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5. 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). P- wave ππ scattering. 5.1. Necessity for vector mesons.

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## 5. Hidden Local Symmetry

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**5. 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).**P-waveππ scattering**5.1. 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**☆ 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**5.2. Model**based on the Hidden Local Symmetry**SU(N ) ×SU(N ) → SU(N )**f f f L R V † U = e → g U g R L a 2iπT /F a π μ † L=tr［∇ U ∇ U］ 2 F g ∈ SU(N ) μ p 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**☆ 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 a f global L f V = VmT・・・ HLS gauge boson R a 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**2**m = ag F 2 2 π ρ ◎ 3 parameters at the leading order Fp・・・ pion decay constant g・・・ gauge coupling of the HLS a = (Fs/Fp)2 ⇔ validity of the vector dominance • Maurer-Cartan 1-forms Vm : HLS gauge field 変換性 ： • Lagrangian**5.3. Phenomenology at tree level**Vm = grm**◎ Predictions for a = 2**・・・ KSRF II relation ・・・ r meson dominance for the EM form factor of p ◎ Low Energy Theorem (a – independent relation) ・・・ KSRF I relation**◎ Pion EM form factor [in the space-like region (p2 < 0)]**= + ggpp g g ρππ a a ρ 2 m π π 2 2 ρ F (p) F (p) = 1 - + V V 2 2 2 2 2 m = ag F m - p m - p 2 2 2 2 π ρ ρ ρ 2 g = agF g = ag/2 g = 1 - a/2 π ρ ρππ gpp cf : a = 2 ⇒ mr = 2 grppFp (KSRF relation) 2 2 2 a = 2 ⇒ vector dominance**☆ KSRF I(low energy theorem) ?**? 15% deviation !!**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 : ☆Predictions (quantitative) g = 5.80 ± 0.91 ; a= 2.07 ± 0.33 ρ– gmixing strength**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**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)**5.4. Vector meson saturation**of the low energy constants - Relation to the chiral perturbation theory - (HLS at tree level)**(V= gρ )**μ μ EOM for V μ Chiral Lagrangianwith O(p )terms 4 ☆ Integrating out vector mesons in the low energy region at tree level identity**2**2 F F † † = 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 4 6 O (p ) = 1 1 1 ［ ［ ］ ］ μν μν - - tr tr V V V V μν μν 2 2 2 g g g 2 2 2 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 // // ◎ ※ ◎ ※ ※**g = 5.80 ± 0.91**G.Ecker, J.Gasser, A.Pich and E.deRafael, NPB 321, 311 (1989)**5.5. Chiral Perturbation Theory with HLS**を記述 ・ HLS at tree ・・・ ・ ChPT with HLS への拡張 loop corrections ⇒**◎ 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) based on chiral symmetry of QCD ρ ・・・ gauge boson of the HLS ◎ Chiral Perturbation Theory with HLS H.Georgi, PRL 63, 1917 (1989); NPB 331, 311 (1990): M.H. and K.Yamawaki, PLB297, 151 (1992) M.Tanabashi, PLB 316, 534 (1993): M.H. and K.Yamawaki, Physics Reports 381, 1 (2003) Systematic low-energy expansion including dynamical r loop expansion ⇔ derivative expansion**☆ Order Counting**・・・ same as ordinary ChPT loop expansion = low-energy expansion ☆ Expansion Parameter ◎ ordinary ChPT for π chiral symmetry breaking scale ◎ ChPT with HLS**◎ 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) based on chiral symmetry of QCD ρ ・・・ gauge boson of the HLS ◎ Chiral Perturbation Theory with HLS H.Georgi, PRL 63, 1917 (1989); NPB 331, 311 (1990): M.H. and K.Yamawaki, PLB297, 151 (1992) M.Tanabashi, PLB 316, 534 (1993): M.H. and K.Yamawaki, Physics Reports 381, 1 (2003) Systematic low-energy expansion including dynamical r loop expansion ⇔ derivative expansion ☆ many parameters ! ・・・ not determined by the chiral symmetry 35個 at O(p4) more experimental data are available**☆ くりこみ群方程式 (leading order parameters)**２次発散の効果も含む NOTE : (g, a) = (0, 1) ・・・ fixed point**5.6 Phase Structure of HLS**くりこみ群方程式を用いた解析例**(RGE for F is solved analytically)**π ☆ Phase change can occur in the HLS ・ illustration with (g, a) = (0,1) ・・・ fixed point ・ at bare level ・ at quantum level The quantum theory can be in the symmetric phase even if the bare theory is written as if it were in the broken phase.**☆ RGEs**◎ on-shell condition ◎ order parameter**☆ Fixed points (line)**・・・ unphysical**☆ Flow diagram on G = 0 plane**symmetric phase VM broken phase**☆ Flow diagram on a = 1 plane**symmetric phase VM ρ decoupled broken phase ・tree level では、a > 0, Fp2 > 0 である限り、パラメータの値に 制限はなかった ・量子補正を考えると、取り得るパラメータの範囲には制限がある**5.8.1. Generalized Hidden Local Symmetry**Bando-Kugo-Yamawaki, NPB 259, 493 (1985); PTP 73, 1541 (1985); Phys.Rept. 164, 217 (1988) Bando-Fujiwara-Yamawaki, PTP 79, 1140 (1988) covariant derivatives**◎ 1-forms**◎ Lagrangian**☆ GHLS から ChPTに reduction する**（rと A1 を integrate out する） ・ HLS と比べると、L10のみが変更を受ける g = 5.80 ± 0.91 exp : L10(mr) = -5.2 ± 0.7 ChPTの low energy constant L1, L2, L3, L9, L10は rと A1 があることでほとんど説明できる**5.8.2. Inclusion of r’**※ これを無限回繰り返せば、 無限個の vector, axial-vector mesons を取り入れ可能 ⇒ より高いエネルギー領域まで使えるようになる**5.8.3. linear (condensed) Moose model**☆HLS R L V [SU(Nf)V]HLS [SU(Nf)R]chiral [SU(Nf)L]chiral ・covariant derivatives ・Lagrangian Harada-Yamawaki, Phys.Rept.383, 1 Harada-Tanabashi-Yamawaki, PLB568, 103**☆GHLS**R L L R [SU(Nf)L]chiral [SU(Nf)L]GHLS [SU(Nf)R]chiral [SU(Nf)R]GHLS ・ Lagrangian Harada-Sasaki, PRD 73, 036001 (2006)**☆無限個への拡張**・・・ より高いエネルギー領域まで使えるように ・linear (condensed) Moose model L∞ V R1 R∞ L R L1 xL0† xR0 xR∞ xL∞† ・ 高エネルギー領域での vector current correlator を再現する Son-Stephanov, PRD69, 065020**・ 5-D models constructed in the large Nc limit**・ Generalized versions of the HLSat tree level 5.8.4. Relation between the HLS and a class of models based on holographic QCD ◎ Models including 5-dimensional gauge field at intermediate step see, e.g., Son-Stephanov, PRD69, 065020 Chivukula-Kurachi-Tanabashi, JHEP0406, 004 Sakai-Sugimoto, PTP113, 843; PTP114, 1083 Erlich-Katz-Son-Stephanov, PRL95, 2612602**z**latticize the 5-th dimension [SU(Nf)R]chiral gauge field [SU(Nf)L]chiral gauge field L R ・linear (condensed) Moose model L R z=z1 z=zi [SU(Nf)]local gauge field ◎ 5-D model to linear (condensed) Moose model M = 0,1,2,3,4 ;μ = 0,1,2,3 ; z: coordinate of 5th dimension z=zL z=zR**lntegrating out vector (axial-vector) mesons leaving the**lightest r meson ☆ Reduction from a holographic model to the HLS model ・linear (condensed) Moose model L R z=z1 z=zi ◎ HLS R L V [SU(Nf)V]HLS [SU(Nf)R]chiral [SU(Nf)L]chiral**☆ Holographic Model by Sakai-Sugimoto**Sakai-Sugimoto, PTP113, 843; PTP114, 1083 ・ Action (5-th dimension is compactified.) ・ Transformation of the gauge field M = 0,1,2,3,4 ;μ = 0,1,2,3 ; ・ Boundary conditions external gauge fields corresponding to the chiral symmetry**◎ Az = 0 gauge**・ Residual gauge symmetry : identified with the HLS : HLS**◎ Reduction to the HLS**HLS の O(p2) + O(p4) ラグランジアン ⇒ 物理量 ・ Pion EM form factor M.H., S. Matsuzaki and K.Yamawaki, in preparation Input : mr = 775.5 MeV － (space-like region) dotted line : Vector meson dominance

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