GPDs, part I

QCD&Hadrons

LC formalism

FFs

PDFs

DIS

NPDs

Models

Generalized Parton DistributionsA.V. Radyushkin

Physics Department, Old Dominion University&

Theory Center, Jefferson Lab

GPDs, part I

QCD&Hadrons

LC formalism

FFs

PDFs

DIS

NPDs

Models

QCD as Fundamental Physics

Example of fundamental physics:search for Higgs boson

Motivation: HB is responsible for mass generation

Observation: By far the largest part of visible mass around usis due to nucleons, mN/me ∼ 2000

⇒ From 940 MeV of mN , only < 30 MeV (currentquark masses) may be related to Higgs particle

⇒ Most part of mass is due to gluons,which are massless!

Situation in hadronic physics:

All relevant fundamental particlesestablished

QCD Lagrangian is known

Need to understand how QCD works

GPDs, part I

QCD&Hadrons

LC formalism

FFs

PDFs

DIS

NPDs

Models

Hadrons in Terms of Quarks and Gluons

How to relate hadronic states |p, s〉to quark and gluon fields q(z1) , q(z2) , . . . ?

Standard way: use matrix elements

〈 0 | qα(z1) qβ(z2) |M(p), s 〉 , 〈 0 | qα(z1) qβ(z2) qγ(z3)|B(p), s 〉

Can be interpreted as hadronic wave functions

GPDs, part I

QCD&Hadrons

LC formalism

FFs

PDFs

DIS

NPDs

Models

Light-cone formalism

Describe hadron by Fock components ininfinite-momentum frame

For nucleon

|P 〉 = ψqqq|q(x1P, k1⊥) q(x2P, k2⊥) q(x3P, k3⊥)〉+ ψqqqG|qqqG〉+ ψqqqqq|qqqqq〉+ ψqqqGG|qqqGG〉+ . . .

xi : momentum fractions∑i

xi = 1

ki⊥: transverse momenta∑i

ki⊥ = 0

GPDs, part I

QCD&Hadrons

LC formalism

FFs

PDFs

DIS

NPDs

Models

Basics of LC Formalism

Take a fast-moving hadron with momentum p inz-direction:

pz →∞ , E =√p2z +m2 →∞ , E2−p2

z = m2 � p2z, E

2

p2 in components:

p2 = p20 − p2

z = (p0 + pz)(p0 − pz) ≡ p+p−

Note: “Plus" component p+ is large, “minus” componentp− = m2/p+ is smallMomentum p is close to a “light cone" vector P , forwhich P− is zero, and P 2 = 0

pµ = Pµ + p2nµ + 0⊥

Light-cone basis vectors:

P 2 = 0 , n2 = 0 , 2(Pn) = 1

GPDs, part I

QCD&Hadrons

LC formalism

FFs

PDFs

DIS

NPDs

Models

Two-particle system in LC Formalism

Take a qq component of a meson, with quarks havingmomenta k and p− k:

k =xP + yn+ k⊥

p− k =(1− x)P + (p2 − y)n− k⊥If both k2 = 0 and (p− k)2 = 0, we have

0 =k2 = xy − k2⊥ ⇒ y = k2

⊥/x

0 =(p− k)2 = (1− x)(p2 − y)− k2⊥

⇒(1− x)(p2 − k2⊥/x)− k2

⊥ = 0

⇒p2 = k2⊥/x+ k2

⊥/(1− x)

For n-body component, the ‘light-cone” energy isn∑i=1

k2⊥i/xi

GPDs, part I

QCD&Hadrons

LC formalism

FFs

PDFs

DIS

NPDs

Models

Problems of LC Formalism

In principle: Solving bound-state equation

H|P 〉 = E|P 〉

one gets |P 〉 which gives complete information abouthadron structureIn practice: Equation (involving infinite number of Fockcomponents) has not been solved and is unlikely to besolved in near futureExperimentally: LC wave functions are not directlyaccessibleWay out: Description of hadron structure in terms ofphenomenological functions

GPDs, part I

QCD&Hadrons

LC formalism

FFs

PDFs

DIS

NPDs

Models

Phenomenological Functions

“Old” functions:Form FactorsUsual Parton DensitiesDistribution Amplitudes

“New” functions:GeneralizedParton Distributions(GPDs)

GPDs = Hybrids ofForm Factors, Parton Densities andDistribution Amplitudes

“Old” functionsare limiting cases of “new” functions

GPDs, part I

QCD&Hadrons

LC formalism

FFs

PDFs

DIS

NPDs

Models

Form Factors

Form factors are defined through matrix elements

of electromagnetic and weak currents between hadronic states

Nucleon EM form factors:

〈 p′, s′ | Jµ(0) | p, s 〉 = u(p′, s′)[γµF1(t) + ∆νσµν

2mNF2(t)

]u(p, s)

(∆ = p− p′, t = ∆2)

Electromagnetic currentJµ(z) =

∑f(lavor) ef ψf (z)γµψf (z)

Helicity non-flip form factorF1(t) =

∑f efF1f (t)

Helicity flip form factorF2(t) =

∑f efF2f (t)

GPDs, part I

QCD&Hadrons

LC formalism

FFs

PDFs

DIS

NPDs

Models

Form Factors, contd.

Limiting Values:Electric charge

F1(t = 0) = eN =∑Nf

f ef

Anomalous magnetic momentF2(t = 0) = κN ≡

∑Nff κf

Nf : number of valence quarks of flavor f

Form Factors are measurablethrough elastic eN scattering

p1 p2

q

FF

k1 k2

GPDs, part I

QCD&Hadrons

LC formalism

FFs

PDFs

DIS

NPDs

Models

Form Factors and Charge Densities

Example of a density in Quantum Mechanics

ρ(z) = ψ∗(z)ψ(z)

Its Fourier transform gives form factor:

F (q) =

∫eiqzρ(z) d~z =

∫Ψ∗(k + q)Ψ(k) d~k

where Ψ(k) is Fourier transform of ψ(z)

Transition from initial state with momentum k to finalstate with momentum k + q

Momentum transfer q is due to a probing currentFourier transform of the form factor gives density

ρ(z) =

∫e−iqzF (q)d~q/(2π)3

GPDs, part I

QCD&Hadrons

LC formalism

FFs

PDFs

DIS

NPDs

Models

Form Factors in LC Formalism

Transition of 2-body system with momentum p into 2-bodysystem with momentum p′ = p+ q in a frame, where q+ = 0:

Initial momenta {xP+, k⊥} , {(1− x)P+,−k⊥}Final momenta {xP+, k⊥ + q⊥} , {(1− x)P+,−k⊥}Momentum of final active quark in terms of final momentumxP+ + k⊥ + q⊥ = x(P+ + q⊥) + k⊥ + (1− x)q⊥

Drell-Yan formula for form factor:

F (q2⊥) =

∫ 1

0dx

∫Ψ∗p′(x, k⊥ + (1− x)q⊥)Ψp(x, k⊥) dk⊥

≡∫ 1

0dxF(x, q2

⊥)

where F(x, q2⊥) is “nonforward parton density”

GPDs, part I

QCD&Hadrons

LC formalism

FFs

PDFs

DIS

NPDs

Models

Usual Parton Densities

Parton Densities are defined throughforward matrix elementsof quark/gluon fields separated bylightlike distances

z/2!z/2

p p

Unpolarized quarks case:

〈 p | ψa(−z/2)γµψa(z/2) | p 〉∣∣z2=0

= 2pµ∫ 1

0

[e−ix(pz)fa(x)− eix(pz)fa(x)

]dx

Momentum spaceinterpretation

xpxp

pp

fa(a)(x) isprobability

to find a (a) quarkwith momentum xp

Local limit z = 0

⇒ sum rule∫ 1

0[fa(x)− fa(x)] dx = Na

for valence quarknumbers

GPDs, part I

QCD&Hadrons

LC formalism

FFs

PDFs

DIS

NPDs

Models

Deep Inelastic Scattering

Classic process to access usual parton densities:deep inelastic scattering γ∗N → X

Spacelike momentumtransfer q2 ≡ −Q2 Im

1

(q + xp)2≈ π

2(pq)δ(x− xBj)

Bjorken variable: xBj = Q2

2(pq)

DIS measures f(xBj)Comparing to form factors:point vertex instead of quarkpropagator and p 6= p′

GPDs, part I

QCD&Hadrons

LC formalism

FFs

PDFs

DIS

NPDs

Models

Parton Densities in LC Formalism

Drell-Yan formula for parton density:

f(x) =

∫Ψ∗p′(x, k⊥)Ψp(x, k⊥) dk⊥

=F(x, q2⊥ = 0)

Usual parton density is a limiting case of(general) nonforward parton density

f(x) = F(x, q2⊥ = 0)

GPDs, part I

QCD&Hadrons

LC formalism

FFs

PDFs

DIS

NPDs

Models

Nonforward Parton Densities(Zero Skewness GPDs)

Combine form factors withparton densities

F1(t) =∑a

F1a(t)

F1a(t) =

∫ 1

0F1a(x, t) dx

Flavor components of form factors

F1a(x, t) ≡ ea[Fa(x, t)−Fa(x, t)]

Forward limit t = 0

Fa(a)(x, t = 0) = fa(a)(x)

GPDs, part I

QCD&Hadrons

LC formalism

FFs

PDFs

DIS

NPDs

Models

Interplay between x and tdependences

Simplest factorized ansatz

Fa(x, t) = fa(x)F1(t)satisfies both forward andlocal constraints

Forward constraintFa(x, t = 0) = fa(x)

Local constraint∫ 10 [Fa(x, t)−Fa(x, t)]dx = F1a(t)

Reality is more complicated:LC wave function withGaussian k⊥ dependence

Ψ(xi, ki⊥) ∼ exp[− 1λ2∑

ik2i⊥xi

]suggests

Fa(x, t) = fa(x)ext/2xλ2

fa(x)=experimental densities

Adjusting λ2 to provide

〈k2⊥〉 ≈ (300MeV)2

GPDs, part I

QCD&Hadrons

LC formalism

FFs

PDFs

DIS

NPDs

Models

Nonforward Parton Densities inGaussian LC Model

Take Drell-Yan formula for NPD:

F(x, q2⊥) =

∫Ψ∗p′(x, k⊥ + (1− x)q⊥)Ψp(x, k⊥) dk⊥

with Gaussian LC wave functions:

Ψ(x, k⊥) = ϕ(x) exp

[− k2

⊥λ2x(1− x)

]

The result is

F(x, q2⊥) =

π

2λ2 x(1− x)ϕ2(x) exp

[−(1− x)q2

⊥2xλ2

]=f(x) exp

[−(1− x)q2

⊥2xλ2

]

GPDs, part I

QCD&Hadrons

LC formalism

FFs

PDFs

DIS

NPDs

Models

Impact Parameter Distributions

NPDs can be treated as Fourier transforms ofimpact parameter b⊥ distributions fa(x, b⊥)

Fa(x, q2⊥) =

∫fa(x, b⊥)ei(q⊥b⊥)d2b⊥

b⊥ = ⊥ distance to center of momentum

IPDs describe nucleonstructure in transverse plane

Distribution fpu(x, b⊥)

GPDs, part I

QCD&Hadrons

LC formalism

FFs

PDFs

DIS

NPDs

Models

Impact Parameter Distributions inGaussian LC Model

Invert Definition of IPD

f(x, b⊥) =

∫F(x, q2

⊥)d2q⊥(2π)2

Using Gaussian Model for NPD:

F(x, q2⊥) =f(x) exp

[−(1− x)q2

⊥2xλ2

]

The result is

f(x, b⊥) =λ2 x

2π(1− x)f(x) exp

[− xb2⊥λ

2

2(1− x)

]

GPDs, part I

QCD&Hadrons

LC formalism

FFs

PDFs

DIS

NPDs

Models

Shape of Impact Parameter Distributionsin Gaussian LC Model

Using the result

f(x, b⊥) =λ2 x

2π(1− x)f(x) exp

[− xb2⊥λ

2

2(1− x)

]

We obtain the dependence of b⊥ profile on x

〈b2⊥(x)〉 ≡(∫

b2⊥ f(x, b⊥) d2b⊥

)/(∫f(x, b⊥) d2b⊥

)=

2

λ2

1− xx

⇒Width goes to zero when x→ 1

GPDs, part I

QCD&Hadrons

LC formalism

FFs

PDFs

DIS

NPDs

Models

Probabilistic Interpretation of IPDs

Distribution in b⊥ plane Distribution in z momentum Combined (x, b⊥) distribution

Defining “center” in ⊥ plane:

Geometric center:∑i bi⊥ = 0

Center of momentum:∑i xibi⊥ = 0

Shape of (x, b⊥) distribution:

Shrinks when x→ 1: leading parton

determines center of momentum

GPDs, part I

QCD&Hadrons

LC formalism

FFs

PDFs

DIS

NPDs

Models

Regge-type models for NPDs

“Regge” improvement:

f(x) ∼ x−α(0)

⇒ F(x, t) ∼ x−α(t)

⇒ F(x, t) = f(x)x−α′t

Accomodating quarkcounting rules:

F(x, t) = f(x)x−α′t(1−x)|x→1

∼ f(x)eα′(1−x)2t

Does not change small-x behavior but provides

f(x)|x→1 vs. F (t)|t→∞ interplay:f(x) ∼ (1− x)n ⇒ F1(t) ∼ t−(n+1)/2

Note: no pQCD involved in these counting rules!

Extra 1/t for F2(t)

can be produced by takingEa(x, t) ∼ (1− x)2Fa(x, t)

for “magnetic” NPDs

More general:

Ea(x, t) ∼ (1−x)ηa Fa(x, t)Fit : ηu = 1.6 , ηd = 1

GPDs, part I

QCD&Hadrons

LC formalism

FFs

PDFs

DIS

NPDs

Models

Fit to All Four NucleonGE,M Form Factors

GPDs, part I

QCD&Hadrons

LC formalism

FFs

PDFs

DIS

NPDs

Models

Fit to F1,2 Proton Form Factors

Modified Regge parametrization describes JLab polarizationtransfer data on GpE/G

pM and F p2 /F

p1

Similar model was constructed by Diehl et al. (right figure)

GPDs, part I

QCD&Hadrons

LC formalism

FFs

PDFs

DIS

NPDs

Models

Summary

1 Why Study QCD and Hadronic Structure

2 Light-cone formalism

3 Form Factors

4 Usual Parton Densities

5 Deep Inelastic Scattering

6 Nonforward Parton Densities

7 Regge-type models for NPDs