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Cédric Lorcé

This Second International Summer School session delves into the correlations between partons in nucleons, exploring various quantum field theory concepts and the internal structure of nucleons through multidimensional perspectives. Topics covered include classical mechanics, statistical mechanics, quantum mechanics, and quantum field theory. Gain insights into how nucleon pictures evolve, delve into phase space tomography, and explore the implications of Special Relativity and Lorentz symmetry on nucleon composition.

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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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