Cédric LorcéSLAC & IFPA Liège
Transversity and orbital angular momentum
January 23, 2015, JLab, Newport News, USA
Outline
• Angular momentum and Relativity • Longitudinal and transverse polarizations• Transversity and orbital angular momentum
Back to basics
Two crucial commutators
RelativisticNon-relativistic
Spin orientation andrelativistic center-of-
mass are frame dependent
Wigner rotation
Special relativity introduces intricate spin-orbit coupling !
Back to basics
Single particle at rest
Total angular
Spin is well-defined and unique
Only upper component matters
Back to basics
Single particle in motion
Total angular
« Spin » is ambiguous and not unique
p-waves are involved
Even for a plane-wave !
Spin vs. Polarization
I will always refer to « spin » as Dirac spin
Dirac states are eigenstates of momentum and polarization operators
but not of spin operator
Pauli-Lubanski four-vector
Polarization four-vector
Spin vs. Polarization
Polarization along z
Total angular momentum is conserved
Spin vs. Polarization
Standard Lorentz transformation defines polarization basis in any frame
Conventional !
Generic Lorentz transformation generates a Wigner rotation of polarization
Changing standard Lorentz transformation results in a Melosh rotation
[Polyzou et al. (2012)]
Popular polarization choices
« Canonical spin »
Advantage : rotations are simple
[Polyzou et al. (2012)]
is a rotationless pure boost
« Light-front helicity » is made of LF boosts
Advantage : LF boosts are simple
Polarization four-vector
Polarization four-vector
Longitudinal vs. Transverse
Longitudinal polarization Helicity !
Reminder
Aka longitudinal spin
Transverse polarization
Transversity !
Helicity vs. Transversity
Chiral odd
Helicity Transversity
Charge odd
Chiral even
Charge even
Many-body system
Axial and tensor charges
Target rest frame quark rest frame
OAM encoded in both WF and spinors
Instant-form and LF wave functions
3Q model of the nucleon
Generalized Melosh rotation
Transfers OAM from spinor to
WF
In many quark models
pure s-wave s-, p- and d-waves
Spherical symmetry !
Not independent !
No gluons, no sea !
Quasi-independent particles in a spherically symmetric potential
Spherical symmetry in quark models
OAM is a pure effect of Generalized Melosh rotation
TMD relations
[Avakian et al. (2010)][C.L., Pasquini (2011)]
[Müller, Hwang (2014)]
[Burkardt (2007)][Efremov et al.
(2008,2010)][She, Zhu, Ma (2009)][Avakian et al. (2010)][C.L., Pasquini (2012)]
[Ma, Schmidt (1998)]
Naive canonical OAM (Jaffe-Manohar)
Transverse spin sum rules
BLT sum rule [Bakker et al. (2004)]
Ambiguous matrix elementsNot related to known
distributions[Leader, C.L. (2014)]
Ji-Leader sum rule
[Leader (2012)][Ji (1997)]
[Ji et al. (2012)][Leader (2013)]
[Harindranath et al. (2013)]
Transverse Pauli-Lubanski sum rule
Spin-orbit correlations
Transverse AM and transversely polarized quark [Burkardt (2006)]
[C.L. (2014)]Longitudinal OAM and longitudinally polarized quark
Summary
• Distinction between « spin » and « polarization » is important
• Helicity and transversity contain complementary information about boosts
• Transversity appears in several sum rules but has no model-independent relation with OAM
