Gluon TMDs and Heavy Quark Production at an EIC

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1 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 Seattle (USA)

2 Quark TMDs Angeles-Martinez et al., Acta Phys, Pol. B46 (25) h q : T -odd distribution of transversely polarized quarks inside an unp. hadron h q T, h q T : helicity flip distributions: T -even and chiral odd Transversity h q hq T + p2 T 2Mp 2 h q T survives under p T integration They are known and can all be accessed in semi-inclusive DIS (SIDIS) 2/27

3 Gluon TMDs Angeles-Martinez et al., Acta Phys, Pol. B46 (25) Mulders, Rodrigues, PRD 63 (2) Meissner, Metz, Goeke, PRD 76 (27) h g : T -even distribution of linearly polarized gluons inside an unp. hadron h g T, h g T : helicity flip distributions like hq T, h q T, but T -odd, chiral even! h g hg T + p2 T 2Mp 2 h g T does not survive under p T integration, unlike transversity In contrast to quark TMDs, gluon TMDs are almost unknown 3/27

4 The distribution of linearly polarized gluons inside an unpolarized proton: h g 4/27

5 Linear polarization of gluons f g ± ± Gluons inside an unpolarized hadron can be linearly polarized It requires nonzero transverse momentum h g ± h g Interference between ± gluon helicity states It does not need ISI/FSI to be nonzero, unlike the Sivers function. However it is affected by them = process dependence 5/27

6 Gluon polarization and the Higgs boson p p H X at the LHC Higgs boson production happens mainly via gg H Pol. gluons affect the Higgs transverse spectrum at NNLO pqcd Catani, Grazzini, NPB 845 (2) The nonperturbative distribution can be present at tree level and would contribute to Higgs production at low q T Boer, den Dunnen, CP, Schlegel, Vogelsang, PRL 8 (22) 6/27

7 Gluon polarization and the Higgs boson p p H X at the LHC q T -distribution of the Higgs boson σ dσ dq 2 T.4 σ - 2 dσ/dq T [GeV -2 ] + R(q 2 T ) lin. pol. > { r = 2/3 r = /3 lin. pol. = q T [GeV] Gaussian Model R = h g h g f g f g R r = 2/3 r=/ q T [GeV] h g (x, p 2 T ) 2M2 p R(QT).3.2. p 2 T f g (x, p2 T ) TMD evolution m H = 26 GeV QT [GeV] Study of H γγ and interference with gg γγ Echevarria, Kasemets, Mulders, CP, JHEP 57 (25) 58 Boer, den Dunnen, CP, Schlegel, PRL (23) 7/27

8 C = + quarkonium production q T -distribution of η Q and χ QJ (Q = c, b) in the kinematic region q T 2M Q dσ(η Q ) σ(η Q ) dqt 2 f g f g [ R(q2 T )] [pseudoscalar] dσ(χ Q ) σ(χ Q ) dqt 2 f g f g [ + R(q2 T )] [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 2 = GeV 2 p T =.25 GeV q T (GeV) q T (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 8/27

9 Heavy quark pair production at an EIC 9/27

10 Heavy quark pair production in DIS Proposal for the EIC Gluon TMDs probed directly in e(l) + p(p, S) e(l ) + Q(K ) + Q(K 2 ) + X the QQ pair is almost back to back in the plane to q and P Boer, Mulders, CP, Zhou, JHEP 68 (26) q l l : four-momentum of the exchanged virtual photon γ S K q T K + K 2 P K φ 2 δφ φ q P K (K K 2 )/2 K 2 K 2 = Correlation limit: q T K, K K K 2 /27

11 Heavy quark pair production in DIS Angular structure of the cross section ± φ T, φ, φ S azimuthal angles of q T, K, S T At LO in pqcd: only γ g QQ contributes ± h g dσ(φ S, φ T, φ ) = dσ U (φ T, φ ) + dσ T (φ S, φ T, φ ) Angular structure of the unpolarized cross section for ep e QQX, q T K dσ U { } d 2 q T d 2 A U K + AU cos φ + A U 2 cos 2φ f g (x, q2 T ) + q2 T Mp 2 h g (x, q 2 T ) { } B U cos 2φ T + B U cos(2φ T φ ) + B U 2 cos 2(φ T φ ) + B U 3 cos(2φ T 3φ ) + B U 4 cos 2(φ T 2φ ) The different contributions can be isolated by defining dφ dφ T W (φ, φ T ) dσ W (φ, φ T ) =, W = cos 2φ T, cos 2(φ φ T ),... dφ dφ T dσ /27

12 h g in ep e QQX Maximal asymmetries Positivity bound for h g : h g (x, p 2 T ) 2M2 p p 2 T f g (x, p2 T ) It can be used to estimate maximal values of the asymmetries Asymmetries usually larger when Q and Q have same rapidities Upper bounds on R cos 2(φ T φ ) and R cos 2φ T at y = 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) CP, Boer, Brodsky, Buffing, Mulders, JHEP 3 (23) Boer, Brodsky, Mulders, CP, PRL 6 (2) 2/27

13 Spin asymmetries in ep e QQX Angular structure of the single polarized cross section for ep e QQX, q T K [ ] [ dσ T sin(φ S φ T ) A T +AT cos φ + A T 2 cos 2φ f g T + cos(φ S φ T ) B T sin 2φ T ] + B T sin(2φ T φ ) + B T 2 sin 2(φ T φ ) + B T 3 sin(2φ T 3φ ) + B T 4 sin(2φ T 4φ ) h g T [ + B T sin(φ S + φ T ) + B T sin(φ S + φ T φ ) + B T 2 sin(φ S + φ T 2φ ) ] +B T 3 sin(φ S + φ T 3φ ) + B T 4 sin(φ S + φ T 4φ ) h g T The φ S dependent terms can be singled out by means of azimuthal moments A W N A W (φ S,φ T ) dφt dφ W (φ S, φ T ) dσ T (φ S, φ T, φ ) N 2 dφt dφ dσ U (φ T, φ ) A sin(φ S φ T ) N f g T f g A sin(φ S +φ T ) N hg f g A sin(φ S 3φ T ) N h g T f g Same modulations as in SIDIS for quark TMDs (φ T φ h ) 3/27

14 Spin asymmetries in ep e QQX Upper bounds Maximal values for A W N, W = sin(φ S + φ T ), sin(φ S 3φ T ) ( K = GeV) W A N max Q 2 = GeV 2 Q 2 = GeV 2 Q 2 = GeV W A N max Q 2 = GeV 2 Q 2 = GeV 2 Q 2 = GeV M Q = M c. M Q = M b y y 4/27

15 Asymmetries in ep e jet jet X Upper bounds Contribution to the denominator also from γ q gq, negligible at small-x Asymmetries much smaller than in c c case for Q 2 GeV 2 Upper bounds for A W N for K 4 GeV W A N max Q 2 = GeV 2 Q 2 = 5 GeV 2 Q 2 = GeV 2 5/ y

16 Process dependence of gluon TMDs 6/27

17 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 γγx (and/or other CS final state) probe GSF with [ ] gauge links Analogue of the sign change of f q T between SIDIS and DY (true also for hg and h g T ) f g [e p e QQ X ] T = f g [p p γ γ X ] T = Boer, Mulders, CP, Zhou (26) We expect to measure the same h g and f g in both processes Motivation to study gluon Sivers effects at both RHIC and the EIC 7/27

18 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 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 6 (26) Boer, Cotogno, Van Daal, Mulders, Signori, Zhou, JHEP 6 (26) 8/27

19 J/ψ-pair production at the LHC Lansberg, CP, Scarpa, Schlegel, in progress 9/27

20 J/ψ-pair production at the LHC J/ψ s are relatively easy to detect. Accessible at the LHC: already studied by LHCb, CMS & ATLAS LHCb PLB 77 (22) CMS JHEP 49 (24) ATLAS EPJC 77 (27) gg fusion dominant, negligible q q contributions even at AFTER@LHC energies Lansberg, Shao, NPB 9 (25) P 2 x2p2 +k2t ρ Φ ρσ g (x 2,k 2T, ζ 2, µ) σ PQ, µ ν xp +kt PQ,2 P Φ µν g (x,k T, ζ, µ) No final state gluon needed for the Born contribution in the Color Singlet Model. Pure colorless final state, hence simple color structure because one has only ISI Lansberg, Shao, PRL (23) 2/27

21 J/ψ-pair production at the LHC Color Singlet vs Color Octet Negligible Color Octet contributions, in particular at low P ΨΨ T [Black/dashed curves vs blue ones] Lansberg, Shao PLB 75 (25) 2/27

22 J/ψ-pair production at the LHC Single vs double parton scattering At low PT ΨΨ, small double parton scattering (DPS) contributions, otherwise required by the CMS, ATLAS data at large rapidity separations y of the J/ψ s Lansberg, Shao, PLB 75 (25) 22/27

23 J/ψ-pair production at the LHC Calculation of the cross section At LO pqcd in the Color Singlet Model, one needs to consider 36 diagrams Qiao, Sun, Sun, JPG 37 (2) 23/27

24 J/ψ-pair production Structure of the cross section dσ dqdy d 2 q T 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 T /Q) Y : rapidity of the J/ψ-pair, along the beam in the hadronic c.m. frame dω = d cos θ CS dφ CS : solid angle for J/ψ-pair in the Collins-Soper frame Analysis similar to the one for pp γγx, pp Jψ γ ( ) X, pp H jet X Qiu, Schlegel, Vogelsang, PRL 7 (2) den Dunnen, Lansberg, CP, Schlegel, PRL 2 (24) Lansberg, CP, Schlegel, NPB 92 (27) Boer, CP, PRD 9 (25) The three contributions can be disentangled by defining the transverse moments cos nφ CS 2π dφ CS cos(n φ CS ) dσ dqdy d 2 q T dω 2π dφ dσ CS dqdy d 2 q T dω dφ CS dσ = f g f g (n = 2, 4) 24/27 cos 2φ CS = f g h g cos 4φ CS = h g h g

25 J/ψ-pair production Extraction of f g at s = 3 TeV We consider q T = P ΨΨ T M ΨΨ /2 in order to have two different scales M - ψψ /2 dσ/dp ψψt [GeV - ] dσ/dp ψψt / Gaussian f g model, <kt 2 > fitted over [ ; M - ψψ /2] LHCb 3 TeV data M - ψψ = 8 GeV <k T 2 > = 4.9 ±.78 GeV 2 Gaussian model: P ψψt [GeV] f g (x, k2 T ) = f g (x) ( ) π kt 2 exp k2 T kt 2 LHCb Coll., JHEP 6 (27) 25/27

26 J/ψ-pair production cos nφ CS at s = 3 TeV At y the modulation C of cos4φ saturates its upper bound C = A The cos 2φ modulation can fix the sign of h g M ΨΨ = 8, 2, 2 GeV relevant for LHCb, CMS, ATLAS cos 2φ CS (%) cos 4φ CS (%) S (2;cos θ CS ) [in %] Model Model 2 <k T 2 > = 4.9 GeV 2 cos θ CS <.25 M ψψ = 8 GeV M ψψ = 2 GeV M ψψ = 2 GeV P ψψt [GeV] S (4;cos θ CS ) [in /%] Model Model 2 <k T 2 > = 4.9 GeV 2 cos θ CS <.25 M ψψ = 2 GeV M ψψ = 8 GeV M ψψ = 2 GeV P ψψt [GeV] Models for h g () h g (x, k 2 T ) = 2M2 k T 2 f g (x, k2 T ) ; (2) h g (x, k 2 T ) = M f g (x) π k T 2 3/2 2e( r) r exp ( r kt 2 ) k T 2 r = 2/3 26/27

27 Conclusions Azimuthal asymmetries in heavy quark pair and dijet production in DIS could probe WW-type gluon TMDs (similar to SIDIS for quark TMDs) Asymmetries maximally allowed by positivity bounds of gluon TMDs can be sizeable in specific kinematic region Different behavior of WW and dipole gluon TMDs accessible at RHIC, and at EIC, overlap of both spin and small-x programs First extraction of unpolarized gluon TMD from LHC data on di-j/ψ 27/27