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) mixingGlobal Fit Scope: V P γ / P γ γ / t vs e+ e-& [e+e-

→ π π /K+K-/KLKS /π γ /η γ /π π π] HLS Estimates of (g-2)μ

Global fits with only scan (NSK) data : e+e- vs τ «t + PDG » Predictions of e+e- → π π vs dataGlobal fit with t, scan and ISR π π data → g-2Conclusions 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 :

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 :

1 1 1 10

1 102 2VR R R R R V RK K Kz Kz

� �

0K Km m

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

12

2( ) (, ) )(m s s s

2

2 21 1

1

1 222

2

2 2

2

( ) ( )( ) ((

( ) ()

( ) ( )

))( )

) (

V

s ss

s sss

mM s m

z sms

s

2 2 20( ) ( ) ( )M s M s M s

12 ( ), ( )ss

6

From Ideal To Physical Fields

( ) ( ) 1R s i R s i 0 01

1

1

11

1

R

R

R

2 22

2 21

2( )1

2( )

( )( )2

2

( )( )

()

)(

V

HK HK

H H

V

K K

s

HK HKm ms

z

m m

m

z

ss

s

s

s

m

s

R1 Fields

Physical Fields

At leading order in ε1(s) , ε2(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 ντ&

PVγ, Pγγ decays & η/η’ γπ π/γγ & ……BHLS :: (almost) an empty shell : [αem, GF , fπ , Vud , Vus , mπ’s, mK’s, mη, , mη’] Present Limits :Up to the ≈ φ mass region ( 1.05 GeV) No scalars mesons, no ρ’, no ρ’’ ……

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 ντ&

PVγ, Pγγ decays & η/η’ γπ π/γγ & ……BHLS :: (almost) an empty shell : [αem, GF , fπ , Vud , Vus ,mπ’s, mK’s, mη, , mη’] Present Limits :Up to the ≈ φ mass region ( 1.05 GeV) No scalars mesons, no ρ’, no ρ’’ ……

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 ντ&

PVγ, Pγγ decays & η/η’ γπ π/γγ & ……BHLS :: (almost) an empty shell : [αem, GF , fπ , Vud , Vus , mπ’s, mK’s, mη , mη’] Present Limits :Up to the ≈ φ mass region ( 1.05 GeV) No scalars mesons, no ρ’, no ρ’’ ……

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) :

3( ) ( ,) )1 (4

cut

th

is

i

s

a ds KH e e H ss

Measured Xsection

Measured Xsection Model Xsection (φ)

11

HLS→Correlated Physics Channels

• Non-Anomalous annihilations : e+e- → π+π- / • Decay spectra : τ ± → π±π0 ντ (later also : η/η’→π+π-γ)• Anomalous Processes : e+e- → π0 γ/ηγ/ π+π- π0

• Radiative decays widths ( VPγ, π0/η/η’ → γ γ)

• φ→ππ (Br ratio and phase)

HLS MODEL & Breaking Schemes:

OVERCONSTRAINED & **DETERMINED** by more than 40 data sets

/K K K KL S

12

HLS→Correlated Physics Channels

• Non-Anomalous annihilations : e+e- → π+π- / • Decay spectra : τ ± → π±π0 ντ (later also : η/η’→π+π-γ)• Anomalous Processes : e+e- → π0 γ/ηγ/ π+π- π0

• Radiative decays widths ( VPγ, π0/η/η’ → γ γ)

• φ→ππ (Br ratio and phase)

New Paradigm : Any information on any channel (π0 γ) is equivalent to improving the statistics on

any other channel (π+π- )

/K K K KL S

13

HLS→Correlated Physics Channels

• Non-Anomalous annihilations : e+e- → π+π- / • Decay spectra : τ ± → π±π0 ντ (later also : η/η’→π+π-γ)• Anomalous Processes : e+e- → π0 γ/ηγ/ π+π- π0

• Radiative decays widths ( VPγ, π0/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

/K K K KL S

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 forπ+π-/ / /π γ /η γ /η’ γ /π π π up to 1.05 GeV

/ L SK K K K

15

Data Set (#data points) Configuration B ( χ2)

Decays (9) 10.4π+π- New Timelike (127) 128.3π+π- Old Timelike (82) 50.2 π0γ (86) 66.9 ηγ (182) 121.0

π+π- π0 112.5/99K+K- (36) 30.3KLKS (119) 122.7τ ALEPH (37) 25.1τ CLEO (29) 38.2τ BELLE (19) 25.6χ2/dofProbability

731.2/801 96.3%

Global Fit Quality with NSK only

Fits : see backup slides

16

Dipion Spectra (τ & e+e-): A puzzle?

spectrum t spectrume e

2χ 88.988Np

2χ 128

127Np(Pr 96%)ob

e+e- vs τ : CMD2/SND Fit Residuals

ΔF

ΔF/F

2χ 128.3127Np

Global fit : 96% Prob

18

e+e- vs τ : Tau Fit Residuals

ΔF

ΔF/F

2χ 88.988Np

19

A « e+e- - τ» Puzzle ?

•

• s-dependent (missing) effect !

M. Davier NP Proc. Supp. 169 (2007) 288

20

A « e+e- - τ» Puzzle ?

• e+e- and τ dipion residuals : simultaneously flat

• |Fπ(e+e-)|2 & |Fπ(τ)|2 yield :simultaneously χ2

π+π- /Nπ+π- & χ2π±π0 /Nπ ± 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 Channel Solution B Direct Estimate

+- 495.40 ± 1.92 498.53 ± 3.73(497.72 ± 2.12)

π0 γ 4.61 ± 0.04 3.35 ± 0.11

η γ 0.64 ± 0.01 0.48 ± 0.01

η' γ 0.01 ± 0.00 ---

π+ π- π0 41.16 ± 0.59 43.24 ± 1.47

KL KS 11.90 ± 0.08 12.31 ± 0.33

K+K- 17.59 ± 0.21 17.88 ± 0.54

Total up to 1.05 GeV

571.30 ± 2.02 575.79 ± 4.06

22

HVP Results with scan & τ data

• (Updated) All uncertainties improved by ≈ 2 Channel Solution B Direct Estimate

+- 495.40 ± 1.92 498.53 ± 3.73(497.72 ± 2.12)

π0 γ 4.61 ± 0.04 3.35 ± 0.11

η γ 0.64 ± 0.01 0.48 ± 0.01

η' γ 0.01 ± 0.00 ---

π+ π- π0 41.16 ± 0.59 43.24 ± 1.47

KL KS 11.90 ± 0.08 12.31 ± 0.33

K+K- 17.59 ± 0.21 17.88 ± 0.54

Total up to 1.05 GeV

571.30 ± 2.02 575.79 ± 4.06

Scan +ISR

23

HVP Results with scan & τ data

• (Updated) Central value shifted by ≈ 3 10-10

Channel Solution B Direct Estimate

+- 495.40 ± 1.92 498.53 ± 3.73(497.72 ± 2.12)

π0 γ 4.61 ± 0.04 3.35 ± 0.11

η γ 0.64 ± 0.01 0.48 ± 0.01

η' γ 0.01 ± 0.00 ---

π+ π- π0 41.16 ± 0.59 43.24 ± 1.47

KL KS 11.90 ± 0.08 12.31 ± 0.33

K+K- 17.59 ± 0.21 17.88 ± 0.54

Total up to 1.05 GeV

571.30 ± 2.02 575.79 ± 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 ντ , e+e-→ /π0 γ/ηγ/π+ π- π0

& VPγ & Pγ γ couplings (from PDG)K K

's

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 ντ , e+e-→ /π0 γ/ηγ/π+ π- π0

& VPγ & Pγ γ couplings (from PDG)K K

26

!!τ + PDG Predictions

Good agreement data versus predictions

Spacelike region well predicted

&Extrapolates much above 1

GeV!!!

27

τ + PDG Predictions

KLOE’s almost perfect!CMD2&SND ≈ OK

BaBar too large below ω

0.85 GeV >mππ> 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 RegionFrom each data set (NSK,KLOE’s,BaBar)

Guesses About Global Fits

29

τ + {BaBar (ρ―ω)} Predictions

0.85 GeV >mππ> 0.70GeV

PION Form Factor

FIT Prediction

Prediction

30

τ + {(ρ―ω) Region from each Exp} Fits

0.85 GeV >mππ> 0.70GeV

PION Form Factor

Fits in «isolation»NSK

KLOE10KLOE 08

BaBar

31

ρ-ω region (0.76→0.82 GeV)/φ(PDG)

Fω = Br(ω→π+π-)*Br(ω→e+e-)DATA SAMPLE Fω X 10-6 Orsay Phase

(degrees)Global Fit Probability

X2/nπ+π-

PDG 1.23±0.07 [104.7±4.1] ------ ---

CMD2+SND (≈ 0σ) 1.22±0.04 106.7±0.3 90% 51/47

KLOE08 (1.9 σ) 1.08±0.04 110.4±1.2 88% 18/11

KLOE10 (3.0 σ) 0.97±0.05 113.6±1.6 93% 11/11

KLOE12 (3.1 σ) 0.92 ±0.07 122.5±2.1 96% 7/11

BaBar (7.7 σ) 1.78±0.01 107.5±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 ντ , e+e-→ K+K-/ K0 K0b/π γ/ηγ/π π π)

KLOE raw & corrected Residuals ΔF

2χ 96.660Np

2χ 63.660Np

2χ 73.775Np

Fits in isolation : Comments

• Good Residuals & Gl. fit Prob. OK & good χ2 : CMD2 & SND & KLOE10 & KLOE12

• (Very) Good Residuals & poor χ2 (1.62) :KLOE08 (correlations in systematics)

• Bad Gl. fit Prob . & at [ρ/ω] peak χ2/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 ντ , e+e-→ K+K-/ K0 K0b/π γ/ηγ/π π π)

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 Channel NSK +KLOE 10&12

+ τ (ABC < 1 GeV) scan only(NSK) + τ (ABC < 1GeV)

Direct Estimate

+- 491.12 ± 1.35 495.40 ± 1.92 498.53 ± 3.73(497.72 ± 2.12)

π0 γ 4.63 ± 0.04 4.61 ± 0.04 3.35 ± 0.11

η γ 0.64 ± 0.01 0.64 ± 0.01 0.48 ± 0.01

η' γ 0.003 ± 0.000 0.003 ± 0.000 ---

π+ π- π0 40.78 ± 0.64 41.16 ± 0.59 43.24 ± 1.47

KL KS 11.94 ± 0.08 11.90 ± 0.08 12.31 ± 0.33

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 ± 2.02 575.79 ± 4.06

Scan +ISR

38

HVP Results up to 1.05 GeV Channel NSK+KLOE10&12

+ τ (ABC < 1 GeV) scan only(NSK) + τ (ABC < 1 GeV)

Direct Estimate

+- 491.12 ± 1.35 495.40 ± 1.92 498.53 ± 3.73(497.72 ± 2.12)

π0 γ 4.63 ± 0.04 4.61 ± 0.04 3.35 ± 0.11

η γ 0.64 ± 0.01 0.64 ± 0.01 0.48 ± 0.01

η' γ 0.003 ± 0.000 0.003 ± 0.000 ---

π+ π- π0 40.78 ± 0.64 41.16 ± 0.59 43.24 ± 1.47

KL KS 11.94 ± 0.08 11.90 ± 0.08 12.31 ± 0.33

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 ± 2.02 575.79 ± 4.06(574.98±2.66)

Scan +ISR

Scan +ISR

39

g-2 HLS Estimates & Others

40

g-2 HLS Estimates & Others

41

g-2 HLS Estimates & Others

aμ(ππ) over mππ=[0.63,0.958] GeV

Model Estimate in Accord with expectations

Uncertainties improved by ≈45%

43

Accord with Exp. expectations

aμ(ππ) over mππ=[0.63,0.958] GeV

≈ No Bias!

44

aμ(ππ) over mππ=[0.63,0.958] GeV

No Mismatch!

45

The Spacelike & threshold Regions

Low mass tail OK!

2

1.00HK

HK

mm

Extrapolation to s<0 perfect

46

Predicted Phase shift (I)

2

1.00HK

HK

mm

47

Predicted Phase shift (II)

2

1.05HK

HK

mm

48

Threshold Behavior

( )2

( )

( )12

sHK HKsm

s

m

2

2 1HK

m HK

m

m

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

51

Residuals : e+e- vs τ

Residuals simultaneously flat

52

Global Fit Residuals : e+e- vs τ

Residuals simultaneously flat

53

From Belle

54

g-2 Estimates & Discrepancy II

55

Dipion Mass Spectrum

•

• s-dependent (missing) effect !

M. Davier NP Proc. Supp. 169 (2007) 288

56

Backup Slides II

57

• Define

• Basic pieces :

• The Hidden Local Symmetry (HLS) Lagrangian expanded in

The HLS Model

†/ // ( )L R L RL R D /

/e

f

R

Pi

L

PS field matrix

/

2 2

4A V

fL Tr L R

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

/ /

22{ }

4A V A V

fL Tr L R X

'FKTUYHL t oV Ho ftS AL L a LLL

/A VX , , // / A VA V A Vq y zM. Hashimoto, Phys. Rev. D54 (1996) 5611

/

22

4A V

fL Tr L R

M.Benayoun et al. EPJ C72 (2012) 1848

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 :

1 1 1 10

1 102 2VR R R R R V RK K Kz Kz

� �

0K Km m

VVP Lagrangian : K*K loops // Yang-Mills term : K*K*loops

→

→

60

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)

( )s

( )s

( )s

f(s)

61

The Mass Matrix Eigen System At one loop:

As :

Solve perturbatively

12

2( ) (, ) )(m s s s

2

2 21 1

1

1 222

2

2 2

2

( ) ( )( ) ((

( ) ()

( ) ( )

))( )

) (

V

s ss

s sss

mM s m

z sms

s

2 2 20( ) ( ) ( )M s M s M s

62

• QqqData Set (#data points) Configuration A ( χ2) Configuration B ( χ2)

Decays (9) 12.9 10.4

New Timelike (127) 127.7 128.3

Old Timelike (82) 50.5 50.2

π0γ (86) 67.4 66.9

ηγ (182) 121.4 121.0

π+π- π0 229.3/179 112.5/99

K+K- (36) 35.8 30.3

KLKS (119) 121.1 122.7

τ ALEPH (37) 24.7 25.1

τ CLEO (29) 38.7 38.2

τ BELLE (19) 25.9 25.6

Χ2/dofProbability

855.3/881 72.7%

731.2/801 96.3%

Global Fit Results with NSK

63

Dipion Spectra (τ & NSK)

spectrum t spectrume e

2χ 88.988Np

2χ 128127Np

(Pr 96%)ob

64

2χ 3036Np

K+ K-

KL KS

2χ 123119Np

e+e- → K Kbar Data

65

3-pion Data

2χ 229179Np

66

e+e- → π0 γ

2χ 6786Np

67

e+e- → ηγ Data

2χ 121182Np

68

Former : Prediction for η/η’→ ππ γ

Г(η’→ ππ γ)=53.11±1.47PDG : 58.3±2.8 keV

Г(η→ ππ γ)=55.82±0.83PDG : 60±4 eV

Lineshapes and yields : tests for g value and ρ0 lineshape

M. Benayoun et al. ArXiv 0907.4047

69

Former : Prediction for η/η’→ ππ γ

Fit in isolation : KLOE 08 Residuals

ΔF

ΔF/F

2χ 96.660Np

Global fit : 56% Prob

Fit in isolation : KLOE 10 Residuals

ΔF

ΔF/F

2χ 73.775Np

Global Fit : 88% Prob

Fit in isolation : KLOE 12 Residuals

ΔF

ΔF/F

2χ 63.660Np

Global fit : 92% Prob

73

Fit in isolation : BaBar Residuals

ΔF

ΔF/F

2χ 341275Np

Global Fit : 22% Prob

Fit in isolation : CMD2/SND Residuals

ΔF

ΔF/F

2χ 128.3127Np

Global fit : 96% Prob

75

Tau Fit Residuals (NSK only)

ΔF

ΔF/F

76

NA7 Residuals (χ2/N≈90/60)!

77

Scale Uncertainty

• Minimize :

• Solve for λ (A= m or M) gives:

• And :

2 1 2 2/Tm M A V m M A

2 12 T TAm M V m MA

12

1 1/T TA V m M A V A

78

V π π Couplings: I=1 part of ω and φ

2 Iiag

�

*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

0(1 ) (1 ) ( )2

( )VVV

i gh sa s

�

Mixing Angles I=1 terms

79

Vector meson couplings to γ

• (γ) V transitions : = +

( ) ( )33

22 13

VVh VVsagf sf z

3 1 22 13

( ) ( )VVf s sVa f hg Vz

( )22 ( )3

s sf ag Vf z

80

ρ couplings to γ/W

• (γ/W) V transitions : = +

Full agreement between data

( ) 2 ( )1 (1 )3 3 3

VV s sf

Wf ag fh VVWf

z

ande e t

81

ρ couplings to γ/W

( )f

WfF s

Re[F(s)]

Im[F(s)]

82

From Belle