1. Low-x gluon TMDs, the dipole picture and diffraction
Diffraction16 (Catania)
Low-x gluon TMDs, the dipole picture and diffraction
Daniel Boer, Sabrina Cotogno, Tom van Daal, Piet J Mulders, Andrea Signori and Yajin Zhou

2. Abstract
Abstract
We discuss the momentum distributions of gluons and consider the dependence of the gluon parton distribution functions (PDFs) on both fractional (longitudinal) momentum x and transverse momentum pT, referred to as the gluon TMDs. Looking at the operator structure of the TMDs, we are able to unify various descriptions at small-x including the dipole picture and the notions of pomeron and odderon exchange. We study the structure of gluon TMDs for unpolarized, vector polarized and tensor polarized targets.
TMD correlators and their operator structure, color gauge invariance
Parametrizations including polarization up to spin 1
Rank of TMD and operator structure
The Wilson loop correlator unifying ideas on diffraction, dipole picture and small-x behavior

3. Introduction PDFs and TMDs to incorporate hadron structurek k
PDFs and TMDs to incorporate hadron structure
High energies (lightlike n = P’/P.P’ and P.n=1)
Polarized targets provide opportunities and challenges
At high energies x linked to scaling variables (e.g. x = Q2/2P.q) and convolutions of transverse momenta are linked to azimuthal asymmetries (noncollinearity) requiring semi-inclusivity and/or polarization
Operator structure PDFs and TMDs can be embedded in a field theoretical framework via Operator Product Expansion (OPE), connecting Mellin moments and transverse moments of (TMD) PDFs with particular QCD matrix elements of operators (spin and twist expansion, gluonic pole matrix elements) k k

4. Matrix elements for TMDs
quark-quark
gluon-gluon
quark-gluon-quark
FA(p-p1,p)

5. TMDs and color gauge invariance
Gauge invariance in a non-local situation requires a gauge link U(0,x)
Introduces path dependence for F(x,pT)
‘Dominant’ paths: along lightcone connected at lightcone infinity (staples)
Reduces to ‘straight line’ for F(x)
x.P xT x
Matrix elements are not gauge invariant
OPE

6. Matrix elements for TMDs
quark-quark
gluon-gluon
… and even single Wilson loop correlator

7. Non-universality because of process dependent gauge links
TMD
path dependent gauge link
Gauge links associated with dimension zero (not suppressed!) collinear An = A+ gluons, leading for TMD correlators to process-dependence:
SIDIS DY
… A+ …
(resummation)
… A+ …
No unique way
Examples: DY & SIDIS
After integration: the same
F[-]
F[+]
Time reversal
Belitsky, Ji, Yuan, 2003; Boer, M, Pijlman, 2003

8. Simplest color flow classes for quarks (in lower hadron)

9. Simplest color flow classes for quarks (in lower hadron)

10. Color flow and gauge-link can become complex
Hadron (+)
Hadron (-)
hard
COLOR or

11. Non-universality because of process dependent gauge links
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:
gg  H
in gg  QQ
Fg[-,-]
Fg[+,+]
Fg[+,-]
Fg[-,+]
Bomhof, M, Pijlman, 2006; Dominguez, Xiao, Yuan, 2011

12. Color flow classes for gluons (in lower hadron)
Gluon correlators at small x related to Wilson loop correlator
Wilson loop correlator linked to dipole picture and diffraction at small x
(depends actually on k2 ~ kT2 in region where k+k- = x k- P+ << |kT2|)

13. A TMD picture for diffractive scattering
GAP
GAP X Y q k2 q’ P P’ p1 p2
GAP
Momentum flow in case of diffraction
x1  MX2/W2  0 and t  p1T2
Picture in terms of TMD and inclusion of gauge links
(including gauge links/collinear gluons in M ~ S – 1)
(Another way of looking at diffraction, cf Dominguez, Xiao, Yuan 2011 or older work of Gieseke, Qiao, Bartels 2000)
More complicated gauge links
Color-entanglement
Explicit calc: some contributions break factorizatn
On the right, a more detailed picture is visible. No factorization

14. Quark correlator Unpolarized target Vector polarized target
Surviving in collinear correlators F(x) and including flavor index
Note: be careful with use of h1T and non-traceless tensor with kT.ST since h1T is not a TMD of definite rank!

15. Definite rank TMDs
Expansion in constant tensors in transverse momentum space
… or traceless symmetric tensors (of definite rank)
Simple azimuthal behavior:
functions showing up in cos(mf) or sin(mf) asymmetries (wrt e.g. fT)
Simple Bessel transform to b-space (relevant for evolution):

16. Structure of quark (8) TMD PDFs in spin ½ target
8 TMDs F…(x,kT2)
Integrated (collinear) correlator: only circled ones survive
Collinear functions are spin-spin correlations
TMDs also momentum-spin correlations (spin-orbit) including also T-odd (single-spin) functions (appearing in single-spin asymmetries)
Existence of T-odd functions because of gauge link dependence!
PARTON SPIN
QUARKS U L T
TARGET SPIN

17. Structure of quark TMD PDFs in spin 1 target
PARTON SPIN
QUARKS U L T
TARGET SPIN LL LT TT
Hoodbhoy, Jaffe & Manohar, NP B312 (1988) 571: introduction of f1LL = b1
Bacchetta & M, PRD 62 (2000) ; h1LT first introduced as T-odd PDF
X. Ji, PRD 49 (1994) 114; introduction of (PFF)

18. Gluon correlators
Unpolarized target
Vector polarized target

19. Gluon correlators
Tensor polarized target

20. Structure of gluon TMD PDFs in spin 1 target
PARTON SPIN
GLUONS U L T
TARGET SPIN LL LT TT
Jaffe & Manohar, Nuclear gluonometry, PL B223 (1989) 218
PJM & Rodrigues, PR D63 (2001)
Meissner, Metz and Goeke, PR D76 (2007)
D Boer, S Cotogno, T van Daal, PJM, A Signori, Y Zhou, ArXiv

21. Untangling 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(Dn) = 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.
x = p.n
Mellin moments are being used for decades.
Just a recap.

22. Transverse moments  operator structure of TMD PDFs
Operator analysis for [U] dependence (e.g. [+] or [-]) TMD functions: in analogy to Mellin moments consider transverse moments  role for quark-gluon m.e.
calculable
T-even
T-odd
T-even (gauge-invariant derivative)
Gauge link has richer structure
3 matrix elements
T-odd (soft-gluon or gluonic pole, ETQS m.e.)
Efremov, Teryaev; Qiu, Sterman;
Brodsky, Hwang, Schmidt; Boer, Teryaev; Bomhof, Pijlman, M

23. Gluonic pole factors are calculable
CG[U] calculable gluonic pole factors (quarks)
Complicates life for ‘double pT’ situation such as Sivers-Sivers in DY, etc.
+ or – gauge links this is just +1 or -1
Also be illustrated by looking at the parametrization in PDFs.
Buffing, Mukherjee, M, PRD86 (2012) , ArXiv
Buffing, Mukherjee, M, PRD88 (2013) , ArXiv
Buffing, M, PRL 112 (2014),

24. Operator classification of quark TMDs (including trace terms)
factor
QUARK TMD RANK FOR VECTOR POL. (SPIN ½) HADRON 1 2 3
Three pretzelocities:
Process dependence also for (T-even) pretzelocity,
Buffing, Mukherjee, M, PRD86 (2012) , ArXiv

25. Operator classification of quark TMDs (including trace terms)
factor
QUARK TMD RANK FOR VECTOR POL. (SPIN ½) HADRON 1 2 3 …
Process dependence in pT dependence of TMDs due to gluonic pole operators
(e.g. affecting <pT2>
with df1[GG c](x) = 0
Boer, Buffing, M, JHEP08 (2015) 053, arXiv:

26. Classifying Polarized Quark TMDs (including tensor pol)
factor
QUARK TMD RANK FOR VECTOR POL. (SPIN ½) HADRON 1 2 3 …
factor
QUARK TMD RANK FOR TENSOR POL. (SPIN 1) HADRON 1 2 3 …

27. Operator classification of gluon TMDs
factor
GLUON TMD PDF RANK FOR SPIN ½ HADRON 1 2 3 …
factor
ADDITIONAL PDFs FOR TENSOR POL. SPIN 1 HADRON 1 2 3 4 …
D Boer, S Cotogno, T van Daal, PJM, A Signori Y Zhou, ArXiv

28. Small x physics in terms of TMDs
The single Wilson-loop correlator G0
Note limit x  0 for gluon TMDs linked to gluonic pole m.e. of G0
RHS depends in fact on t, which for x = 0 becomes pT2
factor
GLUON TMD PDF RANK FOR UNPOL. AND SPIN ½ HADRON 1 2 3 …

29. Small x physics in terms of gluon TMDs
Note limit x  0 for gluon TMDs linked to gluonic pole m.e. of G0
Dipole correlators: at small x only two structures for unpolarized and transversely polarized nucleons: pomeron & odderon structure
Dominguez, Xiao, Yuan 2011
D Boer, MG Echevarria, PJM, J Zhou, PRL 116 (2016) , ArXiv
D Boer, S Cotogno, T van Daal, PJM, A Signori, Y Zhou, ArXiv

30. Conclusion
(Generalized) universality of TMDs studied via operator product expansion, extending the well-known collinear distributions (including polarization 3 for quarks and 2 for gluons) to TMD PDF and PFF functions, ordered into functions of definite rank.
The rank m is linked to specific cos(mf) and sin(mf) azimuthal asymmetries and is important for connection to b-space.
Knowledge of operator structure is important (e.g. in lattice calculations).
Multiple operator possibilities for pretzelocity/transversity
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, even for unpolarized hadrons (with linearly polarized gluons) and possibly in diffractive processes via Wilson loop correlator.