☆ Expansion Parameter

◎ ordinary ChPT for π

chiral symmetry breaking scale

☆ Order Counting

・・・ same as ChPT

◎ ChPT with HLS

☆ Importance of Gauge Invariance

◎ In Matter Field Method

may cause 1/m correctionsρ２

◎ In HLS with R -like gauge fixing

gauge invariance

・・・ well-defined limit of m → 0ρ

ξ

☆ Building blocks

◎ ρ and π fields

h ∈ ［ SU(N ) ］f V local

transform homogeneously

Current quark masses can be included ・・・

S, P ・・・ scalar and pseudoscalar external sources

◎ external fields

L , R ; gauge fields of SU(N )μμ f L,R

transform homogeneously

☆ Lagrangian at O (p )2

π mass term

F = F at leading orderχ π

☆ Lagrangian at O (p )4

○ Identities

○ Equations of motions for π, σ, ρ

◎ Useful Relations → specify independent terms at O(p )4

◎ Terms generating vertices with at least 4-legs

15 independent terms for N ＝ 3f

9 independent terms for N ＝ 2f

◎ Terms with χ^

7 independent terms for N ＝ 2f

◎ Terms with V , V or A＾μν μν μν＾

z , z , z ・・・ contribute to 2-point functions1 2 3

Importance of quadratic divergence in phase transition

☆ NJL model

Model is defined with cutoff Λ●

◎ Auxiliary field method

;

◎ Effective potential in “chain” approximation

◎ Stationary condition (Gap equation)

=

self consistency condition

◎ Phase structure

;

◎ Phase change ・・・ triggered by quadratic divergence

Phase of bare theory ≠ Phase of quantum theory

● at bare level

☆ Background fields

background field

quantum field

background field

quantum field

☆ Background fields including external gauge fields

☆ Transformation properties

☆ Gauge fixing and FP ghost

three or more quantum fields are included

☆ Lagrangian

tree contribution

quantum correction at one loop

equations of motion for backgroud fiels

☆ RGEs for F and zπ 2

1-loop contributions

quadratic divergence

calculated from A - A two point functionμ ν

Renormalization

☆ RGEs for F and zπ 2

effect of quadratic divergences

☆ RGEs for F and zσ １

quadratic divergences

calculated from V - V two point functionμ ν

☆ RGE for g calculated from V - V two point functionμ ν

☆ RGE for z 3calculated from V - V two point functionμ ν

☆ RGEs for F , a and gπ

NOTE : (g, a) = (0, 1) ・・・ fixed point

☆ RGEs for z , z and z1 2 3

parameters of O(p ) Lagrangian4

☆ RGE for F at μ < mπ ρ ρ decouples at μ = m ρ

F , g do not run at μ < m ρσ

F does run by π- loop effectπ

◎ Effect of finite renormalization

◎ Physical Fπ

◎ running of Fπ2

0

ChPT HLS

mρ

μ

(86.4MeV)2

[F (μ)]π(π) 2

F (μ)π2

2

2

◎ running of a

☆ Phase change can occur in the HLS

・ illustration with (g, a) = (0,1) ・・・ fixed point

(RGE for F is solved analytically)π

・ 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

broken phase

VM

☆ Flow diagram on a = 1 plane

symmetric phase

broken phase

ρ decoupled

VM

☆ phase boundary surface

☆ Vector dominance

characterized by

・ In N = 3 QCD ～ real worldf