Teiji Kunihiro 　 (YITP, Kyoto)

Dubna round table session July 7-9, 2005 JINR, Dubna, Russia

Status of the chiral symmetry restoration and sigma-meson physics

Contents

Introduction : vacuum structure v.s. elementary excitations

The sigma meson Chiral restoration and the sigma meson Chiral restoration as seen in other channels including chiral anomaly Summary

A condensed matter physics of vacuum (Y. Nambu; 1960)

Gauge invariance

Axial gauge (chiral) symm.

(Bogoliubov-Anderson)

Inter-deterministic property of the matter and vacuum in QFT

0 0a 0 ; vacuum

In the definition of vacuum in QFT, the definition of the particle picture is pre-requisite.

Equivalence between what is the vacuum and what are the particles to be observed.

Change of the vaccum Change of the particle picture

Eg. Superconductivity

The definition of vacuum:

常伝導 超伝導

gap

E E

Change of vacuum Change in the elementary excitations

22

Dispersion relation of electron-quasi particle in the normal metaland superconducting matterial

and Bogoliubov-Anderson mode

Chiral Transformation

Chirality:

3fNFor , the chiral transformation forms

(left handed)

(right handed)

In the chiral limit (m=0),

qDq

; Chiral invariant

;Chiral invariant!Ｘ

Chiral Invariance of Classical QCD Lagrangian in the chiral limit (m=0)

qq invariant!

L RR Lqq q q q q not invarinat! Dirac mass term

q q order parameter

aQ the generators of a continuous transformation

;0 aj )(xja Noether current )(0 xjdQ aa x

eg. Chiral transformation 　 for

qqdQ aa 2/50

5 x

The two modes of symmetry realization in the vacuum :

a. Wigner mode

0|00 aQ

b. Nambu-Goldstone mode 00 aQ

The notion of Spontaneous Symmetry Breaking

)()()()(, 55 xqxqxqixqiQ abba Notice;

The symmetry is spontaneously broken.

0,000 55 qqQqq aa Now,

00 qq 0 005 aQChiral symmetry is spontaneously broken!

aa

)2()2( RL SUSU

QCD 真空の非摂動的性質（補）

Gell-Mann-Oakes-Renner

using

We have

QCD sum rules for heavy-quark systems,

Non-perturbative properties of QCD vacuum: condensates

Confinement and chiral transion in Lattice QCD

CSC

T

SB

QGPQCD　critical point

meson condensation?

?

0

CFL

QCD phase diagram

H matter?

~150MeV

The sigma mesonThe sigma meson

•Real or not?•If real, what is it?

The significance of the meson in low energy hadron physics and QCD

1. The pole in this mass range observed in the pi-pi S-matrix. As a compilation of the pole positions of the obatined in the modern analyses: Significance of respecting chiral symmetry,unitarity and crossing symmetry to reproduce the phase shifts both in the (s)- and , (t)-channels with a low mass pole;(Igi and Hikasa(1999)).

2. Seen in decay processes from heavy particles; E. M. Aitala et al, Phys. Rev. Lett. (86), 770 (2001)

3. Responsible for the intermediate range attraction in the nuclear force.

4. Accounts for I=1/2 enhancement in K ->2 compared with K+->. E.P. Shabalin (1988); T. Morozumi, C.S. Lim and I. Sanda (1990).

-N sigma term 40-60 MeV (naively » 15 MeV) enhanced by the collectiveness of the (.T.Hatsuda and T.K.(1990)) ; see the next slide.

6. The of the chiral order parameter The Higgs particle in the WSG model

Chiral Transition and the collective modesChiral Transition and the collective modes

0

c.f. Higgs particle in WSH model

; Higgs fieldHiggs particle

K. Igi and K. Hikasa, Phys. Rev. D59, 034005(1999)

The phase shifts in the sigma and rho channel in the N/DMethod; resp. chiral symm., crossing symm and so on.

No but in the t-channeland the in the t-channel

Both with the in the s-

The poles of the S matrix in the complex mass plane forthe sigma meson channel: complied in Z. Xiao and H.Z. Zheng (2001)

G.Colangero, J. Gasser and Leutwyler (2001)

E. M. Aitala et al, Phys. Rev. Lett. (86), 770 (2001)

Without sigma pole

With a sigma pole:

2423

4240

The numbers in ( , ) are those in the naive quark model.(T.K. and T. Hatsuda, Phys. Lett. B240 (1990) 209)

The quark content (or the scalar charge of the quarks) is enhanced by the collective mode in the scalar channel!

flavor mixing

Issues with the low-mass Issues with the low-mass meson in meson in QCDQCD

• In the constituent quark model; 　　 the mass in the 1.2 --- 1.6 GeV region.

Some mechanism needed to down the mass 　 with ~ 600 MeV;

• (i) Color magnetic interaction between the di-quarks? (Jaffe; 1977)

• (ii) The collectiveness of the scalar mode as the ps mode; a superposition of states. Chiral symmetry (NJL)

• (iii) The - molecule as suggested in scatt. . (vi) a mixed state of scalar glue ball and states

• I=0 ;

I=1; a 0 = 0 ++ , a 1 = 1 ++ , a 2 =2 ++ , b 1 = 1 +-

3P0 3P1

3P2 1P1

We need some nontrivial dynamics to down the massas low as 500 – 600 MeV!

in the constituent quark model

They all should be in the same mass range 1.2 – 1.6 GeV

Scalar Mesons in the Di-quark picture

(Jaffe(1977), Alford and Jaffe (2000))

The Scalar mesons on the Lattice

The Scalar Collaboration:S. Muroya,A. Nakamura,C. Nonaka,M. Sekiguchi,H. Wada,T. K.

(Phys. Rev. D70, 034504(2004))

---- A full QCD calculation -----

Simulation parameters

Lattice size : 83 × 16

= 4.8

= 0.1846, 0.1874,

0.1891　CP-PACS well established light meson with large lattice

a = 0.197(2) fm , c = 0.19286(14) ( CP - PACS, Phys. Rev. D60(1999)114508 )

Number of the Z 2 noise = 1000

Wilson Fermions & Plaquette gauge actionDisconnected diagram

Propagator for Propagator for meson (2)meson (2)

Where

))()()((2

),(),(),( 11

yx

xyWyxTrWyxG

)()(),()( 1 xxxxTrWx

Connected diagramq

q

Disconnected diagram

- Vacuum contribution

m_

The meson masses

mm//mm

m/m mcon/m

0.1846 1.583±0.098 2.400±0.018

0.1874 1.336±0.071 2.436±0.025

0.1891 1.112±0.060 2.481±0.031

meson propagatorsmeson propagators 　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　

Connected Part Connected Part & & Disconnected Parts Disconnected Parts (( = 0.1891= 0.1891 ) )

Chiral Transition = a phase transition of QCD 　 vacuum,

being the order parameter. Lattice QCD;eg. F. Karsch, Nucl. Phys. Proc. Suppl. 83, 14 (2000).The wisdom of many-body theory tells us:If a phase transition is of 2nd order or weak 1st order,9 soft modes » the fluctuations of the order parameter For chiral transition,

The meson becomes the soft mode of chiral transition at

T. Hatsuda and T. K. , Phys. Rev. Lett.; Prog. Theor. Phys (1985):

It was also shown that hadronic excitations (para pion and 　 sigma) exisit even in the ``QGP” phase.

Chiral Transition and the collective modesChiral Transition and the collective modes

0

c.f. Higgs particle in WSH model

; Higgs fieldHiggs particle

T. Hatsuda and T. K., Phys. Rev. Lett. 55 (1985), 158

T dependence of the (`para’) sigma and (`para’) pion masses

Large T

What is the significance of the in hadron physics?

the softening ofthe with increasing Tand

Wait!Is the pole observed in the pi-pi phase shift really the as the quantum fluctuation of the order parameter of the chiral transition?

A change of the environment a change of the mode coupled to the order parameter

Production of the -meson in 　 a nuclear mediumUseful for exploring the existence of the and the possible restoration of chiral symmetry at finite density. (T. K., Prog. Theor. Phys. Suppl. 120 (1994), 75)

What is a good observables to see the softening in the sigma channel in nuclear medium?

Notice: A particle might loose it identity when put in a medium.

Need of a calculation of the Spectral function as seen by ,2 and 4 ; lepton pairs at 0

The poles of the S matrix in the complex mass plane forthe sigma meson channel: complied in Z. Xiao and H.Z. Zheng (2001)

G.Colangero, J. Gasser and Leutwyler (2001)

Softening !

T. Hatsuda, H. Shimizu, T.K. ,Phys. Rev. Lett. 82 (1999), 2840

Spectral function in the channel

This ratio represents the net effect of nuclear matter on the interacting 　　　　　　　　　system.

CB: Phys. Rev. Lett. 85, 5539 (2000).

CHAOS:Phys. Rev. C60, 018201 (1999).

P. Camerini et al, Phys. Rev. C64, 067601 (2001). (CHAOS coll.)

A’=2 ! 208CHAOS (1996)

C; A-dependence of A

Oset: Full model of the process, standard nuclear effects discussed, P-wave pionic modes included and the-meson dynamically generated.

Oset and Vicente,PRC60(1999)064621

Muhlich: Model based on Oset’s developed for the and reactions, better treatment of FSI of pions with the nucleus, no medium modifications.

Muhlich et al.,PLB595(2004)216

TAPS CHAOS

I=0

I=1

E~420 MeV

~ 2/30

I~0

I=2

E~420 MeV

~ 1/30

Differential cross sections of the reaction A(, )A'

J.G. Messchendorp et al, Phys. Rev. Lett. 89 (2002), 222302.

----- phase space

TAPS experiment:

L. Roca et al (2002)without softening

A o o X A +/- o X

m(oo) for isoscalar channel only:• drops with increasing A• consistent with isotropic angular distribution

o o angular distributionE = 400 - 460 MeV

J.G.Messchendorp et al., PRL 89, 222302

L. Roca, et a l., PLB 541 (2002) 77, priv. comm.

prel

imin

ary

S.Schadmand

P. Muelich, L. Alvarez-Ruso, O. Buss and U. Mosel,( nucl-th/0401042).

FSI lowers the spectral function in the pi-pi invariant mass.

The spectral enhancememnt in the 　 nonlinear ｒ ealizationD. Jido, T. Hatsuda and T. K.,Phys. Rev. D63} (2000), 011901(R).

In the polar decomposition M=SU,

In the heavy S-field limit, fixed

;

D. Jido et al (2000)

Softening of the in-medium pi-pi cross sectionIn the non-linear realization

The renormalization of the wave function

Due to the new vertex:

C.f. Importance of the w.f. renormalization in other physics:

U. Meissner, J. Oller and A. Wirzba, Ann. Phys. 297 (2002) 27E. Kolomeitzev, N. Kaiser and W. Weise, P.R.L. 90 (2003)092501

Deeply bound pionic nuclei w.f. renormalization

E-dependece of opt. pot.c.f. Freidman and Gal (04)

Deeply-bound pionic nuclei and missing repulsion 　

K. Suzuki et al., Phys. Rev. Lett. 92, 072302 (’04)

Kolomeitsev, Kaiser, & W. Weise, Phys.Rev.Lett. 90 (’03)

○　LO+EE without w.f.r.●　LO+EE with w.f.r.△　NNLO with w.f.r.

P.Kienle and T.Yamazaki,Prog. Part.Nucl. Phys. 52 (2004), 85

The movement of the sigma pole in the complexEnergy plane in the N/D method with MFA

K. Yokokawa, T. Hatsuda, A. Hayashigaki,And T.K.(2002)

A: model

B: model

C: model

D:

The T matrix in the N/D method.

The in-medium - cross sections in I=J=0 channel.The upper (lower) panel shows the case of small (large) restoration corresponding to

K. Yokokawa,T. Hatsuda,A. Hayashigakiand T.K. (2002)

Vector MesonsVector Mesons

QCD sum rules, effective theories as NJL model and Brown-Rho scaling suggest that The vector mesons mass/width, or more precisely, their spectral functions may change in hot and/or dense matter

The softening in the meson channel

K. Yokokawa et al (2002)

Softening of the spectral function in the Vector channel

Softening of the vector mesons in

Nuclear media (H. En’yo et al)

Experiment should be made and is being anallyzed: Spring-8

Vector mesons in nuclei as seen by photon

KEK-PS (Naruki et al): , ,

show a mass drop!, 0.1

0 0( ) (1 / )Vm m

ELSA@Bonn (Metag); CB/TAPS 4

0.15 for

Meson:Meson:

• Related to

( ') (1)AU Anomaly; otherwise

; ideal mixing realized,uu dd ' ss

8 0'

; energy-momentum tensor of QCD

Quantum effects!

( )

Current divergences and Quantum Anomalies

)( fV NSU

)( fA NSU

)1(AU

Dilatation

Chiral Anomaly

Dilatation(scale)Anomaly

)1(AU Problem)3()3( RL UU

)3()1( fV SUU

# of the generators2x(8+1)=18

1+8=9

G=

H=

# of NG-bosons= dim G - dim H = 18 – 9 = 9

(?)

Nambu-Goldstone Theorem

# of the lightest pseudo-scalar mesons

00,, KKK0, 3 + 4 + 1 = 8 9 !

(140) (500) (550) << ' (958)

Why is ' so massive ?

------ UA(1) Problem

Anomaly Operator Equation!0even in the chiral limit!

Also selective coupling of with N*(1535)

chiral dynamics v.s. Chiral doublet (a la DeTar-Kunihiro(‘89) )type?

Near future experiment in GSI (Hayano et al)

( ') or chiral anomaly in the medium at finite T and

T.Kunihiro (1989), Ohnishi et al(1998), Ruivo et al (00)

Jido, Hirenzaki and Nagahiro

Summary

•The meson as the quantum fluctuation of the order parameter of the chiral transition may account for various phenomena in hadron physics which otherwise remain mysterious.

• There have been accumulation of experimental evidence of the pole in the pi-pi scattering matrix. ( chiral symmetry, analyticity and crossing symmetry.

• A full lattice QCD suggests the existence of the •Partial restoration of chiral symmetry in hot and dense mediumas represented by the decreasing f leads to a softeningof the and the pole in the 2nd Riemann sheet in various chiral models.

•Even a slight restoration of chiral symmetry in the hadronic matter leads to a peculiar enhancement in the spectral function in the channel near the 2m threshold.

•Such an enhancement might have been observed in the reaction

•The decrease of the of w.f. renormalization of the pion commonly seen in the deeply bound pionic nuclei, suggesting a strongly coupled system of the pion and nuclear medium.

• Other channels: simultaneous softening of the and c.f. KEK, CERES, STAR N* and parity doublets of other baryons (DeTar and T.K. (1989); Jido, Hosaka and Oka, Hirenzaki, Nagahiro …. )

QCD critical point scalar – vector mode coupling

CSC

T

SB

QGPQCD c.p.

meson condensation?

?

precursoryhadronic modes?

0

CFL

QCD phase diagram

H matter?

(T. Hatsuda and T.K. (’84, ’85)