1. A VMD Estimate of the Muon g-2 A HLS based approach M. Benayoun LPNHE Paris 6/7 1M. Benayoun, VMD & g-2 estimates

2. OUTLINE  Model & Method  Symmetry Breaking : BKY & (ρ,ω,φ) mixing  Global Fit :  vs e + e -  Global Fit : e + e - → π π /KKbar /π γ /η γ /π π π  ππ contribution to g-2  VMD estimate of HVP and g-2  Conclusions M. Benayoun, VMD & g-2 estimates2

3. HLS : A VMD Model (I) The Hidden Local Symmetry model is :  A unified VMD framework including e + e - → π π /KKbar /π γ /η γ /π π π & τ→ππ ν τ & PVγ couplings & η/η’  γπ π/γγ & ……  Few basic parameters : e, f π,V ud,V us, a (≈2), g,…  Present Limits : Up to the ≈ φ mass region ( 1.05 GeV) No scalars mesons, no ρ’, no ρ’’ …… 3M. Benayoun, VMD & g-2 estimates M. Harada & K. Yamawaki Phys. Rep. 381 (2003) 1 M.Bando et al. Phys. Rep. 164 (1988) 217

4. NSK2:: Breaking the HLS Model HLS not enough Flexible to really account for data  Need breaking schemes to become operating:  BKY mechanism  vector meson mixing 4M. Benayoun, VMD & g-2 estimates M.Bando et al. Nucl. Phys. B259 (1985) 493 M.Benayoun et al. EPJ C55 (2008) 199 M.Benayoun et al. Phys. Rev. D58 (1998) 074006 Breaking the HLS Model A. Bramon et al. Phys. Lett. B345 (1995) 263 M.Benayoun et al. EPJ C65 (2010) 211

5. HLS and the BKY Mechanism Define Then Standard HLS Lagrangians BKY breaking : Full HLS Lagrangian : BKY Breaking Matrix : Diag PS field matrix M. Benayoun, VMD & g-2 estimates M. Hashimoto, Phys. Rev. D54 (1996) 5611 M. Benayoun et al, arXiv:1106.1315

6. Isospin Breaking : Vector Field Mixing I tree level:: ideal vector fields ≡ mass eigenstates Vector Mass Term Diagonalization: one loop:: s-dependent transitions among R1 fields 6M. Benayoun, VMD & g-2 estimates C. Wolfe&K. Maltman, Phys. Rev. D80 (2009)114024 F. Jegerlehner & R. Szafron EPJ C71 (2011) 1632

7. Isospin Breaking : Vector Field Mixing II At one loop, the HLS Lagrangian piece induces s-dependent transitions among R1 fields isospin symmetry breaking : 7M. Benayoun, VMD & g-2 estimates

8. Transitions at one loop Define the loops : = K + K -, = K 0 K 0 Then, beside self-masses : → + ~ ε 2 (s) ≠ 0 always → - ≠ 0 by isospin brk → - ~ ε 1 (s) VVP Lagrangian K*K loops // Yang-Mills K*K*loops 8 f(s) M. Benayoun, VMD & g-2 estimates

9. The Mass Matrix Eigen System At one loop: As : Solve perturbatively 9M. Benayoun, VMD & g-2 estimates

10. From Ideal To Physical Fields 10M. Benayoun, VMD & g-2 estimates R 1 Fields Physical Fields

11. Analysis Method GLOBAL FIT to the largest possible data set  Why ?  Check VMD constraints well accepted by DATA? (correct lineshapes & yields with Proba. OK)  IF YES ↔ theoretical correlations fulfilled by DATA →→ improved parameters  HLS form factors & fit parameters values & errors & covariance matrix → contributions to g-2  Procedure fully blind to g-2 M. Benayoun, VMD & g-2 estimates11

12. Channels Considered Non-Anomalous annihilations : e + e - → π π / or decays : τ→ππ ν τ (plays as constraint only!) Anomalous Processes : e + e - → π γ/ηγ/π π π (Anomalous FKTUY Effective Lagrangian pieces) Radiative decays widths ( VPγ, η/η’→ππγ) OVERCONSTRAINED MODEL & Breaking Scheme 12M. Benayoun, VMD & g-2 estimates

13. Channels Considered Non-Anomalous annihilations : e + e - → π π / or decays : τ→ππ ν τ (constraint) Anomalous Processes : e + e - → π γ/ηγ/π π π (Anomalous Effective Lagrangian pieces) Radiative decays widths ( VPγ, η/η’→ππγ) ISR data : NOT INCLUDED OVERCONSTRAINED MODEL & Breaking Scheme Relying on more than 40 (scan) data sets 13M. Benayoun, VMD & g-2 estimates

14. V π π Couplings: I=1 part of ω and φ *At leading order : ρ term unchanged (I=1 part) *small s-dependent ω and φ couplings (I=1 part) * Charged and neutral rho’s: same coupling to pions 14 Mixing Angles I=1 terms M. Benayoun, VMD & g-2 estimates

15. ρ couplings to γ/W (γ/W) V transitions : = + Full agreement between data 15 M. Benayoun, VMD & g-2 estimates One loop corrections

16. ρ couplings to γ/W (γ/W) V transitions : = + Full agreement between data 16 M. Benayoun, VMD & g-2 estimates One loop corrections

17. ρ couplings to γ/W (γ/W) V transitions : = + Full agreement between data 17 M. Benayoun, VMD & g-2 estimates One loop corrections

18. M. Benayoun, VMD & g-2 estimates18 ρ couplings to γ/W Re[F(s)] Im[F(s)]

19. Dipion Spectra M. Benayoun, VMD & g-2 estimates19

20. e + e - Fit Residuals M. Benayoun, e+ e- versus tau20 e + e - Data

21. M. Benayoun, VMD & g-2 estimates21 Tau Fit Residuals

22. M. Benayoun, VMD & g-2 estimates22 Tau Residuals

23. NSK18:: Dipion Mass Spectrum s-dependent (missing) effect ! 23 M. Davier NP Proc. Supp. 169 (2007) 288 M. Benayoun, VMD & g-2 estimates

24. 24 3-pion Data

25. M. Benayoun, VMD & g-2 estimates25 K + K - K L K S e + e - → K Kbar Data

26. M. Benayoun, VMD & g-2 estimates26 K + K - K L K S e + e - → K Kbar Data Ratio

27. e + e - → π 0 γ 27M. Benayoun, VMD & g-2 estimates

28. e + e - → ηγ Data 28M. Benayoun, VMD & g-2 estimates

29. 29 Effect of Global Fitting on a µ (ππ) L.O. Contribution of ππ to g-2 Integrated from 0.630 to 0.958 GeV/c (x10 -10 )  Reconstructed value consistent with exp. average  Global fits provide improved uncertainties  Probabilities provided by the (underlying) global fits FSR Corrected Data Set 10 10 a µ (ππ) χ 2 /dofProbability Exp. Average 360.65 ± 2.55 All NSK  +  - 360.00 ± 1.64 177/208 93% ++  359.80 ± 1.47 263/293 90% ++ (π 0 /η) γ 360.09 ± 1.60 437/549 99% ++ (π + π - π 0 ) 360.91 ± 1.45 661/727 96% ++ K Kbar 362.79 ± 1.43 858/882 71%

30. M. Benayoun, VMD & g-2 estimates30 Data Set 10 10 a µ (ππ) χ 2 /dofProbability Exp. Average 360.65 ± 2.55 All NSK  +  - 360.00 ± 1.64 177/208 93% ++  359.80 ± 1.47 263/293 90% ++ (π 0 /η) γ360.09 ± 1.60 437/549 99% ++ (π + π - π 0 ) 360.91 ± 1.45 661/727 96% ++ K Kbar 362.79 ± 1.43 858/882 71% Effect of Global Fit on a µ (ππ) L.O. Contribution of ππ to g-2 Integrated from 0.630 to 0.958 GeV/c (x10 -10 )  Reconstructed value consistent with exp. average  Global fits provide improved uncertainties  Probabilities provided by the (underlying) global fits

31. M. Benayoun, VMD & g-2 estimates31 Data Set 10 10 a µ (ππ) χ 2 /dofProbability Exp. Average 360.65 ± 2.55 All NSK  +  - 360.00 ± 1.64 177/208 93% ++  359.80 ± 1.47 263/293 90% ++ (π 0 /η) γ 360.09 ± 1.60 437/549 99% ++ (π + π - π 0 ) 360.91 ± 1.45 661/727 96% ++ K Kbar 362.79 ± 1.43 858/882 71% Effect of Global Fit on a µ (ππ) L.O. Contribution of ππ to g-2 Integrated from 0.630 to 0.958 GeV/c (x10 -10 )  Reconstructed value consistent with exp. average  Global fits provide improved uncertainties  Probabilities provided by the (underlying) global fits

32. Contributions to g-2 up to  () 32M. Benayoun, VMD & g-2 estimates ChannelSolution BSolution AExp. Average +-+- 498.54 ± 1.97497.98 ± 1.76498.53 ± 3.73 π0 γπ0 γ 4.64 ± 0.04 4.28 ± 0.04 3.35 ± 0.11 η γη γ 0.65 ± 0.01 0.67 ± 0.01 0.48 ± 0.01 η' γη' γ 0.01 ± 0.00 --- π + π - π 0 42.03 ± 0.60 40.88 ± 0.52 43.24 ± 1.47 K L K S 12.02 ± 0.09 12.07 ± 0.08 12.31 ± 0.33 K+K-K+K- 16.87 ± 0.20 16.93 ± 0.18 17.88 ± 0.54 Total up to 1.05 GeV 574.76 ± 2.10572.82 ± 1.90575.79 ± 4.06 Excl. ISR  Uncertainties uniformly improved by a factor 2

33. Below 1.05 GeV Most of Hadronic VP estimated from our Model (574.76 ± 2.10/ 572.82 ± 1.90) Missing channels (ηπ π, π π π π, etc…) HVP estimated from data (1.55 ± 0.57) Above 1.05 GeV LO HVP estimated from data and pQCD (111.41 ± 4.10) Total LO HVP : B → 687.72 ± 4.63 // A → 685.78 ± 4.55 M. Benayoun, VMD & g-2 estimates33 VMD estimate of LO HVP

34. LO HVP + HO HVP + LBL +QED +EW give a µ = ( 11 659 175.37 ± 5.31) 10 -10 (B) Or a µ = ( 11 659 173.43 ± 5.25) 10 -10 (A) M. Benayoun, VMD & g-2 estimates34 VMD estimate of g-2

35. M. Benayoun, VMD & g-2 estimates35 VMD estimate of g-2

36. Conclusions 1/ HLS with BKY and Vector Meson Mixing provides a good simultaneous fit of e + e - → → π π /K + K - / K L K S / π γ/ηγ/π π π 2/ A simultaneous account of e + e - → → π π & τ→ππ ν τ 3/ VMD physics correlations improve g-2 (Uncertainty : HVP up to  uniformly improved by 2) 36M. Benayoun, VMD & g-2 estimates

37. Backup Slides 37M. Benayoun, VMD & g-2 estimates

38. NSK1:: HLS : A VMD Model (I) The Hidden Local Symmetry model is :  A unified VMD framework including e + e - → π π /Kkbar /π γ /η γ /π π π & τ→ππ ν τ & PVγ couplings & η/η’  γπ π/γγ & ……  Few basic parameters : e, f π,V ud,V us, a (≈2), g,…  Present Limits : Up to the ≈ φ mass region ( 1.05 GeV) No scalars mesons, no ρ’, no ρ’’ …… 38M. Benayoun, VMD & g-2 estimates M. Harada & K. Yamawaki Phys. Rep. 381 (2003) 1 M.Bando et al. Phys. Rep. 164 (1988) 217

39. VMD :: The HLS Framework Hidden Local Symmetry Model (HLS) as a whole L A, L V, L YM, L anomalous (π/K form factors, VPPP, γPPP, VPγ & Pγγ couplings) BKY-like flavor U(3)/SU(3) breaking mechanisms M.Bando, T. Kugo & K. Yamawaki Phys. Rep. 164 (1988) 217 M. Harada & K. Yamawaki Phys. Rep. 381 (2003) 1 M.Benayoun & H.O’Connell PR D 58 (1998) 074006 M. Benayoun et al. PR D 59 (1999) 114027 A. Bramon, A. Grau & G. Pancheri PL B 345 (1995) 263 G. Morpurgo PR D 42 (1990) 1497 M. Benayoun, L. DelBuono & H. O’Connell EPJ C 17 (2000) 593 M. Benayoun et al. EPJ C 55 (2008) 139 M.Bando, T. Kugo & K. Yamawaki Phys. Rep. 164 (1988) 217 39M. Benayoun, VMD & g-2 estimates

40. The Hidden Local Symmetry Model : The Non-Anomalous Sector Define Define covariant derivatives Then and The Full HLS Lagrangian : 40 Expanded form: M.Benayoun & H.O’Connell PR D 58 (1998) 074006 Universal Vector Coupling PS field matrix M. Benayoun, VMD & g-2 estimates

41. The Covariant Derivatives Covariant derivatives ≠ for left- right-ξ fields : With : 41 T ± is CKM matrix reduced to V us and V ud terms M. Benayoun, VMD & g-2 estimates

42. Anomalous Annihilations/Decays Non-Anomalous annihilations : e + e - → π π / or decays : τ→ππ ν τ Anomalous Processes : e + e - → π γ/ηγ/π π π :: Other Effective Lagrangian pieces 42M. Benayoun, VMD & g-2 estimates

43. SIDE RESULT : Prediction for η/η’→ ππ γ Г(η’→ ππ γ)=53.11±1.47 PDG : 60±5 keV Г(η→ ππ γ)=55.82±0.83 PDG : 60±4 eV Lineshapes and yields : tests for g value and ρ 0 lineshape M. Benayoun et al. ArXiv 0907.4047 43M. Benayoun, VMD & g-2 estimates

44. BELLE/ALEPH/CLEO 44M. Benayoun, VMD & g-2 estimates