Hadronic light-by-light from Dyson-Schwinger equations

Transcription

1 Hadronic light-by-light from Dyson-Schwinger equations Christian S. Fischer Justus Liebig Universität Gießen 23rd of October 2014 Together with Richard Williams, Gernot Eichmann, Tobias Goecke, Jan Haas Christian Fischer (University of Gießen) Hadronic light-by-light from DSEs 1 / 23

2 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 2 / 46

3 Experiments BNL,FNAL JPARC Jegerlehner, Nyffeler, Phys. Rept. 477 (2009) ~! a = ~! c ~! s = a µ e ~ B m µ e m µ Talk of T. Mibe, Mainz 2014 apple a µ ~v ~ E BNL,FNAL: magic γ JPARC: ultra-slow muons - no E-field needed Different systematic error! Christian Fischer (University of Gießen) Hadronic light-by-light from DSEs 3 / 23

4 Theory: 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 4 / 46

5 Overview 1.Hadronic vacuum polarisation 2.Hadronic light by light 3.Outlook: towards the photon four-point function Christian Fischer (University of Gießen) Hadronic light-by-light from DSEs 5 / 23

6 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) Hadronic light-by-light from DSEs 6 / 23

7 Nonperturbative QCD: Complementary approach Quarks and gluons Lattice simulations Ab initio Gauge invariant Dyson-Schwinger Equations Physical quark masses Full momentum dependencies Multi-scale problems feasible Hadrons Effective theories and models (χpt, chiral models,...) Dispersive approach Physical degrees of freedom Christian Fischer (University of Gießen) Hadronic light-by-light from DSEs 7 / 23

8 Hadronic vacuum polarization Evaluated at spacelike momenta! Christian Fischer (University of Gießen) Hadronic light-by-light from DSEs 8 / 23

9 Hadronic vacuum polarization Evaluated at spacelike momenta! Need to determine quark propagator and quark-photon vertex: K Christian Fischer (University of Gießen) Hadronic light-by-light from DSEs 8 / 23

10 Hadronic vacuum polarization Evaluated at spacelike momenta! Need to determine quark propagator and quark-photon vertex: K Kernel K uniquely related to quark-dse via axwti Christian Fischer (University of Gießen) Hadronic light-by-light from DSEs 8 / 23

11 Hadronic vacuum polarization Evaluated at spacelike momenta! Need to determine quark propagator and quark-photon vertex: Maris-Tandy model K Two strategies: I. Calc. gluon and vertex from their DSEs II. Combine gluon and vertex in model Christian Fischer (University of Gießen) Hadronic light-by-light from DSEs 8 / 23

12 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! Christian Fischer (University of Gießen) Hadronic light-by-light from DSEs 9 / 23

13 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! NJL-model (k 2 )= ( 2 k 2 ) Christian Fischer (University of Gießen) Hadronic light-by-light from DSEs 9 / 23

14 Quark mass: momentum and flavor dependence Typical solution: Quark Mass Function: M(p 2 ) [GeV] Bottom quark Charm quark Strange quark Up/Down quark Chiral limit p 2 [GeV 2 ] 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) Hadronic light-by-light from DSEs 10 / 23

15 Phenomenology from Maris-Tandy interaction Slide from Pieter Maris Christian Fischer (University of Gießen) Hadronic light-by-light from DSEs 11 / 23

16 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) Hadronic light-by-light from DSEs 12 / 23

17 Transition form factors γ*π γ ωπγ* and ρπγ* good agreement with data rho/omega pole generated dynamically Maris, Tandy, Phys. Rev. C (2002) Christian Fischer (University of Gießen) Hadronic light-by-light from DSEs 13 / 23

18 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) Hadronic light-by-light from DSEs 14 / 23

19 Overview 1.Hadronic vacuum polarisation 2.Hadronic light by light 3.Outlook: towards the photon four-point function Christian Fischer (University of Gießen) Hadronic light-by-light from DSEs 15 / 23

20 Hadronic contributions to a μ : LBL NOT simply related to experiment Non-perturbative QCD-calculation needed Multi-scale problem NJL-model, Vector-meson dominance, largenc Bijnens, Knecht, Melnikov, Nyffeler, Pallante, Prades, de Rafael, Vainshtein... 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) Hadronic light-by-light from DSEs 16 / 23

21 Hadronic Light-by-Light scattering q = μ μ = Derived from gauge invariance! Goecke, CF, Williams, PRD 83 (2011) Eichmann and CF, PRD 87 (2013) 3, = Christian Fischer (University of Gießen) Hadronic light-by-light from DSEs 17 / 23

22 Hadronic Light-by-Light scattering q = +... μ μ = Derived from gauge invariance!... Goecke, CF, Williams, PRD 83 (2011) Eichmann and CF, PRD 87 (2013) 3, = Christian Fischer (University of Gießen) Hadronic light-by-light from DSEs 17 / 23

23 Hadronic Light-by-Light scattering q = +... μ μ No Double-Counting! = Derived from gauge invariance!... Goecke, CF, Williams, PRD 83 (2011) = Eichmann and CF, PRD 87 (2013) 3, Christian Fischer (University of Gießen) Hadronic light-by-light from DSEs 17 / 23

24 Hadronic Light-by-Light scattering Quark-loop q = μ μ Off-shell meson-exchange = +... Derived from gauge invariance! Goecke, CF, Williams, PRD 83 (2011) Eichmann and CF, PRD 87 (2013) 3, = +/ +... Christian Fischer (University of Gießen) Hadronic light-by-light from DSEs 17 / 23

25 Meson-exchange contribution Christian Fischer (University of Gießen) Hadronic light-by-light from DSEs 18 / 23

26 Meson-exchange contribution 2 Components 4 Components 12 Components full momentum dependencies no free parameters Christian Fischer (University of Gießen) Hadronic light-by-light from DSEs 18 / 23

27 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) Hadronic light-by-light from DSEs 19 / 23

28 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) Hadronic light-by-light from DSEs 20 / 23

29 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) Hadronic light-by-light from DSEs 21 / 23

30 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) 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: preliminary! numerical! systematic? GFW T. Goecke, CF, R. Williams, Phys. Rev. D 87 (2013) Christian Fischer (University of Gießen) Hadronic light-by-light from DSEs 22 / 23

31 Overview 1.Hadronic vacuum polarisation 2.Hadronic light by light 3.Outlook: towards the photon four-point function Christian Fischer (University of Gießen) Hadronic light-by-light from DSEs 23 / 23

32 The four-photon amplitude Orthogonal basis constructed: 136 elements (41 after gauge inv.) four-photon amplitude satisfies (exact within RL): Quark-Compton-vertex = + Christian Fischer (University of Gießen) Hadronic light-by-light from DSEs 24 / 23

33 Quark-Compton-vertex = + Compton-vertex: = + + 1PI Quark-Loop incl. meson-exchange basis constructed (128 elements) meson exchange contributions isolated and F recovered 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) Hadronic light-by-light from DSEs 25 / 23

34 Reducing systematic errors in LBL Step by step process: 1. Complete rainbow-ladder calc. of four-photon amplitude Avoid systematic errors of off-shell meson exchange approx. From HVP, form factors, etc.: expect error of <10 % Can be checked systematically by comparison with lattice at selected kinematical points II. Include contributions beyond rainbow-ladder Pion-loop effects Eichmann, CF.,Williams, Haas, work in progress Christian Fischer (University of Gießen) Hadronic light-by-light from DSEs 26 / 23

35 Physics beyond the standard model? Anomalous magnetic moment of myon Experiment: a µ = ( ± 6.3) Bennet et. al., Phys.Rev.D 73, (2006) Our value Theory: (preliminary): a µ = ( ± 4.9) a µ = ( ± 10) Hagiwara et.al., J. Phys. G (2011) no more than a guess... Christian Fischer (University of Gießen) Hadronic light-by-light from DSEs 27 / 23

36 Summary (Euclidean) QCD based approach renormalizable UV IR: dynamical quark mass generation Hadronic vaccuum polarisation agreement with experiment on 5 % level Light-by-light large effects due to momentum dependencies full calculation w.o. off-shell meson approximation under way Highly desirable: Systematic comparison with lattice results at physical and unphysical pion and/or myon masses! Christian Fischer (University of Gießen) Hadronic light-by-light from DSEs 28 / 23