A VMD Estimate of the Muon g-2 A HLS based approach

M. Benayoun LPNHE Paris 6/7

1M. Benayoun, VMD & g-2 estimates

OUTLINE

Model & Method Symmetry Breaking : BKY & (ρ,ω,φ) mixingGlobal 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-2Conclusions

M. Benayoun, VMD & g-2 estimates 2

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 ντ&

PVγ couplings & η/η’ γπ π/γγ & ……Few basic parameters : e, fπ ,Vud,Vus, 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

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

HLS and the BKY Mechanism

• Define

• Then • Standard HLS Lagrangians • BKY breaking :

Full HLS Lagrangian :BKY Breaking Matrix : Diag

/

/e

f

R

Pi

L

†/ // L R L RL R D

/ /

22{ }

4A V A V

fL Tr L R X

'FKTUYHL t oV Ho ftS AL L a LLL

PS field matrix

M. Benayoun, VMD & g-2 estimates

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

M. Benayoun et al, arXiv:1106.1315

/

2 2

4A V

fL Tr L R

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

11

1 1(1 )

RR

R

I

I R

V

V

V

hh

C. Wolfe&K. Maltman, Phys. Rev. D80 (2009)114024

1&

I R ( )

V V Vq y

F. Jegerlehner & R. Szafron EPJ C71 (2011) 1632

Isospin Breaking : Vector Field Mixing II

• At one loop, the HLS Lagrangian piece

induces s-dependent transitions among R1 fields

isospin symmetry breaking :

7

1 1 10 0

1 1 12 2R R R VR RV RK K Kz z K ��

0K Km m

M. Benayoun, VMD & g-2 estimates

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

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

8

f(s)

M. Benayoun, VMD & g-2 estimates

The Mass Matrix Eigen System At one loop:

As :

Solve perturbatively

9

12

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

2

2 21 1

1

1 222

2

2 2

2

( ) ( )

( ) ((

( ) (

)

( ) ( )

)

)

( )

) (

V

s s

s

s s

s

s

m

M s m

z sm

s

s

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

M. Benayoun, VMD & g-2 estimates

From Ideal To Physical Fields

( ) ( ) 1R s i R s i

2( )i

O

0 01

1

1

1

1

1

R

R

R

22

1

2

22

2

2

( )

( ( )2

1

)

( )( )

( )

(

2

(1 ) 2

2

(1 ) (1 2 )

( )

( )

( )

( ))

V

V V

V

V V

s

ss

s

z

z z

z

s

s

sz z

s

m

m

s

s

s

10M. Benayoun, VMD & g-2 estimates

R1 Fields

Physical Fields

Analysis Method

GLOBAL FIT to the largest possible data setWhy ? 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 estimates 11

Channels Considered• Non-Anomalous annihilations : e+e- → π π /

or decays : τ→ππ ντ (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

K K

12M. Benayoun, VMD & g-2 estimates

Channels Considered• Non-Anomalous annihilations : e+e- → π π /

or decays : τ→ππ ντ (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

K K

13M. Benayoun, VMD & g-2 estimates

V π π Couplings: I=1 part of ω and φ

2 I

iag �

*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

0(1 )(1 ) ( )

2( )V

VV

i gh s

as

�

Mixing Angles I=1 terms

M. Benayoun, VMD & g-2 estimates

ρ couplings to γ/W

• (γ/W) V transitions : = +

Full agreement between data

15

M. Benayoun, VMD & g-2 estimates

One loop corrections

( ) 2( )1 (1 )

3 3 3VV s

sf

Wf ag fh VVW

f

z

ande e

ρ couplings to γ/W

• (γ/W) V transitions : = +

Full agreement between data

16

M. Benayoun, VMD & g-2 estimates

One loop corrections

( ) 2( )1 (1 )

3 3 3VV s

sf

Wf ag fh VVW

f

z

ande e

ρ couplings to γ/W

• (γ/W) V transitions : = +

Full agreement between data

17

M. Benayoun, VMD & g-2 estimates

One loop corrections

( ) 2( )1 (1 )

3 3 3VV s

sf

Wf ag fh VVW

f

z

ande e

M. Benayoun, VMD & g-2 estimates 18

ρ couplings to γ/W

( )f

Wf

F s

Re[F(s)]

Im[F(s)]

Dipion Spectra

M. Benayoun, VMD & g-2 estimates 19

spectrum spectrume e

2χ 8688Np

2χ 130127Np

(Pr 98%)ob

e+e- Fit Residuals

M. Benayoun, e+ e- versus tau 20

e+e- Data

M. Benayoun, VMD & g-2 estimates 21

Tau Fit Residuals

M. Benayoun, VMD & g-2 estimates 22

Tau Residuals

NSK18:: Dipion Mass Spectrum

•

• s-dependent (missing) effect !

23

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

M. Benayoun, VMD & g-2 estimates

M. Benayoun, VMD & g-2 estimates 24

3-pion Data

2χ 200179Np

M. Benayoun, VMD & g-2 estimates 25

2χ 3536Np

K+ K-

KL KS

2χ 118119Np

e+e- → K Kbar Data

M. Benayoun, VMD & g-2 estimates 26

2χ 3536Np

K+ K-

KL KS2χ 118

119Np

e+e- → K Kbar Data Ratio

e+e- → π0 γ

27M. Benayoun, VMD & g-2 estimates

2χ 6790Np

e+e- → ηγ Data

28M. Benayoun, VMD & g-2 estimates

2χ 121182Np

M. Benayoun, VMD & g-2 estimates 29

Effect of Global Fitting on aµ(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 1010 aµ(ππ) χ2/dof Probability

Exp. Average 360.65 ± 2.55

All NSK +- 360.00 ± 1.64 177/208 93%

++ 359.80 ± 1.47 263/293 90%

++ (π0/η) γ 360.09 ± 1.60 437/549 99%

++ (π+ π- π0) 360.91 ± 1.45 661/727 96%

++ K Kbar 362.79 ± 1.43 858/882 71%

M. Benayoun, VMD & g-2 estimates 30

Data Set 1010 aµ(ππ) χ2/dof Probability

Exp. Average 360.65 ± 2.55

All NSK +- 360.00 ± 1.64 177/208 93%

++ 359.80 ± 1.47 263/293 90%

++ (π0/η) γ 360.09 ± 1.60 437/549 99%

++ (π+ π- π0) 360.91 ± 1.45 661/727 96%

++ K Kbar 362.79 ± 1.43 858/882 71%

Effect of Global Fit on aµ(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

M. Benayoun, VMD & g-2 estimates 31

Data Set 1010 aµ(ππ) χ2/dof Probability

Exp. Average 360.65 ± 2.55

All NSK +- 360.00 ± 1.64 177/208 93%

++ 359.80 ± 1.47 263/293 90%

++ (π0/η) γ 360.09 ± 1.60 437/549 99%

++ (π+ π- π0) 360.91 ± 1.45 661/727 96%

++ K Kbar 362.79 ± 1.43 858/882 71%

Effect of Global Fit on aµ(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

Contributions to g-2 up to ()

32M. Benayoun, VMD & g-2 estimates

Channel Solution B Solution A Exp. Average

+- 498.54 ± 1.97 497.98 ± 1.76 498.53 ± 3.73

π0 γ 4.64 ± 0.04 4.28 ± 0.04 3.35 ± 0.11

η γ 0.65 ± 0.01 0.67 ± 0.01 0.48 ± 0.01

η' γ 0.01 ± 0.00 0.01 ± 0.00 ---

π+ π- π0 42.03 ± 0.60 40.88 ± 0.52 43.24 ± 1.47

KL KS 12.02 ± 0.09 12.07 ± 0.08 12.31 ± 0.33

K+K- 16.87 ± 0.20 16.93 ± 0.18 17.88 ± 0.54

Total up to 1.05 GeV

574.76 ± 2.10 572.82 ± 1.90 575.79 ± 4.06

Excl. ISR

Uncertainties uniformly improved by a factor 2

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 estimates 33

VMD estimate of LO HVP

• LO HVP + HO HVP + LBL +QED +EW giveaµ= ( 11 659 175.37 ± 5.31) 10-10 (B)

Oraµ= ( 11 659 173.43 ± 5.25) 10-10 (A)

M. Benayoun, VMD & g-2 estimates 34

VMD estimate of g-2

M. Benayoun, VMD & g-2 estimates 35

VMD estimate of g-2

Conclusions

1/ HLS with BKY and Vector Meson Mixing provides a good simultaneous fit of

e+e- → → π π /K+K-/ KLKS/ π γ/ηγ/π π π

2/ A simultaneous account ofe+e- → → π π & τ→ππ ντ

3/ VMD physics correlations improve g-2 (Uncertainty : HVP up to uniformly improved by 2)

36M. Benayoun, VMD & g-2 estimates

Backup Slides

37M. Benayoun, VMD & g-2 estimates

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 ντ&

PVγ couplings & η/η’ γπ π/γγ & ……Few basic parameters : e, fπ ,Vud,Vus, 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

VMD :: The HLS Framework

• Hidden Local Symmetry Model (HLS) as a whole

LA, LV, LYM , Lanomalous

(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) 217M. 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) 114027A. 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

The Hidden Local Symmetry Model :The Non-Anomalous Sector

• Define

• Define covariant derivatives

• Then and

• The Full HLS Lagrangian :

40

/

/e

f

R

Pi

L

,L RD D †

/ // L R L RL R D /

22

4A V

fL Tr L R

HLS A YMVL LL a L Expanded form: M.Benayoun & H.O’Connell PR D 58 (1998) 074006

/ / // /L R L R L R L R L RV GD igi Universal Vector Coupling

PS field matrix

M. Benayoun, VMD & g-2 estimates

The Covariant Derivatives

• Covariant derivatives ≠ for left- right-ξ fields :

• With :

41

/ / // /L R L R L R L R L RV GD igi

2

2LG A WQ T

gTe W

,RG eQA

T± is CKM matrix reduced to Vus and Vud termsM. Benayoun, VMD & g-2 estimates

Anomalous Annihilations/Decays

• Non-Anomalous annihilations : e+e- → π π / or decays : τ→ππ ντ

• Anomalous Processes : e+e- → π γ/ηγ/π π π :: Other Effective Lagrangian pieces

K K

3

1 2 3

4 1 4

2

2

2

2

3

2( )

4

( )4

( ) (3

1 )4

( )1 ( )

V P P

PPP

PPP

V AV

V

AA

c

P A

c

c

c c c

c c c

NL Tr L

NL i Tr

L

gV V g

gV

e eL

P Pf

c

f

P

42M. Benayoun, VMD & g-2 estimates

SIDE RESULT : Prediction for η/η’→ ππ γ

Г(η’→ ππ γ)=53.11±1.47PDG : 60±5 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

43M. Benayoun, VMD & g-2 estimates

BELLE/ALEPH/CLEO

44M. Benayoun, VMD & g-2 estimates