Vector and axial-vector current correlators

within the instanton model of QCD vacuum.

A.E. Dorokhov (JINR, Dubna)

V and A current-current corelators

OPE vs QCD

V-A and V correlator and ALEPH data

Hadronic contribution to muon AMM in Instanton model

Topological susceptibility

Conclusions

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

The two-point correlation functions obey (suitably subtracted) dispersion relations

sQs

dsQ JJ

Im1

02

2

where the imaginary parts of the correlators determine the spectral functions

ALEPH and OPAL measured V and A spectral functions separately and with high

precision from the hadronic -lepton decays

AVJiss TJJ ,,Im4

20, mshadrons

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

ALEPH data

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

pQCD

pQCD

a1 a1

Vector Adler Function (pQCD and QCD).

t

Qt

Qdt

dQ

QdQQD V

VV

022

2

2

222

4

32

2

22

22

02

2

1023

22

2

2

02

2

222 ln

3ln2ln1

4

1; s

sss

pQCDV OQQ

FFQ

FQD

,005.02.1637.6,115.0986.1

,3

38102

16

1,

3

211

4

1

232

10

fff

ff

NNFNF

NN

Adler function is defined as

At short distances pQCD predicts (MSbar)

where

and

2

2

20

12

020

12

0

lnlnQCDss

Q

QQ

At large distances pQCD predicts only

...0'00 22 QDQD VV

D(Q)

?

QCD

Constituent Quarks Current Quarks

pQCD

42 /QG

62

/Qqq

pQCD and QCD predictions for Adler function

86

6

4

2

2

2

322 1

64 QO

Q

O

Q

G

QQDQD D

a

sspQCDVV

Adler Function and ALEPH dataTake an ansatz for the spectral function

32

2

00

48

121

4

1

,

ttDt

stttstt

spQCDV

pQCDJ

pQCDJ

ALEPHJJ

and find the continuum threshold s0 from duality condition

00

00

spQCDJ

sALEPHJ tt

Using the experimental input corresponding to the --decay data and the perturbative expression

0

2214

43

0201023

2

002

02

0

0

0

t

Os

FFs

Fss

tdt

pQCDAV

ssss

spQCDV

where

One find matching from duality condition

pQCD

ALEPHpQCD

ALEPH

00

00

spQCDJ

sALEPHJ tt

Adler function and ALEPH data

24

1

Asymptot Freedom

(N)NLOLO

8)2( 102.7 hvpa

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

4231

4

1

44

54

,,,,2

xXXQXxXQxXXQXxXQxfxdXdG

xqxAxVixqxdS

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

The dressed quark propagator is defined as

The Gap equation

kMk

kMkf

kdpfNGNpM cf 22

24

42

24

has solution pfMpM q2

0.35

7.14 106

M p2

30 p

qq

qGq 2Mconst.

Mcurr.

22

22 22 1,01det

m

m

mq

SMqqmq dq

dJgqGJ

'', 5 kfkfigkk aqq

a

f

Mg q

qq

qq scattering matrix

221ˆˆˆ

qm

qVqV

qGJ

GqTqTqGJGqT

M

ba

q

abPP qkSkSTrqkMkM

kd

M

iqJ

554

4

22

2 with polarization operator

has polesat posiitons of mesonic bound states

The pion vertex

with the quark-pion constant gqq satisfying the Goldberger-Treiman realtion

aaa T

kk

kMkMkkTqkkqk

22'

''',,

Conserved Vector and Axial-Vector currents.

The Vector vertex

NonLocal part

s in pQCD

AF

kSTTkSkqkq Faa

Fa 11 '',, WTI

2q

The iso-triplet Axial-Vector vertex has a pole at 02 q

22

2

2

22

255

'

'''

'',,

kk

kMkM

q

kkqkk

q

kMkMqTqkkqk aa

2q

Pion pole

511

55 '',, kSTTkSkqkq F

aaF

a AWTI

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’

G

G

qJG

kMkMkSkSkqkq

PPFF

'1

'1

'2'',,

25511

550

anomalousAWTI

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

Model parameters and Local matrix elements

Profiles for dynamical quark mass in the Insatanton model

2/

2

11002

kt

qInst tKtItKtIdt

dtMkM

and for the Constrained Instanton it is approximated by Gassian form

222 /2exp kMkM qCI

Parameters of the profile are fixed by the pion weak decay constant

02

222

22 ''

4 uD

uMuuMuuMuMduu

Nf c

and the quark condensate

2

02

where,4

uMuuDuD

uMudu

Nqq c

With the model parameters fixed as

,GeV03.0G,GeV78.1G,GeV96.1G,GeV4.27G

GeV,0.3GeV,1.1GeV,24.02-1

A2-

V2-

V2-

P

a

VPqM

one obtains 3MeV224limit), (ChiralMeV86 qqf

The couplings GV and Ga1

A are fixed by requiring that scattering matrix polescoincide with physical meson masses: MeV.1230MeV,783MeV,770 1 ammm

The (instanton) contribution to the gluon condensate appears through using the gap equation and estimated as

4

0

2

22 GeV012.0

200

uD

uMudu

NG cs

Other condensates are

2

0

2

2

0

22

2

2

GeV2.0'

2

,GeV68.04

1

uD

uMuMuduN

uD

uMudu

N

qqqq

qDq

cs

cq

Averaged quark virtuality in QCDvacuum

<A2> condensate

Current-current correlators in NQMV 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 varify that the Witten inequality is fullfiled 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(

V-A: NQM vs ALEPH

OPE pQCD

QCD

NQM

ALEPH

Low-energy observables and ALEPH-OPAL data

1. E.m. mass difference. By using DGMLY sum rule

MeV2.4QMN0 mm

which is in remarkable agreement with the experimental number (after subtracting md-mu effect)

MeV03.043.4exp0 mm

0

20

22

22

022 3

4ln

4

1

mm

fQdQs

ssds T

AVAV

one has

Electric polarizability of the charged pion is defined as

2

DMO

2

3

f

Ir

mE

Model calculations provides3

QMNDMO2

QMN

2 102.18,fm33.0 Ir

and34

QMNfm109.2

E

A

Q

TAVAV F

rfQQ

Qs

sdsI

3

1

4

12

2

0

222

02DMO

2

with help of the DMO sum rule

While from experiment one has

3

OPALDMO2

PDG

2 108.13.26,fm008.0439.0 Ir

and 34

exp

34

expfm106.09.2fm1088.071.2

PIBETAEOPALE

NQM Adler function and ALEPH data

NJL

ALEPHNQMAS

Quark loop

Quark loop

Mesons

Meson loop

M(p)

LO Hadronic contribution to g

The calculations are based on the spectral representation

1

022

22

4

2hvp)2(

1/,Im

1

3

8

2 txmx

mxdxtKt

t

tKdta

m

which is rewritten via Adler fucntion as

1

0

22

2hvp)2(

1

2/11

3

8 m

x

xD

x

xxdxa

Phenomenological estimates give

,,,10076.0932,6

,,10077.0849,68

8hvp)2(

eee

eeea

and from NQM one gets

,1077.0,1013.0,1033.5

,105.023,68hvp)2(

Mloop,8hvp)2(

mesons-,8hvp)2(

Qloop,

8hvp)2(QMN,

aaa

a

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

Light-by-Light contribution to muon AMM

Vector Meson Dominance like model:(Knecht, Nyffeler)

80LbL)3( 10058.0 a

VMD + OPE(Melnikov, Vainshtein)

8LbL)3(

80LbL)3(

10136.0PVPS,

,100765.0

a

a

LO vs NLO corrections

1% from phenomenology,10% from the model

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

Singlet axial-vector current correlator and the topological susceptebility

Due to anomaly singlet axial-vector current is not conserved

density charge al topologicis~

8/where,2 5550 xGxGxQxQNxJ aa

sf

Longitudinal part of singlet corre;ator is related to topological susceptibility

000 5542 QxQTxediQ iqx by 2

2

2

20, 2Q

Q

NQ fA

L

222

16 QOGQ ass

OPE, SVZ

002 Q Crewther theorem

42

QCD

2 0'0 QOQQ

limit OZIMeV392N

f

Narison Shore, Veneziano,MeV426

Oganesyan Ioffe,MeV648

0'

2

f

2

2

2

Topological susceptibilty in NQM

Model predicts

0'0''

1'

22

10'

,00

,'

12

2

QZI

222

2

22

APf

qff

JG

G

G

Gf

NQ

Q

G

G

G

MNQN

spectrumMeson 1.0',MeV50

caseInstanton ',MeV550'

2

2

GG

GG

Topological susceptibility vs Q2 predicted by NQM

pQCD

216

ass G

Conclusions• Non-local chiral quark model NQM is appropiate for the study of

vacuum and light meson internal structure.• it is consistent with low-energy theorems and its predictions of the

local matrix elements (low-energy constants Li, form factors slopes, etc.) are close to the predictions of the local effective models.

• However, the non-locality allows us effectively resum infinite number of local matrix elements. This property is crucial in attemps to predict the form factors in a wide kinematical region and to extract asymptotic (light-cone) distributions like Distribution Amplitudes, (Generalized) Parton Distributions, etc.

• The nonlocality may be naturally attributed to existance of QCD instantons

• We shown agreement of the model predictions on V-A correlator with ALEPH-OPAL data, pion transtion form factor with CLEO, pion e.-m. form factor with JLAB.