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

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

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

Transveristy in QCD factorization

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

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 pp. Published in Phys.Rev.Lett.87:242001,2001. Leonard P. GambergPennsylvania U.Tufts U.Gary R. GoldsteinTufts U.

“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

Transverse spin and momentum correspondence Similarity of correlators (with opposite parity matrix structures) S T -> k T /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

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: [hep-ph] Xavier ArtruMokhtar ElchikhJean-Marc RichardJacques SofferOleg V. Teryaev Stability of positivity in the course of evolution

Kinetic interpretation of evolution

Master (balance) equation

Positivity vs evolution

Spin-dependent case

Soffer inequality evolution

Positivity preservation

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

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

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 -> f 2 (x,k T ) > h 1 2 (x,k T ) + k T 2 /M 2 h T 2 (x,k T ) Stronger for larger k T Transversity and BM cannot be large simultaneously Similarly – for transversity FF and Collins

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!

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 2 nd moment – twsit 3 Higher – tower of twists Similar to vacuunon-local condensates

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

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

Matching with pQCD results (J. Collins, PRL 42,291,1979) Direct comparison: tan 2 = (k T /Q) 2 New ingredient – expression for Linear in k T Saturates positivity constr a int! Extra probe of transverse momentum

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

Positivity domain with (G)LT relations “Standard” LT Longitudinal GLT

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

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

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?!

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”

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

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

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

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