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
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