HLbl from a Dyson Schwinger Approach

Transcription

1 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 28 th March 4 th / 28

2 Introduction We haven t yet presented our work already there are demands that we: calculate a different contribution to a µ Hadronic Vacuum Polarisation significantly improve the calculation consider rho-pole suppression in quark-loop 2 / 28

3 Introduction Table of contents 1 Introduction Hadronic Vacuum Polarisation Photon Four-Point Function 2 Results HVP Adler function hlbl: pion pole hlbl: quark loop 3 Summary and Outlook 3 / 28

4 Introduction Table of contents 1 Introduction Hadronic Vacuum Polarisation Photon Four-Point Function 2 Results HVP Adler function hlbl: pion pole hlbl: quark loop 3 Summary and Outlook 4 / 28

5 Introduction 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. 5 / 28

6 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 6 / 28

7 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 6 / 28

8 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, 7 / 28

9 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 7 / 28

10 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 8 / 28

11 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 ] 8 / 28

12 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 9 / 28

13 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 9 / 28

14 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 10 / 28

15 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! 11 / 28

16 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! 11 / 28

17 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 11 / 28

18 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) 12 / 28

19 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, / 28

20 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 = 14 / 28

21 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. 15 / 28

22 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) 16 / 28

23 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 16 / 28

24 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 17 / 28

25 Introduction Photon Four-Point Function Quark-loop Calculation (exploit Ward-Takahashi identity) Π (ρ)µναβ = k ρ + permutations Projector a µ = 1 [ tr (ip/ + m µ ) [γ σ, γ ρ ] (ip/ + m µ ) 48m Γ ] σρ µ k 0 with Γ σρ related to muon vertex via ieγ µ = iek ρ Γρµ 18 / 28

26 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! 18 / 28

27 Introduction Photon Four-Point Function 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 19 / 28

28 Introduction Photon Four-Point Function 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 19 / 28

29 Results Table of contents 1 Introduction Hadronic Vacuum Polarisation Photon Four-Point Function 2 Results HVP Adler function hlbl: pion pole hlbl: quark loop 3 Summary and Outlook 20 / 28

30 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) 21 / 28

31 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) 21 / 28

32 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) 21 / 28

33 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! 21 / 28

34 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 22 / 28

35 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 23 / 28

36 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, / 28

37 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? 25 / 28

38 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% 25 / 28

39 Summary and Outlook Table of contents 1 Introduction Hadronic Vacuum Polarisation Photon Four-Point Function 2 Results HVP Adler function hlbl: pion pole hlbl: quark loop 3 Summary and Outlook 26 / 28

40 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) 27 / 28

41 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 28 / 28