A Model Study on
Meson Spectrum
and Chiral Symmetry Transition
Da Huang
@NTU
1
Content
• General Overview
• Dynamically Generated Spontaneous
Chiral Symmetry Breaking In Chiral
Dynamical Model At Zero Temperature
• Chiral Thermodynamic Model of QCD
and its Critical Behavior
2
QCD Lagrangian and Symmetry
Chiral limit: Taking vanishing quark masses mq→ 0.
QCD Lagrangian
(o)
QCD
L
1  
 qL  iD qL  qR  iD qR  G G
4
D     g s  / 2G
u 
 
q  d 
s 
 
1
qR , L  (1   5 )q
2
has maximum global Chiral symmetry :
SUL (3)  SUR (3) U A (1) U B (1)
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The Reality
插入meson spectrum 图片
No so many symmetries in meson spectrum.
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The Reality
'
much heavier than other light pseudoscalars.
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The Problem
1. How are these (exact or
approximate) symmetries in QCD
broken (explicitly or spontaneously)?
2. Can we find some fundamental laws
that govern the seemingly messy
meson spectrum?
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Some Hints
1. Pseudoscalars are much lighter than their
 , K , are Goldstone
chiral partners
bosons of spontaneous chiral symmetry
breaking
SU(3) L  SU(3) R  SU(3)V
2. U(1)_A symmetry is anomalous, so there is
no such symmetry at low energy.
'
is not the Goldstone boson any more.
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Previous Efforts
• Lattice Calculations
Advantage: first principle
Shortcoming: lack transparency;
Nielsen-Ninomiya No-Go theorem to limit the dynamical
fermion
• Model Calculation: NJL model
Advantage: easy to understand chiral symmetry breaking
Shortcoming: difficult to derive it from first principle
• AdS/CFT Models: Model dependent
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Finite Temperature Effects
• Just as magnets
When temperature is high enough, then the
symmetry will be restored.
-- general for any SSB
system
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Finite Temperature Effects
• We expect that chiral symmetry should be
restored at high enough T
• Main Question:
What is the nature of chiral symmetry
restoration?
? Phase transition OR Crossover
? If phase transition, what is the order
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Previous Efforts
• In the chiral limit, general linear sigmamodel predict: Pisarski & Wilczek 84'
2 flavors: 2nd Order;
3 or more flavors: 1st Order (no IR FP)
• current quark mass term complicates the
situation:
Lattice QCD: Crossover Aoki et al 05' Nature
Field Theory Model: mostly Crossover, NJL, PNJL,
Quark-Meson model,... (but Model dependent)
AdS/CFT model: model dependent
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Big Bang Cosmology
• Chiral symmetry breaking happens at 10^-6 s
(T~200 MeV) after the Big Bang
• The net baryon density in Universe is quite
small(<< typical hadron mass), so   0
• Effects:
1. Heat other particles due to entropy conservation;
2. If a strong first-order phase transition, then
bubbles of hadron gas are formed. The bubble
growth, collision and mergence can generate
gravitational waves
3. The initial inhomogeneities have strong effects on
BBN
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Big Bang Cosmology
However, if QCD transition is a crossover,
as indicated by Lattice QCD and many
model calculations, then it is difficult to find
astrophysical evidence for this transition
In summary, the strength of the QCD transition
determines its significance in the evolution of
our Universe.
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Dynamically Generated
Spontaneous Chiral Symmetry Breaking
In Chiral Dynamical Model
At Zero Temperature
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QCD Lagrangian and Symmetry
• QCD Lagrangian with massive light quarks
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Effective Lagrangian
Based on Loop Regularization
Y.B. Dai and Y-L. Wu, Euro. Phys. J. C 39 s1 (2004)
Note that the field Φ has no kinetic terms, so it is an
auxiliary field and can be integrated out.
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The Derivation of Effective Action
Instead, if we integrate out the quark fields, we
can obtain the effective Lagrangian for mesons.
Let us focus on the quark part of the Lagrangian
The path integral method gives
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The Derivation of Effective Action
The Effective Action is
Note that the imaginary part corresponds to
the anomaly part, which we ignore here.
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The Derivation of Effective Action
With the identities:
and
we can obtain:
where
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The Derivation of Effective Action
In fact, we already obtain the desired form of the
action. But a further step is need for generalization
to finite temperature
If we integrate chi-field back, we obtain
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The Derivation of Effective Action
In order to take into account the constitute quark
mass contribution, we can split the Lagrangian
into two parts
Have choosing
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The Derivation of Effective Action
Integrating out the momentum integrals gives
where
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Dynamically Generated
Spontaneous Symmetry Breaking
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Dynamically Generated
Spontaneous Symmetry Breaking
By differentiating the effective potential w.r.t. the VEVs,
we obtain the following Minimal Conditions/Generalized
Gap Equations:
If we take the limit of vanishing instanton effects, we can
recover the usual gap equation
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Dynamically Generated
Spontaneous Symmetry Breaking
Recall that the pseudoscalar mesons, such as π,η, are
much lighter than their scalar chiral partners. This
indicates that we should choose our vacuum expectation
values (VEVs) in the scalar fields:
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Scalars as Partner of Pseudoscalars
According to their quantum numbers, we can reorganize the
mesons into a matrix form.
Scalar mesons:
Pseudoscalar mesons :
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Mass Formula
Pseudoscalar mesons :
Instanton interactions
only affect mass of
SU(3) singlet 27
Mass Formula
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Predictions for Mass Spectra &
Mixings
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Predictions
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Chiral Symmetry Breaking
-- Answer to Our Primary Question



Chiral SUL(3)XSUR(3)
spontaneously broken
Goldstone mesons
π0, η8
Chiral UL(1)XUR(1)
breaking Instanton Effect
of anomaly
Mass of η0
Flavor SU(3) breaking
The mixing of
π0, η and η‫׳‬
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Chiral Thermodynamic Model
of QCD
and its Critical Behavior
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Motivation
Provided that chiral dynamical model
works so well to show chiral symmetry
breaking and to predict the meson
spectrum, we want to see what happens
for this model at finite temperature
-- Chiral symmetry will be restored at high
enough temperature
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Effective Lagrangian
by DH, Y-L Wu, 1110.4491 [hep-ph]
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Closed-Time-Path Formalism
Closed-Time-Path formalism makes the finite
temperature effective Lagrangian rather simple
by just replacing zero-temperature propagator
with its finite temperature counterpart
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Finite Temperature Effective Action
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Dynamically Generated
Spontaneous Symmetry Breaking
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Dynamically Generated
Spontaneous Symmetry Breaking
By differentiating the effective potential with respect to the
VEVs, we can obtain:
After we take the limit of zero current quark masses, the
gap equation can be simplified to
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Critical Temperature
For Chiral Symmetry Restoration
In order to determine the critical temperature for chiral
symmetry restoration, we need to make the following
assumption:
Note that μf 2 (T) is the coupling of the four quark
interaction in our model. In principle this interaction comes
by integrating out the gluon fields. Thus, this assumption
indicates that the low energy gluon dynamics have the
same finite temperature dependence as the quark field.
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Critical Temperature
For Chiral Symmetry Restoration
With the above assumption, we can determine the critical
temperature for chiral symmetry restoration in the chiral
thermodynamic model (CTDM)
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Dynamically Generated
Spontaneous Symmetry Breaking
Recall that the pseudoscalar mesons, such as π,η, are
much lighter than their scalar chiral partners. This
indicates that we should choose our vacuum expectation
values (VEVs) in the scalar fields:
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Scalars as Partner of Pseudoscalars
&
Lightest Composite Higgs Bosons
Scalar mesons:
Pseudoscalar mesons :
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Mass Formula
Pseudoscalar mesons :
Scalar mesons (Lightest Composite Higgs Boson) :
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Predictions for Mass Spectra
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Chiral Symmetry Restoration
at Finite Temperature
Vacuum Expectation Value (VEV)
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Chiral Symmetry Restoration
at Finite Temperature
Pion Decay Constant
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Chiral Symmetry Restoration
at Finite Temperature
Quark Condensate
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Chiral Symmetry Restoration
at Finite Temperature
Mass for Pseudoscalar mesons
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Chiral Symmetry Restoration
at Finite Temperature
All of these quantities have the same scaling
behavior near the critical temperature,
This feature can be traced back to the fact that all of
these quantities are proportional to the VEV near
the critical temperature
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Implications of Our Study
• As discussed before, if the QCD transition
is a true 2nd phase transition, then there
must be some dramatic changes in the
physical quantities, thus it can leave some
imprint in the early universe
• gravitational waves
• inhomegeneity in the BBN
• ...
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THANKS
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Lattice Phase Diagram
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Little Bang: Heavy Ion Collision
HIC provides us the
possibility to test the
order of restoration
experimentally.
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