Process dependence of the gluon Sivers function in inclusive pp collisions: theory Cristian Pisano In collaboration with: U. D’Alesio, C. Flore, F. Murgia, P. Taels
TMD factorization and process dependence 2/24
TMD factorization Two scale processes Q 2 ≫ p 2 T Factorization proven 3/24
TMD factorization Factorization and correlators Hard partonic interactions can be separated from nonperturbative correlators SIDIS Drell-Yan P P h h P B P B ∆ Φ k k k k q q q q p p p p Φ Φ P P P A P A γ ∗ p → h X p p → γ ∗ X Parton correlators Φ and ∆ describe the soft hadron ↔ parton transitions P P h h i j p p ∆ (k;P ,S ) h h Φ (p;P,S) k k P P Parametrized in terms of distribution and fragmentation functions 4/24
TMD factorization Gauge invariance of the correlators Resummation of all gluon exchanges leads to gauge links in the correlators Φ, ∆ P P P P P P P P h h h h h h h h ∆ k k k k k k k q q q q q q q p p−p p p p p−p p p p p 1 1 p−p−p p 1 1 1 2 1 2 Φ P P P P P P P P P P P P h h h h k−k k−k k 1 k k q 1 q k 1 1 p p p p P P P P Boer, Mulders, Pijlman, NPB 667 (2003) � � � U C d s µ A µ ( s ) [0 ,ξ ] = P exp − ig C [0 ,ξ ] The path C depends on the color interactions, i.e. on the specific process 5/24
TMD factorization Gauge invariance of the correlators Gauge invariant definition of Φ (not unique) Φ [ U ] ∝ � � � � � ψ (0) U C P , S [0 ,ξ ] ψ ( ξ ) � P , S � � SIDIS P P P P P P P P h h h h h h h h ∆ k k k k k k k q q q q q q q p p−p p p p p p p−p 1 1 p−p−p p p p 1 1 1 1 2 2 Φ P P P P P P P P Belitsky, Ji, Yuan, NPB 656 (2003) Boer, Mulders, Pijlman, NPB 667 (2003) ξ T ξ T U [+] ≡ [0 ,ξ ] − − ξ ξ Possible effects in transverse momentum observables ( ξ T is conjugate to k T ) 6/24
TMD factorization Process dependence of gauge links SIDIS Drell-Yan P P h h P B P B Φ ∆ k k k k q q q q p p p p Φ Φ P P P A P A ξ T ξ T U [+] U [ − ] [0 ,ξ ] [0 ,ξ ] − − ξ ξ Belitsky, Ji, Yuan, NPB 656 (2003) Boer, Mulders, Pijlman, NPB 667 (2003) Boer, talk at RBRC Synergies workshop (2017) ξ T � d k T − → ξ T = 0 − the same in both cases → − ξ 7/24
TMD factorization The quark Sivers function Fundamental test of TMD theory f ⊥ [ DY ] ( x , k 2 ⊥ ) = − f ⊥ [ SIDIS ] ( x , k 2 h ⊥ [ DY ] ( x , k 2 ⊥ ) = − h ⊥ [ SIDIS ] ( x , k 2 ⊥ ) ⊥ ) 1 T 1 T 1 1 Collins, PLB 536 (2002) FSI in SIDIS ISI in DY ξ T ξ T = − − − ξ ξ ISI/FSI lead to process dependence of TMDs, could even break factorization Collins, Qiu, PRD 75 (2007) Collins, PRD 77 (2007) Rogers, Mulders, PRD 81 (2010) 8/24
Process dependence of gluon TMDs 9/24
Gluon TMDs The gluon correlator p p αβ (p;P,S) Γ P P Gauge invariant definition of Γ µν Γ [ U , U ′ ] µν ∝ � P , S | Tr c [0 ,ξ ] F + µ ( ξ ) U C ′ F + ν (0) U C � � | P , S � [ ξ, 0] Mulders, Rodrigues, PRD 63 (2001) Buffing, Mukherjee, Mulders, PRD 88 (2013) Boer, Cotogno, Van Daal, Mulders, Signori, Zhou, JHEP 1610 (2016) The gluon correlator depends on two path-dependent gauge links ep → e ′ QQX , ep → e ′ jet jet X probe gluon TMDs with [++] gauge links pp → γγ X (and/or other CS final state) probes gluon TMDs with [ −− ] gauge links pp → γ jet X probes an entirely independent gluon TMD: [+ − ] links (dipole) 10/24
The gluon Sivers functions Sign change test Related Processes ep ↑ → e ′ QQX , ep ↑ → e ′ jet jet X probe GSF with [++] gauge links (WW) p ↑ p → γγ X (and/or other CS final state) probe GSF with [ −− ] gauge links Analogue of the sign change of f ⊥ q between SIDIS and DY (true also for h g 1 and h ⊥ g 1 T ) 1 T f ⊥ g [ e p ↑ → e ′ QQ X ] = − f ⊥ g [ p ↑ p → γ γ X ] 1 T 1 T = − Boer, Mulders, CP, Zhou (2016) Motivation to study gluon Sivers effects at both RHIC and the EIC 11/24
The gluon Sivers functions The dipole GSF Complementary Processes ep ↑ → e ′ QQX probes a GSF with [++] gauge links (WW) p ↑ p → γ jet X ( gq → γ q ) probes a gluon TMD with : [+ − ] links (DP) = − At small- x the WW Sivers function appears to be suppressed by a factor of x compared to the unpolarized gluon function, unlike the dipole one The DP gluon Sivers function at small- x is the spin dependent odderon (single spin asymmetries from a single Wilson loop matrix element) Boer, Echevarria, Mulders, Zhou, PRL 116 (2016) Boer, Cotogno, Van Daal, Mulders, Signori, Zhou, JHEP 1610 (2016) 12/24
The Generalized Parton Model The first transverse moments of the WW and DP gluon Sivers functions k 2 � f ⊥ (1) g ( f / d ) f ⊥ g ( f / d ) d 2 k T T ( x , k 2 ( x ) = T ) 1 T 1 T 2 M 2 p related to two different trigluon Qiu-Sterman functions T ( f / d ) , involving the G antisymmetric f abc and symmetric d abc color structures, respectively Bomhof, Mulders, JHEP 0702 (2007) Buffing, Mukherjee, Mulders, PRD 88 (2013) The two distributions have a different behavior under charge conjugation The Burkardt sum rule constraints only the f -type gluon Sivers function � d x f ⊥ (1) a � ( x ) = 0 1 T a = q , ¯ q , g Boer, Lorc´ e, CP, Zhou, AHEP 2015 (2015) 13/24
The Generalized Parton Model The TMD Generalized Parton Model 14/24
The Generalized Parton Model Phenomenological extension of the TMD formalism to processes like X pp → π X pp → jet π X and more ( pp → jet X , pp → γ X ) Single scale processes Anselmino, Boglione, Murgia, PLB 362 (1995), ... Aschenauer, D’Alesio, Murgia, EPJA52 (2016) Transverse Momentum Dependent – Generalized Parton Model (GPM) ◮ Spin & k ⊥ -dependent distribution and fragmentation functions as in TMD scheme ◮ k ⊥ -dependence included in the hard scattering, unlike in the TMD formalism ◮ Universality and TMD factorization: assumption to be tested 15/24
Color Gauge Invariant (CGI) GPM The quark Sivers function The CGI-GPM takes into account the effects of initial and final state interactions Gamberg, Kang, PLB 696 (2011) One-gluon exchange approx.: LO term of of the α S expansion of the gauge link SIDIS q p c p p p k a c a − → p P , S P , S A T A T a f ⊥ q [ SIDIS ] x Hard part 1 T qq ′ → qq ′ p p p p b d b d p p b d k C p I p p p p − → a or k a c a c P , A S T P , S P , S C p p A T A T F c a c f ⊥ q [ SIDIS ] x (CF x Hard part) 1 T f ⊥ q [ SIDIS ] is universal, process dependence absorbed in modified hard functions 1 T 16/24
Color Gauge Invariant (CGI) GPM The quark Sivers function The CGI-GPM recovers the relation f ⊥ [ DY ] = − f ⊥ [ SIDIS ] 1 T 1 T In the CGI-GPM TMDs are process dependent, different predictions w.r.t. GPM Gamberg, Kang, PLB 696 (2011) D’Alesio, Gamberg, Kang, Murgia, CP, PLB 704 (2011) 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 sin φ S sin φ S sin φ S sin φ S sin φ S sin φ S sin φ S sin φ S A N A N A N A N A N A N A N A N GPM 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 CGI A N DY data p ↑ p → jet X 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 ( √ s = 500 GeV) 0 0 0 0 0 0 0 0 -0.02 -0.02 -0.02 -0.02 -0.02 -0.02 -0.02 -0.02 η j = 3.25 η j = 3.25 η j = 3.25 η j = 3.25 η j = 3.25 η j = 3.25 η j = 3.25 η j = 3.25 -0.04 -0.04 -0.04 -0.04 -0.04 -0.04 -0.04 -0.04 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 x x x x x x x x F F F F F F F F Extension of the CGI-GPM to the gluon Sivers function is now completed D’Alesio, Murgia, CP, Taels, PRD 96 (2017) D’Alesio, Flore, Murgia, CP, in preparation Gluon Sivers function constrained from available data on p ↑ p → π 0 X , p ↑ p → J /ψ X , p ↑ p → DX , predictions for p ↑ p → γ X at RHIC 17/24
Gluon Sivers function in p ↑ p → J /ψ X A N in the GPM In the Color Singlet Model, the dominant production channel is gg → J /ψ g p ( P B ) b b c A N ≡ d σ ↑ − d σ ↓ d σ ↑ + d σ ↓ ≡ d ∆ σ 2 d σ a a p ↑ ( P A ) � d x a d ∆ σ GPM = 2 α 3 d x b s x b d 2 k ⊥ a d 2 k ⊥ b δ (ˆ u − M 2 ) s + ˆ t + ˆ s x a � � − k ⊥ a f ⊥ g 1 T ( x a , k ⊥ a ) cos φ a f g / p ( x b , k ⊥ b ) H U s , ˆ × gg → J /ψ g (ˆ t , ˆ u ) M p f ⊥ g 1 T : Gluon Sivers function (one and process independent) 18/24
Gluon Sivers function in p ↑ p → J /ψ X A N in the CGI-GPM GPM CGI-GPM p ( P B ) p ( P B ) p ( P B ) b b b b b b e c c c − → a a a a ′ a a ′ d d p ↑ ( P A ) p ↑ ( P A ) p ↑ ( P A ) C ( f / d ) C ( f / d ) C U [Color Factors] I F c → f ⊥ g ( f ) H Inc ( f ) gg → J /ψ g + f ⊥ g ( d ) H Inc ( d ) [ GPM ] f ⊥ g H U gg → J /ψ g − [ CGI − GPM ] 1 T 1 T 1 T gg → J /ψ g Two independent, universal f ⊥ 1 T ’s, process dependence shifted into new hard parts gg → J /ψ g ≡ C ( f / d ) + C ( f / d ) H Inc ( f / d ) I F c H U gg → J /ψ g C U 19/24
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