Transversity (and TMD friends) Hard Mesons and Photons Productions, ECT*, October 12, 2010

1. Transversity (and TMD friends)Hard Mesons and Photons Productions, ECT*, October 12, 2010 Oleg Teryaev JINR, Dubna

2. Outline • 2 meanings of transversity and 2 ways to transverse spin • Can transversity be probabilistic? • Spin-momentum correspondence – transversity vs TMDs • Positivity constraints for DY: relating transversity to Boer-Mulders function • TMDs in impact parameter space vs exclusive higher twists • Conclusions

3. Transversity in quantum mechanics of spin 1/2 • Rotation –> linear combination (remember poor Schroedinger cat) • New basis • Transversity states - no boost suppression • Spin – flip amplitude -> difference of diagonal ones

4. Transveristy in QCD factorization

5. Light vs Heavy quarks • Free (or heavy) quarks – transverse polarization structures are related • Spontaneous chiral symmetry breaking – light quarks - transversity decouples • Relation of chiral-even and chiral-odd objects – models • Modifications of free quarks • Probabilistic NP ingredient of transversity

6. Transversity as currents interference • DIS with interfering scalar and vector currents – Goldstein, Jaffe, Ji (95) • Application of vast Gary’s experience in Single Spin Asymmetries calculations where interference plays decisive role • Immediately used in QCD Sum Rule calculations by Ioffe and Khodjamirian • Also the issue of the evolution of Soffer inequality raised • Further Gary’s work on transversity includes Flavor spin symmetry estimate of the nucleon tensor charge.Leonard P. Gamberg, (Pennsylvania U. & Tufts U.) , Gary R. Goldstein, (Tufts U.) . TUHEP-TH-01-05, Jul 2001. 4pp. Published in Phys.Rev.Lett.87:242001,2001.

7. “Zavada’s Momentum bag” model – transversity (Efremov,OT,Zavada) • NP stage – probabilistic weighting • Helicity and transversity are defined by the same NP function -> a bit large transversity

8. Transverse spin and momentum correspondence • Similarity of correlators (with opposite parity matrix structures) ST ->kT/M • Perfectly worked for twist 3 contributions in polarized DIS (efremov,OT) and DVCS (Anikin,Pire,OT) • Transversity -> possible to described by dual dual Dirac matrices • Formal similarity of correlators for transversity and Boer-Mulders function • Very different nature – BM-T-odd (effective) • But – produce similar asymmetries in DY

9. Positivity for DY • (SI)DIS – well-studied see e.g. • Spin observables and spin structure functions: inequalities and dynamics.Xavier Artru, Mokhtar Elchikh, Jean-Marc Richard, Jacques Soffer, Oleg V. Teryaev, Published in Phys.Rept.470:1-92,2009. e-Print: arXiv:0802.0164 [hep-ph] • Stability of positivity in the course of evolution

10. Kinetic interpretation of evolution

11. Master (balance) equation

12. Positivity vs evolution

13. Spin-dependent case

14. Soffer inequality evolution

15. Positivity preservation

16. Positivity for dilepton angular distribution • Angular distribution • Positivity of the matrix (= hadronic tensor in dilepton rest frame) • + cubic – det M0> 0 • 1st line – Lam&Tung by SF method

17. Close to saturation – helpful (Roloff,Peng,OT,in preparation)!

18. Constraint relating BM and transversity • Consider proton antiproton (same distribution) double transverse (same angular distributions for transversity and BM) polarized DY at y=0 (same arguments) • Mean value theorem + positivity -> f2(x,kT) > h12(x,kT) + kT2/M2 hT2(x,kT) • Stronger for larger kT • Transversity and BM cannot be large simultaneously • Similarly – for transversity FF and Collins

19. TMD(F) in coordinare impact parameter ) space • Correlator • Dirac structure –projects onto transverse direction • Light cone vector unnecessary (FS gauge) • Related to moment of Collins FF • WW – no evolution!

20. Simlarity to exclusive processes • Similar correlator between vacuum and pion – twist 3 pion DA • Also no evolution for zero mass and genuine twist 3 • Collins 2nd moment – twsit 3 • Higher – tower of twists • Similar to vacuunon-local condensates

21. Conclusions • Transverse sppin – 2 structures • Probabilistic NP approach possible • Transversity enters common positivity bound with BM • Chiral-odd TMD(F) – description in coordinate (impact parameter) space – similar to exclusive processes

22. Kinematic azimuthal asymmetry from polar one Only polar z asymmetry with respect to m! - azimuthal angle appears with new

23. Matching with pQCD results (J. Collins, PRL 42,291,1979) • Direct comparison: tan2 = (kT/Q)2 • New ingredient – expression for • Linear in kT • Saturates positivity constraint! • Extra probe of transverse momentum

24. Generalized Lam-Tung relation (OT’05) Relation between coefficients (high school math sufficient!) Reduced to standard LT relation for transverse polarization ( =1) LT - contains two very different inputs: kinematical asymmetry+transverse polarization

25. Positivity domain with (G)LT relations 2 “Standard” LT Longitudinal GLT 1 -1 -3 -2

26. When bounds are restrictive? • For (BM) – when virtual photon is longitudinal (like Soffer inequality for d-quarks) : kT – factorization - UGPD - nonsense polarization, cf talk of M.Deak) • For (collinear) transverse photon – strong bounds for and • Relevant for SSA in DY

27. SSA in DY TM integrated DY with one transverse polarized beam – unique SSA – gluonic pole (Hammon, Schaefer, OT) Positivity: twist 4 in denominator reqired

28. Contour gauge in DY:(Anikin,OT ) • Motivation of contour gauge – elimination of link • Appearance of infinity – mirror diagrams subtracted rather than added • Field • Gluonic pole appearance • cf naïve expectation • Source of phase?!

29. Phases without cuts • EM GI (experience from g2,DVCS) – 2 contributions • Cf PT – only one diagram for GI • NP tw3 analog - GI only if GP absent • GI with GP – “phase without cut”

30. Analogs/implications • Analogous pole – in gluon GPD • Prescription – also process-dependent: 2-jet diffractive production (Braun et al.) • Analogous diagram for GI – Boer, Qiu(04) • Our work besides consistency proof – factor 2 for asymmetry (missed before) • GI • Naive

31. Sivers function and formfactors • Relation between Sivers function and AMM known on the level of matrix elements (Brodsky, Schmidt, Burkardt) • Phase? • Duality for observables?

32. BG/DYW type duality for DY SSA in exclusive limit • Proton-antiproton DY – valence annihilation - cross section is described by Dirac FF squared • The same SSA due to interference of Dirac and Pauli FF’s with a phase shift • Exclusive large energy limit; x -> 1 : T(x,x)/q(x) -> Im F2/F1

33. Conclusions • General positivity constraints for DY angular distributions • SSA in DY : EM GI brings phases without cuts and factor 2 • BG/DYW duality for DY – relation of Sivers function at at large x to (Im of) time-like magnetic FF