1. 1 Transverse momenta of partons in high-energy scattering processes Piet Mulders International School, Orsay June 2012 mulders@few.vu.nl

2. 2 Introduction What are we after? –Structure of proton (quarks and gluons) –Use of proton as a tool What are our means? –QCD as part of the Standard Model

3. 3 3 colors Valence structure of hadrons: global properties of nucleons d uu proton mass charge spin magnetic moment isospin, strangeness baryon number M p  M n  940 MeV Q p = 1, Q n = 0 s = ½ g p  5.59, g n  -3.83 I = ½: (p,n) S = 0 B = 1 Quarks as constituents

4. 4 A real look at the proton  + N  …. Nucleon excitation spectrum E ~ 1/R ~ 200 MeV R ~ 1 fm

5. 5 A (weak) look at the nucleon n  p + e  +  = 900 s  Axial charge G A (0) = 1.26

6. 6 A virtual look at the proton    N N   + N  N _

7. 7 Local – forward and off-forward m.e. Local operators (coordinate space densities): PP’P’  Static properties: Form factors Examples: (axial) charge mass spin magnetic moment angular momentum

8. 8 Nucleon densities from virtual look proton neutron charge density  0 u more central than d? role of antiquarks? n = n 0 + p   + … ?

9. 9 Quark and gluon operators Given the QCD framework, the operators are known quarkic or gluonic currents such as probed in specific combinations by photons, Z- or W-bosons (axial) vector currents energy-momentum currents probed by gravitons

10. 10 Towards the quarks themselves The current provides the densities but only in specific combinations, e.g. quarks minus antiquarks and only flavor weighted No information about their correlations, (effectively) pions, or … Can we go beyond these global observables (which correspond to local operators)? Yes, in high energy (semi-)inclusive measurements we will have access to non-local operators! L QCD (quarks, gluons) known!

11. 11 Non-local probing Nonlocal forward operators (correlators): Specifically useful: ‘squares’ Momentum space densities of  -ons: Selectivity at high energies: q = p

12. 12 A hard look at the proton Hard virtual momenta (  q 2 = Q 2 ~ many GeV 2 ) can couple to (two) soft momenta   + N  jet    jet + jet

13. 13 Experiments!

14. 14 QCD & Standard Model QCD framework (including electroweak theory) provides the machinery to calculate cross sections, e.g.  *q  q, qq   *,  *  qq, qq  qq, qg  qg, etc. E.g. qg  qg Calculations work for plane waves __

15. 15 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

16. 16 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 INTRODUCTION _

17. 17 PDFs and PFFs Basic idea of PDFs is to get a full factorized description of high energy scattering processes calculable defined (!) & portable INTRODUCTION Give a meaning to integration variables!

18. 18 Example: DIS

19. 19 Instead of partons use correlators Expand parton momenta (for SIDIS take e.g. n= P h /P h.P) Principle for DIS q P ~ Q~ M~ M 2 /Q p

20. 20 (calculation of) cross section in DIS Full calculation + … + + + LEADING (in 1/Q) x = x B =  q 2 /P.q  (x)

21. Lightcone dominance in DIS

22. 22 Result for DIS q P p

23. 23 Parametrization of lightcone correlator Jaffe & Ji NP B 375 (1992) 527 Jaffe & Ji PRL 71 (1993) 2547 leading part M/P + parts appear as M/Q terms in cross section T-reversal applies to  (x)  no T-odd functions

24. 24 Basis of partons  ‘ Good part ’ of Dirac space is 2-dimensional  Interpretation of DF ’ s unpolarized quark distribution helicity or chirality distribution transverse spin distr. or transversity

25. 25  Off-diagonal elements (RL or LR) are chiral-odd functions  Chiral-odd soft parts must appear with partner in e.g. SIDIS, DY Matrix representation for M = [  (x)  + ] T Quark production matrix, directly related to the helicity formalism Anselmino et al. Bacchetta, Boglione, Henneman & Mulders PRL 85 (2000) 712

26. 26 Next example: SIDIS

27. 27 (calculation of) cross section in SIDIS Full calculation + + … + + LEADING (in 1/Q)

28. Lightfront dominance in SIDIS Three external momenta P P h q transverse directions relevant q T = q + x B P – P h /z h or q T = -P h  /z h

29. 29 Result for SIDIS q P PhPh p k

30. 30 Parametrization of  (x,p T ) Also T-odd functions are allowed Functions h 1  (BM) and f 1T  (Sivers) nonzero! Similar functions (of course) exist as fragmentation functions (no T- constraints) H 1  (Collins) and D 1T 

31. Interpretation unpolarized quark distribution helicity or chirality distribution transverse spin distr. or transversity need p T T-odd

32. 32  p T -dependent functions T-odd: g 1T  g 1T – i f 1T  and h 1L   h 1L  + i h 1  (imaginary parts) Matrix representation for M = [   ±] (x,p T )  + ] T Bacchetta, Boglione, Henneman & Mulders PRL 85 (2000) 712

33. 33 Rather than considering general correlator  (p,P,…), one thus integrates over p.P = p  (~M R 2, which is of order M 2 ) and/or p T (which is of order 1) The integration over p  = p.P makes time-ordering automatic. This works for  (x) and  (x,p T ) This allows the interpretation of soft (squared) matrix elements as forward antiquark-target amplitudes (untruncated!), which satisfy particular analyticity and support properties, etc. Integrated quark correlators: collinear and TMD TMD collinear (NON-)COLLINEARITY Jaffe (1984), Diehl & Gousset (1998), … lightfront lightcone

34. 34 Summarizing oppertunities of TMDs 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

35. 35 Twist expansion of (non-local) correlators Dimensional analysis to determine importance of matrix elements (just as for local operators) maximize contractions with n to get leading contributions ‘ Good ’ fermion fields and ‘ transverse ’ gauge fields and in addition any number of n.A(  ) = A n (x) fields (dimension zero!) but in color gauge invariant combinations Transverse momentum involves ‘ twist 3 ’. dim 0: dim 1: OPERATOR STRUCTURE

36. 36 Matrix elements containing   (gluon) fields produce gauge link … essential for color gauge invariant definition Soft parts: gauge invariant definitions OPERATOR STRUCTURE + + … Any path yields a (different) definition

37. 37 Gauge link results from leading gluons Expand gluon fields and reshuffle a bit: OPERATOR STRUCTURE Coupling only to final state partons, the collinear gluons add up to a U + gauge link, (with transverse connection from A T   G n  reshuffling) A.V. Belitsky, X.Ji, F. Yuan, NPB 656 (2003) 165 D. Boer, PJM, F. Pijlman, NPB 667 (2003) 201

38. 38 Even simplest links for TMD correlators non-trivial: Gauge-invariant definition of TMDs: which gauge links? [][] [][] T TMD collinear OPERATOR STRUCTURE These merge into a ‘ simple ’ Wilson line in collinear (p T -integrated) case

39. 39 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 [  ] C Bomhof, PJM, F Pijlman; EPJ C 47 (2006) 147 F Dominguez, B-W Xiao, F Yuan, PRL 106 (2011) 022301 OPERATOR STRUCTURE

40. 40 Gauge invariance for SIDIS COLOR ENTANGLEMENT Strategy: transverse moments Problems with double T-odd functions in DY M.G.A. Buffing, PJM, 1105.4804

41. 41 Complications (example: qq  qq) T.C. Rogers, PJM, PR D81 (2010) 094006 U + [n] [p 1,p 2,k 1 ] modifies color flow, spoiling universality (and factorization) COLOR ENTANGLEMENT

42. 42 Color entanglement COLOR ENTANGLEMENT

43. 43 Featuring: phases in gauge theories COLOR ENTANGLEMENT

44. 44 Conclusions TMDs enter in processes with more than one hadron involved (e.g. SIDIS and DY) Rich phenomenology (Alessandro Bacchetta) Relevance for JLab, Compass, RHIC, JParc, GSI, LHC, EIC and LHeC Role for models using light-cone wf (Barbara Pasquini) and lattice gauge theories (Philipp Haegler) Link of TMD (non-collinear) and GPDs (off-forward) Easy cases are collinear and 1-parton un-integrated (1PU) processes, with in the latter case for the TMD a (complex) gauge link, depending on the color flow in the tree-level hard process Finding gauge links is only first step, (dis)proving QCD factorization is next (recent work of Ted Rogers and Mert Aybat) SUMMARY

45. 45 Rather than considering general correlator  (p,P,…), one integrates over p.P = p  (~M R 2, which is of order M 2 ) and/or p T (which is of order 1) The integration over p  = p.P makes time-ordering automatic. This works for  (x) and  (x,p T ) This allows the interpretation of soft (squared) matrix elements as forward antiquark-target amplitudes (untruncated!), which satisfy particular analyticity and support properties, etc. Thus we need integrated correlators TMD collinear (NON-)COLLINEARITY Jaffe (1984), Diehl & Gousset (1998), … lightfront lightcone

46. 46 Large values of momenta Calculable! etc. Bacchetta, Boer, Diehl, M JHEP 0808:023, 2008 (arXiv:0803.0227) PARTON CORRELATORS p 0T ~ M M << p T < Q hard

47. 47 T-odd  single spin asymmetry with time reversal constraint only even-spin asymmetries the time reversal constraint cannot be applied in DY or in  1-particle inclusive DIS or e  e  In those cases single spin asymmetries can be used to measure T-odd quantities (such as T-odd distribution or fragmentation functions) * *  W  (q;P,S;P h,S h ) =  W  (  q;P,S;P h,S h )  W  (q;P,S;P h,S h ) = W  (q;P,S;P h,S h )  W  (q;P,S;P h,S h ) = W  (q;P,  S;P h,  S h )  W  (q;P,S;P h,S h ) = W  (q;P,S;P h,S h ) _ __ _ _ ___ _ _ _ _ time reversal symmetry structure parity hermiticity * *

48. 48 Relevance of transverse momenta in hadron-hadron scattering At high energies fractional parton momenta fixed by kinematics (external momenta) DY Also possible for transverse momenta of partons 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 Second scale!

49. 49 Lepto-production of pions H 1  is T-odd and chiral-odd

50. 50 Gauge link in DIS In limit of large Q 2 the result of ‘ handbag diagram ’ survives … + contributions from A + gluons ensuring color gauge invariance A + gluons  gauge link Ellis, Furmanski, Petronzio Efremov, Radyushkin A+A+

51. Distribution From  A T (  )  m.e. including the gauge link (in SIDIS) A+A+ One needs also A T G +  =  + A T  A T  (  )= A T  ( ∞ ) +  d  G +  Belitsky, Ji, Yuan, hep-ph/0208038 Boer, M, Pijlman, hep-ph/0303034

52. 52 Generic hard processes C. Bomhof, P.J. Mulders and F. Pijlman, PLB 596 (2004) 277 [hep-ph/0406099]; EPJ C 47 (2006) 147 [hep-ph/0601171] Link structure for fields in correlator 1 E.g. qq-scattering as hard subprocess Matrix elements involving parton 1 and additional gluon(s) A + = A.n appear at same (leading) order in ‘ twist ’ expansion insertions of gluons collinear with parton 1 are possible at many places this leads for correlator involving parton 1 to gauge links to lightcone infinity

53. 53 SIDIS SIDIS   [U + ] =  [+] DY   [U - ] =  [  ]

54. 54 A 2  2 hard processes: qq  qq E.g. qq-scattering as hard subprocess The correlator  (x,p T ) enters for each contributing term in squared amplitude with specific link  [Tr(U □ )U + ] (x,p T ) U □ = U + U  †  [U □ U + ] (x,p T )

55. 55 Gluons Using 3x3 matrix representation for U, one finds in gluon correlator appearance of two links, possibly with different paths. Note that standard field displacement involves C = C ’

56. 56 Integrating  [±] (x,p T )   [±] (x) collinear correlator 

57. 57 Integrating  [±] (x,p T )     [±] (x) transverse moment  G (p,p  p 1 ) T-evenT-odd

58. 58 Gluonic poles Thus:  [U] (x) =  (x)   [U]  (x) =    (x) + C G [U]  G  (x,x) Universal gluonic pole m.e. (T-odd for distributions)  G (x) contains the weighted T-odd functions h 1  (1) (x) [Boer-Mulders] and (for transversely polarized hadrons) the function f 1T  (1) (x) [Sivers]   (x) contains the T-even functions h 1L  (1) (x) and g 1T  (1) (x) For SIDIS/DY links: C G [U ± ] = ±1 In other hard processes one encounters different factors: C G [U □ U + ] = 3, C G [Tr(U □ )U + ] = N c Efremov and Teryaev 1982; Qiu and Sterman 1991 Boer, Mulders, Pijlman, NPB 667 (2003) 201 ~ ~

59. 59 A 2  2 hard processes: qq  qq E.g. qq-scattering as hard subprocess The correlator  (x,p T ) enters for each contributing term in squared amplitude with specific link  [Tr(U □ )U + ] (x,p T ) U □ = U + U  †  [U □ U + ] (x,p T )

60. 60 examples: qq  qq in pp C G [D 1 ] D1D1 = C G [D 2 ] = C G [D 4 ] C G [D 3 ] D2D2 D3D3 D4D4 Bacchetta, Bomhof, Pijlman, Mulders, PRD 72 (2005) 034030; hep-ph/0505268

61. 61 examples: qq  qq in pp D1D1 For N c  : C G [D 1 ]   1 (color flow as DY) Bacchetta, Bomhof, D ’ Alesio,Bomhof, Mulders, Murgia, hep-ph/0703153

62. 62 Gluonic pole cross sections In order to absorb the factors C G [U], one can define specific hard cross sections for gluonic poles (which will appear with the functions in transverse moments) for pp: etc. for SIDIS: for DY: Bomhof, Mulders, JHEP 0702 (2007) 029 [hep-ph/0609206] (gluonic pole cross section) y

63. 63 examples: qg  q  in pp D1D1 D2D2 D3D3 D4D4 Only one factor, but more DY-like than SIDIS Note: also etc. Transverse momentum dependent weigted

64. 64 examples: qg  qg D1D1 D2D2 D3D3 D4D4 D5D5 e.g. relevant in Bomhof, Mulders, Vogelsang, Yuan, PRD 75 (2007) 074019 Transverse momentum dependent weighted

65. 65 examples: qg  qg Transverse momentum dependent weighted

66. 66 examples: qg  qg Transverse momentum dependent weighted

67. 67 examples: qg  qg It is also possible to group the TMD functions in a smart way into two! (nontrivial for nine diagrams/four color-flow possibilities) But still no factorization! Transverse momentum dependent weighted

68. 68 ‘ Residual ’ TMDs We find that we can work with basic TMD functions  [±] (x,p T ) + ‘ junk ’ The ‘ junk ’ constitutes process-dependent residual TMDs The residuals satisfies   (x) = 0 and  G (x,x) = 0, i.e. cancelling k T contributions; moreover they most likely disappear for large k T definite T-behavior no definite T-behavior

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75. 75 1-parton unintegrated Resummation of all phases spoils universality Transverse moments (p T -weighting) feels entanglement Special situations for only one transverse momentum, as in single weighted asymmetries But: it does produces ‘ complex ’ gauge links Applications of 1PU is looking for gluon h 1  g (linear gluon polarization) using jet or heavy quark production in ep scattering (e.g. EIC), D. Boer, S.J. Brodsky, PJM, C. Pisano, PRL 106 (2011) 132001 COLOR ENTANGLEMENT

76. 76 Full color disentanglement? NO! COLOR ENTANGLEMENT Loop 1: X aa  b* b*b* aa M.G.A. Buffing, PJM; in preparation

77. 77 Result for integrated cross section Collinear cross section  (partonic cross section) APPLICATIONS Gauge link structure becomes irrelevant! (1PU)

78. 78 Result for single weighted cross section Single weighted cross section (azimuthal asymmetry) APPLICATIONS (1PU)

79. 79 Result for single weighted cross section APPLICATIONS (1PU)

80. 80 Result for single weighted cross section  universal matrix elements  G (x,x) is gluonic pole (x 1 = 0) matrix element (color entangled!) T-evenT-odd (operator structure) APPLICATIONS Qiu, Sterman; Koike; Brodsky, Hwang, Schmidt, …  G (p,p  p 1 ) (1PU)

81. 81 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 (1PU)

82. 82 Result for single weighted cross section (gluonic pole cross section)  APPLICATIONS (1PU) T-odd part

83. 83 Higher p T moments Higher transverse moments involve yet more functions Important application: there are no complications for fragmentation, since the ‘ extra ’ functions  G,  GG, … vanish. using the link to ‘ amplitudes ’ ; L. Gamberg, A. Mukherjee, PJM, PRD 83 (2011) 071503 (R) In general, by looking at higher transverse moments at tree-level, one concludes that transverse momentum effects from different initial state hadrons cannot simply factorize. SUMMARY

84. 84 DIS q P

85. 85 DIS q P Basis 1 Basis 2

86. 86 Principle for DY Instead of partons use correlators Expand parton momenta (for DY take e.g. n 1 = P 2 /P 1.P 2 ) ~ Q~ M~ M 2 /Q (NON-)COLLINEARITY