Baryons beyond the quark-diquark model Gernot Eichmann (Darmstadt University of Technology, Germany) with Reinhard Alkofer Andreas Krassnigg Diana Nicmorus (University of Graz, Austria) Gernot Eichmann (TU Darmstadt) MENU 2010 18. 3. 2010 1 / 15 Motivation Hadron structure: Covariant bound-state equations Nucleon & delta properties: Dyson-Schwinger equations (DSEs) masses, em. form factors, of QCD provide input: Green functions N transition, . Ab-initio, but truncations necessary “QCD modeling” Are baryons predominantly →훥훾 quark-diquark systems? Chiral limit heavy quarks Complements lattice QCD, Chiral structure, quark models,⟷ effective field theories. role of the pion cloud How are quarks and gluons distributed in hadrons? Mesons: How does the proton spin Bethe-Salpeter equation (BSE) come about? Maris, Roberts ‘97; Maris, Tandy ’99; . Baryons: A comprehensive description of Quark-diquark model hadrons within QCD is needed. Oettel, Hellstern, Alkofer, Reinhardt ‘98, Chiral symmetry breaking, dynamical GE, Alkofer, Krassnigg, Schwinzerl ‘08, . mass generation, confinement Faddeev equation GE, Alkofer, Krassnigg, Nicmorus: PRL 104, 201601 (2010) Gernot Eichmann (TU Darmstadt) MENU 2010 18. 3. 2010 2 / 15 Bound-state equations Mesons: Bound-state amplitudes Bethe-Salpeter equation On-shell: (2) K ( 푃� � �푀� 푃 푞 Covariants quark-spin and OAM 훤 푞, 푃) = �� 푓 (푞�, 푞�∙푃� ) 휏� (푞, 푃) eigenstates in particle’s� restframe: Baryons: ⇔ Three-body equation 0 meson: 4 covariants: (s,p) 1 meson: 8 covariants: (s,p,d) (3) ⁻ 훾� , . K Nucleon: 64 covariants (s,p,d) (Quark-diquark⁻ model: 8) 훾� , . 푝 푃 푞 Dressed quark propagator solve quark DSE ( ( 2- and 3-quark⇒ kernels푀 푝�), 퐴 푝�). the ‘modeling’ part. (2) K ~ , , , . Gernot Eichmann (TU Darmstadt) MENU 2010 18. 3. 2010 3 / 15 Bound-state equations Three-body equation: = + + + Irreducible Quark-quark correlations 3-body assumed as dominant structure in baryons. diagrams Hints: lattice QCD, BSE, hadron spectrum, ... 3-gluon coupling to each quark, ... Gernot Eichmann (TU Darmstadt) MENU 2010 18. 3. 2010 4 / 15 Bound-state equations Faddeev equation: + + Quark-quark correlations � assumed as dominant structure in baryons. Hints: lattice QCD, BSE, hadron spectrum, ... Still numerically expensive, Need to know but we can do this now. Quark propagator Full covariant structure obtained from its Dyson-Schwinger equation of the amplitude! GE, Alkofer, Krassnigg, Nicmorus: 2-quark kernel PRL 104, 201601 (2010) Practical strategy: construct ansätze EPJ Web Conf. 3, 3028 (2010) which satisfy AXWTI ( chiral symmetry) ⇔ Gernot Eichmann (TU Darmstadt) MENU 2010 18. 3. 2010 4 / 15 DSEs Dyson- Quark propagator: Gluon propagator: -1 -1 -1 -1 Schwinger = + = + equations: Quantum equations Ghost propagator: + + of motion for QCD’s -1 -1 Green functions. = + Infinitely coupled system. + + Ghost-gluon vertex: Roberts, Williams: Prog.Part.Nucl.Phys. 33 (1994) Alkofer, von Smekal: = + + + Phys.Rept. 353 (2001) Fischer: J. Phys. G 32 (2006) Quark-gluon vertex: Momentum limits: UV: Perturbation theory = + + + + IR: Infrared exponents Numerical solutions require truncations! + + + + Gernot Eichmann (TU Darmstadt) MENU 2010 18. 3. 2010 5 / 15 DSEs Dyson- Quark propagator: Gluon propagator: Schwinger -1 -1 -1 -1 2 -1 = + = D(k ) equations: Quantum equations of motion for QCD’s Green functions. Infinitely coupled system. “Rainbow truncation” Roberts, Williams: Prog.Part.Nucl.Phys. 33 (1994) Alkofer, von Smekal: Phys.Rept. 353 (2001) Fischer: J. Phys. G 32 (2006) Quark-gluon vertex: Momentum limits: UV: Perturbation theory = Г(k ) IR: Infrared exponents Numerical solutions require truncations! Gernot Eichmann (TU Darmstadt) MENU 2010 18. 3. 2010 5 / 15 Rainbow-Ladder Truncation Quark DSE: “Rainbow” Meson BSE: “Ladder” ( 2 ) 2 = ( ) -1 = -1 + 훼 푘 훼 푘 Why rainbow-ladder? simple exact in the UV respects chiral symmetry & its spontaneous breaking GMOR: = 2 chiral-limit pion is massless ⇒ 푓��푚� 푚��푞푞�� Good description (up to the bottom quark) of pseudoscalar & vector-meson ground states Effective coupling ( 2 ) is model input. 훼 푘 Gernot Eichmann (TU Darmstadt) MENU 2010 18. 3. 2010 6 / 15 Rainbow-Ladder Truncation Quark DSE: “Rainbow” Meson BSE: “Ladder” ( 2 ) 2 = ( ) -1 = -1 + 훼 푘 훼 푘 Why rainbow-ladder? What’s beyond rainbow-ladder? simple Pion cloud: Attractive (~100 MeV) exact in the UV in chiral region; reduces hadron masses, respects chiral symmetry & decay constants; enhances charge radii its spontaneous breaking GMOR: = 2 chiral-limit pion is massless ⇒ 푓��푚� 푚��푞푞�� NJL: Oertel, Buballa, Wambach (2001) Good description (up to the bottom quark) of BSE: Pichowsky, Walawalkar, Capstick (1999) ; pseudoscalar & vector-meson ground states Fischer, Nickel, Wambach (2007); Fischer, Williams (2008) 2 Effective coupling ( ) is model input. Non-resonant corrections: Repulsive (~100 MeV for ); 훼 푘 see talks by Cancellation with pion cloud? Christian Fischer & Richard Williams Fischer, Williams, PRL 103 (2009) 푚� ⇒ Gernot Eichmann (TU Darmstadt) MENU 2010 18. 3. 2010 6 / 15 Rainbow-Ladder Truncation Effective coupling: ‘Fixed strength’ ( ): Maris, Roberts, Tandy, Bhagwat, the only model input! adjust to reproduce experiment Krassnigg, . good description of , ( 2 ) = ( ², ) + ( ²) ground푐 states ⇒ 휋 휌 Maris,훼 푘 Tandy:푐 훼PRC�� 60푘 (1999)휔 훼�� 푘 results insensitive to : depend only on “integrated strength” 휔 0.5 Deep IR: confinement 1.5 15 2 ² [ ²] [ ] IR enhancement: ( ) 0.4 DCSB, mass generation, 1.4 10 hadron(variation physics of ) 푘 0.3 퐺푒푉 UV: perturbation theory, free훼 quarks and gluons 1.3 5 ( 2 ) 0.2 ln / 휔 휋훾� 푒푓푓 푘 훼 푘� 훬� �� 0 0.1 1.2 0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 [ ] [ ] 1.1 1.2 0.22 푘� 퐺푒푉� [ ݉] ࡜ ܩܸ݁ 1.0 0.8 0.18 ݂࡜ ܩܸ݁ 0.9 0.4 푐 0.8 0.14 푚� 0.0 0.7 0.0 0.2 0.4 0.6 0.0 0.2 0.4 0.6 0.0 0.1 0.2 0.3 0.4 0.5 [ ] [ ] [ ] 푚�� 퐺푒푉� Gernot Eichmann (TU Darmstadt) MENU 2010 푚�� 18.퐺푒푉 3.� 2010 7 / 15 Rainbow-Ladder Truncation Effective coupling: ‘Fixed strength’ ( ): Maris, Roberts, Tandy, Bhagwat, the only model input! adjust to reproduce experiment Krassnigg, . good description of , ( 2 ) = ( ², ) + ( ²) ground푐 states ⇒ 휋 휌 Maris,훼 푘 Tandy:푐 훼PRC�� 60푘 (1999)휔 훼�� 푘 ‘Core model’ ( ): GE, Alkofer, Cloet, Krassnigg, adjust to reproduce hadronic Roberts : PRC 77 (2008) quark core w/o pion cloud consistent푐 response in masses, results insensitive to : decay constants, charge radii depend only on “integrated strength” ⇒ 휔 0.5 Deep IR: confinement 1.5 15 2 ² [ ²] [ ] IR enhancement: ( ) 0.4 DCSB, mass generation, 1.4 10 hadron(variation physics of ) 푘 0.3 퐺푒푉 UV: perturbation theory, free훼 quarks and gluons 1.3 5 ( 2 ) 0.2 ln / 휔 휋훾� 푒푓푓 푘 훼 푘� 훬� �� 0 0.1 1.2 0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 [ ] [ ] 1.1 1.2 0.22 푘� 퐺푒푉� [ ݉] ࡜ ܩܸ݁ 1.0 0.8 0.18 ݂࡜ ܩܸ݁ 0.9 0.4 푐 0.8 0.14 푚� 0.0 0.7 0.0 0.2 0.4 0.6 0.0 0.2 0.4 0.6 0.0 0.1 0.2 0.3 0.4 0.5 [ ] [ ] [ ] 푚�� 퐺푒푉� Gernot Eichmann (TU Darmstadt) MENU 2010 푚�� 18.퐺푒푉 3.� 2010 7 / 15 A quick detour: the quark-diquark model Faddeev equations neglect 3q correlations: pairwise qq interaction; involve qq T-matrix as intermediate step T (2) T (2) � � � � � � � 휙� � 휙� 휙� � � � Rainbow-ladder kernel: Diquark ansatz for qq T-matrix: timelike diquark poles in T-matrix T (2) Diquark BSE Im P2 = Total diquark Diquarks� ⎯ momentum P 푃� 푀� Diquark BSE: solve for lightest diquarks: Re P 2 scalar (0 ), �axial-vector (1 ). diquark masses & amplitudes 1⁺ 0⁺ ⁺ ⁺ Quark-diquark BSE nucleon mass & amplitudes ⇒ ⇒ Quark exchange between quark and diquark binds nucleon; gluon exchange between quarks 훷 훷 binds diquarks. Gernot Eichmann (TU Darmstadt) MENU 2010 18. 3. 2010 8 / 15 A quick detour: the quark-diquark model Effective coupling: ‘Fixed strength’ ( ): Quark-diquark model the only model input! adjust to reproduce experiment and masses inherit behavior of good description of , , , meson properties. No baryonic input! 2 푐 푁 훥 ( ) = ( ², ) + ( ²) ground states GE et al., PRC 79 & Nicmorus et al., PRD 80 ⇒ 휋 휌 푁 훥 1.9 훼 푘 푐 훼�� 푘 휔 훼�� 푘 ‘Core model’ ( ): [ ] adjust to reproduce hadronic 1.8 quark core w/o pion cloud 퐺푒푉 consistent푐 response in masses, 1.7 decay constants, charge radii ⇒ 1.6 0.5 Deep IR: 푀� confinement 1.5 15 2 ² [ ²] IR enhancement: ( ) 0.4 DCSB, mass generation, 1.4 10 hadron(variation physics of ) 푘 0.3 UV: perturbation theory, free훼 quarks and gluons 1.3 5 ( 2 ) 0.2 ln / 휔 휋훾� 푒푓푓 푘 훼 푘� 훬� �� 0 0.1 1.2 0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 푀� [ ] [ ] 1.1 1.2 0.22 푘� 퐺푒푉� [ ݉] ࡜ ܩܸ݁ 1.0 0.8 0.18 ݂࡜ ܩܸ݁ 0.9 0.4 푐 0.8 0.14 푚� 0.0 0.7 0.0 0.2 0.4 0.6 0.0 0.2 0.4 0.6 0.0 0.1 0.2 0.3 0.4 0.5 [ ] [ ] [ ] 푚�� 퐺푒푉� Gernot Eichmann (TU Darmstadt) MENU 2010 푚�� 18.퐺푒푉 3.� 2010 9 / 15 Faddeev equation in rainbow-ladder: any pair of 2 quarks bound by gluon ladder exchange ݌Ҡ = ݌Ҡ + + ݌ҟ ݌ ݇ ݌ҟ ݇ ݇ ݌Ҟ ݍ ݌Ҟ Nucleon amplitude: 64 ( , , ) = ( , , , , ) ( , , ) = 1 64 Dirac covariants 훹 푝 푞 푃 � 푓� 푝� 푞� 푝⋅푞 푝⋅푃 푞⋅푃 휏� 푝 푞 푃 푖 Dominant covariants ( quark model ) : ) → ) � ) ↑ ↓ ↓ ↑ ) ↑ 2 푆�� � 훬₊(훾�퐶 훬₊ ~ (푈 푈 � 푈 푈 푈 � ↑ ↓ ↓ ↑ ↑ ↑ ↑ ↓ 퐴�� � 훬₊(훾�퐶 훾�훾�훬₊ ~ (푈 푈 � 푈 푈 푈 � 푈 푈 푈 Gernot Eichmann (TU Darmstadt) MENU 2010 18.