Download Diapositive 1

Spin and Orbital Angular Momentum
of Quarks and Gluons in the Nucleon
ECT* Colloquium:
Introduction to quark and gluon
angular momentum
Cédric Lorcé
IFPA Liège
August 25, 2014, ECT*, Trento, Italy
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 ?
•
Structure of matter
Atom
Nucleus
Nucleons
Quarks
u
Up
Proton
d
Down
Neutron
10-10m
Atomic
physics
10-14m
Nuclear
physics
10-15m
Hadronic
physics
10-18m
Particle
physics
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
Quark
spin
?
?
Lq
Lg
Sq
Dark spin
~ 30 %
?
Lq
?
Sq
Lg
Sg
Lq
Sg
Sq
Jg
Many questions/issues :
• Frame dependence ?
• Gauge invariance ?
• Uniqueness ?
• Measurability ?
•…
Review:
[Leader, C.L. (2014)]
Outline
What is it all about ?
• Why is there a controversy ?
• How can we measure AM ?
•
In short …
Noether’s theorem :
Continuous symmetry
Conserved quantity
Translation invariance
Rotation invariance
Total (linear) momentum
Total 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
Option 1 :
Boost invariance
Uniqueness
Option 2 :
Boost invariance
Uniqueness
Conventional choice :
Option 2 with
The quantity
is boost-invariant BUT
its physical interpretation is simple only in the CM frame !
Back to basics
Classical mechanics
Boost-invariant extension (BIE)
Frame-dependent quantity (e.g.
Frame
)
Back to basics
Classical mechanics
Boost-invariant extension (BIE)
Frame-dependent quantity (e.g.
BIE1
CM
«Natural» frames
(e.g.
Frame
)
)
Back to basics
Classical mechanics
Boost-invariant extension (BIE)
Frame-dependent quantity (e.g.
)
BIE2
BIE1
CM
«Natural» frames
(e.g.
Frame
)
Back to basics
Classical electrodynamics
Charged pointlike particle in external magnetic field
AM conservation ???
Back to basics
Classical electrodynamics
Charged pointlike particle in external magnetic field
«Hidden» kinetic AM
Conserved canonical AM
Kinetic and canonical AM are different
System = matter + radiation
Ambiguous !
Back to basics
Quantum mechanics
Pointlike particle at rest has intrinsic AM (spin)
AM is quantized
All components cannot be simultaneously measured
In general, only
is conserved
Back to basics
Quantum mechanics
Expectation values are in general not quantized
Composite particle at rest
Quantum average
Back to basics
Special relativity
Lorentz boosts do not commute
Rest frame
«Standard» boost
Moving frame
Spin uniquely defined in the rest frame only !
Back to basics
Special relativity
Relativistic mass is frame-dependent
Lorentz contraction
Relativity of simultaneity
No (complete) separation of CM coordinates from internal coordinates !
Back to basics
Special relativity
Lorentz-invariant extension (LIE)
Frame-dependent quantity (e.g.
)
LIE2
LIE1
Rest
«Natural» frames
(e.g.
Frame
)
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
Back to basics
Gauge theory
Analogy with integration
«Gauge» 1
«Gauge» 2
Riemann
Lebesgue
Which one is «physical» ?
Some would say :
None! Only the total area under the curve makes sense
Others would say:
Both! Choosing one or another is a matter of convenience
Back to basics
Gauge theory
3 strategies :
1)
2)
3)
Consider only simple (local) gauge-invariant quantities
Relate these quantities to observables
Try to find an interpretation (optional)
1)
2)
3)
Fix the gauge
Consider quantities with simple interpretation
Try to find the corresponding observables
1)
2)
3)
Define new complicated (non-local) gauge-invariant quantities
Consider quantities with simple interpretation
Try to find the corresponding observables
Back to basics
Gauge theory
Gauge-invariant extension (GIE)
Gauge non-invariant quantity (e.g.
)
GIE2
GIE1
Coulomb
«Natural» gauges
(e.g.
Gauge
)
[Dirac (1955)]
Back to basics
Gauge theory
Infinitely many GIEs
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)]
Some GIEs are nevertheless measurable
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)]
Back to basics
Additional issues
• 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 …
Spin decompositions in a nutshell
uark
uark
luon
luon
Decomposition?
Canonical
Kinetic
Spin decompositions in a nutshell
Canonical
Lq
Lg
Kinetic
Sq
Sg
Gauge non-invariant !
[Jaffe, Manohar (1990)]
Lq
Sq
Jg
« Incomplete »
[Ji (1997)]
Spin decompositions in a nutshell
Canonical
Lq
Lg
Kinetic
Sq
Sg
Lq
Lg
Sq
Sg
Gauge-invariant extension (GIE)
[Chen et al. (2008)]
[Wakamatsu (2010)]
Spin decompositions in a nutshell
Canonical
Lq
Lg
Kinetic
Sq
Sg
Lq
Lg
Sq
Sg
Gauge-invariant extension (GIE)
[Chen et al. (2008)]
[Wakamatsu (2010)]
Stueckelberg symmetry
[Stoilov (2010)]
[C.L. (2013)]
Ambiguous !
Infinitely many possibilities !
Coulomb GIE
Lq
Lq
Sq
Lg
Sg
Lg
Sq
[Chen et al. (2008)]
Lpot
Sg
[Wakamatsu (2010)]
Light-front GIE
Lq
Sq
Lq
Lg
Sg
[Hatta (2011)]
[C.L. (2013)]
Lg
Sq
Sg
Lpot
Outline
What is it all about ?
• Why is there a controversy ?
• How can we measure AM ?
•
Parton correlators
General non-local quark correlator
Parton correlators
Gauge invariant but path dependent
Gauge transformation
Partonic interpretation
Phase-space «density»
Transverse
momentum
Longitudinal
momentum
Transverse
position
2+3D
[Ji (2003)]
[Belitsky, Ji, Yuan (2004)]
[C.L., Pasquini (2011)]
Example : canonical OAM
Spatial distribution of average transverse momentum
« Vorticity »
[C.L., Pasquini (2011)]
[C.L., Pasquini, Xiong, Yuan (2012)]
[Hatta (2012)]
Theoretical tools
Parton distribution zoo
Phase-space (Wigner) distribution
GTMDs
2+3D
[C.L., Pasquini, Vanderhaeghen (2011)]
Theoretical tools
Parton distribution zoo
Phase-space (Wigner) distribution
GTMDs
2+3D
TMDs
2+1D
«Physical» objects
0+3D
GPDs
[C.L., Pasquini, Vanderhaeghen (2011)]
Theoretical tools
Parton distribution zoo
Phase-space (Wigner) distribution
GTMDs
2+3D
GPDs
TMDs
2+1D
«Physical» objects
0+3D
PDFs
FFs
2+0D
0+1D
Charges
[C.L., Pasquini, Vanderhaeghen (2011)]
Theoretical tools
Parton distribution zoo
GTMDs
GPDs
«Physical» objects
TMDs
PDFs
FFs
Charges
[C.L., Pasquini, Vanderhaeghen (2011)]
Asymmetries
Angular modulations of the cross section are sensitive to AM
Example : SIDIS
[Mulders, Tangermann (1996)]
[Boer, Mulders (1998)]
[Bacchetta et al. (2004)]
[Bacchetta et al. (2007)]
[Anselmino et al. (2011)]
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
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
LFWFs
DPDs
GPDs
TMDs
FFs
PDFs
Backup slides
Back to basics
Special relativity
Different foliations of space-time
Light-front components
«Space»
= 3D hypersurface
«Time»
= hypersurface label
Instant-form dynamics
Light-front form dynamics
Time
Space
Energy
Momentum
[Dirac (1949)]
Back to basics
Quantum optics
Photons have only 2 polarization (helicity) states
Twisted light carry OAM
Back to basics
Special relativity
We measure frame-dependent quantities
Then combine them in a frame-independent way
And finally interpret in a special frame
v
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.
[Hoodbhoy, Ji (1999)]
Chen et al. approach
[Chen et al. (2008,2009)]
[Wakamatsu (2010,2011)]
Gauge transformation (assumed)
Pure-gauge covariant derivatives
Field strength
Stueckelberg symmetry
Geometrical interpretation
Fixed reference point
Explicit expressions
Non-local !
[Hatta (2012)]
[C.L. (2013)]
Stueckelberg symmetry
Decomposition is path-dependent !
Non-local !
Path dependence
Stueckelberg non-invariance
?
[Hatta (2012)]
[C.L. (2013)]
Stueckelberg symmetry
Non-local color phase factor
Path-dependent
Path-independent
Path dependence
Stueckelberg non-invariance
[C.L. (2013)]
OAM and path dependence
[Ji, Xiong, Yuan (2012)]
[Hatta (2012)]
[C.L. (2013)]
Quark generalized OAM operator
x-based Fock-Schwinger
Light-front
Lq
Lq
ISI
Drell-Yan
FSI
SIDIS
Coincides locally with kinetic quark OAM
Naive T-even
Stueckelberg symmetry
Degrees of freedom
Classical
Non-dynamical
Quantum
Dynamical
plays the role of a background field !
Active
Passive
[C.L. (2014)]
Stueckelberg symmetry
Quantum Electrodynamics
« Physical »
« Background »
Phase in
internal space
Stueckelberg
Passive
Active
Active x (Passive)-1
Light-front wave functions (LFWFs)
Fock expansion of the nucleon state
Probability associated with the Fock states
Momentum and angular momentum conservation
gauge
Light-front wave functions (LFWFs)
Overlap representation
GTMDs
~
Momentum
Polarization
[C.L., Pasquini, Vanderhaeghen (2011)]
Light-front wave functions (LFWFs)
Light-front quark models
SU(6) spin-flavor
wave function
Wigner rotation
Light-front helicity
Canonical spin
[C.L., Pasquini, Vanderhaeghen (2011)]
Parametrization
Twist-2
GTMDs
Nucleon polarization
Quark polarization
TMDs
GPDs
Complete parametrizations :
Quarks
[Meissner, Metz, Schlegel (2009)]
[C.L., Pasquini (2013)]
Quarks & gluons
Energy-momentum tensor
A lot of interesting physics is contained in the EM tensor
Energy
density
Momentum
density
[Polyakov, Shuvaev (2002)]
[Polyakov (2003)]
[Goeke et al. (2007)]
[Cebulla et al. (2007)]
Shear stress
Normal stress (pressure)
Energy
flux
In rest frame
Momentum
flux
Energy-momentum tensor
In presence of spin density
Belinfante
« improvement »
Spin density gradient
Four-momentum circulation
In rest frame
No « spin » contribution !
Energy-momentum tensor
QCD Energy-momentum operator
Matrix elements
Normalization
Energy-momentum tensor
Energy-momentum FFs
Momentum sum rule
Angular momentum sum rule
Vanishing gravitomagnetic moment !
[Ji (1997)]
Energy-momentum tensor
Energy-momentum FFs
Non-conserved current
Momentum sum rule
Angular momentum sum rule
Vanishing gravitomagnetic moment !
[Ji (1997)]
Energy-momentum tensor
Leading-twist component of
Link with GPDs
Accessible e.g. in DVCS !
[Ji (1997)]