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May , JLab, Newport News, VA, USA

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1 May 7 2013, JLab, Newport News, VA, USA
The proton spin decomposition: Path dependence and gauge symmetry Cédric Lorcé IPN Orsay - LPT Orsay May , JLab, Newport News, VA, USA

2 The decompositions in a nutshell
Jaffe-Manohar (1990) Lq Sq Lg Sg

3 The decompositions in a nutshell
Jaffe-Manohar (1990) Ji (1997) Lq Lq Sq Sq Lg Sg Jg

4 Gauge-invariant extension (GIE)
The decompositions in a nutshell Jaffe-Manohar (1990) Ji (1997) Lq Lq Sq Sq Lg Sg Jg Chen et al. (2008) Lq Sq Lg Sg Gauge-invariant extension (GIE)

5 Gauge-invariant extension (GIE)
The decompositions in a nutshell Jaffe-Manohar (1990) Ji (1997) Lq Lq Sq Sq Lg Sg Jg Chen et al. (2008) Wakamatsu (2010) Lq Lq Sq Sq Lg Sg Lg Sg Gauge-invariant extension (GIE)

6 Gauge-invariant extension (GIE)
The decompositions in a nutshell Canonical Kinetic Jaffe-Manohar (1990) Ji (1997) Lq Lq Sq Sq Lg Sg Jg Chen et al. (2008) Wakamatsu (2010) Lq Lq Sq Sq Lg Lg Sg Sg Gauge-invariant extension (GIE)

7 The Chen et al. approach [Chen et al. (2008,2009)]
[Wakamatsu (2010,2011)]

8 The Chen et al. approach [Chen et al. (2008,2009)]
[Wakamatsu (2010,2011)] Gauge transformation

9 The Chen et al. approach [Chen et al. (2008,2009)]
[Wakamatsu (2010,2011)] Gauge transformation Pure-gauge covariant derivatives

10 The Chen et al. approach [Chen et al. (2008,2009)]
[Wakamatsu (2010,2011)] Gauge transformation Pure-gauge covariant derivatives Field strength

11 The canonical formalism
[C.L. (2013)] Textbook Lagrangian Dynamical variables

12 The canonical formalism
[C.L. (2013)] Textbook Lagrangian Dynamical variables Gauge covariant

13 The canonical formalism
[C.L. (2013)] Textbook Lagrangian Dynamical variables Gauge covariant Gauge invariant Dirac variables [Dirac (1955)] [Mandelstam (1962)] Dressing field Gauge transformation

14 The analogy with General Relativity
[C.L. (2012,2013)] Dual role

15 Physical polarizations
The analogy with General Relativity [C.L. (2012,2013)] Dual role Pure gauge Physical polarizations Degrees of freedom

16 Physical polarizations Geometrical interpretation
The analogy with General Relativity [C.L. (2012,2013)] Dual role Pure gauge Physical polarizations Degrees of freedom Geometrical interpretation Parallelism Curvature

17 The analogy with General Relativity
[C.L. (2012,2013)] Dual role Pure gauge Physical polarizations Degrees of freedom Geometrical interpretation Parallelism Curvature Inertial forces Gravitational forces Analogy with General Relativity

18 The geometrical interpretation
[Hatta (2012)] [C.L. (2012)] Parallel transport

19 The geometrical interpretation
[Hatta (2012)] [C.L. (2012)] Parallel transport

20 The geometrical interpretation
[Hatta (2012)] [C.L. (2012)] Parallel transport

21 The geometrical interpretation
[Hatta (2012)] [C.L. (2012)] Parallel transport Path dependent! Stueckelberg symmetry

22 The gauge symmetry [C.L. (in preparation)] Quantum electrodynamics
« Physical » « Background »

23 The gauge symmetry [C.L. (in preparation)] Quantum electrodynamics
« Physical » « Background » Passive

24 The gauge symmetry [C.L. (in preparation)] Quantum electrodynamics
« Physical » « Background » Passive Active

25 The gauge symmetry [C.L. (in preparation)] Quantum electrodynamics
« Physical » « Background » Passive Active Active x (Passive)-1

26 The gauge symmetry Stueckelberg [C.L. (in preparation)]
Quantum electrodynamics « Physical » « Background » Stueckelberg Passive Active Active x (Passive)-1

27 The semantic ambiguity
Quid ? « physical » « measurable » « gauge invariant »

28 The semantic ambiguity
Quid ? « physical » « measurable » « gauge invariant » Observables Measurable, physical, gauge invariant (active and passive) E.g. cross-sections

29 The semantic ambiguity
Quid ? « physical » « measurable » « gauge invariant » Observables Measurable, physical, gauge invariant (active and passive) E.g. cross-sections Path Stueckelberg Background Expansion scheme dependent E.g. collinear factorization

30 The semantic ambiguity
Quid ? « physical » « measurable » « gauge invariant » Observables Measurable, physical, gauge invariant (active and passive) E.g. cross-sections Path Stueckelberg Background Expansion scheme dependent E.g. collinear factorization Quasi-observables « Measurable », « physical », « gauge invariant » (only passive) E.g. parton distributions

31 The local limit Local limit of quasi-observables Path dependence

32 Parametrized by form factors
The local limit Local limit of quasi-observables Path dependence Genuine local quantities are path independent ! Parametrized by form factors

33 Parametrized by form factors
The local limit Local limit of quasi-observables Path dependence Genuine local quantities are path independent ! Parametrized by form factors Passive and active Passive and path independent Passive and local « True » gauge invariance : [Ji (2009)]

34 The twist-2 OAM [C.L., Pasquini (2011)]
[C.L., Pasquini, Xiong, Yuan (2012)] [Hatta (2012)] Quark Wigner operator

35 The twist-2 OAM [C.L., Pasquini (2011)]
[C.L., Pasquini, Xiong, Yuan (2012)] [Hatta (2012)] Quark Wigner operator Quark OAM operator

36 The twist-2 OAM [C.L., Pasquini (2011)]
[C.L., Pasquini, Xiong, Yuan (2012)] [Hatta (2012)] Quark Wigner operator Quark OAM operator Exact relation « Vorticity »

37 The path dependence [Ji, Xiong, Yuan (2012)] [Hatta (2012)]
[C.L. (2013)] Canonical quark OAM operator

38 x-based Fock-Schwinger Coincides locally with kinetic quark OAM
The path dependence [Ji, Xiong, Yuan (2012)] [Hatta (2012)] [C.L. (2013)] Canonical quark OAM operator x-based Fock-Schwinger Coincides locally with kinetic quark OAM

39 x-based Fock-Schwinger Coincides locally with kinetic quark OAM
The path dependence [Ji, Xiong, Yuan (2012)] [Hatta (2012)] [C.L. (2013)] Canonical quark OAM operator x-based Fock-Schwinger Light-front ISI FSI Drell-Yan SIDIS Coincides locally with kinetic quark OAM Naive T-even

40 The summary Canonical Kinetic Jaffe-Manohar (1990) Not observable
Ji (1997) Observable Lq Lq Sq Sq Lg Sg Jg Chen et al. (2008) Quasi-observable Wakamatsu (2010) Quasi-observable Lq Lq Sq Sq Lg Sg Lg Sg

41 Backup slides

42 Phase-space densities
The parton distributions GTMDs [PRD84 (2011) ] [PRD85 (2012) ] [PRD84 (2011) ] [PLB710 (2012) 486] TMDs TMFFs GPDs [JHEP1105 (2011) 041] TMCs PDFs [PRD79 (2009) ] [Nucl. Phys. A825 (2009) 115] [PRL104 (2010) ] [PRD79 (2009) ] FFs [PRD74 (2006) ] [PRD78 (2008) ] [PRD79 (2009) ] Charges Phase-space densities

43 The spin-spin-orbit correlations
[C.L., Pasquini (2011)]

44 The light-front wave functions
Overlap representation Momentum Polarization Light-front quark models Wigner rotation [PRD74 (2006) ] [PRD78 (2008) ] [PRD79 (2009) ]

45 Phenomenological comparison
The orbital angular momentum OAM Kinetic GPDs Canonical (naive) TMDs Canonical GTMDs Phenomenological comparison but

46 Gauge-variant operator Lorentz-invariant extensions
The gauge-invariant extension (GIE) Gauge-variant operator GIE2 GIE1 Gauge « Natural » gauges Rest Center-of-mass Infinite momentum Lorentz-invariant extensions ~ « Natural » frames

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