INT Workshop INT-12-49W
Orbital Angular Momentum in QCD
February 6 - 17, 2012
OAM in transverse
densities and resonances
Cédric Lorcé
and
09 Feb 2012, INT, Seattle, USA
Outline

Part one
 Electromagnetic
form factors
 Transverse charge densities

Part two
 Decompositions
of total OAM
 How can one access to OAM?
 Physical interpretation
Outline

Part one
 Electromagnetic
form factors
 Transverse charge densities

Part two
 Decompositions
of total OAM
 How can one access to OAM?
 Physical interpretation
Electromagnetic form factors
Textbook interpretation
Breit frame
Spin-1/2
Spatial charge density
[Ernst, Sachs, Wali (1960)]
[Sachs (1962)]
Lorentz contraction
No probabilistic/charge
interpretation
Creation/annihilation of pairs
Electromagnetic form factors
Light-front interpretation
Drell-Yan-West frame
Number
operator
Transverse charge density
[Soper (1977)]
[Burkardt (2000)]
Probabilistic/charge
interpretation
Transverse charge densities
Longitudinally polarized target
Monopole
Proton
+
Neutron
-
Negative core
does not fit
naive picture
[Miller (2007)]
Transverse charge densities
Transversely polarized target
Monopole
Proton
Neutron
+
+
+
Dipole
[Carlson, Vanderhaeghen (2008)]
Transverse charge densities
Induced electric dipole
Light-front « artifact »
[Burkardt (2003)]
d
d
u
Bint
S
X
Helicity flip
Orbital angular
momentum
Anomalous magnetic
moment
Distorted transverse
densities
Induced electric dipole
moment
Transverse charge densities
Deuteron
Dip
Jz=0
Jz=1
Jx=0
Jx=1
[Carlson, Vanderhaeghen (2009)]
Delta(1232)
=
+
Monopole
+
Dipole
+
Quadrupole
Lattice QCD
Octupole
[Alexandrou et al. (2009)]
Transverse charge densities
Arbitrary spin
Orbital angular
momentum
Anomalous moments
2j+1 multipoles
Induced electric
moments
Structureless particle
Standard Model
No OAM
Supergravity
No distortions
No anomalous moments
Natural EM moments
Charge
normalization
Universal
g=2 factor
[C.L. (2009)]
Transverse charge densities
Pion
Based on dispersion integral of imaginary part of timelike pion FF
Singular!
[Miller, Strikman, Weiss (2011)]
Transverse charge densities
N to N* transitions
P11(1440)
S11(1535)
Proton
MAID
Neutron
No probabilistic
interpretation
[Carlson, Vanderhaeghen (2008)]
[Tiator, Vanderhaeghen (2009)]
[Tiator, Drechsel, Kamalov, Vanderhaeghen (2011)]
Transverse charge densities
N to Delta transitions
P33(1232)
D13(1520)
Proton
MAID
Small quadrupole
No probabilistic
interpretation
[Carlson, Vanderhaeghen (2008)]
[Tiator, Vanderhaeghen (2009)]
[Tiator, Drechsel, Kamalov, Vanderhaeghen (2011)]
Outline

Part one
 Electromagnetic
form factors
 Transverse charge densities

Part two
 Decompositions
of total OAM
 How can one access to OAM?
 Physical interpretation
Decompositions of total OAM
Position of the problem
Well-defined, unambiguous, conserved
Position
space
Momentum
space
Phase
space
Species
space
Spin-orbit
space
Crucial importance of the interpretation
Ambiguous!
Gauge
symmetry!
Decompositions of total OAM
Fock expansion of the proton state
Fock states
Simultaneous eigenstates of
Momentum
Light-front
helicity
Decompositions of total OAM
Light-front wave functions
Eigenstates of parton light-front helicity
Eigenstates of total OAM
NB: A+=0 gauge
Proton state
Probability associated with the N,b Fock state
Normalization
Decompositions of total OAM
Naive
Relative
Interparton
distance
Physical interpretation ?
Depends on proton position
Not intrinsic !
Intrinsic
Impact
parameter
Transverse
center of
momentum
Decompositions of total OAM
Equivalence
Intrinsic
Naive
Relative
Due to the momentum constraint
Flavor contribution
Active partons
Flavor contribution
Parton contribution
Fock states
Flavor projector
How can one access to OAM?
TMDs
Quark polarization
But no exact extraction of OAM is known
NB:
No information
on quark position
Nucleon polarization
Most TMDs vanish in absence of OAM
In spherically symmetric
(independent) quark models:
Naive OAM
density
Pretzelosity
TMD
[Burkardt (2007)]
[Efremov , Schweitzer, Teryaev, Zavada (2008,2010)]
[She, Zhu, Ma (2009)]
[Avakian, Efremov , Schweitzer, Yuan (2010)]
[C.L., Pasquini (2011)]
How can one access to OAM?
GPDs
Quark polarization
Ji’s sum rule for quark angular momentum
[Ji (1997)]
Nucleon polarization
Most GPDs vanish in absence of OAM
Divergence
terms
Quark OAM
How can one access to OAM?
GTMDs
Most GTMDs vanish in absence of OAM
Absent in GPDs and TMDs
[Meißner, Metz, Schlegel (2009)]
[C.L., Pasquini, Vanderhaeghen (2011)]
[C.L., Pasquini (2011)]
Spin-orbit
correlation
Nucleon polarization
Quark polarization
OAM
Absent in GPDs and TMDs
Consequence of
axial symmetry
How can one access to OAM?
Standard definition
Canonical quark OAM operator
Expectation value
Singular
normalization
Link with GTMDs
No wave packets
No infinite normalization factors
Wigner or phase-space distributions
GTMD correlator
[C.L., Pasquini (2011)]
[Hatta (2011)]
[C.L., Pasquini, Xiong, Yuan (2011)]
How can one access to OAM?
Overlap representation
[Hägler, Mukherjee, Schäfer (2004)]
[C.L., Pasquini, Xiong, Yuan (2011)]
[C.L., Pasquini (2011)]
Flavor contribution
TMDs
GTMDs
GPDs
Pure quark system
[C.L., Pasquini (2011)]
Conservation of transverse momentum
NB: also valid for N,b Fock states
Conservation of longitudinal momentum
Anomalous
gravitomagnetic
sum rule!
[Brodsky, Hwang, Ma, Schmidt (2001)]
Physical interpretation
Operator representations
Position space
Momentum space
Momentum
Position
Impact
parameter
Interpretation
Naive
Intrinsic
???
Intrinsic?
Question:
Physical interpretation
Models
Chiral quark-soliton model
[Wakamatsu, Tsujimoto (2005)]
[Wakamatsu (2010)]
Non-perturbative
sea contribution
Scalar quark-diquark
[Burkardt, Hikmat (2009)]
Regularization-dependent
3Q light-front wave functions
[C.L., Pasquini (2011)]
Artifacts?
Physical interpretation
Back to theory
OAM
Canonical
Belinfante
Energy-momentum form factors
Total angular momentum
Spin
Physical interpretation
Link with GPDs
Generalized Parton Distributions
QCD energy-momentum tensor
« Skewing »
relies on
QCD
and
Lorentz
covariance
Physical interpretation
Model artifacts?
Models are not QCD
Truncation of Fock space spoils Lorentz covariance
[Carbonell, Desplanques, Karmanov, Mathiot (1998)]
In model calculations, one would expect
The genuine quark OAM is
but
Physical interpretation
Sum rule
GTMD version of Ji sum rule
Canonical decomposition
Based on
QCD
and
Lorentz
covariance
Summary

Part one
 Electromagnetic

2D Fourier transform for physical intepretation
 Transverse



form factors
charge densities
Distortions due to OAM
Natural EM moments for any spin
Part two
 Decompositions

Different types of OAM
 How


can one access to OAM?
TMDs, GPDs, GTMDs
Overlap representation
 Physical

of total OAM
interpretation
Model artifacts
Backup
Complete
picture @
Momentum
space
Position
space
GTMDs
Transverse density in
momentum space
TMDs
Transverse density in
position space
TMFFs
GPDs
Transverse
TMSDs
PDFs
Longitudinal
Charges
FFs
Formalism
Assumption :

in instant form
(automatic w/ spherical symmetry)
More convenient to work in canonical spin basis