[PPT] - Transverse (Spin) Structure of Hadrons Matthias Burkardt PowerPoint Presentation

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Transverse (Spin) Structure of Hadrons

Matthias Burkardt

burkardt@nmsu.edu

New Mexico State University Las Cruces, NM, 88003, U.S.A.

Transverse (Spin) Structure of Hadrons – p.1/138

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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

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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

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Form Factors in nonrel. QM

Potential scattering dσ dΩ = |f( q)|2 Born approx. (“1-photon exchange” for Coulomb interaction) f( q) =

d3re−i

q· rV (

r) Electro-magnetic interactions: V ( r) = eA0( r) with

∇2A0(

r) = −4πρ( r), i.e. f( q) = 4πe

q2
d3re−i

q· rρ(

r) ⇒ dσ dΩ = |f( q)|2 = dσ dΩ

point

×|F( q)|2 with F( q) =

d3re−i

q· rρ(

r) ֒ →

dσ dΩ −

→ F( q) − → ρ( r)

Transverse (Spin) Structure of Hadrons – p.4/138

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Form Factor (Interpretation)

small q2 expansion F( q) = 1 − 1 6r2 q2 + O( q4) Hofstadter (NP 1961): first measurement of F( q) using electron nucleon scattering ֒ →

r2

p ≈ 0.86 fm

Transverse (Spin) Structure of Hadrons – p.5/138

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Issues

Form factor of a “moving target” Relativistic effects Breit frame Infinite momentum frame (IMF)

Transverse (Spin) Structure of Hadrons – p.6/138

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Moving Target (Meson, Nucleon,..)

Fixed target: F( q)

F T

↔ ρ( r) trivial

Transverse (Spin) Structure of Hadrons – p.7/138

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Moving Target (Meson, Nucleon,..)

Fixed target: F( q)

F T

↔ ρ( r) trivial Moving target (nonrel.) Hint = e

d3

r A0( r)j0( r) dσ dΩ ∝ | p′ |Hint| p|2 ֒ → dσ dΩ = dσ dΩ

point

× |F( q)|2 with F( q) ≡

p′
j0(

0)

p
(

q = p − p′)

Transverse (Spin) Structure of Hadrons – p.7/138

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Moving Target (Meson, Nucleon,..)

Fixed target: F( q)

F T

↔ ρ( r) trivial Moving target (nonrel.) Hint = e

d3

r A0( r)j0( r) dσ dΩ ∝ | p′ |Hint| p|2 ֒ → dσ dΩ = dσ dΩ

point

× |F( q)|2 with F( q) ≡

p′
j0(

0)

p
(

q = p − p′) Relation to ρ( r)?

Transverse (Spin) Structure of Hadrons – p.7/138

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Moving Target (Meson, Nucleon,..)

Fixed target: F( q)

F T

↔ ρ( r) trivial Moving target (nonrel.) Hint = e

d3

r A0( r)j0( r) dσ dΩ ∝ | p′ |Hint| p|2 ֒ → dσ dΩ = dσ dΩ

point

× |F( q)|2 with F( q) ≡

p′
j0(

0)

p
(

q = p − p′) Relation to ρ( r)? What is ρ( r)?

Transverse (Spin) Structure of Hadrons – p.7/138

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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 =
≡ N
d3

p | p define charge distribution (for this localized state) ρ( r) ≡

R =
j0(

r)

R =
Transverse (Spin) Structure of Hadrons – p.8/138

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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 =
j0(

r)

R =
= |N|2
d3

p

d3

p′ p′| j0( r) | p = |N|2

d3

p

d3

p′ p′| j0( 0) | pei

r·( p− p′),

= |N|2

d3

p

d3

p′F

− (

p′ − p)2 ei

r·( p− p′)

֒ → ρ( r) =

d3

q (2π)3 F(− q2)ei

q· r

back

Transverse (Spin) Structure of Hadrons – p.9/138

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Form Factors (relativistic)

Lorentz invariance, parity, current conservation ⇒ p′ |jµ(0)| p = pµ + pµ′ F(q2) (spin 0) ¯ u(p′)

γµF1(q2) + iσµνqν

2M F2(q2)

u(p)

(spin 1

2)

with qµ = pµ − pµ′. issues: “energy factors” spoil simple interpretation of form factors as FT of charge distributions q2 depends on p + p′

Transverse (Spin) Structure of Hadrons – p.10/138

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Form Factor vs. Charge Distribution (rel.)

wave packet |Ψ =

d3p ψ(

p)

2E

p(2π)3 |

p , E

p =

M 2 +

p2 and covariant normalization p′| p = 2E

pδ(

p′ − p) Fourier transform of charge distribution in the wave packet ˜ ρ( q) ≡

d3xe−i

q· x Ψ| j0(

x) |Ψ =

d3p

2E

p2E p′ Ψ∗(

p + q)Ψ( p) p′| j0( 0) | p = 1 2

d3pE

p + E p′

E

pE p′ Ψ∗(

p + q)Ψ( p)F(q2).

Transverse (Spin) Structure of Hadrons – p.11/138

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Form Factor vs. Charge Distribution (rel.)

Nonrelativistic case: E

p + E p′

2E

pE p′ = 1

and q2 = − q2 ֒ → Fourier transform of charge distribution in the wave packet ˜ ρ( q) =

d3pΨ∗(

p + q)Ψ( p)F( q2) choose Ψ( p) very localized in position space Ψ∗( p + q) ≈ Ψ∗( p) ֒ → ˜ ρ( q) = F( q2)

Transverse (Spin) Structure of Hadrons – p.12/138

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Form Factor vs. Charge Distribution (rel.)

Relativistic corrections (example rms radius): ˜ ρ( q2) = 1 − R2 6 q2 − R2 6

d3p |Ψ(

p)|2 ( q · p)2 E2

p

+

d3p
q ·

∇Ψ( p)

2

− 1 8

d3p |Ψ(

p)|2 ( q · p)2 E4

p

, [R2 defined as usual: F(q2) = 1 + R2

6 q2 + O(q4)]

If one completely localizes the wave packet, i.e.

d3p
q ·

∇Ψ( p)

2

→ 0, then relativistic corrections diverge (∆x∆p ∼ 1) R2 6

d3p |Ψ(

p)|2 ( q · p)2 E2

p

→ ∞, 1 8

d3p |Ψ(

p)|2 ( q · p)2 E4

p

→ ∞

Transverse (Spin) Structure of Hadrons – p.13/138

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Sachs Form Factors

in rest frame, rel. corrections contribute ∆R2 ∼ λ2

C = 1 M2

identification of charge distribution in rest frame with Fourier transformed form factor only unique down to scale λC standard remedy: interpret F( q) as Fourier transform of charge distribution in “Breit frame” p′ = − p GE(q2) ≡ F1(q2) +

q2 4M2 F2(q2)

GM(q2) ≡ F1(q2) + F2(q2) Physics (Breit “frame”):

q/2
j0(0)
−

q/2

=

2MGE(q2)δs′s

q/2
j(0)
−

q/2

=

GM(q2)χ†

s′i

σ × qχs ֒ → GE

F T

− → charge density; GM

F T

− → 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

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Infinite Momentum Frame Interpretation

rel. corrections governed by

p· q E2

p and

q2 E2

p

consider wave packet Ψ( p⊥) in transverse direction, with sharp longitudinal momentum Pz → ∞ transverse size of wave packet r⊥, with R ≫ r⊥ ≫

1 Pz

take purely transverse momentum transfer ֒ → ˜ ρ( q⊥) = F( q2

⊥)

֒ → form factor can be interpreted as Fourier transform of charge distribution w.r.t. impact parameter in ∞ momentum frame (without λC uncertainties!) impact parameter measured w.r.t. ⊥ center of momentum R⊥ =

i∈q,g xiri ⊥

Transverse (Spin) Structure of Hadrons – p.15/138

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Derivation in LF-Coordinates

light-front (LF) coordinates p+ = 1 √ 2

p0 + p3

p− = 1 √ 2

p0 − p3

form factor for spin 1

2 target (Lorentz invariance, parity, charge

conservation) p′ |jµ(0)| p = ¯ u(p′)

γµF1(q2) + iσµνqν

2M F2(q2)

u(p)

with qµ = pµ − pµ′. If q+ = 0 (Drell-Yan-West frame) then

p′, ↑
j+(0)
p, ↑
= 2p+F1(−q2

⊥)

Transverse (Spin) Structure of Hadrons – p.16/138

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F(q2

⊥) → ρ(r⊥) in LF-Coordinates define state that is localized in ⊥ position:

p+, R⊥ = 0⊥, λ
≡ N
d2p⊥
p+, p⊥, λ
Note: ⊥ boosts in IMF form Galilean subgroup ⇒ this state has

R⊥ ≡

1 P +

dx−d2x⊥ 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

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F(q2

⊥) → ρ(r⊥) in LF-Coordinates use translational invariance to relate to same matrix element that appears in def. of form factor ρ(b⊥) ≡ 1 2p+

p+, R⊥ = 0⊥
j+(0−, b⊥)
p+, R⊥ = 0⊥
=

|N|2 2p+

d2p⊥
d2p′

⊥

p+, p′

⊥

j+(0−, b⊥)
p+, p⊥
=

|N|2 2p+

d2p⊥
d2p′

⊥

p+, p′

⊥

j+(0−, 0⊥)
p+, p⊥
eiq⊥·b⊥

= |N|2

d2p⊥
d2p′

⊥F1(−q2 ⊥)eiq⊥·b⊥

֒ → ρ(b⊥) =

d2q⊥

(2π)2 F1(−q2

⊥)eiq⊥·b⊥

Transverse (Spin) Structure of Hadrons – p.18/138

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Boosts in NRQM

x′ =

x + vt t′ = t purely kinematical (quantization surface t = 0 inv.) ֒ → 1. boosting wavefunctions very simple Ψ

v(

p1, p2) = Ψ

0(

p1 − m1 v, p2 − m2 v).

2. dynamics of center of mass
R ≡
i

xi ri with xi ≡ mi M decouples from the internal dynamics

Transverse (Spin) Structure of Hadrons – p.19/138

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Relativistic Boosts

t′ = γ

t + v

c2 z

,

z′ = γ (z + vt) x′

⊥ = x⊥

generators satisfy Poincaré algebra: [P µ, P ν] = [M µν, P ρ] = i (gνρP µ − gµρP ν)

M µν, M ρλ

= i

gµλM νρ + gνρM µλ − gµρM νλ − gνλM µρ

rotations: Mij = εijkJk, boosts: Mi0 = Ki. [Ki, Pj] = iδijP 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

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Galilean subgroup of ⊥ boosts

introduce generator of ⊥ ‘boosts’: Bx ≡ M +x = Kx + Jy √ 2 By ≡ M +y = Ky − Jx √ 2 Poincaré algebra = ⇒ commutation relations: [J3, Bk] = iεklBl [Pk, Bl] = −iδklP +

P −, Bk
= −iPk

[P +, Bk] = 0 with k, l ∈{x, y}, εxy = −εyx = 1, and εxx = εyy = 0.

Transverse (Spin) Structure of Hadrons – p.21/138

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Galilean subgroup of ⊥ boosts

Together with [Jz, Pk] = iεklPl, as well as

P −, Pk
=
P −, P +

=

P −, Jz
= 0
P +, Pk
=
P +, Bk
=
P +, Jz
= 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 Lz − → rotations around z-axis B⊥ − → generator of boosts in the plane,

Transverse (Spin) Structure of Hadrons – p.22/138

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Consequences of Galilean Subgroup

many results from NRQM carry over to ⊥ boosts in IMF , e.g. ⊥ boosts kinematical Ψ∆⊥(x, k⊥) = Ψ0⊥(x, k⊥ − x∆⊥) Ψ∆⊥(x, k⊥, y, l⊥) = Ψ0⊥(x, k⊥ − x∆⊥, y, l⊥ − y∆⊥) Transverse center of momentum R⊥ ≡

i xir⊥,i plays role

similar to NR center of mass, e.g.

d2p⊥ |p+, p⊥ corresponds to

state with R⊥ = 0⊥.

Transverse (Spin) Structure of Hadrons – p.23/138

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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(q2)

F T

↔ ρ( r) ambigous at scale ∆R ∼ λC =

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

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Deep Inelastic Scattering (DIS)

high-energy lepton (e±, µ±, ν, ) nucleon scattering usually inelastic for Q2 = −q2 = −(p − p′)2 ≫ M 2

p ∼ 1GeV2, probe can resolve

distance scales much smaller than the proton size ֒ → deep inelastic scattering because of high Q2, 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

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Deep Inelastic Scattering (DIS)

X p q e e

ν = E − E′ Q2 ≡ −q2 = 4EE′ sin2 θ

2

d2σ dΩdE′ = 4πα2 MQ4

W2(Q2, ν) cos2 θ

2 + 2W1(Q2, ν) sin2 θ 2

exp. result: Bjorken scaling

Q2 → ∞, ν → ∞ xBj = Q2/2Mν fixed    ⇒ 2M W1(Q2, ν)=F1(xBj) ν W2(Q2, ν)=F2(xBj)

Transverse (Spin) Structure of Hadrons – p.26/138

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Physical Meaning of xBj = Q2

2p·q

p k k q

Go to frame where q⊥ = 0, i.e. Q2 = −q2 = −2q+q− 2p · q = 2q−p+ + 2q+p− Bjorken limit: q− → ∞ , q+ fixed ֒ → xBj = − q+q− q−p+ + q+p− → −q+ p+

Transverse (Spin) Structure of Hadrons – p.27/138

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Physical Meaning of xBj = Q2

2p·q xBj = − q+

p+

LC energy-momentum dispersion relation k− = m2 + k2

⊥

2k+ ֒ → struck quark with k−′ = k− + q− → ∞ can only be on mass shell if k+′ = k+ + q+ ≈ 0 ֒ → k+ = −q+ ⇒ x ≡ k+ p+ = xBj ֒ → xBj 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

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DIS − → light-cone correlations

pt. theorem:

inclusive cross–section ⇔ virtual, forward Compton amplitude

X (ep

e X)

= = +

P P k k k q

q

q

q

q

q
q

2

crossed diagram suppressed

(wavefunction!)

asymptotic freedom ⇒ neglect

interactions of struck quark

struck quark propagates along

light-cone x2 = 0 struck quark carries large momentum: Q2 ≫ Λ2

QCD

Transverse (Spin) Structure of Hadrons – p.29/138

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suppression of crossed diagrams

p p q q a) b) p p q q Flo w

f

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

f

additional propagators.

Transverse (Spin) Structure of Hadrons – p.30/138

SLIDE 34

DIS − → light-cone correlations

pt. theorem:

inclusive cross–section ⇔ virtual, forward Compton amplitude

X (ep

e X)

= = +

P P k k k q

q

q

q

q

q
q

2

crossed diagram suppressed

(wavefunction!)

asymptotic freedom ⇒ neglect

interactions of struck quark

struck quark propagates along

light-cone x2 = 0 struck quark carries large momentum: Q2 ≫ Λ2

QCD

Transverse (Spin) Structure of Hadrons – p.31/138

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DIS − → light-cone correlations

x 3 x x + x

light-cone coordinates:

x+ =

x0 + x3

/ √ 2 x− =

x0 − x3

/ √ 2 DIS related to correlations along light–cone q(xBj) = dx− 2π P|q(0−, 0⊥)γ+q(x−, 0⊥)|P eix−xBjP + Probability interpretation! No information about transverse position of partons!

Transverse (Spin) Structure of Hadrons – p.32/138

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DIS − → light-cone correlations

q(xBj) = dx− 2π P|q(0−, 0⊥)γ+q(x−, 0⊥)|P eix−xBjP + Fourier transform along x− filters out quarks with light-cone momentum k+ = xBjP + 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

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unpolarized ep DIS

σelast

eq

∝ Q2

q

֒ → e + p → e′ + X sensitive to:

4 9

u↑

p(x) + ¯

u↑

p(x) + u↓ p(x) + ¯

u↓

p(x)

+

1 9

d↑

p(x) + ¯

d↑

p(x) + d↓ p(x) + ¯

d↓

p(x)

+

1 9

s↑

p(x) + ¯

s↑

p(x) + s↓ p(x) + ¯

s↓

p(x)

+ ...

where e.g u↑

p(x),u↓ p(x), ¯

u↑

p(x),¯

u↓

p(x), ... are the distribution of u,¯

u, ... in the proton with spin parallel/antiparallel to the proton’s spin neutron target (charge symmetry) ֒ → un(x) = dp(x), dn(x) = up(x), sn(x) = sp(x) ֒ → sensitive to 4

9dp(x) + 1 9up(x) + 1 9sp(x)

֒ → different linear combination of the same distribution functions!

Transverse (Spin) Structure of Hadrons – p.34/138

SLIDE 38

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 g1(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

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Longitudinally Polarized DIS

e− q e− q

e−/q helicity conserved in high-energy interactions ֒ → longitudinally polarized e− preferentially scatter off q with spin opposite to that

f the e−

scattering long. pol. e− off

long. pol. nucleons ⇒ quark/nucleon spin correlation

dσ⇑↑ − dσ⇑↓ ∝ g1(x) with g1(x) =

q

e2

q [q↑(x) − q↓(x)]

q↑(x)/q↓(x) = probability that q has same/opposite spin as N q(x) = q↑(x) + q↓(x) g1(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

SLIDE 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 + Lparton ∆Σ =

q ∆q ≡ q

1

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 Lparton = 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

SLIDE 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 xg ≡ 1 dx xg(x) = 1 −

q

1 dx xq(x) ≈ 0.5 (physics of xg: think of E × B) conceivable that ∆G is of the same order of magnitude as xg (or larger) several ‘explanations’ of the ‘spin crisis’ even suggested ∆G ∼ 4 − 6 ∆G accessible e.g. through ‘QCD-evolution’ of ∆q(x) ALL in

→

p

←

p− → γ+ jet

Transverse (Spin) Structure of Hadrons – p.38/138

SLIDE 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 Q2 of the probe increases resulting Q2 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 Q2 dependence of ∆q need coverage down to small x (g(x) concentrated at very small x) and wide Q2 range (‘QCD evolution’ slow) ֒ → planned EIC (electron ion collider)

Transverse (Spin) Structure of Hadrons – p.39/138

SLIDE 43

∆G from

→

p +

←

p (RHIC-spin)

gg and gq scattering sensitive to (relative) helicity use double-spin asymmetry ALL = σ++−σ+−

σ+++σ+− in

inelastic pp-scattering at RHIC to infer ∆G directly

Transverse (Spin) Structure of Hadrons – p.40/138

SLIDE 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

SLIDE 45

Theoretical Status

perturbative effects in general well understood, e.g. Q2 dependence (=“evolution”) calculable in QCD (Altarelli, Parisi, Gribov, Lipatov eqs.): given q(x, Q2

0) one can calculate q(x, Q2 1) for

Q2

1 > Q2 0 > a few GeV 2

֒ → important applications: compare two experiments at two different Q2 compare low Q2 models or sum rules with experiments at high Q2 nonperturbative effects difficult! power law behavior for x → 0 from Regge phenomenology typically, q(x, Q2

0) from some QCD-inspired models

lowest moments from lattice QCD

Transverse (Spin) Structure of Hadrons – p.42/138

SLIDE 46

Calculating PDFs in lattice QCD

direct evaluation: NO: On a Euclidean lattice, all distances are spacelike (x0 → ix0

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)xn ↔ h| ¯ ψDnψ|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

SLIDE 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”)

nly 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 “Q2 evolution” lattice calculations: lowest moments of PDFs and many “QCD-inspired” models ...

Transverse (Spin) Structure of Hadrons – p.44/138

SLIDE 48

3D imaging of the nucleon

0.2 0.4 0.6 0.8 1

1
0.5

0.5 1 2 4 6 8 2 4

b ? (f m) x

Transverse (Spin) Structure of Hadrons – p.45/138

SLIDE 49

Motivation (GPDs)

X.Ji, PRL 78, 610 (1997): DVCS ⇔ GPDs ⇔

Jq

֒ → 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

SLIDE 50

Outline

0.2 0.4 0.6 0.8 1

1
0.5

0.5 1 2 4 6 2 4

b ? (f m) x

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⊥) E(x, 0, −∆2

⊥) −

→ ⊥ distortion of PDFs when the target is ⊥ polarized Chromodynamik lensing and ⊥ SSAs transverse distortion of PDFs + final state interactions

⇒

⊥ SSA in γN − → π+X Summary

pγ
pN

d u

π+

Transverse (Spin) Structure of Hadrons – p.47/138

SLIDE 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 (xi + xf) of the active quark

dxHq(x, t)

= F q

1 (t)

dx ˜

Hq(x, t) = Gq

A(t)

dxEq(x, t)

= F q

2 (t)

dx ˜

Eq(x, t) = Gq

P (t),

xi and xf 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 ≡ q2 = (P ′ − P)2) P ′, S′| jµ(0) |P, S = ¯ u(P ′, S′)

γµF1(q2) + iσµνqν

2M F2(q2)

u(P, S)

GPDs can be probed in Deeply Virtual Compton Scattering (DVCS)

Transverse (Spin) Structure of Hadrons – p.48/138

SLIDE 52

Deeply Virtual Compton Scattering (DVCS)

virtual Compton scattering: γ∗p − → γp (actually: e−p − → e−γp) ‘deeply’: −q2

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

SLIDE 53

Deeply Virtual Compton Scattering (DVCS)

need γ∗ with several GeV2 for Bjorken scaling DVCS X-section factor α =

1 137 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 Q2

Transverse (Spin) Structure of Hadrons – p.50/138

SLIDE 54

Deeply Virtual Compton Scattering (DVCS)

p p q q

B

j

!

p p q q

T µν = i

d4z ei¯

q·z

p′

TJµ

−z 2

Jν z

2

p
Bj

֒ → gµν

⊥

2 1

−1

dx

1

x−ξ+iε + 1 x+ξ−iε

H(x, ξ, ∆2)¯

u(p′)γ+u(p) + ... ¯ q = (q + q′)/2 ∆ = p′ − p xBj ≡ −q2/2p · q = 2ξ(1 + ξ)

Transverse (Spin) Structure of Hadrons – p.51/138

SLIDE 55

Generalized Parton Distributions (GPDs)

dx− 2π eix− ¯

p+x

p′
¯

q

−x−

2

γ+q

x− 2

p
=

H(x, ξ, ∆2)¯ u(p′)γ+u(p) +E(x, ξ, ∆2)¯ u(p′)iσ+ν∆ν 2M u(p) dx− 2π eix− ¯

p+x

p′
¯

q

−x−

2

γ+γ5q

x− 2

p
=

˜ H(x, ξ, ∆2)¯ u(p′)γ+γ5u(p) ! + ˜ E(x, ξ, ∆2)¯ u(p′)γ5∆+ 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

SLIDE 56

What is Physics of GPDs ?

p′
ˆ

O

p
= H(x, ξ, ∆2)¯

u(p′)γ+u(p) + E(x, ξ, ∆2)¯ u(p′)iσ+ν∆ν 2M u(p) with ˆ O ≡ dx−

2π eix− ¯ p+x¯

q

− x−

2

γ+q
x−

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

SLIDE 57

Form Factors vs. GPDs

perator

¯ qγ+q dx−eixp+x−

4π

¯ q

−x−

2

γ+q
x−

2

forward

matrix elem. Q q(x)

ff-forward

matrix elem. F(t) H(x, ξ, t) position space ρ( r) ?

Transverse (Spin) Structure of Hadrons – p.54/138

SLIDE 58

Form Factors vs. GPDs

perator

¯ qγ+q dx−eixp+x−

4π

¯ q

−x−

2

γ+q
x−

2

forward

matrix elem. Q q(x)

ff-forward

matrix elem. F(t) H(x, 0, t) position space ρ( r) q(x, b⊥) q(x, b⊥) = impact parameter dependent PDF

Transverse (Spin) Structure of Hadrons – p.55/138

SLIDE 59

Impact parameter dependent PDFs

define state that is localized in ⊥ position:

p+, R⊥ = 0⊥, λ
≡ N
d2p⊥
p+, p⊥, λ
Note: ⊥ boosts in IMF form Galilean subgroup ⇒ this state has

R⊥ ≡

1 P +

dx−d2x⊥ x⊥T ++(x) = 0⊥

(cf.: working in CM frame in nonrel. physics) define impact parameter dependent PDF q(x, b⊥) ≡ dx− 4π

p+, R⊥ = 0⊥
¯

ψ(−x− 2 , b⊥)γ+ψ(x− 2 , b⊥)

p+, R⊥ = 0⊥
eixp+x−

Transverse (Spin) Structure of Hadrons – p.56/138

SLIDE 60

Impact parameter dependent PDFs

use translational invariance to relate to same matrix element that appears in def. of GPDs q(x, b⊥) ≡

dx−

p+, R⊥ = 0⊥

¯

ψ(−x− 2 , b⊥)γ+ψ(x− 2 , b⊥)

p+, R⊥ = 0⊥
eixp+x−

= |N|2

d2p⊥
d2p′

⊥

dx−

p+, p′

⊥

¯

ψ(−x− 2 , b⊥)γ+ψ(x− 2 , b⊥)

p+, p⊥
eixp+x−

Transverse (Spin) Structure of Hadrons – p.57/138

SLIDE 61

Impact parameter dependent PDFs

use translational invariance to relate to same matrix element that appears in def. of GPDs q(x, b⊥) ≡

dx−

p+, R⊥ = 0⊥

¯

ψ(−x− 2 , b⊥)γ+ψ(x− 2 , b⊥)

p+, R⊥ = 0⊥
eixp+x−

= |N|2

d2p⊥
d2p′

⊥

dx−

p+, p′

⊥

¯

ψ(−x− 2 , b⊥)γ+ψ(x− 2 , b⊥)

p+, p⊥
eixp+x−

= |N|2

d2p⊥
d2p′

⊥

dx−

p+, p′

⊥

¯

ψ(−x− 2 , 0⊥)γ+ψ(x− 2 , 0⊥)

p+, p⊥
eixp+x−

×eib⊥·(p⊥−p′

⊥)

Transverse (Spin) Structure of Hadrons – p.58/138

SLIDE 62

Impact parameter dependent PDFs

use translational invariance to relate to same matrix element that appears in def. of GPDs q(x, b⊥) ≡

dx−

p+, R⊥ = 0⊥

¯

ψ(−x− 2 , b⊥)γ+ψ(x− 2 , b⊥)

p+, R⊥ = 0⊥
eixp+x−

= |N|2

d2p⊥
d2p′

⊥

dx−

p+, p′

⊥

¯

ψ(−x− 2 , b⊥)γ+ψ(x− 2 , b⊥)

p+, p⊥
eixp+x−

= |N|2

d2p⊥
d2p′

⊥

dx−

p+, p′

⊥

¯

ψ(−x− 2 , 0⊥)γ+ψ(x− 2 , 0⊥)

p+, p⊥
eixp+x−

×eib⊥·(p⊥−p′

⊥)

= |N|2

d2p⊥
d2p′

⊥H

x, 0, − (p′

⊥ − p⊥)2

eib⊥·(p⊥−p′

⊥)

֒ → q(x, b⊥) = d2∆⊥ (2π)2 H(x, 0, −∆2

⊥)e−ib⊥·∆⊥

Transverse (Spin) Structure of Hadrons – p.59/138

SLIDE 63

Impact parameter dependent PDFs

GPDs allow simultaneous determination of longitudinal momentum and transverse position of partons q(x, b⊥) = d2∆⊥ (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

SLIDE 64

Impact parameter dependent PDFs

No relativistic corrections (Galilean subgroup!) ֒ → corollary: interpretation of 2d-FT of F1(Q2) 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 xiri,⊥

֒ → 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

f spectators is related to impact parameter b⊥ via r⊥ =

1 1−xb⊥.

Transverse (Spin) Structure of Hadrons – p.61/138

SLIDE 65

x = 0.5 x = 0.3

bx by bx by bx by

x = 0.1

q(x, b⊥) for unpol. p

0.2 0.4 0.6 0.8 1

1
0.5

0.5 1 2 4 6 8 2 4

b ? (f m) x

x = momentum fraction of the quark

b = ⊥ position of the quark

Transverse (Spin) Structure of Hadrons – p.62/138

SLIDE 66

Summary

form factor

F T

↔ 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

F T

↔ q(x, b⊥) ‘impact parameter dependent PDFs

Transverse (Spin) Structure of Hadrons – p.63/138

SLIDE 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⊥) = d2∆⊥ (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 ∂ ∂by d2∆⊥ (2π)2 E(x,−∆2

⊥)e−ib⊥·∆⊥

Physics: j+ = j0 + j3, and left-right asymmetry from j3

Transverse (Spin) Structure of Hadrons – p.64/138

SLIDE 68

Intuitive connection with Jq

DIS probes quark momentum density in the infinite momentum frame (IMF). Quark density in IMF corresponds to j+ = j0 + j3 component in rest frame ( pγ∗ in −ˆ z direction) ֒ → j+ larger than j0 when quark current towards the γ∗; suppressed when away from γ∗ ֒ → For quarks with positive orbital angular momentum in ˆ x-direction, jz is positive on the +ˆ y side, and negative on the −ˆ y side

pγ

ˆ z ˆ y jz > 0 jz < 0

Details of ⊥ deformation described by Eq(x, −∆2

⊥)

֒ → not surprising that Eq(x, −∆2

⊥) enters Ji relation!

Ji

q

= Si
dx [Hq(x, 0) + Eq(x, 0)] x.

Transverse (Spin) Structure of Hadrons – p.65/138

SLIDE 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) dq

y ≡

dx
d2b⊥q(x, b⊥)by =

1 2M

dxEq(x, 0) = κq/p

2M with κq/p ≡ F u/d

2

(0) contribution from quark flavor q to the proton anomalous magnetic moment κp = 1.793 = 2

3κu/p − 1 3κd/p

κn = −2.033 = 2

3κd/p − 1 3κu/p

֒ → κu/p = 2κp + κn = 1.673 κd/p = 2κn + κp = −2.033. ֒ → dq

y = O(0.2fm)

Transverse (Spin) Structure of Hadrons – p.66/138

SLIDE 70

x = 0.5 x = 0.5 x = 0.3 x = 0.3

bx by bx by bx by bx by bx by bx by

x = 0.1

u(x, b⊥) d(x, b⊥)

pγ

ˆ z ˆ y jz > 0 jz < 0

p polarized in +ˆ x direction

Transverse (Spin) Structure of Hadrons – p.67/138

SLIDE 71

SSAs in SIDIS (γ + p↑− → π+ + X)

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.

e e′ π+

q(x, k⊥) Dπ+

q (z, p⊥)

p q

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

SLIDE 72

GPD ← → SSA (Sivers)

Sivers: distribution of unpol. quarks in ⊥ pol. proton fq/p↑(x, k⊥) = f q

1 (x, k2 ⊥) − f ⊥q 1T (x, k2 ⊥)(ˆ

P × 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

SLIDE 73

GPD ← → SSA (Sivers)

example: γp → πX

pγ
pN

d u

π+

u, d distributions in ⊥ polarized proton have left-right asymmetry in ⊥ position space (T-even!); sign “determined” by κu & κd 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 ֒ → correlation between sign of κp

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 f ⊥u

1T + f ⊥d 1T ≈ 0)

Transverse (Spin) Structure of Hadrons – p.70/138

SLIDE 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

SLIDE 75

Transversity Distribution in Unpolarized Target

Transverse (Spin) Structure of Hadrons – p.72/138

SLIDE 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

SLIDE 77

Boer-Mulders Function

SIDIS: attractive FSI expected to convert position space asymmetry into momentum space asymmetry ֒ → e.g. quarks at negative bx 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 ¯ ET 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 fq↑/p(x, k⊥) = 1 2

f q

1 (x, k2 ⊥) − h⊥q 1 (x, k2 ⊥)(ˆ

P × k⊥) · Sq 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

SLIDE 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

SLIDE 79

probing BM function in tagged SIDIS

Primordial Quark Transversity Distribution ⊥ quark pol.

Transverse (Spin) Structure of Hadrons – p.76/138

SLIDE 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

SLIDE 81

probing BM function in tagged SIDIS

Primordial Quark Transversity Distribution ⊥ quark pol.

Transverse (Spin) Structure of Hadrons – p.78/138

SLIDE 82

probing BM function in tagged SIDIS

Quark Transversity Distribution after γ∗ absorption ⊥ quark pol. quark transversity component in lepton scattering plane flips lepton scattering plane

Transverse (Spin) Structure of Hadrons – p.79/138

SLIDE 83

probing BM function in tagged SIDIS

⊥ momentum due to FSI ⊥ quark pol. kq

⊥ due to FSI

n average, FSI deflects quarks towards the center

lepton scattering plane

Transverse (Spin) Structure of Hadrons – p.80/138

SLIDE 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 3P0 ‘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

SLIDE 85

probing BM function in tagged SIDIS

⊥ momentum due to Collins ⊥ quark pol. k⊥ due to Collins kq

⊥ due to FSI

SSA of π in jet emanating from ⊥ pol. q lepton scattering plane

Transverse (Spin) Structure of Hadrons – p.82/138

SLIDE 86

probing BM function in tagged SIDIS

net ⊥ momentum (FSI+Collins) lepton scattering plane k⊥ due to Collins net kq

⊥

kq

⊥ due to FSI

֒ → in this example, enhancement of pions with ⊥ momenta ⊥ to lepton plane

Transverse (Spin) Structure of Hadrons – p.83/138

SLIDE 87

probing BM function in tagged SIDIS

net kπ

⊥ (FSI + Collins)

lepton scattering plane net kq

⊥

֒ → expect enhancement of pions with ⊥ momenta ⊥ to lepton plane

Transverse (Spin) Structure of Hadrons – p.84/138

SLIDE 88

What is Orbital Angular Momentum

Transverse (Spin) Structure of Hadrons – p.85/138

SLIDE 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

SLIDE 90

example: angular momentum in QED

consider, for simplicity, QED without electrons:

J =
d3r

x ×

E ×

B

=
d3r

x ×

E ×
∇ ×

A

integrate by parts
J =
d3r
Ej
x ×

∇

Aj +
x ×

A

∇ ·

E + E × A

drop 2nd term (eq. of motion

∇ · E = 0), yielding J = L + S with

L =
d3r Ej
x ×

∇

Aj
S =
d3r

E × A note: L and S not separately gauge invariant

Transverse (Spin) Structure of Hadrons – p.87/138

SLIDE 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

ne 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

SLIDE 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

SLIDE 93

The nucleon spin pizza(s)

Ji Jaffe & Manohar

1 2∆Σ 1 2∆Σ

Jg ∆G Lq Lq Lg ‘pizza tre stagioni’ ‘pizza quattro stagioni’

nly 1

2∆Σ ≡ 1 2

q ∆q common to both decompositions!

Transverse (Spin) Structure of Hadrons – p.90/138

SLIDE 94

T µν(x) − → Momentum Operator

energy momentum tensor T µν = T νµ; ∂µT µν = 0 T 00 energy density; T 0i momentum density ˜ P µ ≡

d3xT µ0 conserved

d dt ˜ P µ =

d3x ∂

∂x0 T µ0 ∂µT µν=0 =

d3x ∂

∂xi T µi = 0 T µν contains interactions, e.g. T µν

q

= i

2 ¯

ψ (γµDν + γνDµ) ψ T µ0 contains time derivative (don’t want a Hamiltonian/momentum

perator that contains time derivative!)

֒ → replace by space derivative, using equation of motion, e.g. (iγµDµ − m) ψ = 0 to replace iD0ψ → γ0 iγkDk − m

ψ

some of the resulting space derivatives add up to total derivative terms which do not contribute to volume integral P µ ≡

d3x
T µ0 + ‘eq. of motion terms′ + ‘surface terms′

Transverse (Spin) Structure of Hadrons – p.91/138

SLIDE 95

Angular Momentum Operator

angular momentum tensor M µνρ = xµT νρ − xνT µρ ∂ρM µνρ = 0 ֒ → ˜ Ji = 1

2εijk

d3rM jk0 conserved d dt ˜ Ji = 1 2εijk

d3x∂0M jk0 = 1

2εijk

d3x∂lM 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 xi which then ‘disappears’ ֒ → Ji 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

SLIDE 96

Angular Momentum in QCD (Ji)

following this general procedure, one finds in QCD

J =
d3x
ψ†

Σψ + ψ† x ×

i

∂ − g A

ψ +

x ×

E ×

B

with Σi = i

2εijkγjγk

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

nly 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 Jq = Sq + Lq (-) quark OAM contains interactions (-) only quark spin has partonic interpretation as a single particle density

Transverse (Spin) Structure of Hadrons – p.93/138

SLIDE 97

Ji-decomposition

1 2∆Σ

Jg Lq

Ji (1997) 1 2 =

q

Jq + Jg =

q

1 2∆q + Lq

+ Jg

with (P µ = (M, 0, 0, 1), Sµ = (0, 0, 0, 1)) 1 2∆q = 1 2

d3x P, S| q†(

x)Σ3q( x) |P, S Σ3 = iγ1γ2 Lq =

d3x P, S| q†(

x)

x × i

D 3 q( x) |P, S Jg =

d3x P, S|
x ×
E ×

B 3 |P, S i D = i ∂ − g A

Transverse (Spin) Structure of Hadrons – p.94/138

SLIDE 98

The Ji-relation (poor man’s derivation)

What distinguishes the Ji-decomposition from other decompositions is the fact that Lq can be constrained by experiment: Jq = S 1

−1

dx x [Hq(x, ξ, 0) + Eq(x, ξ, 0)] (nucleon at rest; S is nucleon spin) ֒ → Lz

q = Jz q − 1 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 0y q z = −T 0z q y

֒ → T ++

q

y = T 0y

q z − T 0z q y = Jx q

֒ → relate 2nd moment of ⊥ flavor dipole moment to Jx

q

Transverse (Spin) Structure of Hadrons – p.95/138

SLIDE 99

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 0y q z = −T 0z q y

֒ → T ++

q

y = T 0y

q z − T 0z q y = Jx q

֒ → relate 2nd moment of ⊥ flavor dipole moment to Jx

q

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 2nd moment of ⊥ flavor dipole moment for point-like nucleon ψ =

f(r)
σ·

p E+mf(r)

χ

with χ = 1 √ 2

1

1

Transverse (Spin) Structure of Hadrons – p.96/138

SLIDE 100

The Ji-relation (poor man’s derivation)

derivation (MB-version): T 0z

q

= i¯ q

γ0∂z + γz∂0

q since ψ†∂zψ is even under y → −y, i¯ qγ0∂zq does not contribute to T 0zy ֒ → using i∂0ψ = Eψ, one finds T 0zby = E

d3rψ†γ0γzψy = E
d3rψ†
σz

σz

ψy

= 2E E + M

d3rχ†σzσyχf(r)(−i)∂yf(r)y =

E E + M

d3

consider nucleon state with p = 0, i.e. E = m &

d3rf 2(r) = 1

֒ → 2nd moment of ⊥ flavor dipole moment is

1 2M

֒ → ‘overall shift’ of nucleon COM yields contribution

1 2

dx xHq(x, 0, 0) to T ++

q

y

Transverse (Spin) Structure of Hadrons – p.97/138

SLIDE 101

The Ji-relation (poor man’s derivation)

derivation (MB-version): intrinsic distortion adds 1

2

dx xEq(x, 0, 0) to that

֒ → Jx

q = 1 2

dx x [Hq(x, 0, 0) + Eq(x, 0, 0)]

rotational invariance: should apply to each vector component ֒ → Ji relation

Transverse (Spin) Structure of Hadrons – p.98/138

SLIDE 102

Ji-decomposition

1 2∆Σ

Jg Lq

J =

q 1 2q†

Σq + q†

r × i

D

q +

r ×

E ×

B

applies to each vector component of nucleon

angular momentum, but Ji-decomposition usually applied only to ˆ z component where at least quark spin has parton interpretation as difference between number densities ∆q from polarized DIS Jq = 1

2∆q + Lq from exp/lattice (GPDs)

Lq in principle independently defined as matrix elements of q†

r × i

D

q, but in practice easier by subtraction Lq = Jq − 1

2∆q

Jg in principle accessible through gluon GPDs, but in practice easier by subtraction Jg = 1

2 − Jq

further decomposition of Jg 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

SLIDE 103

Lq for proton from Ji-relation (lattice)

lattice QCD ⇒ moments of GPDs (LHPC; QCDSF) ֒ → insert in Ji-relation

Ji

q

= Si
dx [Hq(x, 0) + Eq(x, 0)] x.

֒ → Lz

q = Jz q − 1 2∆q

Lu, Ld both large! present calcs. show Lu + Ld ≈ 0, but disconnected diagrams ..? m2

π extrapolation

parton interpret.

f Lq...

Transverse (Spin) Structure of Hadrons – p.100/138

SLIDE 104

Angular Momentum in QCD (Jaffe & Manohar)

define OAM on a light-like hypesurface rather than a space-like hypersurface ˜ J3 =

d2x⊥
dx−M 12+

where x− =

1 √ 2

x0 − x−

and M 12+ =

1 √ 2

M 120 + M 123

Since ∂µM 12µ = 0

d2x⊥
dx−M 12+ =
d2x⊥
dx3M 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

SLIDE 105

Jaffe/Manohar decomposition

1 2∆Σ

∆G

q Lq

Lg

in light-cone framework & light-cone gauge A+ = 0 one finds for Jz =

dx−d2r⊥M +xy

1 2 = 1 2∆Σ +

q

Lq + ∆G + Lg where (γ+ = γ0 + γz) Lq =

d3r P, S| ¯

q( r)γ+

r × i

∂ z q( r) |P, S ∆G = ε+−ij

d3r P, S| TrF +iAj |P, S

Lg = 2

d3r P, S| TrF +j
x × i

∂ z Aj |P, S

Transverse (Spin) Structure of Hadrons – p.102/138

SLIDE 106

Jaffe/Manohar decomposition

1 2∆Σ

∆G

q Lq

Lg

1 2 = 1 2∆Σ +

q

Lq + ∆G + Lg ∆Σ =

q ∆q from polarized DIS (or lattice)

∆G from

→

p

←

p or polarized DIS (evolution) ֒ → ∆G gauge invariant, but local operator only in light-cone gauge

dxxn∆G(x) for n ≥ 1 can be described by manifestly gauge inv.

local op. (− → lattice) Lq, Lg independently defined, but no exp. identified to access them not accessible on lattice, since nonlocal except when A+ = 0 parton net OAM L = Lg +

q Lq by subtr. L = 1 2 − 1 2∆Σ − ∆G

in general, Lq = Lq Lg + ∆G = Jg makes no sense to ‘mix’ Ji and JM decompositions, e.g. Jg − ∆G has no fundamental connection to OAM

Transverse (Spin) Structure of Hadrons – p.103/138

SLIDE 107

Lq = Lq

Lq matrix element of q†

r ×
i

∂ − g A z q = ¯ qγ0

r ×
i

∂−g A z q Lz

q matrix element of (γ+ = γ0 + γz)

¯ qγ+

r × i

∂ z q

A+=0

For nucleon at rest, matrix element of Lq same as that of ¯ qγ+

r ×
i

∂−g A z q ֒ → even in light-cone gauge, Lz

q and Lz q still differ by matrix element

f q†
r × g

A z q

A+=0 = q† (xgAy − ygAx) q
A+=0

Transverse (Spin) Structure of Hadrons – p.104/138

SLIDE 108

Summary part 1:

Ji: Jz = 1

2∆Σ + q Lq + Jg

Jaffe: Jz = 1

2∆Σ + q Lq + ∆G + Lg

∆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 Lq = Lq or Jg = ∆G + Lg, but how significant is the difference between Lq and Lq, etc. ?

Transverse (Spin) Structure of Hadrons – p.105/138

SLIDE 109

OAM in scalar diquark model

[M.B. + Hikmat Budhathoki Chhetri (BC), PRD 79, 071501 (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↑

+ 1

2 (x, k⊥) =

M + m

x

φ

ψ↑

− 1

2 = −k1 + ik2

x φ with φ =

c/√1−x M2−

k2 ⊥+m2 x

−

k2 ⊥+λ2 1−x

. quark OAM according to JM: Lq = 1

0 dx

d2k⊥

16π3 (1 − x)

ψ↑

− 1

2

2

quark OAM according to Ji: Lq = 1

2

1

0 dx x [q(x) + E(x, 0, 0)]− 1 2∆q

(using Lorentz inv. regularization, such as Pauli Villars subtraction) both give identical result, i.e. Lq = Lq not surprising since scalar diquark model is not a gauge theory

Transverse (Spin) Structure of Hadrons – p.106/138

SLIDE 110

OAM in scalar diquark model

But, even though Lq = Lq in this non-gauge theory Lq(x) ≡ d2k⊥ 16π3 (1−x)

ψ↑

− 1

2

2

= 1 2 {x [q(x) + E(x, 0, 0)]−∆q(x)} ≡ Lq(x)

0.05 0.1 0.15 0.2 0.2 0.4 0.6 0.8 1 x

L

(x)

L

(x)

֒ → ‘unintegrated Ji-relation’ does not yield x-distribution of OAM

Transverse (Spin) Structure of Hadrons – p.107/138

SLIDE 111

OAM in QED

light-cone wave function in eγ Fock component Ψ↑

+ 1

2 +1(x, k⊥)

= √ 2 k1 − ik2 x(1 − x)φ Ψ↑

+ 1

2 −1(x, k⊥) = −

√ 2k1 + ik2 1 − x Ψ↑

− 1

2 +1(x, k⊥)

= √ 2 m x − m

φ

Ψ↑

− 1

2 +1(x, k⊥) = 0

OAM of e− according to Jaffe/Manohar Le = 1

0 dx

d2k⊥
(1 − x)
Ψ↑

+ 1

2 −1(x, k⊥)

2

−

Ψ↑

+ 1

2 +1(x, k⊥)

2

e− OAM according to Ji Le = 1

2

1

0 dx x [q(x) + E(x, 0, 0)] − 1 2∆q

Le = Le + α

4π = Le

Likewise, computing Jγ from photon GPD, and ∆γ and Lγ from light-cone wave functions and defining ˆ Lγ ≡ Jγ − ∆γ yields ˆ Lγ = Lγ + α

4π = Lγ α 4π appears to be small, but here Le, Le are all of O( α π )

Transverse (Spin) Structure of Hadrons – p.108/138

SLIDE 112

OAM in QCD

֒ → 1-loop QCD: Lq − Lq = αs

3π

recall (lattice QCD): Lu ≈ −.15; Ld ≈ +.15 QCD evolution yields negative correction to Lu and positive correction to Ld ֒ → evolution suggested (A.W.Thomas) to explain apparent discrepancy between quark models (low Q2) and lattice results (Q2 ∼ 4GeV 2) above result suggests that Lu > Lu and Ld > Ld additional contribution (with same sign) from vector potential due to spectators (MB, to be published) ֒ → possible that lattice result consistent with Lu > Ld

Transverse (Spin) Structure of Hadrons – p.109/138

SLIDE 113

Summary

Ji Jaffe & Manohar

1 2∆Σ 1 2∆Σ

Jg ∆G

q Lq
q Lq

Lg

inclusive

→

e

←

p/

→

p

←

p provide access to quark spin 1

2∆q

gluon spin ∆G parton grand total OAM L ≡ Lg +

q Lq = 1 2 − ∆G − q ∆q

DVCS & polarized DIS and/or lattice provide access to quark spin 1

2∆q

Jq & Lq = Jq − 1

2∆q

Jg = 1

2 − q Jq

Jg − ∆G does not yield gluon OAM Lg Lq − Lq = O(0.1 ∗ αs) for O (αs) dressed quark

Transverse (Spin) Structure of Hadrons – p.110/138

SLIDE 114

Quark-Gluon Correlations (Introduction)

(longitudinally) polarized polarized DIS at leading twist − → ‘polarized quark distribution’ gq

1(x) = q↑(x) + ¯

q↑(x) − q↓(x) − ¯ q↓(x)

1 Q2 -corrections to X-section involve ‘higher-twist’ distribution

functions, such as g2(x) σT T ∝ g1 − 2Mx ν g2 g2(x) involves quark-gluon correlations and does not have a parton interpretation as difference between number densities for ⊥ polarized target, g1 and g2 contribute equally to σLT σLT ∝ gT ≡ g1 + g2 ֒ → ‘clean’ separation between higher order corrections to leading twist (g1) and higher twist effects (g2) what can one learn from g2?

Transverse (Spin) Structure of Hadrons – p.111/138

SLIDE 115

Quark-Gluon Correlations (QCD analysis)

g2(x) = gW W

2

(x) + ¯ g2(x), with gW W

2

(x) ≡ −g1(x) + 1

x dy y g1(y)

¯ g2(x) involves quark-gluon correlations, e.g.

dxx2¯

g2(x) = 1 3d2 = 1 6MP +2Sx

P, S
¯

q(0)gG+y(0)γ+q(0)

P, S
√

2G+y ≡ G0y + Gzy = −Ey + Bx matrix elements of ¯ qBxγ+q and ¯ qEyγ+q are sometimes called color-electric and magnetic polarizabilities 2M 2 SχE =

P, S
ja ×

Ea

P, S
& 2M 2

SχB =

P, S
j0

a

Ba

P, S
with d2 = 1

4 (χE + 2χM) — but these names are misleading!

Transverse (Spin) Structure of Hadrons – p.112/138

SLIDE 116

Quark-Gluon Correlations (Interpretation)

¯ g2(x) involves quark-gluon correlations, e.g.

dxx2¯

g2(x) = 1 3d2 = 1 6MP +2Sx

P, S
¯

q(0)gG+y(0)γ+q(0)

P, S
QED: ¯

q(0)eF +y(0)γ+q(0) correlator between quark density ¯ qγ+q and (ˆ y-component of the) Lorentz-force F y = e

E +

v × B y = e (Ey − Bx) = −e

F 0y + F zy

= −e √ 2F +y. for charged paricle moving with v = (0, 0, −1) in the −ˆ z 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 ֒ → d2 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 2d2 (rest frame; Sx = 1)

Transverse (Spin) Structure of Hadrons – p.113/138

SLIDE 117

Quark-Gluon Correlations (Interpretation)

Interpretation of d2 with the transverse FSI force in DIS also consistent with ky

⊥ ≡

1

0 dx

d2k⊥ k2

⊥f ⊥ 1T (x, k2 ⊥) in SIDIS (Qiu,

Sterman) ky

⊥ = − 1

2p+

P, S
¯

q(0) ∞ dx−gG+y(x−)γ+q(0)

P, S
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 d2 same as the integrand (for x− = 0) in the QS-integral: ky

⊥ =

∞ 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

SLIDE 118

Quark-Gluon Correlations (Interpretation)

x2-moment of twist-4 polarized PDF g3(x)

dxx2g3(x)
P, S
¯

q(0)g ˜ Gµν(0)γνq(0)

P, S
∼ f2

֒ → different linear combination f2 = χE − χB of χE and χM ֒ → combine with d2 ⇒ disentangle electric and magnetic force What should one expect (sign)? κp

q −

→ signs of deformation (u/d quarks in ±ˆ y direction for proton polarized in +ˆ x direction − → expect force in ∓ˆ y ֒ → d2 positive/negative for u/d quarks in proton large NC: du/p

2

= −dd/p

2

consistent with f ⊥u

1T + f ⊥d 1T ≈ 0

lattice (Göckeler et al.): du

2 ≈ 0.010 and dd 2 ≈ −0.0056

֒ → (M 2 ≈ 5 GeV

fm

F y

u(0) ≈ −50 MeV fm

F y

d (0) ≈ 28 MeV fm

x2-moment of chirally odd twist-3 PDF e(x) − → transverse force on

transversly polarized quark in unpolarized target (↔ Boer-Mulders h⊥

1 )

Transverse (Spin) Structure of Hadrons – p.115/138

SLIDE 119

Summary

GPDs provide decomposition of form factors w.r.t. the momentum

f the active quark

dx− 2π eixp+x− p′

¯

q

−x−

2

γ+q

x− 2

p
GPDs resemble both PDFs and form factors: defined through

matrix elements of light-cone correlator, but ∆ ≡ p′ − p = 0. t-dependence of GPDs at ξ =0 (purely ⊥ momentum transfer) ⇒ Fourier transform of impact parameter dependent PDFs q(x, b⊥) ֒ → knowledge of GPDs for ξ = 0 provides novel information about nonperturbative parton structure of nucleons: distribution of partons in ⊥ plane q(x, b⊥) = d2∆⊥

(2π)2 H(x, 0, −∆2 ⊥)eib⊥·∆⊥

∆q(x, b⊥) = d2∆⊥

(2π)2 ˜

H(x, 0, −∆2

⊥)eib⊥·∆⊥

q(x, b⊥) has probabilistic interpretation, e.g. q(x, b⊥) > 0 for x > 0

Transverse (Spin) Structure of Hadrons – p.116/138

SLIDE 120

Summary

∆⊥ 2M E(x, 0, −∆2 ⊥) describes how the momentum distribution of

unpolarized partons in the ⊥ plane gets transversely distorted when is nucleon polarized in ⊥ direction. (attractive) final state interaction in semi-inclusive DIS converts ⊥ position space asymmetry into ⊥ momentum space asymmetry ֒ → simple physical explanation for observed Sivers effect in γ∗p → πX physical explanation for Boer-Mulders effect: correlation between quark spins (transversity) and currents OAM in QED: Le = Le and ∆γ = Jγ − Lγ d2 ≡ 3

dxx2¯

g2(x): (average) transverse force on quarks in DIS Fy = −M 2d2 (nucleon polarized in +ˆ x-direction) recommended reading: M.B., A.Miller, and W.-D.Nowak, ‘Spin-Polarized High-Energy Scattering of Charged Leptons on Nucleons’, hep-ph/0812.2208

Transverse (Spin) Structure of Hadrons – p.117/138