Hadronic corrections to muon (g-2) within the Instanton model

1. Hadronic corrections to muon (g-2)within the Instanton model A.E. Dorokhov(JINR, Dubna) Introduction Data from g-2 Collaboration (BNL) vs SM prediction from QED, EW and Strong sectors LO and NLO Hadronic contribution from theory (phenomenology vs models) Conclusions FB18

2. Hadronic Contribution to Muon AMM from BNL FromBNL E821 experiment (1999-2006) From Standard Model plus plus assume that No New Physics at this scale Then the Hadronic Contribution is estimated as The main question how to get such accuracy from theory. FB18

3. a2 The hadronic contributions to the muon AMM  Hadronic Vacuum Polarization contributes 99% and half of error  h   a3 h        e h h h       Light-by-light process contributes 1% and half of error Z*gg*effective coupling FB18

4. How to reach accuracy required? • At present it is not possible to make calculations based on first principles • Phenomenological approach using experimental data to relate with quantities wanted (Leading Order) • Model approaches (VMD,Instanton, Lattice,…) (Next-to-Leading Order) We consider light quark sector where Spontaneous Breaking of Chiral Symmetry is important FB18

5. Instanton Liquid Model The dressed quark propagator is defined as with Quark Mass generated dynamically I Mconst. Interpolate current and constituent masses Mcurr. f(p) is related to the quark zero mode in the instanton field FB18

6. Conserved Vector and Axial-Vector currents. ~as in pQCD The Vector vertex AF q Nonlocalpart r-meson pole k k’ 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) FB18 h’ meson pole

7. Leading Order Contribution LO is expressed via Adler function as Phenomenological approach Model approach Adler function is defined as FB18

8. NcQM Adler function and ALEPH data M(p) AS MasslessQuarks Quark loop ALEPH ILM r,w Quark loop pQCD (NNNLO) Mesons (Nc enhanced) NJL Meson loop (chiral enhanced) cQCD Massive Quarks FB18

9. LO contribution to am From phenomenology one gets M. Davier, S. Eidelman, A. Hocker, and Z. Zhang (03) S. Ghozzi and F. Jegerlehner (04) K. Hagiwara, A. D. Martin, D. Nomura, T. Teubner (04) J. F. de Troconiz and F. J. Yndurain (04) S. Eidelman (ICHEP06) The instanton model estimates (heavy quark sector is not taken) (Dorokhov, 2004; also in B. Holdom, R. Lewis, and R. R. Mendel, 1994) Sensitive to Constituent Quark mass, prefers small values Mq = 200-250 MeV VMD: (M. Perrottet and E. de Rafael,2003) Lattice simulations: (Blum, 2003; Goeckler et.al. QCDSF Coll.,2004) In the Next order there is no phenomenological input FB18

10. The structure of V*AV amplitude For specific kinematics: q2=q is arbitrary, q10 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 This is famous Adler-Bell-Jackiw anomaly FB18

11. V*AV amplitude • 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 (Dorokhov,2005) • Massive corrections (Teryaev, Pasechnik; Jegerliner, Tarasov, 2005; Melnikov,2006) FB18

12. In local theory for quarks with constant mass one gets In local theory for quarks with constant mass one gets In Instanton Model for quarks with momentum dependent mass one gets (Triplet case) Vector Meson Dominance Instanton model FB18

13. Anomalous wL structure (Singlet) In accordance with Anomaly and ‘t Hooft duality principle (no massless states in singlet channel due to UA(1) anomaly) FB18

14. Z*gg*contribution toam Perturbative QCD (Anomaly cancelation) VMD + OPE (Czarnecki, Marciano, Vainshtein, 2003) Instanton model: (Dorokhov, 2005) FB18

15. p0 contribution to Light-by-Light amplitude X p m Calculated for arbitrary virtualities (A. Dorokhov, Lauro Tomio, 2000-2001) CLEO 98 measured at Q2 from 1to 8 GeV2 FB18

16. 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 2006) FB18

17. Conclusions • There are plans to increase accuracy of (g-2) experiment by factor 2 (BNL) or even 10 (J-PARC) • The standard model calculations has to be at the same level of accuracy • The weakest part of calculations is strong sector (BNL-SM discrepancy 3.2s) • In Leading Order Phenomenological analysis is very power • At NLO the effective (Instanton) model provide the approach which competes with other (Lattice QCD, etc) • The first results on LBL corrections with accuracy of calculations at level 10-20% is obtained within the instanton model • The Longitudinal structure of triangle diagram is nonrenormalized by nonperturbative corrections in agreement with ‘t Hooft arguments • Transverse and Longitudinal structures are found at 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. FB18