2. EFT for π ☆ Chiral Perturbation Theory matching to QCD ⇒ Λ ～ 1 GeV P-wave ππ scattering J. Gasser and H. Leutwyler, Annals Phys. 158, 142 (1984); NPB 250, 517 (1985) 1-loop tree

3. ☆ Hidden Local Symmetry EFT for π and ρ 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) based on chiral symmetry of QCD ρgauge boson of HLS works well for E ～ m ρ reduces to chiral Lagrangian for E ≪ m ρ

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

6. ☆ Hidden Local Symmetry U = e = ξ ξ 2 i π/ F π L † R 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) F, F ・・・ Decay constants of π and σ πσ h ∈ ［ SU(N ) ］ fV local g ∈ ［ SU(N ) ］ f L,R global ・ Particles ρ = ρ T ・・・ HLS gauge boson μ μ a a π=π T ・・・ NG boson of ［ SU(N ) × SU(N ) ］ symmetry breaking a a f L R f global σ=σ T ・・・ NG boson of ［ SU(N ) ］ symmetry breaking a a f Vlocal

7. Maurer-Cartan 1-forms Lagrangian

8. 4.3 Phenomenology at tree level

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

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

12. 4.4 Predictions (quantitative) F = 92.42 ± 0.26 MeV π g = 5.80 ± 0.91 ; a = 2.07 ± 0.33 g | = 0.119 ± 0.001 GeV ρ 2 exp cf : ρ– γ mixing strength g = a gF = 0.103 ± 0.023 GeV ρ 2 π 2

13. ☆ Values of Parameters

14. ☆ Electromagnetic Form Factor of pion F (p ) π 2 V = 1 - a 2 + m - p 2 2 ρ g ρ ρππ = 1 - a 2 a 2 + m - p 2 2 ρ m 2 ρ F (p ) π 2 V g = a g/2 ρππ g = a gF ρ π 2 m = a g F π ρ 22 2 F (0) = 1 π V

15. ☆ charge radius of pion F (p ) π 2 V = 1 - a 2 a 2 + m - p 2 2 ρ m 2 ρ = 1 + p 6 + 2 m 2 ρ 3a3a ・・・ m 2 ρ 3a3a = 0.407 ± 0.064 (fm) 〈 r 〉 2 V π = 2 2 V π | exp = 0.455 ± 0.005 ; (1987) 0.439 ± 0.008 ; (1986)

17. ☆ Integrating out vector mesons in the low energy region at tree level a F (V – α ) - (∂ V – i [ V, V ] ) = 0 μ π 2 //μ g 2 1 ν μν ν EOM for V μ (V = gρ ) μ μ identity Chiral Lagrangian with O (p ) terms 4 ; † U = ξ ・ ξ L R 1 2i2i i 2 ＾ α = ξ ・∇ U ・ ξ = ξ ・∇ U ・ ξ † ⊥μ⊥μ L μ R R † L † μ V = α + O (p ) μ //μ m 2 ρ 1 3 ; α = ( D ξ ・ ξ + D ξ ・ ξ ) / (2 i ) //μ μ LR R † L † μ D ξ = ∂ ξ － i R ξ μ R μ R R μ

18. = F tr [ αα ］ 2 π ⊥μ⊥μ ＾＾ ⊥ μ F [ αα ］ 2 π ⊥μ⊥μ ＾＾ ⊥ μ ［ ∇ U ∇ U ］ F π 2 4 μ μ † ［ ∇ U ∇ U ］ F π 2 4 F π 2 4 μ μ † = a F ［ α ］ π 2 ＾ // μ F tr ［ α ］ π 2 μμ α // ＾ α ＾ ＾ μ O (p ) 6 V = α + O (p ) μ //μ m 2 ρ 1 3 ; μ α // ＾ μ α － V // μ = ◎ 1 2i2i i 2 ＾ α = ξ ・∇ U ・ ξ = ξ ・∇ U ・ ξ † ⊥μ⊥μ L μ R R † L † μ ◎ -tr ［ V ］ 2 g 2 1 μν -tr ［ V ］ 2 g 2 1 2 g 2 1 μν ※ ※ ※ L ; O (p ) terms of chiral Larangian 4 V 4

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

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

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

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

25. ☆ Example higher order terms ◎ parameter relations

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

27. ◎ Inequivalence for off-shell ρ

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. ☆ π 0 → γγ* and vector dominance (VD) ◎ π 0 γ* γ* vertices ◎ π 0 γ transition form factor ・ vector meson propagators

32. ◎ determination of (c 3 +c 4 )/2 from experiment