X. Ji, PRL91, 062001 (2003)A. Belitsky, X.Ji, F. Yuan, hep-ph/0307383

OutlineOutline A brief story of the protonA brief story of the proton

The elastic form factors and charge The elastic form factors and charge distributions in spacedistributions in space

The Feynman quark distributions The Feynman quark distributions

Quantum phase-space (Wigner) Quantum phase-space (Wigner) distributiondistribution

Wigner distributions of the quarks in the Wigner distributions of the quarks in the protonproton

Quantum Phase-space tomography Quantum Phase-space tomography

ConclusionsConclusions

A Brief Story of the ProtonA Brief Story of the Proton

Protons, protons, Protons, protons, everywhereeverywhere

The Proton is one of the most abundant The Proton is one of the most abundant particles around us! particles around us! – The sun The sun ☼☼ is almost entirely made of protons... is almost entirely made of protons...

– And all other stars…And all other stars…– And all atomic nuclei…And all atomic nuclei…

The profileThe profile:: – Spin 1/2, Spin 1/2, making MRI (NMR) possiblemaking MRI (NMR) possible

– Mass 938.3 MeV/cMass 938.3 MeV/c22, , making up ½ of our body weightmaking up ½ of our body weight

– Charge +1, Charge +1, making a H-atom by attracting an electronmaking a H-atom by attracting an electron

What’s in A Proton? (Four What’s in A Proton? (Four Nobel Prizes)Nobel Prizes)

It was thought as a point-like particle, like electronIt was thought as a point-like particle, like electron

In 1933, O. Stern measured the magnetic moment In 1933, O. Stern measured the magnetic moment of the proton, finding 2.8of the proton, finding 2.8NN, first evidence that the , first evidence that the proton is not point-like (proton is not point-like (Nobel prize, 1943Nobel prize, 1943))

In 1955, R. Hofstadter measured the charge radius In 1955, R. Hofstadter measured the charge radius of the proton, about 0.8fm.of the proton, about 0.8fm.

(1fm = 10(1fm = 10-13-13 cm, cm, Nobel prize, 1961Nobel prize, 1961))

In 1964, M. Gell-Mann and G. Zweig postulated that In 1964, M. Gell-Mann and G. Zweig postulated that there are three quarks in the proton: two ups and there are three quarks in the proton: two ups and one down (one down (Nobel prize, 1969Nobel prize, 1969))

In 1969, Friedman, Kendall, & Taylor find quarks in In 1969, Friedman, Kendall, & Taylor find quarks in the protonthe proton ( (Nobel prize, 1990Nobel prize, 1990))

QCD and Strong-InteractionsQCD and Strong-Interactions

Building blocksBuilding blocks

– Quarks (u,d,s…, spin-1/2, mQuarks (u,d,s…, spin-1/2, mqq ~ small, 3 colors) ~ small, 3 colors)

– Gluons (spin-1, massless, 3Gluons (spin-1, massless, 32 2 −−1 colors)1 colors)

InteractionsInteractions

In the low-energy region, it represents an In the low-energy region, it represents an extremely extremely relativisticrelativistic, , strongly coupledstrongly coupled, quantum many-body , quantum many-body problem—problem—oneone of the daunting challenges in theoretical physicsof the daunting challenges in theoretical physics

Clay Math. Inst., Cambridge, MAClay Math. Inst., Cambridge, MA

$1M prize to solve QCD! (E. Witten)$1M prize to solve QCD! (E. Witten)

1( )

4a

q a sL i m F F g A

The The ProtonProton in QCD in QCD

We know a lot and we know littleWe know a lot and we know little

2 up quarks (e = 2/3) + 1 down quark (e = 2 up quarks (e = 2/3) + 1 down quark (e = −−1/3)1/3)

+ + any number ofany number of quark-antiquarkquark-antiquark pairspairs

+ any number of+ any number of gluonsgluons

Fundamental questions (Fundamental questions (from quarks to cosmos…from quarks to cosmos…))– Origin of mass?Origin of mass?

~ ~ 90% comes from the motion of quarks & gluons90% comes from the motion of quarks & gluons

~ l0% from Higgs interactions (Tevertron, LHC)~ l0% from Higgs interactions (Tevertron, LHC)– Proton spin budget?Proton spin budget?– How are How are Elements Elements formedformed??

the protons & neutrons interact to form atomic nucleithe protons & neutrons interact to form atomic nuclei

Understanding the ProtonUnderstanding the Proton

Solving QCDSolving QCD– Numerically simulationNumerically simulation, like 4D stat. mech. systems, like 4D stat. mech. systems

Feynman path integral Feynman path integral Wick rotation Wick rotation

Spacetime discretization Spacetime discretization Monte Carlo simulation Monte Carlo simulation

– Effective field theoriesEffective field theories (large N (large Ncc, chiral physics,…), chiral physics,…)

Experimental probesExperimental probes– Study the quark and gluon structure through low and Study the quark and gluon structure through low and

high-energy scatteringhigh-energy scattering

– Require clean reaction mechanism Require clean reaction mechanism

• Photon, electron & perturbative QCDPhoton, electron & perturbative QCD

Elastic Form Factors & Charge Elastic Form Factors & Charge Distributions in SpaceDistributions in Space

Form Factors & Microscopic Form Factors & Microscopic StructureStructure

In studying the microscopic structure of In studying the microscopic structure of matter, the form factor (structure factor) F(qmatter, the form factor (structure factor) F(q22) ) is one of the most fundamental observablesis one of the most fundamental observables– The Fourier Transformation (FT) of the form factor is The Fourier Transformation (FT) of the form factor is

related to the spatial charge (matter) distributions !related to the spatial charge (matter) distributions !

ExamplesExamples– The charge distribution in an atom/molecule The charge distribution in an atom/molecule

– The structure of crystalsThe structure of crystals

– ……

The Proton Elastic Form The Proton Elastic Form FactorsFactors

First measured by Hofstadter First measured by Hofstadter et alet al in the mid in the mid 1950’s 1950’s

Elastic electron scatteringElastic electron scattering

kk’

PP’

q

pUMqiqFqFpUpjp

2'' 2

22

1

What does FWhat does F1,21,2 tell us about the structure of the nucleon? tell us about the structure of the nucleon?

Sachs Interpretation of Form Sachs Interpretation of Form FactorsFactors

According to Sachs, the FT ofAccording to Sachs, the FT of GGEE=F=F11−−ττFF22 and and GGMM=F=F11+F+F22 are related to charge and are related to charge and magnetization distributions.magnetization distributions.

This is obtained by first constructing a wave This is obtained by first constructing a wave packet of the proton (packet of the proton (a spatially-fixed protona spatially-fixed proton))

then measure the charge density relative to the then measure the charge density relative to the centercenter

3

3| ( ) |

(2 )iRPd p

R e p p

0( ) 0 | ( ) | 0r R j r R

Calculate the FT of the charge density, which Calculate the FT of the charge density, which now depends on the wave-packet profilenow depends on the wave-packet profile

Additional assumptionsAdditional assumptions– The wave packet has no dependence on the relative The wave packet has no dependence on the relative

momentum qmomentum q

– ||φφ(P)|(P)|22 ~ ~ δδ(P)(P)

0( ) *( ) ( ) | |2 2 2 2

q q q qF q dP P P P j P

0( ) / 2 | | / 2F q q j q Matrix element In the Breit frame

Sachs Interpretation Sachs Interpretation (Continued)(Continued)

Up-Quark Charge Up-Quark Charge DistributionDistribution

fm

fm

Effects of RelativityEffects of Relativity

Relativistic effectsRelativistic effects– The proton cannot be localized to a distance better than The proton cannot be localized to a distance better than

1/M because of Zitterbewegung1/M because of Zitterbewegung

– When the momentum transfer is large, the proton When the momentum transfer is large, the proton recoils after scattering, generating Lorentz contractionrecoils after scattering, generating Lorentz contraction

The effects are weak ifThe effects are weak if

1/(RM) 1/(RM) « 1 (R is the radius)« 1 (R is the radius)

For the proton, it is ~ 1/4. For the proton, it is ~ 1/4.

For the hydrogen atom, it is ~ 10For the hydrogen atom, it is ~ 10-5-5

Feynman Quark DistributionFeynman Quark Distribution

Momentum DistributionsMomentum Distributions While the form factors provide the static 3D While the form factors provide the static 3D

picture, but they do not yield info about the picture, but they do not yield info about the dynamical motion of the constituentsdynamical motion of the constituents..

To see this, we need to know the To see this, we need to know the momentum momentum space distributionsspace distributions of the particles. of the particles.This can be measured through single-particle knock-out This can be measured through single-particle knock-out

experimentsexperiments Well-known Examples:Well-known Examples:

– Nuclear system: Nuclear system: quasi-elastic scatteringquasi-elastic scattering– Liquid helium & BEC: Liquid helium & BEC: neutron scatteringneutron scattering

Feynman Quark Feynman Quark DistributionsDistributions

Measurable in deep-inelastic scatteringMeasurable in deep-inelastic scattering

Quark distribution as matrix element in QCDQuark distribution as matrix element in QCD

– where where ξξ± ± = (= (ξξ 00± ± ξξ 33)/)/2 are light-cone coordinates.2 are light-cone coordinates.

0

( )1

(0) ( )2 2

ig d Ad

f x P e P

Infinite Momentum Frame Infinite Momentum Frame (IMF)(IMF)

The interpretation is the simplest when the The interpretation is the simplest when the proton travels at the speed of light proton travels at the speed of light (momentum P(momentum P∞). The quantum ∞). The quantum configurations are frozen in time because of configurations are frozen in time because of the Lorentz dilation.the Lorentz dilation.

Density of quarks with longitudinal momentum xPDensity of quarks with longitudinal momentum xP (with (with transverse momentum integrated over) transverse momentum integrated over)

““Feynman momentum” x takes value from –1 to 1, Feynman momentum” x takes value from –1 to 1, Negative x corresponds to antiquark. Negative x corresponds to antiquark.

Rest-Frame InterpretationRest-Frame Interpretation Quark spectral functionQuark spectral function

– Probability of finding a quark in the proton with energy Probability of finding a quark in the proton with energy E=kE=k00, 3-momentum , 3-momentum k, k, defined in the defined in the rest framerest frame of the of the nucleonnucleon

A concept well-known in many-body physicsA concept well-known in many-body physics

Relation to parton distributionsRelation to parton distributions

– Feynman momentum is a linear combination of quark energy Feynman momentum is a linear combination of quark energy and momentum projection in the rest frame. and momentum projection in the rest frame.

4 4 2( ) (2 ) ( ) | | ( ) | |nn

S k P k P n k P

43

4( ) ( ( ) / ) ( )

(2 )

d kf x x E k M S k

Present statusPresent status

GRV, CTEQ, MRS distributionsGRV, CTEQ, MRS distributions

CTEQ6: J. Pumplin et alJHEP 0207, 012 (2002)

Quantum Phase-space Quantum Phase-space (Wigner) Distribution(Wigner) Distribution

Phase-space Distribution?Phase-space Distribution? The state of a classical particle is specified by its The state of a classical particle is specified by its

coordinate and momentum (x,p): coordinate and momentum (x,p): phase-spacephase-space– A state of classical identical particle system can be A state of classical identical particle system can be

described by a phase-space distribution f(x,p). Time described by a phase-space distribution f(x,p). Time evolution of f(x,p) obeys the Boltzmann equation.evolution of f(x,p) obeys the Boltzmann equation.

In quantum mechanics, because of the In quantum mechanics, because of the uncertainty principle, the phase-space uncertainty principle, the phase-space distributions seem useless, but…distributions seem useless, but…

Wigner introduced the first phase-space Wigner introduced the first phase-space distribution in quantum mechanics (1932) distribution in quantum mechanics (1932) – Heavy-ion collisions, quantum molecular dynamics, signal Heavy-ion collisions, quantum molecular dynamics, signal

analysis, quantum info, optics, image processing…analysis, quantum info, optics, image processing…

Wigner functionWigner function

Define as Define as

– When integrated over x (p), one gets the momentum When integrated over x (p), one gets the momentum (probability) density. (probability) density.

– Not positive definite in general, but is in classical limit.Not positive definite in general, but is in classical limit.

– Any dynamical variable can be calculated as Any dynamical variable can be calculated as

),(),(),( pxWpxdxdpOpxO

Short of measuring the wave function, the Wigner functioncontains the most complete (one-body) info about a quantum system.

Simple Harmonic OscillatorSimple Harmonic Oscillator

Husimi distribution: positive definite!Husimi distribution: positive definite!

N=0 N=5

Measuring Wigner function Measuring Wigner function of Quantum Lightof Quantum Light

Measuring Wigner function Measuring Wigner function of the Vibrational State in a of the Vibrational State in a

MoleculeMolecule

Quantum State Tomography of Quantum State Tomography of Dissociateng moleculesDissociateng molecules

Skovsen et al. Skovsen et al. (Denmark) PRL91, 090604(Denmark) PRL91, 090604

Quantum Phase-Space Quantum Phase-Space Distribution for QuarksDistribution for Quarks

Quarks in the ProtonQuarks in the Proton

Wigner operator Wigner operator

Wigner distribution: “Wigner distribution: “densitydensity” for quarks ” for quarks having having position position rr and 4-momentum k and 4-momentum k (off-(off-shell)shell)

No known experiment can measure this!7-dimensional distribtuion

a la Saches

Custom-made for high-Custom-made for high-energy processes energy processes

In high-energy processes, one cannot measure In high-energy processes, one cannot measure kk = (k = (k00–k–kz)z) and therefore, one must integrate and therefore, one must integrate this out. this out.

The reduced Wigner distribution is a function The reduced Wigner distribution is a function of six variables [of six variables [r,k=(r,k=(kk++ kk)]. )].

– After integrating over After integrating over r, r, one gets one gets transverse-momentum transverse-momentum dependent parton distributionsdependent parton distributions

– Alternatively, after integrating over Alternatively, after integrating over kk, one gets a , one gets a

spatial distribution of quarks with fixed Feynman spatial distribution of quarks with fixed Feynman momentummomentum k k++=(k=(k00+k+kzz)=xM. )=xM.

f(r,x)

Proton images at a fixed xProton images at a fixed x

For every choice of x, one can use the Wigner For every choice of x, one can use the Wigner distribution to picture the nucleon; distribution to picture the nucleon; This is This is analogous to viewing the proton through the analogous to viewing the proton through the x x (momentum(momentum) filters!) filters!

The distribution is related to The distribution is related to Generalized Generalized parton distributionsparton distributions (GPD) (GPD) through through

t= – q2

~ qz

What is a GPD?What is a GPD?

A proton matrix element which is a hybrid of A proton matrix element which is a hybrid of elastic form factor and Feynman distributionelastic form factor and Feynman distribution

Depends on Depends on

xx: : fraction of the longitudinal momentum fraction of the longitudinal momentum carried carried

by partonby parton

t=qt=q22: : t-channel momentum transfer squaredt-channel momentum transfer squared

ξξ: : skewness parameterskewness parameter

Charge Density and Current Charge Density and Current in Phase-spacein Phase-space

Quark charge density at fixed xQuark charge density at fixed x

Quark current at fixed x in a spinning nucleonQuark current at fixed x in a spinning nucleon

Mass distributionMass distribution

Gravity plays important role in cosmos and Gravity plays important role in cosmos and Plank scale. In the atomic world, the gravity is Plank scale. In the atomic world, the gravity is too weak to be significant (old view).too weak to be significant (old view).

The phase-space quark distribution allows to The phase-space quark distribution allows to determine the determine the mass distributionmass distribution in the proton in the proton by integrating over x-weighted density, by integrating over x-weighted density,

– Where A, B and C are gravitational form factorsWhere A, B and C are gravitational form factors

Spin of the ProtonSpin of the Proton

Was thought to be carried by the spin of the Was thought to be carried by the spin of the three valence quarksthree valence quarks

Polarized deep-inelastic scattering found that Polarized deep-inelastic scattering found that only 20-30% are in the spin of the quarks.only 20-30% are in the spin of the quarks.

Integrate over the x-weighted phase-space Integrate over the x-weighted phase-space current, one gets the current, one gets the momentum currentmomentum current

One can calculate the total quark (orbital + One can calculate the total quark (orbital + spin) contribution to the spin of the protonspin) contribution to the spin of the proton

How to measure the GPDs?How to measure the GPDs? Compton Scattering Compton Scattering

– Complicated in generalComplicated in general

In the Bjorken limitIn the Bjorken limit

kk’

• Single quark scattering• Photon wind• Non-invasive surgery• Deeply virtual Compton scattering

First Evidence of DVCSFirst Evidence of DVCS

HERA ep Collider inDESY, Hamburg

Zeus detector

Present and Future Present and Future ExperimentsExperiments

HERMES Coll. in DESY and CLAS Coll. in HERMES Coll. in DESY and CLAS Coll. in Jefferson Lab has made further measurements Jefferson Lab has made further measurements of DVCS and related processes.of DVCS and related processes.

COMPASS at CERN, taking dataCOMPASS at CERN, taking data

Jefferson Lab 12 GeV upgradeJefferson Lab 12 GeV upgrade– DVCS and related processes & hadron spectrocopyDVCS and related processes & hadron spectrocopy

Electron-ion collider (EIC)Electron-ion collider (EIC)– 2010? RHIC, JLab? 2010? RHIC, JLab?

Quantum Phase-Quantum Phase-space Tomographyspace Tomography

A GPD or Wigner Function A GPD or Wigner Function ModelModel

A parametrization which satisfies the A parametrization which satisfies the following following Boundary Conditions: Boundary Conditions: (A. Belitsky, (A. Belitsky, X. Ji, and F. Yuan, hep-ph/0307383)X. Ji, and F. Yuan, hep-ph/0307383)– Reproduce measured Feynman distributionReproduce measured Feynman distribution

– Reproduce measured form factorsReproduce measured form factors

– Polynomiality condition Polynomiality condition

– PositivityPositivity

RefinementRefinement– Lattice QCDLattice QCD

– Experimental dataExperimental data

x

y

z

Up-Quark Charge Density at Up-Quark Charge Density at x=0.4x=0.4

Surface of constant charge Surface of constant charge denstiydenstiy

Up-Quark Charge Denstiy at Up-Quark Charge Denstiy at x=0.01x=0.01

Surface of Constant Charge Surface of Constant Charge DensityDensity

Up Quark Density at x=0.7Up Quark Density at x=0.7

Up-Quark Density At x=0.7Up-Quark Density At x=0.7

Surface of Constant Charge Surface of Constant Charge DensityDensity

Charge Denstiy at Negative xCharge Denstiy at Negative x

Charge Denstiy in the MIT Charge Denstiy in the MIT BagBag

CommentsComments

If one puts the pictures at all x together, one If one puts the pictures at all x together, one gets a spherically round nucleon! (Wigner-gets a spherically round nucleon! (Wigner-Eckart theorem)Eckart theorem)

If one integrates over the distribution along the If one integrates over the distribution along the z direction, one gets the 2D impact parameter z direction, one gets the 2D impact parameter space pictures of Burkardt and Soper.space pictures of Burkardt and Soper.

ConclusionsConclusions Form factors provide theForm factors provide the spatialspatial distribution, distribution,

Feynman distribution provide the Feynman distribution provide the momentum-momentum-spacespace density. They do not provide any info on density. They do not provide any info on space-momentum correlation.space-momentum correlation.

The quark and gluon Wigner distributions are The quark and gluon Wigner distributions are the the correlated momentum & coordinatecorrelated momentum & coordinate distributions, allowing us to picture the proton distributions, allowing us to picture the proton at every Feynman x, and are at every Feynman x, and are measurable! measurable!