1. Sivers Function and its Physical Interpretation IWHSS-09, Mainz, March 31, 2009 Oleg Teryaev JINR, Dubna

2. Outline Single Spin Asymmetries in QCD – Sources of helicity flip and (I)FSI Sivers function and time-like Magnetic Formfactors Non-universality of Sivers function: Colour correlations Sivers function evolution : 2 ways Conclusions

3. Single Spin Asymmetries Main properties: – Parity: transverse polarization – helicity flip – Imaginary phase – can be seen T- invariance or technically - from the imaginary i in the (quark) density matrix Various mechanisms – various sources of flip and phases

4. Flips and Phases in QCD Flips and Phases form soft, hard and overlap Transversity – hadron flip - > quark flip at large distances : chirality conservation in hard process - enter in pairs (talk of M. Anselmino) Pair of Chiral-odd DAs and GPDs (talks of M. Polyakov, M. Diehl) - chirality but NOT angular momentum conservation in collinear limit Solution – participation of non-collinear gluons from photon (Impactfactor) – 2 vector mesons production – feasible at COMPASS (Pire et al)

5. Sivers Sivers – hadron flip without quark flip (compensated by orbital momentum/gluon spin) + phase from (soft) gluon exchange Is it something unusual in this effect? Simple formfactor analog at large x!

6. Sivers function and formfactors Relation between Sivers and AMM known on the level of matrix elements (Brodsky, Schmidt, Burkardt) Phase? Duality for observables? Case study: SSA in DY

7. SSA in DY TM integrated DY with one transverse polarized beam – unique SSA – gluonic pole (Hammon, Schaefer, OT,95)

8. SSA in exclusive limit Proton-antiproton – 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 Assume analog of (DYW,BG) duality in (proton-antiproton – to have valence annihilation) DY Exclusive large energy limit; x -> 1 : T(x,x)/q(x) -> Im F2/F1

9. Kinematical domains for SSA’s x Sivers PT Twist 3 FF’s

10. Short+ large overlap– twist 3 Quarks – only from hadrons Various options for factorization – shift of SH separation (prototype of duality) New option for SSA: Instead of 1-loop twist 2 – Born twist 3: Efremov, OT (85, Ferminonc poles); Qiu, Sterman (91, GLUONIC poles)

11. 5 ways from Sivers to twist 3 Twist 3 DY - “Effective” or “non-universal” T-odd quark distribution from GP (Boer, Mulders, OT, 97) Moment of SF – GP (Boer, Mulders, Pijlman, 03) Explicit calculation of SIDVCS for Q >> P T (OT, TRANSVERSITY-05) - compensation of 1/Q suppression by GP) Matching of perturbative SF and twist 3 for DY, SIDIS +… (Ji,Qiu,Vogelsang,Yuan, 06; Bachetta, Boer, Diehl,Mulders,08) SF at large P T (Ratcliffe, OT, 07)-proof of Torino GPM (talk of M. Anselmino, S. Melis) modified by colour factors Follows general line of factorization – all UV to hard part. Also a way to QCD evolution ?!

12. Quark-gluon correlators Non-perturbative NUCLEON structure – physically mean the quark scattering in external gluon field of the HADRON. Depend on TWO parton momentum fractions For small transverse momenta – quark momentum fractions are close to each other- gluonic pole; probed if : Q >> P T >> M

13. Effective Sivers function Follows the expression similar to BMP Up to Colour Factors ! Defined by colour correlation between partons in hadron participating in (I)FSI SIDIS = +1; DY= -1: Collins sign rule Generally more complicated Factorization in terms of twist 3 but NOT SF

14. Colour correlations SIDIS (or PHOTOPRODUCTION) at large pT : -1/6 for mesons from quark, 3/2 from gluon fragmentation (kaons?) DY at large pT: 1/6 in quark antiquark annihilation, - 3/2 in gluon Compton subprocess – Collins sign rule more elaborate – involve crossing of distributions and fragmentations - special role of PION DY (COMPASS). Direct inclusive photons in pp = – 3/2 Hadronic pion production – more complicated – studied for P- exponentials by Amsterdam group + VW IF cancellation – small EFFECTIVE SF Vary for different diagrams – modification of hard part

15. Colour flow Quark at large PT:-1/6 Gluon at large PT : 3/2 Low PT – combination of quark and gluon: 4/3 (absorbed to definition of Sivers function) Similarity to colour transparency phenomenon

16. Evolution – how to extract Gluonic Poles (Ratcliffe,OT) Singular part – GP strength T (x) Helicity conservation – appearance of Natural GI object – Key observation (no bA contribution!)

17. Cornering of T (x) How to make x’s close to each other? Let us make them both close to 1 (val) or -1 (sea) Region close to FF interpretation! y x Symmetric asymptotics of moments L : and T variables

18. Evolution of T(x) (BKL,PGR,BB,BM,BBKT) General twist 3 evolution – complicated

19. Large m=n asymptotics Keep only logarithmically growing terms Multiplicative evolution Similar to unpolarized but numerically enhanced by factor 2 - 1/(2N) and softened by (1-z)/(1-x)

20. Phenomenology? Large x – the region of large SSA’s Evolution – > relative enhancement of T(x) BUT Resummation Twists higher than 3 “Quenching”

21. Another approach to Sivers evolution-BCM equation LLA kernel but corrected kinematics Recently applied to unpolaried TMD PDF evolution (Ceccopieri, Trentadue) First joint attempt (+ Ratcliffe,OT) To attack Sivers function

22. From unpolarized to Sivers Vector correlator Rotational invariance Sivers evoltion

23. Moments Weighted with momentum squared function evolution – similar to unpolarized Second Moment - multiplicative Burkardt SR – preserved due to momentum conservation (cf FF –Schaefer, OT) Singlet and Gluon go to seprate zeros asymptotically

24. Torino (talk by M. Anselmino) parameterization of Sivers

25. Qualitative features – from Gaussian to power-like

26. Matching of two approaches Power tail – should be included into redefinition of coefficient function? Factorization? Colour correlations for BCM? Emission from soft gluon – recovery of twist 3 colour factor CF+CA/2?

27. Conclusions Sivers function at large x is dual to interference of Dirac and (imaginary part of) Pauli FF’s Evolution of 2nd transverse moment at large x may be extracted from generic twist 3 evolution equation Result is similar to unpolarized evolution but modified by colour (“supersymmetric”=CF+CA/2) and kinematical factors BCM equation – also similar to unpolarized Matching of 2 approaches is not obvious

28. Outlook Phenomenological applications : resummations, higher twists Pre-asymptotic terms Theoretical significance (integrability etc)? Exclusive analogs : cornering of H(x,x)?

29. Sum rules EOM + n-independence (GI+rotational invariance) –relation to (genuine twist 3) DIS structure functions

30. Sum rules -II To simplify – low moments Especially simple – if only gluonic pole kept:

31. Gluonic poles and Sivers function Gluonic poles – effective Sivers functions-Hard and Soft parts talk, but SOFTLY Implies the sum rule for effective Sivers function (soft=gluonic pole dominance assumed in the whole allowed x’s region of quark-gluon correlator)

32. Compatibility of SSA and DIS Extractions of and modeling of Sivers function: – “mirror” u and d Second moment at % level Twist -3 - similar for neutron and proton and of the same sign – no mirror picture seen –but supported by colour ordering! Scale of Sivers function reasonable, but flavor dependence differs qualitatively. Inclusion of pp data, global analysis including gluonic (=Sivers) and fermionic poles HERMES, RHIC, E704 –like phonons and rotons in liquid helium; small moment and large E704 SSA imply oscillations JLAB –measure SF and g2 in the same run

33. CONCLUSIONS 5 th way from SF to GP – proof of Torino recipe supplemented by colour correlations Effective SF – small in pp - factorization in terms of twist 3 only Large x – relation between SF, GP and time-like FF’s

34. Outlook (high energies) TMD vs UGPD T-odd UGPD? T-odd (P/O) diffractive distribiutions (analogs - also at small energies) Quark-hadron duality: description of gluon coupling to exotic objects – diffractive production

35. Relation of Sivers function to GPDs Qualitatively similar to Anomalous Magnetic Moment (Brodsky et al) Quantification : weighted TM moment of Sivers PROPORTIONAL to GPD E (hep-ph/0612205 ): Burkardt SR for Sivers functions is now related to Ji SR for E and, in turn, to Equivalence Principle

36. How gravity is coupled to nucleons? Current or constituent quark masses ?– neither! Energy momentum tensor - like electromagnertic current describes the coupling to photons

37. Equivalence principle Newtonian – “Falling elevator” – well known and checked Post-Newtonian – gravity action on SPIN – known since 1962 (Kobzarev and Okun’) – not yet checked Anomalous gravitomagnetic moment iz ZERO or Classical and QUANTUM rotators behave in the SAME way

38. Gravitational formfactors Conservation laws - zero Anomalous Gravitomagnetic Moment : (g=2) May be extracted from high-energy experiments/NPQCD calculations Describe the partition of angular momentum between quarks and gluons Describe interaction with both classical and TeV gravity – similar t-dependence to EM FF

39. Electromagnetism vs Gravity Interaction – field vs metric deviation Static limit Mass as charge – equivalence principle

40. Gravitomagnetism Gravitomagnetic field – action on spin – ½ from spin dragging twice smaller than EM Lorentz force – similar to EM case: factor ½ cancelled with 2 from Larmor frequency same as EM Orbital and Spin momenta dragging – the same - Equivalence principle

41. Sivers function and Extended Equivalence principle Second moment of E – zero SEPARATELY for quarks and gluons –only in QCD beyond PT (OT, 2001) - supported by lattice simulations etc.. -> Gluon Sivers function is small! (COMPASS, STAR, Brodsky&Gardner) BUT: gluon orbital momentum is NOT small: total – about 1/2, if small spin – large (longitudinal) orbital momentum Gluon Sivers function should result from twist 3 correlator of 3 gluons: remains to be proved!

42. Generalization of Equivalence principle Various arguments: AGM 0 separately for quarks and gluons – most clear from the lattice (LHPC/SESAM, confirmed recently)

43. CONCLUSIONS Sivers and other TMD functions contain infinite tower of twists starting from 3 – special role of moments Colour charge of initial/final partons crucial – NO factorization in naive sense Transverse momentum dependence of Sivers SSA in SIDIS and DY (PAX) is a new sensitive test of QCD Relation of Sivers function to twist 3 in DIS: Reasonable magnitude, but problems with flavor dependence.Bochum results with suppressed singlet twist 3 supported! Relation of Sivers to GPD’s – link to Nucleon Spin and Equivalence Principle Problems: evolution (no WW for Sivers) and SR from twist 3.