1. 1 Color Entanglement and Universality of TMDs mulders@few.vu.nl Piet Mulders POETIC V – International Workshop on Physics Opportunities at an ElecTron-Ion Collider Yale, 22 – 26 September 2014

2. TMD issues (remarks at JLAB pre-town meeting) Hadron physics basic part of nuclear physics: Hydrogen atom: full first family of ‘elementary’ particles. Nucleons: smallest asymptotically accessible parts Hadron physics requires interactions theory – experiment (it is a step by step process). ‘Spin puzzle’ is an excellent example of this (Ji’s talk) PDFs and TMD PDFs allow measurement of non-local correlations between constituents (partonic wave functions), rather than global quantities (charge or axial charge densities) f(x) is basic leading input (numbers, chiralities at parton level), very useful as a basic tool (that is under control!) f(x,kT) is next step in non-locality (at parton level) with relevance for understanding spin - orbit structure of nucleon (not OAM!) as a tool for more precise measurements (not yet under control) TMD library: arXiv 1408.3015 [hep-ph] Quarks and gluons! 2 August 2014

3. ABSTRACT TMDs of definite rank Piet Mulders (Nikhef/VU University Amsterdam) Transverse momentum dependent (TMD) parton distribution functions include path- dependent Wilson lines. Using moments in x and pT, it is possible to study the operator structure which is relevant for evolution as well as for modelling of TMDs or lattice calculations. Using the moments it is possible to categorize the TMDs according to their rank which labels the relevant azimuthal behavior in pT. For quark TMDs of rank 2, such as the Pretzelocity TMD, and gluon TMDs of rank 1 and higher, such as the TMD describing linear polarization of gluons, one finds multiple TMDs depending on the color structure of the operators. The explicit appearance of these TMDs in scattering processes, including diffractive scattering, involves factors depends on the color flow in the process in two ways, namely a factor depending on the gluonic rank of the TMD as well as an additional process-dependent factor if multiple TMDs are involved in a process, such as double Sivers asymmetries in the Drell-Yan process. [thanks to: Maarten Buffing, Andrea Signori, Sabrina Cotogno, Cristian Pisano, Thomas Kasemets, Miguel Echevarria, Paul Hoyer and Asmita Mukherjee] 3

4. Starting point TMD’s: color gauge invariant correlators, describing distribution and fragmentation functions including partonic transverse momentum. Nonlocality in field operators including transverse directions Observable in azimuthal dependence, i.e. noncollinearity in hard processes (convolution in kT) Transverse separation complicates gauge link structure TMDs encode novel aspects of hadronic structure, e.g. spin-orbit correlations, such as T-odd transversely polarized quarks or T-even longitudinally polarized gluons in an unpolarized hadron, thus possible applications for precision probing at the LHC, but for sure at a polarized EIC. 4

5. (Un)integrated correlators  = p  integration makes time-ordering automatic. The soft part is simply sliced at the light-front Is already equivalent to a point-like interaction Local operators with calculable anomalous dimension unintegrated TMD (light-front) collinear (light-cone) local 5

6. 6  Gauge links come from dimension zero (not suppressed!) collinear A.n gluons, but leads for TMD correlators to process-dependence: Simplest gauge links for quark TMDs  [-]  [+] Time reversal TMD collinear … A n … AV Belitsky, X Ji and F Yuan, NP B 656 (2003) 165 D Boer, PJM and F Pijlman, NP B 667 (2003) 201 AV Belitsky, X Ji and F Yuan, NP B 656 (2003) 165 D Boer, PJM and F Pijlman, NP B 667 (2003) 201 SIDIS DY

7. 7 Simplest gauge links for gluon TMDs  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:  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 C Bomhof, PJM, F Pijlman; EPJ C 47 (2006) 147 F Dominguez, B-W Xiao, F Yuan, PRL 106 (2011) 022301 gg  H in gg  QQ

8. Including gauge links we have well-defined matrix elements for TMDs but this implies multiple possiblities for gauge links depending on the process and the color flow in the diagram Leading quark TMDs Leading gluon TMDs: Color gauge invariant correlators 8

9. Some details on the gauge links (1) Proper gluon fields (F rather than A, Wilson lines and boundary terms) Resummation of soft n.A gluons (coupling to outgoing color-line) for one correlator produces a gauge-line (along n) Boundary terms give transverse pieces

10. Some details on the gauge links (2) Resummation of soft n.A gluons (coupling to outgoing color-line) for one correlator produces a gauge-line (along n) The lowest order contributions for soft gluons from two different correlators coupling to outgoing color-line resums into gauge-knots: shuffle product of all relevant gauge-lines from that (external initial/final state) line.

11. Which gauge links? With more (initial state) hadrons color gets entangled, e.g. in pp Gauge knot U + [p 1,p 2,…] Outgoing color contributes to a future pointing gauge link in  (p 2 ) and future pointing part of a gauge loop in the gauge link for  (p 1 ) This causes trouble with factorization 11 T.C. Rogers, PJM, PR D81 (2010) 094006

12. Which gauge links? Can be color-detangled if only p T of one correlator is relevant (using polarization, …) but must include Wilson loops in final U 12 MGA Buffing, PJM, JHEP 07 (2011) 065

13. Complications for DY at measured Q T if the transverse momentum of two initial state hadrons is involved Trouble appears already in DY 13

14. Basic strategy: operator product expansion Taylor expansion for functions around zero Mellin transform for functions on [-1,1] interval functions in (transverse) plane 14

15. Operator structure in collinear case (reminder) Collinear functions and x-moments Moments correspond to local matrix elements of operators that all have the same twist since dim(D n ) = 0 Moments are particularly useful because their anomalous dimensions can be rigorously calculated and these can be Mellin transformed into the splitting functions that govern the QCD evolution. 15 x = p.n

16. Operator structure in TMD case For TMD functions one can consider transverse moments Upon integration, these do involve collinear twist-3 multi-parton correlators 16 MGA Buffing, A Mukherjee, PJM, PRD 86 (2012) 074030, Arxiv: 1207.3221 [hep-ph]

17. Operator structure in TMD case For first transverse moment one needs quark-gluon correlators In principle multi-parton, but we need 17  F (p-p 1,p) T-even (gauge-invariant derivative) T-odd (soft-gluon or gluonic pole) Efremov, Teryaev; Qiu, Sterman; Brodsky, Hwang, Schmidt; Boer, Teryaev, M; Bomhof, Pijlman, M

18. Tr c (GG  Tr c (GG) Tr c (  Operator structure in TMD case Transverse moments can be expressed in these particular collinear multi-parton twist-3 correlators (which are not suppressed!) C G [U] calculable gluonic pole factors 18 T-even T-odd T-even T-odd

19. Distribution versus fragmentation functions Operators: 19 T-evenT-odd (gluonic pole) T-even operator combination, but still T-odd functions! out state Collins, Metz; Meissner, Metz; Gamberg, M, Mukherjee, PR D 83 (2011) 071503

20. Collecting right moments gives expansion into full TMD PDFs of definite rank but for TMD PFFs 20 Classifying Quark TMDs

21. Only a finite number needed: rank up to 2(S hadron +s parton ) Rank m shows up as cos(m  ) and sin(m  ) azimuthal asymmetries No gluonic poles for PFFs Classifying Quark TMDs factorTMD PDF RANK 0123 1 21 MGA Buffing, A Mukherjee, PJM, PRD2012, Arxiv: 1207.3221 [hep-ph] factorTMD PFF RANK 0123 1

22. Only a finite number needed: rank up to 2(S hadron +s parton ) Rank m shows up as cos(m  ) and sin(m  ) azimuthal asymmetries Example: quarks in an unpolarized target are described by just 2 functions Classifying Quark TMDs 22 factorQUARK TMD PDF RANK UNPOLARIZED HADRON 0123 1 T-odd T-even [B-M function]

23. Classifying Quark TMDs factorQUARK TMD PDFs RANK SPIN ½ HADRON 0123 1 23 Three pretzelocities: Process dependence in h 1 (broadening)

24. Classifying Quark TMDs factorQUARK TMD PDFs RANK SPIN ½ HADRON 0123 1 24 factorQUARK TMD PFFs RANK SPIN ½ HADRON 0123 1 Just a single ‘pretzelocity’ PFF

25. Classifying Gluon TMDs factorGLUON TMD PDF RANK SPIN ½ HADRON 0123 1 25 factorGLUON TMD PDF RANK UNPOLARIZED HADRON 0123 1

26. Multiple TMDs in cross sections 26

27. Complications if the transverse momentum of two initial state hadrons is involved, resulting for DY at measured Q T in Just as for twist-3 squared in collinear DY Correlators in description of hard process (e.g. DY) 27

28. Classifying Quark TMDs factorTMD RANK 0123 1 28 MGA Buffing, PJM, PRL (2014), Arxiv: 1309.4681 [hep-ph]

29. Example: quarks in an unpolarized target needs only 2 functions Resulting in cross section for unpolarized DY at measured Q T Remember classification of Quark TMDs 29 factorQUARK TMD RANK UNPOLARIZED HADRON 0123 1 D. Boer, PRD 60 (1999) 014012; MGA Buffing, PRL (2014) PJM, Arxiv: 1309.4681 [hep-ph] contains f 1 contains h 1 perp·

30. A TMD picture for diffractive scattering 30 X Y q k2k2 q’ PP’ p1p1 p2p2 GAP Momentum flow in case of diffraction x 1  M X 2 /W 2  0 and t  p 1T 2 Picture in terms of TMD and inclusion of gauge links (including gauge links/collinear gluons in M ~ S – 1) (Work in progress: Hoyer, Kasemets, Pisano, M) (Another way of looking at diffraction, cf Dominguez, Xiao, Yuan 2011 or older work of Gieseke, Qiao, Bartels 2000) GAP

31. A TMD picture for diffractive scattering 31 X Y q k2k2 q’ PP’ p1p1 p2p2 GAP Cross section involving correlators for proton and photon GAP

32. Ingredients Photon PDF Diffractive correlator 32

33. Conclusion with (potential) rewards (Generalized) universality studied via operator product expansion, extending the well-known collinear distributions (including polarization 3 for quarks and 2 for gluons) to novel TMD PDF and PFF functions, ordered into functions of definite rank. Knowledge of operator structure is important for lattice calculations. ! Multiple operator possibilities for pretzelocity/transversity The rank m is linked to specific cos(m  ) and sin(m  ) azimuthal asymmetries. ! The TMD PDFs appear in cross sections with specific calculable factors that deviate from (or extend on) the naïve parton universality for hadron-hadron scattering. ! Applications in polarized high energy processes, but also in unpolarized situations (linearly polarized gluons) and possibly diffractive processes. 33

34. Thank you 34