Cédric LorcéIPN Orsay - LPT Orsay
Parton Wigner Distributions of the
nucleon
June 25 2013, Dipartimento di Fisica, Universita’ di Pavia, Italy
The outline
• Zoo of parton distribution functions• Physical interpretation• Wigner distributions and OAM• Model calculations• Conclusions
The outline
• Zoo of parton distribution functions• Physical interpretation• Wigner distributions and OAM• Model calculations• Conclusions
The charges
Charges
• Polarization
Depends on :
VectorParton number
TensorParton
transversity
AxialParton helicity
DIS
The parton distribution functions (PDFs)
PDFs
Charges
• Polarization• Longitudinal momentum (fraction)
Depends on :
PDFs
Elastic scattering
The form factors (FFs)
FFsPDFs
Charges
• Polarization• Longitudinal momentum (fraction)• Momentum transfer
Depends on :
FFs
DVCS
The generalized PDFs (GPDs)
GPDs
FFsPDFs
Charges
• Polarization• Longitudinal momentum (fraction)• Momentum transfer
Depends on :
GPDs
SIDIS
The transverse momentum-dependent PDFs (TMDs)
• Polarization• Longitudinal momentum (fraction)• Momentum transfer• Transverse momentum
Depends on :
TMDs
No direct connection
TMDs
FFsPDFs
Charges
GPDs
???
The generalized TMDs (GTMDs)
• Polarization• Longitudinal momentum (fraction)• Momentum transfer• Transverse momentum
Depends on :GTMDs
TMDs
FFsPDFs
Charges
GPDs
GTMDs
TMCs
TMFFs
GTMDs
TMDs
???
The complete zoo
FFsPDFs
Charges
• Polarization• Longitudinal momentum (fraction)• Momentum transfer• Transverse momentum
Depends on :
GPDs
GTMDs
[C.L., Pasquini, Vanderhaeghen (2011)]
The double parton scattering
[Thürman, Master thesis (2012)]
DPDFs
DPDFs
• Polarization• Longitudinal momentum (fraction)• Momentum transfer• Transverse momentum• Inter-parton distance
Depends on :
[Diehl, Ostermeier, Schäfer (2012)]
The outline
• Zoo of parton distribution functions• Physical interpretation• Wigner distributions and OAM• Model calculations• Conclusions
The physical interpretation
Initial/final
Position
Momentum
Average/difference
Fourier-conjugated variables
The physical interpretation
Breit frame
Lorentz contraction
Creation/annihilation of pairs
Position w.r.t. the CM
Non-relativistic !
[Ernst, Sachs, Wali (1960)][Sachs (1962)]
The physical interpretation
Drell-Yan frame
Lorentz contraction
Creation/annihilation of pairs
Position w.r.t. the center of momentum
[Soper (1977)][Burkardt (2000)]
The physical interpretation
Quark Wigner operator
Canonical momentum
• Either fix the gauge such that , i.e. work with + boundary condition
Dirac matrix ~ quark
polarization
Wilson line
• Or split the Wilson line to form Dirac variables
The physical interpretation
Quark Wigner operator
Fixed light-front time
No need for time-ordering !
Non-relativistic Wigner distribution
Relativistic Wigner distribution
[Ji (2003)][Belitsky, Ji, Yuan (2004)]
[C.L., Pasquini (2011)][C.L., Pasquini, Xiong, Yuan (2012)]
3+3D
2+3DGTMDs
The phase-space picture
GTMDs
TMDs
FFsPDFs
Charges
GPDs
2+3D
2+1D
2+0D
0+3D
0+1D
The outline
• Zoo of parton distribution functions• Physical interpretation• Wigner distributions and OAM• Model calculations• Conclusions
The phase-space distribution
Wigner distribution
Probabilistic interpretation
Expectation value
Heisenberg’s uncertainty
relations
Position space
Momentum space
Phase space
Galilei covariant
• Either non-relativistic• Or restricted to transverse position
[Wigner (1932)][Moyal (1949)]
The quark orbital angular momentum
GTMD correlator
[C.L., Pasquini (2011)]
Wigner distribution
Orbital angular momentum
[Meißner, Metz, Schlegel (2009)]
Parametrization
Unpolarized quark density
[Meißner, Metz, Schlegel (2009)]
The parametrization @ twist-2 and =0Parametrization :
GTMDs
TMDs GPDs
Monopole Dipole Quadrupole
Nu
cle
on
pola
riza
tion
Quark polarization
FSIISI
The path dependence
Orbital angular momentum
[C.L., Pasquini, Xiong, Yuan (2012)][Hatta (2012)]
[Ji, Xiong, Yuan (2012)][C.L. (2013)]
Drell-Yan
Reference point
SIDIS
Canonical
[Jaffe, Manohar (1990)] [Ji (1997)]Kineti
c
The proton spin decompositions
• Does not satisfy canonical relations• Incomplete decomposition
• Gauge-invariant decomposition• Accessible in DIS and DVCS
Pros:
Cons:
News:[Wakamatsu (2009,2010)]
• Complete decomposition
Pros:
Cons:
• Satisfies canonical relations• Complete decomposition
• Gauge-variant decomposition• Missing observables for the OAM
News:[Chen et al.
(2008)]
• Gauge-invariant extension
• OAM accessible via Wigner distributions [C.L., Pasquini (2012)]
[C.L., Pasquini, Xiong, Yuan(2012)]
[Hatta (2012)]
Canonical
Kinetic[Jaffe, Manohar (1990)] [Ji (1997)]
[C.L. (2013)][Leader, C.L. (in preparation)]
Reviews :
The outline
• Zoo of parton distribution functions• Physical interpretation• Wigner distributions and OAM• Model calculations• Conclusions
Overlap representation
Momentum Polarization
[C.L., Pasquini, Vanderhaeghen (2011)]
Light-front quark models Wigner rotation
The light-front overlap representation
Wigner distribution of unpolarized quark in unpolarized nucleon
[C.L., Pasquini (2011)]
The model results
favored
disfavored
Left-right symmetry No net quark OAM
Distortion induced by the nucleon longitudinal polarization
[C.L., Pasquini (2011)]
The model results
Proton spin
u-quark OAM
d-quark OAM
Average transverse quark momentum in a longitudinally polarized nucleon
[C.L., Pasquini, Xiong, Yuan (2012)]
The model results
« Vorticity »
Distortion induced by the quark longitudinal polarization
[C.L., Pasquini (2011)]
The model results
Quark spin
u-quark OAM
d-quark OAM
Quark spin-nucleon spin correlation
[C.L., Pasquini (2011)]
The model results
Proton spin
u-quark spin
d-quark spin
[C.L., Pasquini (2011)]
The model results
The emerging picture
[C.L., Pasquini (2011)]
[Burkardt (2005)][Barone et al.
(2008)]
Longitudinal
Transverse
The canonical and kinetic OAM
Quark canonical OAM
Quark naive canonical OAM
[Burkardt (2007)][Efremov et al.
(2008,2010)][She, Zhu, Ma (2009)][Avakian et al. (2010)][C.L., Pasquini (2011)]
[C.L., Pasquini (2011)][C.L., Pasquini, Xiong, Yuan (2012)]
[Hatta (2012)]
Model-dependent !
Quark kinetic OAM
[Ji (1997)]
[Penttinen et al. (2000)][Kiptily, Polyakov (2004)]
[Hatta (2012)]
but
No gluons and not QCD EOM !
[C.L., Pasquini (2011)]
Pure twist-3
The conclusions
• Twist-2 parton distributions provide multidimensional pictures of the nucleon
• Relativistic phase-space distributions exist. Open question: how to access them?
• Both kinetic (Ji) and canonical (Jaffe-Manohar) are measurable (twist-2 and twist-3)
• Model calculations can test spin sum rules
Backup slides
OAM and origin dependenceRelative IntrinsicNaive
Transverse center of
momentumPhysical interpretation ?
Depends on proton
position
Equivalence
Intrinsic RelativeNaive
Momentum conservation
Momentum
Fock expansion of the proton state
Fock statesSimultaneous eigenstates
of
Light-front helicity
Overlap representation
Light-front wave functions
Proton state
Eigenstates of parton light-front helicity
Eigenstates of total OAM
Probability associated with the N, Fock state
Normalization
Overlap representation
gauge
Fock-state contributions
Overlap representation
[C.L., Pasquini (2011)]
[C.L. et al. (2012)]
GTMDs
TMDs
GPDsKinetic OAM
Naive canonical OAM
Canonical OAM
Incoherent scatteringDVCS vs. SIDIS
DVCS SIDIS
GPDs TMDs
FFs
Factorization
Compton form factor
Cross section • process dependent• perturbative
• « universal »• non-perturbative
hard soft
GPDs vs. TMDsGPDs TMDs
Correlator
Correlator
Dirac matrix
Wilson line
Off-forward! Forward!
FSIISI
e.g. SIDISe.g. DY
LC helicity and canonical spin
LC helicity Canonical spin
Nu
cleon
pola
riza
tion
Quark polarization Quark polarization
Nu
cleon
pola
riza
tion
[C.L., Pasquini (2011)]
Interesting relationsModel
relations
* **
***
Flavor-dependent
Flavor-independent
Linear relations Quadratic relation
BagLFQSMLFCQM
S DiquarkAV DiquarkCov. Parton
Quark Target
[Jaffe, Ji (1991), Signal (1997), Barone & al. (2002), Avakian & al. (2008-2010)][C.L., Pasquini, Vanderhaeghen (2011)][Pasquini & al. (2005-2008)][Ma & al. (1996-2009), Jakob & al. (1997), Bacchetta & al. (2008)][Ma & al. (1996-2009), Jakob & al. (1997)] [Bacchetta & al. (2008)][Efremov & al. (2009)][Meißner & al. (2007)]
*=SU(6)
*
*
*
*
*
*
Geometrical explanationPreliminaries
Quasi-free quarks Spherical symmetry
[C.L., Pasquini (2011)]
Conditions:
Light-front helicity
Canonical spin
Wigner rotation
(reduces to Melosh rotation in case of FREE
quarks)
Geometrical explanationAxial symmetry about
z
Geometrical explanationAxial symmetry about
z