1. Universality of transverse momentum dependent correlation functions Piet Mulders mulders@few.vu.nl INT Seattle September 16, 2009 Yerevan - 2009

2. Outline Introduction: correlation functions in QCD PDFs (and FFs): from unintegrated to collinear Transverse momenta: collinear and non-collinear parton correlators Distribution functions (collinear, TMD) Observables: azimuthal asymmetries Gauge links Resumming multi-gluon interactions: Initial/final states Color flow dependence The master formula and some applications Universality Conclusions OUTLINE

3. 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 __ INTRODUCTION

4. From hadrons to partons For hard scattering process involving electrons and photons the link to external particles is Confinement leads to hadrons as ‘sources’ for quarks and ‘source’ for quarks + gluons and …. INTRODUCTION

5. Parton p belongs to hadron P: p.P ~ M 2 For all other momenta K: p.K ~ P.K ~ s ~ Q 2 Introduce a generic lightlike 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.P h (or n determined by P and q; doesn’t matter at leading order!) Expand quark momentum Scales in hard processes PARTON CORRELATORS ~ Q~ M~ M 2 /Q Ralston and Soper 79, …

6. The correlator  (p,P;n) contains (in principle) information on hadron structure and can be expanded in a number of ‘Dirac structures’, as yet unintegrated, spectral functions/amplitudes depending on invariants: S i (p.n, p.P, p 2 ) or S i (x, p T 2, M R 2 ) Which (universal) quantities can be measured/isolated (identifying specific observables/asymmetries) in which of (preferably several) processes and in which way (relevant variables) at unintegrated, TMD or collinear level? What about going beyond tree-level (factorization)? Complication: color gauge invariance and colored remnants, higher order calculations in QCD Spectral functions or unintegrated PDFs PARTON CORRELATORS More on this in talk of Ted Rogers (Friday)

7. ‘Physical’ regions Support of (unintegrated) spectral functions ‘physical’ regions with p T 2 0 Distribution region: 0 < x < 1 Fragmentation (z = 1/x): x > 1 PARTON CORRELATORS PP ‘Physical’

8. Large values of momenta Calculable! etc. Bacchetta, Boer, Diehl, M JHEP 0808:023, 2008 (arXiv:0803.0227) PARTON CORRELATORS

9. Leading partonic structure of hadrons With P 1.P 2 ~ P 1.K 1 ~ P 1.K 2 ~ s (large) we get kinematic separation of soft and hard parts Integration  d   … =  d(p 1.P 1 )… leaves  (x,p T ) and  (z,k T ) containing nonlocal operator combinations with  or F INTRODUCTION Fragmentation functions Distribution functions

10. Leading partonic structure of hadrons With P 1.P 2 ~ P 1.K 1 ~ P 1.K 2 ~ s (large) we get kinematic separation of soft and hard parts Integration  d   … =  d(p 1.P 1 )… leaves  (x,p T ) and  (z,k T ) distribution correlator fragmentation correlator PP  (x, p T ) KK  (z, k T ) PARTON CORRELATORS

11. Integrate over p.P = p  (which is of order M 2 ) 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 Integrating quark correlators TMD collinear NON-COLLINEARITY Jaffe (1984), Diehl & Gousset, …

12. Relevance of transverse momenta (can they be measured?) 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 2-particle inclusive hadron-hadron scattering K2K2 K1K1      pp-scattering NON-COLLINEARITY We need more than one hadron and knowledge of hard process(es)! Boer & Vogelsang

13. Twist expansion Not all correlators are equally important Matrix elements expressed in P, p, n allow twist expansion (just as for local operators) (maximize contractions with n to get dominant contributions) Leading correlators involve ‘good fields’ in front form dynamics Above arguments are valid and useful for both collinear and TMD fcts In principle any number of gA.n = gA + can be included in correlators PARTON CORRELATORS Barbara Pasquini,...

14. Gauge link essential for color gauge invariance Arises from all ‘leading’ matrix elements containing      Basic (simplest) gauge links for TMD correlators: TMD correlators: quarks [][] [][] T TMD collinear NON-COLLINEARITY

15. TMD correlators: gluons The most general TMD gluon correlator contains two links, possibly with different paths. Note that standard field displacement involves C = C’ Basic (simplest) gauge links for gluon TMD correlators:  g [  ]  g [  ]  g [  ]  g [  ] NON-COLLINEARITY

16. TMD master formula APPLICATIONS Note that the summation over D is over diagrams and color-flow, e.g. for qq  qq subprocess:

17.  * + quark  gluon + quark Compare this with  *q  q: EXAMPLE Four diagrams, each with two color flow possibilities |M| 2 

18. quark + antiquark  gluon + photon Compare this with qq bar   *: EXAMPLE Four diagrams, each with two similar color flow possibilities |M| 2 

19. Result for integrated cross section Integrate into collinear cross section  (partonic cross section) APPLICATIONS

20. Collinear parametrizations Gauge invariant correlators  distribution functions Collinear quark correlators (leading part, no n-dependence) i.e. massless fermions with 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) i.e. massless gauge bosons with momentum distribution f 1 g (x) = g(x) and polarized distribution g 1 g (x) =  g(x) PARAMETRIZATION

21. 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:

22. Result for weighted cross section Construct weighted cross section (azimuthal asymmetry) APPLICATIONS New info on hadrons (cf models/lattice) Allows T-odd structure (exp. signal: SSA)

23. Result for weighted cross section APPLICATIONS

24. Drell-Yan and photon-jet production Drell-Yan Photon-jet production in hadron-hadron scattering APPLICATIONS q T = p 1T + p 2T

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

26. Drell-Yan and photon-jet production Weighted Drell-Yan Photon-jet production in hadron-hadron scattering APPLICATIONS Consider only p 1T contribution

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

28. Result for 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; …

29. TMD DISTRIBUTION AND FRAGMENTATION FUNCTIONS evenodd U T L  evenodd U T L  evenodd U T L  T odd T even odd U T L  evenodd U T L  evenodd U T L  Gamberg, Mukherjee, Mulders, 2008; Metz 2009, …

30. Gluonic poles (Spectral analysis)  G (x,x-x 1 )  G (x,x-x 1 )  G (x,x) = 0  G (x,x) Lim X 1  0 Gamberg, Mukherjee, M, 2008; Metz 2009, …

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

32. Result for weighted cross section (gluonic pole cross section)  APPLICATIONS

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

34. 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

35. 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, M, JHEP 0702 (2007) 029 [hep-ph/0609206]

36. TMD-universality : qg  qg D1D1 D2D2 D3D3 D4D4 D5D5 e.g. relevant in Bomhof, M, Vogelsang, Yuan, PRD 75 (2007) 074019 UNIVERSALITY Transverse momentum dependent weighted

37. TMD-universality: qg  qg UNIVERSALITY Transverse momentum dependent weighted

38. TMD-universality : qg  qg UNIVERSALITY Transverse momentum dependent weighted

39. TMD-universality : qg  qg It is possible to group the TMD functions in a smart way into two particular TMD functions (nontrivial for eight diagrams/four color-flow possibilities) We get process dependent and (instead of diagram-dependent) But still no factorization! Bomhof, M, NPB 795 (2008) 409 [arXiv: 0709.1390] UNIVERSALITY Transverse momentum dependent weighted

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 k T contributions; moreover it most likely also disappears for large p T Universality for TMD correlators? Bomhof, M, NPB 795 (2008) 409 [arXiv: 0709.1390] UNIVERSALITY

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

42. TMD DISTRIBUTION FUNCTIONS evenodd U T L  even Courtesy: Aram Kotzinian

43. Conclusions Transverse momentum dependence, experimentally important for single spin asymmetries, theoretically challenging (consistency, gauges and gauge links, universality, factorization) For leading integrated and weighted functions factorization is possible, but it requires besides the normal ‘partonic cross sections’ use of ‘gluonic pole cross sections’ and it is important to realize that q T -effects generally come from all partons If one realizes that the hard partonic part including its colorflow is an experimental handle, one recovers universality in terms of the (possibly large) number of possible gauge links, which can be separated into junk and a limited number of TMDs that give nonzero integrated and/or weighted results. CONCLUSIONS Thanks to experimentalists (HERMES, COMPASS, JLAB, RHIC, JPARC, GSI, …) for their continuous support.