1

Su Houng Lee

Quark condensate and the h’ mass

h‘ at finite temperature

2

1. Quark condensate and the h’ meson

1. Some introduction

2. Casher Banks formula

3. Lee-Hatsuda formula

4. Witten – Veneziano formula

5. At finite temperature and den-

sity

3

• Kapusta, Kharzeev, McLerran PRD 96 : Effective U(1)A restoration in medium in SPS data

Some introduction

• Rapp, Wambach, van Hees: SPS dilepton data

• RHIC dilepton puzzle

• Csorgo, Vertesi, Sziklai : Additional contribution from h’ mass reduction

• Experimental and theoretical works on h‘ in nuclear medium

• QCD symmetry: SU(N)Lx SU(N)R SU(N)V and U(1)A is always broken by Anomaly

QCD confinement

4

Finite temperature

qq

T/Tc

/r rn

0/ TT qqqq

18.0/ 0 TT qqss

Tmmss ssT /exp

Quark condensate – Chiral order parameter

Finite density

Lattice gauge theory

Linear density approximation

2

12

1G

mcc

c

5

• Quark condensate

Casher Banks formula - Chiral symmetry breaking (m0)

0

10Tr)0,(Trlim00

0 mDdAexSqq QCDS

x

0 0001

0Tr00 022

m

m

md

mDqq

Phenomenologically need constituent quark mass

Casher Banks formula in the chiral limit

6

• Other order parameters: - s p correlator (mass difference)

),0( )0,( Tr xSxS

00, 00, 1 554 qiqxqixqqqxqxqxdV

aa

0 c 2

1 1

a5 a5

)0,0(Tr),(Tr SxxS

),0( )0,(Tr 55 xSixSi aa

Phenomenologically constituent quark mass terms survives

Generalized Casher Banks formula in the chiral limit

1 1

7

Correlation function for h ‘ (Lee, Hatsuda 96)

• h ‘- p correlator (mass difference)

00,00,1 55554 qiqxqixqqiqxqixqxedV

aaikx

),0()0,(Tr 55 xSixSi

3q

x SU(2)for const qqmq

nspermutatio 0 01

000

40000

4

ysmysydxuddxuxdV s

n=1

)0,0(Tr),(Tr 55 SixxSi

GG~

),0()0,(Tr 55 xSixSi aa

5i 5i

U(1) A symmetry will effectively be restored up to quark mass terms in SU(3)

20

T. Cohen (96)

Lee, Hatsuda (96)

8

• Contributions from glue only from low energy theorem

• When massless quarks are added

• Correlation function

h’ mass? Witten-Veneziano formula - I

0~

,~

e GGxGGdxikP ikx

000 kP

• Large Nc argument

mesons nglueballs n mk

mesonGG

mk

glueballGGkP

22

2

22

2|

~|0

|

~|0

00,e 055 kikx PkkjxjdxikP

GG~

GG~

GG~

GG~

cNOm

mk

GG 1 with

'|~

|02

'2'

2

2

2cN cN

• Need h‘ meson

2

'

2

0

'|~

|00)0(

m

GGPkP c

9

Witten-Veneziano formula – II

• h‘ meson 0

'|~

|002

'

2

Pm

GG

22

2'

2

'2

'

2

11

8

3

424

Gm

fmNNN

FF

MeV 432 11

8

3

2'

22

2'

2'

mG

NNfm F

MeV 411)547()958(' mm

Lee, Zahed (01)

10

• Large Nc counting

Witten-Veneziano formula in medium (Kwon, Morita, Wolf, Lee in prep )

m

ikx GGxGGdxikP 0~

,~

e

2cN

• LET (Novikov, Shifman, Vainshtein, Zhakarov) at finite temperature for S(k): Ellis, Kapusta, Tang (98)

2cN cN

0,e4/1

202

0

GGgxOpdxiOpgd

d ikx

d

T

d

d

TTcOpTc

bgMconstOp ''

8exp

020

2

0

0

22

20

3232

4/1 TTTOp

TTd

bOp

TTd

bOp

gd

d

11

• LET at finite temperature for P(k) : Lee, Zahed (01)

•

22

11

2

3

40 G

TTdkP

'|~

|0 GG

00,00,

2 55

2

4 qiqxqixqqiqxqixqN

Nxedkk

F

ikx

0 phase restored sym chiral

...0|

~|0

0~

,~

2'

2

24

4

mk

GGkGGxGGxedkP ikx

Therefore, 0'|~

|0 GG

12

• W-V formula at finite temperature:

22

11

2

3

4G

TTd

0

2442 GTagG

0'|

~|0

02'

2

Pm

GG

2'

2

m

qq

0

22

22

11

2

3

4

11

2

3

4GdG

TTd

Smoothes out temperature change because if

Therefore , : Observable consequences ?qqm '

13

1. h’ mass is related to quark condensate and thus should reduce in medium

a) Could serve as signature of chiral symmetry restoration

b) Dilepton in Heavy Ion collision

c) Measurements from nuclear targets

Summary

14

• Other order parameters: - s p correlator (mass difference)

xmDmD

x1

001

Tr

00, 00, 1 554 qiqxqixqqqxqxqxdV

aa

0 c 2

a a

a5 a5

01

0Tr1

Tr mD

xmD

x

1

001

Tr 55

xmD

imD

xi aa