Low-x gluon TMDs, the dipole picture and diffraction

Transcription

1 Diffraction16 (Catania) Low-x gluon MDs, the dipole picture and diffraction Daniel Boer, Sabrina Cotogno, om van Daal, Piet J Mulders, Andrea Signori and Yajin Zhou mulders@few.vu.nl 1

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 p, referred to as the gluon MDs. Looking at the operator structure of the MDs, 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 MDs for unpolarized, vector polarized and tensor polarized targets. 1. MD correlators and their operator structure, color gauge invariance 2. Parametrizations including polarization up to spin 1 3. Rank of MD and operator structure 4. he Wilson loop correlator unifying ideas on diffraction, dipole picture and small-x behavior 2

3 Introduction PDFs and MDs to incorporate hadron structure u i (k)u j (k) (/k) ij =) ij (k; P, S) (k) (k) g =) (k; P, S) High energies (lightlike n = P /P.P and P.n=1) k µ = xp µ + k µ +(k2 k 2 )n µ =) (x, k ; P, S) and (x, k ; P, S) Polarized targets provide opportunities and challenges MS µ = S L P µ + MS µ + M 2 S L n µ M 2 S µ = S LL P µ P P {µ S } L M 2 S µ +... ψ i (ξ) ψ j (0) Γ g αβ ( p) At high energies x linked to scaling variables (e.g. x = Q 2 /2P.q) and convolutions of transverse momenta are linked to azimuthal asymmetries (noncollinearity) requiring semi-inclusivity and/or polarization Operator structure PDFs and MDs can be embedded in a field theoretical framework via Operator Product Expansion (OPE), connecting Mellin moments and transverse moments of (MD) PDFs with particular QCD matrix elements of operators (spin and twist expansion, gluonic pole matrix elements) k k k k 3

4 Matrix elements for MDs quark-quark ij(x, p ; n) = gluon-gluon Z d P d 2 (2 ) 3 e ip hp,s j (0) i ( ) P,Si n=0 ψ i (ξ) ψ j (0) µ (x, p ; n) = quark-gluon-quark Z d P d 2 (2 ) 3 e ip hp,s F nµ (0) F n ( ) P,Si n=0 Γ αβ ( p) Φ A (p-p 1,p) Φ α D;ij ( p p 1, p 1 p) = Φ α F;ij ( p p 1, p 1 p) = d 4 ξ d 4 η e i( p p 1 ).ξ+ip 1.η P ψ (2π ) 8 j (0)D α (η) ψ i (ξ) P d 4 ξ d 4 η e i( p p 1 ).ξ+ip 1.η P ψ (2π ) 8 j (0)F nα (η) ψ i (ξ) P 4

5 MDs and color gauge invariance Gauge invariance in a non-local situation requires a gauge link U(0,ξ) ψ(0)ψ(ξ) = n ψ(0)u (0,ξ)ψ(ξ) = 1 n! ξ µ 1... ξ µ N ψ(0)... µ 1 µn ψ(0) n 1 n! ξ µ 1... ξ µ N ψ(0) D... D µ 1 µn ψ(0) Introduces path dependence for Φ(x,p ) µ U (0, ξ ) = P exp ig ds Aµ 0 ξ Τ ξ ξ 0 ξ.p Dominant paths: along lightcone connected at lightcone infinity (staples) Reduces to straight line for Φ(x) Φ (x, p ) Φ(x) 5

6 Matrix elements for MDs quark-quark [U] ij (x, p ; n) = gluon-gluon Z d P d 2 (2 ) 3 e ip hp,s j (0) U [0, ] i ( ) P,Si n=0 [U,U 0 ] µ (x, p ; n) = Z d P d 2 (2 ) 3 e ip hp,s F nµ (0) U [0, ] F n ( ) U 0 [,0] P,Si n=0 and even single Wilson loop correlator (x) [U,U 0 ] 0 (p ; n) = Z d P d 2 (2 ) 3 e ip hp,s U [0, ] U 0 [,0] P,Si n=0 6

7 Non-universality because of process dependent gauge links Φ ij q[c ] (x, p ;n) = d(ξ.p)d 2 ξ e i p.ξ [C P ψ (2π ) 3 j (0)U ] [0,ξ ] ψ i (ξ) P ξ.n=0 MD path dependent gauge link u Gauge links associated with dimension zero (not suppressed!) collinear A n = A + gluons, leading for MD correlators to process-dependence: DY SIDIS A + (resummation) A + Φ [-] Φ [+] ime reversal Belitsky, Ji, Yuan, 2003; Boer, M, Pijlman,

8 Simplest color flow classes for quarks (in lower hadron) q! q h (0) U [+] [0, ] ( ) i [+] (x, k ) qq! h (0) U [ ] [0, ] ( ) i [ ] (x, k ) 8

9 Simplest color flow classes for quarks (in lower hadron) q! q [+] (x, k ) qq! [ ] (x, k ) 9

10 Color flow and gauge-link can become complex Hadron (-) hard COLOR or Hadron (+) qg! q h F (0) U [+] [0, ] F ( ) U [ ] [,0] i qg! qg h F (0) U [+] [+] [0, ] F ( ) U [,0] U ( ) i 10

11 Non-universality because of process dependent gauge links Γ αβ[c,c '] (x, p ;n) = d(ξ.p)d 2 ξ e i p.ξ [C P U ] (2π ) 3 [ξ,0] F nα [C (0)U '] [0,ξ ] F nβ (ξ) P ξ.n=0 u he MD gluon correlators contain two links, which can have different paths. Note that standard field displacement involves C = C αβ F ( ξ) U F ( ξ) U [ ] αβ [ C] [ ηξ, ] [ ξη, ] u Basic (simplest) gauge links for gluon MD correlators: gg è H Φ g [+,+] Φ g [-,-] Φ g [+,-] Φ g [-,+] Bomhof, M, Pijlman, 2006; Dominguez, Xiao, Yuan, 2011 in gg è QQ 11

12 Color flow classes for gluons (in lower hadron) g! g gg! H qg! qg ij[+,+]] ij[, ij[+, (x, k ) ]] (x, k ]] ) (x, k ) Gluon correlators at small x related to Wilson loop correlator q(gg)! q [+, ] 0 (x, k ) (x) [U,U 0 ] 0 (k ; n) = Z d P d 2 (2 ) 3 e ik hp,s U [0, ] U 0 [,0] P,Si n=0 Wilson loop correlator linked to dipole picture and diffraction at small x (depends actually on k 2 ~ k 2 in region where k + k - = x k - P + << k 2 ) 12

13 A MD picture for diffractive scattering q q X [ ξ, ] 1 ψ(ξ 2 ) ψ(0 2 ) [,0 1] p 2 k 2 p 1 P P GAP Y [ +, ξ1][ +, ξ2] [01, + ][02, + ] GAP Momentum flow in case of diffraction x 1 à M X2 /W 2 à 0 and t à p 1 2 Picture in terms of MD and inclusion of gauge links (including gauge links/collinear gluons in M ~ S 1) ξ (Another way of looking at diffraction, cf Dominguez, Xiao, Yuan 2011 or older work of Gieseke, Qiao, Bartels 2000) 13

14 Quark correlator Unpolarized target [U] (x, k )= f [U] 1 (x, k 2 )+ih? [U] 1 (x, k 2 ) /k M Vector polarized target [U] L (x, k )= [U] (x, k )= S L g [U] 1 (x, k2 ) 5 + S L h? [U] g [U] 1 (x, k2 ) k S M 1L (x, k2 ) 5/k M + h [U] 1 (x, k2 ) 5 /S + h? [U] 1 (x, k 2 ) k S 5 /P M 2 2 /P 2 /P 2 Surviving in collinear correlators Φ(x) and including flavor index f q 1 (x) q(x) gq 1 (x) = q(x) hq 1 (x) = q(x) Note: be careful with use of h 1 and non-traceless tensor with k.s since h 1 is not a MD of definite rank! 14

15 Definite rank MDs Expansion in constant tensors in transverse momentum space g µ = gµ P {µ n } µ = Pnµ = +µ or traceless symmetric tensors (of definite rank) k i k ij k ijk = ki k j = k i k j kk 1 2 k2 g ij 1 4 k2 g ij kk + g ik k j + gjk ki Simple azimuthal behavior: k i 1...i m! k e ±im' functions showing up in cos(mφ) or sin(mφ) asymmetries (wrt e.g. φ ) Simple Bessel transform to b-space (relevant for evolution): F m (x, k )= F m (x, b) = Z 1 0 Z 1 0 bdb J m (k b) F m (x, b) k dk J m (k b) F m (x, k ) 15

16 Structure of quark (8) MD PDFs in spin ½ target 8 MDs F (x,k 2 ) PARON SPIN QUARKS γ + γ + γ 5 γ + γ α γ 5 ARGE SPIN U L f 1 f 1 g 1 g 1 h 1 h 1L h 1 h 1 Integrated (collinear) correlator: only circled ones survive Collinear functions are spin-spin correlations MDs also momentum-spin correlations (spin-orbit) including also -odd (single-spin) functions (appearing in single-spin asymmetries) Existence of -odd functions because of gauge link dependence! 16

17 Structure of quark MD PDFs in spin 1 target PARON SPIN QUARKS γ + γ + γ 5 γ + γ α γ 5 U f 1 h 1 L g 1 h 1L ARGE SPIN LL L f 1 f 1LL f 1L g 1 g 1L f 1 h 1L h 1L g 1 h 1 h 1 h 1LL h 1 h 1 Hoodbhoy, Jaffe & Manohar, NP B312 (1988) 571: introduction of f 1LL = b 1 Bacchetta & M, PRD 62 (2000) ; h 1L first introduced as -odd PDF H 1L ĥ 1 X. Ji, PRD 49 (1994) 114; introduction of (PFF) 17

18 Gluon correlators Unpolarized target ij[u] (x, k )= x 2 g ij f [U] 1 (x, k 2 )+ kij M 1 (x, k 2 ) 2 h? [U] Vector polarized target ij[u] L (x, k )= x 2 i ij S Lg [U] 1 (x, k2 )+ {i kj} M 2 S L h? [U] 1L (x, k2 ) ij[u] (x, k )= x g ij 2 M ks f?[u] 1 (x, k 2 ) k{i Sj} + S {i k j} 4M i ij k S M h 1 (x, k 2 ) g[u] 1 (x, k2 ) {i kj} S 2M 3 h? 1 (x, k 2 ) 18

19 ensor polarized target ij[u]] LL (x, k )= x 2 ij[u] L (x, k )= x 2 Gluon correlators g ij S LL f [U] 1LL (x, k2 )+ kij M 2 S LLh?[U] 1LL (x, k2 ) g ij k S L M f [U] 1L (x, k2 )+i ij S L k M g[u] 1L (x, k2 ) ij[u] (x, k )= x 2 + S{i L kj} g ij M k 1L (x, k2 )+ kij h[u] S M 2 S L M 3 f [U] 1 (x, k2 )+i ij h?[u] 1L (x, k2 ) k S M 2 g [U] 1 (x, k2 ) + S ij h[u] 1 (x, k2 )+ S{i + kij S M 4 kj} M 2 h??[u] 1 (x, k2 ) h?[u] 1 (x, k2 ) 19

20 Structure of gluon MD PDFs in spin 1 target PARON SPIN GLUONS g αβ ε αβ p αβ,... U f 1 g f 1 h 1 g L g g 1 g gh 1L ARGE SPIN LL g f 1 g f 1LL g g 1 f 1 g g h 1 1 h 1 g h 1LL L f 1L g g g 1L g h 1L g h 1L f 1 g g g 1 g h 1 g h 1 g h 1 Jaffe & Manohar, Nuclear gluonometry, PL B223 (1989) 218 PJM & Rodrigues, PR D63 (2001) Meissner, Metz and Goeke, PR D76 (2007) D Boer, S Cotogno, van Daal, PJM, A Signori, Y Zhou, ArXiv

21 Untangling operator structure in collinear case (reminder) Collinear functions and x-moments Φ q (x) = d(ξ.p) e i p.ξ [n] P ψ(0)u (2π ) [0,ξ ] ψ(ξ) P ξ.n=ξ =0 x N 1 Φ q (x) = x = p.n = d(ξ.p) e i p.ξ P ψ(0)( n (2π ) ξ ) N 1 [n] U [0,ξ ] ψ(ξ) P ξ.n=ξ =0 d(ξ.p) e i p.ξ [n] P ψ(0)u (2π ) [0,ξ ] (D n ξ ) N 1 ψ(ξ) P ξ.n=ξ =0 Moments correspond to local matrix elements of operators that all have the same twist since dim(d n ) = 0 Φ ( N ) = P ψ(0)(d n ) N 1 ψ(0) P 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

22 ransverse moments à operator structure of MD PDFs Operator analysis for [U] dependence (e.g. [+] or [-]) MD functions: in analogy to Mellin moments consider transverse moments à role for quark-gluon m.e. p α Φ [±] (x, p ;n) = d(ξ.p)d 2 ξ e i p.ξ P ψ(0)u (2π ) 3 [0,± ] id α U [±,ξ ] ψ(ξ) P ξ.n=0 calculable dp p α Φ (x, p ;n) =! Φ α (x) +C G Φ G α (x) -even -odd Φ α (x) = Φ D α (x) Φ A α (x) -even (gauge-invariant derivative) dx Φ α A (x) = PV 1 Φ nα x F (x x 1, x 1 x) 1 Φ α D (x) = dx 1 Φ α D (x x 1, x 1 x) Φ G α (x) = π Φ F nα (x,0 x) -odd (soft-gluon or gluonic pole, EQS m.e.) Efremov, eryaev; Qiu, Sterman; Brodsky, Hwang, Schmidt; Boer, eryaev; Bomhof, Pijlman, M 22

23 Gluonic pole factors are calculable C G [U] calculable gluonic pole factors (quarks) U U [±] U [+] U [ ] 1 N c r c (U [ ] ) U [+] Φ [U] Φ [±] Φ [+ ] Φ [( )+] C [U] G ±1 3 1 C [U] GG, C [U] GG, Complicates life for double p situation such as Sivers-Sivers in DY, etc. Buffing, Mukherjee, M, PRD86 (2012) , ArXiv Buffing, Mukherjee, M, PRD88 (2013) , ArXiv Buffing, M, PRL 112 (2014),

24 Operator classification of quark MDs (including trace terms) factor QUARK MD RANK FOR VECOR POL. (SPIN ½) HADRON [ ] g 1 h 1L [ ] 1 h 1 [ ] C G,c f 1 g 1 h 1 h 1 [G] f 1 [G] C GG,c C GGG,c h 1 [GG1] h 1 [GG 2] hree pretzelocities: Process dependence also for (-even) pretzelocity, [U h ] 1 = h [ ] [U 1 +C ] GG,1 h [GG1] [GG 2] 1 +C GG,2 h 1 [ ]: ψ ψ = r " c # ψψ$ % [GG 1]: r " c # GGψψ$ % [GG 2]: r " c # GG$ % r " ψψ $ c # % Buffing, Mukherjee, M, PRD86 (2012) , ArXiv

25 Operator classification of quark MDs (including trace terms) factor QUARK MD RANK FOR VECOR POL. (SPIN ½) HADRON f 1 g 1 h 1 [ ] g 1 h 1L [ ] h 1 [ ] C G,c h 1 [G] f 1 [G] C GG,c δ f 1 [GG c]... h 1 [GG1] h 1 [GG 2] C GGG,c Process dependence in p dependence of MDs due to gluonic pole operators (e.g. affecting <p 2 > f 1 (x, p 2 ) = f 1 [U + C ] [GG c] GG,c δ f 1 with δf 1 [GG c] (x) = 0 Boer, Buffing, M, JHEP08 (2015) 053, arxiv:

26 Classifying Polarized Quark MDs (including tensor pol) factor QUARK MD RANK FOR VECOR POL. (SPIN ½) HADRON f 1 g 1 h 1 [ ] g 1 h 1L [ ] h 1 [ ] C G,c h 1 [G] f 1 [G] C GG,c δ f 1 [GG c]... h 1 [GG1] h 1 [GG 2] C GGG,c... factor QUARK MD RANK FOR ENSOR POL. (SPIN 1) HADRON f 1LL [ ] f 1L [ ] f 1 C G [.G] δh 1L [G] h 1LL [G] g 1L [G] h 1 [ G] h 1L [ G] g 1 [ G] h 1 C GG,c [GG c] f 1 C GGG,c [GGG c] h 1 [U h ] 1L (x, p 2 [U ) = C ] G δh [.G] 1L (x, p 2 ) with δh [.G] 1L (x) = 0 26

27 Operator classification of gluon MDs factor 1 f 1 g 1 GLUON MD PDF RANK FOR SPIN ½ HADRON [ ] g 1 h 1 [ ] C G,c f 1 [G c] h 1 [G c] h 1L [ G c] h 1 [ G c] [U C ] GG,c C GGG,c δ f 1 [GG c]... h 1 [GG c] h 1 [GGG c] factor 1 ADDIIONAL PDFs FOR ENSOR POL. SPIN 1 HADRON f 1LL h 1 [ ] f 1L [ ] h 1L [ ] f 1 [ ] h 1LL [ ] h 1 [ ] h 1L [ ] h 1 C G,c C GG,c C GGG,c [G c] g 1L [ G c] g 1 [GG f c] [GG 1 h c] [GG c] 1LL h 1 C GGGG,c D Boer, S Cotogno, van Daal, PJM, A Signori Y Zhou, ArXiv [ GG] h 1L [ GG c] h 1 [GGGG c] h 1 27

28 Small x physics in terms of MDs he single Wilson-loop correlator Γ 0 0(k )= 1 2M 2 e(k 2 ks ) M e (k 2 )+... factor 1 C G,c GLUON MD PDF RANK FOR UNPOL. AND SPIN ½ HADRON e, e LL [ ] e L e [G] e [ ] C GG,c e [GG] C GGG,c Note limit x à 0 for gluon MDs linked to gluonic pole m.e. of Γ 0 (2 ) 2 ij [U,U 0] (0,k ) C [U,U 0 ] GG M 2 ij[u,u 0 ] 0 GG (k ) C [U,U 0 ] k i kj GG M 2 RHS depends in fact on t, which for x = 0 becomes p 2 [U,U 0 ] 0 (k ) 28

29 Small x physics in terms of gluon MDs Note limit x à 0 for gluon MDs linked to gluonic pole m.e. of Γ 0 2 [U,U 0] (0,p )=C [U,U 0 ] GG 0 GG (p ) Dipole correlators: at small x only two structures for unpolarized and transversely polarized nucleons: pomeron & odderon structure xf [+, ] 1 (x, k 2 )! k2 2M 2 e[+, ] (k 2 ) xh?[+, ] 1 (x, k 2 )! e [+, ] (k 2 ) xf?[+, ] 1 (x, k 2 )! k2 2M 2 e[+, ] (k 2 ) xh [+, ] 1 (x, k 2 )! k2 2M 2 e[+, ] (k 2 ) xh?[+, ] 1 (x, k 2 )! e [+, ] (k 2 ) Dominguez, Xiao, Yuan 2011 D Boer, MG Echevarria, PJM, J Zhou, PRL 116 (2016) , ArXiv D Boer, S Cotogno, van Daal, PJM, A Signori, Y Zhou, ArXiv

30 Conclusion (Generalized) universality of MDs studied via operator product expansion, extending the well-known collinear distributions (including polarization 3 for quarks and 2 for gluons) to MD PDF and PFF functions, ordered into functions of definite rank. he rank m is linked to specific cos(mφ) and sin(mφ) 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 he MD 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. 30