Understanding Wilsonian Matching for Effective Quantum Field Theories

1. 6. Wilsonian Matching

2. 6.1 Basic Concept

3. ◎ Generating functional in QCD J : external source fields ◎ Generating functional in EFT F : parameters of EFT ☆ Wilsonian matching bare theory

4. high energy QCD quarks and gluons L matching HLS r and p Bare theory bare parameters Quantum effects low energy Quantum theory physical quantities

5. ☆ Axialvector current correlator in space-like region ◎ low energy limit・・・ π dominance

6. ◎ high energy region ・・・Operator Product Expansion (OPE) ・・・ renormalization scale of QCD 2 F (0) π μ 2 Λ ～ 1 GeV

7. ◎ Around Q2～ Λ2～ (1 GeV)2 4 including O(p ) terms 2 2 F (0) F (Λ) π π ・ Integrating out quarks and gluons ・・・ not well-defined degrees of freedom in the low energy region ・ bare HLS Lagrangian Λ

8. ＃ Λ ＞ μ ＞ m ・・・ through RGE ρ HLS OPE 2 2 F (0) F (μ) m π π ρ μ 2 Λ ◎ Λ ＞ μ ・・・ inclusion of quantum corrections from π and ρ

9. ＃ m＞ μ ・・・ through the RGE in ChPT ρ 2 2 F (μ) F (0) m π π ρ μ 2 HLS OPE ChPT Λ effect of finite renormalization

10. 6.2 Wilsonian Matching Conditions

11. ◎ QCD (OPE) Matching ☆Axialvector and Vector Current Correlators ◎ HLS

12. ☆ Wilsonian Matching Conditions ◎ ◎ ◎

13. 6.3 Determination of the Bare Parameters

14. ☆ Bare parameters Λ ＝ 1.0 ～ 1.2 GeV • large enough for validity of OPE • small enough for validity of HLS ☆ Matching Scale Λ ☆ Inputs 3 Wilsonian matching conditions ・ Inputs for OPE

15. ☆ Values of bare parameters

16. ☆ Values of bare parameters (leading order)

17. ☆ Values of bare parameters (next order)

18. 6.4 Results of the Wilsonian Matching

19. ☆ Parameters at m scale ρ bare parameters → (RGE) → parameters at m ρ

21. 4 quantities directly related to experiment ☆ Physical Predictions

22. ◎ ρ- γ mixing strength ・ tree ・ loop through RGE ・ typical prediction cf :

23. ◎ Gasser-Leutwyler’s parameter L 10 ・ typical prediction cf :

24. loop effects through RGE ◎ ρππ coupling ・ bare Lagrangian ・ effective interaction ・ typical prediction cf :

25. ◎ Gasser-Leutwyler’s parameter L 9 ・ typical prediction for cf :

26. ◎ parameter a(0) ・・・ characterize Vector dominance loop effects through RGE ・ bare Lagrangian ・ effective interaction ・ typical prediction ・・・ VD is well satisfied

27. ◎ running of a

28. ◎ Summary of Predictions

30. + 0 ☆ Why π - π mass difference ? ◎ ⇔ vacuum structure M.E. Peskin 80’, J. Preskil 80’ ⇒ stability of U(1)em symmtric vacuum ⇒ instability : U(1)em is broken ◎ ⇔ mass of little Higgs 6.5 π+ - π0 Mass Difference and Wilsonian Matching M.H. M.Tanabashi and K.Yamawaki, Phys. Lett. B 568 103 (2003)

31. ☆ How to calculate ? ◎ A formula from Dashen’s theorem bare parameter improve by RGE

32. ☆ Determination of the bare parameter

33. ☆ Prediction Quantum correction through RGE > 0 in good agreement