Hadronic corrections to muon anomaly within the instanton vacuum model

1. Hadronic corrections to muon anomalywithin the instanton vacuum model • 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: phenomenology • e+e- - annihilation vst – decay (shape discrepancy, as(MZ)) • III. LO and NLO Hadronic contribution from theory (main source of theoretical error) • Conclusions A.E. Dorokhov(JINR, Dubna)

2. Muon Anomaly (Current status) SND 05’

3. Magnetic Anomaly QED Weak Hadronic SUSY... ... or other new physics ?

4. The hadronic contribution to the muon anomaly, where the dominant contribution comes from (a). The hadronic light-by-light contribution is shown in (e) Hadronic polarization Light-by-light process a2 a3 Z*gg*interference

5. Nonlocal Chiral Quark model 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)

6. Quark and Meson Propagators The dressed quark propagator is defined as The Gap equation I has solution Mconst. From existence of derivatives of the quark condensate Mcurr.

7. Conserved Vector and Axial-Vector currents. ~as in pQCD The Vector vertex AF Nonlocalpart The Iso-Triplet Axial-Vector vertex has a pole at Pion pole The Iso-Singlet Axial-Vector vertex has a pole at 1-G’JPP(q2) h’ meson pole

8. III.LO Hadronic contribution to gm-2 Is expressed via Adler function as Adler function is defined as

9. NcQM Adler function and ALEPH data M(p) Quark loop ALEPH ILM AS r,w Quark loop Mesons NJL Meson loop pQCD (NNNLO) cQCD

10. LO contribution to am The instanton model estimates and from phenomenology one gets

11. LBL, OPE and Triangle diagram VAV X X OPE X (Similar to p0g*g*amplitude at large photon virtualities)

12. The structure of V*AV amplitude For specific kinematics Only 2 structures exists in the triangle amplitude The amplitude is transversal with respect to vector current but longitudinal with respect to axial-vector current Thus Adler-Bell-Jackiw anomaly

13. Operator Product Expansion and V*AV amplitude In local theory for massive quarks one gets Which in chiral limit (m=0) becomes • 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

14. Anomalous wL structure (NonSinglet) X X + rest + Diagram with NonLocal Axial vertices Diagram with Local vertices In accordance with Anomaly and ‘t Hooft duality principle (massless Pion states in triplet)

15. Anomalous wL structure (Singlet) X X + rest Diagram with NonLocal Axial vertices Diagram with Local vertices In accordance with Anomaly and ‘t Hooft duality principle (no massless states in singlet)

16. wLT in the Instanton Model (NonSinglet) In perturbative QCD • 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 qO(1/q6) (De Rafael et.al., 2002) • Absence of Power corrections at large q in Instanton model (this work)

17. wLT in the Instanton Model (Singlet)

18. Z*gg*contribution toam Perturbative QCD (Anomaly cancelation) VMD + OPE (Czarnecki, Marciano, Vainshtein, 2003) Instanton model: (Our work, 2005)

19. Light-by-Light contribution to muon AMM • This contribution must be determined by calculation. • the knowledge of this contribution limits knowledge of theoretical value. Vector Meson Dominance like model: (Knecht, Nyffeler 2002) VMD + OPE (Melnikov, Vainshtein 2003) Instanton model: (Dorokhov 2005)

20. 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 cPT to the regions not achievable by these methods.

21. II. Leading OrderHadronic contributions ”Dispersion relation“, uses unitarity (optical theorem) and analyticity (Bouchiat and Michel, 1961) (0) had   had   Still it is impossible to calculate LO contribution from Theory with required accuracy

22. Evaluating the Dispersion Integral use data Agreement between Data (BES) and pQCD (within correlated systematic errors) use QCD Better agreement between exclusive and inclusive (2) data use QCD About 91% of av comes from s<(1.8GeV)2, 73% is covered by final 2p state

23. e+e− → p+p− crosssectionin the SND experiment at theVEPP-2M collider (June,2005) measured at 400 < √s < 1000 MeV The ratio of the e+e− → p+p− crosssection. The shadedarea shows the systematic error of the SND measurements. New SND agrees well with CMD2 and t-data. KLOE is in conflict with SND and CMD2.

24. Comparison with CMD-2 and KLOE in the Energy Range 0.37 <sp<0.93 GeV2 (375.6  0.8stat  4.9syst+theo) 10-10 KLOE (378.6  2.7stat  2.3syst+theo) 10-10 CMD2 SND Data on R(s) 2pcontribution to amhadr • SND has evaluated theDispersions Integral for the 2-Pion-Channel in the Energy Range0.39 <sp<0.97 GeV2 ampp = (488.7  2.6stat  6.6syst) 10-10 (2004) (2003)

25. LO vs NLOHadron corrections to Muon Anomalous magnetic moment (Theory) Light-by-Light scattering Vacuum Polarization Lowest Order (EM and Tau data; Adler function) Higher Order (OPE and Triangle diagram) 1% from phenomenology, 10% from the model NO phenomenology, 50% from the existing model, The aim to get 10% accuracy

26. The hadronic light-by-light contribution is likely to provide the ultimate limit to the precision of the standard-model valueof aμ.

27. Triangle diagram

28. 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

29. 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

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

31. 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.)

32. VAV* correlator It is transverse with respect to vector currents X Triangle diagram

33. wLT structure Exponentially suppressed, no power corrections! Above we used definitions

34. Instanton Vector Meson Dominance

35. ALEPH data on t decays r a1 r a1 pQCD pQCD Inclusive v-a spectral function, measured by the ALEPH collaboration Inclusive v+a spectral function, measured by the ALEPH collaboration