Transverse Momentum Dependent Distribution Functions of Definite Rank

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1 QCD204: QCD Evolution 204 Workshop Santa Fe, 2-6 May 204 ransverse Momentum Dependent Distribution Functions of Definite Rank Piet Mulders

2 ABSRAC MDs of definite rank Piet Mulders (Nikhef/VU University Amsterdam) ransverse momentum dependent (MD) parton distribution functions include pathdependent Wilson lines. Using moments in x and p, it is possible to study the operator structure which is relevant for evolution as well as for modelling of MDs or lattice calculations. Using the moments it is possible to categorize the MDs according to their rank which labels the relevant azimuthal behavior in p. For quark MDs of rank 2, such as the Pretzelocity MD, and gluon MDs of rank and higher, such as the MD describing linear polarization of gluons, one finds multiple MDs depending on the color structure of the operators. he explicit appearance of these MDs 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 MD as well as an additional process-dependent factor if multiple MDs 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, homas Kasemets, Miguel Echevarria, Paul Hoyer and Asmita Mukherjee] 2

3 Starting point! MD 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 k)! ransverse separation complicates gauge link structure! MDs encode novel aspects of hadronic structure, e.g. spin-orbit correlations, such as -odd transversely polarized quarks or -even longitudinally polarized gluons in an unpolarized hadron, thus possible applications for precision probing at the LHC, but for sure at a polarized EIC. 3

4 (Un)integrated correlators Φ(x, p, p.p) = d 4 ξ e i p.ξ P ψ(0) ψ(ξ) P (2π ) 4! unintegrated Φ(x, p ;n) = d(ξ.p)d 2 ξ! MD (light-front) e i p.ξ P ψ(0) ψ(ξ) P (2π ) 3 ξ.n=ξ + =0! σ = p - integration makes time-ordering automatic. he soft part is simply sliced at the light-front Φ(x) = d(ξ.p) e i p.ξ P ψ(0) ψ(ξ) P (2π ) ξ.n=ξ =0 or ξ 2 =0! Is already equivalent to a point-like interaction! collinear (light-cone) Φ = P ψ(0) ψ(ξ) P ξ=0! Local operators with calculable anomalous dimension! local 4

5 Simplest gauge links for quark MDs d( ξ. P) d ξ 2 qc [ ] ip. ξ [ C] ij (, xp; n) e Pψ (0) 3 j U[0, ξ ] ψi( ξ) P ξ. n= 0 (2 π ) Φ = d( ξ. P) (;) xn e Pψ (0) U ψ ( ξ) P Φ = q ip. ξ [ n] ij j [0, ξ ] i ξ. n= ξ = 0 (2 π ) u Gauge links come from dimension zero (not suppressed!) collinear A.n gluons, but leads for MD correlators to process-dependence: DY MD collinear SIDIS A n A n Φ [-] Φ [+] ime reversal AV Belitsky, X Ji and F Yuan, NP B 656 (2003) 65 D Boer, PJM and F Pijlman, NP B 667 (2003) 20 5

6 Simplest gauge links for gluon MDs Φ g 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 [-,+] C Bomhof, PJM, F Pijlman; EPJ C 47 (2006) 47 F Dominguez, B-W Xiao, F Yuan, PRL 06 (20) in gg è QQ 6

7 Color gauge invariant correlators! Including gauge links we have well-defined matrix elements for MDs but this implies multiple possiblities for gauge links depending on the process and the color flow in the diagram! Leading quark MDs Φ [U] (x, p ; n) =! Leading gluon MDs: { f [U] (x, p 2 ) f [U] + h [U] (x, p2 ) γ 5 /S + h [U] s (x, p ) γ 5 /p M 2x Γ µν[u] (x,p )= g µν + iϵ µν ϵp {µ p ν} 2M 2 gg[u] s (x,p )+ f g[u] (x,p 2 )+gµν ( p µ pν M 2 gµν (x, p 2 ) ϵp S M + g[u] s (x, p )γ 5 + ih [U] (x, p 2 ) / p M ϵ p S M f g[u] (x,p 2 ) p 2 2M 2 h g[u] s (x,p ) ϵp {µ S ν} +ϵs {µ p ν} 4M ) h g[u] (x,p 2 ) } / P 2, h g[u] (x,p2 ). 7

8 Some details on the gauge links ()! Proper gluon fields (F rather than A, Wilson lines and boundary terms) P A ( p ) = n. A( p ) + ia ( p ) +... = n. A( p ) p + ig ( p ) +... µ µ µ µ nµ np. p. n! 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

9 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)! he 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.

10 Which gauge links?! With more (initial state) hadrons color gets entangled, e.g. in pp! Gauge knot U + [p,p 2, ]! Outgoing color contributes to a future pointing gauge link in Φ(p 2 ) and future pointing part of a gauge loop in the gauge link for Φ(p )! his causes trouble with factorization.c. Rogers, PJM, PR D8 (200)

11 Which gauge links? ψ(ξ 2 ) ψ(0 2 ) [ ξ, ] [,0 ] [ +, ξ ][ +, ξ ] 2 [ ξ, ] 2 [0, + ][02, + ] [,0 2 ] ψ(ξ ) ψ(0 )! Can be color-detangled if only p of one correlator is relevant (using polarization, ) but must include Wilson loops in final U MGA Buffing, PJM, JHEP 07 (20) 065

12 rouble appears already in DY ] dσ DY r c [Φ(x,p )Γ Φ(x 2,p 2 )Γ = N c Φ(x,p )Γ Φ(x 2,p 2 )Γ,! Complications for DY at measured Q if the transverse momentum of two initial state hadrons is involved 2

13 Basic strategy: operator product expansion! aylor expansion for functions around zero f (z) = f n z n f n = n f n n!! Mellin transform for functions on [-,] interval z n z=0 f (x) = 2πi c+i dn x n M n M n = dx x n c i 0 f (x)! functions in (transverse) plane α p n f α...α n f α =...α n α... αn f ( p ) f ( p ) = p α... n α...α n p =0 3

14 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 Φ q (x) = x = p.n = d(ξ.p) e i p.ξ P ψ(0)( n (2π ) ξ ) N [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 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 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. 4

15 Operator structure in MD case! For MD functions one can consider transverse moments d(ξ.p)d 2 ξ Φ(x, p ;n) = e i p.ξ [±] P ψ(0) U (2π ) 3 [0,ξ ] ψ(ξ) P ξ.n=0 d(ξ.p)d 2 ξ p α Φ [±] (x, p ;n) = e i p.ξ P ψ(0)u (2π ) 3 [0,± ] D α U [±,ξ ] ψ(ξ) P ξ.n=0 p α p α 2 Φ [±] (x, p ;n) = d(ξ.p)d 2 ξ e i p.ξ α P ψ(0)u (2π ) 3 [0,± ] D α D 2 U [±,ξ ] ψ(ξ) P ξ.n=0! Upon integration, these do involve collinear twist-3 multi-parton correlators MGA Buffing, A Mukherjee, PJM, PRD 86 (202) , Arxiv: [hep-ph] 5

16 Operator structure in MD case! For first transverse moment one needs quark-gluon correlators Φ α D (x x, x x) = Φ α F (x x, x x) =! In principle multi-parton, but we need dξ.p dη.p e i( p p ).ξ+ip.η P ψ(0)d α (2π ) 2 (η) ψ(ξ) P ξ.n=ξ =0 dξ.p dη.p e i( p p ).ξ+ip.η P ψ(0)f nα (η) ψ(ξ) P (2π ) 2 ξ.n=ξ =0 Φ α D (x) = dx Φ α D (x x, x x) Φ α A (x) = PV dx Φ nα x F (x x, x x) Φ F (p-p,p) Φ α (x) = Φ D α (x) Φ A α (x) Φ G α (x) = π Φ F nα (x,0 x) -even (gauge-invariant derivative) -odd (soft-gluon or gluonic pole) Efremov, eryaev; Qiu, Sterman; Brodsky, Hwang, Schmidt; Boer, eryaev, M; Bomhof, Pijlman, M 6

17 Operator structure in MD case! ransverse moments can be expressed in these particular collinear multi-parton twist-3 correlators (which are not suppressed!) Φ α (x) = d 2 p p α Φ (x, p ;n) = Φ α (x) +C G Φ G α (x) -even -even -even -odd -odd αβ[u Φ ] (x) = Φ αβ (x) +C GG,c αβ Φ GG,c (x) +C G ( Φ G αβ (x) + Φ αβ (x)) G! C G [U] calculable gluonic pole factors r c (GG ψψ) r c (GG) r c (ψψ) U U [±] U [+] U [ ] N c r c (U [ ] ) U [+] Φ [U] Φ [±] Φ [+ ] Φ [( )+] C [U] G ± 3 C [U] GG, 9 C [U] GG,

18 Distribution versus fragmentation functions! Operators:! Operators: Φ ( p p) ~ P ψ(0)u [0,ξ ] ψ(ξ) P Δ(k k) Φ α (x) = Φ α (x) +C G Φ G α (x) ~ 0 ψ(ξ) K h X K h X ψ(0) 0 X out state Δ G α (x) = π Δ F nα ( Z,0 Z ) = 0 -even -odd (gluonic pole) Φ G α (x) = π Φ F nα (x,0 x) 0 Δ α (x) = Δ α (x) -even operator combination, but still -odd functions! Collins, Metz; Meissner, Metz; Gamberg, M, Mukherjee, PR D 83 (20)

19 Classifying Quark MDs! Collecting right moments gives expansion into full MD PDFs of definite rank Φ (x, p ) = Φ(x, p 2 ) + p! i Φ i (x, p 2 ) + p! ij Φ ij (x, p 2 ) C G,c! but for MD PFFs c + C GG,c c # $ # $ i p i Φ G,c (x, p 2 ) + p! ij Φ{ G},c ij p 2 Φ G.G,c (x, p 2 ij ) p ij Φ GG,c (x, p 2 ) +...% & (x, p 2 ) +...% & Δ (z,k ) = Δ(z,k 2 ) + k! i Δ i (z,k 2 ) + k! ij Δ ij (z,k 2 )

20 Classifying Quark MDs factor C G,c C GG,c C GGG,c! Only a finite number needed: rank up to 2(S hadron +s parton )! Rank m shows up as cos(mφ) and sin(mφ) azimuthal asymmetries! No gluonic poles for PFFs factor MD PDF RANK Φ(x, p 2 ) Φ (x, p 2 ) Φ (x, p 2 ) Φ (x, p 2 ) Φ G,c (x, p 2 ) Φ {G },c (x, p 2 ) Φ{G },c (x, p 2 ) Φ GG,c (x, p 2 ) Φ {GG },c (x, p 2 ) MD PFF RANK Φ GGG,c (x, p 2 ) Δ(z,k 2 Δ Δ (z,k 2 (z,k 2 ) ) ) Δ (z,k 2 ) MGA Buffing, A Mukherjee, PJM, PRD202, Arxiv: [hep-ph] 20

21 Classifying Quark MDs factor C G QUARK MD PDF RANK UNPOLARIZED HADRON f h C GG,c! Only a finite number needed: rank up to 2(S hadron +s parton )! Rank m shows up as cos(mφ) and sin(mφ) azimuthal asymmetries! Example: quarks in an unpolarized target are described by just 2 functions Φ(x, p 2 ) = ( f (x, p 2 )) P 2 -even Φ G α (x, p 2 ) = -odd # ih (x, p 2 ) γ α & % $ M ( P ' 2 [B-M function] 2

22 Classifying Quark MDs factor C G C GG,c QUARK MD PDFs RANK SPIN ½ HADRON f, g, h g, h L h, f h (B) (B2), h h ( A) h (B), h (B2) Process dependence in h (broadening) hree pretzelocities: A: ψ ψ = r " c # ψψ$ % B: r " c # GGψψ$ % B2 : r " c # GG$ % r " ψψ $ c # % 22

23 Classifying Quark MDs factor C G C GG,c QUARK MD PDFs RANK SPIN ½ HADRON f, g, h g, h L h, f h (B) (B2), h h ( A) h (B), h (B2) factor QUARK MD PFFs RANK SPIN ½ HADRON D, G, H D,G, H, H L H Just a single pretzelocity PFF 23

24 Classifying Gluon MDs factor GLUON MD PDF RANK UNPOLARIZED HADRON f h ( A) C GG,c f (Bc) h (Bc) factor GLUON MD PDF RANK SPIN ½ HADRON f, g g h ( A) C G,c g ( A,c) ( f Ac) ( Ac), h h L ( A,c) h ( Ac) C GG,c f (Bc) h (Bc) C GGG,c (Bc) h h (Bc) 24

25 Multiple MDs in cross sections 25

26 Correlators in description of hard process (e.g. DY) ] dσ DY r c [Φ(x,p )Γ Φ(x 2,p 2 )Γ = N c Φ(x,p )Γ Φ(x 2,p 2 )Γ,! Complications if the transverse momentum of two initial state hadrons is involved, resulting for DY at measured Q in [ dσ DY = r c U [p 2 ]Φ(x,p )U [p 2 ]Γ ] U [p ]Φ(x 2,p 2 )U [p ]Γ N c Φ [ ] (x,p )Γ Φ [ ] (x2,p 2 )Γ,! Just as for twist-3 squared in collinear DY 26

27 Classifying Quark MDs factor C G,c C GG,c C GGG,c MD RANK Φ(x, p 2 ) Φ (x, p 2 ) Φ (x, p 2 ) Φ (x, p 2 ) Φ G,c (x, p 2 ) Φ {G },c (x, p 2 ) Φ{G },c (x, p 2 ) Φ GG,c (x, p 2 ) Φ {GG },c (x, p 2 ) Φ GGG,c (x, p 2 ) σ(x,x 2,q ) N c f [U,U 2 ] R G R G2 Φ [U ] (x,p ) R G for Φ [ ] R G for Φ [ ] N 2 c N 2 c +2 (N 2 c 2)(N2 c ) 2 Nc Nc 4 8N2 c 4 (Nc 2 2)(N2 c ) (Nc 2 2)2 (Nc 2 ) MGA Buffing, PJM, PRL (204), Arxiv: [hep-ph] Φ [U 2] (x2,p 2 )ˆσ(x,x 2 ), r c [ a b a b ] r c [ a a ]r c [ b b ] = N 2 c N c. 27

28 Remember classification of Quark MDs factor QUARK MD RANK UNPOLARIZED HADRON f C G h C GG,c! Example: quarks in an unpolarized target needs only 2 functions! Resulting in cross section for unpolarized DY at measured Q σ DY (x, x 2,q ) = N c Φ(x, p ) Φ(x 2, p 2 ) N c N 2 c q αβ Φ α G (x, p ) Φ β G (x 2, p 2 ) D. Boer, PRD 60 (999) 0402; MGA Buffing, PRL (204) PJM, Arxiv: [hep-ph] contains f contains h perp 28

29 A MD picture for diffractive scattering q q X [ ξ, ] ψ(ξ 2 ) ψ(0 2 ) [,0 ] p 2 k 2 p P P GAP Y [ +, ξ][ +, ξ2] [0, + ][02, + ] GAP! Momentum flow in case of diffraction x à M X2 /W 2 à 0 and t à p 2! Picture in terms of MD and inclusion of gauge links (including gauge links/collinear gluons in M ~ S ) ξ 0! (Work in progress: Hoyer, Kasemets, Pisano, M)! (Another way of looking at diffraction, cf Dominguez, Xiao, Yuan 20 or older work of Gieseke, Qiao, Bartels 2000) 29

30 A MD picture for diffractive scattering q q X [ ξ, ] ψ(ξ 2 ) ψ(0 2 ) [,0 ] p 2 k 2 p P P GAP Y [ +, ξ][ +, ξ2] [0, + ][02, + ] GAP! Cross section ξ 0 i d = (p ; P )r c hu [p ]U + [p,p 2 ] U +[p,p 2 ]U [p ] (x 2,p 2 ; q)! involving correlators for proton and photon Φ q/γ[+] (x 2, p 2 ;q) = d(ξ.q)d 2 ξ e i p 2.ξ 2 [+] γ *(q) ψ(0)u (2π ) 3 [0,ξ ] ψ(ξ) γ *(q) ξ.n=0 d 2 ξ Φ [loop] DIF (x, p ;P) = δ(x ) e i p.ξ P U [loop] P (2π ) 2 ξ.n=0 30

31 Ingredients! Photon PDF Φ q/γ[+] (x, p ;q) ~! Diffractive correlator 3α x 2 + ( x) 2 2π 2 p 2 + x Q2 x Φ DIF [loop] (x, p ;P) = δ ( 2 M X ) t f (t) W 2 2 +M GG X 3

32 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 MD 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! he rank m is linked to specific cos(mφ) and sin(mφ) azimuthal asymmetries.!! 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, but also in unpolarized situations (linearly polarized gluons) and possibly diffractive processes. 32

33 hank you 33