Phenomenology of Gluon TMDs

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1 Phenomenology of Gluon TMDs Cristian Pisano

2 Outline Definition of the gluon TMDs of the proton Unpolarized TMD distribution f g linear polarization of gluons inside an unpolarized proton h g Gluon Sivers function f g T Discussion of their experimental determination ep collisions (EIC) pp collisions (RHIC, LHC) 2/29

3 Gluon TMDs The gluon correlator The gluon correlator describes the hadron gluon transition p p P Γ αβ (p;p,s) P Gluon momentum p α = x P α + p α T + p n α, with n 2 = and n P= Definition of Γ αβ for a spin-/2 hadron Γ αβ nρ nσ d(ξ P) d 2 ξ T = (p n) 2 (2π) 3 e ip ξ P, S Tr [ F αρ () U [,ξ] F βσ (ξ) U [ξ,] ] P, S ξ n= U, U : process dependent gauge links Mulders, Rodrigues, PRD 63 (2) 942 transverse projectors: g αβ T g αβ P α n β n α P β, ɛ αβ T ɛαβγδ P γn δ Spin vector: S α ( = S L P α Mh 2 nα) + S T, with SL 2 + S2 T = 3/29

4 Gluon TMDs The gluon correlator Parametrization of Γ αβ (at Leading Twist and omitting gauge links) Γ αβ U (x, p T ) = { ( p g αβ T 2x f g α (x, p2 T ) + T p β T Mh 2 + g αβ p 2 ) } T T 2Mh 2 h g (x, pt 2 ) [unp. hadron] Γ αβ L (x, p T ) = { 2x S L iɛ αβ T g g L (x, p2 T ) + p T ρ ɛ ρ{α } T pβ} T Mh 2 h g L (x, p2 T ) [long. pol. hadron] Γ αβ T (x, p T ) = 2x p T ρɛ ρ{α T Sβ} T { g αβ ɛ ρσ T p T ρ S T σ T M h + S T ρɛ ρ{α T pβ} T 4M h f g T (x, p2 T ) + p T S T iɛαβ T M h ρ{α g g T (x, p2 T ) h g T (x, p2 T ) + p T ρ ɛt pβ} T p T S T 2Mh 2 M h [transv. pol. hadron] h g T (x, p2 T ) } f g : unpolarized TMD gluon distribution h g : distribution of linearly polarized gluons inside an unpol. hadron Mulders, Rodrigues, PRD 63 (2) 942 f g T : T -odd distribution of unp. gluons inside a transversely pol. hadron Sivers, PRD 4 (99) 83 4/29

Gluon TMDs Linear polarization of gluons h g is a T -even, helicity-flip distribution, and a rank-2 tensor in p T h g (x, p 2 T ) in the absence of ISI or FSI, but, as any TMD, it will receive

5 Gluon TMDs Linear polarization of gluons h g is a T -even, helicity-flip distribution, and a rank-2 tensor in p T h g (x, p 2 T ) in the absence of ISI or FSI, but, as any TMD, it will receive contributions from ISI/FSI it can be nonuniversal In the transverse momentum plane (h g taken to be a Gaussian): p y T p y T p x T p x T h g > h g < The ellipsoid axis lengths are proportional to the probability of finding a gluon with a linear polarization in that direction 5/29

6 Extraction of h g at a future EIC Heavy quark pair production in DIS Ideal process: e(l) + p(p) e(l ) + Q(K ) + Q(K 2 ) + X the Q Q pair is almost back to back in the plane to q and P q l l : four-momentum of the exchanged virtual photon γ Tq K K φ l = φ l = φ 2 δφ K 2 φ P K + K 2 K (K K 2 )/2 K 2 = Correlation limit: K, K K K 2 6/29

7 Heavy quark pair production in DIS Angular structure of the cross section ± ± y (y 2 ) rapidities of Q ( Q) in the γ p cms; K + K 2 = (cos φ T, sin φ T ) K (K K 2 )/2 = K (cos φ, sin φ ) x B, y: DIS variables ± h g ± dσ dy dy 2 dy dx B d 2 d 2 K } {A + A cos φ + A 2 cos 2φ f g K + q2 { } T Mp 2 h g B cos 2(φ φ T ) + B cos(φ 2φ T ) + B cos(3φ 2φ T ) + B 2 cos 2φ T + B 2 cos 2(2φ φ T ) Upper bounds on the asymmetries R cos 2(φ φ T ) and R cos 2φ T R Q 2 = GeV 2 Q 2 = GeV 2 Q 2 = GeV R Q 2 = GeV 2 Q 2 = GeV 2 Q 2 = GeV R Q 2 = GeV 2 Q 2 = GeV 2 Q 2 = GeV R Q 2 = GeV 2 Q 2 = GeV 2 Q 2 = GeV M Q = M c. M Q = M b.2 M Q = M c.2 M Q = M b K (GeV) K (GeV) K (GeV) K (GeV) 7/29 CP, Boer, Brodsky, Buffing, Mulders, JHEP 3 (23) 24 Boer, Brodsky, Mulders, CP, PRL 6 (2) 32

8 Polarized gluons and Higgs boson production p p H X At LHC the dominant channel is gg H h g contributes to the Higgs -spectrum at LO 8/29 -distribution of the Higgs boson σ dσ dq 2 T σ - 2 dσ/d [GeV -2 ] + R (qt 2 ) R = h g h g f g f g lin. pol. > {r r = 2/3 = /3 lin. pol. = [GeV] Gaussian Model R h g (x, p 2 T ) 2M2 p p 2 T f g (x, p2 T ) Boer, den Dunnen, CP, Schlegel, Vogelsang, PRL 8 (22) 322 Boer, den Dunnen, CP, Schlegel, PRL (23) 322 Echevarria, Kasemets, Mulders, CP, JHEP 57 (25) 58 r = 2/3 r=/ [GeV] Study of H γγ and interference with gg γγ R(QT).3.2. With TMD evolution m H = 26 GeV QT [GeV]

5 2 σ - 2 dσ / d q (GeV -2 T ) Boer, CP, PRD 86 (22) 947 χ ( g h χ2 = ) η Q.4.5.3 2 p T = GeV 2 2 p T 2 =.25 GeV.2..5 9/29.5.5 2 2.5 3 (GeV).2.4.6.8.2.4 (GeV) Proof of

9 C = + quarkonia -distribution of η Q and χ QJ (Q = c, b) at 2M Q dσ(η Q ) σ(η Q ) dqt 2 f g f g [ R (qt 2 )] [pseudoscalar] dσ(χ Q ) σ(χ Q ) dqt 2 f g f g [ + R (qt 2 )] [scalar] dσ(χ Q2 ) σ(χ Q2 ) dqt 2 f g f g σ - 2 dσ / d q (GeV -2 T ) χ ( g h χ2 = ) η Q σ - 2 dσ / d q (GeV -2 T ) Boer, CP, PRD 86 (22) 947 χ ( g h χ2 = ) η Q p T = GeV 2 2 p T 2 =.25 GeV / (GeV) (GeV) Proof of factorization at NLO for p p η Q X in the Color Singlet Model (CSM) Ma, Wang, Zhao, PRD 88 (23), 427; PLB 737 (24) 3 Study of p p η c X at NLO with TMD evolution (LHCb data) Echevarria, Kasemets, Lansberg, CP, Signori, in preparation

10 Higgs plus jet production Outline of the calculation Advantage: study of the TMD evolution by tuning the hard scale Boer, CP, PRD 9 (25) 7424 TMD Master Formula dσ = d 3 K H d 3 K j dx 2s (2π) 3 2KH (2π) 3 2Kj a dx b d 2 p at d 2 p bt (2π) 4 a,b,c { δ 4 } (p a+p b q)tr Φ g (x a,p at )Φ g (x b,p bt ) M ab Hc 2 Higgs and jet almost back to back in the plane: K = K HT + K jt, K = (K HT K jt )/2 K : evolution scale, only of the pair needs to be small /29

11 Higgs plus jet production Feynman diagrams At LO in pqcd the partonic subprocesses that contribute are g g H g g q H q q q H g (a) (b) (c) (d) (e) (f) Quark masses taken to be zero, except for M t Kauffman, Desai, Risal, PRD 55 (997) 45 No indications that TMD factorization can be broken due to color entanglement Rogers, Mulders, PRD 8 (2) 946 /29

12 Higgs plus jet production Angular structure of the cross section Focus on gg Hg (dominant at the LHC). In the hadronic c.m.s.: = (cos φ T, sin φ T ) K = K (cos φ, sin φ ) φ φ T φ dσ dσ dy H dy j d 2 K d 2 dσ σ dσ q 2 T max dq 2 T 2π dφ dσ Normalized cross section for p p H jet X dσ σ = 2π σ(q2 T ) [ + R (q 2 T ) + R 2(q 2 T ) cos 2φ + R 4(q 2 T ) cos 4φ ] σ (q 2 T ) f g f g q 2 T max dqt 2 f g f g 2/29

13 Higgs plus jet production TMD observables The three contributions can be isolated by defining the observables cos nφ qt 2π dφ cos n φ dσ dσ (n =, 2, 4) such that cos nφ = q 2 T max dq 2 T cos nφ qt σ dσ d 2 qt = + R f g q f g + h g h g T cos 2φ qt = R 2 f g h g cos 4φ qt = R 4 h g h g 3/29

14 Higgs plus jet production Models for the TMD gluon distributions f g : Gaussian + tail f g (x, p2 T ) = f g (x) R2 2 π + p 2 T R2 R = 2 GeV h g : Maximal polarization and Gaussian + tail h g (x, p 2 T ) = 2M2 p p 2 T h g (x, p 2 T ) = 2 f g (x) M2 pr 4 h 2 π f g (x, p2 T ) [max pol.] ( + p 2 T R2 h )2 R h = 3 2 R Boer, den Dunnen, NPB 886 (24) 42 4/29

15 Higgs plus jet production Numerical results -distribution Configuration in which the Higgs and the jet have same rapidities. σ - 2 dσ/d [GeV -2 ]. σ - 2 dσ/d [GeV -2 ].8 K = GeV.8 K = GeV.6 K = GeV lin. pol. =.6 K = GeV lin. pol. = Gaussian + tail, max pol [GeV].2 Gaussian + tail [GeV] Effects largest at small (hard to measure), but model dependent! 5/29

16 Higgs plus jet production Numerical results Azimuthal cos 2φ asymmetries Sensitive to the sign of h g : cos 2φ qt < = h g >.3 cos2φ qt [GeV -2 ].25.2 K = GeV K = GeV.3 cos2φ qt [GeV -2 ].25.2 K = GeV K = GeV.5.5. Gaussian + tail, max pol.. Gaussian + tail [GeV] [GeV] max = M H /2 cos 2φ 2% at K = GeV 6/29

17 Higgs plus jet production Numerical results Azimuthal cos 4φ asymmetries.3 cos4φ qt [GeV -2 ].3 cos4φ qt [GeV -2 ].25.2 K = GeV K = GeV.25.2 K = GeV K = GeV Gaussian + tail, max pol Gaussian + tail [GeV] [GeV] max = M H /2 cos 4φ..2% at K = GeV 7/29

18 Higgs plus jet production Numerical results Gaussian model for the unpolarized TMDs σ - 2 dσ/d [GeV -2 ] K = 3 GeV K = GeV lin. pol. = Gaussian, max pol cos2φ qt [GeV -2 ] K = 3 GeV K = GeV Gaussian, max pol cos4φ qt [GeV -2 ] K = 3 GeV K = GeV Gaussian, max pol [GeV] [GeV] [GeV] max = K /2, cos 2φ 9%, cos 4φ.4% at K = GeV 8/29

19 Quarkonium + photon production Motivation Quarkonium Q Q Q[ 3 S ] and isolated γ produced almost back-to-back den Dunnen,blue Lansberg, CP, Schlegel, PRL 2 (24) 22 Accessible at the LHC : only the transverse momentum of the Q + γ pair needs to be small, not the individual ones Study of TMD evolution by tuning the invariant mass of Q + γ (evolution scale) Color octet (CO) contributions to Q + γ likely smaller than for inclusive Q Kim, Lee, Song, PRD 55 (997) 5429 Li, Wang, PLB 672 (29) 5 Lansberg, PLB 679 (29) 34 CO further suppressed w.r.t. CS contributions when Q γ back-to-back Mathews, Sridhar, Basu, PRD 6 (999) 49 9/29

20 Quarkonium + photon production Details of the calculation TMD factorization approach, in combination with the color-singlet model, for q 2 T Q 2, with = K Q + K γ, Q 2 q 2 = (K Q + K γ) 2 TMD Master Formula dσ = d 3 K Q d 3 K γ dx 2s (2π) 3 2KQ (2π) 3 2Kγ a dx b d 2 p at d 2 p bt (2π) 4 δ 4 (p a+p b q) { Tr Γ g (x a,p at )Γ g (x b,p bt ) } ( A g g Q Q[ ) 3 2 S ] colors Feynman diagrams at LO pqcd: 2/29

21 Quarkonium + photon production Structure of the cross section dσ dqdy d 2 dω A f g f g + B f g h g cos(2φ CS ) + C h g h g cos(4φ CS ) valid up to corrections O ( /Q) Y : rapidity of the Q + γ pair, along the beams in the hadronic c.m. frame dω = d cos θ CS dφ CS : solid angle for Q - γ in the Collins-Soper frame Analysis similar to the one for pp γγx Qiu, Schlegel, Vogelsang, PRL 7 (2) 62 The three contributions can be disentangled by defining the transverse moments S (n) 2π dφ CS cos(n φ CS ) q 2 max T dqt 2 dσ dqdy d 2 dω 2π dφ CS dσ dqdy d 2 dω (n =, 2, 4) q 2 max T = Q2 4 2/29 S q () T = f g f g S q (2) T = f g h g S (4) = h g h g

22 Quarkonium + photon production Color Singlet vs Color Octet Process dominated by gg fusion CS yield is clearly dominant for the Υ, above the CO one for J/ψ at low Q Further suppression of CO contributions by isolating Q (not needed for Υ) Kraan, AIP Conf. Proc. 38 (28) 45 Kikola, NP Proc. Suppl. 24 (2) 77 22/29

23 Quarkonium + photon production Results Q = 2 GeV, Y =, θ CS = π/2 S qt GeV Set B KMR CGC Gaussian GeV. S 2 qt GeV GeV.4 Set B max KMR max CGC Gaussian max S 4 qt GeV Set B max KMR max CGC Gaussian max GeV den Dunnen, Lansberg, CP, Schlegel, PRL 2 (24) 22 f g : Models for Unintegrated Gluon Distributions h g : predictions only in the CGC: otherwise saturated to its upper bound S (2,4) smaller than S () : can be integrated up to = GeV 2.% (KMR) < dq 2 T S (2).3% (CGC) < dq 2 T S (4) < 2.9% (Gauss) <.2% (Gauss) Possible determination of the shape of f g and verification of a non-zero h g 23/29

24 Wilson lines and factorization breaking h g receives contributions from ISI/FSI (gauge links) which make it process dependent and can even break factorization It is possibile to define five independent h g functions with specific color structures. Depending on the process, one extracts different combinations Buffing, Mukherjee, Mulders, PRD 88 (23) 5427 In ep e Q QX and in all the processes with a colorless final state, pp γγx, pp H/η c/χ c/...x, only two h g functions appear (in the same combination) In pp Q QX and pp jet jet X problems due to factorization breaking. Same holds for quarkonium production in pp in the CO and CEM models CP, Boer, Brodsky, Buffing, Mulders, JHEP 3 (23) 24 Mukherjee, Rajesh, arxiv: /29

25 The Gluon Sivers Function Possible measurements at EIC, RHIC, LHC Direct extraction from A N in e p e Q Q X and in e p e jet jet X (EIC) D. Boer, P. Mulders, CP, J. Zhou, in preparation Single spin asymmetries for p p γγx (RHIC) Qiu, Schlegel, Vogelsang, PRL 7 (2) 62 Other observables can probe GSF in future experiments (AFTER@LHC, RHIC): A sin φ S UT in p p η Q X e in p p χ QJ X per 2M Q al LO (CSM) A sin φ S (η UT Q ) = A sin φ S (χ UT Q ) = A sin φ S (χ UT Q2 ) = S T 2( R ) f g f g S T 2( + R ) f g f g S T 2 f g f g f g f g T { f g f g h g T h g T h g h g } T { f g f g + h g T h g T + h g h g } T 25/29 Sezione d urto polarizzata per p p J/ψ + γ X al LO (CSM) [φ φ T φ ] dσ UT dy ψ dy γ d 2 K d 2 sin φ S f g f g T + B { sin(φ S 2φ) f g hg T + sin φ S cos 2φ [f g h g T + h g + cos φ S sin 2φ [f g h g T + h g f g T f g T ] + sin φ S cos 4φ [h g h g T + h g h g T ] } ] + cos φ S sin 4φ [h g h g T + h g h g T ] J. Lansberg, CP, M. Schlegel, in preparation

26 Estimate of the gluon Sivers function The Generalized Parton Model (GPM) Assumption: Factorization and universality of TMDs (Test of the GPM) Motivation: New highly precise mid-rapidity RHIC data for A N in p p π X s = 2 GeV ηπ χ 2 = % χ 2 min DSS - SIDIS2.4 A N P T (GeV) [PHENIX Coll.] PRD 9 (24) 26 Data for A N (p p jet X ) not included in the fit, but consistent with new bound [STAR Coll.] PRD 86 (22) /29

27 Estimate of the gluon Sivers function Analysis A N = A quark N + A gluon N χ 2 = ( A gluon +A quark A exp ) 2 N N N σexp 2 +σ2 quark (4 parameters) σ quark : Estimated theoretical uncertainty on the quark contribution A N A N -.2 χ 2 = % χ 2 min -.2 χ 2 = % χ 2 min -.4 χ 2 = 2% χ 2 min gluon KRE - SIDIS P T (GeV) χ 2 = 2% χ 2 min gluon DSS - SIDIS P T (GeV) Uncertainty bands: envelope of A N curves generated by all parameter sets in the explored phase space giving χ 2 χ 2 min + χ 2, χ 2 = (2%, %) χ 2 min Open issues: factorization, process dependence, relation to twist-three approach Boer, Lorcé, CP, Zhou, AdHEP (25) 27/29

28 Estimate of the gluon Sivers function Results Small, positive GSF, mainly constrained in the range.5 < x <.3 (SIDIS) N f () g/p (x) d 2 k k 4Mp N f g/p (x, k ) = f ()g T (x) χ 2 /χ 2 min =% χ 2 /χ 2 min =% χ 2 /χ 2 min = 2% Ref. [2] χ 2 /χ 2 min = 2% Ref. [2] N f () g (x). N f () g (x).... KRE-SIDIS. DSS - SIDIS2... x... x D Alesio, Murgia, CP, JHEP 59 (25) 9 Ref. [2]: Old bound on the gluon Sivers function Anselmino, D Alesio, Melis, Murgia, PRD74 (26) 94 28/29

29 Summary and conclusions Gluon TMDs could be directly probed by looking at p T distributions and azimuthal asymmetries in e p e Q Q X and e p e jet jet X at a future EIC In pp collisions, h g leads to a modulation of the angular independent transverse momentum distribution of scalar (H, χ c, χ b ) and pseudoscalar (η c, η b ) particles First determination of h g and f g could come from J/ψ(Υ) + γ production at the LHC. Other opportunity offered by J/ψ + J/ψ production at the LHC J. Lansberg, CP, F. Scarpa, in preparation Together with a similar study in the Higgs sector, quarkonium production can be used to extract gluon TMDs and to study their process and scale dependences First extraction of the GSF f T from A N data on pp π X (RHIC) 29/29

Gluon TMDs and Heavy Quark Production at an EIC

Gluon TMDs and Heavy Quark Production at an EIC Cristian Pisano INT-7-3 Workshop Hadron imaging at Jefferson Lab and at a future EIC September 25-29 27 Seattle (USA) Quark TMDs Angeles-Martinez et al.,