1. Cédric Lorcé SLAC & IFPA Liège Transversity and orbital angular momentum January 23, 2015, JLab, Newport News, USA

2. Outline Angular momentum and Relativity Longitudinal and transverse polarizations Transversity and orbital angular momentum

3. Back to basics Two crucial commutators RelativisticNon-relativistic Spin orientation and relativistic center-of-mass are frame dependent Wigner rotation Special relativity introduces intricate spin-orbit coupling !

4. Back to basics Single particle at rest Total angular Spin is well-defined and unique Only upper component matters

5. Back to basics Single particle in motion Total angular « Spin » is ambiguous and not unique p-waves are involved Even for a plane-wave !

6. 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

7. Spin vs. Polarization Polarization along z Total angular momentum is conserved

8. 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)]

9. 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

10. Longitudinal vs. Transverse Longitudinal polarizationHelicity ! Reminder Aka longitudinal spin Transverse polarization Transversity !

11. Helicity vs. Transversity Chiral odd HelicityTransversity Charge odd Chiral even Charge even

12. Many-body system Axial and tensor charges Target rest frame quark rest frame OAM encoded in both WF and spinors

13. 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-waves-, p- and d-waves Spherical symmetry ! Not independent ! No gluons, no sea ! Quasi-independent particles in a spherically symmetric potential

14. 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)

15. Transverse spin sum rules BLT sum rule [Bakker et al. (2004)] Ambiguous matrix elements Not 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

16. Spin-orbit correlations Transverse AM and transversely polarized quark [Burkardt (2006)] [C.L. (2014)] Longitudinal OAM and longitudinally polarized quark

17. 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