Meson in matter
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Transcript Meson in matter
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
SUN 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 i10 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 q0 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 (m0) : order parameter
•
Quark condensate
q 0q0 lim T rS ( 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 0,0i 5
2
iD
/
using
q 0q0 T r 0 0
m
m2 2
where 0 0 |
0
m
0 0 0 0
Chiral symmetry breaking order parameter
qq
TrS 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 qx , q 0q0
q x a i 5 qx , q 0 a i 5 q0
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
TrS ( x, x) TrS (0,0)
1
Tr S ( x,0) S 0, x i 5 S 0, xi 5
ONc
TrS ( x, x) TrS (0,0)
O1
7
•
Other order parameters: V - A correlator (mass difference)
1
4
d
x
V
q x a qx , q 0 a q0
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, xi 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 q0
TrS 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 0i 5 H 0
T r SH ( x,0) S 0, x i 5 S 0, xi 5
•
H x qx , q 0H 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,0T r S ( x,0) S x,0 i 5 S x,0i 5
T
9
Correlators and symmetry
1. Chiral symmetry breaking in Correlator
0
q 0 q0 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 qx , q 0i 5 q0 q x a i 5 qx , q 0 a i 5 q0
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 qx , q 0i 5 q0 q x a i 5 qx , q 0 a i 5 q0
V
For SU(3) :
1
d 4 x u0 x d 0 0 d 0 0u0 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 0u0 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
~
~
Pk i dxeikx GGx , GG0
•
Correlation function
•
Contributions from glue only
•
When massless quarks are added
•
P0 k 0 0
Pk
Large Nc argument
from low energy theorem
0
Pk i dxeikx j5 x , j5 0 k k n Pn 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
m1, 1/Nc , gmmm 1/ Nc1/ 2 , 0 | qq | m Nc1/ 2 , 0 | GG | m Nc1/ 2
•
Glueball
m1, 1/Nc2 , g ggg 1/ Nc , 0 | qq | m Nc , 0 | GG | g Nc
•
Baryon
mNc , gmBB Nc1/ 2 ,
B | qq | B Nc ,
B | GG | B Nc
17
Witten-Veneziano formula – III Nc counting and glueball
•
h‘ meson
ONc1
O1 / Nc1
~
0 | GG | h '
2
mh2'
8
2
4
P0 0
G
3 11N / 3
2
ONc2
h ‘ mass is a large 1/Nc correction
•
glueball
ONc2
0 | GG | g
O1
mg2
2
2
4 18
S 0 0
G
3 11N / 3
2
ONc2
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
Pq 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
•
~
~
Pk i dxeikx GGx , GG0
Large Nc counting
N c2
•
N c2
(Kwon, Morita, Wolf, Lee: PRD 12 )
m
Nc
At finite temperature, only gluonic effect is important
Pk
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
ONc
~
GG
ONc
~
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
O1
~
GG
O1
~
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
GG0
0
2
d 1 / 4 g0
d
Op
d
Op
2
d 1 / 4 g0
•
Lee, Zahed (2001)
T
T
8 2
constM 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
~
~
Pk d 4 xeikx GGx , GG0
4
k k n d 4 xeikx
N F
2
~
0 | GG | h '
k 2 mh2'
2
...
q xi qx, q 0i q0 q xi
5
n
5
qx , q 0i n q0
sym restored phase
chiral
4
k k n d 4 xeikx
N F
2
q xi
5 a
qx , q 0i n 5 a q0 q x i a qx , q 0i n a q0
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 i10 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 q0 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
28