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5. Hidden Local Symmetry

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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  1. 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).

  2. 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

  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. 5.2. 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 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

  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 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

  7. 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

  8. 5.3. Phenomenology at tree level Vm = grm

  9. ◎ 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

  10. ◎ 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

  11. ☆ KSRF I(low energy theorem) ? ? 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 : ☆Predictions (quantitative) g = 5.80 ± 0.91 ; a= 2.07 ± 0.33 ρ– gmixing 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. 5.4. 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 μ Chiral Lagrangianwith O(p )terms 4 ☆ Integrating out vector mesons in the low energy region at tree level identity

  18. 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 // // ◎ ※ ◎ ※ ※

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

  20. 5.5. Chiral Perturbation Theory with HLS を記述 ・ HLS at tree ・・・ ・ ChPT with HLS への拡張 loop corrections ⇒

  21. ◎ 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

  22. ☆ Order Counting ・・・ same as ordinary ChPT loop expansion = low-energy expansion ☆ Expansion Parameter ◎ ordinary ChPT for π chiral symmetry breaking scale ◎ ChPT with HLS

  23. ◎ 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

  24. ☆ 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

  25. 5.6 Phase Structure of HLS くりこみ群方程式を用いた解析例

  26. (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.

  27. ☆ RGEs ◎ on-shell condition ◎ order parameter

  28. ☆ Fixed points (line) ・・・ unphysical

  29. ☆ Flow diagram on G = 0 plane symmetric phase VM broken phase

  30. ☆ 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

  31. 5.7. Generalization of the HLS

  32. 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

  33. ◎ 1-forms ◎ Lagrangian

  34. ☆ Particle Identification

  35. ☆ 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

  36. 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

  37. 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

  38. ☆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)

  39. ☆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

  40. ・ 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

  41. 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

  42. 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

  43. ☆ 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

  44. ◎ Az = 0 gauge ・ Residual gauge symmetry : identified with the HLS : HLS

  45. ◎ 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

  46. The End

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