This document describes the equations of state used to model the phases and phase transitions in neutron stars. It summarizes the relativistic mean field theory used to model the nucleonic phase and parametric equations of state. It also discusses the Maxwell and Glendenning constructions used to model first-order phase transitions from hadronic to quark matter, including the mixed phase region. Key parameters like the bag constant are specified to generate example equations of state with phase transitions.

1. MIXED PHASE IN NEUTRON
STARS
G.B.Alaverdyan, Yu.L.Vartanyan
Yerevan State University, Armenia
1919 – 2009
E
Yerevan 2009

2. Introduction
Neutron Star Hydrostatic Equilibrium Equations of Tolman, Oppenheimer, and Volkov
dP (r ) G[ε (r ) + P (r )][m(r ) + 4π r 3 P(r )]
=−
dr r 2 [1 − 2Gm(r ) / r ]
dm(r )
= 4π r 2ε (r )
dr
Equation of State ?
ε (P )
Pc m(r ), P(r ), ε (r )
P( R) = 0 R, m( R ) = M

3. Introduction
Scalar Vector
RMF-theory: Isoscalar σ ω
J. D. Wale cka, Ann.Phys. 87, 4951, 1974 Isovector δ ρ
σω ρ
BD. Se ro t, J. D. Wale cka, Int.J.Mod.Phys. E6, 515, 1997.
.
Low density asymmetric nuclear matter:
S. Kubis, MKutsche ra, Phys. Lett., B399,191,1997.
.
σω ρ δ BLiu, V. Gre co , V. B
. aran, MCo lo nna, MDi To ro , Phys. Rev. C65, 045201, 2002.
. .
Heavy ion collisions at intermediate energies:
V. Gre co , MCo lo nna, MDi To ro , F. M ra , Phys. Rev. C67, 015203, 2003
. . ate
σω ρ δ T. Gaitano s, MCo lo nna, MDi To ro , H. H. Wo lte r, Phys.Lett. B
. . 595, 209, 2004.
Neutron stars with quark deconfinement phase transitions:
σω ρ δ G. B A rdyan, Astrophysics, 52, 132 , 2009.
. lave
G. B A rdyan, Gravitation and Cosmology, 15, 5, 2009.
. lave
. G. B A rdyan, in Proc.of 8th Int.Conf. " Quark Co nfine me nt and the Hadro n
. lave
Spe ctrum" , Mainz, Germany, 1-6 sept. 2008, PoS (Confinement8)} 171, 2009.

4. Lagrangian density of many-particle system of p,n.σ,ω,ρ,δ
 µ  r r 
 
1 r r
2 
(
L = ψ N γ  i∂ µ − gωωµ ( x) − g ρτ N ×ρ µ ( x) ÷− mN − gσ σ ( x) − gδ τ N × ( x)  ψ N +
δ

)
1 1 2 1
2
( 2
)
+ ∂ µσ ( x) ∂ µσ ( x) − mσ σ ( x)2 − U (σ ( x)) + mω ω µ ( x)ωµ ( x) − Ω µν ( x)Ω µν ( x) +
2 4
r r r 2 1 2 rµ
1
( ) r 1
+ ∂ µ δ ( x) ∂ δ ( x) − mδ δ ( x) + mρ ρ ( x) ρ µ ( x) − Rµν ( x) R µν ( x) ,
2
µ 2
2 4
r r ψ p 
x = xµ = (t , x, y , z ) σ ( x), ωµ ( x ), δ ( x), ρ µ ( x) ψN = ÷
ψ n 
b c
U (σ ) = mN ( gσ σ )3 + ( gσ σ ) 4 ,
3 4
Ω µν ( x ) = ∂ µ ων ( x) − ∂ν ωµ ( x),
ℜ µν ( x ) = ∂ µ ρν ( x) − ∂ν ρ µ ( x).

5. Relativistic mean-field approach
∂L ∂L
− ∂µ =0
∂φ ( x) ∂ (∂ µφ ( x))
m* = mN − g σ σ − g δ δ (3) ,
p
1
e p (k ) = k 2 + m* 2 + g ω ω0 +
p g ρ ρ0 ,
(3)
mn = mN − gσ σ + gδ δ (3) .
*
2
1
e n ( k ) = k 2 + m* 2
n + g ω ω0 − g ρ ρ0 ,
(3)
2 k Fp 3 k Fn3
np = , nn = 2 ,
 dU (σ)  3 π2 3π
mσ 2 σ = g σ  ns p + ns n − ÷,
dσ 
kF p
 1 m*
∫
p
ns p = 2 k 2 dk ,
π
mω ω = g ω ( n p + nn ) ,
2 0 k 2 + m* 2
p
kF n
m*
mδ δ
2 (3)
( )
= g δ ns p − ns n ,
1
ns n = 2
π ∫
0 k +m2
n
*2
k 2 dk .
n
1
mρ 2 ρ0 =
(3)
g ρ ( n p − nn ) ,
2
1
µ p = e p (k F p ) = k F p 2 + m* 2 + g ω ω0 +
p g ρ ρ0 ,
(3)
2
1
µ n = e n ( k F n ) = k F n 2 + m* 2
n + g ω ω0 − gρ ρ0 .
(3)
2

6. Parametric EOS for nuclear matter
gσ σ ≡ σ , gω ω0 ≡ ω , gδ δ (3) ≡ δ , g ρ ρ (3) ≡ ρ ,
2 2 2 2
 gσ  g  g  g  nn − n p
 ÷ ≡ aσ ,  ω ÷ ≡ aω ,  δ ÷ ≡ aδ ,  ρ ÷ ≡ aρ
m ÷ α= , the asymmetry parameter
 mσ   mω   mδ   ρ n
1
kF ( n )(1−α ) 3
1  2 
P ( n, α ) = 2
π ∫
0


k F (n) 2 (1 − α) 3 + (mN − σ − δ) 2 − k 2 + (mN − σ − δ) 2 ÷ k 2 dk +

1
k F ( n )(1+α ) 3
1  2 
+ 2
π ∫
0


k F ( n) 2 (1 + α) 3 + (mN − σ + δ) 2 − k 2 + (mN − σ + δ) 2 ÷ k 2 dk −

% 1  σ 2 ω2 δ 2 ρ 2 
− U ( σ) +  − + − + ÷.
2  aσ aω aδ aρ ÷
 
1
k F ( n )(1−α ) 3
1
ε(n, α) =
π2 ∫
0
k 2 + (mN − σ − δ) 2 k 2 dk +
1
k F ( n )(1+α ) 3
1 1  σ 2 ω2 δ 2 ρ 2 
+ 2 ∫ %
k + (mN − σ + δ) k dk + U (σ) +  + + +
2 2 2
 aσ aω aδ aρ ÷
,
π 2 ÷
0 

7. Parameters of RMF theory
aσ , aω , aδ , aρ , b, c
Symmetric nuclear matter (α =0) Saturation density ( n = n0 )
m* = γ mN ,
N σ 0 = (1 − γ ) mN
d ε(n, α) ε(n0 , 0) B
= = mN + f 0 , f0 = , Binding energy per baryon
dn n = n0 n0 A
α= 0
aω =
1
n0
(mN + f 0 − k F (n0 ) 2 + (mN − σ0 ) 2 )
ω0 = aωn0 = mN + f 0 − k F (n0 ) 2 + (mN − σ 0 ) 2
k F ( n0 )
σ0 2 ( mN − σ 0 )
= 2
aσ π ∫
0 k 2 + ( mN − σ 0 ) 2
k 2 dk − bmN σ 0 2 − cσ 03

8. Parameters of RMF theory
k F ( n0 )
2 b c 4 1  σ0 2 
ε0 = n0 (mN + f 0 ) = 2 ∫ k + (mN − σ0 ) k dk + mN σ0 + σ 0 +  + n0 2 aω ÷
2 2 2 3
π 0
3 4 2  aσ 
d 2 ε(n, α)
K = 9 n0 2 2 ( ) compressibility module
dn n n = n0
α= 0
ε sym 1 d 2 ε(n, α)
= Esym (n) α 2
Esym ( n) =
n 2n d α 2 α=0
Symmetry energy
mN = 938,93 MeV
n0 = 0,153 fm −3
m*
γ = N = 0, 78
mN 0 0,5 1 1,5 2 2.5 3
K = 300 MeV
f 0 = −16,3 MeV 4,794 6,569 8,340 10,104 11,865 13,621 15,372
Esym = 32,5 MeV
(0)

9. Parameters of RMF theory
Parameters σωρ σωρδ
aσ , fm2 9.154 9.154
aω , fm2 4.828 4.828
aδ , fm2 0 2.5
aρ, fm2 4.794 13.621
b , fm-1 1.654 10-2 1.654 10-2
c 1.319 10-2 1.319 10-2

10. Properties of asymmetric nuclear matter

11. Properties of asymmetric nuclear matter

12. Characteristics of β-equilibrium npe- plasma
ε NM (n, α, µ e ) = ε(n, α ) + ε e (µ e ),
1
PNM (n, α, µ e ) = P (n, α) + µ e (µ e 2 − me 2 )3/ 2 − ε e (µ e )
3π2
n p − ne 1 n
q= = (1 − α ) − e
n 2 n

13. Characteristics of charge neutral and β-stable npe- plasma

14. Symmetry energy

15. EOS of neutron star matter in nucleonic phase

16. Two types of deconfinement quark phase transitions
Maxwell’s construction
(Local electroneutrality)
μQ
Glendenning’s mixed phase μN
a,b B
μ0
(Global electroneutrality)
A
ρ = (1 − χ) ρN + χ ρQ
n = (1 − χ) nN + χ nQ
PN P0 PQ P
χ = VQ / V

17. Parameters of Maxwellian phase transition
MFTσωρδ + MIT-bag
µ NM ( P0 ) = µQM ( P0 )
Maxwell’s construction B - bag parameter

18. Parameters of Maxwellian phase transition
εQ
λ=
ε N + P0
λcr = 3 / 2
Seidov criterium
Z Seidov, Ast.Zh., 15, 347,1971

19. EOS with Maxwellian phase transition

20. EOS with Mixed phase transition
P(μb, μel)
Q
b
μb
QH
a N
0
μel

21. Mixed Phase Structure
Nuclear matter Quark matter

22. Mixed Phase Structure
-3
3 3
Mix B=60 MeV/fm + MFTσωρδ B=60 MeV/fm + MFTσ
ω
ρδ
1.00 0
Constituents number density, fm
10
d u
0.75 -1
10 s n
p 1.15
0.50
χ
-2
10 1.10 s
0.25 -3
1.05
10 1.00
d u
e 1.05 1.10 1.15
0.00 0.077 1.083
-4
10
0 400 800 1200 1600 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
3 -3
ε, MeV/fm n , fm
1000 3
B=60 MeV/fm + MFTσωρδ
ω0
Meson fields, MeV
800
600
σ0
400
200
0 δ0
ρ0
-200
0.0 0.2 0.4 0.6 0.8 1.0 1.2
3
n , MeV/fm

23. EOS’ s with Quark Deconfinement Phase transitions
3
10 3
B=60 MeV/fm + MFTσωρδ
2
10
3
P, MeV/fm
1
10
0
10
-1
10 2 3
10 3
10
ε , MeV/fm

24. Neutron stars with mixed quark matter (Hybrid Stars)
2.5
2.0 RMF σωρδ
1.5
M , M
1.0
Maxw B60
0.5
0.0
Mix B60
-2 -1 0 1 2 3
10 10 10 10 10 10
3
Pc , MeV/fm

25. Neutron stars with mixed quark matter (Hybrid Stars)
0.25
Maxw B60
0.20
M , M
0.15
RMF σ
ω
ρ
δ
Mix B60
0.10
1 2
10 10
R , km

26. Neutron stars with mixed quark matter (Hybrid Stars)
0.5
0.4
0.3
Mix B60
zs
RMFσωρδ
0.2
0.1 Maxw B60
0.0
0.0 0.5 1.0 1.5 2.0
M , M

27. Neutron stars with quark core
TOV equations

28. Neutron stars with quark core
λ > 3/ 2 B < 69,3 MeV/fm3
λcr = 3 / 2 B ≈ 69,3 MeV/fm3
λ ≤ 3/ 2 69,3 ≤ B ≤ 90 MeV/fm3
B > 90 MeV/fm3 Unstable QP

29. Catastrophic conversion due to deconfined phase transition
M core ≈ 0.24 M sun M core ≈ 0.087 M sun
Rcore ≈ 4,38 km Econv ≈ 4 ⋅1050 erg
R ≈ 16, 77 km
R ≈ 13,95 km

30. Catastrophic conversion due to deconfined phase transition

31. Summary
 The account of δ-meson field results in reduction of phase
transition parameters,
 The density jamp parameter λ , that has important significance
from the point of view of infinitisimal quark core stability in neutron
star, is increased.
 In case of bag parameter values the condition
λ>3/2 is satisfied, and infinitisimal quark core is unstable.
 For the quark phase is unstable.

32. ÞÜàðвβÈàôÂÚàôÜ
THANK YOU
April 8, Yerevan, 2009