Restoration of chiral symmetry and vector meson in the generalized hidden local symmetry

Restoration of chiral symmetry and vector meson in the generalized

hidden local symmetry

Munehisa Ohtani (RIKEN)Osamu Morimatsu （ KEK ）

Yoshimasa Hidaka(TITech)

Based on Phys. Rev. D73,036004(2006)

2005/11/14 Yoshimasa Hidaka 2

Outline

• Introduction– Vector meson in the medium– Restoration pattern of chiral symmetry

• Chiral restoration: π, ρ and A1 system– Generalized Hidden Local Symmetry(GHLS)– Restoration patterns

• Summary


Introduction

QCD phase diagram Restoration of chiral symmetry

0

l

l

medium

Lepton pair


becomes larger by interaction with medium particle

Restoration of chiral symmetry

1

1

* ***a

a

mmqq f

qq f m m

Rapp-Wambach, Asakawa-Ko, .....

Spe

ctra

l fun

ctio

n

Mass

1a

1a

Spe

ctra

l fun

ctio

n

Mass

Spe

ctra

l fun

ctio

nMass

Dropping Masses

Melting Resonances

1a

1aQSR, B-R scaling….Hatsuda-Lee, Brown-Rho….


Experiment•p-Au, Pb-Au collisions@CERES

Phys. Lett. 405(1998)nucl-ex/0504016

KEK-PS E325

excess

excess


NA60 and our calculation

Y.H. Ph.D Thesis(2005)

Chiral limit

NA60

By E. Scomparin’s talk @QM2005

ρspetrum in the HLS model with OPT


Restoration Scenario of Chiral symmetry

45

Which representation does the pion belong at restoration point ?

Chiral partner of πScalar mesonσ

Chiral partner of πVector meson(longitudinal)ρ

Experiment: (F. J. Gilman and H. Harari (1968), S. Weinberg(1969))

Standard scenario Vector Manifestation scenario

Helicity zero state

(Harada, Yamawaki(2001))

2 2f

* *f f ff fsin c( ( 1,1) (1, 1, ) ( , ,) os )NN N N NN

2 2f f( 1,1) (1, 1)N N

* *f f f f( , ) ( , )N N N N * *

f f f f12 2

f fcos s ( 1,1)in (1, ,1( , ) ( , ) )NN N NA NN

* *f f f f( , ) ( , )N N N N

* *f f f f( , ) ( , )N N N N

2 2f f( 1,1) (1, 1)N N

2 2f f( 1,1) (1, 1)N N

L RSU( ) SU( )f fG N N

0 or 90 ?

90 0


Necessity of A1

1A 1A

It is necessary to incorporate A1.

ρ pairs with A1 in the standard scenario.

mixing

chiral partner

chiral partner

(F. J. Gilman and H. Harari (1968), S. Weinberg(1969))mixing angle 45

Vector Manifestation with Hidden Local Symmetry model(π, ρ)

The STAR Ratio0 /

(Brown, Lee and Rho nurcl-th/0507073)A1 plays important role.

Large-Nf(Harada and Yamawaki)

Finite-T(Harada and Sasaki)

Finite-μ(Harada, Kim and Rho)


Generalized Hidden Local Symmetry (Bando, Kugo, Yamawaki (1985))

L RSU( ) SU( )f fG N N

Global LocalG G

The GHLS model is an effective model based on the non-linear sigma model including π, ρ 　 and A1.

π ： NG bosons associated with global chiral symm. breaking.ρ, A1 : Gauge bosons σ, p ： NG bosons associated with local chiral symm. breaking.　　　　　　　⇒ σ and p are absorbed by the ρ and A1.

VSU( )fN

Chiral symmetry


Generalized Hidden Local Symmetry

2 22 2tr[( ) ](tr[ ] tr[ ] )V MF a L

1

2 2 2AF F

2 2 2M ag F

The Lagrangian at lowest order

1

2 2 2AM g F

2 2F aF masses Decay constants

γ: mixing parameter between　 π and A1

( ) ( )( ) ( )1tr[ ]

1tr[

22]V V A AF FF F

parameters

π-A1 mixing

ρππ coupling 2(1 )2

ag g


Restoration Pattern in the GHLS model

1

1

2

2 2

2(GHLS) 2

22

( )

( )

( )( ) ( )A

AA

Fp z

p

F

M p

2(GHLS) 2

12 2

( )( ) ( ) ,

( )V

Fp z

M p

The vector and axial vector current correlators

Restoration condition(GHLS) 2 (GHLS) 2 2( ) ( ), (0) 0V Ap p F

(GHLS) 2 (GHLS) 2( , ) ( , )V A

d dp p

d d

Renormalization group invariance at restoration point.

This condition should be satisfied for all energy scale.


Renormalization Group Equations

One-loop diagrams RGEs are given by calculating these diagrams.


2 2 42 2

2 2 2 22 2 2 2 2 2 4

2 2 2 2 2 42 2 2

2

2 22 2 2 2 2 2 4

2

[(2 2 (1 )2 2 2 2

3( 2 )]

2

1 3 2 3( 2 2 3 )2 2 2 2

3(1 2 2 )

2

(2 2 2 2

dx a a ax x a

d a

aG a a a a

a

da a a a ax a a a

d

a aGa a a a a

d ax a a

d

2 2

2 4

2 3 22 2 2 2 2 2 2 4

2 2 2 2 2 22 2 2

2 2

2 2 2 2 22 2 2 2 2 2 2 4

2

2)

3( 3 4 2 2 )

2

1 2 22 ( ) 3 ( 1 )

44 1 2 2 2( (5 2 2 2 ))

3 24

aa

aG a a a a a

a

d a a a a a a ax G a a

d a

dG a a aG a a a a

d a

Renormalization Group Equations

Renormalization Group Equations2

2 2( ) ,

(4 ) ( )fNxF

22

( ) ( )(4 )

fNG g


Restoration condition and renormalization invariance

Restoration Pattern

1 1 1

2 2 22

2 2

( ) (0)tan

(0) (0)A A AF M F

F F

Mixing angle

(GHLS) 2 (GHLS) 2( , ) ( , )V A

d dp p

d d

(GHLS) 2 (GHLS) 2( ) ( )V Ap p

Fixed point (μ→0)

(I)VM 1 0 0 0

(II)VM

Standard

Intermediate

1

∞

3/2

0

∞

1/2

0

1

1/3

0

0

arctan1/ 2

/ 2

1

2 2 2/ 1, 0, 0AF F F

1

1

2 2 2

2 2 /

A

A

F F F

F F

2 2/F F 1

2 2/AF F

1

2 21 20, 0,AM M g z z

1

1

2

2 2

2(GHLS) 2

22

( )

( )

( )( ) ( )A

AA

Fp z

p

F

M p

2(GHLS) 2

12 2

( )( ) ( ) ,

( )V

Fp z

M p


Pion form factor and VMD( ') (0) ( ) ( ' ) ( ')a c b abc

Vp p F p p p p V

V

V

22 2 2

V 2 2( ) 1 (1 ) (1 )

2 2

Ma aF p

M p

VMD21 (1 ) 0

2

a

VM fixed point2 1

1 (1 )2 2

a 1 and =0a

Intermediate fixed point 3 1and =

2 3a 2 1

1 (1 )2 3

a

Standard fixed point21 (1 ) 0

2

a 1and =1

1a


Decay width

2(1 )2

ag g

The rho-pi-pi coupling becomes weak due to .

ρ becomes stable.

Dileption emission

A narrow ρ peak but weak magnitude in the pion background will be observed.

0g

The difference of restoration patterns will appear as a quantity of pion background and how the rho meson becomes narrow.


Summary• We have studied the restoration patterns of the chiral symmetry in G

HLS model including A1 in addition to π and ρ.• By including A1 we have found three possibilities of the restoration p

atterns, – the Standard– VM – Intermediate

• Both ρ and A1 masses drop in all scenarios• These three scenarios are different in the violation of the vector mes

on dominance.• Further analysis

– finite temperature and/or density, – Large-Nf.

Restoration of chiral symmetry and vector meson in the generalized hidden local symmetry

Documents

Transcript of Restoration of chiral symmetry and vector meson in the generalized hidden local symmetry