Cédric LorcéIPN Orsay - LPT Orsay

Orbital Angular Momentum in QCD

June 27 2013, Dipartimento di Fisica, Universita’ di Pavia, Italy

The outline

Dark spin

Quark spin?

~ 30 %

• The decompositions in a nutshell• Canonical formalism and Chen et al. approach• Geometrical interpretation of gauge symmetry• Path-dependence and measurability• Conclusions

Basic question

Jaffe-Manohar (1990)

The decompositions in a nutshell

Sq

SgLg

Lq

Noether’s theorem

Ji (1997)Jaffe-Manohar (1990)

The decompositions in a nutshell

Sq

SgLg

Lq Sq

Jg

Lq

Noether’s theorem

Ji (1997)Jaffe-Manohar (1990)

Chen et al. (2008)

The decompositions in a nutshell

Sq

SgLg

Lq

Sq

SgLg

Lq

Sq

Jg

Lq

Gauge-invariant extension (GIE)

Noether’s theorem

Wakamatsu (2010)

Ji (1997)Jaffe-Manohar (1990)

Chen et al. (2008)

The decompositions in a nutshell

Sq

SgLg

Lq

Sq

SgLg

Lq Sq

SgLg

Lq

Sq

Jg

Lq

Gauge-invariant extension (GIE)

Noether’s theorem

Wakamatsu (2010)

Ji (1997)Jaffe-Manohar (1990)

Chen et al. (2008)

Canonical Kinetic

The decompositions in a nutshell

Sq

SgLg

Lq

Sq

SgLg

Lq Sq

SgLg

Lq

Sq

Jg

Lq

Gauge-invariant extension (GIE)

Noether’s theorem

Wakamatsu (2010)

Ji (1997)Jaffe-Manohar (1990)

Chen et al. (2008)

Canonical Kinetic

The decompositions in a nutshell

Sq

SgLg

Lq

Sq

SgLg

Lq Sq

SgLg

Lq

Sq

Jg

Lq

Gauge-invariant extension (GIE)

Noether’s theorem

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

The Chen et al. approach

Gauge transformation (assumed)

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

The Chen et al. approach

Gauge transformation (assumed)

Pure-gauge covariant derivatives

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

The Chen et al. approach

Gauge transformation (assumed)

Field strength

Pure-gauge covariant derivatives

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

The canonical formalism

Textbook

Dynamical variables

Lagrangian

[C.L. (2013)]

The canonical formalism

Textbook

Gauge covariant

Dynamical variables

Lagrangian

[C.L. (2013)]

The canonical formalism

Textbook

Gauge covariant

Gauge invariant

Dynamical variables

Lagrangian

Dirac variables

Dressing field Gauge transformation

[Dirac (1955)][Mandelstam

(1962)]

[C.L. (2013)]

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

Dual role

Pure gauge

Physical polarizations

The analogy with General Relativity

Degrees of freedom

[C.L. (2012,2013)]

Dual role

Pure gauge

Physical polarizations

The analogy with General Relativity

Geometrical interpretationParallelism Curvature

Degrees of freedom

[C.L. (2012,2013)]

Dual role

Pure gauge

Physical polarizations

Analogy with General

Relativity

The analogy with General Relativity

Geometrical interpretationParallelism Curvature

Inertial forces

Gravitational forces

Degrees of freedom

[C.L. (2012,2013)]

Dual role

[Wakamatsu (2010)][Chen et al. (2008)]

The Stueckelberg symmetry

Ambiguous!

[Stoilov (2010)][C.L. (2013)]

Sq

SgLg

Lq Sq

SgLg

Lq

Coulomb GIE

[Hatta (2011)][C.L. (2013)]

Sq

SgLg

Lq

Light-front GIE

Lpot

LpotSq

Sg

Lg

Lq

Infinitely many possibilities!

Gauge

GIE1

GIE2

Gauge-variant operator

« Natural » gauges

Lorentz-invariant extensions~

RestCenter-of-mass

Infinite momentum

« Natural » frames

The gauge-invariant extension (GIE)

The geometrical interpretation[Hatta (2012)]

[C.L. (2012)]Parallel transport

The geometrical interpretation[Hatta (2012)]

[C.L. (2012)]Parallel transport

The geometrical interpretation[Hatta (2012)]

[C.L. (2012)]Parallel transport

Non-local !

The geometrical interpretation[Hatta (2012)]

[C.L. (2012)]Parallel transport

Path dependent !

Stueckelberg symmetry

Non-local !

The path dependence[Ji, Xiong, Yuan (2012)]

[Hatta (2012)][C.L. (2013)]Canonical quark OAM operator

FSIISI

SIDISDrell-Yan

The path dependence[Ji, Xiong, Yuan (2012)]

[Hatta (2012)][C.L. (2013)]

Naive T-even

Canonical quark OAM operator

Light-frontLq

FSIISI

SIDISDrell-Yan

The path dependence[Ji, Xiong, Yuan (2012)]

[Hatta (2012)][C.L. (2013)]

Coincides locally with kinetic quark OAM

Naive T-even

Canonical quark OAM operator

x-based Fock-SchwingerLight-frontLq

Lq

The gauge symmetry

Quantum electrodynamics« Physical »

[C.L. (in preparation)]

« Background »

The gauge symmetry

Quantum electrodynamics

Passive

« Physical »

[C.L. (in preparation)]

« Background »

The gauge symmetry

Quantum electrodynamics

Passive Active

« Physical »

[C.L. (in preparation)]

« Background »

The gauge symmetry

Quantum electrodynamics

Passive Active

« Physical »

[C.L. (in preparation)]

« Background »

Active x (Passive)-1

The gauge symmetry

Quantum electrodynamics

Passive Active

« Physical »

[C.L. (in preparation)]

« Background »

Active x (Passive)-1

Stueckelberg

The semantic ambiguity

« measurable »Quid ? « physical »

« gauge invariant »

The semantic ambiguity

Observables

« measurable »Quid ? « physical »

« gauge invariant »

Measurable, physical, gauge invariant (active and passive)E.g. cross-sections

The semantic ambiguity

PathStueckelbergBackground

Observables

« measurable »Quid ? « physical »

« gauge invariant »

Measurable, physical, gauge invariant (active and passive)

Expansion scheme

E.g. cross-sections

dependentE.g. collinear factorization

The semantic ambiguity

PathStueckelbergBackground

Observables

Quasi-observables

« measurable »Quid ? « physical »

« gauge invariant »

Measurable, physical, gauge invariant (active and passive)

« Measurable », « physical », « gauge invariant » (only passive)

Expansion scheme

E.g. cross-sections

E.g. parton distributions

dependentE.g. collinear factorization

Canonical Kinetic

The observability

Sq

SgLg

Lq

Sq

SgLg

Lq Sq

SgLg

Lq

Sq

Jg

Lq

Not observableObservable Quasi-observable

[Wakamatsu (2010)]

[Ji (1997)][Jaffe-Manohar (1990)]

[Chen et al. (2008)]

The gluon spin

[Jaffe-Manohar (1990)] [Hatta (2011)]

Light-front GIE Light-front gauge

Gluon helicity distribution

Local fixed-gauge interpretation

Non-local gauge-invariant interpretation

« Measurable », gauge invariant but non-local

The kinetic and canonical OAM

Quark naive canonical OAM (Jaffe-Manohar)

[Burkardt (2007)][Efremov et al.

(2008,2010)][She, Zhu, Ma (2009)][Avakian et al. (2010)][C.L., Pasquini (2011)]

Model-dependent !

Kinetic OAM (Ji)

[Ji (1997)]

[Penttinen et al. (2000)][Kiptily, Polyakov (2004)]

[Hatta (2012)]

but

No gluons and not QCD EOM!

[C.L., Pasquini (2011)]

Pure twist-3

Canonical OAM (Jaffe-Manohar) [C.L., Pasquini (2011)][C.L., Pasquini, Xiong, Yuan (2012)]

[Hatta (2012)]

Wakamatsu (2010)

Ji (1997)Jaffe-Manohar (1990)

Chen et al. (2008)

Canonical Kinetic

The conclusion

Sq

SgLg

Lq

Sq

SgLg

Lq Sq

SgLg

Lq

Sq

Jg

Lq

Not observable Observable

Quasi-observableQuasi-observable

Backup slides

[PRD79 (2009) 014507] [Nucl. Phys. A825 (2009) 115]

[PRL104 (2010) 112001][PRD79 (2009) 113011]

GTMDs

TMDs

Charges

PDFs

GPDs

FFsTMCs

TMFFs[PRD84 (2011) 034039]

[PLB710 (2012) 486]

[PRD84 (2011) 014015][PRD85 (2012) 114006]

[JHEP1105 (2011) 041]

[PRD74 (2006) 054019][PRD78 (2008) 034001]

[PRD79 (2009) 074027]

Phase-space densities

The parton distributions

« Vorticity »

The twist-2 OAM

Quark Wigner operator

[C.L., Pasquini (2011)][C.L., Pasquini, Xiong, Yuan (2012)]

[Hatta (2012)]

Quark OAM operator

Exact relation

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

Overlap representation

Momentum Polarization

[PRD74 (2006) 054019][PRD78 (2008) 034001][PRD79 (2009) 074027]

Light-front quark models Wigner rotation

The light-front wave functions

OAM

Canonical (naive)

Kinetic

Canonical GTMDs

TMDs

GPDs

Phenomenological comparison

but

The orbital angular momentum