Hadronic corrections to muon anomaly within the instanton vacuum model

A.E. Dorokhov (JINR, Dubna)

I. Introduction

Data from BNL (g-2) Collaboration (+/-0.5 ppm)

SM prediction from QED, EW and Strong sectors (2.7s)

II. Leading order Hadronic contribution: phenomenologye+e- - annihilation vs – decay (shape discrepancy,

as(MZ))

III. LO and NLO Hadronic contribution from theory (main source of theoretical error)

Conclusions

SND 05’

2

2Sg

a

Muon Anomaly (Current status)

Magnetic AnomalyQED HadronicWeak SUSY... ... or other new

physics ?

QED 10

1

11614098.1 41321.810

3014.2 38.1 0.6

n

n

a

EW 10one two loops 15.4 3 10a

Had 10700 10 10a

The hadronic contribution to the muon anomaly, where the dominant contribution comes from (a). The hadronic light-by-light contribution is shown in (e)

Z*interference

Hadronic polarization Light-by-light process

Nonlocal Chiral Quark model SU(2) nonlocal chirally invariant action describing the interaction of soft quarks

4231

4

1

44

54

,,,,2

xXXQXxXQxXXQXxXQxfxdXdG

xqmxAxVixqxdS

iin

nini

Spin-flavor structure of the interaction is given by matrix products ii

,,),(

)(),('),11(

5555,

555555

1

faa

AV

Taaaa

GGG

iiGiiGiiG

Instanton interaction: G'=-G

For gauge invariance with respect to external fields V and A the delocalized quark fields are defined (with straight line path)

yqzAzVdziTPyxQy

x

aaa

5exp,

Quark and Meson Propagators

1,ˆ1 pZpMppS

kMk

kMkf

kdpfNGNmpM cf 22

24

42

24

22

2

at exp pp

pM

The dressed quark propagator is defined as

The Gap equation

has solution pfmMmpM q2

0.35

7.14 106

M p2

30 p

qq

qGq 2Mconst.

Mcurr.

From existence of derivativesof the quark condensate

pMp

pMp

pdNqDq n

Cn

222

4

42

400

222 at )/exp( pppM

aaa T

kk

kMkMkkTqkkqk

22'

''',,

Conserved Vector and Axial-Vector currents.

The Vector vertex

Nonlocal part

s in pQCD

AF

2q

The Iso-Triplet Axial-Vector vertex has a pole at 02 q

22

2

2

2

255

'

''

1'2',,

kk

kfkfkk

qJG

qJmGMkfkf

q

qTqkkqk

PPP

PPaa

Pion pole

The Iso-Singlet Axial-Vector vertex has a pole at '22mq

522

2

2

2

250

'

''

'1

1'2

'',,

kk

kMkMkk

qJG

qGJ

q

qkMkM

G

Gqkkqk

PP

PP

1-G’JPP(q2)

’ meson pole

III. LO Hadronic contribution to gIs expressed via Adler function as

1

0

22

2hvp)2(

1

2/11

3

8 m

x

xD

x

xxdxa V

, 4 , 2 2V ab iqx V ab ab VTq i d xe x q q g q Q

t

Qt

Qdt

dQ

QdQQD V

VV

022

2

2

222

Adler function is defined as

, 0 0 0J ab a bx T J x J

NQM Adler function and ALEPH data

NJL

ALEPHILMAS

Quark loop

Quark loop

Mesons

Meson loop

M(p)

pQCD(NNNLO)

QCD

The instanton model estimates

,,,10076.0932,6

,,10077.0849,68

8hvp)2(

eee

eeea

and from phenomenology one gets

,1077.0,1013.0,1033.5

,105.023,68hvp)2(

Mloop,8hvp)2(

mesons-,8hvp)2(

Qloop,

8hvp)2(QMN,

aaa

a

LO contribution to a

1q

2q

3q

4q

OPE

22

24

23

21 ,0

qqq

q

1q

2q

LBL, OPE and Triangle diagram VAV

3q

4q

AX

XX

...ˆˆ2

, 52444

21

43

21

43 zjqq

izedyjxjTyedxdi zqqiyiqxiq

(Similar to amplitude at large photon virtualities)

The structure of V*AV amplitudeFor specific kinematics 0h photon wit real is ,arbitrary - 112 qqqq

Only 2 structures exists in the triangle amplitude

21 2 2 1 2 1 2 2 1 2 2 1 2, ,T LT q q w q q q q q q q q w q q q

The amplitude is transversal with respect to vector current

021

TqTq

but longitudinal with respect to axial-vector current

21

222 qqqwTq L Thus Adler-Bell-Jackiw anomaly

Operator Product Expansion and V*AV amplitude

In local theory for massive quarks one gets

1

0221

1

3

22

mqxx

xxdx

Nww C

TL

Which in chiral limit (m=0) becomes

2

1

3

22

q

Nww C

TL

Perturbative nonrenormalization of wL (Adler-Bardeen theorem, 1969)Nonperturbative nonrenormalization of wL (‘t Hooft duality condition, 1980)Perturbative nonrenormalization of wT (Vainshtein theorem, 2003)Nonperturbative corrections to wT at large q are O(1/q6) (De Rafael et.al., 2002)Absence of Power corrections to wT at large q in Instanton model (this work)

NnLO QCD =0 for all n>0

Anomalous wL structure (NonSinglet)

Diagram with Local vertices

5

X

Diagram with NonLocal Axial vertices

X

+ rest

5

2

3 1

3

2

q

Nw C

L

In accordance with Anomalyand ‘t Hooft duality principle(massless Pion states in triplet)

+

Anomalous wL structure (Singlet)

Diagram with Local vertices

5

X

Diagram with NonLocal Axial vertices

X

+ rest

5

0)(0

2022

qL qwq

In accordance with Anomalyand ‘t Hooft duality principle(no massless states in singlet)

wLT in the Instanton Model (NonSinglet)

2

0

2

23

GeV 6.42

q

LT

q

qw

TLLT www 2

Perturbative nonrenormalization of wL (Adler-Bardeen theorem, 1969)Nonperturbative nonrenormalization of wL (‘t Hooft duality condition, 1980)Perturbative nonrenormalization of wT (Vainshtein theorem, 2003)Nonperturbative corrections to wT at large q O(1/q6) (De Rafael et.al., 2002)Absence of Power corrections at large q in Instanton model (this work)

0)3( LTw In perturbative QCD

wLT in the Instanton Model (Singlet)

220 GeV 6.00 qw LT

TLLT www 2

Z*contribution to a

TZ

ZT

Z

ZL

EW

wkm

mw

km

mw

mk

kp

kpk

kdimGa

22

2

22

2

22

2

24

42

21

3

1

2

1

222

VMD + OPE(Czarnecki, Marciano, Vainshtein, 2003)

11EW 1002.2 a

Instanton model:(Our work, 2005)

11EW 1048.1 a

Perturbative QCD(Anomaly cancelation)

0EW a

Light-by-Light contribution to muon AMM

Vector Meson Dominance like model:(Knecht, Nyffeler 2002)

80LbL)3( 10058.0 a

VMD + OPE(Melnikov, Vainshtein 2003)

8LbL)3( 1007.0136.0PVPS, a

• This contribution must be determined by calculation.

• the knowledge of this contribution limits knowledge of theoretical value.

Instanton model:(Dorokhov 2005)

8LbL)3( 1001.0105.0, PVPSa

Conclusions• There are plans to increase accuracy of (g-2) experiment by factor 2

(BNL) or even 10 (Japan)• The standard model calculations has to be at the same level of

accuracy• The weakest part of calculations is strong sector• At the moment effective (Instanton) model provide the approach

which competes with other (Lattice QCD, etc)• The accuracy of calculations is of order 10-20% for NLO corrections• The longitudinal structure of triangle diagram is norenomalized by

nonperturbative corrections in agreement with ‘t Hooft arguments• Transverse structure is calculated for arbitrary q.• At large q the transversal amplitude has exponentially decreasing

corrections, that reflects nonlocal structure of QCD vacuum in terms of instantons

• Instanton model is a way to extrapolate the results of OPE and PT to the regions not achievable by these methods.

2

2had

2

4

( )

( )3

m

a R sK

ss

ds

”Dispersion relation“, uses

unitarity (optical theorem)

and analyticity

(Bouchiat and Michel, 1961)

had

had

Still it is impossible to calculate LO contribution from Theory with required accuracy

II. Leading Order Hadronic contributions

ee

eesR

hadrons

(0)

Better agreement between exclusive and inclusive (2) data

Agreement between Data (BES) and pQCD (within correlated systematic errors)

use QCD

use data

use QCD

Evaluating the Dispersion Integral

About 91% of a comes from s<(1.8GeV)2,

73% is covered by final 2 state

The ratio of the e+e− → +− cross section.

The shaded area shows the systematic error of the SND measurements.

e+e− → p+p− cross section in the SND experiment at theVEPP-2M collider (June,2005)measured at 400 < √s < 1000 MeV

New SND agrees well with CMD2 and -data.KLOE is in conflict with SND and CMD2.

SND Data on R(s)

2 contribution to ahadr

• SND has evaluated the Dispersions Integral for the 2-Pion-Channel in the Energy Range 0.39 <s<0.97 GeV2

a= (488.7 2.6stat 6.6syst ) 10-10

• Comparison with CMD-2 and KLOE in the Energy Range 0.37 <s<0.93 GeV2

(375.6 0.8stat 4.9syst+theo) 10-10

(378.6 2.7stat 2.3syst+theo) 10-10

KLOECMD2

10SND ( ) 10385.6 5 (2005.2 )

(2004)

(2003)

LO vs NLO Hadron corrections to Muon Anomalous magnetic moment (Theory)

1% from phenomenology,10% from the model

NO phenomenology,50% from the existing model,The aim to get 10% accuracy

Lowest Order(EM and Tau data;Adler function)

Higher Order(OPE and Triangle diagram)

Vacuum Polarization Light-by-Light scattering

The hadronic light-by-light contribution is likelyto provide the ultimate limit to the precision of the standard-model value of aμ.

Triangle diagram

Vector and Axial-Vector correlators.V and A correlators are fundamental quantities of the

strong-interaction physics, sensitive to small- and large-distance dynamics. In the limit of exact isospin symmetry they are

000,

2

22,4,

22,4,

baabJ

AL

ab

AT

ababAiqxabA

VT

ababViqxabV

JxJTx

Qqq

Qqgqqxxediq

Qqgqqxxediq

where the QCD currents are

,qTqJ aa ,5

5 qTqJ aa

bJaJ

Current-current correlators

Current-current correlators are sum of dispersive and contact terms

kSkQQkTrkd

MQS

kSkQkkSkQkTrkd

QK

QSQKQQ

Jq

J

JJJ

JJJ

',,,2

2

,,,''',,2

,

4

42

4

42

2222

The transverse and longitudinal part of the correlators are extracted by projectors

22 ,3

1

q

qqP

q

qqgP LT

~

Dispersive term

I

Contact term

Current-current correlators in ILMV correlator

kMkkMkD

kMkdN

kMMMkkkMkkkkMMDD

kdNQQ

C

CV

)2(24

4

2)1(2)1(224

422

3

4'

24

3

4

3

21

22

and the difference of the V and A correlators

22)1()1(2

4

422

3

41

24 kMkMkMkMMMkMM

DD

kdNQQ C

AV

One may explicitly verify that the Witten inequality is fulfilled and that at Q2=0 one gets the results consistent with the first Weinberg sum rule

0,0 2,222 QfQQ AVL

AVT

Above we used definitions

kMkk

MM

kkkM

kk

MMkM '

1, 2222

)2(22

)1(

Other model approaches

eeea ,,10077.0849,6 8hvp)2( Phenomenological estimate:

NQM: 8hvp)2(QMN, 103.053,6 a

Extended Nambu-Iona-Lasinio:(Bijnens, de Rafael, Zheng)

8hvp)2(ENJL, 105.7 a

Minimal hadronic approximation(Local duality):(Peris, Perrottet, de Rafael)

8hvp)2(MHA, 107.17.4 a

Lattice simulations:(Blum;Goeckler et.al. QCDSF Coll.)

8hvp)2(Lattice, 1023.046.4 a

VAV* correlator

kSqkkqkSqkqkqkSkqkTrkd

NqqT VAVC 2221

5114

4

21 ,,,2

2,

It is transverse with respect to vector currents

5 V

Triangle diagram

0,, 212211 qqTqqqTq

V

~

X

wLT structure

kMMkqkMMk

q

kqkMMM

DD

Mkd

q

Nqw C

LT

)1()1(2

22

22

4

22

')(23

4

)(2'2

3

4

Above we used definitions

,

, , ,

22)1(

kk

MMkM

kMMqkkkk

TLLT www 2 0LTPTw

22 qfqwLT Exponentially suppressed, no power corrections!

InstantonVector Meson

Dominance

Inclusive v-a spectral function, measured by the ALEPH collaboration

ALEPH data on decays

Inclusive v+a spectral function, measured by the ALEPH collaboration

pQCD

pQCD

a1 a1