◎ Generating functional in QCD

◎ Generating functional in EFT

☆ Wilsonian matching

J : external source fields

F : parameters of EFT

bare theory

QCD quarks and gluons

Bare theory

HLS and

bare parameters

Quantum effects

Quantum theory

physical quantities

matching

high energy

low energy

☆ Axialvector current correlator

in space-like region

◎ low energy limit ・・・ π dominance

◎ high energy region ・・・ Operator Product Expansion (OPE)

・・・ renormalization scale of QCD

F (0)π2

μ 2

Λ ～ 1 GeV

◎ Around Q2 ～ Λ2 ～ (1 GeV)2

・ Integrating out quarks and gluons ・・・ not well-defined degrees of freedom in the low energy region

・ bare HLS Lagrangian including O(p ) terms4

F (0)π2

F (Λ)π2

Λ

◎ Λ ＞ μ ・・・ inclusion of quantum corrections from π and ρ

＃ Λ ＞ μ ＞ m ・・・ through RGEρ

F (0)π2

Λmρ

OPEHLS

μ 2

F (μ)π2

＃ m ＞ μ ・・・ through the RGE in ChPTρ

mρ Λμ 2

OPE

F (0)π2

F (μ)π2

HLSChPT

[F (μ )]π(π ) 2[F (μ )]π(π ) 2

effect of finite renormalization

☆Axialvector and Vector Current Correlators

◎ HLS

◎ QCD (OPE)

Matching

☆ Wilsonian Matching Conditions

◎

◎

◎

☆ Matching Scale Λ • large enough for validity of OPE• small enough for validity of HLS

Λ ＝ 1.0 ～ 1.2 GeV

☆ Bare parameters

☆ Inputs3 Wilsonian matching conditions

・ Inputs for OPE

☆ Values of bare parameters

☆ Values of bare parameters (leading order)

☆ Values of bare parameters (next order)

☆ Parameters at m scaleρ

bare parameters → (RGE) → parameters at mρ

☆ Physical Predictions

4 quantities directly related to experiment

◎ ρ- γ mixing strength

・ tree

・ loop

・ typical prediction

cf :

through RGE

◎ Gasser-Leutwyler’s parameter L10

・ typical prediction

cf :

◎ ρππ coupling

・ bare Lagrangian

loop effects through RGE

・ effective interaction

・ typical prediction

cf :

◎ Gasser-Leutwyler’s parameter L9

・ typical prediction for

cf :

◎ parameter a(0) ・・・ characterize Vector dominance

・ typical prediction

・ bare Lagrangian

loop effects through RGE

・ effective interaction

・・・ VD is well satisfied

◎ running of a

◎ Summary of Predictions

☆ Why π - π mass difference ? + 0

◎ ⇔ vacuum structure

M.E. Peskin 80’, J. Preskil 80’

⇒ stability of U(1)em symmtric vacuum

⇒ instability : U(1)em is broken

◎ ⇔ mass of little Higgs

M.H. M.Tanabashi and K.Yamawaki, Phys. Lett. B 568 103 (2003)

6.5 π+ - π0 Mass Difference and Wilsonian Matching

☆ How to calculate ?

◎ A formula from Dashen’s theorem

bare parameter improve by RGE

☆ Determination of the bare parameter

☆ Prediction

Quantum correction through RGE

> 0

in good agreement