5. Chiral Perturbation Theory with HLS

44

description

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

Page 1: 5. Chiral Perturbation Theory with HLS
Page 2: 5. Chiral Perturbation Theory with HLS
Page 3: 5. Chiral Perturbation Theory with HLS

☆ Expansion Parameter

◎ ordinary ChPT for π

chiral symmetry breaking scale

☆ Order Counting

・・・ same as ChPT

◎ ChPT with HLS

Page 4: 5. Chiral Perturbation Theory 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

Page 5: 5. Chiral Perturbation Theory with HLS
Page 6: 5. Chiral Perturbation Theory with HLS

☆ Building blocks

◎ ρ and π fields

h ∈ [ SU(N ) ]f V local

transform homogeneously

Page 7: 5. Chiral Perturbation Theory with HLS

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

Page 8: 5. Chiral Perturbation Theory with HLS

☆ Lagrangian at O (p )2

π mass term

F = F at leading orderχ π

Page 9: 5. Chiral Perturbation Theory with HLS

☆ Lagrangian at O (p )4

○ Identities

○ Equations of motions for π, σ, ρ

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

Page 10: 5. Chiral Perturbation Theory with HLS

◎ Terms generating vertices with at least 4-legs

15 independent terms for N = 3f

9 independent terms for N = 2f

Page 11: 5. Chiral Perturbation Theory with HLS

◎ Terms with χ^

7 independent terms for N = 2f

Page 12: 5. Chiral Perturbation Theory with HLS

◎ Terms with V , V or A^μν μν μν^

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

Page 13: 5. Chiral Perturbation Theory with HLS

Importance of quadratic divergence in phase transition

Page 14: 5. Chiral Perturbation Theory with HLS

☆ NJL model

Model is defined with cutoff Λ●

Page 15: 5. Chiral Perturbation Theory with HLS

◎ Auxiliary field method

;

◎ Effective potential in “chain” approximation

Page 16: 5. Chiral Perturbation Theory with HLS

◎ Stationary condition (Gap equation)

=

self consistency condition

Page 17: 5. Chiral Perturbation Theory with HLS

◎ Phase structure

;

Page 18: 5. Chiral Perturbation Theory with HLS

◎ Phase change ・・・ triggered by quadratic divergence

Phase of bare theory ≠ Phase of quantum theory

● at bare level

Page 19: 5. Chiral Perturbation Theory with HLS
Page 20: 5. Chiral Perturbation Theory with HLS

☆ Background fields

background field

quantum field

background field

quantum field

Page 21: 5. Chiral Perturbation Theory with HLS

☆ Background fields including external gauge fields

Page 22: 5. Chiral Perturbation Theory with HLS

☆ Transformation properties

Page 23: 5. Chiral Perturbation Theory with HLS

☆ Gauge fixing and FP ghost

three or more quantum fields are included

Page 24: 5. Chiral Perturbation Theory with HLS

☆ Lagrangian

tree contribution

quantum correction at one loop

equations of motion for backgroud fiels

Page 25: 5. Chiral Perturbation Theory with HLS
Page 26: 5. Chiral Perturbation Theory with HLS

☆ RGEs for F and zπ 2

1-loop contributions

quadratic divergence

calculated from A - A two point functionμ ν

Page 27: 5. Chiral Perturbation Theory with HLS

Renormalization

Page 28: 5. Chiral Perturbation Theory with HLS

☆ RGEs for F and zπ 2

effect of quadratic divergences

Page 29: 5. Chiral Perturbation Theory with HLS

☆ RGEs for F and zσ 1

quadratic divergences

calculated from V - V two point functionμ ν

Page 30: 5. Chiral Perturbation Theory with HLS

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

Page 31: 5. Chiral Perturbation Theory with HLS

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

Page 32: 5. Chiral Perturbation Theory with HLS

☆ RGEs for F , a and gπ

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

Page 33: 5. Chiral Perturbation Theory with HLS

☆ RGEs for z , z and z1 2 3

parameters of O(p ) Lagrangian4

Page 34: 5. Chiral Perturbation Theory with HLS

☆ 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

Page 35: 5. Chiral Perturbation Theory with HLS

◎ running of Fπ2

0

ChPT HLS

μ

(86.4MeV)2

[F (μ)]π(π) 2

F (μ)π2

2

2

Page 36: 5. Chiral Perturbation Theory with HLS

◎ running of a

Page 37: 5. Chiral Perturbation Theory with HLS
Page 38: 5. Chiral Perturbation Theory with HLS

☆ 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.

Page 39: 5. Chiral Perturbation Theory with HLS

☆ RGEs

◎ on-shell condition

◎ order parameter

Page 40: 5. Chiral Perturbation Theory with HLS

☆ Fixed points (line)

・・・ unphysical

Page 41: 5. Chiral Perturbation Theory with HLS

☆ Flow diagram on G = 0 plane

symmetric phase

broken phase

VM

Page 42: 5. Chiral Perturbation Theory with HLS

☆ Flow diagram on a = 1 plane

symmetric phase

broken phase

ρ decoupled

VM

Page 43: 5. Chiral Perturbation Theory with HLS

☆ phase boundary surface

Page 44: 5. Chiral Perturbation Theory with HLS

☆ Vector dominance

characterized by

・ In N = 3 QCD ~ real worldf