Position Space Interpretation for Generalized Parton Distributions

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The physics of E(x, 0, −∆ 2 ⊥ ) So far: only unpolarized (or long. polarized) nucleon In general, use ( ∆ + = 0) ∫ 〈 ( ) ( )∣ dx − x − x 4π eip+ P+∆,↑ ∣ −x − x ∣¯q γ + − ∣∣∣ q P,↑〉 2 2 ∫ 〈 ( ) ( )∣ dx − x − x 4π eip+ P+∆,↑ ∣ −x − x ∣¯q γ + − ∣∣∣ q P,↓〉 2 2 = H(x,0,−∆ 2 ⊥) = − ∆ x−i∆ y 2M E(x,0,−∆2 ⊥). Consider nucleon polarized in x direction (in IMF) |X〉 ≡ |p + ,R ⊥ = 0 ⊥ , ↑〉 + |p + ,R ⊥ = 0 ⊥ , ↓〉. ↩→ unpolarized quark distribution for this state: q X (x,b ⊥ ) = q(x,b ⊥ ) − 1 2M ∂ ∂b y ∫ d 2 ∆ ⊥ (2π) 2 E(x, 0,−∆2 ⊥)e ib ⊥·∆ ⊥ Position Space Interpretation for Generalized Parton Distributions – p.18/55
The physics of E(x, 0, −∆ 2 ⊥ ) q X (x,b ⊥ ) in transversely polarized nucleon is transversely distorted compared to longitudinally polarized nucleons ! mean displacement of flavor q (⊥ flavor dipole moment) d q y ≡ ∫ ∫ dx d 2 b ⊥ q X (x,b ⊥ )b y = 1 2M with κ p u/d ≡ F u/d 2 (0) = O(1 − 2) ⇒ d q y = O(0.2fm) ∫ dxE q (x, 0, 0) = κp q 2M CM for flavor q shifted relative to CM for whole proton by ∫ ∫ dx d 2 b ⊥ q X (x,b ⊥ )xb y = 1 2M ∫ dxxE q (x, 0, 0) ↩→ not surprising to find that second moment of E q is related to angular momentum carried by flavor q Position Space Interpretation for Generalized Parton Distributions – p.19/55
Page 1 and 2: Position Space Interpretation for G
Page 3 and 4: (brief) Motivation generalization t
Page 5 and 6: Generalized Parton Distributions (G
Page 7 and 8: What is Physics of GPDs Definition
Page 9 and 10: Form Factors vs. GPDs operator forw
Page 11 and 12: Impact parameter dependent PDFs q(x
Page 13 and 14: Impact parameter dependent PDFs q(x
Page 15 and 16: Discussion: GPD ↔ q(x,b ⊥ ) GPD
Page 17: Discussion: GPD ↔ q(x,b ⊥ ) b
Page 21 and 22: simple model for E q (x, 0, −∆
Page 23 and 24: connection with ⊥ SSA example: γ
Page 25 and 26: Summary DVCS allows probing GPDS
Page 27 and 28: extrapolating to ξ = 0 bad news:ξ
Page 29 and 30: QCD evolution So far ignored! Can b
Page 31 and 32: suppression of crossed diagrams ¡
Page 33 and 34: use translational invariance to rel
Page 35 and 36: density interpretation of q(x,b ⊥
Page 37 and 38: Boosts in nonrelativistic QM ⃗x
Page 39 and 40: Galilean subgroup of ⊥ boosts int
Page 41 and 42: Consequences many results from NRQM
Page 43 and 44: Poincaré algebra: [P µ , P ν ] =
Page 45 and 46: Together with [J z , P k ] = iε kl
Page 47 and 48: Proof that B ⊥ |p + ,R ⊥ = 0
Page 49 and 50: ¦ ¥ ¡ ¢ £¤ simple model for q
Page 51 and 52: ¡ £ ¡ ¢ back Figure 1: P + P
Page 53 and 54: ¢ ¡ ¡ Figure 2: Schematic view o
Page 55: ¡ ¢ ¢ £ ¡ £ ¤ ¢ ¢ ¥ ¦ §

Position Space Interpretation for Generalized Parton Distributions

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