1. Universality of Transverse Momentum Dependent Distribution Functions Piet Mulders mulders@few.vu.nl INT Program: Gluons and the Quark Sea at High Energies Perturbative & Non-perturbative aspects of QCD at collider energies Seattle, September 14, 2010 INT PROGRAM

2. Abstract Universality of TMDs P.J. Mulders, NIKHEF & VU University, Amsterdam The basic idea of PDFs is achieving a factorized description with soft and hard parts, soft parts being portable and hard parts being calculable. At high energies interference disappears and PDFs are interpreted as probabilities. Beyond the collinear treatment one considers besides the dependence on partonic momentum fractions x also the dependence on the transverse momentum p T of the partons. Experimentally that dependence on transverse momenta shows up in azimuthal asymmetries for produced hadrons or jet-jet asymmetries. In combination with polarization one can explain single spin asymmetries or look at production of polarized Lambdas. Single spin asymmetries are examples of time-reversal odd (T-odd) observables, which rely on phases in a scattering process. In a gauge theory one naturally expects phases, but these need to be detected through interference or, as it turns out in high-energy scattering processes, by considering transverse momenta (p T = iD T – gA T ) in a proper color gaug-invariant theoretical description. The relevant transverse momentum dependent (TMD) distribution functions in the partonic correlators can be interpreted as ‘single spin correlations’ describing transverse spin distributions in an unpolarized hadron or unpolarized quarks in a (transversely) polarized hadron. For gluons similar correlations can be considered. The TMD correlators emerge with specific future or past-pointing gauge links as well as more complex gauge link structures, that depend on the color flow in the hard subprocess, complicating the universality of PDFs. To compare with collinear treatments, one can consider specific p T -weighted cross sections and recover the Qiu-Sterman description of e.g. single spin asymmetries in terms of gluonic pole matrix elements. The study of TMD correlators thus breaks the simple universality of quark and gluon distributions multiplying a hard cross section, but the resulting description offers various new possibilities to employ particular spin-momentum correlations of partons in high energy processes, while such correlations of course also offer new insight into hadron structure. ABSTRACT

3. Issues Basic goals  Try to incorporate small p T (second scale) extending f(x) to f(x,p T 2 )‏  Let’s assume it works (leaving some challenges: factorization)‏ (Theoretical) complications  Operator structure (definition with gauge links) and twist expansion  Gluonic interactions essential (Brodsky, Hwang, Schmidt, …)‏  Effects from soft gluon limit (Qiu, Sterman, Koike, …)‏  Universality (at tree-level) and factorization (Qiu, Yuan, Vogelsang, Collins, …)‏ Opportunities/Applications  New effects (Sivers, Collins, Boer-M)‏  Combination with (transverse) polarization is extremely useful, but also jet effects/broadening/Lambda production (polarimetry)‏ INTRODUCTION

4. Featuring: phases in gauge theories GAUGE THEORIES

5. PDFs and PFFs Basic idea of PDFs is getting a factorized description of high energy scattering processes calculable defined (!)‏ & portable HARD PROCESS

6. Hard part: QCD & Standard Model QCD framework (including electroweak theory) provides the machinery to calculate transition amplitudes, e.g. g*q  q, qq  g*, g*  qq, qq  qq, qg  qg, etc. Example: qg  qg Calculations work for plane waves Use relations such as __ HARD PROCESS

7. Soft part: hadronic matrix elements For hard scattering process involving electrons and photons the link to external particles is, indeed, the ‘one-particle wave function’ Confinement, however, implies hadrons as ‘sources’ for quarks … and also as ‘source’ for quarks + gluons … and also …. PARTON CORRELATORS

8. Thus, the nonperturbative input for calculating hard processes involves [instead of u i (p)u j (p)] forward matrix elements of the form Soft part: hadronic matrix elements quark momentum PARTON CORRELATORS _

9. Matrix elements containing   (gluon) fields produce gauge link … essential for color gauge invariant definition Not calculable like uu, but use parametrization (with ‘spectral functions’)‏ Soft parts: gauge invariant definitions PARTON CORRELATORS ++ … Any path yields a (different) definition

10. Parton p belongs to hadron P: p.P ~ M 2 For all other momenta K: p.K ~ P.K ~ s ~ Q 2 Affects soft physics as a generic light-like vector n satisfying P.n=1, then n ~ 1/Q The vector n gets its meaning in a particular hard process, n = K/K.P e.g. in SIDIS: n = P h /P h.P (or n determined by P and q; doesn’t matter at leading order!)‏ Expand quark momenta Order: partonic momenta at high energies (NON-)COLLINEARITY ~ Q~ M~ M 2 /Q Ralston and Soper 79, … In remainder: retain two scales

11. … rather than considering general correlator  [C] (p,P,…), one integrates over p.P = p  (~M R 2, which is of order M 2 ) and/or p T The integration over p  = p.P makes time-ordering automatic (Jaffe, 1984). This works for  (x) and  (x,p T )‏ This allows the interpretation of soft (squared) matrix elements as forward antiquark-target amplitudes, which satisfy particular support properties, etc. For collinear correlators  (x), this can be extended to off-forward correlators and generalized PDFs Integrating quark correlators TMD collinear (NON-)COLLINEARITY Jaffe (1984), Diehl & Gousset (1998), …

12. Gauge links follow from inclusion of quark-gluon matrix elements Basic (simplest) gauge links for TMD correlators: Gauge-invariant definition of TMDs: which gauge links? [][] [][] T TMD collinear NON-COLLINEARITY These become a unique ‘simple’ Wilson line upon p T -integration

13. Relevance of transverse momenta? In a hard process one probes quarks and gluons Parton momenta fixed by kinematics (external momenta)‏ DIS SIDIS Also possible for transverse momenta of partons SIDIS DY 2-particle inclusive hadron-hadron scattering K2K2 K1K1      pp-scattering NON-COLLINEARITY Care is needed: we need more than one hadron and knowledge of hard process(es)! Boer & Vogelsang Bacchetta, Boer, Diehl, M Second scale! (large q T = collinear)‏ Boer & Vogelsang

14. Relevance of transverse momenta? TMD-correlators are not T-invariant QCD is T-invariant T-odd observables  T-odd TMDs Example of T-odd observable: single spin asymmetry Left-right asymmetry in Collinear hard T-odd contribution zero (~  s 2.  s m q ), p T -contributions remain NON-COLLINEARITY ~ s 3/2 ~ s Qiu & Sterman, 1997 … + ‘normal’ twist three stuff (FF)‏

15. Gauge link calculation for SIDIS Expand gluon fields and reshuffle a bit: GAUGE LINK Coupled to a final state parton, the collinear gluons add up to a U [+] gauge link, (with endpoint from transverse reshuffling)‏

16. Gauge link calculation for SIDIS DIS & SIDIS |M| 2  GAUGE LINK p k For simple color flow, absorb in  (p) and  (k)‏  (p)‏  (k)‏  (p)‏  (k)‏  (p)‏  (k)‏

17. Gauge link calculation for DY DY |M| 2  GAUGE LINK p1p1 p2p2 For simple color flow, absorb in  (p 1 ) and  (p 2 )‏  (p 1 )‏  (p 2 )‏ Gauge link differs from the SIDIS one!

18. Collinear parametrizations Gauge invariant correlators  PDFs Collinear quark correlators (leading part, no n-dependence, no link dependence)‏ Interpretation: quark momentum distribution f 1 q (x) = q(x), chiral distribution g 1 q (x) =  q(x) and transverse spin polarization h 1 q (x) =  q(x) in a spin ½ hadron Collinear gluon correlators (leading part)‏ Interpretation: gluon momentum distribution f 1 g (x) = g(x) and polarized distribution g 1 g (x) =  g(x) PARAMETRIZATION Analogy:

19. COLLINEAR DISTRIBUTION AND FRAGMENTATION FUNCTIONS evenodd U T L  evenodd U T L  Including flavor index one commonly writes flip U T L For gluons one commonly writes:

20. TMD parametrizations Gauge invariant correlators  distribution functions TMD quark correlators (leading part, unpolarized) including T-odd part Interpretation: quark momentum distribution f 1 q (x,p T ) and its transverse spin polarization h 1  q (x,p T ) both in an unpolarized hadron The function h 1  q (x,p T ) is T-odd (momentum-spin correlations!)‏ TMD gluon correlators (leading part, unpolarized)‏ Interpretation: gluon momentum distribution f 1 g (x,p T ) and its linear polarization h 1  g (x,p T ) in an unpolarized hadron (both are T-even)‏ (NON-)COLLINEARITY

21. TMD DISTRIBUTION FUNCTIONS evenodd U T L  even U T L flip

22. TMD correlators: gluons The most general TMD gluon correlator contains two links, which in general can have different paths. Note that standard field displacement involves C = C’ Basic (simplest) gauge links for gluon TMD correlators:  g [  ]  g [  ]  g [  ]  g [  ] NON-COLLINEARITY C Bomhof, PJM, F Pijlman; EPJ C 47 (2006) 147 F Dominguez, B-W Xiao, F Yuan, hep-ph/1009.2141 (today!)‏

23. Problem: color entanglement Illustrated in quark-quark scattering (even without color exchange)‏  (p 1 )‏  (p 2 )‏  (k 2 )‏  (k 1 )‏ aa  b* b*b* aa kinematically decoupled, but color entangled! COLOR ENTANGLEMENT T.C. Rogers, PJM, PR D81:094006,2010/hep-ph 1001.2977 several eikonal factors

24. Color disentanglement? COLOR ENTANGLEMENT Kinematical decoupling:

25. Color disentanglement?  (p 1 )‏  (p 2 )‏  (k 2 )‏  (k 1 )‏ aa  b* b*b* aa COLOR ENTANGLEMENT T.C. Rogers, PJM, PR D81:094006, 2010/hep-ph 1001.2977 ? Ideally:

26. Color disentanglement We would need correlators with more complex gaugelinks, such as If only a TMD for one hadron is involved color can indeed be disentangled in this way (conjecture)‏ This can be used to simplify the integrated cross sections and to study single weighted azimuthal asymmetries COLOR ENTANGLEMENT

27. In situation that just the p T -dependence of a single hadron is relevant, we find a TMD correlator with specific process-dependent gauge links: X… but there is no full TMD factorization TMD treatment (tree level!)‏ APPLICATIONS ‘single-hadron’ TMD factorization + …

28. Single hadron TMD factorized part APPLICATIONS + multi-hadron correlators Particularly useful if multi-hadron correlations disappear (collinear and pT-weighted cases)‏

29. Result for integrated cross section Integrate into collinear cross section  (partonic cross section)‏ APPLICATIONS Multi-hadron correlations irrelevant Gauge link structure becomes irrelevant! + multi-hadron correlators

30. Result for single weighted cross section Construct weighted cross section (azimuthal asymmetry)‏ APPLICATIONS New info on hadrons (cf models/lattice)‏ Can be handled theoretically Allows T-odd structure (explain SSA)‏ Leading cos 2  effects in DY Jet broadening (weighting of cos 2  )‏ In each term, dependence on gauge link remains for only one of the correlators + multi-hadron correlators

31. Result for single weighted cross section APPLICATIONS + multi-hadron correlators

32. Result for single weighted cross section  universal matrix elements  G (x,x) is gluonic pole (x 1 = 0) matrix element T-evenT-odd (operator structure)‏  G (p,p  p 1 ) APPLICATIONS Qiu, Sterman; Koike; Brodsky, Hwang, Schmidt, … + multi-hadron correlators

33. Result for single weighted cross section  universal matrix elements Examples are: C G [U + ] = 1, C G [U  ] = -1, C G [W U + ] = 3, C G [Tr(W)U + ] = N c APPLICATIONS + multi-hadron correlators

34. Result for single weighted cross section (gluonic pole cross section)‏  APPLICATIONS + multi-hadron correlators

35. quark + antiquark  gluon + photon cf qq bar   *: APPLICATION Four diagrams, each with two similar color flow possibilities |M| 2  ‘Single hadron’ TMD factorized part!

36. Drell-Yan and photon-jet production Weighted Drell-Yan Weighted photon-jet production in hadron-hadron scattering APPLICATIONS

37. Drell-Yan and photon-jet production Weighted Drell-Yan Weighted photon-jet production in hadron-hadron scattering APPLICATIONS

38. Drell-Yan and photon-jet production Weighted Drell-Yan Weighted photon-jet production in hadron-hadron scattering Note: color-flow in this case is more SIDIS like than DY APPLICATIONS Gluonic pole cross sections

39. For quark distributions one needs normal hard cross sections For T-odd PDF (such as transversely polarized quarks in unpolarized proton) one gets modified hard cross sections for DIS: for DY: Gluonic pole cross sections y (gluonic pole cross section)‏ APPLICATIONS Bomhof, PJM, JHEP 0702 (2007) 029 [hep-ph/0609206]

40. We can work with basic TMD functions  [±] (x,p T ) + ‘junk’ The ‘junk’ constitutes process-dependent residual TMDs Thus: The junk gives zero after integrating (  (x) = 0) and after weighting (   (x) = 0), i.e. cancelling p T contributions; moreover it most likely also disappears for large p T Junk pieces might be collected in ‘entangled multi-hadron’ part. Universality and factorization for TMDs? Bomhof, PJM, NPB 795 (2008) 409 [arXiv: 0709.1390] UNIVERSALITY

41. (Limited) universality for TMD functions QUARKS GLUONS Bomhof, PJM, NPB 795 (2008) 409 [arXiv: 0709.1390] UNIVERSALITY

42. Outlook (work in progress)‏ A possible way out to include the color entanglement of different hadrons is to include all (or some) hadrons in one soft part (Ted Rogers/Mert Aybat)‏ In p T -averaged situation it reduces to a product of single-hadron collinear soft parts, for single-weighted case to two single-hadron collinear soft parts, … This may have advantages that a single treatment of factorization (higher order QCD contributions) becomes feasible The multi-hadron soft part will most likely involve novel multi-hadron correlations (through color entanglement)‏ OUTLOOK

43. Summarizing: the TMD challenge multi-GeV p T ~ GeV Preference for those observables that vanish at multi-GeV level (e.g. T-odd ones)‏ Theoretical challenges Universality Factorization Experimental considerations High energy is must! Azimuthal asymmetries (particle identification at small p T )‏ Transverse polarization to define directions and select chirality flip CONCLUSIONS

44. Impact on INT programme issues Origin of nucleon spin Spatial nucleon structure Beyond Standard Model Extreme QCD matter TMDs CONCLUSIONS