Cédric Lorcé

Cédric LorcéIFPA Liège

ECT* Colloquium:

Introduction to quark and gluon angular momentum

August 25, 2014, ECT*, Trento, Italy

Spin and Orbital Angular Momentum of Quarks and

Gluons in the Nucleon

Outline

• What is it all about ?

• Why is there a controversy ?

• How can we measure AM ?

Outline




Structure of matter

10-14m 10-15m 10-18m10-10m

d

u

Atom Nucleus Nucleons

Quarks

Atomic physics

Nuclear physics

Hadronic

physics

Particle physics

Proton

Neutron

Up

Down

Structure of nucleons

Our picture/understanding of the nucleon evolves !

But many questions remain unanswered …

• Where does the proton spin come from ?• How are quarks and gluons distributed inside the nucleon ?• What is the proton size ?• Why are quarks and gluons confined ?• How are constituent quarks related to QCD ?• …

Angular momentum decomposition

Sq

SgLg

Lq Sq

SgLg

Lq

Sq

Jg

Lq

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

Dark spin

Quark spin?

~ 30 %

?

?

?

[Leader, C.L. (2014)]

Outline




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

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

Back to basics

AM decomposition is a complicated story

Let’s have a glimpse …

Back to basics

Classical mechanics

Free pointlike particle

Total AM is conserved but not unique !

Back to basics

Classical mechanics

Free composite particle

CM motion can be separated

Back to basics

Classical mechanics

Internal AM

Conventional choice : Option 2 with

Boost invariance

Uniqueness

Option 1 :

Option 2 : Boost invariance

Uniqueness

The quantity is boost-invariant BUT its physical interpretation is simple only in the CM frame !

Frame

Frame-dependent quantity (e.g. )

Boost-invariant extension (BIE)

Back to basics

Classical mechanics

Frame

BIE1


«Natural» frames


Back to basics

Classical mechanics

CM

(e.g. )

Frame

BIE1

BIE2


«Natural» frames


Back to basics

Classical mechanics

CM

(e.g. )

Back to basics

Classical electrodynamics

Charged pointlike particle in external magnetic field

AM conservation ???

Back to basics

Charged pointlike particle in external magnetic field

Kinetic and canonical AM are different

«Hidden» kinetic AM

Conserved canonical AM

System = matter + radiation

Ambiguous !

Classical electrodynamics

Back to basics

Quantum mechanics

Pointlike particle at rest has intrinsic AM (spin)

In general, only is conserved

AM is quantized

All components cannot be simultaneously measured

Back to basics

Composite particle at rest

Quantum average

Expectation values are in general not quantized

Quantum mechanics

Back to basics

Special relativity

Lorentz boosts do not commute

Spin uniquely defined in the rest frame only !

Rest frame

Moving frame

«Standard» boost

Back to basics

Special relativity

Relativistic mass is frame-dependent

No (complete) separation of CM coordinates from internal coordinates !

Lorentz contraction

Relativity of simultaneity


Frame

LIE1

LIE2

«Natural» frames

Lorentz-invariant extension (LIE)

Back to basics

Rest

(e.g. )

Special relativity

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

Gauge theory

Analogy with integration

«Gauge» 1

«Gauge» 2

Riemann Lebesgue

Which one is «physical» ?

Some would say :

Others would say:

None! Only the total area under the curve makes sense

Both! Choosing one or another is a matter of convenience

Back to basics

3 strategies :

1) Consider only simple (local) gauge-invariant quantities2) Relate these quantities to observables3) Try to find an interpretation (optional)

Gauge theory

1) Fix the gauge2) Consider quantities with simple interpretation3) Try to find the corresponding observables

1) Define new complicated (non-local) gauge-invariant quantities2) Consider quantities with simple interpretation3) Try to find the corresponding observables

Gauge non-invariant quantity (e.g. )

Gauge

GIE1

GIE2

«Natural» gauges

Gauge-invariant extension (GIE)

Back to basics

Coulomb

(e.g. )

Gauge theory

[Dirac (1955)]

Infinitely many GIEs

Back to basics

Gauge theory

[…] 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)]

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.

Uniqueness issue

[Hoodbhoy, Ji (1999)]

Some GIEs are nevertheless measurable

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

luonluon

Spin decompositions in a nutshell

Kinetic

uark uarkluonluon

Canonical

uarkuark luon

luon

Decomposition?

uarkuark


[Jaffe, Manohar (1990)]

[Ji (1997)]

Sq

SgLg

Lq Sq

Lq

Jg

Canonical Kinetic

Gauge non-invariant ! « Incomplete »


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

Sq

SgLg

Lq Sq

Lq

Lg

Canonical Kinetic

Sg



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

Sq

SgLg

Lq Sq

Lq

Canonical Kinetic

Sg


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 !

Outline




Parton correlators

General non-local quark correlator

Parton correlators

Gauge transformation

Gauge invariant but path dependent

Partonic interpretation

Phase-space «density»

2+3D

Longitudinal momentum

Transverse momentum

Transverse position

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

[C.L., Pasquini (2011)]

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

[Hatta (2012)]

Example : canonical OAM

« Vorticity »

Spatial distribution of average transverse momentum

Parton distribution zoo

2+3D

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

GTMDsTh

eore

tical

tools

Phase-space (Wigner) distribution


2+1D0+3D

2+3D


GTMDs

TMDs GPDs

«P

hysic

al»

ob

jects

Th

eore

tical

tools



2+1D

2+0D

0+3D

0+1D

2+3D


GTMDs

TMDs

FFsPDFs

Charges

GPDs

«P

hysic

al»

ob

jects

Th

eore

tical

tools




GTMDs

TMDs

FFsPDFs

Charges

GPDs

«P

hysic

al»

ob

jects

Th

eore

tical

tools

Asymmetries

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

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 (2012)]

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

[Kanazawa, C.L., Metz, Pasquini, Schlegel (2014)]

Lattice results

CI DI

[Deka et al. (2013)]

Summary

• We all agree on total angular momentum

• We disagree on its decomposition (matter of convention ?)

• Observables are gauge invariant but physical interpretation need not

• Scattering on nucleon is sensitive to AM

Summary

Nucleon

FFs PDFsTMDsGPDs

GTMDs

LFWFs

DPDs

Backup slides

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

Back to basics

Quantum optics

Photons have only 2 polarization (helicity) states

Twisted light carry OAM

We measure frame-dependent quantities

Then combine them in a frame-independent way

And finally interpret in a special frame

Back to basics

Special relativity

The proper length of a pencil is clearly frame independent. When we say the length of a house in the frame v = 0.9999c is the same as the proper length of the pencil, we are not saying that the length of the house is frame-independent. Rather, we are saying that the length of the house in a special frame can be known from measuring a frame-independent quantity.

v

[Hoodbhoy, Ji (1999)]

Chen et al. approach

Gauge transformation (assumed)

Field strength

Pure-gauge covariant derivatives

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

Explicit expressions


Geometrical interpretation

Non-local !

Fixed reference point

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


Non-local !

Decomposition is path-dependent !

Path dependence Stueckelberg non-invariance

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


Non-local color phase factor

Path dependence Stueckelberg non-invariance

Path-dependent

Path-independent

[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


Degrees of freedom

[C.L. (2014)]

ClassicalNon-dynamical

QuantumDynamical

plays the role of a background field !

PassiveActive

Passive Active

« Physical »

« Background »

Active x (Passive)-1

Stueckelberg


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


~

Overlap representation


GTMDs

Momentum Polarization



Light-front quark models

Wigner rotation

Light-front helicity Canonical spin

SU(6) spin-flavor wave function

Parametrization

GTMDs

TMDs GPDs

Nu

cle

on

pola

riza

tion

Quark polarization

[Meissner, Metz, Schlegel (2009)][C.L., Pasquini (2013)]Quarks & gluons

Complete parametrizations : Quarks

Twist-2

Energy-momentum tensor

A lot of interesting physics is contained in the EM tensor

Energy density

Momentum

density

Energy flux

Momentum flux

Shear stress

Normal stress (pressure)

[Polyakov, Shuvaev (2002)]

[Polyakov (2003)][Goeke et al. (2007)]

[Cebulla et al. (2007)]

In rest frame


In presence of spin density

In rest frame

No « spin » contribution !

Belinfante « improvement »

Spin density gradient Four-momentum circulation

QCD Energy-momentum operator

Matrix elements Normalization


Energy-momentum FFs

Momentum sum rule

Angular momentum sum rule

[Ji (1997)]

Vanishing gravitomagnetic moment !


Energy-momentum FFs

Momentum sum rule

Angular momentum sum rule

[Ji (1997)]

Vanishing gravitomagnetic moment !

Non-conserved current


Leading-twist component of

Link with GPDs

[Ji (1997)]

Accessible e.g. in DVCS !


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Cédric Lorcé