F ( q 2 ⊥ ) → ρ ( r ⊥ ) in LF-Coordinates define state that is localized in ⊥ position: � � p + , R ⊥ = 0 ⊥ , λ d 2 p ⊥ � p + , p ⊥ , λ � � � � ≡ N Note: ⊥ boosts in IMF form Galilean subgroup ⇒ this state has 1 dx − d 2 x ⊥ x ⊥ T ++ ( x ) = 0 ⊥ � R ⊥ ≡ P + (cf.: working in CM frame in nonrel. physics) define charge distribution in impact parameter space 1 � j + (0 − , b ⊥ ) p + , R ⊥ = 0 ⊥ � p + , R ⊥ = 0 ⊥ � � � � ρ ( b ⊥ ) ≡ 2 p + Transverse (Spin) Structure of Hadrons – p.17/138
F ( q 2 ⊥ ) → ρ ( r ⊥ ) in LF-Coordinates use translational invariance to relate to same matrix element that appears in def. of form factor 1 � j + (0 − , b ⊥ ) p + , R ⊥ = 0 ⊥ � � � p + , R ⊥ = 0 ⊥ � � ρ ( b ⊥ ) ≡ 2 p + |N| 2 � � � j + (0 − , b ⊥ ) d 2 p ⊥ d 2 p ′ p + , p ′ � p + , p ⊥ � � � � = ⊥ ⊥ 2 p + |N| 2 � � � j + (0 − , 0 ⊥ ) d 2 p ⊥ d 2 p ′ p + , p ′ � p + , p ⊥ e i q ⊥ · b ⊥ � � � � = ⊥ ⊥ 2 p + � � |N| 2 d 2 p ⊥ d 2 p ′ ⊥ F 1 ( − q 2 ⊥ ) e i q ⊥ · b ⊥ = d 2 q ⊥ � (2 π ) 2 F 1 ( − q 2 ⊥ ) e i q ⊥ · b ⊥ ρ ( b ⊥ ) = → ֒ Transverse (Spin) Structure of Hadrons – p.18/138
Boosts in NRQM x ′ = � t ′ = t � x + � vt purely kinematical (quantization surface t = 0 inv.) ֒ → 1. boosting wavefunctions very simple p 1 − m 1 � p 2 − m 2 � Ψ � v ( � p 1 , � p 2 ) = Ψ � 0 ( � v, � v ) . 2. dynamics of center of mass x i ≡ m i � � R ≡ x i � r i with M i decouples from the internal dynamics Transverse (Spin) Structure of Hadrons – p.19/138
Relativistic Boosts t + v t ′ = γ � � z ′ = γ ( z + vt ) x ′ c 2 z , ⊥ = x ⊥ generators satisfy Poincaré algebra: [ P µ , P ν ] = 0 i ( g νρ P µ − g µρ P ν ) [ M µν , P ρ ] = g µλ M νρ + g νρ M µλ − g µρ M νλ − g νλ M µρ � M µν , M ρλ � � � = i rotations: M ij = ε ijk J k , boosts: M i 0 = 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
Galilean subgroup of ⊥ boosts introduce generator of ⊥ ‘boosts’: B y ≡ M + y = K y − J x B x ≡ M + x = K x + J y √ √ 2 2 Poincaré algebra = ⇒ commutation relations: = − iδ kl P + [ J 3 , B k ] = iε kl B l [ P k , B l ] P − , B k [ P + , B k ] � � = − iP 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
Galilean subgroup of ⊥ boosts Together with [ J z , P k ] = iε kl P l , as well as P − , P k P − , P + � P − , J z � � � � � = = = 0 P + , P k 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 − → momentum in the plane P ⊥ P + − → mass L z − → rotations around z -axis − → generator of boosts in the plane , B ⊥ Transverse (Spin) Structure of Hadrons – p.22/138
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 0 ⊥ ( x, k ⊥ − x ∆ ⊥ , y, l ⊥ − y ∆ ⊥ ) Ψ ∆ ⊥ ( x, k ⊥ , y, l ⊥ ) = Transverse center of momentum R ⊥ ≡ � i x i r ⊥ ,i plays role d 2 p ⊥ | p + , p ⊥ � corresponds to � similar to NR center of mass, e.g. state with R ⊥ = 0 ⊥ . Transverse (Spin) Structure of Hadrons – p.23/138
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 T F ( q 2 ) 1 ↔ ρ ( � r ) ambigous at scale ∆ R ∼ λ C = 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
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 ∼ 1 GeV 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
0 e e Deep Inelastic Scattering (DIS) q X p ν = E − E ′ Q 2 ≡ − q 2 = 4 EE ′ sin 2 θ 2 d Ω dE ′ = 4 πα 2 d 2 σ � W 2 ( Q 2 , ν ) cos 2 θ 2 + 2 W 1 ( Q 2 , ν ) sin 2 θ � MQ 4 2 exp. result: Bjorken scaling Q 2 → ∞ , 2 M W 1 ( Q 2 , ν )= F 1 ( x Bj ) ν → ∞ ⇒ x Bj = Q 2 / 2 Mν ν W 2 ( Q 2 , ν )= F 2 ( x Bj ) fixed Transverse (Spin) Structure of Hadrons – p.26/138
q Physical Meaning of x Bj = Q 2 0 k 2 p · q k p Go to frame where q ⊥ = 0 , i.e. Q 2 = − q 2 = − 2 q + q − 2 p · q = 2 q − p + + 2 q + p − q − → ∞ q + fixed Bjorken limit: , ֒ → q − p + + q + p − → − q + q + q − x Bj = − p + Transverse (Spin) Structure of Hadrons – p.27/138
Physical Meaning of x Bj = Q 2 2 p · q x Bj = − q + p + LC energy-momentum dispersion relation k − = m 2 + k 2 ⊥ 2 k + → struck quark with k −′ = k − + q − → ∞ can only be on mass shell if ֒ k + ′ = k + + q + ≈ 0 → ֒ x ≡ k + k + = − q + ⇒ 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
DIS − → light-cone correlations opt. theorem: inclusive cross–section ⇔ virtual, forward Compton amplitude struck quark carries large k k k momentum: Q 2 ≫ Λ 2 QCD X (ep e X) q -q = P P • crossed diagram suppressed q -q q -q = + (wavefunction!) • asymptotic freedom ⇒ neglect 2 -q interactions of struck quark • struck quark propagates along light-cone x 2 = 0 Transverse (Spin) Structure of Hadrons – p.29/138
q q suppression of crossed diagrams a) p p q q b) p p Flo w of the large momen tum q through t ypical diagrams con- tributing to the forw ard Compton amplitude. a) `handbag' diagrams; b) `cat's ears' diagram. Diagram b) is suppressed at large q due to the presence of additional propagators. Transverse (Spin) Structure of Hadrons – p.30/138
DIS − → light-cone correlations opt. theorem: inclusive cross–section ⇔ virtual, forward Compton amplitude struck quark carries large k k k momentum: Q 2 ≫ Λ 2 QCD X (ep e X) q -q = P P • crossed diagram suppressed q -q q -q = + (wavefunction!) • asymptotic freedom ⇒ neglect 2 -q interactions of struck quark • struck quark propagates along light-cone x 2 = 0 Transverse (Spin) Structure of Hadrons – p.31/138
0 x � DIS − → light-cone correlations x + x 3 x light-cone coordinates: √ x + = x 0 + x 3 � � / 2 √ x − = x 0 − x 3 � � / 2 DIS related to correlations along light–cone � dx − 2 π � P | q (0 − , 0 ⊥ ) γ + q ( x − , 0 ⊥ ) | P � e ix − x Bj P + q ( x Bj ) = Probability interpretation! No information about transverse position of partons! Transverse (Spin) Structure of Hadrons – p.32/138
DIS − → light-cone correlations � dx − 2 π � P | q (0 − , 0 ⊥ ) γ + q ( x − , 0 ⊥ ) | P � e ix − x Bj P + q ( x Bj ) = 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
unpolarized ep DIS σ elast ∝ Q 2 eq q → e + p → e ′ + X sensitive to: ֒ 4 u ↑ u ↑ p ( x ) + u ↓ u ↓ � � p ( x ) + ¯ p ( x ) + ¯ p ( x ) + 9 p ( x ) + ¯ p ( x ) + ¯ 1 � d ↑ d ↑ p ( x ) + d ↓ d ↓ � p ( x ) + 9 1 � s ↑ s ↑ p ( x ) + s ↓ s ↓ � p ( x ) + ¯ p ( x ) + ¯ p ( x ) + ... 9 where e.g u ↑ p ( x ) , u ↓ u ↑ u ↓ p ( x ) , ¯ p ( x ) , ¯ p ( x ) , ... are the distribution of u , ¯ u, ... 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 ) + 1 9 u p ( x ) + 1 ֒ 9 s p ( x ) ֒ → different linear combination of the same distribution functions! Transverse (Spin) Structure of Hadrons – p.34/138
many important results from DIS: discovery of elementary, charged, spin 1 2 constituents in the nucleon → quarks 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 ≡ � 1 � q ↑ ( x ) + ¯ q ↑ ( x ) − q ↓ ( x ) − ¯ q ↓ ( x ) � � ≪ 1 0 dx q = u,d,s (nonrel quark model: 1) ⇒ "spin crisis" Transverse (Spin) Structure of Hadrons – p.35/138
Longitudinally Polarized DIS e − /q helicity conserved in high-energy interactions e − q → longitudinally polarized e − ֒ preferentially scatter off q with spin opposite to that of the e − e − q scattering long. pol. e − off long. pol. nucleons ⇒ quark/nucleon spin correlation dσ ⇑↑ − dσ ⇑↓ ∝ g 1 ( x ) � e 2 q [ q ↑ ( x ) − q ↓ ( x )] with g 1 ( x ) = q 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
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 � 1 ∆Σ = � q ∆ q ≡ � 0 dx [ q ↑ ( x ) − q ↓ ( x )] = fraction of the q 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
∆ 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 � 1 � 1 � � x g � ≡ dx xg ( x ) = 1 − dx xq ( x ) ≈ 0 . 5 0 0 q (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 ) → ← p p − A LL in → γ + jet Transverse (Spin) Structure of Hadrons – p.38/138
∆ 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
→ ← p + p (RHIC-spin) ∆ G from 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
→ ← p + p (RHIC-spin) ∆ G from ‘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
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
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 dxf ( x ) x n ↔ � h | ¯ ψD n ψ | h � 0 ֒ → 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
Summary: DIS Bj − → PDF q ( x ) DIS 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 ¯ u 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
3D imaging of the nucleon 0.2 x 0.4 0.6 0.8 1 8 b ( f m ) ? 6 4 4 2 2 0 0 1 0.5 0 -0.5 -1 Transverse (Spin) Structure of Hadrons – p.45/138
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
0.2 x 0.4 Outline 0.6 0.8 1 b ( f m ) ? 6 4 4 Probabilistic interpretation of GPDs as Fourier 2 2 trafos of impact parameter dependent PDFs 0 0 H ( x, 0 , − ∆ 2 ⊥ ) − → q ( x, b ⊥ ) 1 0.5 0 ˜ -0.5 H ( x, 0 , − ∆ 2 ⊥ ) − → ∆ q ( x, b ⊥ ) -1 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 π + � p γ � p N d Summary u Transverse (Spin) Structure of Hadrons – p.47/138
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 � � F q dx ˜ H q ( x, t ) = G q dxH q ( x, t ) = 1 ( t ) A ( t ) � � F q dx ˜ E q ( x, t ) = G q dxE q ( x, t ) = 2 ( t ) 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 ) , G q A ( t ) , and G q P ( t ) are the Dirac, Pauli, axial, and pseudoscalar formfactors, respectively ( t ≡ q 2 = ( P ′ − P ) 2 ) γ µ F 1 ( q 2 ) + iσ µν q ν � � � P ′ , S ′ | j µ (0) | P, S � = ¯ 2 M F 2 ( q 2 ) u ( P ′ , S ′ ) u ( P, S ) GPDs can be probed in Deeply Virtual Compton Scattering (DVCS) Transverse (Spin) Structure of Hadrons – p.48/138
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
Deeply Virtual Compton Scattering (DVCS) need γ ∗ with several GeV 2 for Bjorken scaling 1 DVCS X-section factor α = 137 smaller than elastic X-section → need high luminosity e − beam with > 10 GeV ֒ ֒ → 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
0 Deeply Virtual Compton Scattering (DVCS) q q � � 0 � q q B j � ! 0 p p 0 p p − z J ν � z � p ′ � �� q · z � � TJ µ � � � d 4 z e i ¯ T µν = i � p � � 2 2 � 1 g µν � 1 1 � Bj H ( x, ξ, ∆ 2 )¯ u ( p ′ ) γ + u ( p ) ⊥ ֒ → dx x − ξ + iε + + ... x + ξ − iε 2 − 1 ∆ = p ′ − p x Bj ≡ − q 2 / 2 p · q = 2 ξ (1 + ξ ) q = ( q + q ′ ) / 2 ¯ Transverse (Spin) Structure of Hadrons – p.51/138
Generalized Parton Distributions (GPDs) � dx − − x − � x − � �� � � � � 2 π e ix − ¯ p + x p ′ � γ + q � H ( x, ξ, ∆ 2 )¯ u ( p ′ ) γ + u ( p ) � ¯ q � p = � � 2 2 u ( p ′ ) iσ + ν ∆ ν + E ( x, ξ, ∆ 2 )¯ u ( p ) 2 M � dx − − x − � x − � �� � � � � 2 π e ix − ¯ p + x ˜ p ′ γ + γ 5 q H ( x, ξ, ∆ 2 )¯ u ( p ′ ) γ + γ 5 u ( p ) � � � ¯ q � p = � � 2 2 u ( p ′ ) γ 5 ∆ + + ˜ E ( x, ξ, ∆ 2 )¯ ! 2 M u ( p ) where ∆ = p ′ − p is the momentum transfer and ξ measures the longi- tudinal momentum transfer on the target ∆ + = ξ ( p + + p + ′ ) . Transverse (Spin) Structure of Hadrons – p.52/138
u ( p ′ ) iσ + ν ∆ ν p ′ � � � � � ˆ = H ( x, ξ, ∆ 2 )¯ u ( p ′ ) γ + u ( p ) + E ( x, ξ, ∆ 2 )¯ O � p u ( p ) � � 2 M What is Physics of GPDs ? � dx − 2 π e ix − ¯ � � � � − x − x − with ˆ p + x ¯ γ + q O ≡ q 2 2 ֒ → 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
Form Factors vs. GPDs forward off-forward operator position space matrix elem. matrix elem. qγ + q ¯ Q F ( t ) ρ ( � r ) � dx − e ixp + x − � � � � − x − x − γ + q q ¯ q ( x ) H ( x, ξ, t ) ? 4 π 2 2 Transverse (Spin) Structure of Hadrons – p.54/138
Form Factors vs. GPDs forward off-forward operator position space matrix elem. matrix elem. qγ + q ¯ Q F ( t ) ρ ( � r ) � dx − e ixp + x − � � � � − x − x − γ + q q ¯ q ( x ) H ( x, 0 , t ) q ( x, b ⊥ ) 4 π 2 2 q ( x, b ⊥ ) = impact parameter dependent PDF Transverse (Spin) Structure of Hadrons – p.55/138
Impact parameter dependent PDFs define state that is localized in ⊥ position: � � p + , R ⊥ = 0 ⊥ , λ d 2 p ⊥ � p + , p ⊥ , λ � � � � ≡ N Note: ⊥ boosts in IMF form Galilean subgroup ⇒ this state has 1 dx − d 2 x ⊥ x ⊥ T ++ ( x ) = 0 ⊥ � R ⊥ ≡ P + (cf.: working in CM frame in nonrel. physics) define impact parameter dependent PDF � dx − ψ ( − x − 2 , b ⊥ ) γ + ψ ( x − � ¯ e ixp + x − p + , R ⊥ = 0 ⊥ � p + , R ⊥ = 0 ⊥ � � � � q ( x, b ⊥ ) ≡ 2 , b ⊥ ) 4 π Transverse (Spin) Structure of Hadrons – p.56/138
Impact parameter dependent PDFs use translational invariance to relate to same matrix element that appears in def. of GPDs ψ ( − x − 2 , b ⊥ ) γ + ψ ( x − � e ixp + x − � ¯ p + , R ⊥ = 0 ⊥ � p + , R ⊥ = 0 ⊥ dx − � � � � q ( x, b ⊥ ) ≡ 2 , b ⊥ ) ψ ( − x − 2 , b ⊥ ) γ + ψ ( x − � � � e ixp + x − = |N| 2 � ¯ d 2 p ⊥ d 2 p ′ p + , p ′ � p + , p ⊥ dx − � � � � 2 , b ⊥ ) ⊥ ⊥ Transverse (Spin) Structure of Hadrons – p.57/138
Impact parameter dependent PDFs use translational invariance to relate to same matrix element that appears in def. of GPDs ψ ( − x − 2 , b ⊥ ) γ + ψ ( x − � e ixp + x − � ¯ p + , R ⊥ = 0 ⊥ � p + , R ⊥ = 0 ⊥ dx − � � � � q ( x, b ⊥ ) ≡ 2 , b ⊥ ) ψ ( − x − 2 , b ⊥ ) γ + ψ ( x − � � � e ixp + x − = |N| 2 � ¯ d 2 p ⊥ d 2 p ′ p + , p ′ � p + , p ⊥ dx − � � � � 2 , b ⊥ ) ⊥ ⊥ ψ ( − x − 2 , 0 ⊥ ) γ + ψ ( x − � � � e ixp + x − = |N| 2 � ¯ d 2 p ⊥ d 2 p ′ p + , p ′ � p + , p ⊥ dx − � � � � 2 , 0 ⊥ ) ⊥ ⊥ × e i b ⊥ · ( p ⊥ − p ′ ⊥ ) Transverse (Spin) Structure of Hadrons – p.58/138
Impact parameter dependent PDFs use translational invariance to relate to same matrix element that appears in def. of GPDs ψ ( − x − 2 , b ⊥ ) γ + ψ ( x − � e ixp + x − � ¯ p + , R ⊥ = 0 ⊥ � p + , R ⊥ = 0 ⊥ dx − � � � � q ( x, b ⊥ ) ≡ 2 , b ⊥ ) ψ ( − x − 2 , b ⊥ ) γ + ψ ( x − � � � e ixp + x − = |N| 2 � ¯ d 2 p ⊥ d 2 p ′ p + , p ′ � p + , p ⊥ dx − � � � � 2 , b ⊥ ) ⊥ ⊥ ψ ( − x − 2 , 0 ⊥ ) γ + ψ ( x − � � � e ixp + x − = |N| 2 � ¯ d 2 p ⊥ d 2 p ′ p + , p ′ � p + , p ⊥ dx − � � � � 2 , 0 ⊥ ) ⊥ ⊥ × e i b ⊥ · ( p ⊥ − p ′ ⊥ ) � � � ⊥ − p ⊥ ) 2 � e i b ⊥ · ( p ⊥ − p ′ = |N| 2 d 2 p ⊥ d 2 p ′ x, 0 , − ( p ′ ⊥ ) ⊥ H � d 2 ∆ ⊥ (2 π ) 2 H ( x, 0 , − ∆ 2 ⊥ ) e − i b ⊥ · ∆ ⊥ q ( x, b ⊥ ) = ֒ → Transverse (Spin) Structure of Hadrons – p.59/138
Impact parameter dependent PDFs GPDs allow simultaneous determination of longitudinal momentum and transverse position of partons � d 2 ∆ ⊥ (2 π ) 2 H ( x, 0 , − ∆ 2 ⊥ ) e − i b ⊥ · ∆ ⊥ q ( x, b ⊥ ) = q ( x, b ⊥ ) has interpretation as density (positivity constraints!) � p + p + � b † ( xp + , b ⊥ ) b ( xp + , b ⊥ ) � � � � ∼ q ( x, b ⊥ ) , 0 ⊥ , 0 ⊥ � 2 ≥ 0 � b ( xp + , b ⊥ ) | p + � �� = , 0 ⊥ q ( x, b ⊥ ) ≥ 0 for x > 0 q ( x, b ⊥ ) ≤ 0 for x < 0 Transverse (Spin) Structure of Hadrons – p.60/138
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 i r 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 1 of spectators is related to impact parameter b ⊥ via r ⊥ = 1 − x b ⊥ . Transverse (Spin) Structure of Hadrons – p.61/138
0.2 x 0.4 0.6 q ( x, b ⊥ ) for unpol. p 0.8 1 b y 8 b ( f m ) ? 6 b x 4 4 x = 0 . 1 2 2 0 0 1 b y 0.5 0 -0.5 -1 b x x = 0 . 3 b y x = momentum fraction of the quark b x � b = ⊥ position of the quark x = 0 . 5 Transverse (Spin) Structure of Hadrons – p.62/138
Summary F T ↔ ρ ( � r ) form factor 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 F T DVCS − → GPDs ↔ q ( x, b ⊥ ) ‘impact parameter dependent PDFs Transverse (Spin) Structure of Hadrons – p.63/138
Transversely Deformed Distributions and E ( x, − ∆ 2 ⊥ ) M.B., Int.J.Mod.Phys.A 18 , 173 (2003) distribution of unpol. quarks in unpol (or long. pol.) nucleon: � d 2 ∆ ⊥ ⊥ ) e − i b ⊥ · ∆ ⊥ ≡ H ( x, b ⊥ ) (2 π ) 2 H ( x, − ∆ 2 q ( x, b ⊥ ) = unpol. quark distribution for nucleon polarized in x direction: � d 2 ∆ ⊥ 1 ∂ (2 π ) 2 E ( x, − ∆ 2 ⊥ ) e − i b ⊥ · ∆ ⊥ q ( x, b ⊥ ) = H ( x, b ⊥ ) − 2 M ∂b y Physics: j + = j 0 + j 3 , and left-right asymmetry from j 3 Transverse (Spin) Structure of Hadrons – p.64/138
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 p γ ∗ in − ˆ component in rest frame ( � z 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 ˆ ֒ x -direction, j z is positive on the +ˆ y side, and negative on the − ˆ y side ˆ y j z > 0 � p γ ˆ z Details of ⊥ deformation described by E q ( x, − ∆ 2 ⊥ ) j z < 0 → not surprising that E q ( x, − ∆ 2 ֒ ⊥ ) enters Ji relation! � J i = S i � � dx [ H q ( x, 0) + E q ( x, 0)] x. q Transverse (Spin) Structure of Hadrons – p.65/138
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) dxE q ( x, 0) = κ q/p 1 � � � d q d 2 b ⊥ q ( x, b ⊥ ) b y = y ≡ dx 2 M 2 M with κ q/p ≡ F u/d (0) contribution from quark flavor q to the proton 2 anomalous magnetic moment κ p = 1 . 793 = 2 3 κ u/p − 1 κ n = − 2 . 033 = 2 3 κ d/p − 1 3 κ d/p 3 κ u/p ֒ → κ u/p = 2 κ p + κ n = 1 . 673 κ d/p = 2 κ n + κ p = − 2 . 033 . → d q y = O (0 . 2 fm ) ֒ Transverse (Spin) Structure of Hadrons – p.66/138
p polarized in +ˆ x direction u ( x, b ⊥ ) d ( x, b ⊥ ) b y b y b x b x x = 0 . 1 ˆ y j z > 0 � p γ z ˆ b y b y j z < 0 b x b x 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
→ π + + X ) SSAs in SIDIS ( γ + p ↑− π + SIDIS = semi-inclusive DIS e ′ Single-Spin-Asymmetry (SSA) = left-right e asymmery in the X-section when only one spin is measured (e.g. target spin) example: nucleon transversely (relative to D π + q q ( z, p ⊥ ) e − beam) polarized − → left-right asymme- try 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
GPD ← → SSA (Sivers) Sivers: distribution of unpol. quarks in ⊥ pol. proton ⊥ )(ˆ P × k ⊥ ) · S f q/p ↑ ( x, k ⊥ ) = f q ⊥ ) − f ⊥ q 1 ( x, k 2 1 T ( x, k 2 M without FSI, f ( x, k ⊥ ) = f ( x, − k ⊥ ) ⇒ f ⊥ q 1 T ( x, k 2 ⊥ ) = 0 with FSI, f ⊥ q 1 T ( x, k 2 ⊥ ) � = 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
GPD ← → SSA (Sivers) example: γp → πX π + p γ � � p N d u u, d distributions in ⊥ polarized proton have left-right asymmetry in ⊥ position space (T-even!); sign “determined” by κ u & κ d attractive FSI deflects active quark towards the center of momentum → FSI translates position space distortion (before the quark is ֒ knocked out) in +ˆ y -direction into momentum asymmetry that favors − ˆ y direction q and sign of SSA: f ⊥ q → correlation between sign of κ p 1 T ∼ − κ p ֒ q f ⊥ q 1 T ∼ − κ p q confirmed by H ERMES data (also consistent with C OMPASS deuteron data f ⊥ u 1 T + f ⊥ d 1 T ≈ 0 ) Transverse (Spin) Structure of Hadrons – p.70/138
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
Transversity Distribution in Unpolarized Target Transverse (Spin) Structure of Hadrons – p.72/138
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
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 +ˆ y get deflected (due to FSI) into +ˆ x direction → (qualitative) connection between Boer-Mulders function h ⊥ ֒ 1 ( x, k ⊥ ) and the chirally odd GPD ¯ E T that is similar to (qualitative) connection between Sivers function f ⊥ 1 T ( x, k ⊥ ) and the GPD E . Boer-Mulders: distribution of ⊥ pol. quarks in unpol. proton � � ⊥ )(ˆ f q ↑ /p ( x, k ⊥ ) = 1 P × k ⊥ ) · S q f q ⊥ ) − h ⊥ q 1 ( x, k 2 1 ( x, k 2 2 M h ⊥ q 1 ( x, k 2 ⊥ ) 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
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(2 φ ) modulation of π distribution relative to lepton scattering ֒ plane ֒ → cos(2 φ ) asymmetry proportional to: Collins × BM Transverse (Spin) Structure of Hadrons – p.75/138
probing BM function in tagged SIDIS Primordial Quark Transversity Distribution ⊥ quark pol. Transverse (Spin) Structure of Hadrons – p.76/138
⊥ 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
probing BM function in tagged SIDIS Primordial Quark Transversity Distribution ⊥ quark pol. Transverse (Spin) Structure of Hadrons – p.78/138
probing BM function in tagged SIDIS Quark Transversity Distribution after γ ∗ absorption ⊥ quark pol. lepton scattering plane quark transversity component in lepton scattering plane flips Transverse (Spin) Structure of Hadrons – p.79/138
probing BM function in tagged SIDIS ⊥ momentum due to FSI ⊥ quark pol. k q ⊥ due to FSI lepton scattering plane on average, FSI deflects quarks towards the center Transverse (Spin) Structure of Hadrons – p.80/138
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: q pair with 3 P 0 ‘vacuum’ struck quark forms pion with ¯ q from q ¯ 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 H ERMES experiment more precise determination of Collins function under way ( KEK ) Transverse (Spin) Structure of Hadrons – p.81/138
probing BM function in tagged SIDIS ⊥ momentum due to Collins k ⊥ due to Collins ⊥ quark pol. 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
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
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
What is Orbital Angular Momentum Transverse (Spin) Structure of Hadrons – p.85/138
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
example: angular momentum in QED consider, for simplicity, QED without electrons: � � � � � � �� � E × � � � ∇ × � � d 3 r � d 3 r � x × x × E × J = B = A integrate by parts � � E j � � A j + � � � � x × � x × � ∇ · � � E + � E × � d 3 r ∇ J = � � A A drop 2 nd term (eq. of motion � ∇ · � E = 0 ), yielding � J = � L + � S with � � d 3 r E j � � � x × � � d 3 r � E × � A j ∇ L = � S = A note: � L and � S not separately gauge invariant Transverse (Spin) Structure of Hadrons – p.87/138
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
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
The nucleon spin pizza(s) Ji Jaffe & Manohar L q 1 1 L q 2 ∆Σ 2 ∆Σ J g L g ∆ G ‘pizza tre stagioni’ ‘pizza quattro stagioni’ only 1 2 ∆Σ ≡ 1 � q ∆ q common to both decompositions! 2 Transverse (Spin) Structure of Hadrons – p.90/138
T µν ( x ) − → Momentum Operator energy momentum tensor T µν = T νµ ; ∂ µ T µν = 0 T 00 energy density; T 0 i momentum density P µ ≡ d 3 xT µ 0 conserved ˜ � d d 3 x ∂ d 3 x ∂ � � ∂x 0 T µ 0 ∂ µ T µν =0 P µ = ∂x i T µi = 0 ˜ = dt T µν contains interactions, e.g. T µν ψ ( γ µ D ν + γ ν D µ ) ψ 2 ¯ = i q T µ 0 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 � T µ 0 + ‘ eq. of motion terms ′ + ‘ surface terms ′ � P µ ≡ d 3 x � Transverse (Spin) Structure of Hadrons – p.91/138
Angular Momentum Operator angular momentum tensor M µνρ = x µ T νρ − x ν T µρ ∂ ρ M µνρ = 0 J i = 1 d 3 rM jk 0 conserved ˜ 2 ε ijk � ֒ → d J i = 1 � d 3 x∂ 0 M jk 0 = 1 � d 3 x∂ l M jkl = 0 ˜ 2 ε ijk 2 ε ijk dt M µνρ contains time derivatives (since T µν does) use eq. of motion to get rid of these (as in T 0 i ) integrate total derivatives appearing in T 0 i 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
Angular Momentum in QCD (Ji) following this general procedure, one finds in QCD � � � � � �� � ψ † � i� ∂ − g � E × � � d 3 x Σ ψ + ψ † � x × x × J = A ψ + � B with Σ i = i 2 ε ijk 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 ˆ z 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
Ji-decomposition L q 1 2 ∆Σ J g Ji (1997) 1 � 1 � � � 2 = J q + J g = 2∆ q + L q + J g q q with ( P µ = ( M, 0 , 0 , 1) , S µ = (0 , 0 , 0 , 1) ) 1 1 � Σ 3 = iγ 1 γ 2 d 3 x � P, S | q † ( � x )Σ 3 q ( � 2∆ q = x ) | P, S � 2 � 3 � � x × i � d 3 x � P, S | q † ( � L q = x ) � D q ( � x ) | P, S � �� 3 � � � E × � � d 3 x � P, S | J g = x × � B | P, S � D = i� i � ∂ − g � A Transverse (Spin) Structure of Hadrons – p.94/138
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: � 1 � � J q � = � S dx x [ H q ( x, ξ, 0) + E q ( x, ξ, 0)] − 1 (nucleon at rest; � S is nucleon spin) q − 1 → L z q = J z ֒ 2 ∆ q derivation (MB-version): consider nucleon state that is an eigenstate under rotation about the ˆ x -axis (e.g. nucleon polarized in ˆ x direction with � p = 0 (wave packet if necessary) for such a state, � T 00 q y � = 0 = � T zz q y � and � T 0 y q z � = −� T 0 z q y � → � T ++ y � = � T 0 y q z − T 0 z q y � = � J x q � ֒ q → relate 2 nd moment of ⊥ flavor dipole moment to J x ֒ q Transverse (Spin) Structure of Hadrons – p.95/138
The Ji-relation (poor man’s derivation) derivation (MB-version): consider nucleon state that is an eigenstate under rotation about the ˆ x -axis (e.g. nucleon polarized in ˆ x direction with � p = 0 (wave packet if necessary) for such a state, � T 00 q y � = 0 = � T zz q y � and � T 0 y q z � = −� T 0 z q y � → � T ++ y � = � T 0 y q z − T 0 z q y � = � J x ֒ q � q → relate 2 nd moment of ⊥ flavor dipole moment to J x ֒ q effect sum of two effects: � T ++ y � for a point-like transversely polarized xnucleon � T ++ y � for a quark relative to the center of momentum of a q transversely polarized nucleon 2 nd moment of ⊥ flavor dipole moment for point-like nucleon � � � � f ( r ) 1 1 √ ψ = χ with χ = � σ · � p E + m f ( r ) 1 2 Transverse (Spin) Structure of Hadrons – p.96/138
The Ji-relation (poor man’s derivation) derivation (MB-version): γ 0 ∂ z + γ z ∂ 0 � T 0 z � = i ¯ q q q since ψ † ∂ z ψ is even under y → − y , i ¯ qγ 0 ∂ z q does not contribute to � T 0 z y ֒ → using i∂ 0 ψ = Eψ , one finds � � σ z 0 � � � T 0 z b y � d 3 rψ † γ 0 γ z ψy = E d 3 rψ † = E ψy σ z 0 2 E � E � d 3 rχ † σ z σ y χf ( r )( − i ) ∂ y f ( r ) y = d 3 = E + M E + M � d 3 rf 2 ( r ) = 1 consider nucleon state with � p = 0 , i.e. E = m & → 2 nd moment of ⊥ flavor dipole moment is 1 ֒ 2 M ֒ → ‘overall shift’ of nucleon COM yields contribution 1 dx xH q ( x, 0 , 0) to � T ++ � y � q 2 Transverse (Spin) Structure of Hadrons – p.97/138
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