Cédric LorcéSLAC & IFPA Liège

How to define and access quark and gluon contributions to the

proton spin

December 2, 2014, IIT Bombay, Bombay, India

INTERNATIONAL WORKSHOP ON FRONTIERS OF QCD IIT BOMBAY DECEMBER 2-5, 2014

Outline

• What is it all about ?

• Why is there a controversy ?

• How can we measure AM ?

Outline

• What is it all about ?

• Why is there a controversy ?

• How can we measure AM ?

Angular momentum decomposition

Sq

SgLg

Lq Sq

SgLg

Lq

Sq

Jg

Lq

Many questions/issues : • Frame dependence ?• Gauge invariance ?• Uniqueness ?• Measurability ?• … Reviews:

Dark spin

Quark spin?

~ 30 %

?

?

?

[Leader, C.L. (2014)][Wakamatsu (2014)]

~ 20 %

[de Florian et al. (2014)]

In short …

Noether’s theorem :

Continuous symmetry

Translation invarianceRotation invariance

Conserved quantity

Total (linear) momentumTotal angular momentum

We all agree on the total quantities

BUT …

We disagree on their decomposition

In short …

3 viewpoints :

• Meaningless, unphysical discussions

No unique definition ill-defined problem

• There is a unique «physical» decomposition

Missing fundamental principle in standard approach

• Matter of convention and convenience

Measured quantities are unique BUT physical interpretation is not unique

Outline

• What is it all about ?

• Why is there a controversy ?

• How can we measure AM ?

Spin decompositions in a nutshell

[Jaffe, Manohar (1990)]

[Ji (1997)]

Sq

SgLg

Lq Sq

Lq

Jg

Canonical Kinetic

Gauge non-invariant ! « Incomplete »

Gluon spin

Gluon helicity distribution

[de Florian et al. (2014)]

« Measurable », gauge invariant but complicated

Gluon spin

[Jaffe-Manohar (1990)]

Light-front gauge

Gluon helicity distribution

Simple fixed-gauge interpretation

« Measurable », gauge invariant but complicated

Chen et al. approach

Gauge transformation (assumed)

Field strength

Pure-gauge covariant derivatives

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

Spin decompositions in a nutshell

[Jaffe, Manohar (1990)]

[Ji (1997)]

Sq

SgLg

Lq Sq

Lq

Jg

Canonical Kinetic

Gauge non-invariant ! « Incomplete »

Spin decompositions in a nutshell

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

Sq

SgLg

Lq Sq

Lq

Lg

Canonical Kinetic

Sg

Gauge-invariant extension (GIE)

Spin decompositions in a nutshell

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

Sq

SgLg

Lq Sq

Lq

Canonical Kinetic

Sg

Gauge-invariant extension (GIE)

Lg

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

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 !

Stueckelberg symmetry

Geometrical interpretation

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

Stueckelberg symmetry

Geometrical interpretation Fixed reference point

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

Stueckelberg symmetry

Geometrical interpretation Fixed reference point

Non-local !

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

Gluon spin

[Jaffe-Manohar (1990)]

Light-front gauge

Gluon helicity distribution

Local fixed-gauge interpretation

« Measurable », gauge invariant but non-local

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

Outline

• What is it all about ?

• Why is there a controversy ?

• How can we measure AM ?

Asymmetries in QCD

Example : SIDIS

[Mulders, Tangermann (1996)][Boer, Mulders (1998)]

[Bacchetta et al. (2004)][Bacchetta et al. (2007)][Anselmino et al. (2011)]

Angular modulations of the cross section are sensitive to AM

Parton distribution zoo

[C.L., Pasquini, Vanderhaeghen (2011)]

TMDs

FFsPDFs

Charges

GPDs

Parton distribution zoo

[C.L., Pasquini, Vanderhaeghen (2011)]

GTMDs

TMDs

FFsPDFs

Charges

GPDs

«P

hysic

al»

ob

jects

Th

eore

tical

tools

LFWFs

Parton distribution zoo

2+1D

2+0D

0+3D

0+1D

2+3D

[C.L., Pasquini, Vanderhaeghen (2011)]

GTMDs

TMDs

FFsPDFs

Charges

GPDs

«P

hysic

al»

ob

jects

Th

eore

tical

tools

Phase-space (Wigner) distribution

Parton correlators

Gauge invariant but path dependent

2+3D

Longitudinal momentum

Transverse momentum

Transverse position

[Ji (2003)][Belitsky, Ji, Yuan (2004)]

[C.L., Pasquini (2011)]

Phase-space «density»

Light-front quark model results

[C.L., Pasquini (2011)]

[C.L., Pasquini (2011)][C.L. et al. (2012)]

[Hatta (2012)]

Example : canonical OAM

« Vorticity »

Spatial distribution of average transverse momentum

Kinetic vs canonical OAM

Quark naive canonical OAM (Jaffe-Manohar)

Model-dependent !

Kinetic OAM (Ji)

but

No gluons and not QCD EOM !

Pure twist-3

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

[Kanazawa et al. (2014)][Mukherjee et al. (2014)]

[Ji (1997)]

[Penttinen et al. (2000)]

[Burkardt (2007)][Efremov et al. (2008,2010)]

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

Lattice results

CI DI

[Deka et al. (2013)]

Summary

• We all agree on total angular momentum• but we disagree on its decomposition (matter of convention ?)

• Observables are gauge invariant but physical interpretation need not

• Information about AM is encoded in• polarized parton distributions

Reviews: [Leader, C.L. (2014)][Wakamatsu (2014)]

Summary

Nucleon

FFs PDFsTMDsGPDs

GTMDs

LFWFs

DPDs

Backup slides

Semantic ambiguity

PathStueckelbergBackground

Observables

Quasi-observables

« measurable »

Quid ?

« physical »

« gauge invariant »

Measurable, physical, gauge invariant and local

« Measurable », « physical », gauge invariant and non-local

Expansion scheme

E.g. cross-sections

E.g. parton distributions

-dependent

E.g. collinear factorization

Gauge fixing

GIE1

GIE2

« Natural » gauges

Lorentz-invariant extensions~

Rest

Center-of-mass

Infinite momentum

« Natural » frames

Stueckelberg symmetry

Gauge non-invariant operator

Stueckelberg fixing

[C.L. (2013)]

Canonical Kinetic

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)]

Two different approaches

Lagrangian Hamiltonian

Time

Space

Lorentz covariance

Physical interpretatio

n

Manifest Not manifest

Complicated Simple

Two different approaches

Stueckelberg invariant

Stueckelberg fixed

Physical dofs

Gauge dof

Gauge invariance

Physical interpretatio

n

Local Non-local

Complicated Simple

Stueckelberg symmetry

Non-local color phase factor

Path dependence Stueckelberg non-invariance

Path-dependent

Path-independent

[C.L. (2013)]

Photon spin and OAM

Should we be happy with ?

Well… for a circularly polarized plane wave travelling along z

Two descriptions related by a non-zero surface term

!

Photon spin and OAM

Should we be happy with ?

[Ghai et al. (2009)]

Single-slit experiment

Photon spin and OAM

Should we be happy with ?

[O’Neil et al. (2002)][Garcés-Chavéz et al. (2003)]

Optically trapped microscopic particle

Back to basics

Gauge theory

Gauge invariant

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)]

A choice of gauge is a choice of basis

Back to basics

• Time dependence and interaction• Forms of dynamics• Scale and scheme dependence• Should Lorentz invariance be manifest ?• Quantum gauge transformation• Surface terms• Evolution equation• How are different GIEs related ?• Should the energy-momentum tensor be symmetric ?• Topological effects ?• Longitudinal vs transverse• …

As promised, it is pretty complicated …

Additional issues

Canonical formalism

Dynamical variables

Lagrangian

[C.L. (2013)]

Starting point

Energy-momentum

Gauge invariant !

Generalized angular

momentum

Conserved tensors

Gauge covariant

Translation invariance

Lorentz invariance

Canonical formalism

Dynamical variables

Lagrangian

Starting point

Energy-momentum

Gauge invariant !

Generalized angular

momentum

Conserved tensors

Gauge invariant

Translation invariance

Lorentz invariance

Dirac variables

Dressing field

[Dirac (1955)][Mandelstam (1962)]

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

FSIISI

SIDISDrell-Yan

OAM and path dependence[Ji, Xiong, Yuan (2012)]

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

Coincides locally with kinetic quark OAM

Naive T-even

x-based Fock-SchwingerLight-front

LqLq

Quark generalized OAM operator

Back to basics

Special relativity

Different foliations of space-time

Instant-form dynamics Light-front form dynamics

[Dirac (1949)]

«Space» = 3D

hypersurface

«Time» = hypersurface

label

Light-front components

Time

Space

Energy

Momentum

Passive Active

« Physical »

« Background »

Active x (Passive)-1

Stueckelberg

Stueckelberg symmetry

Quantum Electrodynamics

Phase in internal space

Light-front wave functions (LFWFs)

Fock expansion of the nucleon state

Probability associated with the Fock states

Momentum and angular momentum conservation

gauge

[C.L., Pasquini, Vanderhaeghen (2011)]

~

Overlap representation

Light-front wave functions (LFWFs)

GTMDs

Momentum Polarization

[C.L., Pasquini, Vanderhaeghen (2011)]

Light-front wave functions (LFWFs)

Light-front quark models

Wigner rotation

Light-front helicity Canonical spin

SU(6) spin-flavor wave function