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Cédric Lorcé

Spin and Orbital Angular Momentum of Quarks and Gluons in the Nucleon. Cédric Lorcé. ECT* Colloquium : Introduction to quark and gluon angular momentum. IFPA Liège. August 25, 2014, ECT*, Trento, Italy. Outline. What is it all about ? Why is there a controversy ?

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Cédric Lorcé

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  1. Spin and Orbital Angular Momentum of Quarks and Gluons in the Nucleon Cédric Lorcé ECT* Colloquium: Introduction to quark and gluon angularmomentum IFPA Liège August 25, 2014, ECT*, Trento, Italy

  2. Outline • Whatisit all about ? • Whyisthere a controversy ? • Howcanwemeasure AM ?

  3. Outline • Whatisit all about ? • Whyisthere a controversy ? • Howcanwemeasure AM ?

  4. Structure of matter Atom Nucleus Nucleons Quarks u d Up Proton Down Neutron 10-10m 10-14m 10-15m 10-18m Atomic physics Nuclear physics Hadronic physics Particle physics

  5. Structure of nucleons Our picture/understanding of the nucleonevolves ! But many questions remainunanswered … • Wheredoes the proton spin come from ? • How are quarks and gluons distributedinside the nucleon ? • Whatis the proton size ? • Why are quarks and gluons confined ? • How are constituent quarks related to QCD ? • …

  6. Angularmomentumdecomposition ~ 30 % Quark spin ? Lq ? ? Lq Sq Sq Dark spin Sg Jg ? Lg Lq Sq Many questions/issues : • Frame dependence ? • Gauge invariance ? • Uniqueness ? • Measurability ? • … Lg Sg [Leader, C.L. (2014)] Review:

  7. Outline • Whatisit all about ? • Whyisthere a controversy ? • Howcanwemeasure AM ?

  8. In short … Noether’stheorem : Continuoussymmetry Translation invariance Rotation invariance Conservedquantity Total (linear) momentum Total angularmomentum We all agree on the totalquantities BUT … Wedisagree on theirdecomposition

  9. In short … 3 viewpoints : • Meaningless, unphysical discussions • No unique definitionill-definedproblem • There is a unique «physical» decomposition • Missingfundamentalprinciple in standard approach • Matter of convention and convenience • Measuredquantities are unique BUT physicalinterpretationisnot unique

  10. In short … 3 viewpoints : • Meaningless, unphysical discussions • No unique definitionill-definedproblem • There is a unique «physical» decomposition • Missingfundamentalprinciple in standard approach • Matter of convention and convenience • Measuredquantities are unique BUT physicalinterpretationisnot unique

  11. Back to basics AM decompositionis a complicated story Let’s have a glimpse …

  12. Back to basics Classicalmechanics Free pointlikeparticle Total AM isconserved but not unique !

  13. Back to basics Classicalmechanics Free composite particle CM motion canbeseparated

  14. Back to basics Classicalmechanics Internal AM Option 1 : Boost invariance Uniqueness Option 2 : Boost invariance Uniqueness Conventionalchoice : Option 2 with The quantityisboost-invariant BUT itsphysicalinterpretationis simple only in the CM frame !

  15. Back to basics Classicalmechanics Boost-invariant extension (BIE) Frame-dependentquantity (e.g. ) Frame

  16. Back to basics Classicalmechanics Boost-invariant extension (BIE) Frame-dependentquantity (e.g. ) (e.g. ) BIE1 Frame CM «Natural» frames

  17. Back to basics Classicalmechanics Boost-invariant extension (BIE) Frame-dependentquantity (e.g. ) BIE2 (e.g. ) BIE1 Frame CM «Natural» frames

  18. Back to basics Classicalelectrodynamics Chargedpointlikeparticle in externalmagneticfield AM conservation ???

  19. Back to basics Classicalelectrodynamics Chargedpointlikeparticle in externalmagneticfield «Hidden» kinetic AM Conserved canonical AM Kinetic and canonical AM are different System = matter + radiation Ambiguous !

  20. Back to basics Quantum mechanics Pointlikeparticleatrest has intrinsic AM (spin) AM isquantized All components cannotbesimultaneouslymeasured In general, onlyisconserved

  21. Back to basics Quantum mechanics Expectation values are in general not quantized Composite particleatrest Quantum average

  22. Back to basics Specialrelativity Lorentz boosts do not commute Rest frame «Standard» boost Moving frame Spin uniquelydefined in the rest frame only !

  23. Back to basics Specialrelativity Relativistic mass is frame-dependent Lorentz contraction Relativity of simultaneity No (complete) separation of CM coordinatesfrominternalcoordinates !

  24. Back to basics Specialrelativity Lorentz-invariant extension (LIE) Frame-dependentquantity (e.g. ) LIE2 (e.g. ) LIE1 Frame Rest «Natural» frames

  25. Back to basics Gauge theory Gauge non-invariant […] in QCD we should make clear what a quark or gluon parton is in an interacting theory. The subtlety here is in the issue of gauge invariance: a pure quark field in one gauge is a superposition of quarks and gluons in another. Different ways of gluon field gauge fixing predetermine different decompositions of the coupled quark-gluon fields into quark and gluon degrees of freedom. [Bashinsky, Jaffe (1998)] Gauge invariant A choice of gauge is a choice of basis

  26. Back to basics Gauge theory Analogywithintegration «Gauge» 1 «Gauge» 2 Riemann Lebesgue Which one is «physical» ? Somewouldsay : Otherswouldsay: None! Only the total area under the curvemakessense Both! Choosing one or anotheris a matter of convenience

  27. Back to basics Gauge theory 3 strategies : Consideronly simple (local) gauge-invariant quantities Relate thesequantities to observables Try to find an interpretation (optional) Fix the gauge Considerquantitieswith simple interpretation Try to find the corresponding observables Define new complicated (non-local) gauge-invariant quantities Considerquantitieswith simple interpretation Try to find the corresponding observables

  28. Back to basics Gauge theory Gauge-invariant extension (GIE) Gauge non-invariant quantity (e.g. ) GIE2 (e.g. ) GIE1 [Dirac (1955)] Gauge Coulomb «Natural» gauges

  29. Back to basics Gauge theory InfinitelymanyGIEs Uniqueness issue […] one can generalize a gauge variant nonlocal operator […] to more than one gauge invariant expressions, raising the problem of deciding which is the “true” one. [Bashinsky, Jaffe (1998)] SomeGIEs are neverthelessmeasurable In other words, the gauge-invariant extension of the gluon spin in light-cone gauge can be measured. Note that one can easily find gauge-invariant extensions of the gluon spin in other gauges. But we may not always find an experimental observable which reduces to the gluon spin in these gauges. [Hoodbhoy, Ji (1999)]

  30. Back to basics Additional issues • Time dependence and interaction • Forms of dynamics • Scaleand schemedependence • Should Lorentz invariance bemanifest ? • Quantum gauge transformation • Surface terms • Evolution equation • How are differentGIEsrelated ? • Should the energy-momentumtensorbesymmetric ? • Topologicaleffects ? • Longitudinal vs transverse • … As promised, itisprettycomplicated …

  31. Spin decompositions in a nutshell uark uark luon luon Decomposition? luon luon uark uark Kinetic luon uark uark luon Canonical

  32. Spin decompositions in a nutshell Canonical Kinetic Lq Lq Sq Sq Sg Lg Jg « Incomplete » Gauge non-invariant ! [Jaffe, Manohar (1990)] [Ji (1997)]

  33. Spin decompositions in a nutshell Canonical Kinetic Lq Lq Sq Sq Sg Sg Lg Lg Gauge-invariant extension (GIE) [Chen et al. (2008)] [Wakamatsu (2010)]

  34. Spin decompositions in a nutshell Canonical Kinetic Lq Lq Sq Sq Sg Sg Lg Lg Gauge-invariant extension (GIE) [Chen et al. (2008)] [Wakamatsu (2010)]

  35. Stueckelberg symmetry [Stoilov (2010)] [C.L. (2013)] Ambiguous ! Infinitely many possibilities ! Coulomb GIE Lq Lq Sq Sq Lpot Lg Sg Sg Lg [Chen et al. (2008)] [Wakamatsu (2010)] Light-front GIE Lq Sq Lq Sq Lpot Lg Lg Sg Sg [Hatta (2011)] [C.L. (2013)]

  36. Outline • Whatisit all about ? • Whyisthere a controversy ? • Howcanwemeasure AM ?

  37. Parton correlators General non-local quark correlator

  38. Parton correlators Gauge invariant but pathdependent Gauge transformation

  39. Partonicinterpretation Phase-space «density» Transverse momentum Longitudinal momentum Transverse position 2+3D [Ji (2003)] [Belitsky, Ji, Yuan (2004)] [C.L., Pasquini (2011)]

  40. Example : canonical OAM Spatial distribution of average transverse momentum « Vorticity » [C.L., Pasquini (2011)] [C.L., Pasquini, Xiong, Yuan (2012)] [Hatta (2012)]

  41. Parton distribution zoo GTMDs Phase-space (Wigner) distribution Theoretical tools 2+3D [C.L., Pasquini, Vanderhaeghen (2011)]

  42. Parton distribution zoo GTMDs TMDs GPDs Phase-space (Wigner) distribution Theoretical tools 2+3D 2+1D 0+3D «Physical» objects [C.L., Pasquini, Vanderhaeghen (2011)]

  43. Parton distribution zoo GTMDs TMDs PDFs FFs GPDs Charges Phase-space (Wigner) distribution Theoretical tools 2+3D 2+1D 0+3D «Physical» objects 2+0D 0+1D [C.L., Pasquini, Vanderhaeghen (2011)]

  44. Parton distribution zoo GTMDs TMDs PDFs FFs GPDs Charges Theoretical tools «Physical» objects [C.L., Pasquini, Vanderhaeghen (2011)]

  45. Asymmetries Angular modulations of the cross section are sensitive to AM Example : SIDIS [Mulders, Tangermann (1996)] [Boer, Mulders (1998)] [Bacchettaet al. (2004)] [Bacchettaet al. (2007)] [Anselminoet al. (2011)]

  46. Kinetic vs canonical OAM Kinetic OAM (Ji) Pure twist-3 Quark naive canonical OAM (Jaffe-Manohar) [C.L., Pasquini (2012)] Model-dependent ! Canonical OAM (Jaffe-Manohar) [C.L., Pasquini (2011)] [C.L., Pasquini, Xiong, Yuan(2012)] [Kanazawa, C.L., Metz, Pasquini, Schlegel (2014)] No gluons and not QCD EOM ! but

  47. Lattice results CI DI [Deka et al. (2013)]

  48. Summary • We all agree on totalangularmomentum • Wedisagree on itsdecomposition(matter of convention ?) • Observables are gauge invariant • but physicalinterpretationneed not • Scattering on nucleonis sensitive to AM

  49. Summary Nucleon GTMDs LFWFs DPDs GPDs TMDs FFs PDFs

  50. Backup slides

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