1. QCD Map of the Proton Xiangdong Ji University of Maryland

2. Outline  An Alternative Formulation of Quantum Mechanics  Wigner parton distributions (WPD) –mother of all distributions!  Transverse-momentum dependent parton distributions and pQCD factorization  GPD & quantum phase-space tomography  Summary

3. Alternative Formulations of Quantum Mechanics  Quantum mechanical wave functions are not directly measurable in experiment. But is it possible to formula quantum mechanics in terms of observables? –Heisenberg’s matrix mechanics (1925) –Wigner’s phase-space distributions (1932) –Feynman path integrals (1948) –…

4. QM with phase-space distribution  Phase-space formulation is based on the statistical nature of quantum mechanics. –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 Boltzmann equation.  Many identical copies of a quantum system can be described by a similar phase-space distribution.

5. Wigner function  Define as  When integrated over p, one gets the coordinate space density ρ(x)=|ψ(x)| 2 –Measurable in elastic scattering  When integrated over x, one gets the coordinate space density n(p)=|ψ(p)| 2 –Measurable in knock-out scattering  Uncertainty principle  Not positive definite in general. But it is in the classical limit!

6. Wigner Distribution  Wigner distributions are physical observables –Real (hermitian) –Super-observable!  Many applications –heavy-ion collisions, –quantum molecular dynamics, –signal analysis, –quantum information, –optics, –image processing…

7. Simple Harmonic Oscillator N=0 N=5 Phase-space distribution gives a vivid “classical” picture. Non-positive definiteness is the key for quantum interference

8. Phase-space tomography   Phase-space distribution (a map) can be constructed from slices with fixed momentum. – –For small p, the oscillator is at the turning point of the oscillator potential. – –For large p, the oscillator is at the middle of the potential – –For every p, we have a topographic picture of the system which give a much detailed map of the system. This information cannot be obtained from the densities in space or momentum alone!

9. Measuring Wigner function of Quantum Light

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

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

12. Wigner distributions for quarks in proton  Wigner operator (X. Ji,PRL91:062001,2003)  Wigner distribution: “density” for quarks having position r and 4-momentum k  (off-shell) No known experiment can measure this! 7-dimensional distributions a la Saches

13. Custom-made for high-energy processes (I)  In high-energy processes, one cannot measure k  = (k 0 –k z) and therefore, one must integrate this out.  The reduced Wigner distribution is a function of 6 variables [r,k=(k + k  )]. Mother of all SP distributions ! Mother of all SP distributions !  Integrating over z, resulting a phase-space distribution q(x, r  k  ) through which parton saturation at small x is easy to see.

14. Custom-made for high-energy processes (II)  Integrating over r , resulting transverse-momentum dependent (TMD) parton distributions! q(x, k  ) q(x, k  ) Measurable in semi-inclusive DIS & Drell-Yan &.. Measurable in semi-inclusive DIS & Drell-Yan &.. A major subject of this meeting… A major subject of this meeting…  Integrating over k , resulting a reduced Wigner distribution The above are not related by Fourier transformation! q(x,r)

15. Wigner parton distributions & offsprings Mother Dis. W(r,p) q(x, r , k  ) TMDPD q (x, k  ) Reduced wigner dis q(x,r) PDF q(x) Density ρ(r)

16. TMD Parton Distribution  Appear in the processes in which hadron transverse-momentum is measured, often together with TMD fragmentation functions.  The leading-twist ones are classified by Boer, Mulders, and Tangerman (1996,1998) –There are 8 of them q(x, k ┴ ), q T (x, k ┴ ), q(x, k ┴ ), q T (x, k ┴ ), Δq L (x, k ┴ ), Δq T (x, k ┴ ), δq(x, k ┴ ), δ L q(x, k ┴ ), δ T q(x, k ┴ ), δ T’ q(x, k ┴ )

17. Factorization for SIDIS with P ┴  For traditional high-energy process with one hard scale, inclusive DIS, Drell-Yan, jet production,…soft divergences typically cancel, except at the edges of phase-space.  At present, we have two scales, Q and P ┴ (could be soft). Therefore, besides the collinear divergences which can be factorized into TMD parton distributions (not entirely as shown by the energy-dependence), there are also soft divergences which can be taken into account by the soft factor. X. Ji, F. Yuan, and J. P. Ma, PRD71:034005,2005

18. Example I  Vertex corrections Four possible regions of gluon momentum k: 1) k is collinear to p (parton dis) 2) k is collinear to p′ (fragmentation) 3) k is soft (wilson line) 4) k is hard (pQCD correction) p p′p′ q k

19. A general reduced diagram

20. Factorization theorem  For semi-inclusive DIS with small p T ~ Hadron transverse-momentum is generated from multiple sources. The soft factor is universal matrix elements of Wilson lines and spin-independent. One-loop corrections to the hard-factor has been calculated

21. Spin-Dependent processes  Ji, Ma, Yuan, PLB597, 299 (2004); PR  Ji, Ma, Yuan, PLB597, 299 (2004); PRD70:074021(2004)

22. Reduced Wigner Distributions and GPDs  The 4D reduced Wigner distribution f(r,x) is related to Generalized parton distributions (GPD) H and E through simple FT, t= – q 2  ~ q z H,E depend only on 3 variables. There is a rotational symmetry in the transverse plane..

23. What is a GPD?  A proton matrix element which is a hybrid of elastic form factor and Feynman distribution  Distributions depending on x: fraction of the longitudinal momentum carried x: fraction of the longitudinal momentum carried by parton by parton t=q 2 : t-channel momentum transfer squared t=q 2 : t-channel momentum transfer squared ξ: skewness parameter (a new variable coming from selection of a light-cone direction) ξ: skewness parameter (a new variable coming from selection of a light-cone direction) Review: M. Diehl, Phys. Rep. 388, 41 (2003) X. Ji, Ann. Rev. Nucl. Part. Sci. 54, 413 (2004) X. Ji, Ann. Rev. Nucl. Part. Sci. 54, 413 (2004)

24. Charge and Current Distributions in Phase-space  Quark charge distributions at fixed x  Quark current at fixed x in a spinning nucleon

25. A GPD or W-Parton Distribution Model  A parametrization which satisfies the following Boundary Conditions: ( A. Belitsky, X. Ji, and F. Yuan, PRD 69,074014,2004 ) –Reproduce measured Feynman distribution –Reproduce measured form factors –Polynomiality condition –Positivity  Refinement –Lattice QCD –Experimental data

26. Imaging quarks at fixed Feynman-x  For every choice of x, one can use the Wigner distributions to picture the nucleon in 3-space;  For every choice of x, one can use the Wigner distributions to picture the nucleon in 3-space; quantum phase-space tomography! z bxbx byby

27. 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.

28. Impact parameter space distribution  Obtained by integrating over z, (Soper, Burkardt)  x and b are in different directions and therefore, there is no quantum mechanical constraint. –It is a true density –Momentum density in the z-direction –Coordinate density in the transverse plane.

30. QCD-Map: how to obtain it?  Data  Parametrizations  Lattice QCD

31. Mass distribution  Gravity plays an important role in cosmos and at 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

32. 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

33. Spin sum rule  One can calculate the total quark (orbital + spin) contribution to the spin of the proton  Amount of proton angular momentum carried by quarks is

34. Summary  One of the central goals for 12 GeV upgrade is to obtain a QCD map of the proton: DNA sequencing in biology  TMD parton distributions: semi-inclusive processes  Quantum phase-space tomography –Mass and spin of the proton