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
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Introduction
QCD phase diagram Restoration of chiral symmetry
0
l
l
medium
Lepton pair
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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….
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Experiment•p-Au, Pb-Au collisions@CERES
Phys. Lett. 405(1998)nucl-ex/0504016
KEK-PS E325
excess
excess
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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
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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
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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)
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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
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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
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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.
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Renormalization Group Equations
One-loop diagrams RGEs are given by calculating these diagrams.
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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
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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
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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
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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.
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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.
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