAM in scalar diquark model But, even though L q = L q in this non-gauge theory L q (x) d 2 k ψ 16π 3 (1 x) {x[q(x) + E(x, 0, 0)] q(x)} L q(x) 0.2 L (x) 0.15 L (x) x unintegrated Ji-relation does not yield x-distribution of OAM Transverse (Spin) Structure of Hadrons p.107/138

111 OAM in QED light-cone wave function in eγ Fock component Ψ + 1+1(x,k ) = 2 k1 ik 2 2 x(1 x) φ Ψ + 1 1(x,k ) = 2 k1 + ik x Ψ 1+1(x,k ) = ( m ) 2 2 x m φ Ψ 1+1(x,k ) = 0 2 OAM of e according to Jaffe/Manohar L e = 1 0 dx d 2 k [(1 x) Ψ + 1 1(x,k ) 2 2 Ψ + 1+1(x,k ) 2 2] e OAM according to Ji L e = dxx[q(x) + E(x, 0, 0)] 1 2 q L e = L e + α 4π L e Likewise, computing J γ from photon GPD, and γ and L γ from light-cone wave functions and defining ˆL γ J γ γ yields ˆL γ = L γ + α 4π L γ α 4π appears to be small, but here L e, L e are all of O( α π ) Transverse (Spin) Structure of Hadrons p.108/138

112 OAM in QCD 1-loop QCD: L q L q = α s 3π recall (lattice QCD): L u.15; L d +.15 QCD evolution yields negative correction to L u and positive correction to L d evolution suggested (A.W.Thomas) to explain apparent discrepancy between quark models (low Q 2 ) and lattice results (Q 2 4GeV 2 ) above result suggests that L u > L u and L d > L d additional contribution (with same sign) from vector potential due to spectators (MB, to be published) possible that lattice result consistent with L u > L d Transverse (Spin) Structure of Hadrons p.109/138

113 Summary Jaffe & Manohar Ji q L q inclusive e p / p p provide access to quark spin 1 2 q gluon spin G q L q L g G 1 2 Σ J g 1 2 Σ parton grand total OAM L L g + q L q = 1 2 G q q DVCS & polarized DIS and/or lattice provide access to quark spin 1 2 q J q & L q = J q 1 2 q J g = 1 2 q J q J g G does not yield gluon OAM L g L q L q = O(0.1 α s ) for O (α s ) dressed quark Transverse (Spin) Structure of Hadrons p.110/138

114 Quark-Gluon Correlations (Introduction) (longitudinally) polarized polarized DIS at leading twist polarized quark distribution g q 1 (x) = q (x) + q (x) q (x) q (x) 1 Q 2 -corrections to X-section involve higher-twist distribution functions, such as g 2 (x) σ TT g 1 2Mx ν g 2 g 2 (x) involves quark-gluon correlations and does not have a parton interpretation as difference between number densities for polarized target, g 1 and g 2 contribute equally to σ LT σ LT g T g 1 + g 2 clean separation between higher order corrections to leading twist (g 1 ) and higher twist effects (g 2 ) what can one learn from g 2? Transverse (Spin) Structure of Hadrons p.111/138

115 Quark-Gluon Correlations (QCD analysis) g 2 (x) = g WW 2 (x) + ḡ 2 (x), with g WW 2 (x) g 1 (x) + 1 x ḡ 2 (x) involves quark-gluon correlations, e.g. dxx 2 ḡ 2 (x) = 1 3 d 2 = 2G +y G 0y + G zy = E y + B x dy y g 1(y) 1 P, S q(0)gg +y (0)γ + q(0) P, S 6MP +2 S x matrix elements of qb x γ + q and qe y γ + q are sometimes called color-electric and magnetic polarizabilities 2M 2 SχE = P, S j a E P, a S & 2M 2 SχB = P, S ja 0 B a P, S with d 2 = 1 4 (χ E + 2χ M ) but these names are misleading! Transverse (Spin) Structure of Hadrons p.112/138

116 Quark-Gluon Correlations (Interpretation) ḡ 2 (x) involves quark-gluon correlations, e.g. dxx 2 ḡ 2 (x) = 1 3 d 2 = 1 P, S q(0)gg +y (0)γ + q(0) P, S 6MP +2 S x QED: q(0)ef +y (0)γ + q(0) correlator between quark density qγ + q and (ŷ-component of the) Lorentz-force F y = e [ E + v B ] y = e (E y B x ) = e ( F 0y + F zy) = e 2F +y. for charged paricle moving with v = (0, 0, 1) in the ẑ direction matrix element of q(0)ef +y (0)γ + q(0) yields γ + density (density relevant for DIS in Bj limit!) weighted with the Lorentz force that a charged particle with v = (0, 0, 1) would experience at that point d 2 a measure for the color Lorentz force acting on the struck quark in SIDIS in the instant after being hit by the virtual photon F y (0) = M 2 d 2 (rest frame; S x = 1) Transverse (Spin) Structure of Hadrons p.113/138

117 Quark-Gluon Correlations (Interpretation) Interpretation of d 2 with the transverse FSI force in DIS also consistent with k y 1 0 dx d 2 k k 2 f 1T (x, k2 ) in SIDIS (Qiu, Sterman) k y = 1 2p + P, S q(0) dx gg +y (x )γ + q(0) P, S 0 semi-classical interpretation: average k in SIDIS obtained by correlating the quark density with the transverse impulse acquired from (color) Lorentz force acting on struck quark along its trajectory to (light-cone) infinity matrix element defining d 2 same as the integrand (for x = 0) in the QS-integral: k y = 0 dtf y (t) (use dx = 2dt) first integration point F y (0) (transverse) force at the begin of the trajectory, i.e. at the moment after absorbing the virtual photon Transverse (Spin) Structure of Hadrons p.114/138