1. Estimates for (g-2) μ using VMD constraints & Global Fits M. Benayoun LPNHE Paris 6/7 ( collaborators : P. David, L. Delbuono, F. Jegerlehner)

2. OUTLINE  The HLS Model & Breaking : BKY & (ρ,ω,φ) mixing  Global Fit Scope: V P γ / P γ γ /  vs e + e - & [e + e - → π π /K + K - /K L K S /π γ /η γ /π π π]  HLS Estimates of (g-2) μ  Global fits with only scan (NSK) data : e + e - vs τ   «  + PDG » Predictions of e + e - → π π vs data  Global fit with  scan and ISR π π data → g-2  Conclusions 2 M.Benayoun et al. EPJ C72 (2012) 1848 M.Benayoun et al. arXiv: 1210.7184// EPJC

3. NSK2:: Breaking the HLS Model The HLS Lagrangian not flexible for data handling  Breaking schemes needed:  BKY mechanism :  vector meson mixing :  Latest Model Status : 3 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 The HLS Model & Breaking M.Benayoun et al. EPJ C65 (2010) 211 M.Benayoun et al. EPJ C72 (2012) 1848 M.Hashimoto Phys. Rev. D54 (1996) 5611 M. Harada & K. Yamawaki Phys. Rep. 381 (2003) 1

4. Isospin Breaking : Vector Field Mixing At one loop, the HLS Lagrangian piece (no IB) s-dependent transitions among vector fields ideal fields no longer mass eigenstates PS mesons :: isospin symmetry breaking : 4 VVP Lagrangian : K*K loops // Yang-Mills term : K*K*loops → →

5. The Mass Matrix Eigen System At one loop: As : Solve perturbatively : (kaon loop) Sum, Difference 5

6. From Ideal To Physical Fields 6 R 1 Fields Physical Fields At leading order in ε 1 (s), ε 2 (s)

7. BHLS : A Global VMD Model The (Broken) Hidden Local Symmetry (BHLS) model :  is a unified VMD framework encompassing e + e - → π π /KKbar /π γ /η γ /π π π & τ→ππ ν τ & PVγ, Pγγ decays & η/η’  γπ π/γγ & ……  BHLS :: (almost) an empty shell : [α em, G F, f π, V ud, V us, m π ’s, m K ’s, m η,, m η’ ]  Present Limits : Up to the ≈ φ mass region ( 1.05 GeV) No scalars mesons, no ρ’, no ρ’’ …… 7

8. HLS : A Global VMD Model The (Broken) Hidden Local Symmetry (BHLS) model :  is a unified VMD framework encompassing e + e - → π π /KKbar /π γ /η γ /π π π & τ→ππ ν τ & PVγ, Pγγ decays & η/η’  γπ π/γγ & ……  BHLS :: (almost) an empty shell : [α em, G F, f π, V ud, V us,m π ’s, m K ’s, m η,, m η’ ] Present Limits : Up to the ≈ φ mass region ( 1.05 GeV) No scalars mesons, no ρ’, no ρ’’ …… 8

9. HLS : A Global VMD Model The (Broken) Hidden Local Symmetry (BHLS) model :  is a unified VMD framework encompassing e + e - → π π /KKbar /π γ /η γ /π π π & τ→ππ ν τ & PVγ, Pγγ decays & η/η’  γπ π/γγ & ……  BHLS :: (almost) an empty shell : [α em, G F, f π, V ud, V us, m π ’s, m K ’s, m η, m η’ ] Present Limits : Up to the ≈ φ mass region ( 1.05 GeV) No scalars mesons, no ρ’, no ρ’’ …… 9

10. g-2 & Global Models/Fits NP Hadronic VP contributions to g-2 (interpolation between energy points) VMD : Underlying physics correlations -> HLS cross-sections derived through a global fit (param. values & error covariance matrix) : 10 Measured Xsection Model Xsection (φ)

11. HLS→Correlated Physics Channels Non-Anomalous annihilations : e + e - → π + π - / Decay spectra : τ ± → π ± π 0 ν τ (later also : η/η’→π + π - γ) Anomalous Processes : e + e - → π 0 γ/ηγ/ π + π - π 0 Radiative decays widths ( VPγ, π 0 /η/η’ → γ γ) φ→ππ (Br ratio and phase) HLS MODEL & Breaking Schemes: OVERCONSTRAINED & **DETERMINED** by more than 40 data sets 11

12. HLS→Correlated Physics Channels Non-Anomalous annihilations : e + e - → π + π - / Decay spectra : τ ± → π ± π 0 ν τ (later also : η/η’→π + π - γ) Anomalous Processes : e + e - → π 0 γ/ηγ/ π + π - π 0 Radiative decays widths ( VPγ, π 0 /η/η’ → γ γ) φ→ππ (Br ratio and phase) New Paradigm : Any information on any channel (π 0 γ) is equivalent to improving the statistics on any other channel (π + π - ) 12

13. HLS→Correlated Physics Channels Non-Anomalous annihilations : e + e - → π + π - / Decay spectra : τ ± → π ± π 0 ν τ (later also : η/η’→π + π - γ) Anomalous Processes : e + e - → π 0 γ/ηγ/ π + π - π 0 Radiative decays widths ( VPγ, π 0 /η/η’ → γ γ) φ→ππ (Br ratio and phase) HOWEVER : Non Perturbative Contributions to HVP above ≈1.05 GeV should be estimated as usual 13

14. VMD Strategy for g-2 Estimate GLOBAL FIT to the largest possible data set  Check:: are VMD constraints well accepted by DATA? (correct lineshapes & yields with Prob.’s OK)  IF YES :  1/ VMD correlations are fulfilled by DATA  2/ THEN :: HLS form factors & fit parameters values & errors covariance matrix should lead to :  Improved estimates of HVP contributions to g-2 for π + π - / / /π γ /η γ /η’ γ /π π π up to 1.05 GeV 14

15. 15 Data Set (#data points)Configuration B ( χ 2 ) Decays (9) 10.4 π + π - New Timelike (127)128.3 π + π - Old Timelike (82)50.2 π 0 γ (86)66.9 ηγ (182) 121.0 π + π - π 0 112.5/99 K + K - (36)30.3 K L K S (119)122.7 τ ALEPH (37)25.1 τ CLEO (29)38.2 τ BELLE (19)25.6 χ 2 /dof Probability 731.2/801 96.3% Global Fit Quality with NSK only Fits : see backup slides

16. Dipion Spectra (τ & e + e - ): A puzzle? 16

17. e + e - vs τ : CMD2/SND Fit Residuals ΔFΔF ΔF/F Global fit : 96% Prob

18. 18 e + e - vs τ : Tau Fit Residuals ΔFΔF ΔF/F

19. A « e + e - - τ» Puzzle ? s-dependent (missing) effect ! 19 M. Davier NP Proc. Supp. 169 (2007) 288

20. 20 A « e + e - - τ» Puzzle ? e + e - and τ dipion residuals : simultaneously flat |F π (e + e - )| 2 & |F π (τ)| 2 yield : simultaneously χ 2 π+π- /N π+π- & χ 2 π±π0 /N π ± π0 ≈1 Global fit Probability > 90% No signal of any e + e - - τ mismatch → nothing puzzling

21. HVP Results with scan & τ data (Updated) All uncertainties improved by ≈ 2 21 ChannelSolution BDirect Estimate +-+- 495.40 ± 1.92498.53 ± 3.73 (497.72 ± 2.12) π0 γπ0 γ 4.61 ± 0.04 3.35 ± 0.11 η γη γ 0.64 ± 0.01 0.48 ± 0.01 η' γη' γ 0.01 ± 0.00 --- π + π - π 0 41.16 ± 0.59 43.24 ± 1.47 K L K S 11.90 ± 0.08 12.31 ± 0.33 K+K-K+K- 17.59 ± 0.21 17.88 ± 0.54 Total up to 1.05 GeV 571.30 ± 2.02575.79 ± 4.06

22. HVP Results with scan & τ data (Updated) All uncertainties improved by ≈ 2 22 ChannelSolution BDirect Estimate +-+- 495.40 ± 1.92498.53 ± 3.73 (497.72 ± 2.12) π0 γπ0 γ 4.61 ± 0.04 3.35 ± 0.11 η γη γ 0.64 ± 0.01 0.48 ± 0.01 η' γη' γ 0.01 ± 0.00 --- π + π - π 0 41.16 ± 0.59 43.24 ± 1.47 K L K S 11.90 ± 0.08 12.31 ± 0.33 K+K-K+K- 17.59 ± 0.21 17.88 ± 0.54 Total up to 1.05 GeV 571.30 ± 2.02575.79 ± 4.06 Scan +ISR

23. HVP Results with scan & τ data (Updated) Central value shifted by ≈ 3 10 -10 23 ChannelSolution BDirect Estimate +-+- 495.40 ± 1.92498.53 ± 3.73 (497.72 ± 2.12) π0 γπ0 γ 4.61 ± 0.04 3.35 ± 0.11 η γη γ 0.64 ± 0.01 0.48 ± 0.01 η' γη' γ 0.01 ± 0.00 --- π + π - π 0 41.16 ± 0.59 43.24 ± 1.47 K L K S 11.90 ± 0.08 12.31 ± 0.33 K+K-K+K- 17.59 ± 0.21 17.88 ± 0.54 Total up to 1.05 GeV 571.30 ± 2.02575.79 ± 4.06 Diff=3.1 units

24. τ Predictions for π + π - Spectra τ dipion spectra : basis (Aleph,Belle&Cleo <1 GeV ) ++ Breaking Info from PDG : Γ(ρ→ e + e - ) & Γ(ω→ππ) & Γ(φ→ππ) (& Orsay ) → → «τ + PDG predictions» for e + e - → π + π - generated by the model using all data for: τ→ππ ν τ, e + e - → /π 0 γ/ηγ/π + π - π 0 & VPγ & Pγ γ couplings (from PDG) 24

25. τ Predictions for π + π - Spectra τ dipion spectra : basis (Aleph,Belle&Cleo < 1 GeV) ++ Breaking Info from PDG : Γ(ρ→ e + e - ) & Γ(ω→ππ) & Γ(φ→ππ) (& Orsay ) → → «τ + PDG predictions» for e + e - → π + π - generated by the model using all data for: τ→ππ ν τ, e + e - → /π 0 γ/ηγ/π + π - π 0 & VPγ & Pγ γ couplings (from PDG) 25

26. !! 26 τ + PDG Predictions Good agreement data versus predictions Spacelike region well predicted & Extrapolates much above 1 GeV!!!

27. 27 τ + PDG Predictions KLOE’s almost perfect! CMD2&SND ≈ OK BaBar too large below ω 0.85 GeV >m ππ > 0.70GeV PION Form Factor

28. Discrepancies between π + π - data sets (oldnews) Clear problem at (ρ,ω) Interference for BaBar HOWEVER PDG IB information : Γ(ρ→ e + e - ) & Γ(ω→ππ) almost equivalent to Experimental IB information : (0.76-0.82) GeV Region From each data set (NSK,KLOE’s,BaBar) 28 Guesses About Global Fits

29. 29 τ + {BaBar (ρ―ω)} Predictions 0.85 GeV >m ππ > 0.70GeV PION Form Factor FIT Prediction

30. 30 τ + {(ρ―ω) Region from each Exp} Fits 0.85 GeV >m ππ > 0.70GeV PION Form Factor Fits in «isolation» NSK KLOE10 KLOE 08 BaBar

31. 31 ρ-ω region (0.76→0.82 GeV)/φ(PDG) F ω = Br(ω→π + π - ) * Br(ω→e + e - ) DATA SAMPLEF ω X 10 -6 Orsay Phase (degrees) Global Fit Probability X 2 /n π+π- PDG1.23±0.07[104.7±4.1] ------ --- CMD2+SND (≈ 0σ)1.22±0.04106.7±0.3 90% 51/47 KLOE08 (1.9 σ)1.08±0.04110.4±1.2 88% 18/11 KLOE10 (3.0 σ)0.97±0.05113.6±1.6 93% 11/11 KLOE12 (3.1 σ)0.92 ±0.07122.5±2.1 96% 7/11 BaBar (7.7 σ)1.78±0.01107.5±0.2 54% 67/35

32. Checking ππ data sets Fits performed with each ππ data set in isolation (scan/KLOE’s/BaBar) All other channels always included in fits (τ→ππ ν τ, e + e - → K + K - / K0 K0b/π γ/ηγ/π π π) 32

33. KLOE raw & corrected Residuals ΔFΔF

34. Fits in isolation : Comments Good Residuals & Gl. fit Prob. OK & good χ 2 : CMD2 & SND & KLOE10 & KLOE12 (Very) Good Residuals & poor χ 2 (1.62) : KLOE08 (correlations in systematics) Bad Gl. fit Prob. & at [ρ/ω] peak χ 2 /N= 1.9 BaBar (hardly consistent with non ππ data) Consistent ππ data sets for Global Treatment : CMD2 & SND & KLOE10 & KLOE12

35. GLOBAL FIT RESULTS Fits performed with combinations of ππ data sets (incl./excl. Spacelike) All other channels always included in fits (τ→ππ ν τ, e + e - → K+K-/ K0 K0b/π γ/ηγ/π π π) 35

36. 36 GLOBAL FIT RESULTS Proves consistency of CMD2/SND/KLOE10/ KLOE12 KLOE’s Fits in «isolation» Global Fit

37. HVP Results up to 1.05 GeV 37 ChannelNSK +KLOE 10&12 + τ (ABC < 1 GeV) scan only(NSK) + τ (ABC < 1GeV) Direct Estimate +-+- 491.12 ± 1.35495.40 ± 1.92498.53 ± 3.73 (497.72 ± 2.12) π0 γπ0 γ 4.63 ± 0.04 4.61 ± 0.04 3.35 ± 0.11 η γη γ 0.64 ± 0.01 0.48 ± 0.01 η' γη' γ 0.003 ± 0.000 --- π + π - π 0 40.78 ± 0.64 41.16 ± 0.59 43.24 ± 1.47 K L K S 11.94 ± 0.08 11.90 ± 0.08 12.31 ± 0.33 K+K-K+K- 17.48 ± 0.21 17.59 ± 0.21 17.88 ± 0.54 Total up to 1.05 GeV 566.58 ± 1.50 571.30 ± 2.02575.79 ± 4.06 Scan +ISR

38. HVP Results up to 1.05 GeV 38 ChannelNSK+KLOE10&12 + τ (ABC < 1 GeV) scan only(NSK) + τ (ABC < 1 GeV) Direct Estimate +-+- 491.12 ± 1.35495.40 ± 1.92498.53 ± 3.73 (497.72 ± 2.12) π0 γπ0 γ 4.63 ± 0.04 4.61 ± 0.04 3.35 ± 0.11 η γη γ 0.64 ± 0.01 0.48 ± 0.01 η' γη' γ 0.003 ± 0.000 --- π + π - π 0 40.78 ± 0.64 41.16 ± 0.59 43.24 ± 1.47 K L K S 11.94 ± 0.08 11.90 ± 0.08 12.31 ± 0.33 K+K-K+K- 17.48 ± 0.21 17.59 ± 0.21 17.88 ± 0.54 Total up to 1.05 GeV 566.58 ± 1.50 571.30 ± 2.02575.79 ± 4.06 (574.98±2.66) Scan +ISR

39. g-2 HLS Estimates & Others 39

40. g-2 HLS Estimates & Others 40

41. g-2 HLS Estimates & Others 41

42. a μ (ππ) over m ππ =[0.63,0.958] GeV Model Estimate in Accord with expectations Uncertainties improved by ≈45%

43. 43 Accord with Exp. expectations a μ (ππ) over m ππ =[0.63,0.958] GeV ≈ No Bias!

44. 44 a μ (ππ) over m ππ =[0.63,0.958] GeV No Mismatch!

45. The Spacelike & threshold Regions 45 Low mass tail OK! Extrapolation to s<0 perfect

46. Predicted Phase shift (I) 46

47. Predicted Phase shift (II) 47

48. Threshold Behavior 48

49. Conclusions 1/ broken HLS model → good simultaneous fit of all data 2/ suitable IB → NO signal for (e + e - vs τ) puzzle 3/ « τ +PDG » predictions → benchmark for π π samples! 4/ Relative inconsistencies of π + π - data sets substantiated 5/ Consistent π + π - data sets : CMD2, SND, KLOE 10 & 12 6/ The discrepancy with BNL g-2 value is ≈ 4.8 to 5.1 σ

50. Backup Slides 50

51. Residuals : e+e- vs τ 51 Residuals simultaneously flat

52. Global Fit Residuals : e + e - vs τ 52 Residuals simultaneously flat

53. From Belle 53

54. g-2 Estimates & Discrepancy II 54

55. Dipion Mass Spectrum s-dependent (missing) effect ! 55 M. Davier NP Proc. Supp. 169 (2007) 288

56. Backup Slides II 56

57. Define Basic pieces : The Hidden Local Symmetry (HLS) Lagrangian expanded in 57 The HLS Model PS field matrix M. Harada & K. Yamawaki Phys. Rep. 381 (2003) 1 M.Benayoun & H.O’Connell PR D 58 (1998) 074006

58. HLS and the BKY Mechanism Basic HLS Lagrangians : Full HLS Lagrangian BKY Broken Lagrangian: Breaking Matrix = Diag M. Hashimoto, Phys. Rev. D54 (1996) 5611 M.Benayoun et al. EPJ C72 (2012) 1848 58 Isospin Brk SU(3) Brk

59. Isospin Breaking : Vector Field Mixing At one loop, the HLS Lagrangian piece (no IB) s-dependent transitions among vector fields ideal fields no longer mass eigenstates PS mesons :: isospin symmetry breaking : 59 VVP Lagrangian : K*K loops // Yang-Mills term : K*K*loops → →

60. 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) 60 f(s)

61. The Mass Matrix Eigen System At one loop: As : Solve perturbatively 61

62. Qqq 62 Data Set (#data points)Configuration A ( χ 2 )Configuration B ( χ 2 ) Decays (9)12.9 10.4 New Timelike (127)127.7128.3 Old Timelike (82)50.550.2 π 0 γ (86)67.466.9 ηγ (182) 121.4121.0 π + π - π 0 229.3/179112.5/99 K + K - (36)35.830.3 K L K S (119)121.1122.7 τ ALEPH (37)24.725.1 τ CLEO (29)38.738.2 τ BELLE (19)25.925.6 Χ 2 /dof Probability 855.3/881 72.7% 731.2/801 96.3% Global Fit Results with NSK

63. Dipion Spectra (τ & NSK) 63

64. 64 K + K - K L K S e + e - → K Kbar Data

65. 65 3-pion Data

66. e + e - → π 0 γ 66

67. e + e - → ηγ Data 67

68. Former : Prediction for η/η’→ ππ γ Г(η’→ ππ γ)=53.11±1.47 PDG : 58.3±2.8 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 68

69. 69 Former : Prediction for η/η’→ ππ γ

70. Fit in isolation : KLOE 08 Residuals ΔFΔF ΔF/F Global fit : 56% Prob

71. Fit in isolation : KLOE 10 Residuals ΔFΔF ΔF/F Global Fit : 88% Prob

72. Fit in isolation : KLOE 12 Residuals ΔFΔF ΔF/F Global fit : 92% Prob

73. 73 Fit in isolation : BaBar Residuals ΔFΔF ΔF/F Global Fit : 22% Prob

74. Fit in isolation : CMD2/SND Residuals ΔFΔF ΔF/F Global fit : 96% Prob

75. 75 Tau Fit Residuals (NSK only) ΔFΔF ΔF/F

76. NA7 Residuals (χ 2 /N≈90/60)! 76

77. Scale Uncertainty Minimize : Solve for λ (A= m or M) gives: And : 77

78. 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 78 Mixing Angles I=1 terms

79. Vector meson couplings to γ (γ) V transitions : = + 79

80. ρ couplings to γ/W (γ/W) V transitions : = + Full agreement between data 80

81. 81 ρ couplings to γ/W Re[F(s)] Im[F(s)]

82. From Belle 82