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Defining TMD correlators

This document presents a detailed discussion on defining Transverse Momentum Distributions (TMDs) in Quantum Chromodynamics (QCD) during the May 2012 workshop at JLAB. Led by Piet Mulders and collaborators, it covers various aspects such as the construction of TMDs, the significance of transverse moments, the integration process, and examples involving unpolarized and polarized targets. The exploration of gauge links in gluon TMD correlators and their impact on higher azimuthal asymmetries are also included, emphasizing their relevance in studying linearly polarized gluons in protons.

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Defining TMD correlators

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  1. QCD workshop, JLAB, May 2012 Defining TMD correlators Piet Mulders m.g.a.buffing@vu.nl mulders@few.vu.nl

  2. Transverse moments • Remember for transverse moments • satisfying • What happens before integration? (at least in cases that a gauge-linked TMD makes sense)

  3. A possible construction of TMDs • Expansion • Weighting

  4. Examples • Expansion • Example 1: unpolarized target

  5. Examples • Expansion • Example 1: unpolarized target

  6. Examples • Expansion • Example 2: TMD PDFs for spin ½ target

  7. Examples • Expansion • Example 3: TMD PFFs for spin ½ fragment

  8. TMDs for gluons • The TMD gluon correlators contain two links, which can have different paths. Note that standard field displacement involves C = C’ • Basic (simplest) gauge links for gluon TMD correlators: Fg[-,-] Fg[+,+] Fg[+,-] Fg[-,+] C Bomhof, PJM, F Pijlman; EPJ C 47 (2006) 147 F Dominguez, B-W Xiao, F Yuan, PRL 106 (2011) 022301

  9. Higher azimuthal asymmetries and gluons • When construtin moments of these TMDs one needs operators <F F>, <F [F,F]>, <F, {F,F}>, <[F,F] [F,F]>, <{F,F} {F,F}> etc. again with increasing twist and several gluonic poles. Relevant in study of linearly polarized gluons in proton • Both Fg[+,+]+Fg[-,-] and Fg[+,-]+Fg[-,+]are T-even with 2nd moments containing <[F,G] [G,F]> and <{F,G} {G,F}> operator terms respectively C Bomhof and PJ Mulders, NPB 795 (2008) 409 F Dominguez, J-W Qiu, B-W Xiao and F Yuan, PR D85 (2012) 045003

  10. Beware! • Our treatment does not show factorization into TMDs with process-dependent gauge links! • It only concerns the study of transverse moments of TMDs and possible full definition of universal TMD distribution functions

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