Gluonic Spin Orbit Correlations Marc Schlegel University of Tuebingen in collaboration with W. Vogelsang, J.-W. Qiu; D. Boer, C. Pisano, W. den Dunnen Orbital Angular Momentum in QCD INT, Seattle, Feb. 10, 2012
Transverse Momentum Dependent PDFs Idea of TMDs: (x, k ) T Implement intrinsic transverse parton momentum k T different kind of factorization opportunity to study different aspects of hadron spin structure (e.g. spin-orbit correlations, overlap rep. etc.)
Transverse Momentum Dependent PDFs Idea of TMDs: (x, k ) T Implement intrinsic transverse parton momentum k T different kind of factorization opportunity to study different aspects of hadron spin structure (e.g. spin-orbit correlations, overlap rep. etc.) (Naive) definition of the quark TMD correlator
Transverse Momentum Dependent PDFs Idea of TMDs: (x, k ) T Implement intrinsic transverse parton momentum k T different kind of factorization opportunity to study different aspects of hadron spin structure (e.g. spin-orbit correlations, overlap rep. etc.) (Naive) definition of the quark TMD correlator Wilson line: process dependence (sign change, color entanglement, etc.)
Transverse Momentum Dependent PDFs Idea of TMDs: (x, k ) T Implement intrinsic transverse parton momentum k T different kind of factorization opportunity to study different aspects of hadron spin structure (e.g. spin-orbit correlations, overlap rep. etc.) (Naive) definition of the quark TMD correlator Wilson line: process dependence (sign change, color entanglement, etc.) correct definition (incl. soft factors, soft gluon resummation, rapidity cut-offs, etc.)
Transverse Momentum Dependent PDFs Idea of TMDs: (x, k ) T Implement intrinsic transverse parton momentum k T different kind of factorization opportunity to study different aspects of hadron spin structure (e.g. spin-orbit correlations, overlap rep. etc.) (Naive) definition of the quark TMD correlator Wilson line: process dependence (sign change, color entanglement, etc.) correct definition (incl. soft factors, soft gluon resummation, rapidity cut-offs, etc.) [Aybat, Rogers, PRD83, 114042; Collins Foundations of pqcd ]
Transverse Momentum Dependent PDFs Idea of TMDs: (x, k ) T Implement intrinsic transverse parton momentum k T different kind of factorization opportunity to study different aspects of hadron spin structure (e.g. spin-orbit correlations, overlap rep. etc.) (Naive) definition of the quark TMD correlator Wilson line: process dependence (sign change, color entanglement, etc.) correct definition (incl. soft factors, soft gluon resummation, rapidity cut-offs, etc.) [Aybat, Rogers, PRD83, 114042; Collins Foundations of pqcd ] Φ ij (x, k T ; S; ξ,µ) evolution equations for ξ, μ
Quark spin projection of correlator on γ +,γ + γ 5,γ + γ γ 5 8 quark TMDs, catagorized by nucleon/quark spin well-studied : [experimentally & theoretically] Sivers function Boer-Mulders function (naive) collinear limits: unpolarized, helicity, transversity wormgear functions pretzelosity quadrupole structure
Quark TMDs in Drell-Yan & SIDIS intrinsic transverse parton momentum through small final state transverse momenta q T Q P ht Q Drell-Yan W µν TMD factorization [Aybat, Rogers, PRD83, 114042; Collins Foundations of pqcd ] d 2 k at d 2 k bt δ (2) ( k at + k bt q T )Tr[ ˆM µ Φ(x a, k at )( ˆM ν ) Φ(xb, k bt )]+Y µν SIDIS W µν q T Q q T Q d 2 k T d 2 p T δ (2) ( k T p T P ht /z)tr[ ˆM µ Φ(x, k T )ˆ(M ν ) (z, p T )]+Y µν
Γ ij (x, k T )= 1 xp + Eight Gluon TMDs dz d 2 z T (2π) 3 e ik z P, S F +i (0) W[0 ; z] F +j (z) P, S z + =0 Γ [T even] (x, k T ) flip Γ [T odd] (x, k T ) flip U L T f g 1 g g 1L g g 1T h g 1 f g 1T h g 1 h g 1L h g 1T gluonic correspondence to Boer-Mulders : T-even unpolarized gluons in transversely pol. proton: gluon Sivers function gluonic transversity / pretzelosity / wormgears: T-odd no chirality two collinear PDFs [Mulders, Rodriues, PRD 63,094021]
Processes sensitive to gluon TMDs Gluon TMDs do not appear in Drell-Yan or SIDIS
Processes sensitive to gluon TMDs Gluon TMDs do not appear in Drell-Yan or SIDIS Jet / Hadron production in pp - collisions Spin dependent processes feasible at RHIC colored final states: problems with TMD factorization
Processes sensitive to gluon TMDs Gluon TMDs do not appear in Drell-Yan or SIDIS Jet / Hadron production in pp - collisions Spin dependent processes feasible at RHIC colored final states: problems with TMD factorization Heavy Quark production in ep - collisions [Boer, Brodsky, Mulders, Pisano, PRL 106, 132001] TMD factorization ok! Spin dependent gluon TMDs: EIC (Nucleon) spin independent gluon TMDs: EIC / HERA(?)
Photon pair production [Qiu, M.S., Vogelsang, PRL 107, 062001 (2011)] quark TMDs gluon TMDs at O(α s 2)
Photon pair production [Qiu, M.S., Vogelsang, PRL 107, 062001 (2011)] quark TMDs gluon TMDs at O(α s 2) no colored final state TMD factorization ok
Photon pair production [Qiu, M.S., Vogelsang, PRL 107, 062001 (2011)] quark TMDs gluon TMDs at O(α s 2) no colored final state TMD factorization ok gauge invariance box finite effectively tree-level
Photon pair production [Qiu, M.S., Vogelsang, PRL 107, 062001 (2011)] quark TMDs gluon TMDs at O(α s 2) no colored final state TMD factorization ok gauge invariance box finite effectively tree-level potentially large gluon distributions
Photon pair production [Qiu, M.S., Vogelsang, PRL 107, 062001 (2011)] quark TMDs gluon TMDs at O(α s 2) no colored final state TMD factorization ok gauge invariance box finite effectively tree-level potentially large gluon distributions new azimuthal observables
Unpolarized pp γγx Cross-Section at q T << Q quark contributions almost identical to DY
Unpolarized pp γγx Cross-Section at q T << Q quark contributions almost identical to DY gluon contributions absent in DY F i (θ) non-trivial functions of cos(θ) and sin(θ) (Logarithms from quark loop)
Unpolarized pp γγx Cross-Section at q T << Q quark contributions almost identical to DY gluon contributions absent in DY F i (θ) non-trivial functions of cos(θ) and sin(θ) (Logarithms from quark loop) Same angular structure in collinear resummation procedure for larger q T [Nadolsky et al. ; Grazzini, Catani et al.] matching of collinear + TMD formalism
Unpolarized pp γγx Cross-Section at q T << Q quark contributions almost identical to DY gluon contributions absent in DY F i (θ) non-trivial functions of cos(θ) and sin(θ) (Logarithms from quark loop) Same angular structure in collinear resummation procedure for larger q T [Nadolsky et al. ; Grazzini, Catani et al.] matching of collinear + TMD formalism cos(4φ) modulation a pure gluonic effect
Unpolarized pp γγx Cross-Section at q T << Q quark contributions almost identical to DY gluon contributions absent in DY F i (θ) non-trivial functions of cos(θ) and sin(θ) (Logarithms from quark loop) Same angular structure in collinear resummation procedure for larger q T [Nadolsky et al. ; Grazzini, Catani et al.] matching of collinear + TMD formalism cos(4φ) modulation a pure gluonic effect Quark contribution similar to DY only ISI / past-pointing Wilson line
Unpolarized pp γγx Cross-Section at q T << Q quark contributions almost identical to DY gluon contributions absent in DY F i (θ) non-trivial functions of cos(θ) and sin(θ) (Logarithms from quark loop) Same angular structure in collinear resummation procedure for larger q T [Nadolsky et al. ; Grazzini, Catani et al.] matching of collinear + TMD formalism cos(4φ) modulation a pure gluonic effect Quark contribution similar to DY only ISI / past-pointing Wilson line requires p T & isolation cuts for the photons
Unpolarized pp γγx Cross-Section at q T << Q quark contributions almost identical to DY gluon contributions absent in DY F i (θ) non-trivial functions of cos(θ) and sin(θ) (Logarithms from quark loop) Same angular structure in collinear resummation procedure for larger q T [Nadolsky et al. ; Grazzini, Catani et al.] matching of collinear + TMD formalism cos(4φ) modulation a pure gluonic effect Quark contribution similar to DY only ISI / past-pointing Wilson line requires p T & isolation cuts for the photons powerful in combination with DY map out quark TMDs in DY gluon TMDs in γγ
Positivity bounds RHIC energy: S = 500 GeV Gaussian ansatz: Gaussian widths: p T -cuts for each photon:
Positivity bounds RHIC energy: S = 500 GeV Gaussian ansatz: Gaussian widths: p T -cuts for each photon: numerical estimates Gaussian Ansatz + saturation of positivity bound [saturation partly supported by models for heavy ions (Metz, Zhou; Dominguez, Qiu, Xiao, Yuan)]
Positivity bounds RHIC energy: S = 500 GeV Gaussian ansatz: Gaussian widths: p T -cuts for each photon: numerical estimates Gaussian Ansatz + saturation of positivity bound [saturation partly supported by models for heavy ions (Metz, Zhou; Dominguez, Qiu, Xiao, Yuan)] cos(4φ) only due to lin. pol. gluons clean, ~1%
Positivity bounds RHIC energy: S = 500 GeV Gaussian ansatz: Gaussian widths: p T -cuts for each photon: numerical estimates Gaussian Ansatz + saturation of positivity bound [saturation partly supported by models for heavy ions (Metz, Zhou; Dominguez, Qiu, Xiao, Yuan)] cos(4φ) only due to lin. pol. gluons clean, ~1% Gluon BM contribution to φ-indep. cross section vanishes upon qt-integration!
Positivity bounds RHIC energy: S = 500 GeV Gaussian ansatz: Gaussian widths: p T -cuts for each photon: numerical estimates Gaussian Ansatz + saturation of positivity bound [saturation partly supported by models for heavy ions (Metz, Zhou; Dominguez, Qiu, Xiao, Yuan)] cos(4φ) only due to lin. pol. gluons clean, ~1% Gluon BM contribution to φ-indep. cross section vanishes upon qt-integration! cos(2φ) determination of sign of h 1 g
Gluon Sivers Effect (Transverse) Spin dependent photon pair cross section:
Gluon Sivers Effect (Transverse) Spin dependent photon pair cross section: Estimates for RHIC 500 GeV
Gluon Sivers Effect (Transverse) Spin dependent photon pair cross section: Estimates for RHIC 500 GeV Gaussian Ansatz + positivity bound for gluon and quark TMDs Flavor cancellation for quark Sivers func.: f,u 1T,d f1t bound only for u-quarks
Gluon Sivers Effect (Transverse) Spin dependent photon pair cross section: Estimates for RHIC 500 GeV Gaussian Ansatz + positivity bound for gluon and quark TMDs Flavor cancellation for quark Sivers func.: f,u 1T,d f1t bound only for u-quarks Sign not fixed by bound quark and gluon Sivers effect could add.
Gluon Sivers Effect (Transverse) Spin dependent photon pair cross section: Estimates for RHIC 500 GeV Gaussian Ansatz + positivity bound for gluon and quark TMDs Flavor cancellation for quark Sivers func.: f,u 1T,d f1t bound only for u-quarks Sign not fixed by bound quark and gluon Sivers effect could add. Gluons dominate at mid-rapidity, quarks at large rapidity
Gluon Sivers Effect (Transverse) Spin dependent photon pair cross section: Estimates for RHIC 500 GeV Gaussian Ansatz + positivity bound for gluon and quark TMDs Flavor cancellation for quark Sivers func.: f,u 1T,d f1t bound only for u-quarks Sign not fixed by bound quark and gluon Sivers effect could add. Gluons dominate at mid-rapidity, quarks at large rapidity Effects by gluon transv. / pretzel. small
Linearly polarized gluons and Higgs production [Boer, den Dunnen, Pisano, M.S., Vogelsang, PRL 108, 032002 (2012)] corresponding collinear resummation studies: [Nadolsky et al., Catani, de Florian, Grazzini; Sun, Xiao, Yuan] Can gluonic TMDs be useful for the LHC? Once a scalar particle (Higgs!?) is found... want to determine its parity. pure Higgs production via top-quark loop f g 1 ±1 ±1 h g 1 ±1 1 ±1 ±1 ±1 1 f g 1 h g 1
Linearly polarized gluons and Higgs production [Boer, den Dunnen, Pisano, M.S., Vogelsang, PRL 108, 032002 (2012)] corresponding collinear resummation studies: [Nadolsky et al., Catani, de Florian, Grazzini; Sun, Xiao, Yuan] Can gluonic TMDs be useful for the LHC? Once a scalar particle (Higgs!?) is found... want to determine its parity. pure Higgs production via top-quark loop f g 1 ±1 ±1 h g 1 ±1 1 ±1 ±1 ±1 1 f g 1 h g 1 q T -behaviour: R = [h g 1 h g 1 ] [f g 1 f g 1 ] 0.35 0.3 0.25 0.2 0.15 0.1 0.05 1 7 2 p T = 1 GeV 2 g f 1 2 2 g p T / (2M ) * h1 7 0 0 0.5 1 1.5 2 2.5 3 3.5 4 p T (GeV) 0.6 0.5 0.4 0.3 0.2 0.1-0.1 sizeable contribution of lin. pol. gluons characteristic double node in q T 0 R 2 p = 7 GeV 2 T 2 p = 1 GeV 2 T -0.2 0 2 4 6 8 10 12 14 16 q T (GeV)
Linearly polarized gluons and Higgs production [Boer, den Dunnen, Pisano, M.S., Vogelsang, PRL 108, 032002 (2012)] corresponding collinear resummation studies: [Nadolsky et al., Catani, de Florian, Grazzini; Sun, Xiao, Yuan] Can gluonic TMDs be useful for the LHC? Once a scalar particle (Higgs!?) is found... want to determine its parity. pure Higgs production via top-quark loop f g 1 ±1 ±1 h g 1 ±1 1 ±1 ±1 ±1 1 q T -behaviour: R = [h g 1 h g 1 ] [f g 1 f g 1 ] 0.35 0.3 0.25 0.2 0.15 0.1 0.05 1 7 2 p T = 1 GeV 2 g f 1 2 2 g p T / (2M ) * h1 7 0 0 0.5 1 1.5 2 2.5 3 3.5 4 p T (GeV) 0.6 0.5 0.4 0.3 0.2 0.1-0.1 sizeable contribution of lin. pol. gluons characteristic double node in q T 0 R 2 p = 7 GeV 2 T 2 p = 1 GeV 2 T f g 1-0.2 0 2 4 6 8 10 12 14 16 q T (GeV) h g 1 linearly polarized gluons sensitive to Higgs parity [f g 1 f g 1 ] ± [h g 1 h g 1 ] +: scalar Higgs -: pseudoscalar Higgs precise q T measurement may offer a way to determine Higgs parity
Including Higgs decay: gg H/A γγ + + +... φ - integrated cross section of Higgs + box: dφ dσgg d 4 qdω F 1 [f g 1 f g 1 ] + F 2 [h g 1 h g 1 ]
Including Higgs decay: gg H/A γγ + + +... φ - integrated cross section of Higgs + box: dφ dσgg d 4 qdω F 1 [f g 1 f g 1 ] + F 2 [h g 1 h g 1 ] F 2 /F 1 1 0.8 0.6 0.4 0.2 0-0.2-0.4 θ = π/2 θ = 3π/8 θ = π/4 θ = π/8 scalar 119.7 m H 120.3 Q [GeV] pseudoscalar 119.7 m H 120.3 Q [GeV] 0.2 0-0.2-0.4-0.6-0.8-1 Q m H : F1 F 2 box dominant Q m H : F 1 F 2 Higgs dominant (pole of the propagator) Sign signature preserved at the pole! small total Higgs width good Q resolution
q 2 T cos(2φ) = May use also azimuthal cos(2φ) modulation... d 2 q T dφ q 2 T cos(2φ) dσ d 4 qdω F 3 (θ, Q 2 ) scalar Higgs contributes to F 3, pseudoscalar doesn t offers alternative determination of Higgs parity. f g 1 h g 1 + h g 1 f g 1 F 3 60 40 20 0-20 scalar θ = π/2 scalar θ = 3π/8 scalar θ = π/4 scalar θ = π/8 pseudoscalar θ = π/2 pseudoscalar θ = 3π/8 pseudoscalar θ= π/4 pseudoscalar θ = π/8 119.8 120 120.2 Q [GeV] cos(2φ)>/<1> [GeV] <q T 2 2 1 0-1 -2 119.7 m H 120.3 θ = π/2 θ = 3π/8 θ = π/4 θ = π/8 scalar Q [GeV] pseudoscalar 119.7 m H 120.3 Q [GeV] 2.5 2 1.5 1 0.5 0
Summary Gluon TMDs from Photon Pair Production Gluon Boer-Mulders effect may offer a way to determine parity of the Higgs boson at LHC Gluon Sivers- and Boer-Mulders effect may be feasible at RHIC (if 0) To-do list Work out other spin observables in photon pair production Work out other final states relevant for LHC, e.g. gg ZZ, WW, Zγ, etc. Work out a more realistic k T -behaviour of h 1 g