HLbl from a Dyson Schwinger Approach

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1 HLbl from a Dyson Schwinger Approach Richard Williams KFUni Graz Tobias Göcke TU Darmstadt Christian Fischer Uni Gießen Dissertantenseminar, March 16 th 2011, Graz 1 / 45

2 Introduction Table of contents 1 Introduction Generalities Hadronic Vacuum Polarisation Photon Four-Point Function 2 Truncation 3 Results HVP Adler function hlbl: pion pole hlbl: quark loop 4 Summary and Outlook 2 / 45

3 Introduction Table of contents 1 Introduction Generalities Hadronic Vacuum Polarisation Photon Four-Point Function 2 Truncation 3 Results HVP Adler function hlbl: pion pole hlbl: quark loop 4 Summary and Outlook 3 / 45

4 Introduction Generalities Introduction Quantum Chromodynamics is a beast DOF are the quarks and gluons non-abelian gauge theory 4 / 45

5 Introduction Generalities Introduction Quantum Chromodynamics is a beast DOF are the quarks and gluons non-abelian gauge theory Features various phenomena Asymptotic freedom Confinement Dynamical Chiral Symmetry Breaking 4 / 45

6 Introduction Generalities Introduction Quantum Chromodynamics is a beast DOF are the quarks and gluons non-abelian gauge theory Features various phenomena Asymptotic freedom Confinement Dynamical Chiral Symmetry Breaking Interested in (via Dyson Schwinger equations) Infrared and hadronic properties Hadronic contributions to physical processes 4 / 45

7 Introduction Generalities The humble electron Sensitive (primarily) to QED 5 / 45

8 Introduction Generalities The humble electron Sensitive (primarily) to QED 5 / 45

9 Introduction Generalities The humble (but fat) muon Sensitive to more than QED 6 / 45

10 Introduction Generalities The humble (but fat) muon Sensitive to more than QED 6 / 45

11 Introduction Generalities The humble (but fat) muon Sensitive to more than QED 6 / 45

12 Introduction Generalities The humble (but fat) muon Sensitive to more than QED 6 / 45

13 Introduction Generalities Introduction: g-2 Leptons: electric/magnetic dipole moments analogy: orbiting particle, mass m and charge e. µ L = e 2mc L ; L = m r v generalise to a quantum mechanical system: µ m = gqµ 0 σ 2 charge Q, Pauli-matrix σ, µ 0 = e /2mc. gyromagnetic ratio g. 7 / 45

14 Introduction Generalities Introduction: g-2 Gyromagnetic ratio Dirac equation = g = 2 (tree-level in QFT) Radiative Corrections g = 2(1 + a l ) a l is the anomalous contribution to the magnetic moment Definition ieu(p ) [F 1 (q 2 )γ α + if 2 (q 2 ) σ αβq β 2m l ] u(p) Anomalous magnetic moment obtained from second form-factor F 2 a l = F 2 (0) = g / 45

15 Introduction Generalities Introduction: g-2 Theoretical Calculation Calculable/Measurable from first-principles (g 2 vanishes at tree-level hence not an input parameter) QED as an Abelian gauge theory + perturbation theory One-loop vertex correction Universal for all leptons (99% of a l ) a = α 2π One of the most precise theoretical predictions Muon heavier: sensitive to QCD, W ±, Z 0, BSM. 9 / 45

Introduction Generalities Introduction: E821 experiment at BNL ω a = e m Directly measures g 2 via the difference between the cyclotron and spin precession frequencies for

16 Introduction Generalities Introduction: E821 experiment at BNL ω a = e m Directly measures g 2 via the difference between the cyclotron and spin precession frequencies for muons in a magnetic storage ring. ( [a µ B a µ 1 ) ] B γ 2 E 1 B E from electric quadrupoles (provide vert. focusing) Vanishes for magic momentum GeV/c when γ = / 45

17 Introduction Generalities Introduction: E821 experiment at BNL Million Events per 149.2ns Directly measures g 2 via the difference between the cyclotron and spin precession frequencies for muons in a magnetic storage ring Time modulo [µs] 100µs ω a = e m [a µ B ( a µ 1 γ 2 1 ) ] B E B E from electric quadrupoles (provide vert. focusing) Vanishes for magic momentum GeV/c when γ = µ decay to e. Extract net spin precession at time t via N(t) exp ( t/γτ µ ) [1 + A cos (ω a t + ϕ)] 10 / 45

18 Introduction Generalities Introduction: E821 experiment at BNL Anomalous mangnetic moment of the muon Experiment: (63.0) 10 11, 11 / 45

19 Introduction Generalities Introduction: E821 experiment at BNL Anomalous mangnetic moment of the muon Experiment: (63.0) 10 11, Theory: (64.6) Comparible errors discrepancy at 3.2 σ confidence level Possible reasons? Unknown systematics in experiment? Physics beyond standard model? Misunderstood errors in theoretical calculation. Errors dominated by hadronic (QCD) effects! 11 / 45

20 Introduction Generalities The muon g 2 Contributions HVP hlbl [ (52.6) 100.3(1.1)] Data: Optical theorem + Dispersion relations + e + e -annihilation Must calculate entirely from theory [116(39)] / 45

21 Introduction Generalities The muon g 2 Photon vacuum polarisation tensors How to proceed? Ideally solve using ONE approach one-scale problem vs. two-scale problem. existing models large N c plus chiral counting / effective description of QCD / scale matching. 13 / 45

22 Introduction Hadronic Vacuum Polarisation Little brother of Π µναβ : Π µν Two-photon vacuum polarisation tensor Extended NJL Dyson-Schwinger or Quarks are different non-renormalisable constituent-like renormalisable momentum dependent 14 / 45

23 Introduction Hadronic Vacuum Polarisation Little brother of Π µναβ : Π µν Two-photon vacuum polarisation tensor Extended NJL Dyson-Schwinger or Rho-poles are the result of ENJL: Bubble sum of constituent-like quarks DSE: QCD corrections to quark-photon vertex (QPV) NOTE QPV satisfies Ward-Takahashi identity. Vector meson contained in structures transverse to photon momentum 14 / 45

24 Introduction Hadronic Vacuum Polarisation Diagrammatic content of Π µν anticipate future truncation large N c rainbow-ladder only planar diagrams no gluon-self interactions Restrict topologies of resummed diagrams. rainbow + + ladder + + neglect/model, 15 / 45

25 Introduction Hadronic Vacuum Polarisation Diagrammatic content of Π µν anticipate future truncation large N c rainbow-ladder only planar diagrams no gluon-self interactions Restrict topologies of resummed diagrams. achieved by appropriate restrictions on the Green s functions: and 15 / 45

26 Introduction Hadronic Vacuum Polarisation Ingredients I Quark Propagator ENJL cf. DSE Parameterisation and Solution S 1 F (p; µ) = ip/ A(p2 ; µ 2 ) + 1 B(p 2 ; µ 2 ) Z f = 1/A, M = B/A: B(p 2 ), A(p 2 ) scalar, vector dressings momentum p, renormalisation point µ DSE Specified by model/truncation: quark-gluon vertex gluon propagator 16 / 45

27 Introduction Hadronic Vacuum Polarisation Ingredients I Quark Propagator ENJL cf. DSE M(p 2 ) [GeV] Z f (p 2 ) s u/d chiral p 2 [GeV 2 ] M(p 2 ) [GeV] Z f (p 2 ) s u/d chiral p 2 [GeV 2 ] 16 / 45

28 Introduction Hadronic Vacuum Polarisation Ingredients II Quark-photon vertex (ladder-truncation) basis decomposition: (1, k/, P/, [k/, P/ ]) (γ µ, k µ, P µ ) 12 Γ µ (P, k) = V µ (i) λ (i) (P, k) = Γ L µ + Γ T µ i=1 covariant tensor V µ (i), scalar dressing function λ (i) (P, k) total momentum P, relative momentum k Γ T µ transverse to P, Γ L µ is non-transverse 17 / 45

29 Introduction Hadronic Vacuum Polarisation Ingredients II Quark-photon vertex (ladder-truncation) solution Ward-Takahashi identity constrains Γ L µ in terms of S 1 F Γ L µ = γ µ Σ A + 2k/ k µ A + i2k µ B Σ A, A functions of A, B function of B Γ T µ solved numerically. Contains dynamical ρ-pole 17 / 45

30 Introduction Hadronic Vacuum Polarisation the story thus far... photon two-point function We have (large N c inspired) truncation of Π µν non-perturbative quark propagator / quark-photon vertex Can determine a µ from the HVP: No scale matching Do not distinguish short/long distances Separate contributions by topology Next: photon four-point function For consistency apply same methods of truncation 18 / 45

31 Introduction Photon Four-Point Function Big brother of Π µν : Π µναβ general diagrammatic content procedure and truncation Order diagrams according to large-n c counting. Neglect non-planar diagrams/gluon self-interactions. Resummation using Dyson-Schwinger equations. classify according to the topology of resummed diagram no separation into short-distance/long-distance! 19 / 45

32 Introduction Photon Four-Point Function Big brother of Π µν : Π µναβ general diagrammatic content (collected by topology)... q procedure and truncation (NB: quark propagators are fully dressed!) Order diagrams according to large-n c counting. Neglect non-planar diagrams/gluon self-interactions. Resummation using Dyson-Schwinger equations. classify according to the topology of resummed diagram no separation into short-distance/long-distance! 19 / 45

33 Introduction Photon Four-Point Function Big brother of Π µν : Π µναβ general diagrammatic content (resummed) q (NB: quark propagators are fully dressed!) necessary ingredients quark propagator quark-photon vertex T-matrix 19 / 45

34 Introduction Photon Four-Point Function T-matrix ladder truncation = + ( P 2 M 2 ) infinite ladder summation of non-perturbative gluons dynamically generates all qq bound-state poles encodes all on- and off-shell information unambiguously T-matrix: hard to calculate in practice (work-in-progress) 20 / 45

35 Introduction Photon Four-Point Function Truncation scheme resonance expansion of T-matrix q + + +/ obtain picture similar to existing approaches. Eduardo de Rafael. Hadronic contributions to the muon g-2 and low-energy QCD. Phys.Lett., B322: , G. Ecker, J. Gasser, H. Leutwyler, A. Pich, and E. de Rafael. Chiral Lagrangians for Massive Spin 1 Fields. Phys.Lett., B223:425, G. Ecker, J. Gasser, A. Pich, and E. de Rafael. The Role of Resonances in Chiral Perturbation Theory. Nucl.Phys., B321:311, / 45

36 Introduction Photon Four-Point Function T-matrix: bound-state poles homogeneous Bethe-Salpeter amplitude postulate existence of particle pole P 2 M 2 obtain equation for amplitude on-shell = 22 / 45

37 Introduction Photon Four-Point Function Pion-pole approximation assume pole dominance form factor analogous to existing approaches Λ πγ γ µν = caveat: pion here is defined on-shell. 23 / 45

38 rough outline Introduction Photon Four-Point Function Off-shell prescription start with axial-vector Ward-Takahashi identity in χ-limit 2P µ Γ a=3 µ5 (k, P) = is 1 (k + )γ 5 + iγ 5 S 1 (k ) Relates S and Γ µ5. Note Γ a=3 µ5 (k, P) P µf π Γ π (k, P) P 2 + M 2 + reg. contains BS amplitude as pseudoscalar pole, with ] Γ π (k; P) = γ 5 [E + S 1 (k) = ip/ A(k) + B(k) 24 / 45

39 rough outline Introduction Photon Four-Point Function Off-shell prescription start with axial-vector Ward-Takahashi identity in χ-limit 2P µ Γ a=3 µ5 (k, P) = is 1 (k + )γ 5 + iγ 5 S 1 (k ) Relates S and Γ µ5. Note Γ a=3 µ5 (k, P) P µf π Γ π (k, P) P 2 + M 2 + reg. contains BS amplitude as pseudoscalar pole, with ] Γ π (k; P) = γ 5 [E + S 1 (k) = ip/ A(k) + B(k) on-shell off-shell E(k, k P) = B(k 2 )/f π, E(k, P) = (B(k+) 2 + B(k ))/2f 2 π E(k, P) = (E(k +, k + P) + E(k, k P)) /2 24 / 45

40 Introduction Photon Four-Point Function Pseudoscalar form-factor form-factor (use full quark-photon vertex) comments similar behaviour to VMD and thus comparable results approximation: pole dominance + off-shell pion BSA 25 / 45

41 Introduction Photon Four-Point Function Quark-loop (a) (b) (c) Projection onto a µ F 2 (k 2 ) = tr { ( (p/ + m) Λ ρ p, p ) ( p/ + m ) Γ ρ (p, p) } Λ ρ (p m 2, p) = [γ k 2 (4m 2 k 2 ρ + k2 + 2m 2 ( p ) m (k 2 4m 2 + p ) ] ) ρ 26 / 45

42 Introduction Photon Four-Point Function Quark-loop Each diagram logarithmically divergent. Sum finite Using Ward Identities: 0 = k µ Π µνλρ (p 1, p 2, p 3, k) = Π µνλρ (p 1, p 2, p 3, k) = k ρ ( / k ρ ) Π µνλρ (p 1, p 2, p 3, k) define Γ ρσ as the muon vertex, Γ σ with / k ρ acting on the photon four-point function. F 2 (0) = 1 48m tr {(p/ + m) [γρ, γ σ ] (p/ + m) Γ ρσ (p, p)} [ M. Samuel, PRD (1992) ] [ S. Laporta and E. Remiddi, Phys. Lett. B (1991) ] [ J. Aldins, T. Kinoshita, S. J. Brodsky and A. J. Dufner, Phys. Rev. D 1 (1970) ] 27 / 45

43 Introduction Photon Four-Point Function Quark-loop Electron loop contribution to a e, a µ Result known analytically in perturbation theory Direct numerical evaluation agrees to high precision units of (α/π) 3 our result analytic a e ± a µ ± (d) (e) (f) (g) 28 / 45

44 Introduction Photon Four-Point Function Quark-loop Calculation (exploit Ward-Takahashi identity) Π (ρ)µναβ = k ρ + permutations Algebraically challenging 2 terms, 12 terms terms Dirac trace algebra before derivative. Full quark-photon vertex Highly non-trivial! 29 / 45

45 Truncation Table of contents 1 Introduction Generalities Hadronic Vacuum Polarisation Photon Four-Point Function 2 Truncation 3 Results HVP Adler function hlbl: pion pole hlbl: quark loop 4 Summary and Outlook 30 / 45

46 Truncation Axial-vector WTI + = connects quark Σ to Bethe-Salpeter K truncation must consider this important for chiral properties of pseudoscalars - Gell-Mann Oakes-Renner / massless pion in the chiral limit difficult and intricate to implement in practice. m πf 2 π 2m q qq 31 / 45

47 Truncation Rainbow-ladder truncation = K = Amounts to: Iterated one-gluon exchange 32 / 45

48 Truncation Rainbow-ladder truncation = λ 1 ( q 2 ) K = λ 1 ( q 2 ) Amounts to: Iterated one-gluon exchange ( Also: dress γ µ vertex with function λ ) 1 q 2 Reduces twelve components of k 2, p 2, q 2 to one with only q 2 dependence. Combination λ 1 ( q 2 ) D µν (q) modelled as eff. interaction 32 / 45

49 Truncation Rainbow-ladder truncation Success depends on generating DχSB consistent with axwti Get it right correct (light) pseudoscalar observables 33 / 45

50 Truncation Rainbow-ladder truncation Success depends on generating DχSB consistent with axwti Get it right correct (light) pseudoscalar observables Phenomenological Effective interaction (γ µ γ µ ) tuned to generate DCSB Quantities tend to be insensitive to the model details. Cannot learn precise details of interaction from observables. BUT can model additional effects (spin-orbit int.) 33 / 45

51 Truncation Rainbow-ladder truncation Success depends on generating DχSB consistent with axwti Get it right correct (light) pseudoscalar observables Phenomenological Effective interaction (γ µ γ µ ) tuned to generate DCSB Quantities tend to be insensitive to the model details. Cannot learn precise details of interaction from observables. BUT can model additional effects (spin-orbit int.) Philosophical improvements Use calculated Lattice/DSE/FRG gluon model quark-gluon vertex Model quark-mass dependence 33 / 45

52 Truncation Specifics: model interaction effective quark-gluon interaction (rainbow-ladder) phenomenologically successful model: gluon and quark-gluon vertex meson and baryon spectroscopy EM form-factors, pion charge radius decay constants, widths. Parameters of model are tuned for meson phenomenology. use values from literature without fine-tuning 34 / 45

53 Truncation Specifics: model interaction effective quark-gluon interaction (rainbow-ladder) 10 1 Maris-Tandy Z(p 2 ) Γ YM (p 2 ) p 2 [GeV 2 ] IR enhancement provides dynamical chiral symmetry breaking. UV tail matches perturbation theory 34 / 45

54 Truncation Specifics: model interaction effective quark-gluon interaction (rainbow-ladder) 10 1 Maris-Tandy Soft-divergent Z(p 2 ) Γ YM (p 2 ) p 2 [GeV 2 ] Qualitatively similar to modern effective interactions (from solutions of QGV and gluon propagator) 34 / 45

55 Results Table of contents 1 Introduction Generalities Hadronic Vacuum Polarisation Photon Four-Point Function 2 Truncation 3 Results HVP Adler function hlbl: pion pole hlbl: quark loop 4 Summary and Outlook 35 / 45

56 Results HVP Adler function Results: Adler function (N f = 5) preliminary/unpublished D(Q) /4π [ 1 3q 2 T µν(q) Dispersion Relation DSE MT ] Q [GeV] = Π(q 2 ), D(q) = q 2 dπ(q2 ) dq 2 (dispersion data from Jegerlehner, 2008) 36 / 45

57 Results HVP Adler function Results: Adler function (N f = 5) preliminary/unpublished D(Q) /4π [ 1 3q 2 T µν(q) Dispersion Relation DSE MT ] Q [GeV] = Π(q 2 ), D(q) = q 2 dπ(q2 ) dq 2 (dispersion data from Jegerlehner, 2008) 36 / 45

58 Results HVP Adler function Results: Adler function (N f = 5) preliminary/unpublished D(Q) /4π a HVP µ = α π Dispersion Relation DSE MT Q [GeV] [ ( )] x dx (1 x) e 2 2 Π 1 x m2 µ (de Rafael, 1994) 36 / 45

59 Results HVP Adler function Results: Adler function (N f = 5) preliminary/unpublished D(Q) /4π Dispersion Relation DSE MT Q [GeV] a HVP,LO µ,dse = cf. a HVP,LO µ,expt = (52.6) approach gives consistent results for HVP! 36 / 45

60 Results HVP Adler function Results: Adler function (N f = 5) Dependence on vertex truncation D(Q) /4π bare vertex 1BC vertex BC vertex full vertex Dispersion Relation pqcd asymptotic limit N f = Q [GeV] vertex a µ bare BC BC full vector meson pole necessary importance depends on kinematics of problem at hand one-scale problem vs. two-scale problem for Π µναβ 37 / 45

61 Results HVP Adler function Now: Hadronic Light-by-Light Scattering photon two-point function Approach gives reasonable prediction for HVP leading-order result at 5%-level! No additional fine tuning of model parameters Justifies application of DSE approach to the four-point function 38 / 45

62 pion exchange Results hlbl: pion pole Results: pseudoscalar-pole leading π 0 -pole a LBL µ = 58(7) phenomenological mixing for η, η Total result for pseudoscalar-pole a LBL µ = 80.7(12) Marc Knecht and Andreas Nyffeler. Hadronic light by light corrections to the muon g-2: The Pion pole contribution. Phys.Rev., D65:073034, / 45

63 Results hlbl: pion pole g-2: results for pion-pole contribution Group Model aµ LBL (π 0 -pole) Bijnens, Prades, Pallante ENJL 59(11) Hayakawa and Kinoshita HLS 57(4) Knecht and Nyffeler LMD+V 58(10) Melnikov and Vainshtein LMD+V 77(5) Dorokhov and Broniowski NLχQM 65(2) Nyffeler LMD+V 72(12) Our Result DSE 58(7) Table: π 0 -pole contribution to hlbl scattering. [ C. S. Fischer, T. Goecke and RW, PRD arxiv: ] [ T. Goecke, C. S. Fischer and RW, EPJA arxiv: ] 40 / 45

64 Results hlbl: quark loop Results: quark-loop quark loop bare vertex a LBL µ = ( 61 ± 2) BC aµ LBL = (107 ± 2) full BC aµ LBL = (176 ± 4) Dressing effects from Ward-Identity enhance: A 1, B M. Size of suppression from rho-pole? 41 / 45

65 quark loop Results hlbl: quark loop Results: quark-loop bare vertex a LBL µ = ( 61 ± 2) BC aµ LBL = (107 ± 2) full BC aµ LBL = (176 ± 4) full vertex (unpublished) aµ LBL = (106 ± 5) Dressing effects from Ward-Identity enhance: A 1, B M. Size of suppression from rho-pole? 40% 41 / 45

66 Results hlbl: quark loop Results: in summary DSE/BSE ENJL [BPP] HLS [HKS] π, η, η 81 ± ± ± 6.4 quark loop 106 ± 5 21 ± ± 11.1 [units of ] comments Quark-loop dominates. Large discrepency. Dressing of quark-photon vertex important. PS exchange result compatible off-shell suppression? 42 / 45

67 Results hlbl: quark loop Results: in summary DSE/BSE ENJL [BPP] HLS [HKS] π, η, η 81 ± ± ± 6.4 quark loop 106 ± 5 21 ± ± 11.1 π, K-loops 19 ± ± 8.1 axial-vector 2.5 ± ± 0 scalar 6.8 ± 2.0 total 187 ± 17 ± ± ± 15.4 [units of ] comments Quark-loop dominates. Large discrepency. Dressing of quark-photon vertex important. PS exchange result compatible off-shell suppression? 42 / 45

68 Summary and Outlook Table of contents 1 Introduction Generalities Hadronic Vacuum Polarisation Photon Four-Point Function 2 Truncation 3 Results HVP Adler function hlbl: pion pole hlbl: quark loop 4 Summary and Outlook 43 / 45

69 Summary Summary and Outlook Summary and Outlook First DSE calculation of g-2 for HVP predicted within our approach! (cf 6903 expt) hlbl pseudoscalar-pole 81(12) quark-loop 106(5) Find: vertex dressing important transverse dominated by rho-pole non-transverse constrained by Ward-Identity preliminary estimate: a µ = (71.0)10 11 (2.4σ) 44 / 45

70 Summary and Outlook Summary and Outlook Outlook Check quality of pion-pole approximation via DSEs Melnikov Vainshtein constraint calculate pion propagator calculate quark-antiquark scattering matrix T 45 / 45

71 Summary and Outlook Summary and Outlook Outlook Check quality of pion-pole approximation via DSEs Melnikov Vainshtein constraint calculate pion propagator calculate quark-antiquark scattering matrix T 45 / 45

HLbl from a Dyson Schwinger Approach

HLbl from a Dyson Schwinger Approach Richard Williams KFUni Graz Tobias Göcke TU Darmstadt Christian Fischer Uni Gießen INT Workshop on Hadronic Light-by-Light contribution to the Muon Anomaly February