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A Model Study on Meson Spectrum

and Chiral Symmetry Transition

Da Huang

@NTU

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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

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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 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 Hints1. Pseudoscalars are much lighter than their chi

ral partners are Goldstone bosons of spontaneous chiral symmetry breaking

,,K

VRL SUSUSU )3()3()3(

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 sigma-model predict: Pisarski & Wilczek 84'

2 flavors: 2nd Order; 3 or more flavors: 1st Order (no IR FP)• current quark mass term complicates the s

ituation: Lattice QCD: Crossover Aoki et al 05' Nature Field Theory Model: mostly Crossover, NJL, PNJL, Quar

k-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

• Effects:1. Heat other particles due to entropy conservation;2. If a strong first-order phase transition, then bubble

s of hadron gas are formed. The bubble growth, collision and mergence can generate gravitational waves

3. The initial inhomogeneities have strong effects on BBN

0

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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 ModelAt Zero Temperature

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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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The Derivation of Effective Action

Let us focus on the quark part of the Lagrangian

Instead, if we integrate out the quark fields, we can obtain the effective Lagrangian for mesons.

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 ActionIn fact, we already obtain the desired form of the action. But a further step is need for generalizationto finite temperature

If we integrate chi-field back, we obtain

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The Derivation of Effective ActionIn order to take into account the constitute quarkmass contribution, we can split the Lagrangian into two parts

Have choosing

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The Derivation of Effective ActionIntegrating 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

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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Closed-Time-Path FormalismClosed-Time-Path formalism makes the finitetemperature effective Lagrangian rather simpleby 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 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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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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THANKSTHANKS

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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.