1

Su Houng Lee

Theme:

1. Will UA(1) symmetry breaking effects remain at high T/ r

2. Relation between Quark condensate and the h’ mass

Ref:

SHL, T. Hatsuda, PRD 54, R1871 (1996)

Y. Kwon, SHL, K. Morita, G. Wolf, PRD86,034014 (2012)

SHL, S. Cho, IJMP E 22 (2013) 1330008

Meson in matter

2

QCD Lagrangian

h‘ mass , Chiral symmetry restoration and UA(1) effect ?

FF NUNU

Usual vacuum

RL

RL

RL

RL

d

uU

d

u

,

,

,

,

1UNSU F 1UNSUNSU FF

GGNqq sf

~

45

0qq

Chiral sym restored

a1

r

p

h‘ ?

?

mass

3

CBELSA/TAPS coll

Experimental evidence of property change of h‘ in matter ?

6' 00

MeV 5.210 MeV 101037 iV

Nanova et al.

1. Imaginary part: Transparency ratio

2. Real part: Excitation function + momentum distribution of the

meson

4

Correlators and symmetry

1. Chiral symmetry breaking in Correlator

2. UA(1) breaking effects in Correlators

factor form or 00 000

mAAVVqq

mode zerofactor form '' 200

0

fNm m

Cohen 96

Hatsuda, Lee 96

5

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

6

• Quark condensate

Chiral symmetry breaking (m0) : order parameter

0

10Tr)0,(Trlim00

0 mDdAexSqq QCDS

x

00 00Tr00 000

22

m

m

mqq

Casher Banks formula: |00 where using Di

55 0,00,0Tr2

1 iSiSdAe QCDS

0055 ,0,0

2

1 ,0Tr

ixSixSxSqq

Chiral symmetry breaking order parameter

7

• Other order parameters: - s p correlator

),0( )0,( Tr xSxS

00, 00, 1 554 qiqxqixqqqxqxqxdV

aa

1 1

a5 a5

)0,0(Tr),(Tr SxxS

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

1 1

55 ,0,0)0,( Tr ixSixSxS )0,0(Tr),(Tr SxxS

cNO 1O

8

• Other order parameters: V - A correlator (mass difference)

),0( )0,( Tr xSxS aa

00, 00, 1 554 qiqxqixqqqxqxqxdV

aaaa

a

a 5 a 5

)0,0(Tr),(Tr SxxS aa

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

55 ,0,0)0,( Tr ixSixSxS

0055 ,0,0 ,0Tr

ixSixSxSqq

a a a

9

• Meson with one heavy quark : S-P

00, 00, 1 554 HiqxqixHHqxqxHxdV

55 ,0,0)0,( Tr ixSixSxSH

• Baryon sector : L – L*

0, 0, 1 554 HdCuxHCduHdCiuxHCdiuxdV

TTTT

55 0,0,)0,( Tr0, ixSixSxSxSH

10

Correlators and symmetry

1. Chiral symmetry breaking in Correlator

2. UA(1) breaking effects in Correlators

factor form or 00 000

mAAVVqq

mode zerofactor form '' 200

0

fNm m

Cohen 96

Hatsuda, Lee 96

11

UA(1) effect : effective order parameter (Lee, Hatsuda 96)

• h ‘- p correlator : n = 0 part

00,00,1 55554 qiqxqixqqiqxqixqxedV

aaikx

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

GG~

),0()0,(Tr 55 xSixSi aa 5i 5i

20

T. Cohen (96)

• Topologically nontrivial contributions

.....10 ZZZ

: 0

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

12

• h ‘- p correlator : n nonzero part

00,00,1 55554 qiqxqixqqiqxqixqxedV

aaikx

3q

x const qqmq

nspermutatio 0 01

000

40000

4

ysmysydxuddxuxdV s

n=1

Lee, Hatsuda (96)

Lu

Ru

Ld

Rd

Ls Rs

For SU(3) :

const 0 01

000004

xuddxuxdV

n=1

Lu

Ru

Ld

Rd

For SU(2) : Always non zero

For N-point function: U(1)A will be restored with chiral symmetry for N > NF

but always broken for N < NF

13

• Recent Lattice results ?

1. S. Aoki et al. (PRD 86 11451) : no UA(1) effect above Tc

2. M. Buchoff et al. (PRD89 054514): UA(1) effect survives Tc in SU(2) in susceptibilities

But what happens to the h‘ mass?

What is the relation to chrial symmetry

Chiral symmetry restoration UA(1) symmetry restoration ?

aa ,,

,, chiral

UA(1)

14

Correlators and h’ meson mass

1. Witten – Veneziano formula

2. At finite temperature and den-

sity

15

• 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

0'|

~|0

0)0( 2

'

2

0

m

GGPkP

16

Witten-Veneziano formula – II

• h‘ meson 0'|

~|0

02'

2

Pm

GG

22

2'

2

'2

'

2

3/11

8

3

4

14

GNm

fmNN F

F

MeV 432 MeV 250 11

8'

22'

2'

mG

Nfm

MeV 411)547()958(' mm

Lee, Zahed (01)

Should be related to

at m 0 limit

17

Few Formula in Large Nc

• Meson

2/12/12/1c ||0 ,||0 ,/1 ,1/ ,1 cccmmm NmGGNmqqNgNm

• Glueball

cccgggc NgGGNmqqNgNm ||0 ,||0 ,/1 ,1/ ,1 2

• Baryon

cccmBBc NBGGBNBqqBNgNm || ,|| , , 2/1

18

Witten-Veneziano formula – III Nc counting and glueball

• h‘ meson

22

02'

2

3/11

8

3

40

'|~

|0G

NP

m

GG

h ‘ mass is a large 1/Nc correc-tion

• glueball

22

02

2

3/11

18

3

40

||0G

NS

m

gGG

g

2cNO

1cNO

1/1 cNO

2cNO

2cNO

1O

19

Witten-Veneziano formula – IV

• Low energy theorem is a Non-perturbative effect

h ‘ mass is a large 1/Nc correc-tion

2

222

11

80

~

4

3

~

4

3e

11

180

4

3

4

3e

GGGxGGdxiqP

GGxGdxiqS

iqx

iqx

20

• Large Nc counting

Witten-Veneziano formula at finite T (Kwon, Morita, Wolf, Lee: PRD 12 )

m

ikx GGxGGdxikP 0~

,~

e

2cN

• At finite temperature, only gluonic effect is important

2cN cN

Term Scattering |

~|0

|

~|0

22

2

22

2

mesons nglueballs n mk

mesonGG

mk

glueballGGkP

? scattering '|

~|0

0)0( 2

'

2

0

m

GGPkP c

Glue Nc2

Quark Nc Quark Nc2 ?

21

• Large Nc argument for Nucleon Scattering Term

GG~

GG~

Nucleon

cNO

ccc

c NNN

N

2

2/1

1GG~

densityNm

nGGn

2|

~|

Witten

That is, scattering terms are of order Nc and can be safely ne-glected

cNO

Nucleon

cc

c NN

N

12

22

• Large Nc argument for Meson Scattering Term

GG~

GG~

Meson

1O 1O

111

2

2/1

2

2/12

ccc NN

NGG~

2

'

2

0

'|~

|00

m

GGP

Witten

That is, scattering terms are of order 1 and can be safely ne-glected

WV relation remains the same

23

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

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

cTnear even dependence T Weak

(2012) al.et Morita

20

20 G

TTd

bP

• Lee, Zahed (2001)

2

'

2

0

'|~

|00

m

GGP c

24

• at finite temperature '|~

|0 GG

00,00,

4 55

2

4 qiqxqixqqiqxqixqN

xedkkF

ikx

phase restored sym chiral

...'|

~|0

0~

,~

2'

2

2

4

mk

GGGGxGGxedkP ikx

Therefore, when chiral symmetry gets restored 0'|~

|0 GG

00,00,

4 55

2

4 qiqxqixqqiqxqixqN

xedkk aaaa

F

ikx

restored issymmetry Chiral when ,any for 0 k

25

• W-V formula at finite temperature:

22

11

2

3

4G

TTd

0

'|~

|002

'

2

Pm

GG

2qq

Smooth temperature dependence even near Tc

Therefore ,

eta’ mass should decrease at finite temperature

qqmm '

26

CBELSA/TAPS coll

Experimental evidence of property change of h‘ in matter ?

6' 00

MeV 5.210 MeV 101037 iV

10 % reduction of mass from around 400 MeV from chiral symmetry break-

ing

27

1. h’ correlation functions should exhibit symmetry breaking from N-point function in SU(N) flavor even when chiral symmetry is restored.

For SU(2), UA(1) effect will be broken in the two point function

Summary

2. In W-V formula h’ mass is related to quark condensate and thus should reduce at finite temperature independent of flavor due to chiral symmetry restoration

a) Could serve as signature of chiral symmetry restoration

b) Dilepton in Heavy Ion collision

c) Measurements from nuclear targets seems to support it ?

28

Summary

1. Chiral symmetry breaking in Correlator

2. UA(1) breaking effects in Correlators

Restored in SU(3) and real world

3. WV formula suggest mass of h ‘ reduces in medium

and at finite temperature: due to chiral symmetry

restoration

4. Renewed interest in Theory and Experiments both for

nuclear matter and at may be at finite T

factor form or 00 000

mAAVVqq

mode zerofactor form '' 200

0

fNm m