Outline a brief story of the proton

Outline

• A brief story of the proton

• The elastic form factors and charge distributions in space

• The Feynman quark distributions

• Quantum phase-space (Wigner) distribution

• Wigner distributions of the quarks in the proton

• Quantum Phase-space tomography

• Conclusions

A Brief Story of the Proton

• A Brief Story of the Proton

Protons, protons, everywhere

• The Proton is one of the most abundant particles around us!

• The sun ☼ is almost entirely made of protons...
• And all other stars…
• And all atomic nuclei…

• The profile:

• Spin 1/2, making MRI (NMR) possible
• Mass 938.3 MeV/c2, making up ½ of our body weight
• Charge +1, making a H-atom by attracting an electron

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

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

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

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

• (1fm = 10-13 cm, Nobel prize, 1961)

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

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

QCD and Strong-Interactions

• Building blocks

• Quarks (u,d,s…, spin-1/2, mq ~ small, 3 colors)
• Gluons (spin-1, massless, 32 −1 colors)

• Interactions

• In the low-energy region, it represents an extremely relativistic, strongly coupled, quantum many-body problem—one of the daunting challenges in theoretical physics
• Clay Math. Inst., Cambridge, MA
• $1M prize to solve QCD! (E. Witten)

The Proton in QCD

• We know a lot and we know little

• 2 up quarks (e = 2/3) + 1 down quark (e = −1/3)
• + any number of quark-antiquark pairs
• + any number of gluons

• Fundamental questions (from quarks to cosmos…)

• Origin of mass?
• ~ 90% comes from the motion of quarks & gluons
• ~ l0% from Higgs interactions (Tevertron, LHC)
• Proton spin budget?
• How are Elements formed?
• the protons & neutrons interact to form atomic nuclei

Understanding the Proton

• Solving QCD

• Numerically simulation, like 4D stat. mech. systems
•  Feynman path integral  Wick rotation
•  Spacetime discretization  Monte Carlo simulation
• Effective field theories (large Nc, chiral physics,…)

• Experimental probes

• Study the quark and gluon structure through low and high-energy scattering
• Require clean reaction mechanism
• Photon, electron & perturbative QCD

Elastic Form Factors & Charge Distributions in Space

• Elastic Form Factors & Charge Distributions in Space

Form Factors & Microscopic Structure

• In studying the microscopic structure of matter, the form factor (structure factor) F(q2) is one of the most fundamental observables

• The Fourier Transformation (FT) of the form factor is related to the spatial charge (matter) distributions !

• Examples

• The charge distribution in an atom/molecule
• The structure of crystals
• …

The Proton Elastic Form Factors

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

• Elastic electron scattering

Sachs Interpretation of Form Factors

• According to Sachs, the FT of GE=F1−τF2 and GM=F1+F2 are related to charge and magnetization distributions.

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

• then measure the charge density relative to the center

Sachs Interpretation (Continued)

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

• Additional assumptions

• The wave packet has no dependence on the relative momentum q
• |φ(P)|2 ~ δ(P)

Up-Quark Charge Distribution

Effects of Relativity

• Relativistic effects

• The proton cannot be localized to a distance better than 1/M because of Zitterbewegung
• When the momentum transfer is large, the proton recoils after scattering, generating Lorentz contraction

• The effects are weak if

• 1/(RM) « 1 (R is the radius)

• For the proton, it is ~ 1/4.

• For the hydrogen atom, it is ~ 10-5

Feynman Quark Distribution

• Feynman Quark Distribution

Momentum Distributions

• While the form factors provide the static 3D picture, but they do not yield info about the dynamical motion of the constituents.

• To see this, we need to know the momentum space distributions of the particles.

• This can be measured through single-particle knock-out experiments

• Well-known Examples:

• Nuclear system: quasi-elastic scattering
• Liquid helium & BEC: neutron scattering

Feynman Quark Distributions

• Measurable in deep-inelastic scattering

• Quark distribution as matrix element in QCD

• where ξ± = (ξ 0± ξ 3)/2 are light-cone coordinates.

Infinite Momentum Frame (IMF)

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

• Density of quarks with longitudinal momentum xP (with transverse momentum integrated over)
• “Feynman momentum” x takes value from –1 to 1, Negative x corresponds to antiquark.

Rest-Frame Interpretation

• Quark spectral function

• Probability of finding a quark in the proton with energy E=k0, 3-momentum k, defined in the rest frame of the nucleon
• A concept well-known in many-body physics

• Relation to parton distributions

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

Present status

• GRV, CTEQ, MRS distributions

Quantum Phase-space (Wigner) Distribution

• Quantum Phase-space (Wigner) Distribution

Phase-space Distribution?

• The state of a classical particle is specified by its coordinate and momentum (x,p): phase-space

• A state of classical identical particle system can be described by a phase-space distribution f(x,p). Time evolution of f(x,p) obeys the Boltzmann equation.

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

• Wigner introduced the first phase-space distribution in quantum mechanics (1932)

• Heavy-ion collisions, quantum molecular dynamics, signal analysis, quantum info, optics, image processing…

Wigner function

• Define as

• When integrated over x (p), one gets the momentum (probability) density.
• Not positive definite in general, but is in classical limit.
• Any dynamical variable can be calculated as

Simple Harmonic Oscillator

Measuring Wigner function of Quantum Light

Measuring Wigner function of the Vibrational State in a Molecule

Quantum State Tomography of Dissociateng molecules

Quantum Phase-Space Distribution for Quarks

• Quantum Phase-Space Distribution for Quarks

Quarks in the Proton

• Wigner operator

• Wigner distribution: “density” for quarks having position r and 4-momentum k (off-shell)

Custom-made for high-energy processes

• In high-energy processes, one cannot measure k = (k0–kz) and therefore, one must integrate this out.

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

• After integrating over r, one gets transverse-momentum dependent parton distributions
• Alternatively, after integrating over k, one gets a spatial distribution of quarks with fixed Feynman momentum k+=(k0+kz)=xM.

Proton images at a fixed x

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

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

What is a GPD?

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

• Depends on

• x: fraction of the longitudinal momentum carried

• by parton

• t=q2: t-channel momentum transfer squared

• ξ: skewness parameter

Charge Density and Current in Phase-space

• Quark charge density at fixed x

• Quark current at fixed x in a spinning nucleon

Mass distribution

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

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

• Where A, B and C are gravitational form factors

Spin of the Proton

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

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

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

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

How to measure the GPDs?

• Compton Scattering

• Complicated in general

• In the Bjorken limit

First Evidence of DVCS

Present and Future Experiments

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

• COMPASS at CERN, taking data

• Jefferson Lab 12 GeV upgrade

• DVCS and related processes & hadron spectrocopy

• Electron-ion collider (EIC)

• 2010? RHIC, JLab?

Quantum Phase-space Tomography

• Quantum Phase-space Tomography

A GPD or Wigner Function Model

• A parametrization which satisfies the following Boundary Conditions: (A. Belitsky, X. Ji, and F. Yuan, hep-ph/0307383)

• Reproduce measured Feynman distribution
• Reproduce measured form factors
• Polynomiality condition
• Positivity

• Refinement

• Lattice QCD
• Experimental data

A Mini-Movie

Up Quark Density at x=0.7

Comments

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

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

Conclusions

• Form factors provide the spatial distribution, Feynman distribution provide the momentum-space density. They do not provide any info on space-momentum correlation.

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