Gluon TMDs and Opportunities at EIC Cristian Pisano INT-18-3 Workshop Transverse Spin and TMDs October 8-12 2018 Seattle (USA)
2/35 TMD factorization and color gauge invariance
TMD factorization Two scale processes Q 2 p 2 T Factorization proven 3/35
TMD factorization Gauge invariant definition of Φ (not unique) Φ [U] P, S ψ(0) U C [0,ξ] ψ(ξ) P, S P h P h P h P h SIDIS P h P h P h P h q k k q q k k q q k k q k q p p p p 1 p 1 p p p p p 1 1 p p p 1 2 p 1 p 2 p P Φ P P P P P P P Belitsky, Ji, Yuan, NPB 656 (2003) Boer, Mulders, Pijlman, NPB 667 (2003) ξ T ξ ξ T U [+] [0,ξ] ξ Possible effects in transverse momentum observables (ξ T is conjugate to k T ) 4/35
TMD factorization Process dependence of gauge links SIDIS Drell-Yan P h P h P B Φ P B q k k q k q q k p p p p P Φ P P A Φ P A U [+] [0,ξ] ξ T ξ ξ T ξ U [ ] [0,ξ] dk T ξ T = 0 ξ T ξ Belitsky, Ji, Yuan, NPB 656 (2003) Boer, Mulders, Pijlman, NPB 667 (2003) Boer, talk at RBRC Synergies workshop (2017) the same in both cases 5/35
TMD factorization The quark Sivers function Fundamental test of TMD theory f [DY ] 1T (x, k ) 2 = f [SIDIS] 1T (x, k ) 2 h [DY ] 1 (x, k ) 2 = h [SIDIS] 1 (x, k ) 2 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) 6/35
7/35 Process dependence of gluon TMDs
Gluon TMDs The gluon correlator p p P Γ αβ (p;p,s) P Gauge invariant definition of Γ µν Γ [U,U ]µν [ P, S Tr c F +ν (0) U[0,ξ] C F +µ ] (ξ) U[ξ,0] C P, S 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 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) 8/35
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 1T between SIDIS and DY (true also for hg 1 f g [e p e QQ X ] 1T = f g [p p γ γ X ] 1T and h g 1T ) = Boer, Mulders, CP, Zhou (2016) Motivation to study gluon Sivers effects at both RHIC and the EIC 9/35
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 116 (2016) Boer, Cotogno, Van Daal, Mulders, Signori, Zhou, JHEP 1610 (2016) 10/35
Gluon TMDs Angeles-Martinez et al., Acta Phys, Pol. B46 (2015) Mulders, Rodrigues, PRD 63 (2001) Meissner, Metz, Goeke, PRD 76 (2007) h g 1 : T -even distribution of linearly polarized gluons inside an unp. hadron h g 1T, h g 1T : helicity flip distributions like hq 1T, h q 1T, but T -odd, chiral even! h g 1 hg 1T + p2 T 2Mp 2 h g 1T does not survive under p T integration, unlike transversity In contrast to quark TMDs, gluon TMDs are almost unknown 11/35
12/35 The distribution of linearly polarized gluons inside an unpolarized proton: h g 1
Linear polarization of gluons f g 1 ±1 ±1 1 Gluons inside an unpolarized hadron can be linearly polarized It requires nonzero transverse momentum 1 1 h g 1 ±1 h g 1 1 Interference between ±1 gluon helicity states It does not need ISI/FSI to be nonzero, unlike the Sivers function. However it is affected by them = process dependence 13/35
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 (2011) The nonperturbative distribution can be present at tree level and would contribute to Higgs production at low q T Sun, Xiao, Yuan, PRD 84 (2011) Boer, den Dunnen, CP, Schlegel, Vogelsang, PRL 108 (2012) 14/35
Gluon polarization and the Higgs boson p p H X at the LHC q T -distribution of the Higgs boson 1 σ 0.12 0.1 0.08 0.06 0.04 0.02 dσ dq 2 T 0.14 σ -1 2 dσ/dq T [GeV -2 ] 1 + R(q 2 T ) lin. pol. > 0{ r = 2/3 r = 1/3 lin. pol. = 0 0 0 2 4 6 8 10 q T [GeV] Gaussian Model 1 0.8 0.6 0.4 0.2 0-0.2 R = h g 1 h g 1 f g 1 f g 1 R 0 r = 2/3 r=1/3 0 2 4 6 8 10 12 14 q T [GeV] h g 1 (x, p 2 T ) 2M2 p 0.06 0.05 0.04 R(QT) 0.03 0.02 0.01 p 2 T f g 1 (x, p2 T ) TMD evolution m H = 126 GeV 0.00 0 5 10 15 20 QT [GeV] Study of H γγ and interference with gg γγ Echevarria, Kasemets, Mulders, CP, JHEP 1507 (2015) 158 Boer, den Dunnen, CP, Schlegel, PRL 111 (2013) 15/35
C = +1 quarkonium production q T -distribution of η Q and χ QJ (Q = c, b) in the kinematic region q T 2M Q 1 dσ(η Q ) σ(η Q ) dqt 2 f g 1 f g 1 [1 R(q2 T )] [pseudoscalar] 1 dσ(χ Q0 ) σ(χ Q0 ) dqt 2 f g 1 f g 1 [1 + R(q2 T )] [scalar] 1 dσ(χ Q2 ) σ(χ Q2 ) dqt 2 f g 1 f g 1 0.8 0.7 0.6 0.5 σ -1 2 dσ / d q (GeV -2 T ) χ 0 ( g h χ2 1 = 0) η Q 3 2.5 2 σ -1 2 dσ / d q (GeV -2 T ) Boer, CP, PRD 86 (2012) 094007 χ 0 ( g h χ2 1 = 0) η Q 0.4 1.5 0.3 2 p T 2 = 1 GeV 1 2 p T = 0.25 GeV 2 0.2 0.1 0.5 0 0 0.5 1 1.5 2 2.5 3 q T (GeV) 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 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 (2013), 014027; PLB 737 (2014) 103 16/35
17/35 J/ψ-pair production at the LHC
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 707 (2012) CMS JHEP 1409 (2014) ATLAS EPJC 77 (2017) gg fusion dominant, negligible q q contributions even at AFTER@LHC energies Lansberg, Shao, NPB 900 (2015) P 2 x2p2 +k2t ρ Φ ρσ g (x 2,k 2T, ζ 2, µ) PQ,1 σ x1p1 +k1t µ ν PQ,2 P 1 Φ µν g (x 1,k 1T, ζ 1, 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 111 (2013) Negligible Color Octet contributions, in particular at low P ΨΨ T 18/35
J/ψ-pair production Structure of the cross section dσ dqdy d 2 q T dω A f g 1 f g 1 + B f g 1 h g 1 cos(2φ CS ) + C h g 1 h g 1 cos(4φ CS ) Lansberg, CP, Scarpa, Schlegel, PLB 784 (2018) 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 107 (2011) den Dunnen, Lansberg, CP, Schlegel, PRL 112 (2014) Lansberg, CP, Schlegel, NPB 920 (2017) Boer, CP, PRD 91 (2015) The three contributions can be disentangled by defining the transverse moments cos nφ CS 2π 0 dφ CS cos(n φ CS ) dσ dqdy d 2 q T dω 2π 0 dφ dσ CS dqdy d 2 q T dω dφ CS dσ = f g 1 f g 1 (n = 2, 4) 19/35 cos 2φ CS = f g 1 h g 1 cos 4φ CS = h g 1 h g 1
J/ψ-pair production Extraction of f g 1 at s = 13 TeV We consider q T = P ΨΨ T M ΨΨ /2 in order to have two different scales dσ/dp ψψt / 0 <M ψψ > /2 dσ/dpψψt [GeV -1 ] Gaussian model: 0.4 0.3 0.2 0.1 Gaussian f 1 g, <kt 2 > fit over [0 ;<M ψψ >/2] LHCb data without DPS <M ψψ > = 8 GeV <k T 2 > = 3.3 ± 0.8 GeV 2 reduced χ 2 = 0.36 0 0 2 4 6 8 10 12 (b) P ψψt [GeV] Lansberg, CP, Scarpa, Schlegel, PLB 784 (2018) LHCb Coll., JHEP 06 (2017) f g 1 (x, k2 T ) = f g 1 (x) ( ) π kt 2 exp k2 T kt 2 20/35
21/35 Heavy quark pair production at an EIC
Heavy quark pair production in DIS Proposal for the EIC Gluon TMDs probed directly in e(l) + p(p, S) e(l ) + Q(K 1 ) + Q(K 2 ) + X the QQ pair is almost back to back in the plane to q and P Boer, Mulders, CP, Zhou, JHEP 1608 (2016) q l l : four-momentum of the exchanged virtual photon γ S K 1 q T K 1 + K 2 P K 1 φ 2 δφ φ 1 q P K (K 1 K 2 )/2 K 2 K 2 = Correlation limit: q T K, K K 1 K 2 22/35
Heavy quark pair production in DIS Angular structure of the cross section ±1 1 φ T, φ, φ S azimuthal angles of q T, K, S T At LO in pqcd: only γ g QQ contributes ±1 1 h g 1 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 0 K + AU 1 cos φ + A U 2 cos 2φ f g 1 (x, q2 T ) + q2 T Mp 2 h g 1 (x, q 2 T ) { } B U 0 cos 2φ T + B U 1 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 23/35
h g 1 in ep e QQX Maximal asymmetries Positivity bound for h g 1 : h g 1 (x, p 2 T ) 2M2 p p 2 T f g 1 (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 = 0.01 0.3 0.25 0.2 R Q 2 = 100 GeV 2 Q 2 = 10 GeV 2 Q 2 = 1 GeV 2 0.3 0.25 0.2 R Q 2 = 100 GeV 2 Q 2 = 10 GeV 2 Q 2 = 1 GeV 2 0.5 0.4 0.3 R Q 2 = 100 GeV 2 Q 2 = 10 GeV 2 Q 2 = 1 GeV 2 0.5 0.4 0.3 R Q 2 = 100 GeV 2 Q 2 = 10 GeV 2 Q 2 = 1 GeV 2 0.15 0.15 0.1 M Q = M c 0.1 M Q = M b 0.2 M Q = M c 0.2 M Q = M b 0.05 0.05 0.1 0.1 0 5 10 15 20 25 K (GeV) 0 5 10 15 20 25 K (GeV) 0 5 10 15 20 25 K (GeV) 0 5 10 15 20 25 K (GeV) CP, Boer, Brodsky, Buffing, Mulders, JHEP 1310 (2013) Boer, Brodsky, Mulders, CP, PRL 106 (2011) 24/35
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 0 +AT 1 cos φ + A T 2 cos 2φ f g 1T + cos(φ S φ T ) B T 0 sin 2φ T ] + B T 1 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 1T [ + B T 0 sin(φ S + φ T ) + B T 1 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 1T Boer, Mulders, CP, Zhou, JHEP 1608 (2016) 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 1T f g 1 A sin(φ S +φ T ) N hg 1 f g 1 A sin(φ S 3φ T ) N h g 1T f g 1 Same modulations as in SIDIS for quark TMDs (φ T φ h ) 25/35
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 = 1 GeV) 0.6 0.6 0.5 0.4 W A N max Q 2 = 100 GeV 2 Q 2 = 10 GeV 2 Q 2 = 1 GeV 2 0.5 0.4 W A N max Q 2 = 100 GeV 2 Q 2 = 10 GeV 2 Q 2 = 1 GeV 2 0.3 0.3 0.2 0.2 0.1 M Q = M c 0.1 M Q = M b 0 0 0.2 0.4 0.6 0.8 1 y 0 0 0.2 0.4 0.6 0.8 1 y 26/35
Spin asymmetries in ep e QQX Sivers asymmetry The Sivers asymmetry in open heavy quark production is bounded by 1 A sin(φ S φ T ) = q T M p f g 1T (x, q2 T ) f g 1 (x, q2 T ) ) ks A UT (φ 0.08 0.06 0.04 10% pos 0.02 0 L int = 10 fb 1 0.02 0.04 0.06 10% pos parton level 0.08 0 1 2 3 4 5 6 φ [rad] ks Zheng, Aschenauer, Lee, Xiao, Yin, PRD 98 (2018) If f g 1T is 10% of the positivity bound, then the measurement of A sin(φ S φ T ) N is problematic because of statistics The situation for dijets is more promising, but theoretically less clean 27/35
Asymmetries in e p 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 10 GeV 2 Upper bounds for A W N for K 4 GeV 0.18 0.16 0.14 0.12 W A N max Q 2 = 100 GeV 2 Q 2 = 50 GeV 2 Q 2 = 10 GeV 2 0.1 0.08 0.06 0.04 0.02 28/35 0 0 0.2 0.4 0.6 0.8 1 y
Asymmetries in e A e jet jet X small-x framework Also in a e A collisions polarization shows itself through a cos 2φ distribution Dumitru, Lappi, Skokov, PRL 115 (2015) cos 2φ has opposite signs for L and T γ -polarization, large effects 6 10 0.14 dn/dφ (a.u.) 0.12 0.1 0.08 0.06 Pythia6 Signal dijets Background dijets s = 90 GeV 4 < Q 2 < 12 GeV 2 1.25 < q < 1.75 GeV/c 3.00 < P < 3.50 GeV/c 1 < η < 2.5 T T 0.04 S/B = 11.28 ± 0.16 0.02 0 0 1 2 3 4 5 6 φ (rad) Dumitru, Skokov, Ullrich, arxiv:1809.02615 29/35 Monte-Carlo Generator: measurament feasible at the EIC
Quarkonium production at the EIC Bacchetta, Boer, CP, Taels, arxiv:1809.02056 30/35
Quarkonium production at the EIC Color Octet production mechanism e p e Q X with Q either a J/ψ or a Υ meson, with P 2 QT M 2 Q Q 2 Color octect (CO) production dominates q QQ [ 2S+1 L (8) J ] P Q p Theoretically described by Color Evaporation Model or NRQCD Godbole, Misra, Mukherjee, Rawoot, PRD 85 (2012) Godbole, Kaushik, Misra, Rawoot, PRD 91 (2015) Mukherjee, Rajesh, EPJC 77 (2017) Rajesh, Kishore, Mukherjee, PRD 98 (2018) Results depend on the quite uncertain CO 1 S 0 and 3 P J LDMEs 31/35
Quarkonium production at the EIC Upper bounds of the asymmetries Upper bounds for cos 2φ T and A W N with W = sin(φ S + φ T ), sin(φ S 3φ T ) 0.5 0.5 0.4 0.3 W A N max (J/ψ) CMSWZ Q 2 = 1 GeV 2 Q 2 = 5 GeV 2 Q 2 = 10 GeV 2 0.4 0.3 W A N max (Υ) Q 2 = 1 GeV 2 Q 2 = 5 GeV 2 Q 2 = 10 GeV 2 Q 2 = 100 GeV 2 0.2 0.2 SV 0.1 SV 0.1 0 0 0.2 0.4 0.6 0.8 1 y 0 0 0.2 0.4 0.6 0.8 1 y cos 2φ T still sizeable at small x, using the MV model as a starting input for h g 1 and f g 1 at x = 10 2 and evolve them with the JIMWLK equations Bacchetta, Boer, CP, Taels, arxiv:1809.02056 Marquet, Rosnel, Taels, PRD 97 (2018) 32/35
Quarkonium production at the EIC Upper bounds of the asymmetries The Sivers asymmetry does not depend on the CO NRQCD LDMEs A sin(φ S φ T ) = q T M p f g 1T (x, q2 T ) f g 1 (x, q2 T ) The other asymmetries depend on them, but one can consider ratios of asymmetries to cancel them out A cos 2φ T A sin(φ S +φ T ) = q2 T M 2 p h g 1 (x, q 2 T ) h g 1 (x, q2 T ) A cos 2φ T A = 1 h g 1 (x, qt 2 ) sin(φ S 3φ T ) 2 h g 1T (x, q2 T ) A sin(φ S 3φ T ) A sin(φ S +φ T ) = q2 T 2M 2 p h g 1T (x, q2 T ) h g 1 (x, q2 T ) Same relations hold for e p e Q Q X 33/35
Quarkonium production at the EIC CO NRQCD LDMEs @EIC One can consider ratios where the TMDs cancel out and one can obtain new experimental information on the CO NRQCD LDMEs It requires comparison with e p e Q Q X R cos 2φ dφt cos 2φ T dσ Q (φ S, φ T ) = dφt dφ cos 2φ T dσ QQ (φ S, φ T, φ ) R= dφt dσ Q (φ S, φ T ) dφt dφ dσ QQ (φ S, φ T, φ ) Two observables depending on two unknowns: O8 S 0 O Q 8 ( 1 S 0) 0 O8 P 0 O Q 8 ( 3 P 0) 0 [ R cos 2φ = 27π2 1 O8 S 1 ] O P 4 M Q MQ 2 8 R= 27 π2 4 1 M Q [1 + (1 y) 2 ] O S 8 + (10 10y + 3y 2 ) O P 8 /M 2 Q 26 26y + 9y 2 Plus similar (but different) equations for polarized quarkonium production 34/35
Conclusions Azimuthal asymmetries in heavy quark pair and dijet production in DIS could probe WW-type gluon TMDs (similar to SIDIS for quark TMDs) Quarkonia are also good probes for gluon TMDs: first extraction of unpolarized gluon TMD from LHC data on di-j/ψ production 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, AFTER@LHC and at EIC, overlap of both spin and small-x programs 35/35