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Cédric Lorcé

Explore the internal structure of nucleons and the correlations between partons through lectures on phase space, deep inelastic scattering, form factors, generalized parton distributions, and energy-momentum tensor.

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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 (2/3) IFPA Liège June 30-July 4, 2014, LPT, Paris-Sud University, Orsay, France

  2. Reminder 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

  3. Outline • Introduction • Tour in phase space • Galileo vs Lorentz • Photon point of view Lecture 1 • Nucleon 1D picture • Nucleon 2D picture • Nucleon 2+1D picture • Energy-momentum tensor Lecture 2

  4. Deep inelastic scattering 1/31 Coherent scattering

  5. Deep inelastic scattering 2/31 Coherent scattering Incoherent scattering

  6. Deep inelastic scattering 3/31 Coherent scattering Incoherent scattering Hard/perturbative Factorization Soft/non-perturbative

  7. Deep inelastic scattering 4/31 Parton model Probability to find parton with momentum fraction QCD corrections + virtual diagrams Perturbative Process-dependent Non-perturbative Process-independent

  8. Deep inelastic scattering 5/31 Cross-section must not depend on DGLAP evolution equations Splitting functions Non-perturbative

  9. Parton distribution functions (PDFs) 6/31 Optical theorem Final-state cut PDFs PDF correlator Parametrization Constrained by Lorentz and discrete space-time symmetries

  10. Parton distribution functions (PDFs) PDFs Charges 7/31 Vector Quark number Axial Quark helicity Tensor Quark transversity

  11. Elastic scattering 8/31 Diffraction pattern Constructive interference (Bragg’s law) Crystal Scattered amplitude Scatterer distribution Form factor Reconstructed charge distribution Let’s replace the crystal by a nucleon !

  12. Form factors (FFs) 9/31 Pointlike Spatial structure FFs FF correlator Parametrization (electromagnetic case) Sachs FFs

  13. Form factors (FFs) PDFs FFs Charges 10/31 Electric charge Anomalous magnetic moment

  14. Form factors (FFs) 11/31 Longitudinally polarized nucleon Monopole Proton Neutron Negative core does not fit naive picture ! + - [Miller (2007)]

  15. Form factors (FFs) 12/31 Transversely polarized nucleon Monopole Dipole Proton Neutron Apparent electric dipole ! P & T violations ??? - + + No !!! [Carlson, Vanderhaeghen (2008)]

  16. Form factors (FFs) u d d 13/31 Transversely polarized nucleon Origin of EDM : light-front artifact [Burkardt (2003)] Orbital angular momentum Anomalous moment Induced electric moment [Lorcé (2009)]

  17. Compton scattering 14/31 Real Compton Scattering Recoiling target Compton wavelength Question : why are there 2 peaks? Virtual Compton scattering (VCS) interferes with VCS Bethe-Heitler

  18. Generalized parton distributions (GPDs) 15/31 Deeply virtual Compton scattering (DVCS) Compton FF [Collins, Freund (1999)] [Belitsky et al. (2000)] Factorization Evolution @ NLO GPDs GPD correlator Reviews : [Diehl (2003)] [Belitsky, Radyushkin (2005)] [Boffi, Pasquini (2008)]

  19. Generalized parton distributions (GPDs) PDFs FFs GPDs Charges 16/31 Nucleon tomography/imaging [GPD] PDF [FF]

  20. Generalized parton distributions (GPDs) 17/31 [Weiss(2009)] Flat in t Narrow in b Steep in t Wide in b [Guidal et al. (2013)]

  21. Generalized parton distributions (GPDs) 18/31 Explicit Lorentz invariance is broken by the preferred light-front direction Lorentz invariance and dependences constrained by Lorentz invariance !

  22. Generalized parton distributions (GPDs) FFs GPDs 19/31 Lowest moment

  23. Generalized parton distributions (GPDs) 20/31 Generic moment in LF gauge

  24. Generalized parton distributions (GPDs) 21/31 Generic moment using Derivative of Wilson line Covariant derivatives

  25. Generalized parton distributions (GPDs) 22/31 Generic local operator Symmetrization + trace removal Generic FFs Constrained by Lorentz and discrete space-time symmetries Polynomiality of GPDs

  26. Generalized parton distributions (GPDs) 23/31 How to ensure polynomiality property? « Covariantization » Lorentz-invariant variables

  27. Generalized parton distributions (GPDs) 24/31 GPD parametrization Constraint : Double distribution (DD) parametrization Support : [Müller et al. (1994)] [Radyushkin (1999)] [Polyakov, Weiss (1999)]

  28. Generalized parton distributions (GPDs) 25/31 Relation between GPDs and DDs Polynomiality property and dependences not constrained by Lorentz invariance ! [Müller et al. (1994)] [Radyushkin (1999)] [Polyakov, Weiss (1999)]

  29. Energy-momentum tensor 26/31 A lot of interesting physics is contained in the EM tensor [Polyakov, Shuvaev (2002)] [Polyakov (2003)] [Goeke et al. (2007)] [Cebulla et al. (2007)] Energy density Momentum density Shear stress Normal stress (pressure) Momentum flux Energy flux In rest frame

  30. Energy-momentum tensor 27/31 In presence of spin density Belinfante « improvement » Spin density gradient Four-momentum circulation In rest frame No « spin » contribution !

  31. Energy-momentum tensor 28/31 QCD Energy-momentum operator Matrix elements Normalization

  32. Energy-momentum tensor 29/31 Energy-momentum FFs Momentum sum rule Angular momentum sum rule Vanishing gravitomagnetic moment ! [Ji (1997)]

  33. Energy-momentum tensor 30/31 Energy-momentum FFs Non-conserved current Momentum sum rule Angular momentum sum rule Vanishing gravitomagnetic moment ! [Ji (1997)]

  34. Energy-momentum tensor 31/31 Leading-twist component of Link with GPDs Accessible e.g. in DVCS ! [Ji (1997)]

  35. Summary Lecture 2 • PDFs provide 1D pictures of the nucleon • FFs provide 2D pictures of the nucleon • GPDs generalize both PDFs and FFs and give access to the EMT Energy density Momentum density Shear stress [GPD] Normal stress (pressure) Energy flux Momentum flux PDF [FF]

  36. Backup slides

  37. Nuclear charge densities and FFs

  38. Nucleon FFs [Perdrisat et al. (2006)]

  39. Nucleon FFs Pun05 Gay02 green : Rosenbluth data (SLAC, JLab) JLab/HallA recoil polarization data

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