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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
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
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