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Oleg Teryaev JINR, Dubna

Chiral-even and odd faces of transverse Sum Rule Trieste(+Dubna), November 24 2006. Oleg Teryaev JINR, Dubna. Outline. Even chirality of spin and angular momentum operators Comparing longitudinal and transverse sum rules: g_1 -> g_T

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Oleg Teryaev JINR, Dubna

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  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

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