su houng lee theme: 1.will u a (1) symmetry breaking effects remain at high t/ 2.relation between...
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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
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
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