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Cédric Lorcé

Second International Summer School of the GDR PH-QCD “Correlations between partons in nucleons”. N. Cédric Lorcé. Multidimensional pictures of the nucleon (1/3). IFPA Liège. June 30-July 4, 2014, LPT, Paris-Sud University, Orsay, France. Outline. Introduction Tour in phase space

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Cédric Lorcé

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  1. Second International Summer School of the GDR PH-QCD “Correlations between partons in nucleons” N Cédric Lorcé Multidimensional pictures of the nucleon (1/3) IFPA Liège June 30-July 4, 2014, LPT, Paris-Sud University, Orsay, France

  2. Outline • Introduction • Tour in phase space • Galileo vs Lorentz • Photon point of view Lecture 1

  3. Introduction 1/33 Structure of matter Atom Nucleus Nucleons Quarks u d Up Proton Down Neutron 10-10m 10-14m 10-15m 10-18m Atomic physics Nuclear physics Hadronic physics Particle physics

  4. Introduction 2/33 Elementary particles Matter Interaction Graviton ?

  5. Introduction 3/33 Degrees of freedom « Resolution » Relevant degrees of freedom depend on typical energy scale

  6. Introduction 4/33 Nucleon pictures « Naive » Realistic 3 non-relativistic heavy quarks Indefinite # of relativistic light quarks and gluons MN ~ Σ mconst MN = Σ mconst(~2%) + Eint (~98% !) Brout-Englert-Higgs mechanism QCD Quantum Mechanics + Special Relativity = Quantum Field Theory

  7. Introduction 5/33 Goal : understanding the nucleon internal structure Why ? At the energy frontier LHC Quark & gluon distributions ? Proton Proton

  8. Introduction 6/33 Goal : understanding the nucleon internal structure Why ? E.g. At the intensity frontier Electron Nucleon Nucleon spin structure ? Nucleon shape & size ? …

  9. Introduction 7/33 Goal : understanding the nucleon internal structure How ? Deep Inelastic Scattering Elastic Scattering Proton « tomography » Semi-Inclusive DIS Deeply Virtual Compton Scattering Phase-space distribution

  10. Phase space 8/33 Classical Mechanics State of the system Momentum Particles follow well-defined trajectories Position [Gibbs (1901)]

  11. Phase space 9/33 Statistical Mechanics Phase-space density Position-space density Momentum Momentum-space density Phase-space average Position [Gibbs (1902)]

  12. Phase space 10/33 Quantum Mechanics Wigner distribution Position-space density Momentum Momentum-space density Phase-space average Position [Wigner (1932)] [Moyal (1949)]

  13. Phase space 11/33 Quantum Mechanics Wigner distribution Momentum Position Hermitian (symmetric) derivative [Wigner (1932)] [Moyal (1949)] Fourier conjugate variables

  14. Phase space 12/33 Wigner distributions have applications in: Harmonic oscillator • Nuclear physics • Quantum chemistry • Quantum molecular dynamics • Quantum information • Quantum optics • Classical optics • Signal analysis • Image processing • Quark-gluon plasma • … Quasi-probabilistic Heisenberg’s uncertainty relations

  15. Phase space 13/33 In quantum optics, Wigner distributions are « measured » using homodyne tomography [Lvovski et al. (2001)] [Bimbard et al. (2014)] Idea : measuring projections of Wigner distributions from different directions Find the hidden 3D picture 3D 2D 2D Binocular vision in phase space !

  16. Phase space 14/33 Quantum Field Theory Covariant Wigner operator Time ordering ? Scalar fields Equal-time Wigner operator Phase-space/Wigner distribution [Carruthers, Zachariasen (1976)] [Ochs, Heinz(1997)]

  17. Phase space 15/33 Interesting Not so interesting Nucleons are composite systems ~ internal motion center-of-mass motion

  18. Phase space 16/33 CoM momentum Localized state in momentum space in position space CoM position Phase-space compromise Intrinsic phase-space/Wigner distribution Breit frame Identified with intrinsic variables [Ji (2003)] [Belitsky, Ji, Yuan (2004)]

  19. Phase space 17/33 CoM momentum Localized state in momentum space in position space CoM position Phase-space compromise Intrinsic phase-space/Wigner distribution Breit frame Same energy ! Identified with intrinsic variables Time translation [Ji (2003)] [Belitsky, Ji, Yuan (2004)]

  20. Galilean symmetry 18/33 All is fine as long as space-time symmetry is Galilean Position operator can be defined

  21. Lorentz symmetry 19/33 But in relativity, space-time symmetry is Lorentzian No separation of CoM and internal coordinates Position operator is ill-defined ! Further issues : Creation/annihilation of pairs Lorentz contraction Spoils (quasi-) probabilistic interpretation

  22. Forms of dynamics 20/33 Space-time foliation Light-front components « Space » = 3D hypersurface « Time » = hypersurface label Light-front form dynamics Instant-form dynamics Time Space Energy Momentum [Dirac (1949)]

  23. Instant form vs light-front form 21/33 Ordinary point of view t1, z1 t3, z3 t2, z2 t4, z4 t5, z5 t6, z6 t7, z7

  24. Instant form vs light-front form 22/33 Photon point of view x+ x+ x+ x+ x+ x+ x+

  25. Instant form vs light-front form 23/33 Initial frame

  26. Instant form vs light-front form 24/33 Boosted frame

  27. Instant form vs light-front form 25/33 (Quasi) infinite-momentum frame

  28. Light-front operators 26/33 Transverse space-time symmetry is Galilean Transverse position operator can be defined ! Longitudinal momentum plays the role of mass in the transverse plane Transverse boost [Kogut, Soper (1970)]

  29. Quasi-probabilistic interpretation 27/33 What about the further issues with Special Relativity ? Transverse boosts are Galilean No transverse Lorentz contraction ! No sensitivity to longitudinal Lorentz contraction ! Particle number is conserved in Drell-Yan frame Drell-Yan frame is conserved and positive Conclusion : quasi-probabilistic interpretation is possible !

  30. Non-relativistic phase space 28/33 CoM momentum Localized state in momentum space in position space CoM position Equal-time Wigner operator Intrinsic phase-space/Wigner distribution Identified with intrinsic variables Time translation [Ji (2003)] [Belitsky, Ji, Yuan (2004)]

  31. Relativistic phase space 29/33 « CoM » momentum Localized state in momentum space in position space « CoM » position [Soper (1977)] [Burkardt (2000)] [Burkardt (2003)] Equal light-front time Wigner operator [Ji (2003)] [Belitsky, Ji, Yuan (2004)] Intrinsic relativistic phase-space/Wigner distribution Boost invariant ! Space-time translation Normalization [C.L., Pasquini (2011)]

  32. Relativistic phase space 30/33 Our intuition is instant form and not light-front form NB : is invariant under light-front boosts It can be thought as instant form phase-space/Wigner distribution in IMF ! Transverse momentum Longitudinal momentum Transverse position 2+3D In IMF, the nucleon looks like a pancake [C.L., Pasquini (2011)]

  33. Link with parton correlators 31/33 Phase-space/Wigner distributions are Fourier transforms Space-time translation General parton correlator

  34. Link with parton correlators 32/33 Adding spin to the picture Dirac matrix ~ quark polarization Leading twist

  35. Link with parton correlators 33/33 Adding spin and color to the picture Wilson line Gauge transformation Very very very complicated object not directly measurable Let’s look for simpler measurable correlators

  36. Summary Lecture 1 • Understanding nucleon internal structure is essential • Concept of phase space can be generalized to QM and QFT • Relativistic effects force us to abandon 1D in phase space Transverse momentum Transverse position 2+3D

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