Diffraction 16 Catania Lowx gluon TMDs the dipole

Diffraction 16 (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 mulders@few. vu. nl 1

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 p T, 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. 1. 2. 3. 4. 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 2

Introduction PDFs and TMDs to incorporate hadron structure k High energies (lightlike n = P’/P. P’ and P. n=1) k k k Polarized targets provide opportunities and challenges At high energies x linked to scaling variables (e. g. x = Q 2/2 P. 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) 3

Matrix elements for TMDs quark-quark gluon-gluon quark-gluon-quark FA(p-p 1, p) 4

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

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

Non-universality because of process dependent gauge links TMD path dependent gauge link u Gauge links associated with dimension zero (not suppressed!) collinear A n = A+ gluons, leading for TMD correlators to process-dependence: SIDIS DY … A+ … (resummation) … A+ … F[-] F[+] Time reversal Belitsky, Ji, Yuan, 2003; Boer, M, Pijlman, 2003 7

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

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

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

Non-universality because of process dependent gauge links u The TMD gluon correlators contain two links, which can have different paths. Note that standard field displacement involves C = C’ u Basic (simplest) gauge links for gluon TMD correlators: Fg[+, +] Fg[-, -] Fg[+, -] Fg[-, +] gg H in gg QQ Bomhof, M, Pijlman, 2006; Dominguez, Xiao, Yuan, 2011 11

Color flow classes for gluons (in lower hadron) Gluon correlators at small x related to Wilson loop correlator linked to dipole picture and diffraction at small x (depends actually on k 2 ~ k. T 2 in region where k+k- = x k- P+ << |k. T 2|) 12

A TMD picture for diffractive scattering q’ q p 2 P p 1 X k 2 P’ GAP Y Momentum flow in case of diffraction x 1 MX 2/W 2 0 and t p 1 T 2 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) 13

Quark correlator Unpolarized target Vector polarized target Surviving in collinear correlators F(x) and including flavor index Note: be careful with use of h 1 T and non-traceless tensor with k. T. ST since h 1 T is not a TMD of definite rank! 14

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. f. T) Simple Bessel transform to b-space (relevant for evolution): 15

Structure of quark (8) TMD PDFs in spin ½ target 8 TMDs F…(x, k. T 2) PARTON SPIN TARGET SPIN QUARKS U L T Integrated (collinear) correlator: only circled ones survive Collinear functions are spin-spin correlations TMDs also momentum-spin correlations (spin-orbit) including also -odd (single-spin) functions (appearing in single-spin asymmetries) Existence of T-odd functions because of gauge link dependence! T 16

Structure of quark TMD PDFs in spin 1 target PARTON SPIN QUARKS U TARGET SPIN L T LL LT TT Hoodbhoy, Jaffe & Manohar, NP B 312 (1988) 571: introduction of f 1 LL = b 1 Bacchetta & M, PRD 62 (2000) 114004; h 1 LT first introduced as T-odd PDF X. Ji, PRD 49 (1994) 114; introduction of (PFF) 17

Gluon correlators Unpolarized target Vector polarized target 18

Gluon correlators Tensor polarized target 19

Structure of gluon TMD PDFs in spin 1 target PARTON SPIN GLUONS U TARGET SPIN L T LL LT TT Jaffe & Manohar, Nuclear gluonometry, PL B 223 (1989) 218 PJM & Rodrigues, PR D 63 (2001) 094021 Meissner, Metz and Goeke, PR D 76 (2007) 034002 D Boer, S Cotogno, T van Daal, PJM, A Signori, Y Zhou, Ar. Xiv 1607. 01654 20

Untangling operator structure in collinear case (reminder) Collinear functions and x-moments x = p. n 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. 21

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) T-odd (soft-gluon or gluonic pole, ETQS m. e. ) Efremov, Teryaev; Qiu, Sterman; Brodsky, Hwang, Schmidt; Boer, Teryaev; Bomhof, Pijlman, M 22

Gluonic pole factors are calculable CG[U] calculable gluonic pole factors (quarks) Complicates life for ‘double p. T’ situation such as Sivers-Sivers in DY, etc. Buffing, Mukherjee, M, PRD 86 (2012) 074030, Ar. Xiv 1207. 3221 Buffing, Mukherjee, M, PRD 88 (2013) 054027, Ar. Xiv 1306. 5897 Buffing, M, PRL 112 (2014), 092002 23

Operator classification of quark TMDs (including trace terms) factor QUARK TMD RANK FOR VECTOR POL. (SPIN ½) HADRON 0 1 2 3 1 Three pretzelocities: Process dependence also for (T-even) pretzelocity, Buffing, Mukherjee, M, PRD 86 (2012) 074030, Ar. Xiv 1207. 3221 24

Operator classification of quark TMDs (including trace terms) factor QUARK TMD RANK FOR VECTOR POL. (SPIN ½) HADRON 0 1 2 3 1 … … Process dependence in p. T dependence of TMDs due to gluonic pole operators (e. g. affecting <p. T 2> with df 1[GG c](x) = 0 Boer, Buffing, M, JHEP 08 (2015) 053, ar. Xiv: 1503. 03760 25

Classifying Polarized Quark TMDs (including tensor pol) factor QUARK TMD RANK FOR VECTOR POL. (SPIN ½) HADRON 0 1 2 3 1 … factor QUARK TMD RANK FOR TENSOR POL. (SPIN 1) HADRON 0 1 … … 2 3 1 … 26

Operator classification of gluon TMDs factor GLUON TMD PDF RANK FOR SPIN ½ HADRON 0 1 2 3 1 … … factor … ADDITIONAL PDFs FOR TENSOR POL. SPIN 1 HADRON 0 1 … … 2 3 4 1 … … … D Boer, S Cotogno, T van Daal, PJM, A Signori Y Zhou, Ar. Xiv 1607. 01654 … 27

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

Small x physics in terms of gluon TMDs Note limit x 0 for gluon TMDs linked to gluonic pole m. e. of G 0 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) 122001, Ar. Xiv 1511. 03485 D Boer, S Cotogno, T van Daal, PJM, A Signori, Y Zhou, Ar. Xiv 1607. 01654 29

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 scattering. Applications in polarized high energy processes, even for unpolarized hadrons (with linearly polarized gluons) and possibly in diffractive processes via Wilson loop correlator. 30