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).

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) P-wave ππ scattering 1-loop tree

Anti-symmetric tensor field Massive Yang-Mills Hidden local symmetry ☆ 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

the Hidden Local Symmetry 4.2 Model based on the Hidden Local Symmetry

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

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

Maurer-Cartan 1-forms 変換性 ： Lagrangian

4.3 Phenomenology at tree level

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

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

☆ Values of Parameters

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

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

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

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

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

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

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

4.6 Relation to other models of vector mesons

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

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

☆ Correspondence between parts of MFM and those of HLS ▽ HLS in the unitary gauge ・・・ σ＝0 ξ ＝ξ ＝ξ L R † ρ ＝ζ(α － V ) ＝ζα (C) 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.

☆ Example higher order terms ◎ parameter relations

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

◎ In-equivalence for off-shell ρ

4.7 Anomalous Processes

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

☆ VVπ, Vγπ, γγπ vertices

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

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