Gluon TMDs: Universality and Process Dependence Cristian Pisano INT Program Spatial and Momentum Tomography of Hadrons and Nuclei August 28 - September 29 2017 Seattle (USA) UNIVERSITÀ DI PAVIA MANUALE D’USO DEL MARCHIO Colore 01 Colore 02 Colore 03 Progettazione Logo 3D Spin GDR Juljan Rushaj [email protected] TMD factorization and color gauge invariance UNIVERSITÀ DI PAVIA 2/35 MANUALE D’USO DEL MARCHIO Colore 01 Colore 02 Colore 03 Progettazione Logo 3D Spin GDR Juljan Rushaj [email protected] TMD factorization 2 2 Two scale processes Q pT Factorization proven UNIVERSITÀ DI PAVIA 3/35 MANUALE D’USO DEL MARCHIO Colore 01 Colore 02 Colore 03 Progettazione Logo 3D Spin GDR Juljan Rushaj [email protected] TMD factorization Factorization and correlators Hard partonic interactions can be separated from nonperturbative correlators SIDIS Drell-Yan Ph Ph PB PB ∆ Φ k k q k k q q q p p p p Φ Φ PP PA PA γ∗p ! h X p p ! γ∗X Parton correlatorsΦ and∆ describe the soft hadron $ parton transitions Ph Ph i j p p ∆ (k;Ph ,S h ) Φ(p;P,S) k PP k Parametrized in terms of distribution and fragmentation functions UNIVERSITÀ DI PAVIA 4/35 MANUALE D’USO DEL MARCHIO Colore 01 Colore 02 Colore 03 Progettazione Logo 3D Spin GDR Juljan Rushaj [email protected] TMD factorization Gauge invariance of the correlators Resummation of all gluon exchanges leads to gauge links in the correlatorsΦ,∆ P P P P h h h h Ph Ph Ph Ph k k k ∆ q k k q q q q p p−p p−p p p p 1 1 1 1k p−p−p p p p q k q 1 2 1 2 PP PP PP P P P P p h p h h h Φ k−k k−k q 1 k1 k q k k1 1 PP p p p p PP PP Boer, Mulders, Pijlman, NPB 667 (2003) Z ! C µ U[0,ξ] = Pexp −ig dsµA (s) C[0,ξ] The path C depends on the color interactions, i.e. on the specific process UNIVERSITÀ DI PAVIA 5/35 MANUALE D’USO DEL MARCHIO Colore 01 Colore 02 Colore 03 Progettazione Logo 3D Spin GDR Juljan Rushaj [email protected] TMD factorization Gauge invariance of the correlators Gauge invariant definition of Φ (not unique) D E [U]P P C P P P P P ΦP h / P; Sh (0) U h (ξ) P;h S h h h h [0,ξ] k k k ∆ q k k q q q SIDIS q p p−p p−p p p p 1 1 1 1k p−p−p p p p q k q 1 2 1 2 PP PP PP p p Φ PP Belitsky, Ji, Yuan, NPB 656 (2003) Boer, Mulders, Pijlman, NPB 667 (2003) ξ ξ T T [+] ≡U [0,ξ] ξ − ξ − Possible effects in transverse momentum observables (ξT is conjugate to kT ) UNIVERSITÀ DI PAVIA 6/35 MANUALE D’USO DEL MARCHIO Colore 01 Colore 02 Colore 03 Progettazione Logo 3D Spin GDR Juljan Rushaj [email protected] TMD factorization Process dependence of gauge links SIDIS Drell-Yan Ph Ph PB PB ∆ Φ k k q k k q q q p p p p Φ Φ PP P P A ξ A T [+] [−] U[0,ξ] U[0,ξ] ξ − ξ T Belitsky, Ji, Yuan, NPB 656 (2003) Boer, Mulders, Pijlman, NPB 667 (2003) Boer, talk at RBRC Synergies workshop (2017) ξ − T ξ Z dkT −! ξT = 0 −! the same in both cases ξ − UNIVERSITÀ DI PAVIA 7/35 MANUALE D’USO DEL MARCHIO Colore 01 Colore 02 Colore 03 Progettazione Logo 3D Spin GDR Juljan Rushaj [email protected] Quark TMDs Leading Twist TMDs Nucleon Spin Quark Spin Quark Polarization Un-Polarized Longitudinally Polarized Transversely Polarized (U) (L) (T) ┴ — h1 = U ƒ1 = Boer-Mulders — L g1L = ┴ — h1L = Helicity h1 = — T ┴ — ┴ ƒ1T = — Transversity Nucleon Polarization Nucleon g1T = Sivers ┴ — h1T = UNIVERSITÀ DI PAVIA 8/35 MANUALE D’USO DEL MARCHIO Colore 01 Colore 02 Colore 03 Progettazione Logo 3D Spin GDR Juljan Rushaj [email protected] T -odd distributions The Sivers and Boer-Mulders functions Beyond the unpolarized f1, helicity g1L and transversity h1 surving the collinear ? ? limit, we have five more. In particular the Sivers (f1T ) and Boer-Mulders (h1 ): S · (p × k?) sq · (p × k?) Sivers effect Boer-Mulders effect Correlations between (proton or quark) spin and quark transverse momentum The Sivers effect is expected to give rise to transverse single spin asymmetries Sivers, PRD 41 (1990) UNIVERSITÀ DI PAVIA 9/35 MANUALE D’USO DEL MARCHIO Colore 01 Colore 02 Colore 03 Progettazione Logo 3D Spin GDR Juljan Rushaj [email protected] T -odd distributions The Sivers and Boer-Mulders functions Fundamental test of TMD theory ? [DY ] 2 ? [SIDIS] 2 ? [DY ] 2 ? [SIDIS] 2 f1T (x; k?) = −f1T (x; k?) h1 (x; k?) = −h1 (x; k?) Collins, PLB 536 (2002) FSI in SIDIS ISI in DY ξ T = − ξ − ξ T ISI/FSI lead to process dependence of TMDs, could even break factorization ξ − Collins, Qiu, PRD 75 (2007) Collins, PRD 77 (2007) Rogers, Mulders, PRD 81 (2010) UNIVERSITÀ DI PAVIA 10/35 MANUALE D’USO DEL MARCHIO Colore 01 Colore 02 Colore 03 Progettazione Logo 3D Spin GDR Juljan Rushaj [email protected] Process dependence of gluon TMDs UNIVERSITÀ DI PAVIA 11/35 MANUALE D’USO DEL MARCHIO Colore 01 Colore 02 Colore 03 Progettazione Logo 3D Spin GDR Juljan Rushaj [email protected] Gluon TMDs The gluon correlator p p αβ Γ (p;P,S) PP Gauge invariant definition of Γµν [U;U0]µν +ν C +µ C0 Γ / hP; Sj Trc F (0) U[0,ξ] F (ξ) U[ξ;0] jP; Si 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 ! e0QQX , ep ! e0 jet jet X probe gluon TMDs with [++] gauge links pp ! γγX (and/or other CS final state) probes gluon TMDs with [−−] gauge links pp ! γ jet X probes an entirely independent gluon TMD: [+−] links (dipole) UNIVERSITÀ DI PAVIA 12/35 MANUALE D’USO DEL MARCHIO Colore 01 Colore 02 Colore 03 Progettazione Logo 3D Spin GDR Juljan Rushaj [email protected] Gluon TMDs Angeles-Martinez et al., Acta Phys, Pol. B46 (2015) Mulders, Rodrigues, PRD 63 (2001) Meissner, Metz, Goeke, PRD 76 (2007) ? g I h1 : T -even distribution of linearly polarized gluons inside an unp. hadron ? g I f1T : T -odd gluon Sivers function In contrast to quark TMDs, gluon TMDs are almost unknown UNIVERSITÀ DI PAVIA 13/35 MANUALE D’USO DEL MARCHIO Colore 01 Colore 02 Colore 03 Progettazione Logo 3D Spin GDR Juljan Rushaj [email protected] Unpolarized gluon distributions WW vs DP Even unpolarized gluon TMDs are process dependent: two relevant types This was first realized in the small-x framework: Dominguez, Marquet, Xiao, Yuan, PRD (2011) I Weizs¨acker-Williams distribution (WW) I Dipole distribution (DP) Unpolarized (and in general T -even) gluon TMDs [+ +] = [− −] (WW) [+ −] = [− +] (DP) In general they can differ in magnitude and width. Only constraint: Z Z 2 [++] g 2 2 [+−] g 2 d kT f1 (x; kT ) = d kT f1 (x; kT ) Different processes can probe either types or a mixture of them UNIVERSITÀ DI PAVIA 14/35 MANUALE D’USO DEL MARCHIO Colore 01 Colore 02 Colore 03 Progettazione Logo 3D Spin GDR Juljan Rushaj [email protected] The gluon Sivers functions Sign change test Related Processes ep" ! e0QQX , ep" ! e0 jet jet X probe GSF with [++] gauge links (WW) p"p ! γγX (and/or other CS final state) probe GSF with [−−] gauge links ? q g ? g Analogue of the sign change of f1T between SIDIS and DY (true also for h1 and h1T ) ? g [e p"!e0 QQX ] ? g [p" p!γ γ X ] f1T = −f1T = − Boer, Mulders, CP, Zhou (2016) Motivation to study gluon Sivers effects at both RHIC and the EIC UNIVERSITÀ DI PAVIA 15/35 MANUALE D’USO DEL MARCHIO Colore 01 Colore 02 Colore 03 Progettazione Logo 3D Spin GDR Juljan Rushaj [email protected] The gluon Sivers functions The dipole GSF Complementary Processes ep" ! e0QQX probes a GSF with [++] gauge links (WW) p"p ! γ jet X (gq ! γq) 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) UNIVERSITÀ DI PAVIA 16/35 MANUALE D’USO DEL MARCHIO Colore 01 Colore 02 Colore 03 Progettazione Logo 3D Spin GDR Juljan Rushaj [email protected] The gluon Sivers functions f and d-type gluon distributions The first transverse moments of the WW and DP gluon Sivers functions Z 2 ?(1)g (f =d) 2 kT ? g (f =d) 2 f1T (x) = d kT 2 f1T (x; kT ) 2Mp (f =d) related to two different trigluon Qiu-Sterman functions TG , involving the antisymmetric fabc and symmetric dabc 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 Z X ?(1)a dx f1T (x) = 0 a=q;q¯;g Boer, Lorc´e,CP, Zhou, AHEP 2015 (2015) UNIVERSITÀ DI PAVIA 17/35 MANUALE D’USO DEL MARCHIO Colore 01 Colore 02 Colore 03 Progettazione Logo 3D Spin GDR Juljan Rushaj [email protected] The distribution of linearly polarized gluons ? g inside an unpolarized proton: h1 UNIVERSITÀ DI PAVIA 18/35 MANUALE D’USO DEL MARCHIO Colore 01 Colore 02 Colore 03 Progettazione Logo 3D Spin GDR Juljan Rushaj [email protected] Linear polarization of gluons Gluons inside an unpolarized hadron can be linearly polarized It requires nonzero transverse momentum Interference between ±1 gluon helicity states Like the unpolarized gluon TMD, it is T -even and exists in different versions: I [++] = [−−] (WW) I [+−] = [−+] (DP) UNIVERSITÀ DI PAVIA 19/35 MANUALE D’USO DEL MARCHIO Colore 01 Colore 02 Colore 03 Progettazione Logo 3D Spin GDR Juljan Rushaj [email protected] Linear polarization of gluons Visualization of the gluon polarization in the transverse momentum plane ? g h1 is taken to be a Gaussian y y pT pT x x pT pT ? g ? g h1 > 0 h1 < 0 The ellipsoid axis lengths are proportional to the