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3.1. Formalism 15
content to the mean-field theory of Walecka, except that one is now working in a
finite system. Hence, it can be shown that, when starting from the langrangian of
Eq. (3.2), the following (static) Dirac equation for the baryon field Ψ α results [16] :
[
i⃗α · ⃗∇
]
+ γ 0 M + γ 0 Σ H (⃗r) ψ α (⃗r) ≡ Ĥψ α(⃗r) = ɛ α ψ α (⃗r) , (3.3)
where the self-energy Σ H (⃗r) is defined according to
Σ H (⃗r) = −g s φ 0 (⃗r) + g v γ µ V µ (⃗r) + g π γ 5 τ α π α (⃗r) + 1 2 g ργ µ τ α b µα (⃗r)
+ 1 2 γ µ(1 + τ 3 )A µ (⃗r) , (3.4)
and again only static field solutions were taken into consideration. Assuming further
that the nuclear ground state is spherically symmetric and a parity eigenstate, it can
be shown that the pion field does not enter in the Hartree approximation. Furthermore,
the meson fields only depend on the radius r, and only the time component
of the vector fields survives [16].
The general solutions to a Dirac equation with spherically symmetric potentials
have the form
[ ]
ıGnκt (r)/r Y
ψ α (⃗r) ≡ ψ nκmt (⃗r) =
κm η t
, (3.5)
−F nκt (r)/r Y −κm η t
where n denotes the principal, κ and m the generalized angular momentum and t the
isospin quantum numbers. The Y ±κm are the well-known spin spherical harmonics
and determine the angular and spin parts of the wavefunction :
Y κm = ∑ 1
< lm l
m l m s
2 m s|l 1 2 jm > Y l,m l
χ 1
2 ms ,
j = |κ| − 1 2 , l = {
κ, κ > 0
−(κ + 1), κ < 0 .
(3.6)
The Hartree approximation yields the following set of coupled equations for the
different fields :
d 2
dr 2 φ 0(r) + 2 d
r dr φ 0(r) − m 2 sφ 0 (r) = −g s ρ s (r)
∑
( ) 2jα + 1 [
≡ −g s
α occ
4πr 2 |G α (r)| 2 − |F α (r)| 2] ,
d 2
dr 2 V 0(r) + 2 d
r dr V 0(r) − m 2 vV 0 (r) = −g s ρ B (r)
∑
( ) 2jα + 1 [
≡ −g v
α occ
4πr 2 |G α (r)| 2 + |F α (r)| 2] ,