1. VMD/HLS Approach to (g-2)μ : A Solution to the (τ - e+e- ) Puzzle
M. Benayoun
LPNHE Paris 6/7

2. OUTLINE HLS Model & Breaking : BKY & (ρ,ω,φ) mixing
The π π channel in e+e- annihilation & τ decay
The τ (+ PDG) Predictions for e+e- → π+ π-
Global Fit : [t vs e+ e-] [t ± →π± π0υτ//e+e-→ π+π-/K+K-/KLKS/π γ /η γ /π π π]
Updated HLS Estimates of (g-2)μ
Final Results & Conclusions

3. NSK2:: Breaking the HLS Model
The HLS Model & Breaking
The HLS Lagrangian + FKTUY (anomalous) pieces
Breaking schemes are needed:
BKY mechanism :
vector meson mixing :
Latest Model Status :
M. Harada & K. Yamawaki Phys. Rep. 381 (2003) 1
M.Bando et al. Nucl. Phys. B259 (1985) 493
M.Hashimoto Phys. Rev. D54 (1996) 5611
M.Benayoun et al. Phys. Rev. D58 (1998)
M.Benayoun et al. EPJ C55 (2008) 199
M.Benayoun et al. EPJ C65 (2010) 211
M.Benayoun et al. EPJ C72 (2012) 1848

4. The HLS Model & BKY Breaking
Define
The covariant derivatives
matrices ->
BKY breaking
Breaking Matrix = Diag
M. Harada & K. Yamawaki Phys. Rep. 381 (2003) 1
PS field matrix

5. 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 : →
VVP Lagrangian : K*K loops // Yang-Mills term : K*K*loops →

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

7. From Ideal To Physical Fields
BKY renorm. Fields
At leading order in ε1(s) , ε2(s)

8. V π π Couplings: I=1 part of ω and φ
Mixing Angles I=1 terms
*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

9. Vector meson couplings to γ
(γ) V transitions become s-dependent:
= [ ]

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

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

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

13. Isospin Breaking in the τ Sector
Short Range & Long Range IB corrections
π±π0 Phase space factor
≈ No sensitivity to ρ± − ρ0 mass/width diff.
W. Marciano & A. Sirlin, PRL 71 (1993) 3629.
V. Cirigliano et al. PL B 513 (2001) & JHEP 08 (2002) 002
IB in τ & IB in e+e- :: unrelated but both involved in global fits

14. HLS : A Global VMD Model
The (Broken) HLS model (BHLS)
is a unified VMD framework encompassing
e+e- → π π /KKbar /π γ /η γ /π π π & τ→ππ ντ & PVγ, Pγγ decays & η/η’  γπ π
Valid up to the ≈ φ mass region ( 1.05 GeV)
A nearly empty shell [αem,GF ,fπ ,Vud ,Vus ] fed via :
a Global Fit (≈ 50 data samples)

15. 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) :
Measured Xsection
Measured Xsection
Model Xsection (φ)

16. π+ π- Spectra : NSK, KLOE, BaBar
Several measurements of the π+ π- spectrum
CMD2, SND
KLOE
BaBar
exhibit conflicting behaviors within BHLS
CMD2: Phys. Lett. B648 (2007) 28, JETP Lett. 84 (2006) 413
SND: JETP 103 (2006) 380
KLOE08 : AIP Conf. Proc (2009) *
KLOE10: Phys. Lett. B700 (2011) 102
KLOE12: Phys. Lett. B720 (2013)336
BaBar : Phys. Rev. Lett (2009) *
Phys. Rev . D86 (2012)
M. Benayoun et al EPJ C73 (2013)2453

17. π+ π- Spectra : τ + PDG Predictions
Perform a global fit within BHLS with:
t → ππ υτ (Aleph, CLEO, Belle)
e+e- → K+K-/KLKS /π γ /η γ /π π π (scan from NSK)
All e+e- → π π samples discarded
& Isospin Breaking information from RPP
Г(ρ → e+e-)
Г(ω → π π) & ORSAY phase (~104○)
Г(φ → π π) & Г(φ → π π) X Г(φ → e+e-)

18. π+ π- Spectra : τ + PDG Predictions
In spacelike & timelike regions
π π samples NOT FITTED
Average χ2 distance to
τ + PDG prediction displayed

19. π+ π- Spectra : τ + PDG Predictions
π π samples NOT FITTED :
Average χ2 distance to
τ + PDG prediction

20. π+ π- Spectra : τ + PDG Predictions
NO τ – e+e- Puzzle within BHLS
Strong evidence for
Sample dependent Behavior

21. Global Fit & Scale Uncertainty
M. Benayoun et al EPJ C73 (2013)2453
Minimize :
Solve for λ :
Choice of A : σexp (bias) , [σth ≈ σiter](unbiased)
the
G. D’Agostini NIM A346 (1994)306
V. Blobel eConf C MOET002 (2003)
M. Benayoun et al , to be published
R.D.Ball et al JHEP 1005 (2010)075// Nucl. Phys. B838 (2010) 136

22. Fitting π+π- Data using τ Samples
- Fit Cond. (χ2/N π+π - )
KLOE08(60)
KLOE10(75)
KLOE12(60)
NSK(127/209)
BaBar(250)
(trunc)
Single
(χ2/N π+π - ) % % %
0.96 [0.83] % [99%]
1.15
74%
Comb 1 χ2/N : 1.28(11%)
1.02
1.48
1.18[0.96]
1.35
Comb 2 χ2/N: 1.06(97%)
1.05
1.10[0.89]
Comb 3 χ2/N: 0.98 (96%)
0.97
1.00

23. Global Fit Results with τ & NSK&KLOE
Channel (#data points)
χ2 (NSK+τ)
Χ2 (NSK+KLOE)+τ
Decays (10)
8.4
9.2
New Timelike (127)
122.3
139.7
Old Timelike (82)
50.4
46.2
π0γ (86)
64.0
64.2
ηγ (182)
120.1
120.8
π+π- π0 (99)
102.3
101.8
K+K- (36)
29.9
KLKS (119)
119.3
119.1
τ ALEPH (37)
19.5
19.3
τ CLEO (29)
35.6
36.4
τ Belle (19)
28.3
30.9
Χ2/dof // Probability
701/801 //99.5%
857/936// 97%

24. Dipion Spectra (τ & NSK+KLOE)
TOTO
Dipion Spectra (τ & NSK+KLOE)

25. e+e- vs τ : Fit Residuals
TOTO
e+e- vs τ : Fit Residuals
e+e- & τ Residuals :
Simultaneously Flat
IB in e+e- & τ channels :
Consistent

26. From Belle

27. aµ (π+π-+τ, √s=[0.630,0.958] GeV) Data and BHLS estimates
Amp. : ( GeV)
Green : A = σexp (it=0)
Black : A = σiter (it=1)
It=0 ≈ It=1

28. HVP contribution (E≤ 1.05 GeV)
Including/excluding τ spectra : improved r.m.s.
Channel
A=M0 (excl. τ)
A=M0 (incl. τ)
Direct Estimate
p+p-
± 1. 39
± 1.11
( ± 3.73)scan
( ± 3.13)isr
π0 γ
4.50 ± 0.05
4.54 ± 0.04
3.35 ± 0.11
η γ
0.64 ±
0.48 ± 0.01
π+ π- π0
40.84 ± 0.62
40.85 ± 0.58
43.24 ± 1.47
KL KS
11.53 ±
11.56 ±
12.31 ± 0.33
K+K-
16.88 ± 0.22
16.77 ± 0.21
17.88 ± 0.54
Total up to GeV
± 1.63
± 1.27
( ± 4.06)scan ( ± 3.52)isr

29. HVP contribution (E≤ 1.05 GeV)
p+p- : A=σexp (« biased ») A= σiter (« unbiased »)
Channel
A=m
A=M0
Direct Estimate
p+p-
± 1.48
± 1.11
( ± 3.73)scan
( ± 3.13)isr
π0 γ
4.53 ± 0.04
4.54 ± 0.04
3.35 ± 0.11
η γ
0.64 ±
0.48 ± 0.01
π+ π- π0
± 0.58
40.85 ± 0.58
43.24 ± 1.47
KL KS
11.56 ±
12.31 ± 0.33
K+K-
16.78 ± 0.21
16.77 ± 0.21
17.88 ± 0.54
Total up to GeV
± 1.60
± 1.27
( ± 4.06)scan ( ± 3.52)isr

30. g-2 Estimates & Discrepancy
qqq

31. The Spacelike & threshold Regions
Extrapolation to s<0 perfect

32. Predicted Phase shift (I)

33. Predicted Phase shift (II)

34. Final Result + Additional Systematics
significance for Δ aμ > 4.5/3.9 σ
NSK+KLOE (B)
NSK+KLOE+ BaBar (trunc)

35. Conclusions 1/ broken HLS model → good simultaneous fit of all data
2/ IB → NO signal of a (e+e- vs τ) puzzle within BHLS
3/ Relative inconsistencies of π π data sets substantiated
4/ Consistent π π data sets : CMD2, SND, KLOE 10 & 12
5/ The discrepancy with BNL g-2 value is ≈ 4.5/3.9 σ
6/ Error on LO-HVP dominated by [1.05, 2] GeV data
7/ Experimental Estimate vs Global Fit Estimate

36. Backup Slides

37. Scale Uncertainty(ies)& Biases
The function to be minimized :
A : σexp (bias) , or [σtruth ≈ σiter](unbiased)

38. Scale Uncertainty(ies)& Biases
The function to be minimized :
A : σexp (bias) , or [σtruth ≈ σiter](unbiased)

39. e+e- : Fit Residuals (scale corr.)
TOTO
e+e- : Fit Residuals (scale corr.)
e+e- Residuals :
Scale corrected
e+e- Residuals :
Raw Distribution

40. Scale Uncertainty(ies)& Biases
The function to be minimized :
A : σexp (bias) , or [σtruth ≈ σiter](unbiased)

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

42. Transitions at one loop
Define the loops :
= K+ K- , = K0 K0
Then, beside self-masses :
→ ~ ε2(s) ≠ 0 always →
≠ 0 by isospin brk
→ ~ ε1(s)
f(s)

43. 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-/K+K-/ KLKS /π γ /η γ /η’ γ /π π π up to 1.05 GeV

44. Global Fit & Scale Uncertainty
M. Benayoun et al EPJ C73 (2013)2453
Minimize :
Solve for λ :
Choice of A : m (bias) , [Mtruth, Miter](unbiased)
the
G. D’Agostini NIM A346 (1994)306
V. Blobel eConf C MOET002 (2003)
M. Benayoun et al , to be published
R.D.Ball et al JHEP 1005 (2010)075// Nucl. Phys. B838 (2010) 136

45. An Effect of Scale Uncertainty
Experimental quantity M ?
m? or m-λA ?
Contribution to HVP should be corrected
∫K(s) M(s) = ∫K(s) m(s) - λ ∫K(s) A(s)
Correction Evaluation requires λ & A(s) → but fits provide M(s) directly

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

47. Scale Uncertainty(ies)
M. Benayoun et al EPJ C73 (2013)2453
Minimize :
Solving for λ ( ) leads to:
with :
How to choose A ?
the
G. D’Agostini NIM A346 (1994)306
V. Blobel eConf C MOET002 (2003)
R.D.Ball et al JHEP 1005 (2010)075

48. NA7 Residuals (χ2/N≈90/60)!

49. Backup Slides II

50. e+e- → K Kbar Data
K+ K-
KL KS

51. 3-pion Data

52. e+e- → π0 γ

53. e+e- → ηγ Data

54. Former : Prediction for η/η’→ ππ γ
M. Benayoun et al. ArXiv
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 ρ0 lineshape

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