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Multidimensional View of Nucleon Structures: Summer School Discussions and Lectures

Dive into the depths of nucleon structures with discussions and lectures including topics on nucleon internal views, deep inelastic scattering, parton models, form factors, and more. Join the Second International Summer School of the GDR PH-QCD to explore correlations between partons in nucleons. Learn about q quantum mechanics and quantum field theory as we navigate multidimensional pictures of the nucleon at IFPA Liège, June 30 - July 4, 2014, LPT at Paris-Sud University, Orsay, France.

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Multidimensional View of Nucleon Structures: Summer School Discussions and Lectures

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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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