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Transversity and TMD Friends: Hard Mesons and Photons Productions

Explore the significance of transversity in quantum mechanics of spin 1/2 and its relation to TMDs in impacting parameter space. Learn about transversity in QCD factorization, application in Single Spin Asymmetries calculations, and the evolution of Soffer inequality. Discover insights from theoretical models and experimental data.

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Transversity and TMD Friends: Hard Mesons and Photons Productions

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  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

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