5. Chiral Perturbation Theory with HLS
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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
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
