Meson in matter
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
1
h‘ mass , Chiral symmetry restoration and UA(1) effect ?
QCD Lagrangian
Chiral sym restored
Usual vacuum
U  N F  U  N F 
SU N F  SU N F U 1
SUN F U 1
 uL,R 
 uL,R 

  U 

 d L,R 
 d L,R 
mass

s ~
  q   q  N f
GG
4
5
qq  0
a1
?
h‘
r
?

2
Experimental evidence of property change of h‘ in
matter ?
CBELSA/TAPS coll

h '   0 0h  6
V  37  10  10 MeV  i10  2.5 MeV
Nanova et al.
1. Imaginary part: Transparency ratio
2. Real part: Excitation function + momentum distribution of the meson
3
Correlators and symmetry
1. Chiral symmetry breaking in Correlator
0
q 0 q0  or VV  AA m
  0  0  form factor
Cohen 96
2.
UA(1) breaking effects in Correlators
h 'h '
0
m
  0  0  form factor
m
N f 2
zero mode 
Hatsuda, Lee 96
4
Quark condensate – Chiral order parameter
Finite temperature
Lattice gauge theory
 qq T /  qq T 0
 ss T /  qq T 0  0.8
cc  
1  2
G
12mc 
Finite density
1
 ss T  ms exp ms / T 
T/Tc
Linear density approximation
 qq r
r/rn
5
Chiral symmetry breaking (m0) : order parameter
•
Quark condensate
q 0q0   lim T rS ( x,0)   dAe
 SQCD
x 0
   dAe
 Casher Banks formula:
 S QCD


1
T r 0
0 
/ m 
 D

1
Tr S 0,0  i 5 S 0,0i 5
2
iD
/     
using
q 0q0   T r  0  0

m
m2  2

where   0  0 | 
0
m
  0 0  0 0
 Chiral symmetry breaking order parameter
qq
 TrS 0, x  

1
S 0, x   i 5 S 0, x i 5
2

  0  0
6
•
Other order parameters:    correlator
1
4
d
x

V


q x qx , q 0q0 
q x  a i 5 qx , q 0 a i 5 q0
 T r S ( x,0) S (0, x)
1

 T r ai 5 S ( x,0)  ai 5 S (0, x)
1
1

 5 a
 5 a

TrS ( x, x) TrS (0,0)

1


  Tr S ( x,0) S 0, x  i 5 S 0, xi 5
ONc 


TrS ( x, x) TrS (0,0)
O1
7
•
Other order parameters: V - A correlator (mass difference)
1
4
d
x

V


q x   a qx , q 0  a q0 

 T r   a S ( x,0)   a S (0, x)




  a

 5  a
 5  a

 T r   S ( x,0)  S 0, x  i 5 S 0, xi 5
qq
 
  a
 Tr ai 5  S ( x,0)  ai 5  S (0, x)


Tr   a S ( x, x)  Tr   a S (0,0)

  a
  a

q  x  a i 5  q x , q 0 a i 5  q0
 TrS 0, x  
S 0, x   i

5
S 0, x i 5

  0  0
8
•
Meson with one heavy quark : S-P

1
d 4x

V

H x i 5 q x , q 0i 5 H 0

 T r SH ( x,0) S 0, x  i 5 S 0, xi 5

•
H x qx , q 0H 0 


Baryon sector : L – L*
1
4
d
x
V
 u i Cd H x, u i Cd H 0
T

5
5

T


u Cd H x , u Cd H 0 
T
 SH x,0T r S ( x,0) S x,0  i 5 S x,0i 5
T

9
Correlators and symmetry
1. Chiral symmetry breaking in Correlator
0
q 0 q0  or VV  AA m
  0  0  form factor
Cohen 96
2.
UA(1) breaking effects in Correlators
h 'h '
0
m
  0  0  form factor
m
N f 2
zero mode 
Hatsuda, Lee 96
10
UA(1) effect : effective order parameter (Lee, Hatsuda 96)
•
Topologically nontrivial contributions
Z  Zn 0  Zn 1  .....
•
h ‘  correlator : n = 0 part

1
d 4 xe ikx q  x i 5 qx , q 0i 5 q0  q x  a i 5 qx , q 0 a i 5 q0

V

n  0 :   T r i 5 S ( x,0)i 5 S (0, x)


 T r i S ( x,0) i S (0, x)

a
5

a

5



Tr i 5 S ( x, x)  Tr i 5 S (0,0)



 
i 5
~
GG
Tr i 5 S ( x, x)  Tr i 5 S (0,0)

i 5
 r  0
2
T. Cohen
(96)
11
•
h ‘  correlator : n nonzero part
Lee, Hatsuda (96)

1
d 4 xe ikx q  x i 5 qx , q 0i 5 q0  q x  a i 5 qx , q 0 a i 5 q0

V

For SU(3) :
1
  d 4 x u0 x d 0 0 d 0 0u0 x   d 4 ys0  y ms s0  y   permutatio ns
V
uL
dL
n=1
n 0
uR
dR
sL
 const x  mq q q
sR
q 3
For SU(2) : Always non zero

1
d 4 x u0  x d 0 0  d 0 0u0  x 

V
n 0
 const 
uL
dL
n=1
uR
dR
For N-point function: U(1)A will be restored with chiral symmetry for N > NF
but always broken for N < NF
12
•
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
 ,   a , a
chiral
 ,  h,h
UA(1)
Chiral symmetry restoration
UA(1) symmetry restoration ?

But what happens to the h‘ mass?

What is the relation to chrial symmetry
13
Correlators and h’ meson mass
1. Witten – Veneziano formula
2. At finite temperature and density
14
h’ mass?
Witten-Veneziano formula - I
~
~
Pk   i  dxeikx GGx , GG0
•
Correlation function
•
Contributions from glue only
•
When massless quarks are added
•
P0 k  0  0
Pk  
Large Nc argument
from low energy theorem
0
Pk   i  dxeikx   j5  x ,   j5 0   k  k n Pn k
0
2
~
0 | GG | glueball

k 2  mn2
glueballs
~
GG


mesons
~
GG
•
Need h‘ meson


2
~
GG
Nc
 1 

with mh2'  O
N
 c
k 2  mh2'
P(k  0)  P0 0 
k 2  mn2
~
GG
N c2
~
0 | GG | h '
2
~
0 | GG | m eson
~
0 | GG | h '
mh2'
2
0
15
Witten-Veneziano formula – II
•
~
0 | GG | h '
h‘ meson
2
mh '
2
 P0 0
2

 4   1
N F mh2' fh ' 

 
2
8
 2
    NF
   4 
G


2
mh '
 3  11N / 3 
2
Lee, Zahed (01)
mh2' fh ' 
2
8  2
G
11N 
Should be related to
 250 MeV  mh '  432 MeV
at m  0 limit
mh ' (958)  mh (547)  411MeV
16
Few Formula in Large Nc
•
Meson





m1, 1/Nc , gmmm 1/ Nc1/ 2 , 0 | qq | m Nc1/ 2 , 0 | GG | m Nc1/ 2
•
Glueball



m1,  1/Nc2 , g ggg 1/ Nc , 0 | qq | m Nc , 0 | GG | g Nc 
•
Baryon


mNc , gmBB Nc1/ 2 ,
B | qq | B Nc ,
B | GG | B Nc 
17
Witten-Veneziano formula – III Nc counting and glueball
•
h‘ meson
ONc1 
O1 / Nc1 
~
0 | GG | h '
2
mh2'
8
 2
 4 
 P0 0  
G

 3  11N / 3 
2
ONc2 
h ‘ mass is a large 1/Nc correction
•
glueball
ONc2 
0 | GG | g
O1
mg2
2
 2
 4  18
 S 0 0  
G

 3  11N / 3 
2
ONc2 
18
Witten-Veneziano formula – IV
•
Low energy theorem is a Non-perturbative effect
S q   i  dxeiqx
3 2
3 2
18  2
G x 
G 0 
G
4
4
11 
Pq   i  dxeiqx
3 ~
3 ~
8  2
GG x 
GG 0  
G
4
4
11 
h ‘ mass is a large 1/Nc correction
19
Witten-Veneziano formula at finite T
•
~
~
Pk   i  dxeikx GGx , GG0
Large Nc counting
N c2
•
N c2
(Kwon, Morita, Wolf, Lee: PRD 12 )
m
Nc
At finite temperature, only gluonic effect is important
Pk  
2
~
0 | GG | glueball

k m
2
glueballs
2
n

mesons
Glue Nc2

P(k  0)  P0 0 

2
~
0 | GG | m eson
k m
2
Quark Nc
~
0 | GG | h '
2
mh '
2
n
 ScatteringT erm
Quark Nc2 ?
2
c
 scattering?
20
•
Large Nc argument for Nucleon Scattering Term
ONc 
~
GG
ONc 
~
GG
Nucleon
Witten
Nucleon
~
GG
2
 1 
N c  1/ 2  N c  N c
 Nc 
That is, scattering terms are of order Nc and can be safely neglected

~
n | GG | n
mN
2
r density
 1
N c2 
 Nc

  N c

21
•
Large Nc argument for Meson Scattering Term
O1
~
GG
O1
~
GG
Meson
Witten
~
GG
 1 
N c2  1/ 2 
 Nc 
2
2
 1 
 1/ 2   1
 Nc 
That is, scattering terms are of order 1 and can be safely neglected

P0 0 
~
0 | GG | h '
2
2
WV relation remains the same
mh '
22
•
LET (Novikov, Shifman, Vainshtein, Zhakarov) at finite temperature : Ellis, Kapusta, Tang (98)
d
ikx
2


Op


i
dx
e
Op
x
,
g
GG0
0
2

d 1 / 4 g0


d
Op
d
Op
2
d  1 / 4 g0

•
Lee, Zahed (2001)

T
T

 8 2 
 constM 0 exp  2   c' T d  Op
 bg0 

32 2 
 

d

T

 Op
b 
T 
T
d

c
'
T
T0
32 2 
 

d

T

 Op
b 
T 
T0
2
   2
P0 0    d  T
 G
b
T  
 Moritaet al. (2012)
WeakT dependenceeven near Tc

P0 0 
~
0 | GG | h '
2
c
2
mh '
23
•
~
0 | GG | h ' at finite temperature

~
~
Pk    d 4 xeikx GGx , GG0
 4
 k  k n  d 4 xeikx 
 N F



2
~
0 | GG | h '

k 2  mh2'
2
 ...
 q xi  qx, q 0i  q0  q xi

5
n
5

qx , q 0i n q0

sym restored phase
chiral


 4
 k  k n  d 4 xeikx 
 N F



2
 q xi  

5 a
qx , q 0i n  5 a q0  q x i  a qx , q 0i n  a q0

 0 for any k  , when Chiral symmetryis restored
Therefore,
~
0 | GG | h '  0
when chiral symmetry gets restored
24
•
W-V formula at finite temperature:
qq
2
~
0 | GG | h '
mh2'
2
 P0 0
   2
 4  2 
 
d T
 G
3

11

T
 

 
2
Smooth temperature dependence even near Tc
Therefore ,
mh '  mh  q q
 eta’ mass should decrease at finite temperature
25
Experimental evidence of property change of h‘ in
matter ?
CBELSA/TAPS coll

h '   0 0h  6
V  37  10  10 MeV  i10  2.5 MeV
10 % reduction of mass from around 400 MeV from chiral symmetry breaking
26
Summary
1.
2.
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
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 ?
27
Summary
1. Chiral symmetry breaking in Correlator
0
q 0 q0  or VV  AA m
  0  0  form factor
2.
UA(1) breaking effects in Correlators
h 'h '
0
m
  0  0  form factor
m
N f 2
zero mode 
 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
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