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. 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

3. 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 (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. the Hidden Local Symmetry
4.2 Model
based on
the Hidden Local Symmetry

5. ☆ 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

6. ☆ 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

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

8. 4.3 Phenomenology at tree level

10. ? ☆ 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 ?

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

12. ☆ Values of Parameters

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

14. 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

15. | ☆ 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
= ± (fm)
〈r 〉 V π =
〈r 〉 2 V π |
exp =
0.452 ± ; (PDG2006)

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

17. 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

18. L ; O (p ) terms of chiral Larangian1 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

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. ［ [ ］ 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 †

23. ☆ 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

24. ☆ 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.

25. ☆ Example
higher order terms
◎ parameter relations

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