4. Hidden Local Symmetry

1. 4. Hidden Local Symmetry Effective (Field) Theory including vector mesons in addition to pseudoscalar mesons • M.Bando, T.Kugo and K.Yamawaki, Phys. Rept. 164,217 (1988). • M.Harada and K.Yamawaki, Phys. Rept. 381, 1 (2003).

2. P-waveππ scattering 4.1 Necessity for vector mesons ☆ Chiral Perturbation Theory EFT for π J. Gasser and H. Leutwyler, Annals Phys. 158, 142 (1984); NPB 250, 517 (1985) 1-loop tree

3. ☆ What EFT do we need to include r and p ? ◎ several ways to include r • Matter field • Anti-symmetric tensor field • Massive Yang-Mills • Hidden local symmetry These are all equivalent at tree level. A difference appears at loop level. ◎ Hidden Local Symmetry Theory ・・・ EFT for r and p M. Bando, T. Kugo, S. Uehara, K. Yamawaki and T. Yanagida, PRL 54 1215 (1985) M. Bando, T. Kugo and K. Yamawaki, Phys. Rept. 164, 217 (1988) M.H. and K.Yamawaki, Physics Reports 381, 1 (2003 based on chiral symmetry of QCD ρ ・・・ gauge boson of the HLS

4. 4.2 Model based on the Hidden Local Symmetry

5. SU(N ) ×SU(N ) → SU(N ) f f f L R V † U = e → g U g R L a 2iπ T /F a π μ † L=tr［∇ U ∇ U］ 2 F g ∈ SU(N ) μ π f L,R L,R 4 ∇ U ≡∂ U－ iLU + iUR μ μ μ μ L, R ; gauge fields of SU(N ) μ μ f L,R ☆ Chiral Lagrangian Non-Linear Realization of Chiral Symmetry ◎ Basic Quantity ; ◎ Lagrangian

6. ☆ Hidden Local Symmetry † U=e= ξ ξ L R 2iπ/ F π F , F・・・ Decay constants of π and σ π σ h ∈ ［SU(N ) ］ a f V π=π T・・・ NG boson of ［SU(N ) × SU(N ) ］ symmetry breaking local a R a f L f global ρ = ρ T・・・ HLS gauge boson a μ μ g ∈ ［SU(N ) ］ a f L,R global L,R σ=σ T・・・ NG boson of ［SU(N ) ］ symmetry breaking a f V local M.Bando, T.Kugo, S.Uehara, K.Yamawaki and T.Yanagida, PRL 54, 1215 (1985) M.Bando, T.Kugo and K.Yamawaki, Phys. Rept. 164, 297 (1988) ・ Particles

7. Maurer-Cartan 1-forms 変換性 ： • Lagrangian

8. 4.3 Phenomenology at tree level

10. Exact in the low energy limit ; p = 0 2 ρ ２ ☆ KSRF I (on-shell ; p = m ) ? ２ ρ ρ ☆ Low Energy Theorem gauge boson --- well-defined off-shell ?

11. ２ ２ ☆ KSRF I(on-shell ; p = m ) ? ρ ρ ? 15% deviation !!

12. ☆ Values of Parameters

13. F = 92.42 ± 0.26 MeV π 2 2 g = agF = 0.103 ± 0.023 GeV π ρ 2 g | = 0.119 ± 0.001 GeV ρ exp cf : 4.4 Predictions (quantitative) g = 5.80 ± 0.91 ; a= 2.07 ± 0.33 ρ– γ mixing strength

14. g g ρππ a a a ρ 2 m π π π 2 2 = 1 - + ρ F (0)= 1 F (p) F (p) = 1 - + V V V 2 2 2 2 2 m = ag F m - p m - p 2 2 2 2 2 π ρ ρ ρ 2 g = agF g = ag/2 π ρ ρππ ☆ Electromagnetic Form Factor of pion

15. 3a 3a a a 2 2 2 m m m π 2 π π 2 ρ ρ ρ F (p) = 1 - + 2 2 = = 0.407 ± 0.064(fm) 〈r 〉 〈r 〉 V 2 2 m - p 2 2 V V ρ | = exp ☆ charge radius of pion 2 p + ・・・ = 1 + 6 0.452 ± 0.011; (PDG2006)

16. 4.5 Vector meson saturation of the low energy constants - Relation to the chiral perturbation theory - (HLS at tree level)

17. (V= gρ ) μ μ ; † EOM for V α = (Dξ ・ξ + D ξ ・ξ )/(2i) † ν 2 ν aF (V – α ) - (∂ V – i[ V , V ] ) = 0 μ 2 m i μ π //μ //μ μ μν R μ μν L R L ρ 2 D ξ = ∂ ξ － iR ξ μ μ R R R μ 1 3 V = α + O(p ) μ //μ † † g † 2 † ＾ U = ξ・ξ α = ξ・∇ U・ξ = ξ ・∇U・ξ μ L ⊥μ 1 R μ R L R L ; 1 2i Chiral Lagrangianwith O(p )terms 4 ☆ Integrating out vector mesons in the low energy region at tree level identity

18. 1 ◎ † † † ＾ α = ξ・∇ U・ξ = ξ ・∇U・ξ 2 2 μ F F ⊥μ R μ L R L 2i † † = π π μ μ ［ ［ ］ ］ U U U U 2 2 ∇ ∇ ∇ ∇ μ μ tr tr [ [ ］ ］ tr tr F F ＾ ＾ ＾ ＾ α α α α μ μ π π 4 4 ⊥μ ⊥μ ⊥ ⊥ 2 F π i 4 2 ◎ ; ＾ α = α － V μ // μ μ // 6 O (p ) = 2 m ρ 3 V = α + O(p ) 1 1 1 μ //μ ［ ［ ］ ］ μν μν - - tr tr V V V V μν μν 2 2 2 g g g 2 2 2 1 2 2 V a L ; O (p ) terms of chiral Larangian ［ ［ ］ ］ μ μ 4 tr tr F F ＾ ＾ α α α α ＾ ＾ 4 // // μ μ π π // // ※ ※ ※

20. g = 5.80 ± 0.91 G.Ecker, J.Gasser, A.Pich and E.deRafael, NPB 321, 311 (1989)

21. 4.6 Relation to other models of vector mesons

22. 2 2 F F † † † † α = (Dξ・ξ - Dξ・ξ )/(2i) π π μ μ ［ ［ ］ ］ U U U U ∇ ∇ ∇ ∇ tr tr ⊥μ μ μ μ μ 2iπ/F 4 4 π U = ξ・ξ= e 2 F π 2 4 μ [ ］ tr F α α π L ⊥μ ⊥ = † † † D ξ = ∂ ξ＋i ξR D ξ = ∂ ξ ＋i ξL † μ μ μ μ μ † μ α = (Dξ・ξ + Dξ・ξ )/(2i) μ //μ μ iπ/F π † † ξ= e → h(π, g , g ) ・ξ・ g = g・ξ・ h (π, g , g ) † L † α → h・α ・ h R L R L α → h・α ・ h + ∂ h・ h / i R ⊥μ ⊥μ //μ μ //μ = ★ Matter field method ☆ CCWZ Lagrangianforπ ◎ Bulding blocks ; ; ◎ transformaion properties † ; ◎ Lagrangian with least derivatives

23. † (C) (C) ρ → h(π,g ,g ) ・ρ・ h (π,g ,g ) μ μ R L R L (C) (C) (C) (C) (C) ρ ≡ D ρ －D ρ μ ν μ μν ν (C) (C) (C) (C) D ρ ≡∂ ρ － i［α , ρ ］ ＾ † † μ ν μ ν V ≡（ξR ξ + ξ Lξ ）/2 //μ ν μν μν μν ＾ † † A ≡（ξR ξ － ξ Lξ ）/2 μν μν μν L, R ; gauge fields of SU(N ) μ μ f L,R ☆ vector meson field・・・ matter field ◎ transformation property ◎ Building blocks ; ; ◎ Lagrangian with vector meson

24. † (C) ＾ ξ ＝ξ ＝ξ ρ ＝ζ(α － V ) ＝ζα R L μ //μ μ //μ All the building blocks of the MFM are expressed by those of the HLS. For any Lagrangian of the MFM, whatever the form it takes, we can construct the equivalent Lagrangian of the HLS. ☆ Correspondence between parts of MFM and those of HLS ▽HLS in the unitary gauge ・・・ σ＝0 ;

25. ☆ Example higher order terms ◎ parameter relations

26. ☆ MFM ＝ HLS ・・・ true only at on-shell !! ◎ parameter relations ◎ Relations of physical quantities ◎ Differences appear at off-shell

27. ◎ In-equivalence for off-shell ρ

28. 4.7 Anomalous Processes

29. ☆ Generalization of Wess-Zumino action ・・・ inclusion of vector mesons based on the HLS ◎ Wess-Zumino anomaly equation ◎ general solution

30. ☆ VVπ, Vγπ, γγπ vertices

31. ☆ π0 → γγ* and vector dominance (VD) ◎ π0 γ* γ* vertices ◎ π0 γ transition form factor ・ vector meson propagators

32. ◎ determination of (c3+c4)/2 from experiment