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 2001. 4pp. Published in Phys.Rev.Lett.87:242001,2001.

“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) 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

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

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 M0> 0 1st 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 -> f2(x,kT) > h1 2(x,kT) + kT

2/M2 hT 2(x,kT) Stronger for larger kT 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 2nd 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

zasymmetry with respect to m! - azimuthal

angle appears with new

0

n m

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

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

2

-2

1-1-3

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

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