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Oleg Teryaev JINR, Dubna, Russia

Comparing T-odd and T-even spin sum rules Diffraction-2014 Primosten, Croatia, September 13, 2014. Oleg Teryaev JINR, Dubna, Russia. Main Topics. Sum rules from limited region: extrapolation with generalized cut moments EMTSRs and coupling to gravity

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Oleg Teryaev JINR, Dubna, Russia

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  1. Comparing T-odd and T-even spin sum rulesDiffraction-2014Primosten, Croatia, September 13, 2014 Oleg Teryaev JINR, Dubna, Russia

  2. Main Topics • Sum rules from limited region: extrapolation with generalized cut moments • EMTSRs and coupling to gravity • Burkardt sum rule for T-odd Sivers function and energy momentum tensor • Deuteron tensor polarization and T-even EP-related sum rule (HERMES and JLab)

  3. Extrapolation to 0 (case of Bjorken sum rule) • Interval of x’s limited from below (especially important for JLab) • Continuation – substantial part pf SRs check (~ 50%!) • New method (Kotlorz,Mikhailov,OT) – generalized cut moments – bunch of function with the same 1st moment – extra degree of freedom to constrain continuation • Contains extra (smooth) test function; powers – the same DGLAP evolution as original one

  4. Continuation to zero for Bjorken SR • Continuation from x~0.1 seems possible

  5. Energy-momentum tensor and sum rules • Related to (angular) momentum conservation • Simplest case – partons momentum conservation (gluons discovery) • More involved Ji’s sum rules for GPDs (talks of W.-D. Nowak, D. Mueller, O. Selyugin) • Relevant also for coupling of hadrons to gravity (OT’99)

  6. 1-st moments - EM, 2-nd -Gravitational Formfactors Describe the partition of angular momentum between quarks and gluons • Conservation laws - zero Anomalous Gravitomagnetic Moment : (g=2) • Describe interaction with both classical and TeV gravity (and Pomeron?? – talk of J. Soffer) – all smallness in gravity coupling • May be extracted from high-energy experiments/NPQCD calculations

  7. Equivalence principle • Newtonian – “Falling elevator” – well known and checked • Post-Newtonian – gravity action on SPIN – known since 1962 (Kobzarev and Okun’) – not checked on purpose but in fact checked in atomic spins experiments at % level (Reanalysis - Silenko,OT’07) • Anomalous gravitomagnetic moment iz ZERO (= Ji SR) or • Classical and QUANTUM rotators behave in the SAME way

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

  9. Recent lattice study (M. Deka et al. arXiv:1312.4816) • Sum of u and d for Dirac (T1) and Pauli (T2) FFs

  10. Extended Equivalence Principle=Exact EquiPartition In pQCD – violated Reason – in the case of ExEP- 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 Supported by generic smallness of E (isoscalar AMM)

  11. T-odd quantities • P-conserving Single Spin asymmetries: Spin pseudovector accompanied by Levi-Civita tensor – change sign under the time reflection • Hermiticity of density matrix: accompanied by i (like photon circular polarization) • May appear either due to real T (CP) violation (complex couplings) or due to non-trivial phases (FSI)

  12. T-odd TMDs • Transverse momentum - like in UGDF in kT factorization (but soft kT – sum of higher twists) – talks of P. Zavada,O. Kouznetsov • T-odd (ST-kT correlation) Sivers TMD due to FSI (Brodsky, Hwang, Schmidt) – relation to AMM • Burkardt sum rule • Conjecture (OT’06) AMM ->AGM: Ji SR <-> Burkardt SR • Direct derivation of Burkardt SR from EMT?

  13. Burkardt SR and Energy-Momentum tensor • Consider twist 3 (Bukhvostov, Kuraev,Lipatov;Efremov,OT’83) – related to relevant moment of Sivers function being infinite tower of twists • Specific vector qqG(zero momentum) correlator – gluonic pole • EMT forward matrix element: no spin-dependent structure • Naively: Valid identically due to symmetry properties implied by T-invariance) • However: EOM – generated pole should get imaginary part due to EMGI and related contour gauge(DY process: Anikin, OT, PLB2010) – analog of the choice of Wilson line • Cut in Q2: cf cuts in pion dissociation to dijet (D. Ivanov et al.)

  14. Electromagnetric Gauge Invariance in DY process • Extra diagram – factor 2 in transverse (TM integrated) asymmetry • Absent in pQCD

  15. Pole prescription and Burkardt SR • Pole prescription (dynamics!) provides (“T-odd”) symmetric part! • SR: (but relation of gluon Sivers to twist 3 still not found – prediction!) • Can it be valid separately for each quark flavour: nodes (related to “sign problem”)? • Valid if structures forbidden for TOTAL EMT do not appear for each flavour • Structure contains besides S gauge vector n: If GI separation of EMT – forbidden: SR valid separately!

  16. Can one have T-even inclusive spin SR? • Spin pseudovector – calls for either GPDs or TMDs (extra momentum required for Levi-Civita tensor) • But! Tensor polarization (talk of K. Slifer) is P-even – no T-oddness as well!

  17. Another manifestation of post-Newtonian (Ex)EP for spin 1 hadrons (SRs - Efremov,OT’82; EP- OT’09) Tensor polarization -coupling of gravity to spin in forward matrix elements - inclusive processes Second moments of tensor distributions should sum to zero =0 for ExEP

  18. HERMES – data on tensor spin structure function Isoscalar target – proportional to the sum of u and d quarks – combination required by EMT Second moments – compatible to zero better than the first one (collective glue << sea) – for valence:

  19. Are more accurate data possible? • HERMES – unlikely • JLab (talks of O.Rondon,K.Slifer) may provide information about collective sea and glue in deuteron and indirect new test of Equivalence Principle

  20. CONCLUSIONS Generalized cut moment may be useful for SRs extrapolation T-odd Burkardt SR is related to EMT conservation and peculiar cut in soft gluon emission EMTSR for tensor polarization testing deuteron collective sea/glue and Equivalence Principle – may be tested at JLab?

  21. BACKUP SLIDES

  22. Cut Mellin moments (Kotlorz, Mikhailov )

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

  24. 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

  25. Equivalence principle for moving particles • Compare gravity and acceleration: gravity provides EXTRA space components of metrics • Matrix elements DIFFER • Ratio of accelerations: - confirmed by explicit solution of Dirac equation (Silenko, O.T.)

  26. EEP and AdS/QCD • Recent development – calculation of Rho formfactors in Holographic QCD (Grigoryan, Radyushkin) • Provides g=2 identically! • Experimental test at time –like region possible

  27. Preprint (in Russian) with first derivation of sum rules for 1st (=Close-Kumano) and 2nd moments

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