Vector and axial-vector current correlators within the instanton model of QCD vacuum.

1. Vector and axial-vector current correlators within the instanton model of QCD vacuum. V and A current-current corelators OPE vs cQCD V-A and V correlator and ALEPH data Hadronic contribution to muon AMM in Instanton model Topological susceptibility Conclusions A.E. Dorokhov(JINR, Dubna)

2. 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 where the QCD currents are

3. The two-point correlation functions obey (suitably subtracted) dispersion relations where the imaginary parts of the correlators determine the spectral functions ALEPH and OPALmeasured V and Aspectral functions separately and with high precision from the hadronic t-lepton decays

4. ALEPH data r a1 r a1 Inclusive v+a spectral function, measured by the ALEPH collaboration pQCD pQCD Inclusive v-a spectral function, measured by the ALEPH collaboration

5. Vector Adler Function (pQCD and cQCD). Adler function is defined as At short distances pQCD predicts (MSbar) where and At large distances pQCD predicts only

6. pQCD and cQCD predictions for Adler function D(Q) pQCD ? cQCD Constituent Quarks Current Quarks

7. Adler Function and ALEPH data Take an ansatz for the spectral function where and find the continuum threshold s0 from duality condition: Using the experimental input corresponding to the t--decay data and the perturbative expression

8. One find matching from duality condition pQCD pQCD ALEPH ALEPH

9. Adler function and ALEPH data (N)NLO LO Asymptot Freedom

10. Nonlocal Chiral Quark model (cNQM) SU(2) nonlocal chirally invariant action describing the interaction of soft quarks I Spin-flavor structure of the interaction is given by matrix products (Instanton interaction: G'=-G) For gauge invariance with respect to external fields V and A the delocalizedquark fields are defined (with straight line path)

11. Quark and Meson Propagators The dressed quark propagator is defined as The Gap equation I has solution Mconst. Mcurr.

12. qq scattering matrix with polarization operator haspolesat posiitons of mesonic bound states The pion vertex with the quark-pion constant gpqq satisfying the Goldberger-Treiman realtion

13. Conserved Vector and Axial-Vector currents. The Vector vertex ~as in pQCD AF NonLocal part WTI

14. The iso-triplet Axial-Vector vertex has a pole at Pion pole AWTI The iso-singlet Axial-Vector vertex has a pole at 1-G’JPP(q2) h’ meson pole’ anomalous AWTI

15. Current-current correlators Current-current correlators are sum of dispersive and contact terms The transverse and longitudinal part of the correlators are extracted by projectors I Contact term Dispersive term

16. Model parameters and Local matrix elements Profiles for dynamical quark massin the Insatanton model and for the Constrained Instanton it is approximated by Gassian form Parameters of the profile are fixed by thepion weak decay constant and the quark condensate

17. With the model parameters fixed as one obtains The couplings Gr,wV and Ga1A are fixed by requiring that scattering matrix poles coincide with physical meson masses: The (instanton) contribution to the gluon condensate appears through using the gap equation and estimated as Other condensates are Averaged quark virtuality in QCD vacuum <A2> condensate

18. Current-current correlators in cNQM V correlator and the difference of the V and A correlators 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 Above we used definitions

19. V-A: cNQM vs ALEPH cQCD cNQM OPE pQCD ALEPH

20. Low-energy observables and ALEPH-OPAL data • E.m.p mass difference. By using DGMLY sum rule one has which is in remarkable agreement with the experimental number (after subtracting md-mu effect)

21. 2. Electric polarizability of the charged pion is defined as with help of the DMO sum rule Model calculations provides and While from experiment one has and

22. NcQM Adler function and ALEPH data M(p) Quark loop ALEPH NcQM AS r,w Quark loop Mesons NJL Meson loop

23. LO Hadronic contribution to gm-2 The calculations are based on the spectral representation which is rewritten via Adler fucntion as Phenomenological estimates give and from NcQM one gets

24. Other model approaches Phenomenological estimate: NcQM: Extended Nambu-Iona-Lasinio: (Bijnens, de Rafael, Zheng) Minimal hadronic approximation (Local duality): (Peris, Perrottet, de Rafael) Lattice simulations: (Blum; Goeckler et.al. QCDSF Coll.)

25. Light-by-Light contribution to muon AMM Vector Meson Dominance like model: (Knecht, Nyffeler) VMD + OPE (Melnikov, Vainshtein)

26. LO vs NLO corrections 1% from phenomenology, 10% from the model NO phenomenology, 50% from the existing model, The aim to get 10% accuracy

27. Singlet axial-vector current correlator and the topological susceptebility Due to anomaly singlet axial-vector current is not conserved Longitudinal part of singlet corre;ator is related to topological susceptibility by OPE, SVZ Crewther theorem

28. Topological susceptibilty in NcQM Model predicts

29. Topological susceptibility vs Q2 predicted by NcQM pQCD

30. Conclusions • Non-local chiral quark model NcQM 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.