Dynamical Mass Generation in Hadrons and Leptons

Transcription

1 Dynamical Mass Generation in Hadrons and Leptons Christian S. Fischer Justus Liebig Universität Gießen with Gernot Eichmann, Helios Sanchis-Alepuz, Richard Williams and Walter Heupel, Stanislav Kubrak Christian Fischer (University of Gießen) Dynamical mass generation 1 / 46

2 Overview 1.Introduction 1 = 1 2.Gluons, quarks and mesons + pion cloud 3.Pion cloud effects in baryons + pion cloud 4.Hadronic effects in (g-2) μ 5.Tetraquarks Christian Fischer (University of Gießen) Dynamical mass generation 2 / 46

3 Facility for Antiproton Ion Reactions: FAIR PANDA: Hadron physics CBM: QCD phase diagram } Nonperturbative QCD Universities of Darmstadt, Frankfurt, Giessen + FIAS Continued funding by State of Hesse approved on 06/20/2014 Christian Fischer (University of Gießen) Dynamical mass generation 3 / 46

4 Properties of QCD: Dynamical mass generation Dynamical quark masses via weak and strong force Yoichiro Nambu, Nobel prize 2008 [MeV/c 2 ] u d s c b t Mweak [MeV/c 2 ] Mstrong Mtotal [MeV/c 2 ] S 1 (p) =[ip/ + M(p 2 )]/Z f (p 2 ) Christian Fischer (University of Gießen) Dynamical mass generation 4 / 46

5 Properties of QCD: Dynamical mass generation Dynamical quark masses via weak and strong force Yoichiro Nambu, Nobel prize 2008 [MeV/c 2 ] u d s c b t Mweak [MeV/c 2 ] Input parameters in Nf=2+1 QCD Mstrong Mtotal [MeV/c 2 ] S 1 (p) =[ip/ + M(p 2 )]/Z f (p 2 ) Christian Fischer (University of Gießen) Dynamical mass generation 4 / 46

6 Properties of QCD: Confinement Color confinement: We are not detecting quarks and gluons, but baryons, mesons, tetraquarks, glueballs, hybrids... Strategies to deal with this situation: Effective theories in terms of hadrons Nonperturbative QCD: Lattice, Functional methods Christian Fischer (University of Gießen) Dynamical mass generation 5 / 46

7 QCD in covariant gauge Quarks and Gluons Z QCD = Z D[,A] exp Z d 4 x 1 (id/ m) 4 F µ a 2 o +gauge fixing Landau gauge propagators in momentum space, D Gluon µ (p) = µ p µ p p 2 Z(p 2 ) p 2 S Quark (p) = Z f (p 2 )[ ip/ + M(p 2 )] 1 The Goal: gauge invariant information in a gauge fixed approach. Christian Fischer (University of Gießen) Dynamical mass generation 6 / 46

8 Nonperturbative QCD: Complementary approach Quarks and gluons Hadrons Lattice simulations Ab initio Effective theories and models (χpt, chiral models,. ) Physical degrees of freedom Functional approaches (DSE, FRG, Hamilton): Chiral symmetry: physical quark masses Infinite volume and continuum limit Multi-scale problems feasible (e.g. (g-2) μ ) Chemical potential: no sign problem CF, Luecker, PLB 718 (2013) 1036 Christian Fischer (University of Gießen) Dynamical mass generation 7 / 46

9 Overview 1.Introduction 1 = 1 2.Gluons, quarks and mesons + pion cloud 3.Pion cloud effects in baryons + pion cloud 4.Hadronic effects in (g-2) μ 5.Tetraquarks Christian Fischer (University of Gießen) Dynamical mass generation 8 / 46

10 DSEs and Bethe-Salpeter equation Kernel K uniquely related to quark-dse via axwti π, ρ,... Pion is bound state and Goldstone boson Maris, Roberts, Tandy, PLB 420 (1998) 267 Two strategies: I. use rainbow-ladder model for quark-gluon interaction ok for some phenomenological applications II. calculate gluon and vertex from their DSEs Christian Fischer (University of Gießen) Dynamical mass generation 9 / 46

11 Strategie I: Model for quark-gluon interaction (k 2 )= 7 k 2 2 e 2 k UV (k 2 ) Maris, Tandy, 1999 two (related) parameters and from f π UV from perturbation theory masses mu=md from m π or m ρ Renormalizable and momentum dependent - in contrast to (cf. (g-2) μ ) Christian Fischer (University of Gießen) Dynamical mass generation 10 / 46

12 Strategie I: Model for quark-gluon interaction (k 2 )= 7 k 2 2 e 2 k UV (k 2 ) Maris, Tandy, 1999 two (related) parameters and from f π UV from perturbation theory masses mu=md from m π or m ρ Renormalizable and momentum dependent - in contrast to NJL-model (k 2 )= ( 2 k 2 ) (cf. (g-2) μ ) Christian Fischer (University of Gießen) Dynamical mass generation 10 / 46

13 Quark mass: flavor dependence [S(p)] 1 =[ ip/ + M(p 2 )]/Z f (p 2 ) Typical solution: Quark Mass Function: M(p 2 ) [GeV] Bottom quark Charm quark Strange quark Up/Down quark Chiral limit p 2 [GeV 2 ] CF, JPG 32 (2006) R253 M(p 2 ): momentum dependent! Dynamical mass: Mstrong 350 MeV Flavour dependence because of Mweak Chiral condensate: h i (250 MeV) 3 Christian Fischer (University of Gießen) Dynamical mass generation 11 / 46

14 Quark mass: flavor dependence [S(p)] 1 =[ ip/ + M(p 2 )]/Z f (p 2 ) Typical solution: Quark Mass Function: M(p 2 ) [GeV] Not directly measurable! Bottom quark Charm quark Strange quark Up/Down quark Chiral limit p 2 [GeV 2 ] CF, JPG 32 (2006) R253 M(p 2 ): momentum dependent! Dynamical mass: Mstrong 350 MeV Flavour dependence because of Mweak Chiral condensate: h i (250 MeV) 3 Christian Fischer (University of Gießen) Dynamical mass generation 11 / 46

15 Phenomenology from Maris-Tandy interaction Slide from Pieter Maris Christian Fischer (University of Gießen) Dynamical mass generation 12 / 46

16 Rainbow-ladder: spectrum good channels (ground state): 0 -+, 1 -- acceptable channels (ground state) : 2 ++, 3 --,... clear deficiencies in other channels deficiencies for excited states CF, Kubrak, Williams, arxiv: Christian Fischer (University of Gießen) Dynamical mass generation 13 / 46

17 Rainbow-ladder: Regge-trajectory 4,0 3,0 ρ 3 (1690) M 2 [GeV 2 ] 2,0 a 2 (1320) 1,0 ρ(770) isovector nn 0, J PC CF, Kubrak, Williams, arxiv: naive and intuitive explanation for Regge (spinning flux tube) is not the mechanism at work here... Christian Fischer (University of Gießen) Dynamical mass generation 14 / 46

18 Strategy II: Beyond rainbow ladder π, ρ,... treat Yang-Mills sector explicitly include gauge effects in vertex anomalous quark magnetic moment Recent improvements beyond rainbow-ladder: include gluon self-interaction effects include pion cloud effects CF, Watson, Cassing, PRD 72 (2005) Chang, Roberts, PRL 103 (2009) Heupel, Goecke, CF, EPJA 50 (2014), [ ] Chang, Liu, Roberts, PRL 106 (2011) CF, Williams, PRL 103 (2009) CF, Nickel, Wambach PRD 76 (2007) Christian Fischer (University of Gießen) Dynamical mass generation 15 / 46

19 Strategy II: Landau gauge gluon propagator 1 = Sternbeck et. al. (2006) Funktionalzugang Z(p 2 ) = 1 D µ (p) = µ p [GeV] Z(p 2 ) p µ p p 2 p 2 CF, Maas, Pawlowski, Annals Phys. 324 (2009) Huber and von Smekal, JHEP 1304 (2013) 149 spacelike momenta: excellent agreement with lattice spectral function: positivity violations spectral function Cornwall, Papavassiliou,... Gluon cannot appear in detector! Strauss, CF, Kellermann, Phys. Rev. Lett. 109, (2012) Christian Fischer (University of Gießen) Dynamical mass generation 16 / 46

20 Strategy II: Pion effects in quark-gluon interaction quark-gluon vertex: = quark: Christian Fischer (University of Gießen) Dynamical mass generation 17 / 46

21 Strategy II: Pion effects in quark-gluon interaction quark-gluon vertex: = + N c 2 2 N c + π +... quark: Christian Fischer (University of Gießen) Dynamical mass generation 17 / 46

22 Strategy II: Pion effects in quark-gluon interaction quark-gluon vertex: = + N c 2 2 N c + π π = YM quark: Christian Fischer (University of Gießen) Dynamical mass generation 17 / 46

23 Strategy II: Pion effects in quark-gluon interaction quark-gluon vertex: = + N c 2 2 N c + π π = YM Bethe-Salpeter equation: π = π + π π Christian Fischer (University of Gießen) Dynamical mass generation 17 / 46

24 Unquenching effects: Light mesons CF, Williams, PRL 103 (2009), PRD 78 (2008) Attractive effects of pion cloud Scalar too large or... too low! Ebert, Faustov, Galkin, PRD 79 (2009) Parganlija, Kovacs, Wolf, Giacosa and Rischke, PRD 87 (2013) tetraquarks!? see later Christian Fischer (University of Gießen) Dynamical mass generation 18 / 46

25 Unquenching effects: Light mesons CF, Williams, PRL 103 (2009), PRD 78 (2008) Attractive effects of pion cloud Scalar too large or... too low! Ebert, Faustov, Galkin, PRD 79 (2009) Parganlija, Kovacs, Wolf, Giacosa and Rischke, PRD 87 (2013) tetraquarks!? see later Christian Fischer (University of Gießen) Dynamical mass generation 18 / 46

26 Unquenching effects: Light mesons CF, Williams, PRL 103 (2009), PRD 78 (2008) Attractive effects of pion cloud Scalar too large or... too low! Ebert, Faustov, Galkin, PRD 79 (2009) Parganlija, Kovacs, Wolf, Giacosa and Rischke, PRD 87 (2013) tetraquarks!? see later Christian Fischer (University of Gießen) Dynamical mass generation 18 / 46

27 Overview 1.Introduction 1 = 1 2.Gluons, quarks and mesons + pion cloud 3.Pion cloud effects in baryons + pion cloud 4.Hadronic effects in (g-2) μ 5.Tetraquarks Christian Fischer (University of Gießen) Dynamical mass generation 19 / 46

28 Faddeev - equation = neglect irreducible three-body forces (three-gluon interaction!) approximate two-body interactions by RL-gluon exchange one-parameter-model (MT) 64 tensor structures for nucleon: s, p, d - wave numerically expensive but manageable! Eichmann, Alkofer, Krassnigg, Nicmorus, PRL 104 (2010) Eichmann, PRD 84 (2011) Sanchis-Alepuz,Eichmann, Villalba-Chavez, Alkofer, PRD (2012) Christian Fischer (University of Gießen) Dynamical mass generation 20 / 46

29 Faddeev - equation = neglect irreducible three-body forces (three-gluon interaction!) approximate two-body interactions by RL-gluon exchange one-parameter-model (MT) 64 tensor structures for nucleon: s, p, d - wave numerically expensive but manageable! Eichmann, Alkofer, Krassnigg, Nicmorus, PRL 104 (2010) Eichmann, PRD 84 (2011) Sanchis-Alepuz,Eichmann, Villalba-Chavez, Alkofer, PRD (2012) Christian Fischer (University of Gießen) Dynamical mass generation 20 / 46

30 Baryon masses first covariant threebody calculations! grosso modo: consistent description of mesons and baryons masses dominated by s-waves Eichmann, Alkofer, Krassnigg, Nicmorus, PRL 104 (2010) Eichmann, PRD 84 (2011) Sanchis-Alepuz,Eichmann, Villalba-Chavez, Alkofer, PRD (2012) Christian Fischer (University of Gießen) Dynamical mass generation 21 / 46

31 Nucleon EM form factors ( ) Crawford Ron Zhan Paolone Punjabi ( ) Geis Passchier, Herberg, Zhu, Bermuth, Warren Glazier Plaster/Madey Riordan / / 2 ( ) 1 ( ) Anderson Kubon Anklin Lachniet Lung Rock / / missing pion cloud effects Eichmann, PRD 84 (2011) similar for axial form factors Eichmann and CF, Eur. Phys. J. A48 (2012) 9 Christian Fischer (University of Gießen) Dynamical mass generation 22 / 46

32 Magnetic moments missing pion cloud effects in isovector moment κv no pion cloud effects in isoscalar moment κs Eichmann, PRD 84 (2011) Christian Fischer (University of Gießen) Dynamical mass generation 23 / 46

33 Pion cloud effects in baryons M (GeV) Sanchis-Alepuz, CF, Kubrak, PLB 733 (2014) [ ] 2 fix Λ by fπ, vary η s.t. fπ still ok k (k 2 ) = 7 e 2 effects of the order of MeV missing: gluon self-interaction effects work in progress: mass splitting of Λ and Σ Christian Fischer (University of Gießen) Dynamical mass generation 2 k2 2 + UV (k 2 ) 24 / 46

34 Pion cloud effects in baryons 1 1 = π YM M (GeV) Sanchis-Alepuz, CF, Kubrak, PLB 733 (2014) [ ] 2 fix Λ by fπ, vary η s.t. fπ still ok k (k 2 ) = 7 e 2 effects of the order of MeV missing: gluon self-interaction effects work in progress: mass splitting of Λ and Σ Christian Fischer (University of Gießen) Dynamical mass generation 2 k2 2 + UV (k 2 ) 24 / 46

35 Pion cloud effects in baryons: structure Nucleon Delta s-wave p-wave d-wave s-wave p-wave d-wave f-wave pure rainbow ladder quark core , ,2 quark core plus pion cloud ,2 pion cloud does not change shape of nucleon: uniform skin pion cloud does affect sigma-term but still too small... Christian Fischer (University of Gießen) Dynamical mass generation 25 / 46

36 Nucleon Compton scattering Nonperturbative description of hadron-photon and hadron-meson scattering on quark-gluon basis Eichmann and CF, PRD 85 (2012) Technical/conceptual progress: Derive fermion-two-photon vertex consistent with gauge invariance free of kinematic singularities transverse part: on-shell nucleon Compton amplitude Reproduce πυυ transition form factor on t-channel pole Eichmann and CF, PRD 87 (2013) Next steps: Two-photon contributions to EM form factor Polarisabilities PANDA: p p! Christian Fischer (University of Gießen) Dynamical mass generation 26 / 46

37 Overview 1.Introduction 1 = 1 2.Gluons, quarks and mesons + pion cloud 3.Pion cloud effects in baryons + pion cloud 4.Hadronic effects in (g-2) μ 5.Tetraquarks Christian Fischer (University of Gießen) Dynamical mass generation 27 / 46

38 Physics beyond the standard model (SM)? Anomalous magnetic moment of muon µ = g l e~ 2mc S a l = g l 2 2 Experiment: Theorie (SM): a µ = ( ± 6.3) a µ = ( ± 4.9) Bennet et. al., Phys.Rev.D 73, (2006) Hagiwara et.al., J.Phys.G G (2011) Discrepancy of three standard deviations... Christian Fischer (University of Gießen) Dynamical mass generation 28 / 46

39 Hadronic contributions to a μ Hadronic vacuum polarisation: Extracted from e + e - and τ-decay: a HV P(LO) µ = (692.1 ± 5.6) Light-by-light scattering: Hagiwara et.al., J.Phys.G G (2011) Model calculation: a LBL µ = (10.5 ± 2.6) J. Prades, E. de Rafael and A. Vainshtein, arxiv: dominate theory error! a µ = ( ± 4.9) Christian Fischer (University of Gießen) Dynamical mass generation 29 / 46

40 Hadronic vacuum polarization Need to determine quark propagator and quark-photon vertex: Maris-Tandy model Kernel K uniquely related to quark-dse via axwti Bethe-Salpeter equation: meson mass and wave function π, ρ,... Christian Fischer (University of Gießen) Dynamical mass generation 30 / 46

41 Quark-photon vertex and pion form factors Pion form factor: Q 2 π π Krassnigg, Schladming 2011; Maris, Tandy NPPS 161, 2006 Vector meson poles dynamically generated! Christian Fischer (University of Gießen) Dynamical mass generation 31 / 46

42 Results: Hadronic vacuum polarisation a µ [10-10 ] RBC-UKQCD (N f =2+1) Mainz CLS (N f =2+1) Mainz CLS (N f =2) ETMC (N f =2) DSE (N f =2+1) DSE (N f =2) u/d s c b a µ M ρ [GeV] Experiment: DSE: a had.(1) (m π ) µ = (744.0 ± 2) Goecke, CF, Williams, PLB 704 (2011) a had.(1) µ = (676.0 ± 2) a had.(1) µ = (692.1 ± 5.6) Very reasonable agreement! (m ρ ) Christian Fischer (University of Gießen) Dynamical mass generation 32 / 46

43 Hadronic contributions to a μ : LBL Multi-scale problem NJL-model, Vector-meson dominance, largenc Lattice gauge theory: exploratory... Dispersion theory: just started... Bijnens, Knecht, Melnikov, Nyffeler, Pallante, Prades, de Rafael, Vainshtein... Blum, Izubuchi,... Colangelo, Vanderhaeghen,... Theorie (LBL): a LBL µ = ( 10.5 ± 2.6) Theorie (SM): a µ = ( ± 4.9) Experiment: a µ = ( ± 6.3) Challenge for non-perturbative QCD-calculation! Christian Fischer (University of Gießen) Dynamical mass generation 33 / 46

44 Hadronic Light-by-Light scattering q = μ μ = Dictated by gauge invariance! Goecke, CF, Williams, PRD 83 (2011) Eichmann and CF, PRD 87 (2013) 3, = Christian Fischer (University of Gießen) Dynamical mass generation 34 / 46

45 Hadronic Light-by-Light scattering q = +... μ μ = Dictated by gauge invariance!... Goecke, CF, Williams, PRD 83 (2011) Eichmann and CF, PRD 87 (2013) 3, = Christian Fischer (University of Gießen) Dynamical mass generation 34 / 46

46 Hadronic Light-by-Light scattering q = +... μ μ No Double-Counting! = Dictated by gauge invariance!... Goecke, CF, Williams, PRD 83 (2011) = Eichmann and CF, PRD 87 (2013) 3, Christian Fischer (University of Gießen) Dynamical mass generation 34 / 46

47 Hadronic Light-by-Light scattering Quark-loop q = μ μ Off-shell meson-exchange = +... Dictated by gauge invariance! Goecke, CF, Williams, PRD 83 (2011) Eichmann and CF, PRD 87 (2013) 3, = +/ +... Christian Fischer (University of Gießen) Dynamical mass generation 34 / 46

48 Meson-exchange contribution Christian Fischer (University of Gießen) Dynamical mass generation 35 / 46

49 Meson-exchange contribution 2 Components 4 Components 12 Components full momentum dependencies no free parameters Christian Fischer (University of Gießen) Dynamical mass generation 35 / 46

50 Meson exchange contribution to LBL F F (,, 0 ),, 0 F πγγ overall similar to VMD-models our value: comparable with model calculations short distance constraints satisfied! (a,, 0 µ ) DSE =8.1(1.3) see also Dorokhov, Broniowski, PRD 78 (2008) numerical: 0.2 systematic: 1.1 Goecke, CF, Williams, PRD 83 (2011) Christian Fischer (University of Gießen) Dynamical mass generation 36 / 46

51 Quark-loop contribution to LBL q Numerically demanding due to superficial divergency Vertex contains 12 tensor structures µ X = BC i L µ X i + F i T µ i i=1,4 i=1,8 gauge part WTI transverse part vector-mesons ENJL-model: (2.1 ± 0.3) Bijnens, Pallante and Prades PRL 75, (1995) DSE: bare vertex (6.1 ± 0.2) BC1 only (11.1 ± 0.2) BC1 + F1 (10.7 ± 0.3) Calculation not yet completed! CF, Goecke, Williams, EPJA 47 (2011) 28 Goecke, CF, Williams, PRD 87 (2013) Christian Fischer (University of Gießen) Dynamical mass generation 37 / 46

52 Hadronic LBL contributions - DSE + Models Group Tool π quark-loop π SUM BPP ENJL 8.5(1.3) 2.1(0.3) -1.9(1.3) 8.3(3.2) HKS HLS 8.3(0.6) 1.0(1.1) -0.4(0.8) 8.9(1.6) PdRV 11.4(1.3) 0,2-1.9(1.3) 10.5(2.6) GdR QM 6.8(0.3) 8.2 (0.7) 12.2 (1.8) ( GFW DSE 8.1(0.2) 10.7(0.3) 18.8(0.5) BPP J. Bijnens, E. Pallante and J. Prades Phys. Rev. Lett. 75, 1447 (1995) HKS M. Hayakawa, T. Kinoshita and A. I. Sanda, Phys. Rev. Lett. 75, 790 (1995) PdRV J. Prades, E. de Rafael and A. Vainshtein, arxiv: GdR D. Greynat, E. de Rafael, JHEP 1207 (2012) 020 GFW T. Goecke, CF, R. Williams, Phys. Rev. D 87 (2013) preliminary! numerical! systematic? Christian Fischer (University of Gießen) Dynamical mass generation 38 / 46

53 Prospects for the future Goals: Reliable theoretical error Reduce theoretical error down to 1-2 x level Tools: Lattice gauge theory: increasing activities (cf. Lattice2014) Dispersive approach: formalism outlined, experimental input needed DSE/BSE: aim at LBL on 10-20% level Christian Fischer (University of Gießen) Dynamical mass generation 39 / 46

54 The four-photon amplitude Orthogonal basis: 136 elements (41 four-photon amplitude satisfies (exac Quark-Compton-vertex = + Christian Fischer (University of Gießen) Dynamical mass generation 40 / 46

55 Quark-Compton-vertex = + Compton-vertex: = + + basis constructed (128 elements) Quark-Loop meson exchange incl. meson-exchange contributions isolat 1PI F Next steps: calc. full quark-loop contribution calc. full 1PI contribution and systematically compare with off-shell meson approximation Eichmann and CF, PRD 87 (2013) 3, Eichmann, CF, Williams, work in progress Christian Fischer (University of Gießen) Dynamical mass generation 41 / 46

56 Overview 1.Introduction 1 = 1 2.Gluons, quarks and mesons + pion cloud 3.Pion cloud effects in baryons + pion cloud 4.Hadronic effects in (g-2) μ 5.Tetraquarks Christian Fischer (University of Gießen) Dynamical mass generation 42 / 46

57 Tetraquarks from DSEs/BSEs Hadrons Quark configurations: Diquark-Antidiquark Meson-Meson Christian Fischer (University of Gießen) Dynamical mass generation 43 / 46

58 Tetraquarks from DSEs/BSEs Hadrons Quark configurations: Diquark-Antidiquark Meson-Meson Light meson sector: Scalars! Christian Fischer (University of Gießen) Dynamical mass generation 43 / 46

59 Tetraquarks from DSEs/BSEs Hadrons Quark configurations: Diquark-Antidiquark Meson-Meson Light meson sector: Scalars! Christian Fischer (University of Gießen) Dynamical mass generation 43 / 46

60 Strategie II: Tetraquark-BSEs Meson Diquark Input: Covariant Quark-Gluon interaction - Maris-Tandy model 1 = 1 k (k 2 )= 7 2 Mesons and Diquarks via Bethe-Salpeter equation Dynamical decision between Meson- and Diquark-configurations Christian Fischer (University of Gießen) Dynamical mass generation 44 / 46 2 e 2 k UV (k 2 )

61 Results: scalar tetraquarks Heupel, Eichman, CF, PLB 718 (2012) up/down charm Pion-Pion-contribution dominates! m(0 ++ ) = 403 MeV } f 0(500) Narrow scalar cccc: m(0 ++ ) = 5.3 ± (0.5) GeV see also Caprini, Colangelo and Leutwyler, PRL. 96 (2006) Parganlija, Kovacs, Wolf, Giacosa and Rischke, PRD 87 (2013) Christian Fischer (University of Gießen) Dynamical mass generation 45 / 46

62 Summary and outlook Light and heavy mesons Explore quark-gluon interaction: consequences for pheno current work: masses and EM-properties of charmonia Baryons form factors in rainbow-ladder masses including pion cloud effects current work: multipletts, excited states and Compton scattering Hadronic contributions to (g-2) μ HVP: in agreement with lattice and experiment current work: determine LBL beyond simple models Tetraquarks σ as tetraquark in π-π-configuration current work: include other channels; four body equation Christian Fischer (University of Gießen) Dynamical mass generation 46 / 46

63 Backup Slides Christian Fischer (University of Gießen) Dynamical mass generation 47 / 46

64 Delta and Omega form factors technically demanding natural scale: M Δ compare two RL models no pion cloud yet... Results: Oblate shape of Δ and Ω agreement with lattice Sanchis-Alepuz, Williams and Alkofer, PRD 87 (2013) Lattice: Alexandrou et al. NPA 825 (2009) 115, PoS CD 09 (2009) 092 Christian Fischer (University of Gießen) Dynamical mass generation 48 / 46

65 Form factor ratios Eichmann, PRD 84 (2011) Christian Fischer (University of Gießen) Dynamical mass generation 49 / 46

66 Axial and pseudoscalar form factors Lattice: [ ] RBC/UKQCD (Yamazaki 09) LHPC (Bratt 10) ETMC (Alexandrou 11) QCDSF (Ali Khan 06) 1.0 DSE/Faddeev ChPT (Procura 07) [ ] [ ] ( ) Lattice: RBC/UKQCD (Yamazaki 09) = 329 MeV = 416 MeV ETMC (Alexandrou 11) = 298 MeV / ( ) DSE/Faddeev Phenomenology Lattice: Alexandrou 07 ( = 411 MeV) / Eichmann, CF, EPJA 48 (2012) 9 GT-relation satisfied missing pion cloud ma = 1.28 (6) GeV sizeable deviations from dipole form Christian Fischer (University of Gießen) Dynamical mass generation 50 / 46

67 Comparison ENJL vs. DSE NJL: no momentum dependence in quark NJL: no relative momenta in quark-photon vertex artificial suppression of quark-loop contribution µ(p, p, p.p ) ENJL-model: DSE (BC1 + F1): Nf = 2 Nf = Goecke, CF, Williams, Phys. Rev. D 87 (2013) Christian Fischer (University of Gießen) Dynamical mass generation 51 / 46