fields and their corresponding inhomogeneous part can be expanded as $$\Lambda(r)=\sum_{n=1}^{n_B} \Lambda_n R_{n0}(r), \quad
S_\Lambda(r)=\sum_{n=1}^{n_B} S_n^\Lambda R_{n0}(r),
\label{ans}$$ where $\Lambda(r)=\phi(r),V_0(r),b_0(r)$ and $$R_{n l}(r)=\frac{N_{nl}}{b_0^{3/2}} x^l L^{l+1/2}_{n-1}(x^2)exp(-x^2/2),$$ where $x=r/b_0$ is the radius measured in units of the oscillator length, $$N_{nl}=\sqrt{2(n-1)!/(l+n-1/2)!}
\label{norm}$$ is the normalization constant and $L^m_n(x^2)$ are the associated Laguerre polynomials. For the calculation of the meson fields $l=0$ in the expressions given below. Once the ansatz given by eqs.(\[ans\]) are substituted into eqs.(\[elphi\]- \[elb0\]), a set of inhomogeneous equations is obtained: $$\sum_{n'=1}^{n_B} {\cal H}_{n n'} \Lambda_{n'}=S_n^\Lambda$$ where $${\cal H}_{n n'}= \delta_{n n'}\left( b_B^{-2} (2(n-1)+3/2) + m^2_\Lambda
\right)$$ $$+ \delta_{n n'+1} b_B^{-2} \sqrt{n(n+1/2)}+ \delta_{n+1 n'} b_B^{-2}
\sqrt{n'(n'+1/2)}.$$ Only the massive fields can be calculated with this method because the convergence of the Coulomb field, which has a long range, is very slow. The Green’s function method is then chosen to describe the electromagnetic interaction:

$$A_{0}(r)=e~\int~r'^{2}dr'\rho_{p}(r')G_c(r,r'), \label{green}$$

with $$G_c(r,r')= \biggl\{ \begin{array}{c}
1/r {~~~~\rm for} ~~r>r'  \\
1/r' {~~~~\rm for} ~r'>r. \end{array}$$

Results
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Parity Violating Electron Scattering and the Neutron Radius
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We start this section by defining the asymmetry for polarized electron scattering of a hadronic target as $$\mathcal{A}=\frac{d\sigma_{+}/d\Omega-d\sigma_{-}/d\Omega}{d\sigma_{+}/d\Omega+d\sigma_{-}/d\Omega},$$ where $d\sigma_{\pm}/d\Omega$ is the differential cross section for initially polarized electrons with 