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Universality of transverse momentum dependent correlation functions

Yerevan - 2009. INT Seattle September 16, 2009. Universality of transverse momentum dependent correlation functions. Piet Mulders. mulders@few.vu.nl. OUTLINE. Outline. Introduction: correlation functions in QCD PDFs (and FFs): from unintegrated to collinear

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Universality of transverse momentum dependent correlation functions

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

  2. OUTLINE 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

  3. INTRODUCTION 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 _ _

  4. INTRODUCTION 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 ….

  5. PARTON CORRELATORS ~ Q ~ M ~ M2/Q Scales in hard processes • Parton p belongs to hadron P: p.P ~ M2 • For all other momenta K: p.K ~ P.K ~ s ~ Q2 • 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 = Ph/P.Ph (or n determined by P and q; doesn’t matter at leading order!) • Expand quark momentum Ralston and Soper 79, …

  6. PARTON CORRELATORS Spectral functions or unintegrated PDFs • The correlator F(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: Si(p.n, p.P, p2) or Si(x, pT2, MR2) • 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 More on this in talk of Ted Rogers (Friday)

  7. PARTON CORRELATORS P P ‘Physical’ regions • Support of (unintegrated) spectral functions • ‘physical’ regions with pT2 < 0 and MR2 > 0 • Distribution region: 0 < x < 1 • Fragmentation (z = 1/x): x > 1 ‘Physical’

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

  9. INTRODUCTION Leading partonic structure of hadrons With P1.P2 ~ P1.K1 ~ P1.K2 ~ s (large) we get kinematic separation of soft and hard parts Integration ds1… =  d(p1.P1)… leaves F(x,pT) and D(z,kT) containing nonlocal operator combinations with f = y or F Distribution functions Fragmentation functions

  10. PARTON CORRELATORS fragmentation correlator distribution correlator K K D(z, kT) F(x, pT) P P Leading partonic structure of hadrons With P1.P2 ~ P1.K1 ~ P1.K2 ~ s (large) we get kinematicseparation of soft and hard parts Integration ds1… =  d(p1.P1)… leaves F(x,pT) and D(z,kT)

  11. NON-COLLINEARITY Integrating quark correlators TMD collinear • Integrate over p.P = p- (which is of order M2) • The integration over p- = p.P makes time-ordering automatic (Jaffe, 1984). This works for F(x) andF(x,pT) • This allows the interpretation of soft (squared) matrix elements as forward antiquark-target amplitudes, which satisfy particular support properties, etc. • For collinear correlators F(x), this can be extended to off-forward correlators Jaffe (1984), Diehl & Gousset, …

  12. NON-COLLINEARITY f2 - f1 K1 df K2 pp-scattering 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 We need more than one hadron and knowledge of hard process(es)! Boer & Vogelsang

  13. PARTON CORRELATORS 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 Barbara Pasquini, ...

  14. NON-COLLINEARITY TMD correlators: quarks TMD collinear • Gauge link essential for color gauge invariance • Arises from all ‘leading’ matrix elements containing y A+...A+ y • Basic (simplest) gauge links for TMD correlators: F[-] F[+] T

  15. NON-COLLINEARITY 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: Fg[+,+] Fg[-,-] Fg[-,+] Fg[+,-]

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

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

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

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

  20. PARAMETRIZATION Collinear parametrizations • Gauge invariant correlators  distribution functions • Collinear quark correlators (leading part, no n-dependence) • i.e. massless fermions with momentum distribution f1q(x) = q(x), chiral distribution g1q(x) = Dq(x) and transverse spin polarization h1q(x) = dq(x) in a spin ½ hadron • Collinear gluon correlators (leading part) • i.e. massless gauge bosons with momentum distributionf1g(x) = g(x) and polarized distribution g1g(x) = Dg(x)

  21. COLLINEAR DISTRIBUTION AND FRAGMENTATION FUNCTIONS Including flavor index one commonly writes For gluons one commonly writes: c even odd flip U U L L T T c even odd U L T

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

  23. APPLICATIONS Result for weighted cross section

  24. APPLICATIONS Drell-Yan and photon-jet production qT = p1T + p2T • Drell-Yan • Photon-jet production in hadron-hadron scattering

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

  26. APPLICATIONS Drell-Yan and photon-jet production • Weighted Drell-Yan • Photon-jet production in hadron-hadron scattering Consider only p1T contribution

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

  28. APPLICATIONS FG(p,p-p1) Result for weighted cross section   T-even universal matrix elements T-odd (operator structure) FG(x,x) is gluonic pole (x1 = 0) matrix element Qiu, Sterman; Koike; …

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

  30. Gamberg, Mukherjee, M, 2008; Metz 2009, … Gluonic poles (Spectral analysis) DG(x,x-x1) FG(x,x-x1) Lim X1  0 FG(x,x) DG(x,x) = 0

  31. APPLICATIONS Result for weighted cross section   universal matrix elements Examples are: CG[U+] = 1, CG [U- ] = -1, CG[U□ U+] = 3, CG [Tr(U□)U+] = Nc

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

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

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

  35. APPLICATIONS (gluonic pole cross section) y Gluonic pole cross sections • 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: Bomhof, M, JHEP 0702 (2007) 029 [hep-ph/0609206]

  36. UNIVERSALITY TMD-universality : qg qg weighted Transverse momentum dependent D1 D2 D3 D4 D5 e.g. relevant in Bomhof, M, Vogelsang, Yuan, PRD 75 (2007) 074019

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

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

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

  40. UNIVERSALITY Universality for TMD correlators? • We can work with basic TMD functions F[±](x,pT) + ‘junk’ • The ‘junk’ constitutes process-dependent residual TMDs • Thus: • The junk gives zero after integrating (dF(x) = 0) and after weighting (dF(x) = 0), i.e. cancelling kT contributions; moreover it most likely also disappears for large pT Bomhof, M, NPB 795 (2008) 409 [arXiv: 0709.1390]

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

  42. TMD DISTRIBUTION FUNCTIONS c even odd even odd U L T Courtesy: Aram Kotzinian

  43. CONCLUSIONS 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 qT-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. Thanks to experimentalists (HERMES, COMPASS, JLAB, RHIC, JPARC, GSI, …) for their continuous support.

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