1. Chiral-even and odd faces of transverse Sum Rule Trieste(+Dubna), November 24 2006 Oleg Teryaev JINR, Dubna

2. Outline Even chirality of spin and angular momentum operators Comparing longitudinal and transverse sum rules: g_1 -> g_T Non-zero contribution of Gluon Spin to transverse SR Burkhatrdt-Cottingham sum rule: similarity of longitudinal and transverse spin structure Chiral-odd BLT SR – test of dynamical picture of nucleon, relations of even and odd – POSSIBLE! Belinfante invariance and equivalence principle – possible violation in the case of transversity appearance in Spin Sun Rule Relation to Sivers functions, Burkardt SR and Brodsky/Gardner conjecture

3. Free vs bounded particles – longitudinal case

4. Free vs bounded particles – transverse case Density matrix – 2 terms responsible for 1 2 transverse polarization - chiral odd twist 2 transversity (1) and chiral even twist –3 g_T SAME for free particles – independent for quarks bounded in a nucleon

5. Field-tyheoretical origin of parton model sum rules

6. Momentum and Spin sum rules

7. How derive SR for longitudinal and transverse spin? Different components of angular momentum tensor and Pauli-Lubanski vector do not commute – one needs yo specify projection onto space-like vector n, (nP)=0. Different projections (T vs L)– lead to appearance of different parton distributions – but ALWAYS chiral-even

8. Quarks: Various projections of axial current: Related by EOM to quark-gluon correlations

9. Gluons No gluonic transversity for spin-1/2 BUT transverse twist 3 distribution analogous to quark case: May contribute to jet double transverse asymmetries at RHIC

10. Transverse sum rule Similar to longitudinal Twist 3 not suppressed in SR – no Q Spins same as L due to BC SR Orbital -?

11. Different L and T orbital momenta – natural from the point of view of Brodsky-Gardner conjecture Sivers function similar to (transverse) L and AMM Small singlet Sivers -> Small singlet AMM -> EQUIPARTION of momentum and TOTAL angular momentum + small gluon spin -> large (longitudinal) orbital momentum

12. Transversity and BLT sum rule Should imply some relation of even and odd operators May test DYNAMICAL picture of nucleon which ay be surprisingly simple Say, transversity may be quite well understood kinematically(Efremov, OT, Zavada) relating even and odd terms – may justify (implicit) notion of free particles – explains larger values of transversity than helicity

13. Fractional sum rule for transversity (Pire, Soffer, OT) First moment is not conserved, but May be a candidate for models/NPQCD

14. Chiral-even transverse SR – supported by EQUIVALENCE principle Belinfante invariance -> spin in (chiral- even) orbital form Momentum+Angular momentum conservation -> JI SR

15. Equivalence principle Newtonian – “Falling elevator” + Anomalous gravitomagnetic moment iz ZERO or Classical and QUANTUM rotators behave in the SAME way

16. Electromagnetism vs Gravity Interaction – field vs metric deviation Static limit Mass as charge – equivalence principle

17. Gravitational formfactors Conservation laws - zero Anomalous Gravitomagnetic Moment : (g=2) Moments of GPD’s (X. Ji)- may be extracted from high-energy experiments/NPQCD calculations Describe the partition of angular momentum between quarks and gluons Valid for any spin projection! Appearance of chiral- odd term in angular momentum conservation may violate EP – unless it is related to chiral-even

18. Gravitomagnetism Gravitomagnetic field – action on spin – ½ from spin dragging twice smaller than EM Lorentz force – similar to EM case: factor ½ cancelled with 2 from Larmor frequency same as EM Orbital and Spin momenta dragging – the same - Equivalence principle

19. Generalization of Equivalence principle Various arguments: AGM 0 separately for quarks and gluons – most clear from the lattice (LHPC/SESAM)

20. Extended Equivalence Principle=Exact EquiPartition In pQCD – violated Reason – in the case of EEP- no smooth transition for zero fermion mass limit (Milton, 73) Conjecture (O.T., 2001 – prior to lattice data) – valid in NP QCD – zero quark mass limit is safe due to chiral symmetry breaking

21. Another arguments in favour of EEP J=1/2 -> J=1. QCD SR calculation of Rho’s AMM gives g close to 2. Maybe because of similarity of moments. Gluons momentum fraction sizable. Direct calculation for AGM are desired! “Valence” Parametrization of E (GPV) – remarakble relations between valence quantities - physical input – EQUIPARTITION Relation: E -> Sivers; EP -> Burkardt SR; EEP -> Brodsky/Gardner conjecture

22. Conclusions Standard derivation -> chiral-even transverse SR Longitudinal and transverse quark and gluon spins – same if BCSR is valid L and T Orbital momenta – related to Brodsky et al conjectures Chiral-odd sum rules – may test dynamical picture of nucleon Spin (L and T)sum rules – related to equivalence principle; independent chiral-odd terms may violate it