equation of state of baryonic matter.

(1)

Density dependent magnetic field and the equation of state of baryonic matter. Rudiney Casali Profa. Dra. D´ebora P. Menezes

Density dependent magnetic field and the

equation of state of baryonic matter.

Rudiney Casali

Profa. Dra. D´

ebora P. Menezes

Universidade Federal de Santa Catarina / Universidade de Coimbra

Centro de Ciˆ

encias F´ısica e Matem´

aticas / Centro de F´ısica Computacional

Departamento de F´ısica / Departamento de F´ısica

(2)

Coopera¸c˜

ao Brasil-Portugal

Work developed in cooperation

with University of Coimbra,

under supervision of Professor

Constan¸ca Providˆ

encia.

Supported by:

(3)

Outlook

Used model.

Equations.

Chosen parameterization.

(4)

Lagrangian:

L

b

=

X

b=n,p,hyp

ψ

_b

[i γ

µ

∂

µ

+ q

b

γ

µ

A

µ

− g

ρb

τ

3b

γ

µ

~

ρ

µ

− m

∗

b

(1)

−g

ωb

γ

µ

ω

µ

−

1

2 µ

N

k

b

σ

µν

F

µν

_]ψ

b

L

m

=

1

2 (∂

µ

σ∂

µ

_{σ − m}

2 σ

σ

2 _{) −}

1

3 bm

n

(g

σN

σ)

3 ₋

1

4 c(g

σN

σ)

4 ₍₂₎

+

1

2 m

2 ω

ω

µ

ω

µ

−

1

4 Ω

µν

_Ω

µν

+

1

2 m

2 ρ

~

ρ

µ

· ~

ρ

µ

−

1

4 F

µν

_F

µν

−

1

4 P

µν

_P

µν

+

1 4!

ξg

4 ω

(ω

µ

ω

µ

)

2 + Λ

ω

(g

ρ

2 ρ

~

µ

· ~

ρ

µ

)(g

ω

2 ω

µ

ω

µ

).

L

l

=

X

l =e,µ

ψ

l

[i γ

µ

∂

µ

+ q

l

γ

µ

A

µ

− m

l

]ψ

l

(3)

L = L

b

+ L

m

+ L

l

(4)

(5)

Lagrangian.

(ψ

b

) - Baryon field

(ψ

l

) - Lepton field

(σ) - Scalar meson

(ω

µ

) - Vectorial meson

(ρ

µ

) - Isovetorial meson

g

σ

, g

ω

, g

ρ

- Meson-baryon

coupling constants

γ

µ

- Dirac Matrices

A

µ

= (V , ~

A) - Lorentz

tensor

k

p

= 1.79285,

k

n

= −1.91315

-Anomalous magnetic

moment (AMM)

τ

3 - Isospin projection

q

b

- Electrical charge of

the particle b

µ

N

- Nuclear magneton.

(6)

Lagrangian.

m

∗

_b

= (m

b

− g

σb

σ)

-Baryon effective mass

Ω

µν

= ∂

µ

ω

ν

− ∂

ν

ω

µ

P

µν

= ∂

µ

ρ

ν

− ∂

ν

ρ

µ

F

µν

= ∂

µ

A

ν

− ∂

ν

A

µ

σ

µν

=

₂

i

[γ

µ

, γ

ν

]

-Electromagn´

etic tensor

ξ - Omega meson

self-interaction parameter

Λ

ω

- Parameter that

changes the density

dependence of the

symmetry energy

b and c - Parameters

related with the weights of

the non-linear scalar terms

(7)

Magnetic field.

B(n) = B

surf

+ B

0 1 − exp

− β

n

0 γ

(5)

B

surf

= 10

15 G , n

0 = 0.153 fm

−3

Fast: γ = 4, β = 1.83 Slow: γ = 1, β = 1.92

The unit of the magnetic field is the critical field

B

c

(8)

Magnetic field.

10

15

10

16

10

17

10

18

10

19

0

0.1

0.2

0.3

0.4

0.5

0.6 B (G)

n (fm

-3

)

B

Fast

B

Slow

(9)

Magnetic field.

Dividing the work in two types:

Fixed magnetic field (FIX): The magnetic field is considerate

constant on the equations of state.

Variable density dependent magnetic field (VAR): The magnetic

field varies on the equations of state, see eq. (??).

Both cases consider a density dependent variable magnetic field

B term on the total energy density and pressure, eq. (??).

(10)

Mesonic field equations:

Utilizing the mean field approximation and the Euler-Lagrange

equations.:

m

2 _σ

σ

0 = −cg

σ

4 σ

3

0 − bm

n

g

σ

3 σ

2

0 + g

σ

n

s

(6)

m

∗2

_ω

ω

0 = −

ξ

6 g

4 ω

ω

3

0 + g

ω

n

b

m

∗2

ρ

03 = τ

3b

g

ρ

n

3 ,

m

∗2

_ρ

= m

2 ρ

+ 2Λ

ω

g

ρ

2 g

ω

2 ω

2

0 m

∗2

_ω

= m

2 ω

+ 2Λ

ω

g

ρ

2 g

ω

2 ρ

2

03 g

iH

= X

iH

g

iN

X

σH

= 0.700,X

ωH

= 0.738,

X

ρH

= 0.600

n

s

_{= n}

s

p

+ n

s

n

b

= n

p

+ n

n

3 =

(n

p

−n

n

)

2

(11)

Densities:

Scalar

1 :

n

_b

s

=

q

p

Bm

∗

p

2π

2 ν

max

X

ν

X

s

q

m

∗2

p

+ 2νq

p

B − sµ

N

k

p

B

q

m

∗2

p

+ 2νq

p

B

(7)

× ln

p

_{F ,ν,s}

p

+ E

_F

p

q

m

∗2

p

+ 2νq

p

B − sµ

N

k

p

B

n

_b

s

=

m

∗

n

4π

2 X

s

E

_F

n

p

_{F ,s}

n

− m

2 n

ln

p

_{F ,s}

n

+ E

_F

n

m

n

(12)

Densities:

Vectorial:

n

b

=

q

p

B

2π

2 ν

max

X

ν

X

s

p

_{F ,ν,s}

(8)

n

b

=

1 2π

2 X

s

1

3 (p

n

F ,s

)

3 −

1

2 sµ

N

k

n

B

×

m

n

p

n

F ,s

+ E

F

n2

arcsin

m

n

E

n

F

−

π

2 n

l

=

|q

l

|B

2π

2 ν

max

X

ν

X

s

p

_{F ,ν,s}

l

n

B

=

X

b=n,p,hyp

n

b

+ n

l

(9)

(13)

Baryons:

b Mb(MeV) qb µb/µN kb p 938.3 1 2.79 1.79 n 939.6 0 -1.91 -1.91 Λ0 _1115.7 ₀ _-0.61 _-0.61 Σ+ _1189.4 ₁ _2.46 _1.67 Σ0 1189.4 0 -1.61 -1.61 Σ− 1189.4 -1 -1.16 -0.38 Ξ0 1314.9 0 -1.25 -1.25 Ξ− 1314.9 -1 -0.65 0.06

Tabela:

Masses, charges, magnetic moments e anomalous magnetic

moments for baryons.

Where k

b

=

_µ

µ

b

N

− q

b

m

p

m

b

.

(14)

Parametrizations:

Modelo mσ mω mρ gσ gω gρ GM1 512.0 783.0 770.0 8.910 10.610 8.196

Tabela:

Masses and coupling constants for mesons.

Modelo c b ξ Λω

GM1 -0.001070 0.002947 0.00 0.00

(15)

Energy spectra:

E

_ν,s

b

=

r

p

2 z

+ (

q

m

∗

p

+ 2νq

p

B − sµ

N

k

p

B)

2 + g

ω

0 +

1

2 g

ρ

0 ₍₁₀₎

E

s

b

=

r

p

2 z

+ (

q

m

∗

n

+ p

⊥

2 − sµ

N

k

n

B)

2 + g

ω

0 −

1

2 g

ρ

0 _,

ν = n +

1

2 −sgn(q

b

)

s

2 = 0, 1, 2, ... - landau levels for fermions

with electrical charge q

s is the spin, which assumes values +1 for spin up and −1 for

spin down

p

2

(16)

Momenta:

p

_{F ,ν,s}

b

= E

_F

b2

− [

q

m

∗2

b

+ 2νq

b

B − sµ

N

k

b

B]

2 (11)

p

b

_{F ,s}

= E

_F

b

− m

2 b

p

_{F ,ν,s}

l

= E

_F

l

− m

2 l

Where p

b

F ,ν,s

is the Fermi momentum for the charged baryons,

p

_{F ,s}

b

for the neutron and p

l

_{F ,ν,s}

for the leptons.

m

b

= m

∗

− sµ

N

k

b

B, m

l

= m

l

2 − 2ν|q

l

B|.

(17)

Landau levels:

ν

max

=

(E

b

F

+ sµ

N

k

b

B)

2 − m

b

∗

2|q

b

|B

(12)

ν

max

=

(E

l 2

F

) − m

l

2 2|q

l

|B

Landau levels are summed up to its maximum value, for which

the square of the Fermi moment of the particle is still positive.

(18)

Chemical Potencials:

µ

p

= E

F

p

+ g

ω

0 +

1

2 g

ρ

0 ₍₁₃₎

µ

n

= E

F

n

+ g

ω

0 −

1

2 g

ρ

0 µ

p

= µ

Σ

+

= µ

_n

− µ

_e

µ

Λ

= µ

Σ

0

= µ

_Ξ

0

= µ

n

µ

Σ

−

= µ

_Ξ

−

= µ

_n

+ µ

_e

µ

= µ

e

X

b=n,p,hyp

q

b

n

b

+

X

l

q

l

n

l

= 0

(19)

Total energy density:

ε

m

=

X

b=p,n,hyp

ε

b

+

X

l =e,µ

ε

l

+

1

2 m

σ

2

0 +

1

2 m

ω

2

0 +

1

2 m

ρ

2

0 (14)

+

1

8 ξ(g

ω

0 )

4 ₊

1

3 bm

n

(g

σ

0 )

3 ₊

1

4 c(g

σ

0 )

4 +3Λ

ω

(g

ω

0 )

2 (g

ρ

0 )

2 ,

(20)

Energy density:

ε

b

=

n

max

X

n=0

X

s

|q

b

|B

4π

2 p

b

_{F ,n,s}

E

_F

b

+ (

q

m

2 b

+ 2q

b

Bn − sµ

N

k

b

B)

2 (15)

×ln

p

_{F ,n,s}

b

+ E

_F

b

(pm

2 b

+ 2q

b

Bn − sµ

N

k

b

B)

,

ε

b

=

1 4π

2 X

s

1

2 E

b3

F

p

b

F ,n,s

−

2

3 sµ

N

k

b

BE

b3

F

arc sin

m

E

b

F

−

π

2 −

1

3 sµ

N

k

b

B +

1

4 m

mp

F ,n,s

b

E

F

b

+ m

3 ln

E

b

F

+ p

b

F ,n,s

m

ε

l

=

n

max

X

n=0

X

s

|q

l

|B

4π

2 p

_{F ,n,s}

l

E

_F

l

+ m

2 _l

× ln

p

_{F ,n,s}

l

+ E

_F

l

m

l

(21)

Pressure and total energy densities:

P

m

=

X

i =e,µ,n,p,hyp

µ

i

n

i

− ε

m

(16)

Attributing the contributions of magnetic fields, we find the total

energy density and pressure:

ε

T

= ε

m

+

B

2

2 P

T

= P

m

+

B

2

(22)

Symmetry energy and slope:

a

sym

=

1 2n

B

∂

2 _E

∂t

2 (18)

L

= 3n

0 ∂a

sym

∂n

B

_n

B

→n

0

n

0 = 0.153 fm

−3

- Nuclear saturation density

t =

n

−n

p

(23)

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