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5. Chiral Perturbation Theory with HLS. 5.1 Derivative Expansion in the HLS. ☆ Expansion Parameter. ◎ ordinary ChPT for π. chiral symmetry breaking scale. ◎ ChPT with HLS. ☆ Order Counting. ・・・ same as ChPT. ２. may cause 1/ m corrections. ρ. ・・・ well-defined limit of m → 0. - PowerPoint PPT Presentation

### Transcript of 5. Chiral Perturbation Theory with HLS   ☆ 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

calculated from A - A two point functionμ ν Renormalization ☆ RGEs for F and zπ 2 ☆ RGEs for F and zσ １

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

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