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16 Chapter 3. Relativistic Bound State Wave Functions
Meson Mass
Coupling constant
m s = m σ = 520 MeV gs 2 = 109.6
m v = m ω = 783 MeV gv 2 = 190.4
m ρ = 770 MeV gρ 2 = 65.23
Table 3.1 Meson masses and coupling constants, as taken from Ref. [16].
dr 2 b 0(r) + 2 d
r dr b 0(r) − m 2 ρφ 0 (r) = − 1 2 g ρρ 3 (r)
≡ − 1 2 g ∑
( ) 2jα + 1 [
ρ
α occ
4πr 2 |G α (r)| 2 + |F α (r)| 2] (−1) tα−1/2 ,
d 2
dr 2 A 0(r) + 2 d
r dr A 0(r) = −eρ P (r)
≡ −e ∑ ( ) 2jα + 1 [
α occ
4πr 2 |G α (r)| 2 + |F α (r)| 2] (t α + 1 2 ) ,
d 2
d
dr G α(r) + κ r G α(r) − [E α − g v V 0 (r) − t α g ρ b 0 (r)
−(t α + 1 2 )eA 0(r) + M − g s φ 0 (r)]F α (r) = 0 ,
d
dr F α(r) − κ r F α(r) + [E α − g v V 0 (r) − t α g ρ b 0 (r)
∫ ∞
0
−(t α + 1 2 )eA 0(r) − M + g s φ 0 (r)]G α (r) = 0 ,
dr (|G α | 2 + |F α | 2 ) = 1 , (3.7)
where ρ s (r), ρ B (r), ρ 3 (r) and ρ P (r) are the scalar, baryon, rho and proton densities,
respectively. The above set of equations constitute the basis of the relativistic
Hartree approach to the lagrangian of Eq. (3.2). This set depends on the meson
and nucleon masses and coupling constants. Some of these can be taken directly
from experiments (M p , M n , m v = m ω , m ρ , and e 2 ), others can be calculated by
requiring that when the Dirac-Hartree equations are solved in the limit of infinite
matter, relevant empirical equilibrium observables are reproduced. For all bound
state wave functions that will be used in this work we have adopted the values
quoted by Horowitz et al. in Ref. [16]. Table 3.1 shows the values for the coupling
constants and meson masses quoted in this work.
3.2 Results
A new computer program to solve the set of coupled nonlinear differential equations
of Eq. (3.7) was developed. Starting from an initial guess for the scalar and vector