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Masayasu Harada (Nagoya Univ.)
based on M.H. and K.Yamawaki, Phys. Rept. 381, 1 (2003) M.H., T.Fujimori and C.Sasaki, in preparation
@ KIAS-Hanyang Joint Workshop on Multifaceted Skyrmions and Effective Field Theory
(October 26, 2004 KIAS)
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Q C D
Low Energy hadron Phenomena
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E
αs
Asymptotic freedom
☆ Difficulty ?
QCD ・・・ Strong Coupling Gauge Theory
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☆ QCD → Effective Field Theories
Chiral Symmetry
E
αsEffective
Field
Theory
based on
chiral symmetry
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☆ Effective Field Theories based on the chiral symmetry of QCD
EFT for π ◎ Chiral Perturbation TheoryJ. Gasser and H. Leutwyler, Annals Phys. 158, 142 (1984); NPB 250, 517 (1985)
・ leading order Lagrangian + Skyrme term stable soliton⇒
◎ Hidden Local Symmetry Theory ・・・ EFT for and 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)
ρ ・・・ gauge boson of HLS
Systematic low-energy expansion including dynamical loop expansion ⇔ derivative expansion
e.g., M.H. and K.Yamawaki, Physics Reports 381, 1 (2003)
・ leading order Lagrangian stable soliton⇒
☆ many parameters ! ・・・ not determined by the chiral symmetry
should be detemined from QCD
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Outline
1. Introduction
2. Hidden Local Symmetry
3. Wilsonian Matching in the Chiral Limit
4. Wilsonian matching with Current Quark Masses
5. Summary
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2. Hidden Local Symmetry
[SU(N ) ×SU(N ) ] ×[SU(N ) ] → [SU(N ) ]f f fL R Vglobal local Vf global
[SU(N ) ×SU(N ) ] ×[SU(N ) ] → [SU(N ) ]f f fL R Vglobal local Vf global
U = e = ξ ξ2iπ/ F πL†
R
ξ = e e → h ξ g±iπ / Fπiσ / FσL,R L,R L,R
†ξ = e e → h ξ g±iπ / Fπ±iπ / Fπiσ / Fσiσ / FσL,R L,R L,R
†
F , F ・・・ Decay constants of π and σπ σ
h ∈ [ SU(N ) ]f V local
g ∈ [ SU(N ) ]fL,R L,R global
・ Particles
in the leading order Lagrangian
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QCD quarks and gluons
HLS and
high energy
low energy
Bare theory
bare parameters
Quantum effects
Quantum theory
physical quantities
(perturbative treatment)
matching
~ 1 GeVBoth (perturbative) QCD andHLS are applicable
☆ Basic Concept of Wilsonian matching
integrateout
M.H. and K.Yamawaki, PRD 64, 014023 (2001)
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◎ Generating functional in EFT
J : external source fields
F : parameters of EFT
◎ Generating functional in QCD
☆ Basic Concept of Wilsonian matching
◎ Wilsonian matching
bare theory bare parameter
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☆Matching of axial-vector and vector current correlators
◎ QCD (OPE)
◎ HLS
Matching
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☆ Wilsonian Matching Conditions
◎
◎
◎
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☆ A typical prediction of Wilsonian Matching
・ bare parameters
• M.H. and K.Yamawaki, Phys. Rept. 381, 1 (2003)
◎ ρ - γ mixing strength
good agreement !
+ quantum corrections improved by RGEs
+ + ・・・
π
π
ρ γ
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M.H., T.Fujimori and C.Sasaki, in preparation
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☆ Inclusion of current quark masses in the HLS
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☆ Axial-vector current correlator
・ 2 components because of the explicit chiral symmetry breaking
π-channel
matching
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☆ Matching at Q2 = Λ2
◎ Wilsonian matching conditions
cf: Gelman-Oaks-Renar relation
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☆ Typical predictions M.H., T.Fujimori and C.Sasaki, in preparation
bare parameters
ρ
π
ρ
K
+ quantum corrections improved by RGEs
+ + ・・・
+ + ・・・
very good agreement !
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◎ Wilsonian matching between the HLS and QCD with current quark masses included
Matching of axial-vector current correlator
Determination of the bare parameters
+ quantum corrections improved by RGEs
Physical predictions
very good agreement !
◎ future direction Application of the WM to hot and/or dense QCD with the effect of current quark masses included