The gluon Sivers function and its process dependence from RHIC data - PowerPoint PPT Presentation

The gluon Sivers function and its process dependence from RHIC data Cristian Pisano In collaboration with: U. D’Alesio, C. Flore, F. Murgia, P. Taels

The TMD Generalized Parton Model 2/17

TMD factorization Two scale processes Q 2 ≫ p 2 T Factorization proven 3/17

TMD factorization 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 4/17

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 5/17

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, PRD 99 (2019) Gluon Sivers function constrained from available data on p ↑ p → π 0 X and p ↑ p → DX ; predictions for p ↑ p → J /ψ X and p ↑ p → γ X at RHIC Talk by F. Murgia, WG7 6/17

Color Gauge Invariant (CGI) GPM Two independent gluon Sivers functions with first transverse moments 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) 7/17

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) 8/17

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 9/17

Gluon Sivers function in p ↑ p → J /ψ X A N in the CGI-GPM p ( P B ) b b c C ( f ) F c = C ( d ) c ¯ c pair in a color singlet state, no FSIs: F d = 0 F. Yuan, PRD 78 (2003) a d a ′ p ↑ ( P A ) p ( P B ) b b e c C ( f ) = − 1 C ( d ) 2 C U = 0 I I a d a ′ p ↑ ( P A ) Only f ⊥ g ( f ) contributes to A N 1 T � d x a d ∆ σ CGI = 2 α 3 d x b s d 2 k ⊥ a d 2 k ⊥ b δ (ˆ s + ˆ u − M 2 ) t + ˆ s x a x b � − k ⊥ a � f ⊥ g ( f ) ( x a , k ⊥ a ) cos φ a f g / p ( x b , k ⊥ b ) H Inc ( f ) s , ˆ gg → J /ψ g (ˆ t , ˆ u ) × 1 T M p gg → J /ψ g = − 1 H Inc ( f ) 2 H U gg → J /ψ g 10/17

Gluon Sivers function in p ↑ p → D X A N in the CGI-GPM LO channels are gg → c ¯ c and q ¯ q → c ¯ c . Color factors for gg → c ¯ c : b b 0 b b 0 b b 0 b b 0 e a a a a a 0 a 0 a 0 a 0 c c c C ( f / d ) C ( f / d ) C ( f / d ) C U I F c F d Agreement with gluonic pole strenghts calculated for p ↑ p → h h X C ( f / d ) + C ( f / d ) + C ( f / d ) C ( f / d ) I F c F d ≡ G C U Bomhof, Mulders, JHEP 0702 (2007) Agreement with twist-three results for p ↑ p → D X Kang, Qiu, Vogelsang, Yuan, PRD 78 (2008) contribute to A N ( p ↑ p → D X ) Both f ⊥ g ( f ) and f ⊥ g ( d ) 1 T 1 T 11/17

Gluon Sivers function in p ↑ p → D X C Inc ( f / d ) ≡ C ( f / d ) + C ( f / d ) Color factors for gg → c ¯ c I I F c C ( f ) C ( f ) C ( f ) C ( d ) C ( d ) C ( d ) C Inc ( f ) C Inc ( d ) D C U I F c F d I F c F d 2 N 2 1 N c 1 1 1 N c 1 1 c − 1 − − c − 1) − 4 N c 8( N 2 c − 1) 8 N c 8 N c ( N 2 8 N c ( N 2 c − 1) 8( N 2 c − 1) 8 N c 8 N c ( N 2 c − 1) 8 N c ( N 2 c − 1) N 2 c +1 N 2 c +1 1 N c 1 1 N c 1 1 − c − 1) − − c − 1) − c − 1) − − − 8( N 2 8 N c ( N 2 8 N c ( N 2 8( N 2 8 N c ( N 2 8 N c ( N 2 4 N c c − 1) 8 N c c − 1) 8 N c c − 1) N c N c N c N c N c N c N c N c 0 − − − 2( N 2 c − 1) 4( N 2 c − 1) 8( N 2 c − 1) 8( N 2 c − 1) 8( N 2 c − 1) 8( N 2 c − 1) 8( N 2 c − 1) 8( N 2 c − 1) N c N c N c N c N c N c 0 0 0 − 4( N 2 c − 1) 8( N 2 c − 1) 8( N 2 c − 1) 8( N 2 c − 1) 8( N 2 c − 1) 4( N 2 c − 1) N c N c N c N c N c N c 0 0 0 − 4( N 2 c − 1) 8( N 2 c − 1) 8( N 2 c − 1) 8( N 2 c − 1) 8( N 2 c − 1) 4( N 2 c − 1) N c N c N c N c N c N c N c 0 0 − − 4( N 2 c − 1) 8( N 2 c − 1) 8( N 2 c − 1) 8( N 2 c − 1) 8( N 2 c − 1) 8( N 2 c − 1) 8( N 2 c − 1) N c N c N c N c N c N c N c 0 0 − − 4( N 2 c − 1) 8( N 2 c − 1) 8( N 2 c − 1) 8( N 2 c − 1) 8( N 2 c − 1) 8( N 2 c − 1) 8( N 2 c − 1) 1 1 1 1 1 1 1 0 0 − − c − 1) − c − 1) − − c − 1) − 4 N c ( N 2 c − 1) 8 N c ( N 2 8 N c ( N 2 8 N c ( N 2 c − 1) 8 N c ( N 2 c − 1) 8 N c ( N 2 8 N c ( N 2 c − 1) 1 1 1 1 1 1 1 0 0 − − c − 1) − c − 1) − − c − 1) − 4 N c ( N 2 c − 1) 8 N c ( N 2 8 N c ( N 2 8 N c ( N 2 c − 1) 8 N c ( N 2 c − 1) 8 N c ( N 2 8 N c ( N 2 c − 1) D’Alesio, Murgia, Pisano, Taels, PRD 96 (2017) Modified hard functions H Inc ( f / d ) are not simply proportional to H U Similar tables and results for all the channels in p ↑ p → π X D’Alesio, Flore, Murgia, Pisano, Taels, PRD 99 (2019) 12/17

Comparison with data and results 13/17

Gluon Sivers function in p ↑ p → π 0 X Upper bounds Assumption: the GSFs have a factorized form in x - k ⊥ , Gaussian k ⊥ -dependence p ↑ p → π 0 X 0.1 GPM x F = 0 √ s = 200 GeV 0.08 N g (x) = + 1 0.06 CGI f-type A N (max) 0.04 Maximized GSFs ( N g = +1) 0.02 CGI d-type 0 CGI quark ( fi t) -0.02 -0.04 1 2 3 4 5 6 7 8 9 10 p T (GeV) PHENIX Collaboration, PRD 90 (2014) The f -type GSF is dominant in the CGI-GPM approach 14/17

Gluon Sivers function in p ↑ p → π 0 X Conservative scenario Reduced f -type GSF ( N ( f ) = 0 . 1), negative saturated d -type GSF ( N ( d ) = − 1) g g p ↑ p → π 0 X 0.01 x F = 0 √ s = 200 GeV 0.005 A N 0 -0.005 CGI (f) =0.1, N g (d) =-1] full [N g -0.01 1 2 3 4 5 6 p T (GeV) Shaded area represents a ± 20% uncertainty on N ( f ) g 15/17