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ρ = ρ
s
T = 0
E/ρ
P × ρ/ρ
0
m
∗
= M
∗
0
/M K
0
T = 0
ρ/ρ
0
m
∗
= M
∗
0
/M K
0
T = 0
M
∗
/M ρ/ρ
0
m
∗
= 0, 54 K
0
= 551 MeV
T = 0
M
∗
/M ρ/ρ
0
m
∗
= 0, 56 K
0
= 450 MeV
T = 0
M
∗
/M ρ/ρ
0
m
∗
= 0, 58 K
0
= 350 MeV
T = 0
M
∗
/M ρ/ρ
0
m
∗
= 0, 60 K
0
= 250 MeV
T = 0
M
∗
/M ρ/ρ
0
m
∗
= 0, 62 K
0
= 220 MeV
T = 0
ρ/ρ
0
m
∗
=
M
∗
0
/M
K
0
T = 5 MeV
ρ/ρ
0
m
∗
=
M
∗
0
/M
K
0
T = 10 MeV
ρ/ρ
0
m
∗
=
M
∗
0
/M K
0
T = 15 MeV
ρ/ρ
0
m
∗
= M
∗
0
/M K
0
T = 5 MeV
ρ/ρ
0
m
∗
= M
∗
0
/M K
0
T = 10 MeV
ρ/ρ
0
m
∗
= M
∗
0
/M K
0
T = 15 MeV
M
∗
/M
ρ = ν = 0
M
∗
/M
ρ = ν = 0
ρ/ρ
0
m
∗
= M
∗
0
/M K
0
T = 5 MeV
ρ/ρ
0
m
∗
= M
∗
0
/M K
0
T = 10MeV
ρ/ρ
0
m
∗
= M
∗
0
/M K
0
T = 15 MeV
ρ
s
/ρ
0
ρ/ρ
0
m
∗
K
0
P × ρ/ρ
0
¯
M
0
/M = 0, 54 K
0
= 551 MeV T = 0
ρ = ρ
s
P × ρ/ρ
0
¯
M
0
/M = 0, 56
K
0
= 450 MeV T = 0
ρ = ρ
s
P × ρ/ρ
0
¯
M
0
/M = 0, 58 K
0
= 350 MeV T = 0
ρ = ρ
s
P × ρ/ρ
0
¯
M
0
/M = 0, 60 K
0
= 250 MeV T = 0
ρ = ρ
s
P × ρ/ρ
0
¯
M
0
/M = 0, 62 K
0
= 220 MeV T = 0
ρ = ρ
s
E/ρ × ρ/ρ
0
¯
M
0
/M = 0, 54 K
0
= 551 MeV T = 0
ρ = ρ
s
E/ρ × ρ/ρ
0
¯
M
0
/M = 0, 56 K
0
= 450 MeV T = 0
ρ = ρ
s
E/ρ × ρ/ρ
0
¯
M
0
/M = 0, 58 K
0
= 350 MeV T = 0
ρ = ρ
s
E/ρ × ρ/ρ
0
¯
M
0
/M = 0, 60 K
0
= 250 MeV T = 0
ρ = ρ
s
E/ρ × ρ/ρ
0
¯
M
0
/M = 0, 62 K
0
= 220 MeV T = 0
ρ = ρ
s
m
∗
×ρ/ρ
0
T = 0
m
∗
= M
∗
0
/M ρ = ρ
s
m
∗
=
¯
M
0
/M m
∗
= 0, 54 K
0
= 551 MeV
m
∗
×ρ/ρ
0
T = 0
m
∗
= M
∗
0
/M ρ = ρ
s
m
∗
=
¯
M
0
/M m
∗
= 0, 56 K
0
= 450 MeV
m
∗
×ρ/ρ
0
T = 0
m
∗
= M
∗
0
/M ρ = ρ
s
m
∗
=
¯
M
0
/M m
∗
= 0, 58 K
0
= 350 MeV
m
∗
×ρ/ρ
0
T = 0
m
∗
= M
∗
0
/M ρ = ρ
s
m
∗
=
¯
M
0
/M m
∗
= 0, 60 K
0
= 250 MeV
m
∗
×ρ/ρ
0
T = 0
m
∗
= M
∗
0
/M ρ = ρ
s
m
∗
=
¯
M
0
/M m
∗
= 0, 62 K
0
= 220 MeV
(K
0
/m
∗
)
1/2
ρ
1/3
0
K = 180 200 220 240
250 260 280 300 MeV
m
∗
c
P
c
/ρ
c
T
c
(K
0
/m
∗
)
1/2
ρ
1/3
0
T
1
< T
2
< T
3
< T
c
I
2
π
2
/6 k
F
T = 1 5 10 MeV
ρ/ρ
0
T = 1 MeV
ρ
0
= 0, 15 fm
−3
ρ/ρ
0
T = 5 MeV
ρ
0
= 0, 15 fm
−3
m
∗
K
0
G
′
V
2
G
′
s
2
M
2
m
∗
K
0
G
2
V
G
2
s
M
2
m
∗
K
0
E
′
c
B
′
= 0 E
0
= E
′
1
+ E
′
2
+ E
′
3
+
E
′
c
−M
b = 0
E
0
= E
1
+ E
2
+ E
3
+ E
c
− M
E
0
= E
′
1
+ E
′
2
+ E
′
3
+ E
′
4
+ E
′
c
−M
E
0
= E
1
+ E
2
+ E
3
+ E
4
+ E
c
− M
0, 58 ≤ m
∗
≤ 0 , 6 2
ρ = 0, 01 fm
−3
T = 1 MeV
ν = 913 MeV
ρ = 0, 01 fm
−3
x
T = 1 MeV
ν = 913 MeV
π
σ
ω
550 MeV
230 ± 5 MeV
K = 225 MeV
ρ = ρ
s
g
s
g
V
m
s
m
V
R
R ∼
c
mc
2
m → ∞
m → ∞
L =
¯
ψ(iγ
µ
∂
µ
−M)ψ +
1
2
G
′
s
2
(
¯
ψψ)
2
−
1
2
G
′
V
2
(
¯
ψγ
µ
ψ)
2
+
1
3
A
′
(
¯
ψψ)
3
+
1
4
B
′
(
¯
ψψ)
4
(
¯
ψψ)
3
(
¯
ψψ)
4
G
′
s
2
G
′
V
2
A
′
B
′
(
¯
ψψ)
2
A
′
= B
′
= 0
L
W
=
¯
ψ(iγ
µ
∂
µ
− M)ψ +
1
2
∂
µ
φ∂
µ
φ −
1
2
m
2
s
φ
2
−
1
4
F
µν
F
µν
+
1
2
m
2
V
V
µ
V
µ
− g
s
¯
ψφψ − g
V
¯
ψγ
µ
V
µ
ψ
F
µν
≡ ∂
µ
V
ν
−∂
ν
V
µ
1
◦
2
◦
3
◦
G
′
s
2
G
′
V
2
MeV
−2
A
′
MeV
−5
B
′
MeV
−8
g
s
g
V
m
s
m
V
MeV
4
◦
5
◦
6
◦
7
◦
m
s
m
V
W [J, A
µ
, η, ¯η] = N
[Dψ][D
¯
ψ][DV
µ
][Dφ]
iS
F
N =
1
d
4
x
iS
S =
d
4
x L
W
S
F
=
d
4
x
L
W
+ A
µ
(x)V
µ
(x) + J(x)φ(x) + ¯η(x)ψ(x) + η(x)
¯
ψ(x)
A
µ
(x) J(x) η(x) ¯η(x) V
µ
φ
¯
ψ
ψ S
S
F
V
′µ
≡ m
V
V
µ
φ
′
≡ m
s
φ
G
′
s
≡
g
s
m
s
G
′
V
≡
g
V
m
V
A
′
µ
≡
A
µ
m
V
J
′
≡
J
m
s
L
W
=
¯
ψ(iγ
µ
∂
µ
− M)ψ +
1
2m
2
s
∂
µ
φ
′
∂
µ
φ
′
−
1
2
φ
′2
−
1
4m
2
V
F
′µν
F
′
µν
+
1
2
V
′
µ
V
′µ
− G
′
s
¯
ψφ
′
ψ − G
′
V
¯
ψγ
µ
V
′
µ
ψ
L
W
= L
′
W
+
1
2m
2
s
∂
µ
φ
′
∂
µ
φ
′
−
1
4m
2
V
F
′µν
F
′
µν
≡ L
′
W
+ U(φ
′
, V
′µ
)
F
′µν
= ∂
µ
V
′ν
− ∂
ν
V
′µ
L
′
W
=
¯
ψ(iγ
µ
∂
µ
−M)ψ −
1
2
φ
′2
+
1
2
V
′
µ
V
′µ
−G
′
s
¯
ψφ
′
ψ − G
′
V
¯
ψγ
µ
V
′
µ
ψ
W [J
′
, A
′
µ
, η, ¯η] = N
[Dψ][D
¯
ψ][DV
′µ
][Dφ
′
]
i
d
4
x U(φ
′
, V
′µ
) + S
′
F
S
′
F
=
d
4
x
L
′
W
+ A
′
µ
(x)V
′µ
(x) + J
′
(x)φ
′
(x) + ¯η(x)ψ(x) + η(x)
¯
ψ(x)
1
2m
2
s
1
4m
2
V
U(φ
′
, V
′µ
)
U(φ
′
, V
′µ
) =
1
2m
2
s
∂
µ
φ
′
∂
µ
φ
′
−
1
4m
2
V
F
′µν
F
′
µν
=
1
2m
2
s
∂
µ
δ
δJ
′
2
−
1
4m
2
V
∂
ν
δ
δA
′
µ
− ∂
µ
δ
δA
′
ν
2
= U
δ
δJ
′
,
δ
δA
′
µ
W [J
′
, A
′
µ
, η, ¯η] = N
i
d
4
x U
δ
δJ
′
,
δ
δA
′
µ
[Dψ][D
¯
ψ][DV
′µ
][Dφ
′
]
iS
′
F
i
d
4
x U
δ
δJ
′
,
δ
δA
′
µ
≃ 1
W [J
′
, A
′
µ
, η, ¯η] = N
[Dψ][D
¯
ψ][DV
′µ
][Dφ
′
]
iS
′
F
φ
′
V
′µ
ψ
¯
ψ
J
′
(x) = A
′
µ
(x) = 0
−
1
2
φ
′2
−G
′
s
¯
ψφ
′
ψ = −
1
2
(φ
′
+ G
′
s
¯
ψψ)
2
+
1
2
G
′
s
2
(
¯
ψψ)
2
1
2
V
′µ
V
′
µ
− G
′
V
¯
ψγ
µ
V
′
µ
ψ =
1
2
(V
′µ
− G
′
V
¯
ψγ
µ
ψ)
2
−
1
2
G
′
V
2
(
¯
ψγ
µ
ψ)
2
S
′
F
=
d
4
x
¯
ψ(iγ
µ
∂
µ
− M)ψ −
1
2
(φ
′
+ G
′
s
¯
ψψ)
2
+
1
2
G
′
s
2
(
¯
ψψ)
2
+
1
2
(V
′µ
− G
′
V
¯
ψγ
µ
ψ)
2
−
1
2
G
′
V
2
(
¯
ψγ
µ
ψ)
2
+ ¯η(x)ψ(x) + η(x)
¯
ψ(x)
λ = φ
′
+ G
′
s
¯
ψψ
R
µ
= V
′µ
−G
′
V
¯
ψγ
µ
ψ
[Dφ
′
]
−i
d
4
x
1
2
(φ
′
+ G
′
s
¯
ψψ)
2
=
[Dλ]
−i
d
4
x
1
2
λ
2
[DV
′µ
]
i
d
4
x
1
2
(V
′µ
−G
′
V
¯
ψγ
µ
ψ)
2
=
[DR
µ
]
i
d
4
x
1
2
R
µ
R
µ
dy
dx f (x ± y)g(y) =
dy
dx
′
f(x
′
)g(y)
x
′
= x ±y 1
W [η, ¯η] = N
[Dψ][D
¯
ψ]
iS
′′
F
[Dλ]
−i
R
d
4
x
1
2
λDλ
= π
2
[D
−1/2
] D
D = 1
N =
[Dλ]
−i
d
4
x
1
2
λ
2
[DR
µ
]
i
d
4
x
1
2
R
µ
R
µ
[Dλ]
−i
d
4
x
1
2
λ
2
[DR
µ
]
i
d
4
x
1
2
R
µ
R
µ
[Dψ][D
¯
ψ]
iS
′′
=
1
[Dψ][D
¯
ψ]
iS
′′
S
′′
=
d
4
x
¯
ψ(iγ
µ
∂
µ
− M)ψ +
1
2
G
′
s
2
(
¯
ψψ)
2
−
1
2
G
′
V
2
(
¯
ψγ
µ
ψ)
2
S
′′
F
=
d
4
x
¯
ψ(iγ
µ
∂
µ
− M)ψ +
1
2
G
′
s
2
(
¯
ψψ)
2
−
1
2
G
′
V
2
(
¯
ψγ
µ
ψ)
2
+ ¯ηψ + η
¯
ψ
L = (iγ
µ
∂
µ
−M)ψ +
1
2
G
′
s
2
(
¯
ψψ)
2
−
1
2
G
′
V
2
(
¯
ψγ
µ
ψ)
2
L
M
=
¯
ψ(iγ
µ
∂
µ
− M)ψ +
1
2
∂
µ
φ∂
µ
φ −
1
2
m
2
s
φ
2
−
1
4
F
µν
F
µν
+
1
2
m
2
V
V
µ
V
µ
− g
s
¯
ψφψ − g
V
¯
ψγ
µ
V
µ
ψ + L
3
+ L
4
L
3
= −
A
′
3
m
2
s
g
s
3
φ
3
+ 3
m
2
s
g
s
2
φ
2
¯
ψψ + 3
m
2
s
g
s
φ(
¯
ψψ)
2
L
4
= −
B
′
4
m
2
s
g
s
4
φ
4
+ 4
m
2
s
g
s
3
φ
3
¯
ψψ + 6
m
2
s
g
s
2
φ
2
(
¯
ψψ)
2
+ 4
m
2
s
g
s
φ(
¯
ψψ)
3
W [J, A
µ
, η, ¯η] = N
[Dψ][D
¯
ψ][DV
µ
][Dφ]
iS
F
N =
1
d
4
x
iS
S =
d
4
x L
M
S
F
=
d
4
x
L
M
+ A
µ
(x)V
µ
(x) + J(x)φ(x) + ¯η(x)ψ(x) + η(x)
¯
ψ(x)
L
M
=
¯
ψ(iγ
µ
∂
µ
− M)ψ +
1
2m
2
s
∂
µ
φ
′
∂
µ
φ
′
−
1
2
φ
′2
−
1
4m
2
V
F
′µν
F
′
µν
+
1
2
V
′
µ
V
′µ
− G
′
s
¯
ψφ
′
ψ − G
′
V
¯
ψγ
µ
V
′
µ
ψ + L
′
3
+ L
′
4
≡ L
′
M
+ L
′
3
+ L
′
4
+ U(φ
′
, V
′µ
)
L
′
M
=
¯
ψ(iγ
µ
∂
µ
−M)ψ −
1
2
φ
′2
+
1
2
V
′
µ
V
′µ
−G
′
s
¯
ψφ
′
ψ − G
′
V
¯
ψγ
µ
V
′
µ
ψ
L
′
3
= −
A
′
3
1
G
′
s
3
φ
′3
+
3
G
′
s
2
φ
′2
¯
ψψ +
3
G
′
s
φ
′
(
¯
ψψ)
2
L
′
4
= −
B
′
4
1
G
′
s
4
φ
′4
+
4
G
′
s
3
φ
′3
¯
ψψ +
6
G
′
s
2
φ
′2
(
¯
ψψ)
2
+
4
G
′
s
φ
′
(
¯
ψψ)
3
U(φ
′
, V
′µ
)
W [J
′
, A
′
µ
, η, ¯η] = N
[Dψ][D
¯
ψ][DV
′µ
][Dφ
′
]
i
d
4
x U(φ
′
, V
′µ
) + S
′
F
S
′
F
=
d
4
x
L
′
M
+ L
′
3
+ L
′
4
+ A
′
µ
(x)V
′µ
(x) + J
′
(x)φ
′
(x) + ¯η(x)ψ(x) + η(x)
¯
ψ(x)
U(φ
′
, V
′µ
)
W [J
′
, A
′
µ
, η, ¯η] = N
[Dψ][D
¯
ψ][DV
′µ
][Dφ
′
]
iS
′
F
J
′
(x) = A
′
µ
(x) = 0
−
A
′
3
1
G
′
s
3
φ
′3
+
3
G
′
s
2
φ
′2
¯
ψψ +
3
G
′
s
φ
′
(
¯
ψψ)
2
= −
A
′
3G
′
s
3
(φ
′
+G
′
s
¯
ψψ)
3
+
A
′
3
(
¯
ψψ)
3
−
B
′
4
1
G
′
s
4
φ
′4
+
4
G
′
s
3
φ
′3
¯
ψψ +
6
G
′
s
2
φ
′2
(
¯
ψψ)
2
+
4
G
′
s
φ
′
(
¯
ψψ)
3
= −
B
′
4G
′
s
4
(φ
′
+ G
′
s
¯
ψψ)
4
+
B
′
4
(
¯
ψψ)
4
S
′
F
=
d
4
x
¯
ψ(iγ
µ
∂
µ
− M)ψ −
1
2
(φ
′
+ G
′
s
¯
ψψ)
2
+
1
2
G
′
s
2
(
¯
ψψ)
2
+
1
2
(V
′µ
− G
′
V
¯
ψγ
µ
ψ)
2
−
1
2
G
′
V
2
(
¯
ψγ
µ
ψ)
2
−
A
′
3G
′
s
3
(φ
′
+ G
′
s
¯
ψψ)
3
+
A
′
3
(
¯
ψψ)
3
−
B
′
4G
′
s
4
(φ
′
+ G
′
s
¯
ψψ)
4
+
B
′
4
(
¯
ψψ)
4
+ ¯η(x)ψ(x) + η(x)
¯
ψ(x)
λ = φ
′
+ G
′
s
¯
ψψ
R
µ
= V
′µ
−G
′
V
¯
ψγ
µ
ψ
[Dφ
′
]
−i
d
4
x
1
2
(φ
′
+ G
′
s
¯
ψψ)
2
+
A
′
3G
′
s
3
(φ
′
+ G
′
s
¯
ψψ)
3
+
B
′
4G
′
s
4
(φ
′
+ G
′
s
¯
ψψ)
4
=
=
[Dλ]
−i
d
4
x
1
2
λ
2
+
A
′
3G
′
s
3
λ
3
+
B
′
4G
′
s
4
λ
4
[DV
′µ
]
i
d
4
x
1
2
(V
′µ
− G
′
V
¯
ψγ
µ
ψ)
2
=
[DR
µ
]
i
d
4
x
1
2
R
µ
R
µ
W [η, ¯η] = N
[Dψ][D
¯
ψ]
iS
′′
F
N =
1
[Dψ][D
¯
ψ]
iS
′′
S
′′
=
d
4
x
¯
ψ(iγ
µ
∂
µ
− M)ψ +
1
2
G
′
s
2
(
¯
ψψ)
2
−
1
2
G
′
V
2
(
¯
ψγ
µ
ψ)
2
+
A
′
3
(
¯
ψψ)
3
+
B
′
4
(
¯
ψψ)
4
S
′′
F
=
d
4
x
¯
ψ(iγ
µ
∂
µ
− M)ψ +
1
2
G
′
s
2
(
¯
ψψ)
2
−
1
2
G
′
V
2
(
¯
ψγ
µ
ψ)
2
+
A
′
3
(
¯
ψψ)
3
+
B
′
4
(
¯
ψψ)
4
+ ¯η(x)ψ(x) + η(x)
¯
ψ(x)
208
∂
µ
∂L
∂(∂
µ
Q
i
)
−
∂L
∂Q
i
= 0
Q
i
=
¯
ψ ψ
γ
µ
i∂
µ
−G
′
V
2
¯
ψγ
µ
ψ
−
M − G
′
s
2
¯
ψψ − A
′
(
¯
ψψ)
2
− B
′
(
¯
ψψ)
3
ψ = 0
¯
ψγ
µ
ψ
¯
ψψ
¯
ψψ
¯
ψγ
µ
ψ
ρ
s
=
¯
ψψ
ρ =
¯
ψγ
0
ψ
ρ
s
ρ
[iγ
µ
∂
µ
− γ
0
G
′
V
2
ρ − (M − G
′
s
2
ρ
s
− A
′
ρ
2
s
− B
′
ρ
3
s
)]ψ = 0
M
∗
≡ M − G
′
s
2
ρ
s
− A
′
ρ
2
s
−B
′
ρ
3
s
ρ =
γ
2π
2
k
F
0
k
2
dk
=
γ
6π
2
k
3
F
ρ
s
=
γ
2π
2
k
F
0
M
∗
(k
2
+ M
∗
2
)
1/2
k
2
dk
=
γM
∗
4π
2
k
F
(k
2
F
+ M
∗
2
)
1/2
− M
∗
2
ln
k
F
+ (k
2
F
+ M
∗
2
)
1/2
M
∗
ψ(x, t)
k
3
F
k
F
k γ
γ = 4
γ = 2
T
µν
= −g
µν
L +
i
∂L
∂(∂
µ
Q
i
)
∂
ν
Q
i
g
µν
T
µν
= −g
µν
¯
ψ(iγ
α
∂
α
− M)ψ +
1
2
G
′
s
2
(
¯
ψψ)
2
−
1
2
G
′
V
2
(
¯
ψγ
α
ψ)
2
+
A
′
3
(
¯
ψψ)
3
+
B
′
4
(
¯
ψψ)
4
+ i
¯
ψγ
µ
∂
ν
ψ
T
(ACM )
µν
= −g
µν
¯
ψ(iγ
α
∂
α
−M)ψ +
1
2
G
′
s
2
(
¯
ψψ)
2
−
1
2
G
′
V
2
(
¯
ψγ
0
ψ)
2
+
A
′
3
(
¯
ψψ)
3
+
B
′
4
(
¯
ψψ)
4
+ i
¯
ψγ
µ
∂
ν
ψ
E = T
00
¯
ψ(iγ
α
∂
α
− M)ψ =
¯
ψγ
0
ψG
′
V
2
ρ − G
′
s
2
ρ
s
¯
ψψ − A
′
ρ
2
s
¯
ψψ − B
′
ρ
3
s
¯
ψψ
¯
ψ(iγ
α
∂
α
− M)ψ
= G
′
V
2
ρ
2
−G
′
s
2
ρ
2
s
− A
′
ρ
3
s
− B
′
ρ
4
s
µ = ν = 0
E =
1
2
G
′
s
2
ρ
2
s
−
1
2
G
′
V
2
ρ
2
+
2
3
A
′
ρ
3
s
+
3
4
B
′
ρ
4
s
+ i
¯
ψγ
0
∂
0
ψ
i
¯
ψγ
0
∂
0
ψ
ψ
ψ(x, t) = u(k, λ)
−ik
µ
x
µ
i
¯
ψγ
0
∂
0
ψ = k
0
¯
ψγ
0
ψ
k
0
= G
′
V
2
¯
ψγ
0
ψ ± (k
2
+ M
∗
2
)
1/2
= G
′
V
2
ρ ± (k
2
+ M
∗
2
)
1/2
k
0
i
¯
ψγ
0
∂
0
ψ = [G
′
V
2
ρ + (k
2
+ M
∗
2
)
1/2
]
¯
ψγ
0
ψ
i
¯
ψγ
0
∂
0
ψ
= G
′
V
2
ρ
¯
ψγ
0
ψ
+
(k
2
+ M
∗
2
)
1/2
¯
ψγ
0
ψ
= G
′
V
2
ρ
ψ
†
ψ
+
(k
2
+ M
∗
2
)
1/2
ψ
†
ψ
ψ(x, t)
ψ(x, t) =
1
(2π)
3
d
3
k [a
k,λ
u(k, λ)
−ik
µ
x
µ
+ b
†
k,λ
v(k, λ)
ik
µ
x
µ
]
i
¯
ψγ
0
∂
0
ψ
= G
′
V
2
ρ
2
+
γ
2π
2
k
F
0
(k
2
+ M
∗
2
)
1/2
k
2
dk
E =
1
2
G
′
V
2
ρ
2
+
1
2
G
′
s
2
ρ
2
s
+
2
3
A
′
ρ
3
s
+
3
4
B
′
ρ
4
s
+
γ
2π
2
k
F
0
(k
2
+ M
∗
2
)
1/2
k
2
dk
P =
1
3
T
ii
= −
g
ii
3
¯
ψ(iγ
α
∂
α
−M)ψ +
1
2
G
′
s
2
(
¯
ψψ)
2
−
1
2
G
′
V
2
(
¯
ψγ
α
ψ)
2
+
A
′
3
(
¯
ψψ)
3
+
B
′
4
(
¯
ψψ)
4
+
i
3
¯
ψγ
i
∂
i
ψ
=
1
2
G
′
V
2
ρ
2
−
1
2
G
′
s
2
ρ
2
s
−
2
3
A
′
ρ
3
s
−
3
4
B
′
ρ
4
s
+
i
3
¯
ψγ
i
∂
i
ψ
i
¯
ψγ
i
∂
i
ψ
i
¯
ψγ
i
∂
i
ψ
= i
¯
ψγ
0
∂
0
ψ
− i
¯
ψγ
µ
∂
µ
ψ
= G
′
V
2
ρ
2
+
γ
2π
2
k
F
0
(k
2
+ M
∗
2
)
1/2
k
2
dk −G
′
V
2
ρ
2
+ M
∗
ρ
s
=
γ
2π
2
k
F
0
(k
2
+ M
∗
2
)
1/2
−
M
∗
2
(k
2
+ M
∗
2
)
1/2
k
2
dk
=
γ
2π
2
k
F
0
k
4
(k
2
+ M
∗
2
)
1/2
dk
P =
1
2
G
′
V
2
ρ
2
−
1
2
G
′
s
2
ρ
2
s
−
2
3
A
′
ρ
3
s
−
3
4
B
′
ρ
4
s
+
γ
6π
2
k
F
0
k
4
(k
2
+ M
∗
2
)
1/2
dk
G
′
s
2
G
′
V
2
A
′
B
′
E
0
ρ
0
K
0
M
0
E
0
= −16 MeV
E
0
K
0
M
0
ρ
0
= 0, 15 fm
−3
210 ± 30 MeV
230 ±5 MeV
E/ρ ρ K
0
∼ 500−220 MeV
0, 58 ≤ M
∗
0
/M ≤ 0, 62
K = 9
∂(E + P )
∂ρ
−
E + P
ρ
E + P = G
′
V
2
ρ
2
+ (k
2
F
+ M
∗
2
)
1/2
ρ
∂(E + P )
∂ρ
=
∂(E + P )
∂k
F
∂k
F
∂ρ
+
∂(E + P )
∂M
∗
∂M
∗
∂ρ
k
F
= k
F
(ρ) M
∗
= M
∗
(ρ
s
) ρ
s
= ρ
s
(k
F
) = ρ
s
(ρ)
∂(E + P )
∂k
F
∂k
F
∂ρ
=
G
′
V
2
ρ
γk
2
F
π
2
+ (k
2
F
+ M
∗
2
)
1/2
γk
2
F
2π
2
+
ρk
F
(k
2
F
+ M
∗
2
)
1/2
2π
2
γk
2
F
= 2G
′
V
2
ρ + (k
2
F
+ M
∗
2
)
1/2
1 +
k
2
F
3(k
2
F
+ M
∗
2
)
K = 9
G
′
V
2
ρ +
k
2
F
3(k
2
F
+ M
∗
2
)
1/2
+
∂(E + P )
∂M
∗
∂M
∗
∂ρ
= 9
G
′
V
2
ρ +
k
2
F
3(k
2
F
+ M
∗
2
)
1/2
+
ρM
∗
(k
2
F
+ M
∗
2
)
1/2
∂M
∗
∂ρ
∂M
∗
∂ρ
= −(G
′
s
2
+ 2A
′
ρ
s
+ 3B
′
ρ
2
s
)
∂ρ
s
∂ρ
ρ
s
= ρ
s
(k
F
, M
∗
)
∂ρ
s
∂ρ
=
∂ρ
s
∂k
F
∂k
F
∂ρ
+
∂ρ
s
∂M
∗
∂M
∗
∂ρ
∂ρ
s
∂k
F
=
γM
∗
2π
2
k
2
F
(k
2
F
+ M
∗
2
)
1/2
∂ρ
s
∂ρ
=
M
∗
(k
2
F
+ M
∗
2
)
1/2
+
∂ρ
s
∂M
∗
∂M
∗
∂ρ
∂M
∗
∂ρ
= −(G
′
s
2
+ 2A
′
ρ
s
+ 3B
′
ρ
2
s
)
M
∗
(k
2
F
+ M
∗
2
)
1/2
+
∂ρ
s
∂M
∗
∂M
∗
∂ρ
= −
M
∗
(k
2
F
+ M
∗
2
)
1/2
G
′
s
2
+ 2A
′
ρ
s
+ 3B
′
ρ
2
s
1 + (G
′
s
2
+ 2A
′
ρ
s
+ 3B
′
ρ
2
s
)
∂ρ
s
∂M
∗
∂ρ
s
∂M
∗
γk
F
2π
2
(k
2
F
+ M
∗
2
)
1/2
∂ρ
s
∂M
∗
= 3
ρ
s
M
∗
−
ρ
(k
2
F
+ M
∗
2
)
1/2
K = 9G
′
V
2
ρ +
3k
2
F
E
∗
F
−
9M
∗
2
E
∗
F
2
(G
′
s
2
+ 2A
′
ρ
s
+ 3B
′
ρ
2
s
)ρ
1 + 3(G
′
s
2
+ 2A
′
ρ
s
+ 3B
′
ρ
2
s
)
ρ
s
M
∗
−
ρ
E
∗
F
E
∗
F
= (k
2
F
+ M
∗
2
)
1/2
K
0
ρ = ρ
0
P (ρ
0
) = 0
E(ρ
0
)
ρ
0
−M = −E
0
K(ρ
0
) = K
0
M
∗
(ρ
0
) = M
∗
0
∂(E/ρ)
∂ρ
ρ=ρ
0
= 0
−E
0
K
0
M
∗
0
Z
G
= [
−β(
b
H−µ
b
N)
]
=
n
1
···
n
∞
¯n
1
···
¯n
∞
× n
1
, ··· , n
∞
; ¯n
1
, ··· , ¯n
∞
|
−β(
b
H−µ
b
N)
|n
1
, ··· , n
∞
; ¯n
1
, ··· , ¯n
∞
H
N µ
β ≡
1
T
n
1
¯n
1
n
2
¯n
2
k
ˆ
H =
d
3
x T
00
¯
ψψ
≡ σ
0
¯
ψγ
0
ψ
≡ v
0
T
(ACM )
00
T
(ACM )
00
= i
¯
ψγ
0
∂
0
ψ +
1
2
G
′
s
2
σ
2
0
−
1
2
G
′
V
2
v
2
0
+
2
3
A
′
σ
3
0
+
3
4
B
′
σ
4
0
ψ ψ
†
ψ =
1
√
V
k
a
k,λ
u(k, λ)
−iǫ
+
(k)t+ik·x
+ b
†
k,λ
v(k, λ)
−iǫ
−
(k)t−ik·x
ψ
†
=
1
√
V
k
a
†
k,λ
u
†
(k, λ)
iǫ
+
(k)t−ik·x
+ b
k,λ
v
†
(k, λ)
iǫ
−
(k)t+ik·x
V a
†
k,λ
a
k,λ
b
†
k,λ
b
k,λ
i
¯
ψγ
0
∂
0
ψ = iψ
†
∂
0
ψ
=
1
V
k,k
′
a
†
k,λ
a
k
′
,λ
u
†
(k, λ)u(k
′
, λ)
i[ǫ
+
(k)−ǫ
+
(k
′
)]t −i(k−k
′
)·x
ǫ
+
(k
′
)
+ a
†
k,λ
b
†
k
′
,λ
u
†
(k, λ)v(k
′
, λ)
i[ǫ
+
(k)−ǫ
−
(k
′
)]t −i(k+k
′
)·x
ǫ
−
(k
′
)
+ b
k,λ
a
k
′
,λ
v
†
(k, λ)u(k
′
, λ)
i[ǫ
−
(k)−ǫ
+
(k
′
)]t i(k+k
′
)·x
ǫ
+
(k
′
)
+ b
k,λ
b
†
k
′
,λ
v
†
(k, λ)v(k
′
, λ)
i[ǫ
−
(k)−ǫ
−
(k
′
)]t i(k−k
′
)·x
ǫ
−
(k
′
)
ˆ
H =
d
3
x iψ
†
∂
0
ψ +
1
2
G
′
s
2
σ
2
0
+
4
3
A
′
σ
3
0
+
3
2
B
′
σ
4
0
−G
′
V
2
v
2
0
d
3
x
=
d
3
x iψ
†
∂
0
ψ +
1
2
G
′
s
2
σ
2
0
+
4
3
A
′
σ
3
0
+
3
2
B
′
σ
4
0
−G
′
V
2
v
2
0
V
δ
k,k
′
=
1
V
d
3
x
i(k−k
′
)·x
u
†
(k, λ)u(k, λ) = v
†
(k, λ)v(k, λ) = 1
ˆ
H =
k
(a
†
k,λ
a
k,λ
ǫ
+
+ b
k,λ
b
†
k,λ
ǫ
−
) +
1
2
G
′
s
2
σ
2
0
+
4
3
A
′
σ
3
0
+
3
2
B
′
σ
4
0
− G
′
V
2
v
2
0
V
ǫ
±
= G
′
V
2
v
0
±
k
2
+ M
∗
2
1/2
b
k,λ
, b
†
k
′
,λ
′
= δ
k,k
′
δ
λ,λ
′
ˆ
H =
k
k
2
+ M
∗
2
1/2
(a
†
k,λ
a
k,λ
+ b
†
k,λ
b
k,λ
) + G
′
V
2
v
0
ˆ
N
+
1
2
G
′
s
2
σ
2
0
+
4
3
A
′
σ
3
0
+
3
2
B
′
σ
4
0
− G
′
V
2
v
2
0
V
k
(k
2
+M
∗
2
)
1/2
G
′
V
2
v
0
k
1
ˆ
N =
k
(a
†
k,λ
a
k,λ
−b
†
k,λ
b
k,λ
)
|n
1
, ··· , n
∞
; ¯n
1
, ··· , ¯n
∞
≡ |n
i
; ¯n
i
,
ˆ
H |n
i
; ¯n
i
=
k
E
∗
k
n
k
+
k
′
E
∗
k
′
¯n
k
′
+ G
′
V
2
v
0
k
n
k
− G
′
V
2
v
0
k
′
¯n
k
′
+
1
2
G
′
s
2
σ
2
0
+
4
3
A
′
σ
3
0
+
3
2
B
′
σ
4
0
− G
′
V
2
v
2
0
V
|n
i
; ¯n
i
µ
ˆ
N |n
i
; ¯n
i
= µ
k
n
k
−
k
′
¯n
k
′
|n
i
; ¯n
i
Z
G
= n
1
|
−β(E
∗
1
+G
′
V
2
v
0
−µ)n
1
| n
1
. . . n
∞
|
−β(E
∗
∞
+G
′
V
2
v
0
−µ)n
∞
| n
∞
×
× ¯n
1
|
−β(E
∗
1
−G
′
V
2
v
0
+µ)¯n
1
| ¯n
1
. . . ¯n
∞
|
−β(E
∗
∞
−G
′
V
2
v
0
+µ)¯n
∞
| ¯n
∞
×
×
−
βV
2
(
G
′
s
2
σ
2
0
+
4
3
A
′
σ
3
0
+
3
2
B
′
σ
4
0
−G
′
V
2
v
2
0
)
Z
G
=
−
βV
2
(
G
′
s
2
σ
2
0
+
4
3
A
′
σ
3
0
+
3
2
B
′
σ
4
0
−G
′
V
2
v
2
0
)
∞
i=0
∞
n
i
=1
n
i
|
−β(E
∗
i
−ν)n
i
| n
i
×
×
∞
j=0
∞
¯n
j
=1
¯n
j
|
−β(E
∗
j
+ν)¯n
j
| ¯n
j
ν ≡ µ −G
′
V
2
v
0
n
i
, ¯n
i
= 0, 1
Z
G
=
−
βV
2
(
G
′
s
2
σ
2
0
+
4
3
A
′
σ
3
0
+
3
2
B
′
σ
4
0
−G
′
V
2
v
2
0
)
i
[1 +
−β(E
∗
i
−ν)
]
j
[1 +
−β(E
∗
j
+ν)
]
Ω(T, V, µ) = −
1
β
ln Z
G
Ω(T, V, µ) =
1
2
G
′
s
2
σ
2
0
+
4
3
A
′
σ
3
0
+
3
2
B
′
σ
4
0
−G
′
V
2
v
2
0
V −
1
β
i
ln [1 +
−β(E
∗
i
−ν)
]
−
1
β
j
ln [1 +
−β(E
∗
j
+ν)
]
Ω(T, V, µ)
ρ = −
1
V
∂Ω
∂µ
T,V,σ
0
,v
0
ρ = −
1
V
∂Ω
∂ν
T,V,σ
0
,v
0
=
1
βV
i
β
−β(E
∗
i
−ν)
1 +
−β(E
∗
i
−ν)
+
1
βV
j
−β
−β(E
∗
j
+ν)
1 +
−β(E
∗
j
+ν)
=
1
V
i
1
1 +
β(E
∗
i
−ν)
−
1
V
j
1
1 +
β(E
∗
j
+ν)
1
V
−→
γ
(2π)
3
d
3
k
ρ =
γ
(2π)
3
d
3
k [n
k
(T ) − ¯n
k
(T )]
=
γ
2π
2
∞
0
k
2
[n
k
(T ) − ¯n
k
(T )] dk
n
k
(T ) =
1
(
√
k
2
+M
∗
2
−ν)/T
+ 1
¯n
k
(T ) =
1
(
√
k
2
+M
∗
2
+ν)/T
+ 1
ˆρ =
1
V
d
3
x ψ
†
ψ
n
k
(T ) = a
†
k,λ
a
k,λ
¯n
k
(T ) = b
†
k,λ
b
k,λ
ρ =
¯
ψγ
0
ψ = ψ
†
ψ = v
0
1
V
∂Ω
∂M
T,V,µ,σ
0
,v
0
=
1
V
∂Ω
∂M
∗
T,V,µ,σ
0
,v
0
=
1
V
i
M
∗
E
∗
i
−β(E
∗
i
−ν)
1 +
−β(E
∗
i
−ν)
+
1
V
i
M
∗
E
∗
j
−β(E
∗
j
+ν)
1 +
−β(E
∗
j
+ν)
=
1
V
k
M
∗
E
∗
(k)
[n
k
(T ) + ¯n
k
(T )]
ˆρ
s
=
1
V
d
3
x
¯
ψψ
ρ
s
=
¯
ψψ = σ
0
ρ
s
=
γ
(2π)
3
d
3
k
M
∗
E
∗
[n
k
(T ) + ¯n
k
(T )]
=
γ
2π
2
∞
0
M
∗
(k
2
+ M
∗
2
)
1/2
k
2
[n
k
(T ) + ¯n
k
(T )] dk
Ω(T, V, µ) = −P V = E − T S − µN
P = −Ω/V
P =
1
2
G
′
V
2
v
2
0
−
1
2
G
′
s
2
σ
2
0
−
2
3
A
′
σ
3
0
−
3
4
B
′
σ
4
0
+
1
β
γ
(2π)
3
d
3
k
ln [1 +
−β(E
∗
−ν)
] + ln [1 +
−β(E
∗
+ν)
]
P =
1
2
G
′
V
2
ρ
2
−
1
2
G
′
s
2
ρ
2
s
−
2
3
A
′
ρ
3
s
−
3
4
B
′
ρ
4
s
+
γ
6π
2
∞
0
k
4
(k
2
+ M
∗
2
)
1/2
[n
k
(T ) + ¯n
k
(T )] dk
S = −
∂Ω
∂T
= β
2
∂Ω
∂β
E =
1
V
∂(βΩ)
∂β
+ µρ
E = −
1
2
G
′
V
2
v
2
0
+
1
2
G
′
s
2
σ
2
0
+
2
3
A
′
σ
3
0
+
3
4
B
′
σ
4
0
+ (µ − ν)ρ
+
γ
(2π)
3
d
3
k E
∗
[n
k
(T ) + ¯n
k
(T )]
u = ln [1 +
−β(E
∗
−ν)
]
v =
1
3
k
3
E =
1
2
G
′
V
2
ρ
2
+
1
2
G
′
s
2
ρ
2
s
+
2
3
A
′
ρ
3
s
+
3
4
B
′
ρ
4
s
+
γ
2π
2
∞
0
k
2
(k
2
+ M
∗
2
)
1/2
[n
k
(T ) + ¯n
k
(T )] dk
L
M
=
¯
ψ (iγ
µ
∂
µ
− M) ψ +
1
2
∂
µ
φ∂
µ
φ −
1
4
F
µν
F
µν
+
1
2
m
2
V
V
µ
V
µ
−g
V
¯
ψγ
µ
V
µ
ψ
−
1
2
m
s
φ +
g
s
m
s
¯
ψψ
2
+
g
2
s
2m
2
s
¯
ψψ
2
−
A
′
3
m
s
g
s
φ +
¯
ψψ
3
+
A
′
3
¯
ψψ
3
−
B
′
4
m
2
s
g
s
φ +
¯
ψψ
4
+
B
′
4
¯
ψψ
4
λ ≡
m
2
s
g
s
φ +
¯
ψψ
L
M
=
¯
ψ (iγ
µ
∂
µ
− M) ψ −
1
4
F
µν
F
µν
+
1
2
m
2
V
V
µ
V
µ
− g
V
¯
ψγ
µ
V
µ
ψ −
1
2
G
′
s
2
λ
2
+
1
2
G
′
s
2
¯
ψψ
2
−
A
′
3
λ
3
+
A
′
3
¯
ψψ
3
−
B
′
4
λ
4
+
B
′
4
¯
ψψ
4
+
G
′
s
2m
2
s
∂
µ
λ − ∂
µ
¯
ψψ
∂
µ
λ − ∂
µ
¯
ψψ
G
′
s
=
g
s
m
s
λ → λ ≡ λ
V
µ
→ V
µ
≡ δ
µ0
V
0
¯
ψψ
λ
¯
ψψ
L
(ACM )
M
=
¯
ψ (iγ
µ
∂
µ
−M) ψ +
1
2
m
2
V
V
2
0
−g
V
¯
ψγ
0
V
0
ψ −
1
2
G
′
s
2
λ
2
+
1
2
G
′
s
2
¯
ψψ
2
−
A
′
3
λ
3
+
A
′
3
¯
ψψ
3
−
B
′
4
λ
4
+
B
′
4
¯
ψψ
4
λ V
µ
λ
¯
ψ
ψ V
0
λ(G
′
s
2
+ A
′
λ + B
′
λ
2
) = 0
V
0
=
g
V
m
2
V
¯
ψγ
0
ψ
[iγ
µ
∂
µ
− g
V
γ
0
V
0
− (M − G
′
s
2
(
¯
ψψ) −A
′
(
¯
ψψ)
2
−B
′
(
¯
ψψ)
3
)]ψ = 0
ψ
¯
ψψ
¯
ψγ
µ
ψ
V
0
=
g
V
m
2
V
¯
ψγ
0
ψ
=
g
V
m
2
V
ρ
[iγ
µ
∂
µ
−γ
0
G
′
V
2
ρ − (M − G
′
s
2
ρ
s
− A
′
ρ
2
s
− B
′
ρ
3
s
)]ψ = 0
G
′
V
=
g
V
m
V
ρ
s
=
¯
ψψ
ψ
M
∗
= M − G
′
s
2
ρ
s
− A
′
ρ
2
s
− B
′
ρ
3
s
ψ
ρ ρ
s
T
(ACM )
µν
= −g
µν
¯
ψ(iγ
α
∂
α
− g
V
γ
0
V
0
− M)ψ +
1
2
m
2
V
V
2
0
−
1
2
G
′
s
2
λ
2
+
1
2
G
′
s
2
(
¯
ψψ)
2
−
A
′
3
λ
3
+
A
′
3
(
¯
ψψ)
3
−
B
′
4
λ
4
+
B
′
4
¯
ψψ
4
+ i
¯
ψγ
µ
∂
ν
ψ
= −g
µν
−
1
2
G
′
s
2
(
¯
ψψ)
2
−
2A
′
3
(
¯
ψψ)
3
−
3B
′
4
(
¯
ψψ)
4
+
1
2
m
2
V
V
2
0
−
1
2
G
′
s
2
λ
2
−
A
′
3
λ
3
−
B
′
4
λ
4
+ i
¯
ψγ
µ
∂
ν
ψ
E =
T
(ACM )
00
=
1
2
G
′
s
2
ρ
2
s
+
2A
′
3
ρ
3
s
+
3B
′
4
ρ
4
s
−
1
2
G
′
V
2
ρ
2
+
1
2
G
′
s
2
λ
2
+
A
′
3
λ
3
+
B
′
4
λ
4
+ i
¯
ψγ
0
∂
0
ψ
i
¯
ψγ
0
∂
0
ψ
k
0
= g
V
V
0
+ (k
2
+ M
∗
2
)
1/2
= G
′
V
2
ρ + (k
2
+ M
∗
2
)
1/2
E =
1
2
G
′
V
2
ρ
2
+
1
2
G
′
s
2
ρ
2
s
+
2A
′
3
ρ
3
s
+
3B
′
4
ρ
4
s
+
1
2
G
′
s
2
λ
2
+
A
′
3
λ
3
+
B
′
4
λ
4
+
γ
2π
2
k
F
0
(k
2
+ M
∗
2
)
1/2
k
2
dk
P =
1
3
T
ii
=
1
2
G
′
V
2
ρ
2
−
1
2
G
′
s
2
ρ
2
s
−
2A
′
3
ρ
3
s
−
3B
′
4
ρ
4
s
−
1
2
G
′
s
2
λ
2
−
A
′
3
λ
3
−
B
′
4
λ
4
+
1
3
i
¯
ψγ
i
∂
i
ψ
i
¯
ψγ
i
∂
i
ψ
P =
1
2
G
′
V
2
ρ
2
−
1
2
G
′
s
2
ρ
2
s
−
2A
′
3
ρ
3
s
−
3B
′
4
ρ
4
s
−
1
2
G
′
s
2
λ
2
−
A
′
3
λ
3
−
B
′
4
λ
4
+
γ
6π
2
k
F
0
k
4
(k
2
+ M
∗
2
)
1/2
dk
1
2
G
′
s
2
λ
2
+
A
′
3
λ
3
+
B
′
4
λ
4
= 0
E =
1
2
G
′
V
2
ρ
2
+
1
2
G
′
s
2
ρ
2
s
+
2
3
A
′
ρ
3
s
+
3
4
B
′
ρ
4
s
+
γ
2π
2
k
F
0
(k
2
+ M
∗
2
)
1/2
k
2
dk
P =
1
2
G
′
V
2
ρ
2
−
1
2
G
′
s
2
ρ
2
s
−
2
3
A
′
ρ
3
s
−
3
4
B
′
ρ
4
s
+
γ
6π
2
k
F
0
k
4
(k
2
+ M
∗
2
)
1/2
dk
ρ = 0
λ =
m
2
s
gs
φ +
¯
ψψ A
′
= B
′
= 0
λ
λ
φ φ
φ
¯
ψψ
λ λ
λ
λ
A
′
B
′
λ
λ ρ
λ
ρ
λ
A
′
= B
′
= 0
G
′
s
2
A
′
B
′
ρ λ
G
′
s
=
g
s
m
s
G
′
V
=
g
V
m
V
L
W NL
=
¯
ψ(iγ
µ
∂
µ
−M)ψ +
1
2
∂
µ
φ∂
µ
φ −
1
2
m
2
s
φ
2
−
1
4
F
µν
F
µν
+
1
2
m
2
V
V
µ
V
µ
− g
s
¯
ψφψ − g
V
¯
ψγ
µ
V
µ
ψ −
A
3
φ
3
−
B
4
φ
4
F
µν
≡ ∂
µ
V
ν
−∂
ν
V
µ
g
s
g
V
B A
MeV m
s
m
V
MeV
φ V
µ
ψ
(∂
µ
∂
µ
+ m
2
s
)φ + Aφ
2
+ Bφ
3
= −g
s
¯
ψψ
∂
ν
F
µν
−m
2
V
V
µ
= −g
V
¯
ψγ
µ
ψ
[γ
µ
(i∂
µ
−g
V
V
µ
) − (M + g
s
φ)] ψ = 0
A = B = 0
¯
ψψ
¯
ψγ
µ
ψ
M
∗
= M + g
s
φ
T
µν
= −g
µν
1
2
∂
α
φ∂
α
φ −
1
4
F
αβ
F
αβ
+
1
2
m
2
V
V
α
V
α
−
1
2
m
2
s
φ
2
−
A
3
φ
3
−
B
4
φ
4
+ ∂
µ
φ∂
ν
φ + i
¯
ψγ
µ
∂
ν
ψ + F
αµ
∂
ν
V
α
φ → φ ≡ φ
0
V
µ
→ V
µ
≡ δ
µ0
V
0
L
(ACM )
W NL
=
¯
ψ(iγ
µ
∂
µ
− M)ψ −
1
2
m
2
s
φ
2
0
+
1
2
m
2
V
V
2
0
− g
s
¯
ψφ
0
ψ − g
V
¯
ψγ
0
V
0
ψ
−
A
3
φ
3
0
−
B
4
φ
4
0
T
(ACM )
µν
= −g
µν
1
2
m
2
V
V
2
0
−
1
2
m
2
s
φ
2
0
−
A
3
φ
3
0
−
B
4
φ
4
0
+ i
¯
ψγ
µ
∂
ν
ψ
φ
0
= −
g
s
m
2
s
ρ
s
−
A
m
2
s
φ
2
0
−
B
m
2
s
φ
3
0
V
0
=
g
V
m
2
V
ρ
iγ
µ
∂
µ
−g
V
γ
0
V
0
−M
∗
ψ = 0
ρ
s
=
¯
ψψ
ρ =
¯
ψγ
0
ψ
ψ
ρ ρ
s
ρ =
γ
2π
2
k
F
0
k
2
dk =
γ
6π
2
k
3
F
ρ
s
=
γ
2π
2
k
F
0
M
∗
(k
2
+ M
∗
2
)
1/2
k
2
dk
=
γM
∗
4π
2
k
F
(k
2
F
+ M
∗
2
)
1/2
− M
∗
2
ln
k
F
+ (k
2
F
+ M
∗
2
)
1/2
M
∗
M
∗
= M −
g
2
s
m
2
s
ρ
s
−
Ag
s
m
2
s
φ
2
0
−
Bg
s
m
2
s
φ
3
0
= M −
g
2
s
m
2
s
ρ
s
−
A
m
2
s
(M
∗
− M)
2
g
s
−
B
m
2
s
(M
∗
−M)
3
g
2
s
E =
T
(ACM )
00
P =
1
3
T
(ACM )
ii
E = −
g
2
V
2m
2
V
ρ
2
+
1
2
m
2
s
φ
2
0
+
A
3
φ
3
0
+
B
4
φ
4
0
+ i
¯
ψγ
0
∂
0
ψ
k
0
= g
V
V
0
± (k
2
+ M
∗
2
)
1/2
=
g
2
V
m
2
V
ρ ± (k
2
+ M
∗
2
)
1/2
i
¯
ψγ
0
∂
0
ψ
=
g
2
V
m
2
V
ρ
2
+
γ
2π
2
k
F
0
(k
2
+ M
∗
2
)
1/2
k
2
dk
E =
g
2
V
2m
2
V
ρ
2
+
1
2
m
2
s
φ
2
0
+
A
3
φ
3
0
+
B
4
φ
4
0
+
γ
2π
2
k
F
0
(k
2
+ M
∗
2
)
1/2
k
2
dk
=
g
2
V
2m
2
V
ρ
2
+
m
2
s
(∆M)
2
2g
2
s
+
A(∆M)
3
3g
3
s
+
B(∆M)
4
4g
4
s
+
γ
2π
2
k
F
0
(k
2
+ M
∗
2
)
1/2
k
2
dk
∆M ≡ M
∗
− M
P =
g
2
V
2m
2
V
ρ
2
−
1
2
m
2
s
φ
2
0
−
A
3
φ
3
0
−
B
4
φ
4
0
+
1
3
i
¯
ψγ
i
∂
i
ψ
i
¯
ψγ
i
∂
i
ψ
1
3
i
¯
ψγ
i
∂
i
ψ
=
γ
6π
2
k
F
0
k
4
(k
2
+ M
∗
2
)
1/2
dk
P =
g
2
V
2m
2
V
ρ
2
−
1
2
m
2
s
φ
2
0
−
A
3
φ
3
0
−
B
4
φ
4
0
+
γ
6π
2
k
F
0
k
4
(k
2
+ M
∗
2
)
1/2
dk
=
g
2
V
2m
2
V
ρ
2
−
m
2
s
(∆M)
2
2g
2
s
−
A(∆M)
3
3g
3
s
−
B(∆M)
4
4g
4
s
+
γ
6π
2
k
F
0
k
4
(k
2
+ M
∗
2
)
1/2
dk
K = 9
∂(E + P )
∂ρ
−
E + P
ρ
E + P =
g
2
V
m
2
V
ρ
2
+ (k
2
F
+ M
∗
2
)
1/2
ρ
∂(E + P )
∂ρ
=
∂(E + P )
∂k
F
∂k
F
∂ρ
+
∂(E + P )
∂M
∗
∂M
∗
∂ρ
k
F
= k
F
(ρ) M
∗
= M
∗
(ρ
s
) = M
∗
(ρ)
∂(E + P )
∂k
F
∂k
F
∂ρ
= 2
g
2
V
m
2
V
ρ + (k
2
F
+ M
∗
2
)
1/2
1 +
k
2
F
3(k
2
F
+ M
∗
2
)
K = 9
g
2
V
m
2
V
ρ +
k
2
F
3(k
2
F
+ M
∗
2
)
1/2
+
∂(E + P )
∂M
∗
∂M
∗
∂ρ
= 9
g
2
V
m
2
V
ρ +
k
2
F
3(k
2
F
+ M
∗
2
)
1/2
+
ρM
∗
(k
2
F
+ M
∗
2
)
1/2
∂M
∗
∂ρ
∂M
∗
∂ρ
= −
g
2
s
m
2
s
∂ρ
s
∂ρ
−
∂φ
0
∂ρ
2Ag
s
m
2
s
φ
0
+
3Bg
s
m
2
s
φ
2
0
= −
g
2
s
m
2
s
∂ρ
s
∂ρ
−
∂M
∗
∂ρ
2A
m
2
s
φ
0
+
3B
m
2
s
φ
2
0
ρ
s
ρ
s
ρ
∂ρ
s
∂ρ
=
M
∗
(k
2
F
+ M
∗
2
)
1/2
+ 3
ρ
s
M
∗
−
ρ
(k
2
F
+ M
∗
2
)
1/2
∂M
∗
∂ρ
∂M
∗
∂ρ
∂M
∗
∂ρ
= −
g
2
s
M
∗
m
2
s
E
∗
F
1 +
2A
m
2
s
φ
0
+
3B
m
2
s
φ
2
0
+
3g
2
s
m
2
s
ρ
s
M
∗
−
ρ
E
∗
F
E
∗
F
= (k
2
F
+ M
∗
2
)
1/2
K = 9
g
2
V
m
2
V
ρ +
3k
2
F
E
∗
F
−
9M
∗
2
E
∗
F
2
g
2
s
ρ
m
2
s
1 +
2A∆M
m
2
s
g
s
+
3B(∆M)
2
m
2
s
g
2
s
+
3g
2
s
m
2
s
ρ
s
M
∗
−
ρ
E
∗
F
M
∗
= M − G
2
s
ρ
s
+ a(∆M)
2
+ b(∆M)
3
E =
1
2
G
2
V
ρ
2
+
(∆M)
2
2G
2
s
+
a(∆M)
3
3
+
b(∆M)
4
4
+
γ
2π
2
k
F
0
(k
2
+ M
∗
2
)
1/2
k
2
dk
P =
1
2
G
2
V
ρ
2
−
(∆M)
2
2G
2
s
−
a(∆M)
3
3
−
b(∆M)
4
4
+
γ
6π
2
k
F
0
k
4
(k
2
+ M
∗
2
)
1/2
dk
K = 9G
2
V
ρ +
3k
2
F
E
∗
F
−
9M
∗
2
ρ
E
∗
F
2
1
G
2
s
+ 2a∆M + 3b(∆M)
2
+ 3
ρ
s
M
∗
−
ρ
E
∗
F
G
2
s
≡
g
2
s
m
2
s
G
2
V
≡
g
2
V
m
2
V
a ≡
A
g
3
s
b ≡
B
g
4
s
P (ρ
0
) = 0
E(ρ
0
)
ρ
0
−M = −E
0
K(ρ
0
) = K
0
M
∗
(ρ
0
) = M
∗
0
G
2
s
G
2
V
MeV
−2
a MeV b
ρ
0
= 0, 15 fm
−3
E
0
= −16 MeV
K
0
∼ 500 − 220 MeV
0, 58 ≤ M
∗
0
/M ≤ 0, 62
T
00
T
(ACM )
00
= i
¯
ψγ
0
∂
0
ψ +
1
2
m
2
s
φ
2
0
−
1
2
m
2
V
V
2
0
+
1
3
Aφ
3
0
+
1
4
Bφ
4
0
ψ ψ
†
ˆ
H =
d
3
x T
(ACM )
00
=
d
3
x iψ
†
∂
0
ψ +
1
2
m
2
s
φ
2
0
+
2
3
Aφ
3
0
+
1
2
Bφ
4
0
− m
2
V
V
2
0
V
=
k
(a
†
k,λ
a
k,λ
ǫ
+
+ b
k,λ
b
†
k,λ
ǫ
−
) +
1
2
m
2
s
φ
2
0
+
2
3
Aφ
3
0
+
1
2
Bφ
4
0
−m
2
V
V
2
0
V
ǫ
±
= g
V
V
0
±
k
2
+ M
∗
2
1/2
ˆ
H =
k
k
2
+ M
∗
2
1/2
(a
†
k,λ
a
k,λ
+ b
†
k,λ
b
k,λ
) + g
V
V
0
ˆ
N
+
1
2
m
2
s
φ
2
0
+
2
3
Aφ
3
0
+
1
2
Bφ
4
0
− m
2
V
V
2
0
V
ˆ
N =
k
(a
†
k,λ
a
k,λ
−b
†
k,λ
b
k,λ
)
φ
0
V
0
ˆ
H |n
i
; ¯n
i
=
k
E
∗
k
n
k
+
k
′
E
∗
k
′
¯n
k
′
+ g
V
V
0
k
n
k
−g
V
V
0
k
′
¯n
k
′
+
1
2
m
2
s
φ
2
0
+
2
3
Aφ
3
0
+
1
2
Bφ
4
0
− m
2
V
V
2
0
V
|n
i
; ¯n
i
µ
ˆ
N |n
i
; ¯n
i
= µ
k
n
k
−
k
′
¯n
k
′
|n
i
; ¯n
i
Z
G
=
−
βV
2
(
m
2
s
φ
2
0
+
2
3
Aφ
3
0
+
1
2
Bφ
4
0
−m
2
V
V
2
0
)
∞
i=0
∞
n
i
=1
n
i
|
−β(E
∗
i
−ν)n
i
| n
i
×
×
∞
j=0
∞
¯n
j
=1
¯n
j
|
−β(E
∗
j
+ν)¯n
j
| ¯n
j
=
−
βV
2
(
m
2
s
φ
2
0
+
2
3
Aφ
3
0
+
1
2
Bφ
4
0
−m
2
V
V
2
0
)
i
[1 +
−β(E
∗
i
−ν)
]
j
[1 +
−β(E
∗
j
+ν)
]
ν ≡ µ −g
V
V
0
k
(k
2
+ M
∗
2
)
1/2
g
V
V
0
k
1
Ω(T, V, µ) =
1
2
m
2
s
φ
2
0
+
2
3
Aφ
3
0
+
1
2
Bφ
4
0
−m
2
V
V
2
0
V −
1
β
i
ln [1 +
−β(E
∗
i
−ν)
]
−
1
β
j
ln [1 +
−β(E
∗
j
+ν)
]
Ω
ρ = −
1
V
∂Ω
∂µ
T,V,φ
0
,V
0
= −
1
V
∂Ω
∂ν
T,V,φ
0
,V
0
=
γ
2π
2
∞
0
k
2
[n
k
(T ) − ¯n
k
(T )] dk
ρ =
¯
ψγ
0
ψ = ψ
†
ψ =
m
2
V
g
V
V
0
ρ
s
=
1
V
∂Ω
∂M
T,V,µ,φ
0
,V
0
=
1
V
∂Ω
∂M
∗
T,V,µ,φ
0
,V
0
=
γ
2π
2
∞
0
M
∗
(k
2
+ M
∗
2
)
1/2
k
2
[n
k
(T ) + ¯n
k
(T )] dk
¯
ψψ
P = −Ω/V
P =
1
2
m
2
V
V
2
0
−
1
2
m
2
s
φ
2
0
−
1
3
Aφ
3
0
−
1
4
Bφ
4
0
+
1
β
γ
(2π)
3
d
3
k
ln [1 +
−β(E
∗
−ν)
] + ln [1 +
−β(E
∗
+ν)
]
P =
1
2
G
2
V
ρ
2
−
(∆M)
2
2G
2
s
−
a(∆M)
3
3
−
b(∆M)
4
4
+
γ
6π
2
∞
0
k
4
(k
2
+ M
∗
2
)
1/2
[n
k
(T ) + ¯n
k
(T )] dk
E =
1
V
∂(βΩ)
∂β
+ µρ
= −
1
2
m
2
V
V
2
0
+
1
2
m
2
s
φ
2
0
+
1
3
Aφ
3
0
+
1
4
Bφ
4
0
+ (µ − ν)ρ
+
γ
(2π)
3
d
3
k E
∗
[n
k
(T ) + ¯n
k
(T )]
E =
1
2
G
2
V
ρ
2
+
(∆M)
2
2G
2
s
+
a(∆M)
3
3
+
b(∆M)
4
4
+
γ
2π
2
∞
0
k
2
(k
2
+ M
∗
2
)
1/2
[n
k
(T ) + ¯n
k
(T )] dk
a = b = 0
0 0.2 0.4
0.6
0.8 1
ρ/ρ
0
-1
-0.5
0
0.5
1
P (MeV/fm
3
)
m*=0.54 K
0
=551 MeV
m*=0.56 K
0
=450 MeV
m*=0.58 K
0
=350 MeV
m*=0.60 K
0
=250 MeV
m*=0.62 K
0
=220 MeV
T = 0 MeV
P ×ρ/ρ
0
m
∗
= M
∗
0
/M K
0
T = 0
ρ
0
= 0.15 fm
−3
E
0
= −16 MeV
m
∗
≡ M
∗
0
/M K
0
0
0.25 0.5 0.75
1
1.25 1.5 1.75
2
ρ/ρ
0
-16
-8
0
8
16
24
ε/ρ - M (MeV)
m*=0.54 K
0
=551 MeV
m*=0.56 K
0
=450 MeV
m*=0.58 K
0
=350 MeV
m*=0.60 K
0
=250 MeV
m*=0.62 K
0
=220 MeV
T = 0 MeV
ρ/ρ
0
m
∗
= M
∗
0
/M K
0
T = 0
E
0
= −16 MeV
M
∗
/M ρ/ρ
0
0
0.5
1
1.5
2
2.5
3
3.5
4
ρ/ρ
0
0
0.2
0.4
0.6
0.8
1
M
*
/M
m*=0.54 K
0
=551 MeV
M
∗
/M
ρ/ρ
0
m
∗
= 0, 54 K
0
= 551 MeV
T = 0
0
0.5
1
1.5
2
2.5
3
3.5
4
ρ/ρ
0
0
0.2
0.4
0.6
0.8
1
M
*
/M
m*=0.56 K
0
=450 MeV
M
∗
/M
ρ/ρ
0
m
∗
= 0, 56 K
0
= 450 MeV
T = 0
0
0.5
1
1.5
2
2.5
3
3.5
4
ρ/ρ
0
0
0.2
0.4
0.6
0.8
1
M
*
/M
m*=0.58 K
0
=350 MeV
M
∗
/M ρ/ρ
0
m
∗
= 0, 58 K
0
= 350 MeV
T = 0
0
0.5
1
1.5
2
2.5
3
3.5
4
ρ/ρ
0
0
0.2
0.4
0.6
0.8
1
M
*
/M
m*=0.60 K
0
=250 MeV
M
∗
/M ρ/ρ
0
m
∗
= 0, 60 K
0
= 250 MeV
T = 0
0
0.5
1
1.5
2
2.5
3
3.5
4
ρ/ρ
0
0
0.2
0.4
0.6
0.8
1
M
*
/M
m*=0.62 K
0
=220 MeV
M
∗
/M ρ/ρ
0
m
∗
= 0, 62 K
0
= 220 MeV
T = 0
T = 0 T = 5
10 15 MeV
0 0.2 0.4
0.6
0.8 1
ρ/ρ
0
-1
-0.5
0
0.5
1
P (MeV/fm
3
)
m*=0.54 K
0
=551 MeV
m*=0.56 K
0
=450 MeV
m*=0.58 K
0
=350 MeV
m*=0.60 K
0
=250 MeV
m*=0.62 K
0
=220 MeV
T = 5 MeV
ρ/ρ
0
m
∗
= M
∗
0
/M
K
0
T = 5 MeV
0 0.2 0.4
0.6
0.8 1
ρ/ρ
0
-1
-0.5
0
0.5
1
P (MeV/fm
3
)
m*=0.54 K
0
=551 MeV
m*=0.56 K
0
=450 MeV
m*=0.58 K
0
=350 MeV
m*=0.60 K
0
=250 MeV
m*=0.62 K
0
=220 MeV
T = 10 MeV
ρ/ρ
0
m
∗
= M
∗
0
/M
K
0
T = 10 MeV
0 0.2 0.4
0.6
0.8 1
ρ/ρ
0
-1
-0.5
0
0.5
1
P (MeV/fm
3
)
m*=0.54 K
0
=551 MeV
m*=0.56 K
0
=450 MeV
m*=0.58 K
0
=350 MeV
m*=0.60 K
0
=250 MeV
m*=0.62 K
0
=220 MeV
T = 15 MeV
ρ/ρ
0
m
∗
= M
∗
0
/M
K
0
T = 15 MeV
0
0.25 0.5 0.75
1
1.25 1.5 1.75
2
ρ/ρ
0
-16
-8
0
8
16
24
ε/ρ - M (MeV)
m*=0.54 K
0
=551 MeV
m*=0.56 K
0
=450 MeV
m*=0.58 K
0
=350 MeV
m*=0.60 K
0
=250 MeV
m*=0.62 K
0
=220 MeV
T = 5 MeV
ρ/ρ
0
m
∗
= M
∗
0
/M K
0
T = 5 MeV
0
0.25 0.5 0.75
1
1.25 1.5 1.75
2
ρ/ρ
0
-16
-8
0
8
16
24
ε/ρ - M (MeV)
m*=0.54 K
0
=551 MeV
m*=0.56 K
0
=450 MeV
m*=0.58 K
0
=350 MeV
m*=0.60 K
0
=250 MeV
m*=0.62 K
0
=220 MeV
T = 10 MeV
ρ/ρ
0
m
∗
= M
∗
0
/M K
0
T = 10 MeV
0
0.25 0.5 0.75
1
1.25 1.5 1.75
2
ρ/ρ
0
-16
-8
0
8
16
24
ε/ρ - M (MeV)
m*=0.54 K
0
=551 MeV
m*=0.56 K
0
=450 MeV
m*=0.58 K
0
=350 MeV
m*=0.60 K
0
=250 MeV
m*=0.62 K
0
=220 MeV
T = 15 MeV
ρ/ρ
0
m
∗
= M
∗
0
/M K
0
T = 15 MeV
M
∗
/M
ν = 0
V
0
ρ
160 165
170
175
180
185
190
195
200
T (MeV)
0
0.2
0.4
0.6
0.8
1
M
*
/M
m*=0.54 K
0
=551 MeV
m*=0.56 K
0
=450 MeV
m*=0.58 K
0
=350 MeV
m*=0.60 K
0
=250 MeV
m*=0.62 K
0
=220 MeV
M
∗
/M
ρ = ν = 0
160 165
170
175
180
185
190
195
200
T (MeV)
0
0.2
0.4
0.6
0.8
1
M
*
/M
m*=0.54 K
0
=551 MeV
m*=0.56 K
0
=450 MeV
m*=0.58 K
0
=350 MeV
m*=0.60 K
0
=250 MeV
m*=0.62 K
0
=220 MeV
M
∗
/M
ρ = ν = 0
m
∗
K
0
F = E − T S
S
T
V N ρ
F 5 10
15 MeV
f/ρ f = F/V
0
0.25 0.5 0.75
1
1.25 1.5 1.75
2
ρ/ρ
0
-40
-30
-20
-10
0
f/ρ -M (MeV)
m*=0.54 K
0
=551 MeV
m*=0.56 K
0
=450 MeV
m*=0.58 K
0
=350 MeV
m*=0.60 K
0
=250 MeV
m*=0.62 K
0
=220 MeV
T = 5 MeV
ρ/ρ
0
m
∗
= M
∗
0
/M K
0
T = 5 MeV
0
0.5
1
1.5
2
ρ/ρ
0
-40
-30
-20
-10
0
f/ρ -M (MeV)
m*=0.54 K
0
=551 MeV
m*=0.56 K
0
=450 MeV
m*=0.58 K
0
=350 MeV
m*=0.60 K
0
=250 MeV
m*=0.62 K
0
=220 MeV
T = 10 MeV
ρ/ρ
0
m
∗
= M
∗
0
/M K
0
T = 10MeV
0
0.5
1
1.5
2
ρ/ρ
0
-40
-30
-20
-10
0
f/ρ -M (MeV)
m*=0.54 K
0
=551 MeV
m*=0.56 K
0
=450 MeV
m*=0.58 K
0
=350 MeV
m*=0.60 K
0
=250 MeV
m*=0.62 K
0
=220 MeV
T = 15 MeV
ρ/ρ
0
m
∗
= M
∗
0
/M K
0
T = 15 MeV
−
A
′
3
3
m
2
s
g
s
2
φ
2
¯
ψψ + 3
m
2
s
g
s
φ(
¯
ψψ)
2
−
B
′
4
4
m
2
s
g
s
3
φ
3
¯
ψψ + 6
m
2
s
g
s
2
φ
2
(
¯
ψψ)
2
+ 4
m
2
s
g
s
φ(
¯
ψψ)
3
−
A
′
3
m
2
s
g
s
3
φ
3
−
B
′
4
m
2
s
g
s
4
φ
4
G
′
s
2
= G
2
s
G
′
V
2
= G
2
V
A
′
m
2
s
g
s
3
= A
B
′
m
2
s
g
s
4
= B
A
′
m
2
s
g
s
3
B
′
m
2
s
g
s
4
A
′
B
′
A B a =
A
gs
3
b =
B
gs
4
m
2
s
g
s
12
=
A
4
A
′
4
m
2
s
g
s
12
=
B
3
B
′
3
m
2
s
g
2
s
12
=
A
4
A
′
4
1
g
12
s
=
a
4
A
′
4
m
2
s
g
2
s
12
=
B
3
B
′
3
1
g
12
s
=
b
3
B
′
3
a
4
b
3
=
A
′
4
B
′
3
G
′
V
2
M
2
G
′
s
2
M
2
A
′
4
B
′
3
(10
8
MeV
4
)
m
∗
= 0, 54 K
0
= 551 MeV 273, 58 357, 91
m
∗
= 0, 56 K
0
= 450 MeV 260, 61 354, 54 2, 94
m
∗
= 0, 58 K
0
= 350 MeV 247, 55 351, 04 5, 43
m
∗
= 0, 60 K
0
= 250 MeV 234, 40 347, 49 7, 74
m
∗
= 0, 62 K
0
= 220 MeV 221, 18 335, 71 9, 58
m
∗
K
0
G
′
V
2
G
′
s
2
M
2
G
2
V
M
2
G
2
s
M
2
a
4
b
3
(10
8
MeV
4
)
m
∗
= 0, 54 K
0
= 551 MeV 273, 58 357, 91
m
∗
= 0, 56 K
0
= 450 MeV 260, 61 355, 16 −2, 94
m
∗
= 0, 58 K
0
= 350 MeV 247, 55 353, 59 −5, 43
m
∗
= 0, 60 K
0
= 250 MeV 234, 40 353, 56 −7, 73
m
∗
= 0, 62 K
0
= 220 MeV 221, 18 343, 98 −9, 58
m
∗
K
0
G
2
V
G
2
s
M
2
(G
2
s
−G
′
s
2
)M
2
m
∗
0, 54
G
′
V
2
G
2
V
G
′
s
2
G
2
s
A
′
4
/B
′
3
a
4
/b
3
m
∗
= 0, 54 K
0
= 551 MeV 1, 0000 1, 0000
m
∗
= 0, 56 K
0
= 450 MeV 1, 0000 0, 99825 −1, 00
m
∗
= 0, 58 K
0
= 350 MeV 1, 0000 0, 99279 −1, 00
m
∗
= 0, 60 K
0
= 250 MeV 1, 0000 0, 98283 −1, 00
m
∗
= 0, 62 K
0
= 220 MeV 1, 0000 0, 97596 −1, 00
m
∗
K
0
E(ρ
0
)/ρ
0
−M
E
′
1
=
G
′
V
2
2
ρ
0
E
′
2
=
G
′
s
2
ρ
2
s
2ρ
0
E
′
3
=
2A
′
ρ
3
s
3ρ
0
E
′
c
(ρ
0
) E
0
(MeV ) (MeV ) (MeV ) (MeV ) (MeV )
m
∗
= 0, 54 177, 18 201, 17 0 544, 00 −15, 92
m
∗
= 0, 56 168, 78 197, 32 −5, 37 561, 54 −15, 99
m
∗
= 0, 58 160, 32 192, 68 −9, 87 579, 15 −15, 99
m
∗
= 0, 60 151, 81 187, 54 −13, 90 596, 83 −15, 99
m
∗
= 0, 62 143, 25 181, 97 −17, 51 614, 57 −15, 99
E
′
c
B
′
= 0
E
0
= E
′
1
+ E
′
2
+ E
′
3
+ E
′
c
− M
E
1
=
G
2
V
2
ρ
0
E
2
=
δ
2
2G
2
s
ρ
0
E
3
=
aδ
3
3ρ
0
E
c
(ρ
0
) E
0
(MeV ) (M eV ) (MeV ) (MeV ) (MeV )
m
∗
= 0, 54 177, 18 201, 17 0 544, 00 −15, 92
m
∗
= 0, 56 168, 78 189, 68 2, 68 561, 54 −15, 99
m
∗
= 0, 58 160, 32 177, 87 4, 94 579, 15 −15, 99
m
∗
= 0, 60 151, 81 166, 69 6, 95 596, 83 −15, 99
m
∗
= 0, 62 143, 25 155, 70 8, 76 614, 57 −15, 99
b = 0 E
0
= E
1
+ E
2
+ E
3
+ E
c
− M
B
′
b
E
′
1
=
G
′
V
2
2
ρ
0
E
′
2
=
G
′
s
2
ρ
2
s
2ρ
0
E
′
3
=
2A
′
ρ
3
s
3ρ
0
E
′
4
=
3B
′
ρ
4
s
4ρ
0
E
′
c
(ρ
0
) E
0
(MeV ) (MeV ) (M eV ) (MeV ) (MeV ) (MeV )
m
∗
= 0, 54 K
0
= 551 MeV 177, 18 201, 17 0 0 544, 00 −15, 92
m
∗
= 0, 56 K
0
= 450 MeV 168, 78 201, 34 −21, 46 12, 07 561, 54 −15, 99
m
∗
= 0, 58 K
0
= 350 MeV 160, 32 201, 01 −43, 21 25, 00 579, 15 −15, 99
m
∗
= 0, 60 K
0
= 250 MeV 151, 81 200, 47 −65, 63 38, 80 596, 83 −15, 99
m
∗
= 0, 62 K
0
= 220 MeV 143, 25 192, 99 −61, 63 33, 09 614, 57 −15, 99
E
0
= E
′
1
+ E
′
2
+ E
′
3
+ E
′
4
+ E
′
c
− M
E
1
=
G
2
V
2
ρ
0
E
2
=
δ
2
2G
2
s
ρ
0
E
3
=
aδ
3
3ρ
0
E
4
=
bδ
4
4ρ
0
E
c
(ρ
0
) E
0
(MeV ) (MeV ) (MeV ) (MeV ) (MeV ) (MeV )
m
∗
= 0, 54 K
0
= 551 MeV 177, 18 201, 17 0 0 544, 00 −15, 92
m
∗
= 0, 56 K
0
= 450 MeV 168, 78 185, 25 10, 73 −4, 02 561, 54 −15, 99
m
∗
= 0, 58 K
0
= 350 MeV 160, 32 169, 54 21, 59 −8, 33 579, 15 −15, 99
m
∗
= 0, 60 K
0
= 250 MeV 151, 81 153, 79 32, 75 −12, 89 596, 83 −15, 99
m
∗
= 0, 62 K
0
= 220 MeV 143, 25 142, 66 34, 84 −13, 04 614, 57 −15, 99
E
0
= E
1
+ E
2
+ E
3
+ E
4
+ E
c
−M
T = T
c
T
c
ρ
c
P
c
m
∗
c
=
M
∗
c
M
P
c
ρ
c
T
c
T
c
(MeV ) ρ
c
(fm
−3
) P
c
(MeV/fm
3
) m
∗
c
P
c
ρ
c
T
c
m
∗
= 0, 54 K
0
= 551 MeV 18, 35 0, 065 0, 43 0, 79 0, 36
m
∗
= 0, 56 K
0
= 450 MeV 16, 90 0, 059 0, 34 0, 82 0, 33
m
∗
= 0, 58 K
0
= 350 MeV 15, 64 0, 052 0, 26 0, 84 0, 32
m
∗
= 0, 60 K
0
= 250 MeV 14, 75 0, 044 0, 20 0, 87 0, 31
m
∗
= 0, 62 K
0
= 220 MeV 14, 80 0, 043 0, 20 0, 88 0, 31
T
c
(MeV ) ρ
c
(fm
−3
) P
c
(MeV/fm
3
) m
∗
c
P
c
ρ
c
T
c
m
∗
= 0, 54 K
0
= 551 MeV 18, 35 0, 065 0, 43 0, 79 0, 36
m
∗
= 0, 56 K
0
= 450 MeV 16, 86 0, 060 0, 33 0, 81 0, 33
m
∗
= 0, 58 K
0
= 350 MeV 15, 45 0, 053 0, 25 0, 84 0, 31
m
∗
= 0, 60 K
0
= 250 MeV 14, 27 0, 045 0, 19 0, 87 0, 30
m
∗
= 0, 62 K
0
= 220 MeV 14, 12 0, 043 0, 18 0, 88 0, 29
T = T
c
ρ = ρ
s
ρ = ρ
s
ρ ≤ ρ
0
0 0.2 0.4
0.6
0.8 1
ρ/ρ
0
0
0.2
0.4
0.6
0.8
1
ρ
s
/ρ
0
ρ/ρ
0
m*=0.54 K=551 MeV
m*=0.56 K=450 MeV
m*=0.58 K=350 MeV
m*=0.60 K=250 MeV
m*=0.62 K=220 MeV
T=0 MeV
ρ
s
/ρ
0
ρ/ρ
0
m
∗
K
0
ρ = ρ
s
m
∗
= 0, 54 K
0
= 551 MeV
A
′
= B
′
= 0
ρ ρ
s
6%
ρ = ρ
s
ρ =
¯
ψγ
0
ψ = ψ
†
ψ
ρ
s
=
¯
ψψ = ψ
†
γ
0
ψ
¯
ψ = ψ
†
γ
0
ψ =
φ
χ
φ χ
ρ ρ
s
ρ = φ
2
+ χ
2
ρ
s
= φ
2
− χ
2
ρ = ρ
s
χ = 0
ψ = φ
k
2
≪ M
∗
2
ρ = ρ
s
ψ
(iγ
µ
∂
µ
− M + G
′
s
2
ρ
s
− γ
0
G
′
V
2
ρ + A
′
ρ
2
s
+ B
′
ρ
3
s
)ψ = 0
(iγ
0
∂
0
+ iγ
a
∂
a
− M + G
′
s
2
ρ
s
− γ
0
G
′
V
2
ρ + A
′
ρ
2
s
+ B
′
ρ
3
s
)ψ = 0
γ
0
= β =
1
2×2
0
2×2
0
2×2
−1
2×2
1
2×2
0
2×2
2 ×2
i∂
0
+ iγ
0
γ
a
∂
a
+ γ
0
(G
′
s
2
ρ
s
− M + A
′
ρ
2
s
+ B
′
ρ
3
s
) − G
′
V
2
ρ
ψ = 0
−iα
a
∂
a
+ β(M − G
′
s
2
ρ
s
− A
′
ρ
2
s
− B
′
ρ
3
s
) + G
′
V
2
ρ
ψ = i
∂ψ
∂t
[α · k + β(M + S
′
) + V
′
] ψ = i
∂ψ
∂t
[α · k + β(M + S
′
) + V
′
] ψ = Eψ
α
a
= γ
0
γ
a
k
a
= i∂
a
S
′
≡ −G
′
s
2
ρ
s
− A
′
ρ
2
s
− B
′
ρ
3
s
V
′
≡ G
′
V
2
ρ
α · k =
0
2×2
σ · k
σ · k 0
2×2
ρ = ρ
s
M + S
′
+ V
′
σ · k
σ · k −M − S
′
+ V
′
φ
χ
= E
φ
χ
σ · k χ + (M + S
′
+ V
′
)φ = Eφ
σ · k φ − (M + S
′
− V
′
)χ = Eχ
χ (4.13) (4.12)
(M + S
′
+ V
′
− E)φ +
(σ · k)
2
M + S
′
−V
′
+ E
φ = 0
(M + S
′
− V
′
+ E)(M + S
′
+ V
′
− E)φ + (σ · k)
2
φ = 0
(M + S
′
)
2
−(E − V
′
)
2
+ (σ · k)
2
φ = 0
(M + S
′
)
2
− (E − V
′
)
2
+ k
2
φ = 0
M
∗
= M + S
′
E = V
′
±
k
2
+ (M + S
′
)
2
1/2
Σ = V
′
+ S
′
∆ = V
′
− S
′
σ · k χ + (M + Σ)φ = Eφ
σ · k φ − (M − ∆)χ = Eχ
ρ = ρ
s
χ
(M + Σ)φ +
(σ · k)
2
M − ∆ + E
φ = E φ
k
2
M − ∆ + E
+ Σ
φ = (E − M)φ
k
2
2M(1 −
ǫ+∆
2M
)
+ Σ
φ = −ǫφ
ǫ = M − E
ˆ
H =
k
2
2
¯
M
+ Σ
¯
M = M
1 −
ǫ + ∆
2M
M
∗
= M + S
′
¯
M =
1
2
[M
∗
+ (M − V − ǫ)]
ρ = ρ
s
ρ = ρ
s
Σ ∆
Σ = V
′
+ S
′
= G
′
V
2
ρ − G
′
s
2
ρ
s
− A
′
ρ
s
2
−B
′
ρ
s
3
= (G
′
V
2
−G
′
s
2
)ρ − A
′
ρ
2
−B
′
ρ
3
≡ Σ
0
ρ
ρ
0
+ α
ρ
2
ρ
0
2
+ β
ρ
3
ρ
0
3
∆ = V
′
− S
′
= G
′
V
2
ρ + G
′
s
2
ρ
s
+ A
′
ρ
s
2
+ B
′
ρ
s
3
= (G
′
V
2
+ G
′
s
2
)ρ + A
′
ρ
2
+ B
′
ρ
3
≡ ∆
0
ρ
ρ
0
− α
ρ
2
ρ
0
2
−β
ρ
3
ρ
0
3
ρ
0
T = 0 Σ
0
∆
0
α β
4.24
¯
M
¯
M M
4.24
ǫ/2M
¯
M = M
1 −
∆
2M
= M
1 −
∆
0
ρ
ρ
0
− α
ρ
2
ρ
2
0
− β
ρ
3
ρ
3
0
2M
E
1
= Σ +
k
2
2
¯
M
ǫ = 16 M eV M = 939 M eV
ǫ
2M
≪ 1
ρ = ρ
s
E = NΣ +
γ
2
¯
M
k
F
i=0
k
2
i
k
F
γ
N
k
k k =
2πn
L
n
L
1
L
k
F
i=0
k
2
i
=
1
2π
k
F
i=0
2π
L
k
2
i
=
1
2π
k
F
i=0
∆k k
2
i
L → ∞
1
L
k
F
i=0
k
2
i
=
1
2π
k
F
0
dk k
2
1
L
3
k
F
i=0
k
2
i
=
1
(2π)
3
d
3
k k
2
L
3
= V
E = NΣ +
γV
(2π)
3
k
2
2
¯
M
d
3
k
V
E = Σρ +
γ
(2π)
3
1
2
¯
M
k
F
0
4πk
4
dk
= Σρ +
γ
4π
2
¯
M
k
5
F
5
k
F
ρ =
γ
6π
2
k
3
F
ρ = ρ
s
E = Σρ +
γ
20π
2
¯
M
6π
2
γ
5/3
ρ
5/3
= Σρ +
3
10
6π
2
γ
−1
6π
2
γ
5/3
ρ
5/3
¯
M
= Σρ +
3
10M
6π
2
γ
2/3
1
1 −
∆
2M
ρ
5/3
Σ ∆ (4.27) (4.28)
E = Σ
0
ρ
2
ρ
0
+ α
ρ
3
ρ
0
2
+ β
ρ
4
ρ
0
3
+
3
10M
6π
2
γ
2/3
1
1 −
∆
0
ρ
ρ
0
−α
ρ
2
ρ
0
2
−β
ρ
3
ρ
0
3
2M
ρ
5/3
µ =
∂E
∂ρ
= 2Σ
0
ρ
ρ
0
+ 3α
ρ
2
ρ
0
2
+ 4β
ρ
3
ρ
0
3
+
1
2M
6π
2
γ
2/3
1 −
∆
0
ρ
5Mρ
0
−
αρ
2
10Mρ
0
2
−
2βρ
3
5Mρ
0
3
1 −
∆
0
ρ
ρ
0
−α
ρ
2
ρ
0
2
−β
ρ
3
ρ
0
3
2M
2
ρ
2/3
µ =
∂E
∂ρ
=
P + E
ρ
f = E
P = Σ
0
ρ
2
ρ
0
+ 2α
ρ
3
ρ
0
2
+ 3β
ρ
4
ρ
0
3
+
1
5M
6π
2
γ
2/3
1 +
∆
0
ρ
4Mρ
0
−
αρ
2
Mρ
0
2
−
7βρ
3
4Mρ
0
3
1 −
∆
0
ρ
ρ
0
−α
ρ
2
ρ
0
2
−β
ρ
3
ρ
0
3
2M
2
ρ
5/3
ρ = ρ
s
K = 9
∂P
∂ρ
K = 18Σ
0
ρ
ρ
0
+ 54α
ρ
2
ρ
0
2
+ 108β
ρ
3
ρ
0
3
+
3
M
6π
2
γ
2/3
c
1
1 −
∆
0
ρ
ρ
0
−α
ρ
2
ρ
2
0
−β
ρ
3
ρ
3
0
2M
+
6
5
c
2
c
3
1 −
∆
0
ρ
ρ
0
−α
ρ
2
ρ
2
0
−β
ρ
3
ρ
3
0
2M
3
ρ
2/3
c
1
= 1 +
2∆
0
ρ
5Mρ
0
−
11αρ
2
5Mρ
2
0
−
49βρ
3
10Mρ
3
0
c
2
= 1 +
∆
0
ρ
4Mρ
0
−
αρ
2
Mρ
2
0
−
7βρ
3
4Mρ
3
0
c
3
=
∆
0
ρ
2Mρ
0
−
αρ
2
Mρ
2
0
−
3βρ
3
2Mρ
3
0
∆
0
Σ
0
α β
E
0
ρ
0
¯
M
0
K
0
P (ρ
0
) = 0
E(ρ
0
)
ρ
0
= −E
0
K(ρ
0
) = K
0
¯
M(ρ
0
) =
¯
M
0
ρ = ρ
s
ρ = ρ
s
ρ = ρ
s
0 0.2 0.4
0.6
0.8 1
ρ/ρ
0
-1
-0.5
0
0.5
1
P (MeV/fm
3
)
exato
aproximação
T = 0 MeV
M
0
/M = 0.54 K
0
= 551 MeV
P ×ρ/ρ
0
¯
M
0
/M = 0, 54
K
0
= 551 MeV T = 0
ρ = ρ
s
0 0.2 0.4
0.6
0.8 1
ρ/ρ
0
-1
-0.5
0
0.5
1
P (MeV/fm
3
)
exato
aproximação
T = 0 MeV
M
0
/M = 0.56 K
0
= 450 MeV
P ×ρ/ρ
0
¯
M
0
/M = 0, 56 K
0
= 450 MeV T = 0
ρ = ρ
s
ρ = ρ
s
0 0.2 0.4
0.6
0.8 1
ρ/ρ
0
-1
-0.5
0
0.5
1
P (MeV/fm
3
)
exato
aproximação
T = 0 MeV
M
0
/M = 0.58 K
0
= 350 MeV
P ×ρ/ρ
0
¯
M
0
/M = 0, 58 K
0
= 350 MeV T = 0
ρ = ρ
s
0 0.2 0.4
0.6
0.8 1
ρ/ρ
0
-1
-0.5
0
0.5
1
P (MeV/fm
3
)
exato
aproximação
T = 0 MeV
M
0
/M = 0.60 K
0
= 250 MeV
P ×ρ/ρ
0
¯
M
0
/M = 0, 60 K
0
= 250 MeV T = 0
ρ = ρ
s
ρ = ρ
s
0 0.2 0.4
0.6
0.8 1
ρ/ρ
0
-1
-0.5
0
0.5
1
P (MeV/fm
3
)
exato
aproximação
T = 0 MeV
M
0
/M = 0.62 K
0
= 220 MeV
P ×ρ/ρ
0
¯
M
0
/M = 0, 62 K
0
= 220 MeV T = 0
ρ = ρ
s
ρ/ρ
0
¯
M
0
/M
K
0
0 0.2 0.4
0.6
0.8 1 1.2 1.4
1.6
ρ/ρ
0
-20
-15
-10
-5
0
5
ε/ρ - M (MeV)
exato
aproximação
T = 0 MeV
M
0
/M = 0.54 K
0
= 551 MeV
E/ρ × ρ/ρ
0
¯
M
0
/M = 0, 54 K
0
= 5 51 MeV T = 0
ρ = ρ
s
ρ = ρ
s
0 0.2 0.4
0.6
0.8 1 1.2 1.4
1.6
ρ/ρ
0
-20
-15
-10
-5
0
5
ε/ρ - M (MeV)
exato
aproximação
T = 0 MeV
M
0
/M = 0.56 K
0
= 450 MeV
E/ρ × ρ/ρ
0
¯
M
0
/M = 0, 56 K
0
= 4 50 MeV T = 0
ρ = ρ
s
0 0.2 0.4
0.6
0.8 1 1.2 1.4
1.6
ρ/ρ
0
-20
-15
-10
-5
0
5
ε/ρ - M (MeV)
exato
aproximação
T = 0 MeV
M
0
/M = 0.58 K
0
= 350 MeV
E/ρ × ρ/ρ
0
¯
M
0
/M = 0, 58 K
0
= 3 50 MeV T = 0
ρ = ρ
s
ρ = ρ
s
0 0.2 0.4
0.6
0.8 1 1.2 1.4
1.6
ρ/ρ
0
-20
-15
-10
-5
0
5
ε/ρ - M (MeV)
exato
aproximação
T = 0 MeV
M
0
/M = 0.60 K
0
= 250 MeV
E/ρ × ρ/ρ
0
¯
M
0
/M = 0, 60 K
0
= 250 MeV T = 0
ρ = ρ
s
0 0.2 0.4
0.6
0.8 1 1.2 1.4
1.6
ρ/ρ
0
-20
-15
-10
-5
0
5
ε/ρ - M (MeV)
exato
aproximação
T = 0 MeV
M
0
/M = 0.62 K
0
= 220 MeV
E/ρ × ρ/ρ
0
¯
M
0
/M = 0, 62 K
0
= 220 MeV T = 0
ρ = ρ
s
¯
M
0
/M
ρ = ρ
s
¯
M
0
/M = 1
ρ = ρ
s
M
¯
M
0
/M = 1
∆
0
= α + β
¯
M/M ρ/ρ
0
¯
M
0
/M
K
0
0
0.5
1
1.5
2
2.5
3
3.5
4
ρ/ρ
0
0
0.2
0.4
0.6
0.8
1
m*
exato
aproximação
M
0
/M = 0.54 K
0
= 551 MeV
m
∗
× ρ/ρ
0
T = 0
m
∗
= M
∗
0
/M ρ = ρ
s
m
∗
=
¯
M
0
/M
m
∗
= 0, 54 K
0
= 551 MeV
0
0.5
1
1.5
2
2.5
3
3.5
4
ρ/ρ
0
0
0.2
0.4
0.6
0.8
1
m*
exato
aproximação
M
0
/M = 0.56 K
0
= 450 MeV
m
∗
× ρ/ρ
0
T = 0
m
∗
= M
∗
0
/M ρ = ρ
s
m
∗
=
¯
M
0
/M
m
∗
= 0, 56 K
0
= 450 MeV
ρ = ρ
s
0
0.5
1
1.5
2
2.5
3
3.5
4
ρ/ρ
0
0
0.2
0.4
0.6
0.8
1
m*
exato
aproximação
M
0
/M = 0.58 K
0
= 350 MeV
m
∗
× ρ/ρ
0
T = 0
m
∗
= M
∗
0
/M
ρ = ρ
s
m
∗
=
¯
M
0
/M m
∗
= 0, 58
K
0
= 350 MeV
0
0.5
1
1.5
2
2.5
3
3.5
4
ρ/ρ
0
0
0.2
0.4
0.6
0.8
1
m*
exato
aproximação
M
0
/M = 0.60 K
0
= 250 MeV
m
∗
× ρ/ρ
0
T = 0
m
∗
= M
∗
0
/M
ρ = ρ
s
m
∗
=
¯
M
0
/M m
∗
= 0, 60
K
0
= 250 MeV
0
0.5
1
1.5
2
2.5
3
3.5
4
ρ/ρ
0
0
0.2
0.4
0.6
0.8
1
m*
exato
aproximação
M
0
/M = 0.62 K
0
= 220 MeV
m
∗
× ρ/ρ
0
T = 0
m
∗
= M
∗
0
/M ρ = ρ
s
m
∗
=
¯
M
0
/M
m
∗
= 0, 62 K
0
= 220 MeV
ρ/ρ
0
ρ = ρ
s
m
∗
= 0, 54
K
0
= 551 MeV ρ/ρ
0
= 2, 13
ρ/ρ
0
≃ 2.52
m
∗
= 0, 62 K
0
= 220 MeV
T = 0
16 MeV
∼ 0, 6 f m
T = 0
MeV
6 12 MeV
T
c
13 24 MeV
T = 0
T
c
ρ
s
ρ
µ m
∗
T m
∗
P
1
= P
2
µ
1
= µ
2
T
1
= T
2
T = 0
1 2
T = 0
T
c
= C
K
0
m
∗
1/2
ρ
1/3
0
K
0
m
∗
ρ
0
C
T = 0
T = 0
T
c
× (K
0
/m
∗
)
1/2
ρ
1/3
0
ρ
0
= 0, 15 fm
−3
180 − 300 MeV
32 34
36
38 40 42
(K
0
/m*)
1/2
ρ
0
-1/3
(MeV
1/2
fm)
13
13.5
14
14.5
15
15.5
T
c
(MeV)
(K
0
/m
∗
)
1/2
ρ
1/3
0
K = 180 200 22 0 240 250 260 280 300 MeV
m
∗
= 0, 58 m
∗
= 0 , 62
(K
0
/m
∗
)
1/2
ρ
1/3
0
ρ
0
K
0
m
∗
T = 0
ρ
c
P
c
m
∗
c
=
M
∗
c
M
P
c
ρ
c
T
c
T = 0
K
0
MeV m
∗
MeV ρ
c
fm
−3
P
c
MeV.fm
−3
m
∗
c
P
c
ρ
c
T
c
m
∗
c
P
c
ρ
c
T
c
180 0, 58 0, 0398 0, 152 0, 879 0, 290 0, 255
0, 62 0, 0405 0, 161 0, 886 0, 290 0, 257
200 0, 58 0, 0406 0, 160 0, 875 0, 291 0, 255
0, 62 0, 0420 0, 171 0, 882 0, 292 0, 258
220 0, 58 0, 0421 0, 169 0, 871 0, 292 0, 255
0, 62 0, 0437 0, 182 0, 878 0, 294 0, 258
240 0, 58 0, 0437 0, 179 0, 867 0, 294 0, 255
0, 62 0, 0453 0, 194 0, 874 0, 297 0, 259
250 0, 58 0, 0446 0, 184 0, 865 0, 294 0, 254
0, 62 0, 0461 0, 200 0, 872 0, 298 0, 260
260 0, 58 0, 0453 0, 190 0, 863 0, 296 0, 255
0, 62 0, 0470 0, 207 0, 870 0, 299 0, 260
280 0, 58 0, 0471 0, 202 0, 858 0, 297 0, 255
0, 62 0, 0488 0, 222 0, 866 0, 303 0, 262
300 0, 58 0, 0488 0, 216 0, 854 0, 301 0, 257
0, 62 0, 0505 0, 238 0, 862 0, 307 0, 264
0, 58 ≤ m
∗
≤ 0, 62
T = 0
m
∗
0, 58 m
∗
c
T = 0 m
∗
c
P
c
ρ
c
T
c
m
∗
c
P
c
ρ
c
T
c
(K
0
/m
∗
)
1/2
ρ
1/3
0
T = 0
32 34
36
38 40 42
(K
0
/m*)
1/2
ρ
0
-1/3
(MeV
1/2
fm)
0.255
0.2575
0.26
0.2625
0.265
m*
c
P
c
/ρ
c
T
c
m
∗
c
P
c
/ρ
c
T
c
(K
0
/m
∗
)
1/2
ρ
1/3
0
m
∗
c
P
c
ρ
c
T
c
0, 257 m
∗
c
P
c
ρ
c
T
c
•
•
φ
¯
ψψ
•
•
•
ρ/ρ
0
ρ/ρ
0
T = 0
ρ/ρ
0
T
•
•
• T
c
ρ
c
P
c
m
∗
c
P
c
ρ
c
T
c
•
•
¯
M/M
m
∗
ρ/ρ
0
≃ 2 , 5 2
m
∗
= 0, 62 K
0
= 220 MeV
•
0, 58 < m
∗
< 0, 62
m
∗
c
P
c
ρ
c
T
c
•
T < 5 MeV
128 22
T < 5 MeV
= c = k
B
= 1
x
µ
= (x
0
, x
1
, x
2
, x
3
) = (t, x, y, z)
x
µ
= (x
0
, x
1
, x
2
, x
3
) = (t, −x, −y, −z)
x
µ
= g
µν
x
ν
g
µν
g
µν
=
1 0 0 0
0 −1 0 0
0 0 −1 0
0 0 0 −1
µ, ν = {0, 1, 2, 3}
i, j = {1, 2, 3}
∂
µ
=
∂
∂x
µ
= (∂
0
, ∂
i
)
∂
µ
=
∂
∂x
µ
= (∂
0
, −∂
i
)
∂
µ
∂
µ
=
∂
2
∂t
2
− △ =
a · b = a
µ
b
µ
= g
µν
a
µ
b
µ
= a
0
b
0
− a · b
∂
µ
J
µ
= ∂
0
J
0
+ ∂
i
J
i
=
∂J
0
∂t
+ ∇ · J = 0
M
(α · p + βM)ψ(x, t) =
∂ψ(x, t)
∂t
ψ(x, t) α β
α =
0 σ
σ 0
β =
I 0
0 −I
σ 2 ×2 I
σ
1
=
0 1
1 0
σ
2
=
0 −i
i 0
σ
3
=
1 0
0 −1
I =
1 0
0 1
α
i
β
α
i
α
j
+ α
j
α
i
=
α
i
, α
j
= 2g
ij
βα
i
+ α
i
β =
β, α
i
= 2δ
ij
(α
i
)
2
= (β)
2
= 1
(iγ
µ
∂
µ
− M)ψ(x, t) = 0
γ
µ
∂
µ
= γ
0
∂
0
+ γ
i
∂
i
γ
µ
γ
0
= β
γ
i
= βα
i
γ
µ
γ
ν
= {γ
µ
, γ
ν
} = 2g
µν
(γ
µ
)
†
= γ
0
γ
µ
γ
0
γ
0
=
I 0
0 −I
γ
i
=
0 σ
i
−σ
i
0
[γ
µ
(i∂
µ
−av
µ
) − (M −bσ)]ψ(x, t) = 0
v
µ
σ a b
v
µ
bσ
v
µ
σ
v
µ
= v
0
σ
[iγ
µ
∂
µ
− aγ
0
v
0
− M
∗
]ψ = 0
M
∗
≡ M − σ
ψ(x, t) = u(k, λ)
−ik
µ
x
µ
v
µ
u(k, s) k λ
(γ
µ
k
µ
−aγ
0
v
0
−M
∗
)u(k, λ) = 0
β α
i
= βγ
i
(α · k + βM
∗
)u = (k
0
−av
0
)u
(α · k + βM
∗
)u ≡ E
∗
u
E
∗
= ±(k
2
+ M
∗
2
)
1/2
k
0
= E
E = av
0
± (k
2
+ M
∗
2
)
1/2
v
0
¯
ψψ
¯
ψγ
µ
ψ
ρ
s
=
¯
ψψ
ρ =
¯
ψγ
0
ψ
u
†
(α · k + βM
∗
) = E
∗
u
†
α
†
i
= α
i
β
†
= β
u
†
β βu
u
†
βα · k u + u
†
uM
∗
= u
†
βuE
∗
u
†
α · kβu + u
†
uM
∗
= u
†
βuE
∗
{β, α
i
} = 0 u
†
βu = u
†
γ
0
u = ¯uu
¯uu =
M
∗
E
∗
u
†
u
ψ
ψ(x, t) = v(k, λ)
ik
µ
x
µ
¯vv = −
M
∗
E
∗
v
†
v
ψ(x, t)
ψ(x, t) =
1
(2π)
3
d
3
k [a
k,λ
u(k, λ)
−ik
µ
x
µ
+ b
†
k,λ
v(k, λ)
ik
µ
x
µ
]
ψ
†
(x, t) =
1
(2π)
3
d
3
k [a
†
k,λ
u
†
(k, λ)
ik
µ
x
µ
+ b
k,λ
v
†
(k, λ)
−ik
µ
x
µ
]
ρ =
¯
ψγ
0
ψ
=
0|ψ
†
ψ|0
= 0|
d
3
x ψ
†
ψ | 0
= 0|
1
(2π)
3
d
3
k (a
†
k,λ
a
k,λ
+ b
k,λ
b
†
k,λ
) |0
= 0|
1
(2π)
3
d
3
k (a
†
k,λ
a
k,λ
−b
†
k,λ
b
k,λ
) |0
=
1
(2π)
3
d
3
k (N
p
− N
a
)
N
p
N
a
ρ =
γ
(2π)
3
d
3
k
=
γ
2π
2
k
F
0
k
2
dk
=
γ
6π
2
k
3
F
γ
γ = 4
u
†
u = v
†
v = 1
ρ
s
=
¯
ψψ
=
0|
¯
ψψ|0
= 0|
d
3
x
¯
ψψ |0
= 0|
1
(2π)
3
d
3
k
M
∗
E
∗
(a
†
k,λ
a
k,λ
− b
k,λ
b
†
k,λ
) |0
= 0|
1
(2π)
3
d
3
k
M
∗
E
∗
(a
†
k,λ
a
k,λ
+ b
†
k,λ
b
k,λ
) |0
=
1
(2π)
3
d
3
k
M
∗
E
∗
(N
p
+ N
a
)
ρ
s
=
γ
(2π)
3
M
∗
E
∗
d
3
k
=
γ
2π
2
k
F
0
M
∗
(k
2
+ M
∗
2
)
1/2
k
2
dk
=
γM
∗
4π
2
k
F
(k
2
F
+ M
∗
2
)
1/2
− M
∗
2
ln
k
F
+ (k
2
F
+ M
∗
2
)
1/2
M
∗
E/ρ
f =
F
V
F = E − T S
∂f
∂ρ
=
∂(F/V )
∂(N/V )
=
∂(F/V )
∂V
∂V
∂(N/V )
= −
1
V
∂F
∂V
−
F
V
2
V
2
N
=
V
N
−
∂F
∂V
+
F
N
=
P + f
ρ
P = −
∂F
∂V
P (ρ
0
) = 0
∂f
∂ρ
ρ=ρ
0
=
f(ρ
0
)
ρ
0
∂(f /ρ)
∂ρ
=
1
ρ
∂f
∂ρ
−
f
ρ
2
µ =
∂F
∂N
∂(f /ρ)
∂ρ
ρ=ρ
0
= 0
f/ρ
P (ρ
0
) = 0
F = E
∂E
∂ρ
=
P + E
ρ
∂(E/ρ)
∂ρ
=
1
ρ
∂E
∂ρ
−
E
ρ
2
P (ρ
0
) = 0
∂(E/ρ)
∂ρ
ρ=ρ
0
= 0
E/ρ
ρ
0
= 0, 15 fm
−3
E/ρ−M
M −16 MeV
K
T
= 9
ρ
2
∂
2
(f/ρ)
∂ρ
2
+ 2ρ
∂(f /ρ)
∂ρ
∂(f /ρ)
∂ρ
=
1
ρ
∂f
∂ρ
−
f
ρ
2
∂
2
(f/ρ)
∂ρ
2
=
1
ρ
∂
2
f
∂ρ
2
−
2
ρ
2
∂f
∂ρ
+
2f
ρ
3
2ρ ρ
2
K
T
= 9ρ
∂
2
f
∂ρ
2
∂
2
f
∂ρ
2
=
∂
∂ρ
∂f
∂ρ
=
∂
∂ρ
P + f
ρ
=
1
ρ
∂(P + f)
∂ρ
−
P + f
ρ
2
K
T
= 9
∂(P + f )
∂ρ
−
P + f
ρ
∂
2
f
∂ρ
2
=
∂
∂ρ
P + f
ρ
=
1
ρ
∂P
∂ρ
−
P
ρ
2
+
1
ρ
∂f
∂ρ
−
f
ρ
2
=
1
ρ
∂P
∂ρ
−
1
ρ
P + f
ρ
−
∂f
∂ρ
P + f
ρ
−
∂f
∂ρ
= 0
∂
2
f
∂ρ
2
=
1
ρ
∂P
∂ρ
K
T
= 9
∂P
∂ρ
f = E
ρ ρ
s
I
±
=
∞
0
f(k) [n
k
(T ) ± ¯n
k
(T )] dk
n
k
(T ) =
1
(
√
k
2
+M
∗
2
−ν)/T
+ 1
¯n
k
(T ) =
1
(
√
k
2
+M
∗
2
+ν)/T
+ 1
.
I
−
ρ I
+
ρ
s
x ≡
k
M
y ≡
M
∗
M
ν
′
≡
ν
M
T
′
≡
T
M
I
±
= M
∞
0
f(Mx) [η
x
(T ) ± ¯η
x
(T )] dx
η
x
(T ) =
1
(
√
x
2
+y
2
−ν
′
)/T
′
+ 1
¯η
x
(T ) =
1
(
√
x
2
+y
2
+ν
′
)/T
′
+ 1
x
1
−1
F (t)dt =
n
i=1
A
i
F (t
i
) .
T < 5 MeV
80 90
T = 5 MeV x 5 6 T = 5 MeV
x
0 x
x x
I
±
= M
x
0
f(Mx) [η
x
(T ) ± ¯η
x
(T )] dx +
+
x
x
f(Mx) [η
x
(T ) ± ¯η
x
(T )] dx
x
F (x) =
1
(
√
x
2
+y
2
−ν
′
)/T
′
+ 1
1/2
x =
ν
′
2
− y
2
x
1/
21
∼ 10
−10
1
21
≃
1
(
√
x
2
+y
2
−ν
′
)/T
′
x =
(21T + ν
′
)
2
−y
2
.
x x
x = x
1 +
(21T + ν
′
)
2
− ν
′
2
x
2
1/2
.
x x
80 90 20
28
P =
1
2
G
′
V
2
ρ
2
−
1
2
G
′
s
2
ρ
2
s
−
2
3
A
′
ρ
3
s
−
3
4
B
′
ρ
4
s
+
γ
6π
2
∞
0
k
4
(k
2
+ M
∗
2
)
1/2
[n
k
(T ) + ¯n
k
(T )] dk
ρ =
γ
2π
2
∞
0
k
2
[n
k
(T ) − ¯n
k
(T )] dk
ρ
s
=
γ
2π
2
∞
0
M
∗
k
2
(k
2
+ M
∗
2
)
1/2
[n
k
(T ) + ¯n
k
(T )] dk
P =
1
2
G
′
V
2
ρ
2
−
1
2
G
′
s
2
ρ
2
s
−
2
3
A
′
ρ
3
s
−
3
4
B
′
ρ
4
s
+
γM
4
6π
2
x
0
x
4
(x
2
+ y
2
)
1/2
[η
x
(T ) + ¯η
x
(T )] dx +
+
x
x
x
4
(x
2
+ y
2
)
1/2
[η
x
(T ) + ¯η
x
(T )] dx
ρ =
γM
3
2π
2
x
0
x
2
[η
x
(T ) − ¯η
x
(T )] dx +
x
x
x
2
[η
x
(T ) − ¯η
x
(T )] dx
ρ
s
=
γM
3
y
2π
2
x
0
x
2
(x
2
+ y
2
)
1/2
[η
x
(T ) + ¯η
x
(T )] dx +
+
x
x
x
2
(x
2
+ y
2
)
1/2
[η
x
(T ) + ¯η
x
(T )] dx
T = 1 MeV
ν
ν
y
1 − y −
1
M
(G
′
s
2
ρ
s
+ A
′
ρ
s
2
+ B
′
ρ
s
3
) = 0
ρ
s
y
ρ = 0, 01 fm
−3
P (MeV/fm
3
)
8 8 −0, 00891435784
8 10 −0, 00891500657
8 12 −0, 00891494199
8 14 −0, 00891494728
8 16 −0, 00891494691
10 8 −0, 00891445989
10 10 −0, 00891510864
10 12 −0, 00891504406
10 14 −0, 00891504936
10 16 −0, 00891504898
12 8 −0, 008914457
12 10 −0, 00891510576
12 12 −0, 00891504118
12 14 −0, 00891504647
12 16 −0, 0089150461
ρ = 0, 01 f m
−3
T = 1 M eV ν = 913 MeV
8
8 10
x
P (MeV/fm
3
) MeV/fm
3
88 −0, 00888745404 136 −0, 00891455106
92 −0, 0089254596 140 −0, 00891546396
96 −0, 00891469677 144 −0, 00891475101
100 −0, 00890984879 148 −0, 00891521826
104 −0, 00892181319 152 −0, 0089149696
108 −0, 0089088173 156 −0, 0089150575
112 −0, 00891973255 160 −0, 0089150683
116 −0, 00891208637 162 −0, 00891498915
120 −0, 00891652978 166 −0, 00891507339
124 −0, 00891460128 170 −0, 00891503794
128 −0, 008914873 174 −0, 0089150417
132 −0, 0089154927 178 −0, 00891505475
ρ = 0, 01 fm
−3
x
T = 1 M eV ν = 913 MeV
156 20
∂P
∂ρ
= 0
T
1
T
c
ρ
P
T
c
T
3
T
2
T
1
T
1
< T
2
< T
3
< T
c
∆P
∆ρ
=
P
i+1
− P
i
ρ
i+1
− ρ
i
ρ
i
ρ
i+1
ν = µ − G
2
V
ρ
J =
∞
0
φ(x)dx
1 +
(
√
x
2
+y
2
−ν)/T
x
2
+ y
2
x
2
y
2
ν
z =
x
2
+ y
2
dz =
xdx
x
2
+ y
2
dx =
zdz
z
2
−y
2
J =
∞
y
φ(
z
2
−y
2
)zdz
[
z
2
− y
2
][1 +
(z−ν)/T
]
ω =
z − ν
T
g(ω) =
zφ(
z
2
− y
2
)
z
2
− y
2
J = T
∞
y−ν
T
g(ω)
1 +
ω
dω
J
J = T ψ(ω)f(ω)
∞
y−ν
T
− T
∞
y−ν
T
ψ(ω)f
′
(ω)dω
f(ω) =
1
1 +
ω
f
′
(ω) = −
ω
(1 +
ω
)
2
ψ(ω) =
ω
0
g(ω
′
)dω
′
ψ(ω) ω = 0
ψ(ω) = ψ(0) +
dψ
dω
ω = 0
ω +
d
2
ψ
dω
2
ω = 0
ω
2
2
+ ···
= ψ(0) + g(0)ω +
dg
dω
ω = 0
ω
2
2
+ ···
J = T
g(0)ω +
dg
dω
ω = 0
ω
2
2
+ ···
f(ω)
∞
y−ν
T
− T
∞
y−ν
T
g(0)ω +
dg
dω
ω = 0
ω
2
2
+ ···
f
′
(ω)dω
J
1
= T g(0)
∞
y−ν
T
ω
ω
(1 +
ω
)
2
dω
J
2
=
T
2
dg
dω
ω = 0
∞
y−ν
T
ω
2
ω
(1 +
ω
)
2
dω
J = T
g(0)ω +
dg
dω
ω = 0
ω
2
2
+ ···
f(ω)
∞
y−ν
T
+ J
1
+ J
2
+ ···
J
n
=
T
n!
d
n−1
g
dω
n−1
ω = 0
I
n
I
n
=
∞
y−ν
T
ω
n
ω
(1 +
ω
)
2
dω
ν =
x
2
f
+ y
2
−T Ln
1 −
y
2
T
x
2
F
√
x
2
F
+y
2
1 +
y
2
T
x
2
F
√
x
2
F
+y
2
f(x) =
1
1+e
(
√
x
2
+y
2
−ν)/T
x
I
n
J
n
n
I
n
E =
1
2
G
2
V
ρ
2
+
M
2
2G
2
s
(1 − y)
2
+
γM
4
2π
2
∞
0
x
2
(y
2
+ x
2
)
1/2
dx(η
x
+ ¯η
x
)
P =
1
2
G
2
V
ρ
2
−
M
2
2G
2
s
(1 − y)
2
+
γM
4
6π
2
∞
0
x
4
dx
(x
2
+ y
2
)
1/2
(η
x
+ ¯η
x
)
x =
k
M
y =
M
∗
M
k M M
∗
η
x
¯η
x
k
F
φ(x)
J
2
φ(x) J
2
x
2
(2x
2
F
+y
2
)
2x
F
T
2
I
2
x
4
(x
2
+ y
2
)
−1/2
3x
F
(x
2
F
+y
2
)
1/2
2
T
2
I
2
x
2
(x
2
+ y
2
)
1/2
(3x
2
F
+y
2
)(x
2
F
+y
2
)
1/2
2x
F
T
2
I
2
x
2
(x
2
+ y
2
)
−1/2
(x
2
F
+y
2
)
1/2
2x
F
T
2
I
2
I
n
I
2
k
F
0 0,2 0,4
0,6
0,8 1 1,2 1,4
1,6
1,8
k
F
(MeV)
0,5
0,6
0,7
0,8
0,9
1
6I
2
/π
2
T = 10 MeV
T = 5 MeV
T = 1 MeV
I
2
π
2
/6 k
F
T = 1 5 10 MeV
0 0,2 0,4
0,6
0,8 1
ρ/ρ
0
-1
-0,8
-0,6
-0,4
-0,2
0
0,2
P (MeV.fm
-3
)
T = 1 MeV
ρ/ρ
0
T = 1 MeV
ρ
0
= 0, 15 fm
−3
0 0,2 0,4
0,6
0,8 1
ρ/ρ
0
-1
-0,8
-0,6
-0,4
-0,2
0
0,2
P (MeV.fm
-3
)
T = 5 MeV
ρ/ρ
0
T = 5 MeV
ρ
0
= 0, 15 fm
−3
ρ = 0 ρ = ρ
0
T
x
F
≪ 1
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