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Second International Summer School of the GDR PH-QCD “Correlations between partons in nucleons” N Multidimensional pictures of the nucleon (1/3) Cédric Lorcé IFPA Liège June 30-July 4, 2014, LPT, Paris-Sud University, Orsay, France Outline Lecture 1 • Introduction • Tour in phase space • Galileo vs Lorentz • Photon point of view Introduction 1/33 Structure of matter Atom Nucleus Nucleons Quarks u Up Proton d Down Neutron 10-10m Atomic physics 10-14m Nuclear physics 10-15m Hadronic physics 10-18m Particle physics Introduction 2/33 Elementary particles Matter Interaction Graviton ? Introduction Degrees of freedom « Resolution » Relevant degrees of freedom depend on typical energy scale 3/33 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 Introduction 5/33 Goal : understanding the nucleon internal structure Why ? At the energy frontier LHC Quark & gluon distributions ? Proton Proton Introduction 6/33 Goal : understanding the nucleon internal structure Why ? E.g. At the intensity frontier Electron Nucleon Nucleon spin structure ? Nucleon shape & size ? … 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 Phase space 8/33 Classical Mechanics Momentum State of the system Particles follow well-defined trajectories Position [Gibbs (1901)] Phase space 9/33 Statistical Mechanics Phase-space density Momentum Position-space density Momentum-space density Phase-space average Position [Gibbs (1902)] Phase space 10/33 Quantum Mechanics Wigner distribution Momentum Position-space density Momentum-space density Phase-space average Position [Wigner (1932)] [Moyal (1949)] Phase space 11/33 Quantum Mechanics Momentum Wigner distribution Position Hermitian (symmetric) derivative Fourier conjugate variables [Wigner (1932)] [Moyal (1949)] Phase space Wigner distributions have applications in: 12/33 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 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 3D 2D 2D Binocular vision in phase space ! Find the hidden 3D picture 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)] Phase space Nucleons are composite systems ~ 15/33 Interesting Not so interesting internal motion center-of-mass motion Phase space Localized state in momentum space in position space 16/33 CoM momentum CoM position Phase-space compromise Intrinsic phase-space/Wigner distribution Breit frame Identified with intrinsic variables [Ji (2003)] [Belitsky, Ji, Yuan (2004)] 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)] Galilean symmetry All is fine as long as space-time symmetry is Galilean Position operator can be defined 18/33 Lorentz symmetry 19/33 But in relativity, space-time symmetry is Lorentzian Position operator is ill-defined ! No separation of CoM and internal coordinates Further issues : Lorentz contraction Creation/annihilation of pairs Spoils (quasi-) probabilistic interpretation Forms of dynamics Space-time foliation 20/33 Light-front components « Space » = 3D hypersurface « Time » = hypersurface label Instant-form dynamics Light-front form dynamics Time Space Energy Momentum [Dirac (1949)] Instant form vs light-front form Ordinary point of view t1, z1 t3, z3 t2, z2 t4, z4 t5, z5 t7, z7 t6, z6 21/33 Instant form vs light-front form Photon point of view x+ x+ x+ x+ x+ x+ x+ 22/33 Instant form vs light-front form Initial frame 23/33 Instant form vs light-front form Boosted frame 24/33 Instant form vs light-front form (Quasi) infinite-momentum frame 25/33 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)] 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 ! Non-relativistic phase space Localized state in momentum space in position space 28/33 CoM momentum CoM position Equal-time Wigner operator Intrinsic phase-space/Wigner distribution Identified with intrinsic variables Time translation [Ji (2003)] [Belitsky, Ji, Yuan (2004)] Relativistic phase space 29/33 « CoM » momentum Localized state in momentum space in position space « CoM » position Equal light-front time Wigner operator [Soper (1977)] [Burkardt (2000)] [Burkardt (2003)] [Ji (2003)] [Belitsky, Ji, Yuan (2004)] Intrinsic relativistic phase-space/Wigner distribution Boost invariant ! Space-time translation Normalization [C.L., Pasquini (2011)] 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)] Link with parton correlators 31/33 Phase-space/Wigner distributions are Fourier transforms Space-time translation General parton correlator Link with parton correlators 32/33 Adding spin to the picture Dirac matrix ~ quark polarization Leading twist Link with parton correlators 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 33/33 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