Aspects of Quark Orbital Angular Momentum

33 Pages·2006·0.83 MB·English

Aspects of Quark Orbital
Angular Momentum
Matthias Burkardt
[email protected]
New Mexico State University
Las Cruces, NM, 88003, U.S.A.
AspectsofQuarkOrbitalAngularMomentum–p.1/33
Outline
GPDs: probabilistic interpretation as Fourier transforms of impact
parameter dependent PDFs
2
H(x, 0, −∆ ) −→ q(x, b )
⊥ ⊥
H˜ (x, 0, −∆2 ) −→ ∆q(x, b )
⊥ ⊥
E(x, 0, −∆2 )
⊥
֒→ ⊥ deformation of unpol. PDFs in ⊥ pol. target
physics: orbital motion of the quarks
֒→ Sivers effect
˜
2H + E −→ ⊥ deformation of ⊥ pol. PDFs in unpol. target
T T
correlation between quark angular momentum and quark
transversity
֒→ Boer-Mulders function h⊥(x, k )
1 ⊥
?
κ −→ Lq
Summary
AspectsofQuarkOrbitalAngularMomentum–p.2/33
Generalized Parton Distributions (GPDs)
GPDs: decomposition of form factors at a given value of t, w.r.t. the
average momentum fraction x = 1 (x + x ) of the active quark
i f
2
q ˜ q
dxH (x, ξ, t) = F (t) dxH (x, ξ, t) = G (t)
q 1 q A
Z Z
q ˜ q
dxE (x, ξ, t) = F (t) dxE (x, ξ, t) = G (t),
q 2 q P
Z Z
x and x are the momentum fractions of the quark before and
i f
after the momentum transfer
2ξ = x − x
f i
GPDs can be probed in deeply virtual Compton scattering (DVCS)
AspectsofQuarkOrbitalAngularMomentum–p.3/33
Generalized Parton Distributions (GPDs)
formal definition (unpol. quarks):
dx− x− x−
− +
ix p¯ x ′ + 2 ′ +
e p q¯ − γ q p = H(x, ξ, ∆ )u¯(p )γ u(p)
2π 2 2
Z (cid:28) (cid:12) (cid:18) (cid:19) (cid:18) (cid:19)(cid:12) (cid:29)
(cid:12) (cid:12)
iσ+ν∆
(cid:12) (cid:12) 2 ′ ν
+E(x, ξ, ∆ )u¯(p ) u(p)
(cid:12) (cid:12)
2M
in the limit of vanishing t and ξ, the nucleon non-helicity-flip GPDs
must reduce to the ordinary PDFs:
˜
H (x, 0, 0) = q(x) H (x, 0, 0) = ∆q(x).
q q
GPDs are form factor for only those quarks in the nucleon carrying
a certain fixed momentum fraction x
֒→ t dependence of GPDs for fixed x, provides information on the
position space distribution of quarks carrying a certain momentum
fraction x
AspectsofQuarkOrbitalAngularMomentum–p.4/33
Impact parameter dependent PDFs
define state that is localized in ⊥ position:
[D.Soper,PRD15, 1141 (1977)]
p+, R = 0 , λ ≡ N d2p p+, p , λ
⊥ ⊥ ⊥ ⊥
Z
(cid:12) (cid:11) (cid:12) (cid:11)
(cid:12) (cid:12)
Note: ⊥ boosts in IMF form Galilean subgroup ⇒ this state has
R ≡ 1 dx−d2x x T++(x) = x r = 0
⊥ P+ ⊥ ⊥ i i i,⊥ ⊥
(cf.: working in CM frame in nonrel. physics)
R P
define impact parameter dependent PDF
dx− x− x−
+ −
q(x, b ) ≡ p+, R = 0 q¯(− , b )γ+q( , b ) p+, R = 0 eixp x
⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥
4π 2 2
Z
(cid:10) (cid:12) (cid:12) (cid:11)
(cid:12) (cid:12)
AspectsofQuarkOrbitalAngularMomentum–p.5/33
GPDs ←→ q(x, b )
⊥
֒→ relate GPDs to distribution of partons in ⊥ plane
q(x, b ) = d2∆⊥ ei∆⊥·b⊥H(x, 0, −∆2 ),
⊥ (2π)2 ⊥
∆q(x, b ) = d2∆⊥ ei∆⊥·b⊥H˜ (x, 0, −∆2 ),
⊥ R (2π)2 ⊥
R
no rel. corrections to this result! (Galilean subgroup of ⊥ boosts)
q(x, b ) has probabilistic interpretation, e.g.
⊥
q(x, b ) ≥ |∆q(x, b )| ≥ 0 for x > 0
⊥ ⊥
q(x, b ) ≤ |∆q(x, b )| ≤ 0 for x < 0
⊥ ⊥
Note that x already measures longitudinal momentum of quarks
֒→ no simultaneous measurement of long. position of quarks
(Heisenberg) −→ no FT w.r.t. ξ
AspectsofQuarkOrbitalAngularMomentum–p.6/33
q(x, b ) in a simple model
⊥
0.2
0.4
0.6
0.8
1
8
6
44
22
x
00
1
0.5
0
-0.5 b (fm)
?
-1
AspectsofQuarkOrbitalAngularMomentum–p.7/33
Transversely Deformed Distributions and E(x, 0, −∆2 )
⊥
M.B., Int.J.Mod.Phys.A18, 173 (2003)
So far: only unpolarized (or long. pol.) nucleon! In general (ξ = 0):
− + −
dx eip x x hP+∆,↑|q¯(0) γ+q(x−)| P,↑i = H(x,0,−∆2 )
4π ⊥
R dx− eip+x−x hP+∆,↑|q¯(0) γ+q(x−)| P,↓i = −∆x−i∆y E(x,0,−∆2 ).
4π 2M ⊥
R
Consider nucleon polarized in x direction (in IMF)
|Xi ≡ |p+, R = 0 , ↑i + |p+, R = 0 , ↓i.
⊥ ⊥ ⊥ ⊥
֒→ unpolarized quark distribution for this state:
1 ∂ d2∆
q(x,b ) = H(x,b ) − ⊥ E(x, 0,−∆2 )e−ib⊥·∆⊥
⊥ ⊥ ⊥
2M ∂b (2π)2
y
Z
Physics: j+ = j0 + j3, and left-right asymmetry from j3 !
[X.Ji, PRL 91, 062001 (2003)]
AspectsofQuarkOrbitalAngularMomentum–p.8/33
~
Intuitive connection with L
q
Electromagnetic interaction couples to vector current. Due to
kinematics of the DIS-reaction (and the choice of coordinates —
zˆ-axis in direction of the momentum transfer) the virtual photons
“see” (in the Bj-limit) only the j+ = j0 + jz component of the quark
current
If up-quarks have positive orbital angular momentum in the
xˆ-direction, then jz is positive on the +yˆ side, and negative on the
−yˆ side
yˆ
jz > 0
p~
γ
zˆ
jz < 0
AspectsofQuarkOrbitalAngularMomentum–p.9/33
~
Intuitive connection with L
q
Electromagnetic interaction couples to vector current. Due to
kinematics of the DIS-reaction (and the choice of coordinates —
zˆ-axis in direction of the momentum transfer) the virtual photons
“see” (in the Bj-limit) only the j+ = j0 + jz component of the quark
current
If up-quarks have positive orbital angular momentum in the
xˆ-direction, then jz is positive on the +yˆ side, and negative on the
−yˆ side
֒→ j+ is deformed not because there are more quarks on one side
than on the other but because the DIS-photons (coupling only to
j+) “see” the quarks on the +yˆ side better than on the −yˆ side.
⊥ deformation described by E (x, 0, t)
q
֒→ not surprising to find that E (x, 0, t) enters the Ji relation
q
i i
J = S dx [H (x, 0, 0) + E (x, 0, 0)] x.
q q q
Z
(cid:10) (cid:11)
AspectsofQuarkOrbitalAngularMomentum–p.10/33