Transverse momenta of partons in high-energy scattering processes

1. International School “Enrico Fermi” Varenna, June 2011 Transverse momenta of partons in high-energy scattering processes Piet Mulders mulders@few.vu.nl

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. d u u proton 3 colors Valence structure of hadrons: global properties of nucleons • mass • charge • spin • magnetic moment • isospin, strangeness • baryon number • Mp Mn 940 MeV • Qp = 1, Qn = 0 • s = ½ • gp 5.59, gn -3.83 • I = ½: (p,n) S = 0 • B = 1 Quarks as constituents

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

5. A (weak) look at the nucleon n  p + e- + n • = 900 s  Axial charge GA(0) = 1.26

6. A virtual look at the proton _ g* N N g*+ N  N

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

8. Nucleon densities from virtual look neutron proton • charge density  0 • u more central than d? • role of antiquarks? • n = n0 + pp- + … ?

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

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! • LQCD (quarks, gluons) known!

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

12. A hard look at the proton • Hard virtual momenta ( q2 = Q2 ~ many GeV2) can couple to (two) soft momenta g* + N  jet g* jet + jet

13. Experiments!

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

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. Soft part: hadronic matrix elements Thus, the nonperturbative input for calculating hard processes involves [instead of ui(p)uj(p)]forward matrix elements of the form _ quark momentum INTRODUCTION

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

18. Example: DIS

19. q ~ Q ~ M ~ M2/Q P Principle for DIS • Instead of partons use correlators • Expand parton momenta (for SIDIS take e.g. n= Ph/Ph.P) p

20. F(x) LEADING (in 1/Q) x = xB = -q2/P.q (calculation of) cross section in DIS Full calculation + + + … +

21. Lightcone dominance in DIS

22. q P Result for DIS p

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

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

26. Next example: SIDIS

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

28. Lightfront dominance in SIDIS Three external momenta P Ph q transverse directions relevant qT = q + xB P – Ph/zh or qT = -Ph^/zh

29. Ph q P Result for SIDIS k p

30. Parametrization of F(x,pT) • Also T-odd functions are allowed • Functions h1^ (BM) and f1T^ (Sivers) nonzero! • Similar functions (of course) exist as fragmentation functions (no T-constraints) H1^ (Collins) and D1T^

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

32. pT-dependent functions Matrix representationfor M = [F[±](x,pT)g+]T T-odd: g1T g1T – i f1T^ and h1L^  h1L^ + i h1^(imaginary parts) Bacchetta, Boglione, Henneman & Mulders PRL 85 (2000) 712

33. Jaffe (1984), Diehl & Gousset (1998), … Integrated quark correlators: collinear and TMD • Rather than considering general correlator F(p,P,…), one thus integrates over p.P = p- (~MR2, which is of order M2) • and/or pT (which is of order 1) • The integration over p- = p.P makes time-ordering automatic. This works for F(x) andF(x,pT) • This allows the interpretation of soft (squared) matrix elements as forward antiquark-target amplitudes (untruncated!), which satisfy particular analyticity and support properties, etc. TMD lightfront collinear lightcone (NON-)COLLINEARITY

34. Summarizing oppertunities of TMDs • TMD quark correlators (leading part, unpolarized) including T-odd part • Interpretation: quark momentum distribution f1q(x,pT) and its transverse spin polarization h1q(x,pT) both in an unpolarized hadron • The function h1q(x,pT) is T-odd (momentum-spin correlations!) • TMD gluon correlators (leading part, unpolarized) • Interpretation: gluon momentum distribution f1g(x,pT) and its linear polarization h1g(x,pT) in an unpolarized hadron (both are T-even) (NON-)COLLINEARITY

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(x) = An(x) fields (dimension zero!) but in color gauge invariant combinations • Transverse momentum involves ‘twist 3’. dim 0: dim 1: OPERATOR STRUCTURE

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

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

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

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

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

41. T.C. Rogers, PJM, PR D81 (2010) 094006 Complications (example: qq  qq) U+[n] [p1,p2,k1] modifies color flow, spoiling universality (and factorization) COLOR ENTANGLEMENT

42. Color entanglement COLOR ENTANGLEMENT

43. Featuring: phases in gauge theories COLOR ENTANGLEMENT

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