Quantum Phase-space “Tomography” of the Quarks in the Proton

1. Quantum Phase-space “Tomography” of the Quarks in the Proton X. Ji, PRL91, 062001 (2003) A. Belitsky, X.Ji, F. Yuan, hep-ph/0307383

2. 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

3. A Brief Story of the Proton

4. 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

5. 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)

6. 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—oneof the daunting challenges in theoretical physics Clay Math. Inst., Cambridge, MA $1M prize to solve QCD! (E. Witten)

7. 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 ofquark-antiquarkpairs + any number ofgluons • 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

8. 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

9. Elastic Form Factors & Charge Distributions in Space

10. 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 • …

11. k’ q P’ P The Proton Elastic Form Factors • First measured by Hofstadter et al in the mid 1950’s Elastic electron scattering k What does F1,2 tell us about the structure of the nucleon?

12. Sachs Interpretation of Form Factors • According to Sachs, the FT ofGE=F1−τF2 and GM=F1+F2are 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

13. 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) Matrix element In the Breit frame

14. Up-Quark Charge Distribution fm fm

15. 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

16. Feynman Quark Distribution

17. 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

18. Feynman Quark Distributions • Measurable in deep-inelastic scattering • Quark distribution as matrix element in QCD • where ξ± = (ξ0± ξ3)/2 are light-cone coordinates.

19. 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.

20. 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.

21. Present status • GRV, CTEQ, MRS distributions CTEQ6: J. Pumplin et al JHEP 0207, 012 (2002)

22. Quantum Phase-space (Wigner) Distribution

23. 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…

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

25. Simple Harmonic Oscillator N=5 N=0 Husimi distribution: positive definite!

26. Measuring Wigner function of Quantum Light

27. Measuring Wigner function of the Vibrational State in a Molecule

28. Quantum State Tomography of Dissociateng molecules Skovsen et al. (Denmark) PRL91, 090604

29. Quantum Phase-Space Distribution for Quarks

30. Quarks in the Proton • Wigner operator • Wigner distribution: “density” for quarks having position r and 4-momentum k(off-shell) a la Saches 7-dimensional distribtuion No known experiment can measure this!

31. 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. f(r,x)

32. 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 t= – q2  ~ qz

33. 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

34. Charge Density and Current in Phase-space • Quark charge density at fixed x • Quark current at fixed x in a spinning nucleon

35. 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

36. 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

37. How to measure the GPDs? • Compton Scattering • Complicated in general • In the Bjorken limit k’ k • Single quark scattering • Photon wind • Non-invasive surgery • Deeply virtual Compton scattering

38. First Evidence of DVCS HERA ep Collider in DESY, Hamburg Zeus detector

39. 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?

40. Quantum Phase-space Tomography

41. 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

42. Up-Quark Charge Density at x=0.4 z y x

43. Surface of constant charge denstiy

44. Up-Quark Charge Denstiy at x=0.01

45. Surface of Constant Charge Density

46. Up Quark Density at x=0.7

47. Up-Quark Density At x=0.7

48. Surface of Constant Charge Density

49. Charge Denstiy at Negative x

50. Charge Denstiy in the MIT Bag