1. Defining TMD correlators
QCD workshop, JLAB, May 2012
Defining TMD correlators
Piet Mulders

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)

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)

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