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Chiral Symmetry Breaking and Restoration in QCD

Da Huang

Institute of Theoretical Physics, Chinese Academy of Sciences@Chongqing

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Content

• Dynamically Generated Spontaneous Chiral Symmetry Breaking In Chiral Dynamical Model At Zero Temperature

• Chiral Thermodynamic Model of QCD and its Critical Behavior

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Dynamically Generated Spontaneous Chiral Symmetry Breaking

In Chiral Dynamical ModelAt Zero Temperature

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Chiral limit: Taking vanishing quark masses mq→ 0.

QCD Lagrangian

qq

s

d

u

q

GgD

GGqiDqqiDqL

LR

s

RRLLoQCD

)1(2

1

2/

4

1

5,

)(

has maximum global Chiral symmetry :

)1()1()3()3( BARL UUSUSU

QCD Lagrangian and SymmetryQCD Lagrangian and Symmetry

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The Problem

However, in the low-energy meson spectrum, we have not seen such symmetry patterns.

The natural question is how these (exact or approximate) symmetries are broken (explicitly or spontaneously)?

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QCD Lagrangian and SymmetryQCD 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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Dynamically Generated Spontaneous Symmetry Breaking

Instead, if we integrate out the quark fields, we can obtain the following effective potential.

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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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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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Low Energy Dynamics Of QCD

g

q q

q q

Dynamically Generated Higgs Potential For Scalar Mesons

V() = - 2 2 + 4

Spontaneous Symmetry Breaking

Pseudoscalaris

Goldstone

Scalar field is

Higgs

quark

Mesons

Composite Higgs FieldsComposite Higgs Fields

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Scalars as Partner of Pseudoscalars

Scalar mesons:

Pseudoscalar mesons :

According to their quantum numbers, we can reorganize the mesons into a matrix form.

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Mass Formula Pseudoscalar mesons :

Instanton interactions only affect mass of

SU(3) singlet

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Mass Formula

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Predictions for Mass Spectra & Mixings

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Predictions

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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׳

Chiral Symmetry BreakingChiral Symmetry Breaking-- Answer to Our Primary Question -- Answer to Our Primary Question

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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 Lagrangianby DH, Y-L Wu, 1110.4491 [hep-ph]

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Dynamically Generated Spontaneous Symmetry Breaking

By integrating out the quark fields with Closed Time Path (CTP) formalism, we obtain the following effective potential at finite temperature

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Dynamically Generated Spontaneous Symmetry Breaking

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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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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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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 Restorationat Finite Temperature

Vacuum Expectation Value (VEV)

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Chiral Symmetry Restorationat Finite Temperature

Pion Decay Constant

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Chiral Symmetry Restorationat Finite Temperature

Quark Condensate

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Chiral Symmetry Restorationat Finite Temperature

Mass for Pseudoscalar mesons

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Chiral Symmetry Restorationat Finite Temperature

This feature can be traced back to the fact that all of these quantities are proportional to the VEV near the critical temperature

All of these quantities have the same scaling behavior near the critical temperature,

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Summary and Outlook• In the simple chiral dynamical model, Prof. Da

i and Prof. Wu derived a consistent spectrum of lowest lying meson states.

• Within this model, we discuss the chiral symmetry restoration at finite temperature by determining the critical temperature and critical behavior of some characteristic quantities.

• In the future work, we need to consider finite chemical potential effects. Also, we also want to extend the current work into three flavor case and consider instanton finite temperature effects.

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THANKSTHANKS

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Back up

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Dynamically Generated Spontaneous Symmetry Breaking