Sivers Function and its Physical Interpretation IWHSS-09 , Mainz, March 31, 2009

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 >> PT (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 PT (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 samesign – nomirror 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 • 5th 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, confirmedrecently)

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.