\section{Coleman-Weinberg analysis of the Higgs potential in the NMSSM and the nMSSM}\label{apCW}
The two models under consideration essentially differ, at tree-level, by their Higgs sectors.
Additionally, one should require the limit $\kappa^2\to0$ in the nMSSM neutralino sector, 
with respect to that of the NMSSM.

{\bf SM-fermion contributions:}\newline\noi
In the base of Dirac-fermions $(u,d,\nu_e,e)$, the squared mass-matrix of SM-fermions in terms of neutral Higgs fields 
reads (we omit color and generation indices):
\begin{equation}
 {\cal M}_{f}^2(H_{u,d}^0)=\begin{bmatrix}
  Y_u^2\left|H_u^0\right|^2&0&0&0\\0&Y_d^2\left|H_d^0\right|^2&0&0\\0&0&0&0\\0&0&0&Y_e^2\left|H_d^0\right|^2
 \end{bmatrix}
\end{equation}\noi
leading to the potential:
{\begin{align}
 \delta{\cal V}_{\mbox{\small eff}}^{\Lambda,f}(H_u^0,H_d^0,S)&=-\frac{1}{16\pi^2}\sum_{f}{Y_f^4\left|H_f^0\right|^4\left[\ln\left(\frac{Y_f^2\left|H_f^0\right|^2}{\Lambda^2}\right)-\frac{3}{2}\right]}\nonumber\\
&\simeq-\frac{1}{16\pi^2}\sum_{f}Y_f^4\left|H_f^0\right|^4\ln\left(\frac{Y_f^2\mbox{\small \em v}_f^2}{\Lambda^2}\right)+\ldots
\end{align}}\noi
where we have kept only the leading, $SU(2)_L$-invariant, logarithmic terms. We deduce the corresponding contributions to the Higgs potential:
\begin{equation}\begin{cases}
\lambda_1^f\simeq-\frac{1}{8\pi^2}\sum_{f=d,e}Y_f^4\ln\left(\frac{m_f^2}{\Lambda^2}\right)\\
\lambda_2^f\simeq-\frac{1}{8\pi^2}\sum_{u}Y_u^4\ln\left(\frac{m_u^2}{\Lambda^2}\right)
\end{cases}\end{equation}

{\bf SM-Gauge-boson contributions:}\newline\noi
In the base of real vector fields $(\gamma^0,W^1,W2,Z^0)$, the squared mass-matrix of SM-Gauge-bosons in terms of neutral Higgs fields reads:
\begin{equation}
 {\cal M}_{G}^2(H_{u,d}^0)=\frac{1}{2}\begin{bmatrix}
  0&0&0&0\\0&g^2&0&0\\0&0&g^2&0\\0&0&0&g^2+g'^2
 \end{bmatrix}\left(\left|H_u^0\right|^2+\left|H_d^0\right|^2\right)
\end{equation}\noi
leading to the potential (note that in the $SU(2)_L$-conserving limit, these fields are massless):
{\small\begin{align}
 \delta{\cal V}_{\mbox{\small eff}}^{\Lambda,f}(H_u^0,H_d^0,S)&=\frac{3}{64\pi^2}\left\{\frac{g^4}{2}\left(\left|H_u^0\right|^2+\left|H_d^0\right|^2\right)^2\left[\ln\left(\frac{g^2\left(\left|H_u^0\right|^2+\left|H_d^0\right|^2\right)}{2\Lambda^2}\right)-\frac{3}{2}\right]\right.\nonumber\\
& \left.+\frac{(g^2+g'^2)^2}{4}\left(\left|H_u^0\right|^2+\left|H_d^0\right|^2\right)^2\left[\ln\left(\frac{(g^2+g'^2)\left(\left|H_u^0\right|^2+\left|H_d^0\right|^2\right)}{2\Lambda^2}\right)-\frac{3}{2}\right]\right\}\\
&\simeq\frac{3}{256\pi^2}\left[2g^4\ln\left(\frac{M_W^2}{\Lambda^2}\right)+(g^2+g'^2)^2\ln\left(\frac{M_Z^2}{\Lambda^2}\right)\right]\left(\left|H_u^0\right|^2+\left|H_d^0\right|^2\right)^2\ldots\nonumber
\end{align}}\noi
providing us with the couplings:
\begin{equation}\begin{cases}
\lambda_1^G\simeq\frac{3}{128\pi^2}\left[2g^4\ln\left(\frac{M_W^2}{\Lambda^2}\right)+(g^2+g'^2)^2\ln\left(\frac{M_Z^2}{\Lambda^2}\right)\right]\\
\lambda_2^G\simeq\frac{3}{128\pi^2}\left[2g^4\ln\left(\frac{M_W^2}{\Lambda^2}\right)+(g^2+g'^2)^2\ln\left(\frac{M_Z^2}{\Lambda^2}\right)\right]\\
\lambda_3^G+\lambda_4^G\simeq\frac{3}{128\pi^2}\left[2g^4\ln\left(\frac{M_W^2}{\Lambda^2}\right)+(g^2+g'^2)^2\ln\left(\frac{M_Z^2}{\Lambda^2}\right)\right]\\
\end{cases}\end{equation}

{\bf Sfermion contributions:}\newline\noi
The Sfermion squared mass-matrix, in the base $(\tilde{F}_L,\tilde{F}_R^{c\,*})$ for a flavour $f$, is given by:
{\small \begin{equation}
 {\cal M}^2_{\tilde{F}}(S,H_u^0,H_d^0)=\begin{bmatrix}
m^2_{\tilde{F}_L}+|Y_fH_f^0|^2+\frac{g'^2{\cal Y}_L^f-2g^2I_3^f}{4}(|H_u^0|^2-|H_d^0|^2)&Y_f(A_fH_f^0-\lambda S^*H_{\tilde{f}}^{0\,*})\\
Y_f(A_fH_f^{0\,*}-\lambda SH_{\tilde{f}}^{0})&m^2_{\tilde{F}_R}+|Y_fH_f^0|^2+\frac{g'^2{\cal Y}_R^f}{4}(|H_u^0|^2-|H_d^0|^2)
                        \end{bmatrix}
\end{equation}}\noi
Defining $T_{\tilde{F}}=({\cal M}^2_{\tilde{F}})_{11}+({\cal M}^2_{\tilde{F}})_{22}$ and $R_{\tilde{F}}^2=\left[({\cal M}^2_{\tilde{F}})_{11}-({\cal M}^2_{\tilde{F}})_{22}\right]^2+4|{\cal M}^2_{\tilde{F}}|^2_{12}$, 
we obtain the eigenvalues $m_{\tilde{F}_{\pm}}^2=\frac{1}{2}\left[T_{\tilde{F}}\pm\sqrt{R_{\tilde{F}}^2}\right]$ and the Higgs potential:
\begin{equation}
 \delta{\cal V}_{\mbox{\small eff}}^{\Lambda,\tilde{F}}(H_u^0,H_d^0,S)=\frac{1}{128\pi^2}\sum_{f}\left\{(T_{\tilde{F}}^2+R_{\tilde{F}}^2)\left[\ln\left(\frac{T_{\tilde{F}}^2-R_{\tilde{F}}^2}{4\Lambda^4}\right)-3\right]+2\,T_{\tilde{F}}\cdot R_{\tilde{F}}\ln\left(\frac{T_{\tilde{F}}+R_{\tilde{F}}}{T_{\tilde{F}}-R_{\tilde{F}}}\right)\right\}
\end{equation}\noi
One derives the couplings:
{\small\begin{equation}\begin{cases}
\left(m_{H_{u}}^2\right)^{\tilde{F}}=\frac{1}{16\pi^2}\sum_f\left\{\delta_{fu}Y_f^2\left[{\cal A}^{\tilde{F}}_L+{\cal A}^{\tilde{F}}_R+A_f^2{\cal B}^{\tilde{F}}_{L,R}\right]+\frac{g'^2{\cal Y}_L^f-2g^2I_3^f}{4}{\cal A}^{\tilde{F}}_L+\frac{g'^2{\cal Y}_R^f}{4}{\cal A}^{\tilde{F}}_R\right\}\\
\left(m_{H_{d}}^2\right)^{\tilde{F}}=\frac{1}{16\pi^2}\sum_f\left\{\delta_{f(d,e)}Y_f^2\left[{\cal A}^{\tilde{F}}_L+{\cal A}^{\tilde{F}}_R+A_f^2{\cal B}^{\tilde{F}}_{L,R}\right]+\frac{g'^2{\cal Y}_R^f}{4}{\cal A}^{\tilde{F}}_L+\frac{g'^2{\cal Y}_L^f-2g^2I_3^f}{4}{\cal A}^{\tilde{F}}_R\right\}\\
A_{ud}^{\tilde{F}}=\frac{1}{16\pi^2}\sum_f\lambda Y_f^2A_f{\cal B}^{\tilde{F}}_{L,R}\\
\lambda_1^{\tilde{F}}=\frac{1}{16\pi^2}\sum_f\left\{\delta_{f(d,e)}\left[\left(Y_f^2+\frac{g'^2{\cal Y}^f_R}{4}\right)^2\ln\left(\frac{m_{\tilde{F}_L}^2}{\Lambda^2}\right)+\left(Y_f^2+\frac{g'^2{\cal Y}^f_L-2g^2I_3^f}{4}\right)^2\ln\left(\frac{m_{\tilde{F}_R}^2}{\Lambda^2}\right)\right.\right.\\
\null\hspace{1.8cm}\left.\left.+Y_f^2A_f^2\left[\left(2Y_f^2+\frac{g'^2({\cal Y}^f_L+{\cal Y}_R^f)-2g^2I_3^f}{4}\right){\cal C}^{\tilde{F}}_{L,R}-\left(Y_f^2A_f^2-\frac{g'^2({\cal Y}^f_L-{\cal Y}_R^f)-2g^2I_3^f}{4}\Delta m^{2\,\tilde{F}}_{L,R}\right){\cal D}^{\tilde{F}}_{L,R}\right]\right]\right.\\
\null\hspace{1.7cm}\left.+\delta_{fu}\left[\left(\frac{g'^2{\cal Y}^f_R}{4}\right)^2\ln\left(\frac{m_{\tilde{F}_L}^2}{\Lambda^2}\right)+\left(\frac{g'^2{\cal Y}^f_L-2g^2I_3^f}{4}\right)^2\ln\left(\frac{m_{\tilde{F}_R}^2}{\Lambda^2}\right)\right.\right.\\
\null\hspace{2cm}\left.\left.+Y_f^2\lambda^2s^2\left[\frac{g'^2({\cal Y}^f_L+{\cal Y}_R^f)-2g^2I_3^f}{4}{\cal C}^{\tilde{F}}_{L,R}-\left(Y_f^2\lambda^2s^2-\frac{g'^2({\cal Y}^f_L-{\cal Y}_R^f)-2g^2I_3^f}{4}\Delta m^{2\,\tilde{F}}_{L,R}\right){\cal D}^{\tilde{F}}_{L,R}\right]\right]\right\}\\
\lambda_2^{\tilde{F}}=\frac{1}{16\pi^2}\sum_f\left\{\delta_{fu}\left[\left(Y_f^2+\frac{g'^2{\cal Y}^f_L-2g^2I_3^f}{4}\right)^2\ln\left(\frac{m_{\tilde{F}_L}^2}{\Lambda^2}\right)+\left(Y_f^2+\frac{g'^2{\cal Y}^f_R}{4}\right)^2\ln\left(\frac{m_{\tilde{F}_R}^2}{\Lambda^2}\right)\right.\right.\\
\null\hspace{1.8cm}\left.\left.+Y_f^2A_f^2\left[\left(2Y_f^2+\frac{g'^2({\cal Y}^f_L+{\cal Y}_R^f)-2g^2I_3^f}{4}\right){\cal C}^{\tilde{F}}_{L,R}-\left(Y_f^2A_f^2+\frac{g'^2({\cal Y}^f_L-{\cal Y}_R^f)-2g^2I_3^f}{4}\Delta m^{2\,\tilde{F}}_{L,R}\right){\cal D}^{\tilde{F}}_{L,R}\right]\right]\right.\\
\null\hspace{1.7cm}\left.+\delta_{f(d,e)}\left[\left(\frac{g'^2{\cal Y}^f_L-2g^2I_3^f}{4}\right)^2\ln\left(\frac{m_{\tilde{F}_L}^2}{\Lambda^2}\right)+\left(\frac{g'^2{\cal Y}^f_R}{4}\right)^2\ln\left(\frac{m_{\tilde{F}_R}^2}{\Lambda^2}\right)\right.\right.\\
\null\hspace{2cm}\left.\left.+Y_f^2\lambda^2s^2\left[\frac{g'^2({\cal Y}^f_L+{\cal Y}_R^f)-2g^2I_3^f}{4}{\cal C}^{\tilde{F}}_{L,R}-\left(Y_f^2\lambda^2s^2+\frac{g'^2({\cal Y}^f_L-{\cal Y}_R^f)-2g^2I_3^f}{4}\Delta m^{2\,\tilde{F}}_{L,R}\right){\cal D}^{\tilde{F}}_{L,R}\right]\right]\right\}\\
(\lambda_3+\lambda_4)^{\tilde{F}}=\frac{1}{16\pi^2}\sum_f\left\{\delta_{fu}\left[\frac{g'^2{\cal Y}^f_R}{4}\left(Y_f^2+\frac{g'^2{\cal Y}^f_L-2g^2I_3^f}{4}\right)\ln\left(\frac{m_{\tilde{F}_L}^2}{\Lambda^2}\right)+\frac{g'^2{\cal Y}^f_L-2g^2I_3^f}{4}\left(Y_f^2+\frac{g'^2{\cal Y}^f_R}{4}\right)\ln\left(\frac{m_{\tilde{F}_L}^2}{\Lambda^2}\right)\right.\right.\\
\null\hspace{0.5cm}\left.\left.+Y_f^2\lambda^2s^2\left[\left(Y_f^2+\frac{g'^2({\cal Y}^f_L+{\cal Y}_R^f)-2g^2I_3^f}{8}\left(1+\frac{A_f^2}{\lambda^2s^2}\right)\right){\cal C}^{\tilde{F}}_{L,R}-\left(2A_f^2+\frac{g'^2({\cal Y}^f_L-{\cal Y}_R^f)-2g^2I_3^f}{8}(1-\frac{A_f^2}{\lambda^2s^2})\Delta m^{2\,\tilde{F}}_{L,R}\right){\cal D}^{\tilde{F}}_{L,R}\right]\right]\right.\\
\null\hspace{3cm}\left.+\delta_{f(d,e)}\left[\frac{g'^2{\cal Y}^f_L-2g^2I_3^f}{4}\left(Y_f^2+\frac{g'^2{\cal Y}^f_R}{4}\right)\ln\left(\frac{m_{\tilde{F}_L}^2}{\Lambda^2}\right)+\frac{g'^2{\cal Y}^f_R}{4}\left(Y_f^2+\frac{g'^2{\cal Y}^f_L-2g^2I_3^f}{4}\right)\ln\left(\frac{m_{\tilde{F}_L}^2}{\Lambda^2}\right)\right.\right.\\
\null\hspace{0.5cm}\left.\left.+Y_f^2\lambda^2s^2\left[\left(Y_f^2+\frac{g'^2({\cal Y}^f_L+{\cal Y}_R^f)-2g^2I_3^f}{8}\left(1+\frac{A_f^2}{\lambda^2s^2}\right)\right){\cal C}^{\tilde{F}}_{L,R}-\left(2A_f^2+\frac{g'^2({\cal Y}^f_L-{\cal Y}_R^f)-2g^2I_3^f}{8}(\frac{A_f^2}{\lambda^2s^2}-1)\Delta m^{2\,\tilde{F}}_{L,R}\right){\cal D}^{\tilde{F}}_{L,R}\right]\right]\right\}\\
\lambda_5^{\tilde{F}}=-\frac{1}{16\pi^2}\sum_{f}Y_f^4A_f^2\lambda^2s^2{\cal D}^{\tilde{F}}_{L,R}\\
\lambda_6^{\tilde{F}}=\frac{1}{16\pi^2}\sum_f\left\{\delta_{fu}Y_f^2A_f\lambda s\left[\left(Y_f^2+\frac{g'^2({\cal Y}^f_L+{\cal Y}_R^f)-2g^2I_3^f}{8}\right){\cal C}^{\tilde{F}}_{L,R}-\left(Y_f^2A_f^2+\frac{g'^2({\cal Y}^f_L-{\cal Y}_R^f)-2g^2I_3^f}{8}\Delta m^{2\,\tilde{F}}_{L,R}\right){\cal D}^{\tilde{F}}_{L,R}\right]\right.\\
\null\hspace{2.3cm}\left.+\delta_{f(d,e)}Y_f^2A_f\lambda s\left[\frac{g'^2({\cal Y}^f_L+{\cal Y}_R^f)-2g^2I_3^f}{8}{\cal C}^{\tilde{F}}_{L,R}-\left(Y_f^2\lambda^2s^2+\frac{g'^2({\cal Y}^f_L-{\cal Y}_R^f)-2g^2I_3^f}{8}\Delta m^{2\,\tilde{F}}_{L,R}\right){\cal D}^{\tilde{F}}_{L,R}\right]\right\}\\
\lambda_7^{\tilde{F}}=\frac{1}{16\pi^2}\sum_f\left\{\delta_{f(d,e)}Y_f^2A_f\lambda s\left[\left(Y_f^2+\frac{g'^2({\cal Y}^f_L+{\cal Y}_R^f)-2g^2I_3^f}{8}\right){\cal C}^{\tilde{F}}_{L,R}-\left(Y_f^2A_f^2-\frac{g'^2({\cal Y}^f_L-{\cal Y}_R^f)-2g^2I_3^f}{8}\Delta m^{2\,\tilde{F}}_{L,R}\right){\cal D}{\tilde{F}}_{L,R}\right]\right.\\
\null\hspace{2.3cm}\left.+\delta_{fu}Y_f^2A_f\lambda s\left[\frac{g'^2({\cal Y}^f_L+{\cal Y}_R^f)-2g^2I_3^f}{8}{\cal C}^{\tilde{F}}_{L,R}-\left(Y_f^2\lambda^2s^2-\frac{g'^2({\cal Y}^f_L-{\cal Y}_R^f)-2g^2I_3^f}{8}\Delta m^{2\,\tilde{F}}_{L,R}\right){\cal D}^{\tilde{F}}_{L,R}\right]\right\}\\
(\lambda_P^u)^{\tilde{F}}=\frac{1}{16\pi^2}\sum_{f=d,e}Y_f^2\lambda^2{\cal B}^{\tilde{F}}_{L,R}\\
(\lambda_P^d)^{\tilde{F}}=\frac{1}{16\pi^2}\sum_{f=u}Y_f^2\lambda^2{\cal B}^{\tilde{F}}_{LR}
\end{cases}\end{equation}}\noi
where we have used the notations:
{\small\begin{align}
 &{\cal A}^{\tilde{F}}_L\equiv m_{\tilde{F}_L}^2\left[\ln\left(\frac{m_{\tilde{F}_L}^2}{\Lambda^2}\right)-1\right]\hspace{0.5cm};\hspace{0.5cm}{\cal A}^{\tilde{F}}_R\equiv m_{\tilde{F}_R}^2\left[\ln\left(\frac{m_{\tilde{F}_R}^2}{\Lambda^2}\right)-1\right]\hspace{0.5cm};\hspace{0.5cm}\Delta m^{2\,\tilde{F}}_{L,R}\equiv m_{\tilde{F}_L}^2-m_{\tilde{F}_R}^2\nonumber\\
 &{\cal B}^{\tilde{F}}_{L,R}\equiv \frac{1}{m_{\tilde{F}_L}^2-m_{\tilde{F}_R}^2}\left[{\cal A}^{\tilde{F}}_L-{\cal A}^{\tilde{F}}_R\right]\xrightarrow{m_{\tilde{F}_{L,R}}^2\rightarrow m^2}\ln\left(\frac{m^2}{\Lambda^2}\right)\\
 &{\cal C}^{\tilde{F}}_{L,R}\equiv \frac{1}{m_{\tilde{F}_L}^2-m_{\tilde{F}_R}^2}\ln\left(\frac{m_{\tilde{F}_L}^2}{m_{\tilde{F}_R}^2}\right)\xrightarrow{m_{\tilde{F}_{L,R}}^2\rightarrow m^2}\frac{1}{m^2}\nonumber\\
 &{\cal D}^{\tilde{F}}_{L,R}\equiv \frac{1}{(m_{\tilde{F}_L}^2-m_{\tilde{F}_R}^2)^3}\left[m_{\tilde{F}_L}^2\left(\ln\left(\frac{m_{\tilde{F}_L}^2}{m_{\tilde{F}_R}^2}\right)-2\right)+m_{\tilde{F}_R}^2\left(\ln\left(\frac{m_{\tilde{F}_L}^2}{m_{\tilde{F}_R}^2}\right)+2\right)\right]\xrightarrow{m_{\tilde{F}_{L,R}}^2\rightarrow m^2}\frac{1}{6m^4}\nonumber
\end{align}}

{\bf Chargino contributions:}\newline
The chargino squared mass-matrix, in a base of Dirac (winos,higgsinos), is given by:
\begin{equation}
 {\cal M}^2_{\chi^{\pm}}(S,H_u^0,H_d^0)=\begin{bmatrix}
M_2^2+g^2|H_d^0|^2&g(M_2H_u^{0\,*}+\lambda S H_d^0)\\
g(M_2H_u^{0}+\lambda S^* H_d^{0\,*})&\lambda^2|S|^2+g^2|H_u^0|^2
                        \end{bmatrix}
\end{equation}\noi
Defining $T_{\chi^{\pm}}=({\cal M}^2_{\chi^{\pm}})_{11}+({\cal M}^2_{\chi^{\pm}})_{22}$ and $R_{\tilde{F}}^2=\left[({\cal M}^2_{\chi^{\pm}})_{11}-({\cal M}^2_{\chi^{\pm}})_{22}\right]^2+4|{\cal M}^2_{\chi^{\pm}}|^2_{12}$, 
we obtain the eigenvalues $m_{(\chi^{\pm})_{\pm}}^2=\frac{1}{2}\left[T_{\chi^{\pm}}\pm\sqrt{R_{\chi^{\pm}}^2}\right]$ and the Higgs potential:
\begin{equation}
 \delta{\cal V}_{\mbox{\small eff}}^{\Lambda,\chi^{\pm}}(H_u^0,H_d^0,S)=-\frac{1}{64\pi^2}\left\{(T_{\chi^{\pm}}^2+R_{\chi^{\pm}}^2)\left[\ln\left(\frac{T_{\chi^{\pm}}^2-R_{\chi^{\pm}}^2}{4\Lambda^4}\right)-3\right]+2\,T_{\chi^{\pm}}\cdot R_{\chi^{\pm}}\ln\left(\frac{T_{\chi^{\pm}}+R_{\chi^{\pm}}}{T_{\chi^{\pm}}-R_{\chi^{\pm}}}\right)\right\}
\end{equation}\noi
One can derive the couplings:
{\small\begin{equation}\begin{cases}
\left(m_{H_{u,d}^0}^2\right)^{\chi^{\pm}}=-\frac{g^2}{8\pi^2(M_2^2-\lambda^2s^2)^3}\left\{M_2^8\left[\ln\left(\frac{M_2^2}{\Lambda^2}\right)-1\right]-3M_2^6\lambda^2s^2\left[\ln\left(\frac{M_2^2}{\Lambda^2}\right)-1\right]\right.\\
\hspace{3.5cm}\left.+3M_2^4\lambda^4s^4\left[2\ln\left(\frac{M_2^2}{\Lambda^2}\right)-\ln\left(\frac{\lambda^2s^2}{\Lambda^2}\right)-\frac{8}{3}\right]-M_2^2\lambda^6s^6\left[\ln\left(\frac{\lambda^2s^2}{\Lambda^2}\right)-7\right]-\lambda^8s^8\right\}\\
\left(m_{12}^2\right)^{\chi^{\pm}}=-\frac{g^2M_2\lambda^3s^3}{8\pi^2(M_2^2-\lambda^2s^2)^3}\left\{M_2^4\left[\ln\left(\frac{M_2^2}{\lambda^2s^2}\right)-3\right]+3M_2^2\lambda^2s^2\left[\ln\left(\frac{M_2^2}{\lambda^2s^2}\right)+\frac{2}{3}\right]+\lambda^4s^4\right\}\\
\left(A_{(u,d)s}\right)^{\chi^{\pm}}=\frac{g^2\lambda^4s^3}{4\pi^2(M_2^2-\lambda^2s^2)^3}\left\{2M_2^4\left[\ln\left(\frac{M_2^2}{\lambda^2s^2}\right)-\frac{3}{2}\right]+4M_2^2\lambda^2s^2-\lambda^4s^4\right\}\\
\left(\tilde{A}_{ud}\right)^{\chi^{\pm}}=-\frac{g^2M_2\lambda^3s^2}{8\pi^2(M_2^2-\lambda^2s^2)^3}\left\{M_2^4\left[\ln\left(\frac{M_2^2}{\lambda^2s^2}\right)-3\right]+3M_2^2\lambda^2s^2\left[\ln\left(\frac{M_2^2}{\lambda^2s^2}\right)+\frac{2}{3}\right]+\lambda^4s^4\right\}\\
\left(A_{ud}\right)^{\chi^{\pm}}=\frac{g^2\lambda M_2}{8\pi^2(M_2^2-\lambda^2s^2)^3}\left\{M_2^6\left[\ln\left(\frac{M_2^2}{\Lambda^2}\right)-1\right]-M_2^4\lambda^2s^2\left[5\ln\left(\frac{M_2^2}{\Lambda^2}\right)-2\ln\left(\frac{\lambda^2s^2}{\Lambda^2}\right)-8\right]\right.\\
\null\hspace{3.5cm}\left.+3M_2^2\lambda^4s^4\left[\ln\left(\frac{\lambda^2s^2}{\Lambda^2}\right)-3\right]-\lambda^6s^6\left[\ln\left(\frac{\lambda^2s^2}{\Lambda^2}\right)-2\right]\right\}\\
\left(\lambda_{1,2}\right)^{\chi^{\pm}}=-\frac{g^4}{8\pi^2(M_2^2-\lambda^2s^2)^3}\left\{M_2^6\ln\left(\frac{M_2^2}{\Lambda^2}\right)-3M_2^4\lambda^2s^2\left[\ln\left(\frac{M_2^2}{\Lambda^2}\right)-\frac{2}{3}\right]+3M_2^2\lambda^4s^4\left[\ln\left(\frac{\lambda^2s^2}{\Lambda^2}\right)-\frac{2}{3}\right]-\lambda^6s^6\ln\left(\frac{\lambda^2s^2}{\Lambda^2}\right)\right\}\\
\left(\lambda_{3}+\lambda_4\right)^{\chi^{\pm}}=-\frac{g^4}{8\pi^2(M_2^2-\lambda^2s^2)^3}\left\{M_2^6-2M_2^4\lambda^2s^2\left[\ln\left(\frac{M_2^2}{\lambda^2s^2}\right)-\frac{1}{2}\right]-2M_2^2\lambda^4s^4\left[\ln\left(\frac{M_2^2}{\lambda^2s^2}\right)+\frac{1}{2}\right]-\lambda^6s^6\right\}\\
\left(\lambda_{5}\right)^{\chi^{\pm}}=\frac{g^4M_2^2\lambda^2s^2}{8\pi^2(M_2^2-\lambda^2s^2)^3}\left\{M_2^2\left[\ln\left(\frac{M_2^2}{\lambda^2s^2}\right)-2\right]+\lambda^2s^2\left[\ln\left(\frac{M_2^2}{\lambda^2s^2}\right)+2\right]\right\}\\
\left(\lambda_{M}\right)^{\chi^{\pm}}=\frac{g^2\lambda^3sM_2}{8\pi^2(M_2^2-\lambda^2s^2)^3}\left\{2M_2^4\left[\ln\left(\frac{M_2^2}{\lambda^2s^2}\right)-\frac{3}{2}\right]+4M_2^2\lambda^2s^2-\lambda^4s^4\right\}\\
\left(\lambda^P_{M}\right)^{\chi^{\pm}}=-\frac{g^2\lambda^3sM_2}{16\pi^2(M_2^2-\lambda^2s^2)^3}\left\{M_2^4-2M_2^2\lambda^2s^2\ln\left(\frac{M_2^2}{\lambda^2s^2}\right)-\lambda^4s^4\right\}\\
\left(\tilde{\lambda}^P_{M}\right)^{\chi^{\pm}}=\frac{g^2\lambda^3sM_2}{16\pi^2(M_2^2-\lambda^2s^2)^3}\left\{2M_2^4\left[\ln\left(\frac{M_2^2}{\lambda^2s^2}\right)-\frac{3}{2}\right]+4M_2^2\lambda^2s^2-\lambda^4s^4\right\}\\
\left(\lambda^P_{u,d}\right)^{\chi^{\pm}}=-\frac{g^2\lambda^2}{8\pi^2(M_2^2-\lambda^2s^2)^3}\left\{M^6_2\left[\ln\left(\frac{M_2^2}{\Lambda^2}\right)-1\right]+M_2^4\lambda^2s^2\left[\ln\left(\frac{M_2^2}{\Lambda^2}\right)+2\ln\left(\frac{\lambda^2s^2}{\Lambda^2}\right)-1\right]\right.\\
\null\hspace{3.5cm}\left.+3M_2^2\lambda^4s^4\left[\ln\left(\frac{\lambda^2s^2}{\Lambda^2}\right)+1\right]-\lambda^6s^6\left[\ln\left(\frac{\lambda^2s^2}{\Lambda^2}\right)+1\right]\right\}\\
\left(\tilde{\lambda}^P_{u,d}\right)^{\chi^{\pm}}=-\frac{g^2\lambda^3sM_2}{16\pi^2(M_2^2-\lambda^2s^2)^3}\left\{2M_2^4\left[\ln\left(\frac{M_2^2}{\lambda^2s^2}\right)-\frac{3}{2}\right]+4M_2^2\lambda^2s^2-\lambda^4s^4\right\}\\
\end{cases}\end{equation}\noi
\begin{displaymath}
 \begin{cases}
  \left(\lambda_T\right)^{\chi^{\pm}}=\frac{\lambda^4s^3}{12\pi^2}\hspace{1.cm};\hspace{1.cm}
\left(m_{S}^2\right)^{\chi^{\pm}}=-\frac{\lambda^4s^2}{4\pi^2}\hspace{1.cm};\hspace{1.cm}
\left(\mu_{S}^2\right)^{\chi^{\pm}}=-\frac{\lambda^4s^2}{8\pi^2}\hspace{1.cm};\hspace{1.cm}
\left(\tilde{A}_{S}^2\right)^{\chi^{\pm}}=\frac{3\lambda^4s}{4\pi^2}\\
 \left(\kappa^2\right)^{\chi^{\pm}}=-\frac{\lambda^4}{16\pi^2}\left[\ln\left(\frac{M_2^2}{\Lambda^2}\right)-\frac{3}{2}\right]\hspace{1.2cm};\hspace{1.2cm}
\left(\kappa_S^2\right)^{\chi^{\pm}}=\frac{\lambda^4}{48\pi^2}\hspace{1.2cm};\hspace{1.2cm}
\left(\tilde{\kappa}_S^2\right)^{\chi^{\pm}}=-\frac{\lambda^4}{6\pi^2}
\end{cases}
\end{displaymath}
}

{\bf Neutralino contributions:}\newline
The (hermitian) neutralino squared mass-matrix is determined by its entries in the base of Weyl spinors $(-\imath\tilde{b}^0,
-\imath\tilde{w}^0_3,\tilde{h}^0_u,\tilde{h}^0_d,\tilde{h}^0_s)$:
{\small\begin{align}
 \left({\cal M}^2_{\chi^0}\right)_{11}(S,H_u^0,H_d^0)&=M_1^2+\frac{g'^2}{2}\left(|H_u^0|^2+|H_d^0|^2\right)\nonumber\\
 \left({\cal M}^2_{\chi^0}\right)_{22}(S,H_u^0,H_d^0)&=M_2^2+\frac{g^2}{2}\left(|H_u^0|^2+|H_d^0|^2\right)\nonumber\\
 \left({\cal M}^2_{\chi^0}\right)_{12}(S,H_u^0,H_d^0)&=-\frac{gg'}{2}\left(|H_u^0|^2+|H_d^0|^2\right)\nonumber\\
 \left({\cal M}^2_{\chi^0}\right)_{33}(S,H_u^0,H_d^0)&=\lambda^2(|S|^2+|H_d^0|^2)+\frac{g^2+g'^2}{2}|H_u^0|^2\nonumber\\
 \left({\cal M}^2_{\chi^0}\right)_{44}(S,H_u^0,H_d^0)&=\lambda^2(|S|^2+|H_u^0|^2)+\frac{g^2+g'^2}{2}|H_d^0|^2\nonumber\\
 \left({\cal M}^2_{\chi^0}\right)_{34}(S,H_u^0,H_d^0)&=\left(\lambda^2-\frac{g^2+g'^2}{2}\right)H_u^0H_d^{0\,*}\nonumber\\
 \left({\cal M}^2_{\chi^0}\right)_{55}(S,H_u^0,H_d^0)&=4\kappa^2|S|^2+\lambda^2(|H_u^0|^2+|H_d^0|^2)\\
 \left({\cal M}^2_{\chi^0}\right)_{35}(S,H_u^0,H_d^0)&=\lambda^2S^*H_u^0-2\lambda\kappa SH_d^{0\,*}\nonumber\\
 \left({\cal M}^2_{\chi^0}\right)_{45}(S,H_u^0,H_d^0)&=\lambda^2S^*H_d^0-2\lambda\kappa SH_u^{0\,*}\nonumber\\
 \left({\cal M}^2_{\chi^0}\right)_{13}(S,H_u^0,H_d^0)&=\frac{g'}{\sqrt{2}}(M_1H_u^{0\,*}+\lambda SH_d^0)\nonumber\\
 \left({\cal M}^2_{\chi^0}\right)_{14}(S,H_u^0,H_d^0)&=-\frac{g'}{\sqrt{2}}(M_1H_d^{0\,*}+\lambda SH_u^0)\nonumber\\
 \left({\cal M}^2_{\chi^0}\right)_{23}(S,H_u^0,H_d^0)&=-\frac{g}{\sqrt{2}}(M_2H_u^{0\,*}+\lambda SH_d^0)\nonumber\\
 \left({\cal M}^2_{\chi^0}\right)_{24}(S,H_u^0,H_d^0)&=\frac{g}{\sqrt{2}}(M_1H_d^{0\,*}+\lambda SH_u^0)\nonumber
\end{align}}\noi
One can expand its eigenvalues in terms of doublet fields:
\begin{multline}
 m_i^2(S,H_u^0,H_d^0)=m_i^{2\,(0)}(S)+m_i^{2\,(1)}(S,H_u^0,H_d^0)+m_i^{2\,(2)}(S,H_u^0,H_d^0)+O\left((H^0_{u,d})^5\right)\\
\begin{cases} m_i^{2\,(1)}(S,H_u^0,H_d^0)=m_i^{2\,(1u)}(S)|H_u^0|^2+m_i^{2\,(1d)}(S)|H_d^0|^2+\left(m_i^{2\,(1ud)}(S)H_u^0H_d^0+h.c.\right)\\
m_i^{2\,(2)}(S,H_u^0,H_d^0)=m_i^{2\,(2d)}(S)|H_d^0|^4+m_i^{2\,(2u)}(S)|H_u^0|^4+m_i^{2\,(2ud)}(S)|H_u^0|^2|H_d^0|^2+\ldots\end{cases}
\end{multline}\noi
The associated potential is then given by:
{\small\begin{multline}
 \delta{\cal V}_{\mbox{\small eff}}^{\Lambda,\chi^{0}}(H_u^0,H_d^0,S)=-\frac{1}{32\pi^2}\sum_im_i^4(S,H_u^0,H_d^0)\left[\ln\left(\frac{m_i^2(S,H_u^0,H_d^0)}{\Lambda^2}\right)-\frac{3}{2}\right]\\
=-\frac{1}{32\pi^2}\sum_i\left\{m_i^{4\,(0)}(S)\left[\ln\left(\frac{m_i^{2\,(0)}(S)}{\Lambda^2}\right)-\frac{3}{2}\right]+2m_i^{2\,(0)}(S)m_i^{2\,(1)}(S,H_u^0,H_d^0)\left[\ln\left(\frac{m_i^{2\,(0)}(S)}{\Lambda^2}\right)-1\right]\right.\\\left.+\left(2m_i^{2\,(0)}(S)m_i^{2\,(2)}(S,H_u^0,H_d^0)\left[\ln\left(\frac{m_i^{2\,(0)}(S)}{\Lambda^2}\right)-1\right]+m_i^{4\,(1)}(S,H_u^0,H_d^0)\ln\left(\frac{m_i^{2\,(0)}(S)}{\Lambda^2}\right)\right)+O(H^5_{u,d})\right\}
\end{multline}}

\noi${\cal M}^2_{\chi^0}(S,H_{u,d}^0=0)$ is already diagonal with the eigenstates $\{M_1^2,\left|E_1\right>\}$, $\{M_2^2,\left|E_2\right>\}$, 
$\{\lambda^2|S|^2,\left|E_3\right>,\left|E_4\right>\}$ and $\{4\kappa^2|S|^2,\left|E_5\right>\}$ ($\{\left|E_i\right>\}$ stands for the canonical base of $\mathbb{C}^5$), from which one obtains easily the pure-singlet parameters:
{\small\begin{equation}
 \begin{cases}
  \left(\lambda_T\right)^{\chi^{0}}=\frac{1}{12\pi^2}(\lambda^4+8\kappa^4)s^3\\
\left(m^2_S\right)^{\chi^{0}}=-\frac{1}{4\pi^2}(\lambda^4+8\kappa^4)s^2\\
\left(\mu_S^2\right)^{\chi^{0}}=-\frac{1}{8\pi^2}(\lambda^4+8\kappa^4)s^2\\
\left(\tilde{A}_S\right)^{\chi^{0}}=\frac{3}{4\pi^2}(\lambda^4+8\kappa^4)s\\
\left(\kappa^2\right)^{\chi^{0}}=-\frac{1}{16\pi^2}\left\{\lambda^4\left[\ln\left(\frac{\lambda^2s^2}{\Lambda^2}\right)+\frac{3}{2}\right]+8\kappa^4\left[\ln\left(\frac{4\kappa^2s^2}{\Lambda^2}\right)+\frac{3}{2}\right]\right\}\\
\left(\kappa^2_S\right)^{\chi^{0}}=\frac{1}{48\pi^2}(\lambda^4+8\kappa^4)\\
\left(\tilde{\kappa}^2_S\right)^{\chi^{0}}=-\frac{1}{6\pi^2}(\lambda^4+8\kappa^4)
 \end{cases}
\end{equation}}

\noi The $O(H^2)$ masses also come without much effort:
{\small\begin{align}
 &m_1^{2\,(1)}=\frac{g'^2}{M_1^2-\lambda^2|S|^2}\left[M_1^2(|H_u^0|^2+|H_d^0|^2)+2\lambda M_1\mbox{Re}(SH_u^0H_d^0)\right]\nonumber\\
 &m_2^{2\,(1)}=\frac{g^2}{M_2^2-\lambda^2|S|^2}\left[M_2^2(|H_u^0|^2+|H_d^0|^2)+2\lambda M_2\mbox{Re}(SH_u^0H_d^0)\right]\\
 &m_3^{2\,(1)}+m_4^{2\,(1)}=\frac{g'^2}{\lambda^2|S|^2-M_1^2}\left[\lambda^2|S|^2(|H_u^0|^2+|H_d^0|^2)+2\lambda M_1\mbox{Re}(SH_u^0H_d^0)\right]\nonumber\\
&\null\hspace{2.6cm}+\frac{g^2}{\lambda^2|S|^2-M_2^2}\left[\lambda^2|S|^2(|H_u^0|^2+|H_d^0|^2)+2\lambda M_2\mbox{Re}(SH_u^0H_d^0)\right]\nonumber\\
&\null\hspace{2.6cm}+\frac{2\lambda^2}{(\lambda^2-4\kappa^2)|S|^2}\left[\lambda^2|S|^2(|H_u^0|^2+|H_d^0|^2)-4\kappa\lambda\mbox{Re}(S^{*\,2}H_u^0H_d^0)\right]\nonumber\\
 &m_5^{2\,(1)}=\frac{8\lambda^2}{(4\kappa^2-\lambda^2)|S|^2}\left[\kappa^2|S|^2(|H_u^0|^2+|H_d^0|^2)-\kappa\lambda\mbox{Re}(S^{*\,2}H_u^0H_d^0)\right]\nonumber
\end{align}}\noi
from which we can derive the couplings (we focus on the logarithmic terms):
{\small\begin{equation}
 \begin{cases}
\left(m^2_{H_{u,d}}\right)^{\chi^{0}}=-\frac{g'^2}{16\pi^2(M_1^2-\lambda^2s^2)^3}\left\{M_1^4\left[M_1^4-3M^2_1\lambda^2s^2+6\lambda^4s^4\right]\ln\left(\frac{M_1^2}{\Lambda^2}\right)-\lambda^4s^4\left[3M_1^2+\lambda^2s^2\right]\ln\left(\frac{\lambda^2s^2}{\Lambda^2}\right)\right\}\\
\null\hspace{2cm}-\frac{g^2}{16\pi^2(M_2^2-\lambda^2s^2)^3}\left\{M_2^4\left[M_2^4-3M^2_2\lambda^2s^2+6\lambda^4s^4\right]\ln\left(\frac{M_2^2}{\Lambda^2}\right)-\lambda^4s^4\left[3M_2^2+\lambda^2s^2\right]\ln\left(\frac{\lambda^2s^2}{\Lambda^2}\right)\right\}\\
\left(m^2_{12}\right)^{\chi^{0}}=\frac{g'^2M_1^3\lambda^3s^3}{8\pi^2(M_1^2-\lambda^2s^2)^3}\left[M_1^2+3\lambda^2s^2\right]\ln\left(\frac{M_1^2}{\lambda^2s^2}\right)+\frac{g^2M_2^3\lambda^3s^3}{8\pi^2(M_2^2-\lambda^2s^2)^3}\left[M_2^2+3\lambda^2s^2\right]\ln\left(\frac{M_2^2}{\lambda^2s^2}\right)\\
\left(A_{(u,d)s}\right)^{\chi^{0}}=\frac{g'^2M_1^4\lambda^4s^3}{4\pi^2(M_1^2-\lambda^2s^2)^3}\ln\left(\frac{M_1^2}{\lambda^2s^2}\right)+\frac{g^2M_2^4\lambda^4s^3}{4\pi^2(M_2^2-\lambda^2s^2)^3}\ln\left(\frac{M_2^2}{\lambda^2s^2}\right)\\
\left(A_{ud}\right)^{\chi^{0}}=\frac{g'^2\lambda M_1}{16\pi^2(M_1^2-\lambda^2s^2)^3}\left\{M_1^4\left[M_1^2-5\lambda^2s^2\right]\ln\left(\frac{M_1^2}{\Lambda^2}\right)-\lambda^2s^2\left[2M_1^4+3M_1^2\lambda^2s^2-\lambda^4s^4\right]\ln\left(\frac{\lambda^2s^2}{\Lambda^2}\right)\right\}\\
\null\hspace{1.8cm}+\frac{g^2\lambda M_2}{16\pi^2(M_2^2-\lambda^2s^2)^3}\left\{M_2^4\left[M_2^2-5\lambda^2s^2\right]\ln\left(\frac{M_2^2}{\Lambda^2}\right)-\lambda^2s^2\left[2M_2^4+3M_2^2\lambda^2s^2-\lambda^4s^4\right]\ln\left(\frac{\lambda^2s^2}{\Lambda^2}\right)\right\}\\
\left(\tilde{A}_{ud}\right)^{\chi^{0}}=-\frac{g'^2M_1^3\lambda^3s^2}{16\pi^2(M_1^2-\lambda^2s^2)^3}\left[M_1^2+3\lambda^2s^2\right]\ln\left(\frac{M_1^2}{\lambda^2s^2}\right)-\frac{g^2M_2^3\lambda^3s^2}{16\pi^2(M_2^2-\lambda^2s^2)^3}\left[M_2^2+3\lambda^2s^2\right]\ln\left(\frac{M_2^2}{\lambda^2s^2}\right)\\
\left(\lambda^P_{u,d}\right)^{\chi^{0}}=-\frac{g'^2\lambda^2}{16\pi^2(M_1^2-\lambda^2s^2)^3}\left\{M_1^4\left[M_1^2+\lambda^2s^2\right]\ln\left(\frac{M_1^2}{\Lambda^2}\right)-\lambda^2s^2\left[4M_1^4-3M_1^2\lambda^2s^2+\lambda^4s^4\right]\ln\left(\frac{\lambda^2s^2}{\Lambda^2}\right)\right\}\\
\null\hspace{1.8cm}-\frac{g^2\lambda^2}{16\pi^2(M_2^2-\lambda^2s^2)^3}\left\{M_2^4\left[M_2^2+\lambda^2s^2\right]\ln\left(\frac{M_2^2}{\Lambda^2}\right)-\lambda^2s^2\left[4M_2^4-3M_2^2\lambda^2s^2+\lambda^4s^4\right]\ln\left(\frac{\lambda^2s^2}{\Lambda^2}\right)\right\}\\
\null\hspace{1.8cm}+\frac{\lambda^2}{8\pi^2(4\kappa^2-\lambda^2)}\left\{\lambda^4\ln\left(\frac{\lambda^2s^2}{\Lambda^2}\right)-16\kappa^4\ln\left(\frac{4\kappa^2s^2}{\Lambda^2}\right)\right\}\\
\left(\tilde{\lambda}^P_{u,d}\right)^{\chi^{0}}=-\frac{g'^2M_1^4\lambda^4s^2}{16\pi^2(M_1^2-\lambda^2s^2)^3}\ln\left(\frac{M_1^2}{\lambda^2s^2}\right)-\frac{g^2M_2^4\lambda^4s^2}{16\pi^2(M_2^2-\lambda^2s^2)^3}\ln\left(\frac{M_2^2}{\lambda^2s^2}\right)\\
\left(\lambda^P_M\right)^{\chi^{0}}=\frac{g'^2M_1^3\lambda^5s^3}{16\pi^2(M_1^2-\lambda^2s^2)^3}\ln\left(\frac{M_1^2}{\lambda^2s^2}\right)+\frac{g^2M_2^3\lambda^5s^3}{16\pi^2(M_2^2-\lambda^2s^2)^3}\ln\left(\frac{M_2^2}{\lambda^2s^2}\right)\\
\null\hspace{1.8cm}+\frac{\kappa\lambda^3}{4\pi^2(4\kappa^2-\lambda^2)}\left\{\lambda^2\ln\left(\frac{\lambda^2s^2}{\Lambda^2}\right)-4\kappa^2\ln\left(\frac{4\kappa^2s^2}{\Lambda^2}\right)\right\}\\
\left(\lambda_M\right)^{\chi^{0}}=\frac{g'^2M_1^5\lambda^3s}{8\pi^2(M_1^2-\lambda^2s^2)^3}\ln\left(\frac{M_1^2}{\lambda^2s^2}\right)+\frac{g^2M_2^5\lambda^3s}{8\pi^2(M_2^2-\lambda^2s^2)^3}\ln\left(\frac{M_2^2}{\lambda^2s^2}\right)\\
\left(\tilde{\lambda}^P_M\right)^{\chi^{0}}=\frac{g'^2M_1^5\lambda^3s}{16\pi^2(M_1^2-\lambda^2s^2)^3}\ln\left(\frac{M_1^2}{\lambda^2s^2}\right)+\frac{g^2M_2^5\lambda^3s}{16\pi^2(M_2^2-\lambda^2s^2)^3}\ln\left(\frac{M_2^2}{\lambda^2s^2}\right)\\
 \end{cases}
\end{equation}}\noi
The limit $\kappa^2\to0$ for the nMSSM is straightforward.

{\bf Charged-Higgs contributions -- NMSSM:}\newline
In the base $(H_u^-,H_d^-)$, the hermitian squared mass-matrix of Charged-Higgs bosons in terms of neutral Higgs fields reads (we use the general notation 
of a $\mathbb{Z}_3$-conserving potential; those parameters should be replaced, in practice, by their tree-level value; we also 
define $\lambda_P^.$, replacing $\lambda_P^{u,d}$, which coincide at tree-level; same thing for $\lambda_.$, replacing 
$\lambda_{1,2}$):
{\small\begin{align}
 &({\cal M}_{H^{\pm}}^2)_{11}(S,H_{u,d}^0)=M_{EW}^2+(A_{ud}+\lambda_P^Ms)\frac{s}{\tan\beta}+\lambda_.|H_u^0|^2+\lambda_3|H_d^0|^2+\lambda_P^.(|S|^2-s^2)\nonumber\\
 &({\cal M}_{H^{\pm}}^2)_{22}(S,H_{u,d}^0)=M_{EW}^2+(A_{ud}+\lambda_P^Ms)s\tan\beta+\lambda_.|H_d^0|^2+\lambda_3|H_u^0|^2+\lambda_P^.(|S|^2-s^2)\\
 &({\cal M}_{H^{\pm}}^2)_{12}(S,H_{u,d}^0)=A_{ud}S+\lambda_P^MS^{*\,2}-\lambda_4(H_u^0H_d^0)^*\nonumber
\end{align}}\noi
We have introduced $M_{EW}^2$ to replace constant terms generated by the electroweak v.e.v.\ and regularizing the (otherwise-vanishing)
Goldstone mass (which does not correspond to a Goldstone boson since $SU(2)_L$ is conserved in our approach). In practice, this 
$M_{EW}^2$ should be chosen as $M_W^2$, typically, since it replaces the longitudinal component of $W$-bosons. Now, defining 
$T\equiv\mbox{Tr}{\cal M}_{H^{\pm}}^2(S,H_{u,d}^0)$ and $R^2\equiv T^2-4\mbox{det}{\cal M}_{H^{\pm}}^2(S,H_{u,d}^0)$, we obtain
the two eigenvalues $m^2_{h/H}(S,H_{u,d}^0)=\frac{1}{2}\left[T-/+R\right]$, as well as the potential:
\begin{equation}\label{CHpot}
 \delta{\cal V}_{\mbox{\small eff}}^{\Lambda,H^{\pm}}(H_u^0,H_d^0,S)=\frac{1}{128\pi^2}\left\{(T^2+R^2)\left[\ln\left(\frac{T^2-R^2}{4\Lambda^4}\right)-3\right]+2\,T\cdot R\ln\left(\frac{T+R}{T-R}\right)\right\}
\end{equation}\noi
Then we focus on the logarithms (the notation $\left<\cdot\right>$ means that Higgs fields are replaced by their v.e.v.'s):
\begin{equation}\begin{cases}
 \ln\left(\frac{T^2-R^2}{4\Lambda^2}\right)\simeq\ln\left[\frac{M_{EW}^2}{\Lambda^4}\left(M_{EW}^2+\frac{2(A_{ud}+\lambda_P^M s)s}{\sin{2\beta}}\right)\right]+\ldots\equiv\ln\left<\frac{m_{h}^2m_{H}^2}{\Lambda^{4}}\right>+\ldots\\
\ln\left(\frac{T+R}{T-R}\right)\simeq\ln\left[1+\frac{2(A_{ud}+\lambda_P^M s)s}{\sin{2\beta}M_{EW}^2}\right]+\ldots\equiv-\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>+\ldots
\end{cases}\end{equation}\noi
Expanding their coefficients, we obtain the leading charged-Higgs contributions to the Higgs-potential parameters. Note that the 
coefficients multiplying $\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>$ are in general very complicated. Here, for simplicity, we give only the leading term in 
$\sin{2\beta}\to0$ ($\tan\beta\to\infty$)
{\small\begin{equation}\begin{cases}
\left(\lambda_T\right)^{H^{\pm}}\simeq\frac{\lambda_P^.\lambda_P^Ms^3\sin{2\beta}}{32\pi^2\left(A_{ud}+\lambda_P^Ms\right)}(5A_{ud}+12\lambda_P^Ms)\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\\
\left(m_{H_{u}}^2\right)^{H^{\pm}}\simeq\frac{1}{32\pi^2}\left\{\left[\frac{2(A_{ud}+\lambda_P^M s)}{\sin{2\beta}}\left(\lambda_.s\cos^2\beta+\lambda_3s\sin^2\beta\right)+(\lambda_.+\lambda_3)(M_{EW}^2-\lambda_P^.s^2)\right]\ln\left<\frac{m_{h}^2m_{H}^2}{\Lambda^{4}}\right>\right.\\
\null\hspace{6cm}\left.+\frac{2\lambda_3s(A_{ud}+\lambda_P^Ms)}{\sin{2\beta}}\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\right\}\\
\left(m_{H_{d}}^2\right)^{H^{\pm}}\simeq\frac{1}{32\pi^2}\left\{\left[\frac{2(A_{ud}+\lambda_P^M s)}{\sin{2\beta}}\left(\lambda_.s\sin^2\beta+\lambda_3s\cos^2\beta\right)+(\lambda_.+\lambda_3)(M_{EW}^2-\lambda_P^.s^2)\right]\ln\left<\frac{m_{h}^2m_{H}^2}{\Lambda^{4}}\right>\right.\\
\null\hspace{6cm}\left.+\frac{2\lambda_.s(A_{ud}+\lambda_P^Ms)}{\sin{2\beta}}\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\right\}\\
\left(m_{S}^2\right)^{H^{\pm}}\simeq\frac{1}{32\pi^2}\left\{\left[A_{ud}^2+2\lambda_P^.\left(M_{EW}^2-\lambda_P^.s^2+\frac{s(A_{ud}+\lambda_P^Ms)}{\sin{2\beta}}\right)\right]\ln\left<\frac{m_{h}^2m_{H}^2}{\Lambda^{4}}\right>-\frac{2\lambda_P^.s(A_{ud}+\lambda_P^Ms)}{\sin{2\beta}}\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\right\}\\
\left(m_{12}^2\right)^{H^{\pm}}\simeq-\frac{\lambda_4\lambda_P^.s^2\sin{2\beta}}{16\pi^2\left(A_{ud}+\lambda_P^Ms\right)}(A_{ud}+3\lambda_P^Ms)\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\\
\left(\mu_S^2\right)^{H^{\pm}}\simeq-\frac{3\lambda_P^M\lambda_P^.s^2\sin{2\beta}}{16\pi^2\left(A_{ud}+\lambda_P^Ms\right)}(2A_{ud}+3\lambda_P^Ms)\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\\
\left(A_{ud}\right)^{H^{\pm}}\simeq\frac{\lambda_4A_{ud}}{32\pi^2}\left\{\ln\left<\frac{m_{h}^2m_{H}^2}{\Lambda^{4}}\right>-\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\right\}\\
\left(\tilde{A}_{ud}\right)^{H^{\pm}}\simeq\frac{\lambda_4\lambda_P^.s\sin{2\beta}}{32\pi^2\left(A_{ud}+\lambda_P^Ms\right)}(A_{ud}+6\lambda_P^Ms)\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\\
\left(A_{us}\right)^{H^{\pm}}\simeq\frac{\lambda_.\lambda_P^Ms\sin{2\beta}}{32\pi^2\left(A_{ud}+\lambda_P^Ms\right)}(3A_{ud}+4\lambda_P^Ms)\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\\
\left(A_{ds}\right)^{H^{\pm}}\simeq\frac{\lambda_3\lambda_P^Ms\sin{2\beta}}{32\pi^2\left(A_{ud}+\lambda_P^Ms\right)}(3A_{ud}+4\lambda_P^Ms)\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\\
\left(A_{S}\right)^{H^{\pm}}\simeq\frac{3\lambda_P^MA_{ud}}{32\pi^2}\left\{\ln\left<\frac{m_{h}^2m_{H}^2}{\Lambda^{4}}\right>-\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\right\}\\
\left(\tilde{A}_S\right)^{H^{\pm}}\simeq\frac{9\lambda_P^.\lambda_P^Ms\sin{2\beta}}{16\pi^2\left(A_{ud}+\lambda_P^Ms\right)}(A_{ud}+3\lambda_P^Ms)\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>
\end{cases}\end{equation}
\begin{displaymath}\begin{cases}
\left(\lambda_1\right)^{H^{\pm}}\simeq\frac{1}{32\pi^2}\left\{(\lambda_.^2+\lambda_3^2)\ln\left<\frac{m_{h}^2m_{H}^2}{\Lambda^{4}}\right>+(\lambda_3^2-\lambda_.^2)\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\right\}\\
\left(\lambda_2\right)^{H^{\pm}}\simeq\frac{1}{32\pi^2}\left\{(\lambda_.^2+\lambda_3^2)\ln\left<\frac{m_{h}^2m_{H}^2}{\Lambda^{4}}\right>+(\lambda_.^2-\lambda_3^2)\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\right\}\\
(\lambda_3+\lambda_4)^{H^{\pm}}\simeq\frac{1}{32\pi^2}\left\{(2\lambda_.\lambda_3+\lambda_4^2)\ln\left<\frac{m_{h}^2m_{H}^2}{\Lambda^{4}}\right>-\lambda_4^2\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\right\}\\
\left(\lambda_5\right)^{H^{\pm}}\simeq\frac{\lambda^2_4}{32\pi^2}\sin{2\beta}^2\\
\left(\lambda_6\right)^{H^{\pm}}\simeq\frac{\lambda_4\lambda_3}{32\pi^2}\sin{2\beta}\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\\
\left(\lambda_7\right)^{H^{\pm}}\simeq\frac{\lambda_4\lambda_.}{32\pi^2}\sin{2\beta}\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\\
(\lambda_P^u)^{H^{\pm}}\simeq\frac{1}{32\pi^2}\left\{\lambda_P^.(\lambda_.+\lambda_3)\ln\left<\frac{m_{h}^2m_{H}^2}{\Lambda^{4}}\right>+\lambda_P^.(\lambda_.-\lambda_3)\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\right\}\\
(\lambda_P^d)^{H^{\pm}}\simeq\frac{1}{32\pi^2}\left\{\lambda_P^.(\lambda_.+\lambda_3)\ln\left<\frac{m_{h}^2m_{H}^2}{\Lambda^{4}}\right>+\lambda_P^.(\lambda_3-\lambda_.)\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\right\}\\
(\tilde{\lambda}_P^u)^{H^{\pm}}\simeq-\frac{\lambda_.\lambda_P^M\sin{2\beta}}{32\pi^2\left(A_{ud}+\lambda_P^Ms\right)}(3A_{ud}+\lambda_P^Ms)\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\\
(\tilde{\lambda}_P^d)^{H^{\pm}}\simeq-\frac{\lambda_3\lambda_P^M\sin{2\beta}}{32\pi^2\left(A_{ud}+\lambda_P^Ms\right)}(3A_{ud}+\lambda_P^Ms)\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\\
\left(\lambda_M\right)^{H^{\pm}}\simeq-\frac{\lambda_4\lambda_P^.\sin{2\beta}}{32\pi^2(A_{ud}+\lambda_P^Ms)}(2A_{ud}+3\lambda_P^Ms)\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\\
(\lambda_P^M)^{H^{\pm}}\simeq\frac{\lambda_4\lambda_P^M}{32\pi^2}\left\{\ln\left<\frac{m_{h}^2m_{H}^2}{\Lambda^{4}}\right>-\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\right\}\\
\left(\tilde{\lambda}^P_M\right)^{H^{\pm}}\simeq-\frac{\lambda_4\lambda_P^.}{32\pi^2}\frac{A_{ud}\sin{2\beta}}{(A_{ud}+\lambda_P^Ms)}\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\\
(\kappa^2)^{H^{\pm}}\simeq\frac{1}{32\pi^2}\left\{\left(\lambda_P^{M\,2}+\lambda_P^{.\,2}\right)\ln\left<\frac{m_{h}^2m_{H}^2}{\Lambda^{4}}\right>-\lambda_P^{M\,2}\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\right\}\\
\left(\kappa^2_S\right)^{H^{\pm}}\simeq-\frac{\lambda_P^M\lambda_P^.}{8\pi^2}\frac{A_{ud}\sin{2\beta}}{(A_{ud}+\lambda_P^Ms)}\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\\
\left(\tilde{\kappa}^2_S\right)^{H^{\pm}}\simeq-\frac{\lambda_P^M\lambda_P^.}{8\pi^2}\frac{\sin{2\beta}}{(A_{ud}+\lambda_P^Ms)}(4A_{ud}+3\lambda_P^Ms)\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>
\end{cases}\end{displaymath}}

{\bf Charged-Higgs contributions -- nMSSM:}\newline
The Charged-Higgs boson squared mass-matrix now reads :
{\small\begin{align}
 &({\cal M}_{H^{\pm}}^2)_{11}(S,H_{u,d}^0)=M_{EW}^2+(A_{ud}s-m_{12}^2)\tan^{-1}\beta+\lambda_.|H_u^0|^2+\lambda_3|H_d^0|^2+\lambda_P^.(|S|^2-s^2)\nonumber\\
 &({\cal M}_{H^{\pm}}^2)_{22}(S,H_{u,d}^0)=M_{EW}^2+(A_{ud}s-m_{12}^2)\tan\beta+\lambda_.|H_d^0|^2+\lambda_3|H_u^0|^2+\lambda_P^.(|S|^2-s^2)\\
 &({\cal M}_{H^{\pm}}^2)_{12}(S,H_{u,d}^0)=A_{ud}S-m_{12}^2-\lambda_4(H_u^0H_d^0)^*\nonumber
\end{align}}\noi
Applying the same recipe as in the previous paragraph, we obtain the corrections:
{\small\begin{equation}\begin{cases}
\left(\lambda_T\right)^{H^{\pm}}\simeq-\frac{A_{ud}m_{12}^2}{32\pi^2}\left\{\ln\left<\frac{m_{h}^2m_{H}^2}{\Lambda^{4}}\right>-\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\right\}\\
\left(m_{H_{u}}^2\right)^{H^{\pm}}\simeq\frac{1}{32\pi^2}\left\{\left[\frac{2(A_{ud}s-m_{12}^2)}{\sin{2\beta}}\left(\lambda_.\cos^2\beta+\lambda_3\sin^2\beta\right)+(\lambda_.+\lambda_3)(M_{EW}^2-\lambda_P^.s^2)\right]\ln\left<\frac{m_{h}^2m_{H}^2}{\Lambda^{4}}\right>\right.\\
\null\hspace{6cm}\left.+\frac{2\lambda_3(A_{ud}s-m_{12}^2)}{\sin{2\beta}}\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\right\}\\
\left(m_{H_{d}}^2\right)^{H^{\pm}}\simeq\frac{1}{32\pi^2}\left\{\left[\frac{2(A_{ud}s-m_{12}^2)}{\sin{2\beta}}\left(\lambda_.\sin^2\beta+\lambda_3\cos^2\beta\right)+(\lambda_.+\lambda_3)(M_{EW}^2-\lambda_P^.s^2)\right]\ln\left<\frac{m_{h}^2m_{H}^2}{\Lambda^{4}}\right>\right.\\
\null\hspace{6cm}\left.+\frac{2\lambda_.(A_{ud}s-m_{12}^2)}{\sin{2\beta}}\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\right\}\\
\left(m_{S}^2\right)^{H^{\pm}}\simeq\frac{1}{32\pi^2}\left\{\left[A_{ud}^2+2\lambda_P^.\left(M_{EW}^2-\lambda_P^.s^2+\frac{A_{ud}s-m_{12}^2}{\sin{2\beta}}\right)\right]\ln\left<\frac{m_{h}^2m_{H}^2}{\Lambda^{4}}\right>+\frac{2\lambda_P^.(A_{ud}s-m_{12}^2)}{\sin{2\beta}}\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\right\}\\
\left(m_{12}^2\right)^{H^{\pm}}\simeq-\frac{\lambda_4m_{12}^2}{16\pi^2}\left\{\ln\left<\frac{m_{h}^2m_{H}^2}{\Lambda^{4}}\right>-\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\right\}\\
\left(\mu_S^2\right)^{H^{\pm}}\simeq\frac{A_{ud}^2m_{12}^2\sin^2{2\beta}}{64\pi^2\left(A_{ud}s-m_{12}^2\right)}\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\\
\left(A_{ud}\right)^{H^{\pm}}\simeq\frac{\lambda_4A_{ud}}{32\pi^2}\left\{\ln\left<\frac{m_{h}^2m_{H}^2}{\Lambda^{4}}\right>-\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\right\}\\
\left(\tilde{A}_{ud}\right)^{H^{\pm}}\simeq\frac{\lambda_4\lambda_P^.A_{ud}s^2\sin{2\beta}}{32\pi^2\left(A_{ud}s-m_{12}^2\right)}\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\\
\left(A_{us}\right)^{H^{\pm}}\simeq\frac{\lambda_.A_{ud}m_{12}^2\sin{2\beta}}{32\pi^2\left(A_{ud}s-m_{12}^2\right)}\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\\
\left(A_{ds}\right)^{H^{\pm}}\simeq\frac{\lambda_3A_{ud}m_{12}^2\sin{2\beta}}{32\pi^2\left(A_{ud}s-m_{12}^2\right)}\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\\
\left(A_{S}\right)^{H^{\pm}}\simeq-\frac{3\lambda_P^.A^3_{ud}s^2\sin^3{2\beta}}{64\pi^2(A_{ud}s-m_{12}^2)^2}\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\\
\left(\tilde{A}_S\right)^{H^{\pm}}\simeq-\frac{3\lambda_P^.A_{ud}m_{12}^2\sin{2\beta}}{32\pi^2\left(A_{ud}s-m_{12}^2\right)}\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>
\end{cases}\end{equation}
\begin{displaymath}\begin{cases}
\left(\lambda_1\right)^{H^{\pm}}\simeq\frac{1}{32\pi^2}\left\{(\lambda_.^2+\lambda_3^2)\ln\left<\frac{m_{h}^2m_{H}^2}{\Lambda^{4}}\right>+(\lambda_3^2-\lambda_.^2)\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\right\}\\
\left(\lambda_2\right)^{H^{\pm}}\simeq\frac{1}{32\pi^2}\left\{(\lambda_.^2+\lambda_3^2)\ln\left<\frac{m_{h}^2m_{H}^2}{\Lambda^{4}}\right>+(\lambda_.^2-\lambda_3^2)\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\right\}\\
(\lambda_3+\lambda_4)^{H^{\pm}}\simeq\frac{1}{32\pi^2}\left\{(2\lambda_.\lambda_3+\lambda_4^2)\ln\left<\frac{m_{h}^2m_{H}^2}{\Lambda^{4}}\right>-\lambda_4^2\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\right\}\\
\left(\lambda_5\right)^{H^{\pm}}\simeq\frac{\lambda^2_4}{32\pi^2}\sin{2\beta}^2\\
\left(\lambda_6\right)^{H^{\pm}}\simeq\frac{\lambda_4\lambda_3}{32\pi^2}\sin{2\beta}\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\\
\left(\lambda_7\right)^{H^{\pm}}\simeq\frac{\lambda_4\lambda_.}{32\pi^2}\sin{2\beta}\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\\
(\lambda_P^u)^{H^{\pm}}\simeq\frac{1}{32\pi^2}\left\{\lambda_P^.(\lambda_.+\lambda_3)\ln\left<\frac{m_{h}^2m_{H}^2}{\Lambda^{4}}\right>+\lambda_P^.(\lambda_--\lambda_3)\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\right\}\\
(\lambda_P^d)^{H^{\pm}}\simeq\frac{1}{32\pi^2}\left\{\lambda_P^.(\lambda_.+\lambda_3)\ln\left<\frac{m_{h}^2m_{H}^2}{\Lambda^{4}}\right>+\lambda_P^.(\lambda_3-\lambda_.)\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\right\}\\
(\tilde{\lambda}_P^u)^{H^{\pm}}\simeq\frac{(\lambda_3-\lambda_.)A_{ud}\lambda_P^.s\sin^2{2\beta}}{64\pi^2\left(A_{ud}s-m_{12}^2\right)}\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\\
(\tilde{\lambda}_P^d)^{H^{\pm}}\simeq\frac{(\lambda_.-\lambda_3)A_{ud}\lambda_P^.s\sin^2{2\beta}}{64\pi^2\left(A_{ud}s-m_{12}^2\right)}\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\\
\left(\lambda_M\right)^{H^{\pm}}\simeq-\frac{\lambda_4\lambda_P^.\sin{2\beta}}{32\pi^2(A_{ud}s-m_{12}^2)}(2A_{ud}s-m_{12}^2)\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\\
(\lambda_P^M)^{H^{\pm}}\simeq-\frac{\lambda_4\lambda_P^.sA_{ud}\sin{2\beta}}{32\pi^2(A_{ud}s-m_{12}^2)}\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\\
\left(\tilde{\lambda}^P_M\right)^{H^{\pm}}\simeq\frac{\lambda_4\lambda_P^.sA_{ud}\sin^3{2\beta}}{64\pi^2(A_{ud}s-m_{12}^2)}\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\\
(\kappa^2)^{H^{\pm}}\simeq\frac{1}{32\pi^2}\left\{\lambda_P^{.\,2}\ln\left<\frac{m_{h}^2m_{H}^2}{\Lambda^{4}}\right>-\frac{\lambda_P^.A_{ud}^2\sin{2\beta}}{A_{ud}s-m_{12}^2}\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\right\}\\
\left(\kappa^2_S\right)^{H^{\pm}}\simeq-\frac{A_{ud}^3\lambda_P^.s\sin^5{2\beta}}{64\pi^2(A_{ud}s-m_{12}^2)^2}\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\\
\left(\tilde{\kappa}^2_S\right)^{H^{\pm}}\simeq\frac{\lambda_P^.A_{ud}^2\sin^3{2\beta}}{32\pi^2(A_{ud}s-m_{12}^2)^2}(3A_{ud}s-m_{12}^2)\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>
\end{cases}\end{displaymath}}

{\bf Neutral-Higgs contributions - NMSSM:}\newline
The $6\times6$ symmetric squared mass-matrix is given by its entries, in the base {\small$(h_u^0,h_d^0,h_S^0,a_u^0,a_d^0,a_S^0)$}:
{\small\begin{align}
 ({\cal M}_{H^0}^2)_{11}&=M_{EW}^2+(A_{ud}+\lambda_P^Ms)\frac{s}{\tan\beta}+\lambda_P^.(|S|^2-s^2)+\lambda_.(2|H_u^0|^2+\mbox{Re}(H_u^{0\,2}))+(\lambda_3+\lambda_4)|H_d^0|^2\nonumber\\
 ({\cal M}_{H^0}^2)_{22}&=M_{EW}^2+(A_{ud}+\lambda_P^Ms)s\tan\beta+\lambda_P^.(|S|^2-s^2)+\lambda_.(2|H_d^0|^2+\mbox{Re}(H_d^{0\,2}))+(\lambda_3+\lambda_4)|H_u^0|^2\nonumber\\
 ({\cal M}_{H^0}^2)_{12}&=-A_{ud}\mbox{Re}(S)-\lambda_P^M\mbox{Re}(S^2)+2(\lambda_3+\lambda_4)\mbox{Re}(H_u^0)\mbox{Re}(H_d^0)\nonumber\\
 ({\cal M}_{H^0}^2)_{33}&=(A_S+4\kappa^2s)s+2A_S(\mbox{Re}(S)-s)+2\kappa^2(2|S|^2+\mbox{Re}(S^2)-3s^2)+\lambda_P^.(|H_u^0|^2+|H_d^0|^2)-2\lambda_P^M\mbox{Re}(H_u^0H_d^0)\nonumber\\
 ({\cal M}_{H^0}^2)_{13}&=-A_{ud}\mbox{Re}(H_d^0)-2\lambda_P^M\mbox{Re}(S^*H_d^0)+2\lambda_P^.\mbox{Re}(S)\mbox{Re}(H_u^0)\nonumber\\
 ({\cal M}_{H^0}^2)_{23}&=-A_{ud}\mbox{Re}(H_u^0)-2\lambda_P^M\mbox{Re}(S^*H_u^0)+2\lambda_P^.\mbox{Re}(S)\mbox{Re}(H_d^0)\nonumber\\
 ({\cal M}_{H^0}^2)_{44}&=M_{EW}^2+(A_{ud}+\lambda_P^Ms)\frac{s}{\tan\beta}+\lambda_P^.(|S|^2-s^2)+\lambda_.(2|H_u^0|^2-\mbox{Re}(H_u^{0\,2}))+(\lambda_3+\lambda_4)|H_d^0|^2\nonumber\\
 ({\cal M}_{H^0}^2)_{55}&=M_{EW}^2+(A_{ud}+\lambda_P^Ms)s\tan\beta+\lambda_P^.(|S|^2-s^2)+\lambda_.(2|H_d^0|^2-\mbox{Re}(H_d^{0\,2}))+(\lambda_3+\lambda_4)|H_u^0|^2\nonumber\\
 ({\cal M}_{H^0}^2)_{45}&=A_{ud}\mbox{Re}(S)+\lambda_P^M\mbox{Re}(S^2)+2(\lambda_3+\lambda_4)\mbox{Im}(H_u^0)\mbox{Im}(H_d^0)\nonumber\\
 ({\cal M}_{H^0}^2)_{66}&=-3A_Ss-2A_S(\mbox{Re}(S)-s)+2\kappa^2(2|S|^2-\mbox{Re}(S^2)-s^2)+\lambda_P^.(|H_u^0|^2+|H_d^0|^2)+2\lambda_P^M\mbox{Re}(H_u^0H_d^0)\nonumber\\
 ({\cal M}_{H^0}^2)_{46}&=A_{ud}\mbox{Re}(H_d^0)-2\lambda_P^M\mbox{Re}(S^*H_d^0)+2\lambda_P^.\mbox{Im}(S)\mbox{Im}(H_u^0)\nonumber\\
 ({\cal M}_{H^0}^2)_{56}&=A_{ud}\mbox{Re}(H_u^0)-2\lambda_P^M\mbox{Re}(S^*H_u^0)+2\lambda_P^.\mbox{Im}(S)\mbox{Im}(H_d^0)\nonumber\\
 ({\cal M}_{H^0}^2)_{14}&=\lambda_.\mbox{Im}(H_u^{0\,2})\\
 ({\cal M}_{H^0}^2)_{15}&=A_{ud}\mbox{Im}(S)-\lambda_P^M\mbox{Im}(S^2)+2(\lambda_3+\lambda_4)\mbox{Re}(H_u^0)\mbox{Im}(H_d^0)\nonumber\\
 ({\cal M}_{H^0}^2)_{16}&=A_{ud}\mbox{Im}(H_d^0)+2\lambda_P^M\mbox{Im}(SH_d^{0\,*})+2\lambda_P^.\mbox{Im}(S)\mbox{Re}(H_u^0)\nonumber\\
 ({\cal M}_{H^0}^2)_{24}&=A_{ud}\mbox{Im}(S)-\lambda_P^M\mbox{Im}(S^2)+2(\lambda_3+\lambda_4)\mbox{Im}(H_u^0)\mbox{Re}(H_d^0)\nonumber\\
 ({\cal M}_{H^0}^2)_{25}&=\lambda_.\mbox{Im}(H_d^{0\,2})\nonumber\\
 ({\cal M}_{H^0}^2)_{26}&=A_{ud}\mbox{Im}(H_u^0)+2\lambda_P^M\mbox{Im}(SH_u^{0\,*})+2\lambda_P^.\mbox{Im}(S)\mbox{Re}(H_d^0)\nonumber\\
 ({\cal M}_{H^0}^2)_{34}&=A_{ud}\mbox{Im}(H_d^0)+2\lambda_P^M\mbox{Im}(S^*H_d^{0})+2\lambda_P^.\mbox{Re}(S)\mbox{Im}(H_u^0)\nonumber\\
 ({\cal M}_{H^0}^2)_{35}&=A_{ud}\mbox{Im}(H_u^0)+2\lambda_P^M\mbox{Im}(S^*H_u^{0})+2\lambda_P^.\mbox{Re}(S)\mbox{Im}(H_d^0)\nonumber\\
 ({\cal M}_{H^0}^2)_{36}&=-2A_S\mbox{Im}(S)+2\kappa^2\mbox{Im}(S^2)-2\lambda_P^M\mbox{Im}(H_u^0H_d^0)\nonumber
\end{align}}\noi
One can expand its eigenvalues in terms of doublet fields:
{\small\begin{equation}
 m_i^2(S,H_u^0,H_d^0)=m_i^{2\,(0)}(S)+m_i^{2\,(1)}(S,H_u^0,H_d^0)+m_i^{2\,(2)}(S,H_u^0,H_d^0)+O\left((H^0_{u,d})^5\right)
\end{equation}\noi}
The associated potential is then given by:
{\small\begin{multline}\label{RHpot}
 \delta{\cal V}_{\mbox{\small eff}}^{\Lambda,H^{0}}(H_u^0,H_d^0,S)=\frac{1}{64\pi^2}\sum_im_i^4(S,H_u^0,H_d^0)\left[\ln\left(\frac{m_i^2(S,H_u^0,H_d^0)}{\Lambda^2}\right)-\frac{3}{2}\right]\\
=\frac{1}{64\pi^2}\sum_i\left\{m_i^{4\,(0)}(S)\left[\ln\left(\frac{m_i^{2\,(0)}(S)}{\Lambda^2}\right)-\frac{3}{2}\right]+2m_i^{2\,(0)}(S)m_i^{2\,(1)}(S,H_u^0,H_d^0)\left[\ln\left(\frac{m_i^{2\,(0)}(S)}{\Lambda^2}\right)-1\right]\right.\\\left.+\left(2m_i^{2\,(0)}(S)m_i^{2\,(2)}(S,H_u^0,H_d^0)\left[\ln\left(\frac{m_i^{2\,(0)}(S)}{\Lambda^2}\right)-1\right]+m_i^{4\,(1)}(S,H_u^0,H_d^0)\ln\left(\frac{m_i^{2\,(0)}(S)}{\Lambda^2}\right)\right)+O(H^5_{u,d})\right\}
\end{multline}}\noi
The large logarithms are then those terms multiplying $\ln\left(\frac{m_i^{2\,(0)}(S)}{\Lambda^2}\right)\simeq\ln\left(\frac{m_i^{2\,(0)}(s)}{\Lambda^2}\right)+\ldots$
Thence consists our primary task in diagonalizing ${\cal M}_{H^0}^2$ perturbatively with respect to the doublet fields.

\noi We first consider ${\cal M}_{H^0}^2(S,H_{u,d}^0=0)$ in order to obtain $m_i^{2\,(0)}(S)$. We denote as $\left|E_i\right>\equiv(\delta_{ij})_{j=1,\ldots,6}$
the elements of the canonical base of $\mathbb{R}^6$. The subspaces $\mbox{Vec}\{\left|E_1\right>,\left|E_2\right>,\left|E_4\right>,\left|E_5\right>\}$
and $\mbox{Vec}\{\left|E_3\right>,\left|E_6\right>\}$ obviously decouple in ${\cal M}_{H^0}^2(S,H_{u,d}^0=0)$. In the doublet sector, one notices that the eigenstate
equation $\left({\cal M}_{H^0}^2(S,H_{u,d}^0=0)-m^2\right)\sum_{i=1,2,4,5}x_i\left|E_i\right>=0$ is equivalent to $\left(\tilde{\cal M}^2-m^2\right)(z_1,z_2)^T=0$,
where $z_1\equiv x_1+\imath x_3$, $z_2=x_2-\imath x_4$ and $\tilde{\cal M}^2$ is the $2\times2$ (complex) hermitian matrix
determined by the following entries:
\begin{align}
 \tilde{\cal M}^2_{11}&=M_{EW}^2+(A_{ud}+\lambda_P^Ms)\frac{s}{\tan\beta}+\lambda_P^.(|S|^2-s^2)\nonumber\\
 \tilde{\cal M}^2_{22}&=M_{EW}^2+(A_{ud}+\lambda_P^Ms)s\tan\beta+\lambda_P^.(|S|^2-s^2)\\
 \tilde{\cal M}^2_{12}&=-A_{ud}S^*-\lambda_P^MS^{2}\nonumber
\end{align}\noi
One recognises ${\cal M}_{H^{\pm}}^2(S,H_{u,d}^0=0)$, up to the sign of the off-diagonal terms. $\tilde{\cal M}^2$ is diagonalized by the eigenstates:
{\small\begin{multline}
 m_{h^0/H^0}^2=M_{EW}^2+\frac{(A_{ud}+\lambda_P^Ms)s}{\sin{2\beta}}+\lambda_P^{.}(|S|^2-s^2)-/+\left[\left(\frac{(A_{ud}+\lambda_P^Ms)s}{\sin{2\beta}}\right)^2+\left(\left|A_{ud}S^*+\lambda_P^MS^2\right|^2-(A_{ud}+\lambda_P^Ms)^2s^2\right)\right]^{1/2}\\
(z_1)_{h^0}=\frac{m_{H^0}^2-\tilde{\cal M}^2_{11}}{\sqrt{(m_{H^0}^2-\tilde{\cal M}^2_{11})^2+|\tilde{\cal M}^2_{12}|^2}}\equiv x_D\ \ \ ;\ \ \ (z_2)_{h^0}=\frac{-\tilde{\cal M}^{2\,*}_{12}}{\sqrt{(m_{H^0}^2-\tilde{\cal M}^2_{11})^2+|\tilde{\cal M}^2_{12}|^2}}\equiv -y^*_D\\
(z_1)_{H^0}=\frac{\tilde{\cal M}^{2}_{12}}{\sqrt{(m_{H^0}^2-\tilde{\cal M}^2_{11})^2+|\tilde{\cal M}^2_{12}|^2}}\equiv y_D\ \ \ ;\ \ \ (z_2)_{H^0}=\frac{m_{H^0}^2-\tilde{\cal M}^2_{11}}{\sqrt{(m_{H^0}^2-\tilde{\cal M}^2_{11})^2+|\tilde{\cal M}^2_{12}|^2}}\equiv x_D\hspace{0.4cm}\null
\end{multline}}\noi
The following relations will proove useful later:
\begin{equation}\label{relD}
 x_D^2+|y_D|^2=1\ \ \ ;\ \ \ x_D^2-|y_D|^2=\frac{2}{\tan{2\beta}}\frac{(A_{ud}+\lambda_P^Ms)s}{m_{h^0}^2(S)-m_{H^0}^2(S)}\ \ \ ;\ \ \ x_Dy_D=\frac{A_{ud}S^*+\lambda_P^MS^2}{m_{h^0}^2(S)-m_{H^0}^2(S)}
\end{equation}\noi
$Z_1\equiv\left((z_1)_{h^0},(z_2)_{h^0}\right)^T$ and $Z_2\equiv\left((z_1)_{H^0},(z_2)_{H^0}\right)^T$ are eigenvectors of $\tilde{\cal M}^2$ in
the complex sense. In the real sense, $\imath Z_1$ and $\imath Z_2$ form two other linearly-independant (and degenerate to $Z_1$, $Z_2$)
eigenstates. We thus obtain the doublet eigenstates of ${\cal M}_{H^0}^2(S,H_{u,d}^0=0)$:
{\small\begin{align}
 m_{h^0}^2\ \ \ ;&\ \ \begin{cases}\left|h^0_1\right>=x_D\left|E_1\right>-\mbox{Re}(y_D)\left|E_2\right>-\mbox{Im}(y_D)\left|E_4\right>\\ \left|h^0_2\right>=-\mbox{Im}(y_D)\left|E_2\right>+x_D\left|E_3\right>+\mbox{Re}(y_D)\left|E_4\right> \end{cases}\\
 \nonumber m_{H^0}^2\ \ \ ;&\ \ \begin{cases}\left|H^0_1\right>=\mbox{Re}(y_D)\left|E_1\right>+x_D\left|E_2\right>+\mbox{Im}(y_D)\left|E_3\right>\\ \left|H^0_2\right>=-\mbox{Im}(y_D)\left|E_1\right>+\mbox{Re}(y_D)\left|E_3\right>-x_D\left|E_4\right> \end{cases}
\end{align}}

\noi For the remaining singlet-states, ${\cal M}_{H^0}^2(S,H_{u,d}^0=0)$ is diagonalized by:
{\small\begin{align}
& m_{h_S^0/a_S^0}=(-A_S+2\kappa^2s)s+4\kappa^2(|S|^2-s^2)+/-2\left[|A_SS+\kappa^2S^{*\,2}|^2\right]^{1/2}\ \ \ ;\ \ \ \begin{cases}
\left|h_S^0\right>=x_S\left|E_3\right>-y_S\left|E_6\right>\\\left|a_S^0\right>=y_S\left|E_3\right>+x_S\left|E_6\right>\end{cases}\\
\nonumber&x_S\equiv\frac{{\cal M}_{H^0}^2(S,0)_{66}-m^2_{h_S^0}}{\sqrt{\left({\cal M}_{H^0}^2(S,0)_{66}-m^2_{h_S^0}\right)^2+|{\cal M}_{H^0}^2(S,0)_{36}|^2}}\ \ \ ;\ \ \ y_S\equiv\frac{{\cal M}_{H^0}^2(S,0)_{36}}{\sqrt{\left({\cal M}_{H^0}^2(S,0)_{66}-m^2_{h_S^0}\right)^2+|{\cal M}_{H^0}^2(S,0)_{36}|^2}}
\end{align}}\noi
It is convenient to introduce the notation $z_S=x_S+\imath y_S$ and note the following relations:
\begin{equation}\label{relS}
 |z_S|^2=1\ \ \ \ \ \ ;\ \ \ \ \ \ z_S^2=4\frac{A_SS+\kappa^2S^{*\,2}}{m^2_{h_S^0}(S)-m^2_{a_S^0}(S)}
\end{equation}

\noi At this stage, one can already determine the pure-singlet parameters of the potential. Similarly to the charged case (eq. \ref{CHpot}),
one can formulate the first term in eq. \ref{RHpot} in terms of $T_D=\frac{1}{2}\left(m_{H^0}^2+m_{h^0}^2\right)$, $R_D=\frac{1}{2}\left(m_{H^0}^2-m_{h^0}^2\right)$,
$T_S=m_{h^0_S}^2+m_{a^0_S}^2$ and $R_S=m_{h^0_S}^2-m_{a^0_S}^2$. Moreover, the contributions from the doublet are trivially 
identical to those in the charged case and here, as well, we give only the leading term in $\sin{2\beta}\to0$ for the coefficient multiplying 
$\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>$. The logarithmicaly-enhanced parameters are then:
{\small\begin{equation}\begin{cases}
\left(\lambda_T\right)^{H^{0}}\simeq\frac{\lambda_P^.\lambda_P^Ms^3\sin{2\beta}}{32\pi^2\left(A_{ud}+\lambda_P^Ms\right)}(5A_{ud}+12\lambda_P^Ms)\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>-\frac{A_Ss(A_S+2\kappa^2s)}{128\pi^2}\ln\left<\frac{m_{h_S^0}^2}{m_{a_S^0}^2}\right>\\
\left(m_{S}^2\right)^{H^{0}}\simeq\frac{1}{32\pi^2}\left\{\left[A_{ud}^2+2\lambda_P^.\left(M_{EW}^2-\lambda_P^.s^2+\frac{s(A_{ud}+\lambda_P^Ms)}{\sin{2\beta}}\right)\right]\ln\left<\frac{m_{h}^2m_{H}^2}{\Lambda^{4}}\right>-\frac{2\lambda_P^.s(A_{ud}+\lambda_P^Ms)}{\sin{2\beta}}\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\right.\\
\null\hspace{1cm}+\left.\frac{1}{2}\left[A_S^2-2 A_S\kappa^2s-4 \kappa^4 s^2\right]\ln\left<\frac{m_{h_S^0}^2m_{a_S^0}^2}{\Lambda^4}\right>-\frac{3A_S^5+10A_S^4\kappa^2s+2A_S^3\kappa^4s^2-60A_S^2\kappa^6s^3-20A_S\kappa^8s^4+8\kappa^{10}s^5}{8\left(A_S+\kappa^2s\right)^3}\ln\left<\frac{m_{h_S^0}^2}{m_{a_S^0}^2}\right>\right\}\\
\left(\mu_S^2\right)^{H^{0}}\simeq-\frac{3\lambda_P^M\lambda_P^.s^2\sin{2\beta}}{16\pi^2\left(A_{ud}+\lambda_P^Ms\right)}(2A_{ud}+3\lambda_P^Ms)\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>+\frac{3A_S}{256\pi^2}\frac{A_S^4+2A_S^3\kappa^2s-6A_S^2\kappa^2s^2-32A_S\kappa^3s^3-16\kappa^4s^4}{(A_S+\kappa^2s)^3}\ln\left<\frac{m_{h_S^0}^2}{m_{a_S^0}^2}\right>\\
\left(A_{S}\right)^{H^{0}}\simeq\frac{3\lambda_P^MA_{ud}}{32\pi^2}\left\{\ln\left<\frac{m_{h}^2m_{H}^2}{\Lambda^{4}}\right>-\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\right\}\\
\null\hspace{1cm}+\frac{3\kappa^2A_S}{64\pi^2}\left\{\ln\left<\frac{m_{h_S^0}^2m_{a_S^0}^2}{\Lambda^4}\right>-\frac{A_S}{16}\frac{3A_S^4+19A_S^3\kappa^2s+30A_S^2\kappa^4s^2-120A_S\kappa^6s^3-80\kappa^8s^4}{\kappa^2s(A_S+\kappa^2s)^3}\ln\left<\frac{m_{h_S^0}^2}{m_{a_S^0}^2}\right>\right\}\\
\left(\tilde{A}_S\right)^{H^{0}}\simeq\frac{9\lambda_P^.\lambda_P^Ms\sin{2\beta}}{16\pi^2\left(A_{ud}+\lambda_P^Ms\right)}(A_{ud}+3\lambda_P^Ms)\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>+\frac{9\kappa^2A_S}{1024\pi^2}\frac{A_S^4+17A_S^3\kappa^2s+42A_S^2\kappa^4s^2-8A_S\kappa^6s^3-16\kappa^8s^4}{\kappa^2s(A_S+\kappa^2s)^3}\ln\left<\frac{m_{h_S^0}^2}{m_{a_S^0}^2}\right>\\
(\kappa^2)^{H^{0}}\simeq\frac{1}{32\pi^2}\left\{\left(\lambda_P^{M\,2}+\lambda_P^{.\,2}\right)\ln\left<\frac{m_{h}^2m_{H}^2}{\Lambda^{4}}\right>-\lambda_P^{M\,2}\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\right\}\\
\null\hspace{1cm}+\frac{\kappa^4}{64\pi^2}\left\{5\ln\left<\frac{m_{h_S^0}^2m_{a_S^0}^2}{\Lambda^4}\right>-\frac{1}{64}\frac{A_S^5+14A_S^4\kappa^2s+32A_S^3\kappa^4s^2+256A_S^2\kappa^6s^3-640A_S\kappa^8s^4-256\kappa^{10}s^5}{\kappa^4s^2(A_S+\kappa^2s)^3}\ln\left<\frac{m_{h_S^0}^2}{m_{a_S^0}^2}\right>\right\}\\
\left(\kappa^2_S\right)^{H^{0}}\simeq-\frac{\lambda_P^M\lambda_P^.}{8\pi^2}\frac{A_{ud}\sin{2\beta}}{(A_{ud}+\lambda_P^Ms)}\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>+\frac{\kappa^4A_S}{2048\pi^2}\frac{5A_S^4+22A_S^3\kappa^2s+144A_S^2\kappa^4s^2-352A_S\kappa^6s^3-128\kappa^8s^4}{\kappa^4s^2(A_S+\kappa^2s)^3}\ln\left<\frac{m_{h_S^0}^2}{m_{a_S^0}^2}\right>\\
\left(\tilde{\kappa}^2_S\right)^{H^{0}}\simeq-\frac{\lambda_P^M\lambda_P^.}{8\pi^2}\frac{\sin{2\beta}}{(A_{ud}+\lambda_P^Ms)}(4A_{ud}+3\lambda_P^Ms)\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>-\frac{\kappa^4A_S}{512\pi^2}\frac{A_S^4+2A_S^3\kappa^2s-36A_S^2\kappa^4s^2-344A_S\kappa^6s^3-64\kappa^8s^4}{\kappa^4s^2(A_S+\kappa^2s)^3}\ln\left<\frac{m_{h_S^0}^2}{m_{a_S^0}^2}\right>
\end{cases}\end{equation}}

\noi The next step consists in considering $O(H^2)$ corrections to the neutral eigenvalues. Note that, for the eigenvalue $m_i^{2\,(0)}(S)$
of ${\cal M}_{H^0}^2(S,H_{u,d}^0=0)$, with $\mbox{Tr}_i$ the trace on the corresponding eigenspace,
\begin{equation}
 \mbox{Tr}_i[m_i^{2\,(2)}(S,H_u^0,H_d^0)]=\mbox{Tr}_i\left\{{\cal M}^{2\,(2)}_{H^0}+\sum_{j\neq i}\frac{{\cal M}^{2\,(1)}_{H^0}P_j{\cal M}^{2\,(1)}_{H^0}}{m_i^{2\,(0)}(S)-m_j^{2\,(0)}(S)}\right\}
\end{equation}\noi
where ${\cal M}^{2\,(1,2)}_{H^0}$ stand for the matrices with terms of $O(H^{1,2})$ in ${\cal M}^{2}_{H^0}$, $P_j$ corresponds to 
the projector on the eigenspace of the eigenvalue $m_j^{2\,(0)}$. Defining $\eta=1;-1$ for $D=H^0;h^0$ and $\epsilon=1;-1$ for 
$S=h_S^0;a_S^0$ and using the relations \ref{relD}, \ref{relS}, we obtain the following matrix elements:
{\small\begin{align}
&\mbox{Tr}_D\left<D\right|{\cal M}^{2\,(2)}_{H^0}\left|D\right>\equiv\left<D_S\right>+\frac{\eta}{m_{H^0}^2-m_{h^0}^2}\left<D_A\right>\nonumber\\
&\null\hspace{0.5cm}\left<D_S\right>=\left[2\lambda_.+\lambda_3+\lambda_4\right](|H_u^0|^2+|H_d^0|^2)\nonumber\\
&\null\hspace{0.5cm}\left<D_A\right>=\frac{2(A_{ud}+\lambda_P^Ms)s}{\tan{2\beta}}\left[2\lambda_.-\lambda_3-\lambda_4\right](|H_u^0|^2-|H_d^0|^2)-4(\lambda_3+\lambda_4)\left[A_{ud}\mbox{Re}(SH_u^0H_d^0)+\lambda_P^M\mbox{Re}(S^{*\,2}H_u^0H_d^0)\right]\nonumber\\
&\left<S\right|{\cal M}^{2\,(2)}_{H^0}\left|S\right>\equiv\left<S_S\right>+\frac{\epsilon}{m_{h_S^0}^2-m_{a_S^0}^2}\left<S_A\right>\nonumber\\
&\null\hspace{0.5cm}\left<S_S\right>=\lambda_P^.(|H_u^0|^2+|H_d^0|^2)\nonumber\\
&\null\hspace{0.5cm}\left<S_S\right>=-8\lambda_P^M\left[A_S\mbox{Re}(SH_u^0H_d^0)+\kappa^2\mbox{Re}(S^{*\,2}H_u^0H_d^0)\right]\nonumber\\
&\left<S\right|{\cal M}^{2\,(1)}_{H^0}P_D{\cal M}^{2\,(1)}_{H^0}\left|S\right>\equiv\left<1;1\right>+\frac{\eta}{m_{H^0}^2-m_{h^0}^2}\left<-1;1\right>+\frac{\epsilon}{m_{h_S^0}^2-m_{a_S^0}^2}\left<1;-1\right>+\frac{\eta}{m_{H^0}^2-m_{h^0}^2}\frac{\epsilon}{m_{h_S^0}^2-m_{a_S^0}^2}\left<-1;-1\right>\nonumber\\
&\null\hspace{0.5cm}\left<1;1\right>=\frac{1}{2}\left\{A_{ud}^2+4\lambda_P^{M\,2}|S|^2+2\lambda_P^{.\,2}|S|^2\right\}(|H_u^0|^2+|H_d^0|^2)-2\lambda_P^.\left[A_{ud}\mbox{Re}(SH_u^0H_d^0)+2\lambda_P^M\mbox{Re}(S^{*\,2}H_u^0H_d^0)\right]\nonumber\\
&\null\hspace{0.5cm}\left<-1;1\right>=2\lambda_P^.\left[A_{ud}^2|S|^2+2\lambda_P^{M\,2}|S|^4+3\lambda_P^MA_{ud}\mbox{Re}(S^3)\right](|H_u^0|^2+|H_d^0|^2)\\
&\null\hspace{1cm}-\frac{(A_{ud}+\lambda_P^Ms)s}{\tan{2\beta}}\left[A_{ud}^2+4\lambda_P^{M\,2}|S|^2-2\lambda_P^{.\,2}|S|^2\right](|H_u^0|^2-|H_d^0|^2)\nonumber\\
&\null\hspace{1cm}-4\lambda_P^{.\,2}|S|^2\left[A_{ud}\mbox{Re}(SH_u^0H_d^0)+\lambda_P^M\mbox{Re}(S^{*\,2}H_u^0H_d^0)\right]-8\lambda_P^MA_{ud}\left[\lambda_P^M|S|^2\mbox{Re}(SH_u^0H_d^0)+A_{ud}\mbox{Re}(S^{*\,2}H_u^0H_d^0)\right]\nonumber\\
&\null\hspace{0.5cm}\left<1;-1\right>=\left[8\lambda_P^MA_{ud}(A_S|S|^2+\kappa^2\mbox{Re}(S^3))+4\lambda_P^{.\,2}\left(A_S\mbox{Re}(S^3)+\kappa^2|S|^4\right)\right](|H_u^0|^2+|H_d^0|^2)\nonumber\\
&\null\hspace{2cm}-8\lambda_P^.\left[(\kappa^2A_{ud}+2\lambda_P^MA_S)|S|^2\mbox{Re}(SH_u^0H_d^0)+(A_SA_{ud}+2\lambda_P^M\kappa^2|S|^2)\mbox{Re}(S^{*\,2}H_u^0H_d^0)\right]\nonumber\\
&\null\hspace{0.5cm}\left<-1;-1\right>=8\lambda_P^.\left[A_{ud}(\kappa^2A_{ud}+3\lambda_P^MA_S)|S|^4+2\lambda_P^{M\,2}\kappa^2|S|^6\right.\nonumber\\
&\null\hspace{5cm}\left.+(A_SA_{ud}^2+3\lambda_P^MA_{ud}\kappa^2|S|^2+2\lambda_P^{M\,2}A_S|S|^2)\mbox{Re}(S^3)\right](|H_u^0|^2+|H_d^0|^2)\nonumber\\
&\null\hspace{1cm}-\frac{8(A_{ud}+\lambda_P^Ms)s}{\tan{2\beta}}\left[2\lambda_P^MA_{ud}\left(A_S|S|^2+\kappa^2\mbox{Re}(S^3)\right)-\lambda_P^{.\,2}\left(A_S\mbox{Re}(S^3)+\kappa^2|S|^4\right)\right](|H_u^0|^2-|H_d^0|^2)\nonumber\\
&\null\hspace{1cm}-16\lambda_P^{.\,2}\left(A_S\mbox{Re}(S^3)+\kappa^2|S|^4\right)\left[A_{ud}\mbox{Re}(SH_u^0H_d^0)+\lambda_P^M\mbox{Re}(S^{*\,2}H_u^0H_d^0)\right]\nonumber\\
&\null\hspace{1cm}-8\left[\left(\lambda_P^MA_S(A_{ud}^2+4\lambda_P^{M\,2}|S|^2)+\kappa^2A_{ud}^3\right)|S|^2\mbox{Re}(SH_u^0H_d^0)\right.\nonumber\\
&\null\hspace{3cm}\left.+A_{ud}\left(A_S(A_{ud}^2+4\lambda_P^{M\,2}|S|^2)+4\lambda_P^{M\,3}\kappa^2|S|^4\right)\mbox{Re}(S^{*\,2}H_u^0H_d^0)\right.\nonumber\\
&\null\hspace{5cm}\left.+\lambda_P^MA_{ud}^2\kappa^2\mbox{Re}(S^{4}H_u^0H_d^0)+4\lambda_P^{M\,2}A_{ud}\kappa^2\mbox{Re}(S^{*\,5}H_u^0H_d^0)\right]\nonumber
\end{align}}

\noi Getting back to \ref{RHpot}, one obtains the $O(H^2)$ coefficients:
{\small\begin{equation}
 \null\hspace{0cm}\begin{cases}
\left(m_{H_{u}}^2\right)^{H^{0}}\simeq\frac{1}{64\pi^2}\left\{\left[A_{ud}^2+4\lambda_.\left(M_{EW}^2+\lambda_P^.s^2+2\cos^2\beta\frac{(A_{ud}+\lambda_P^Ms)s}{\sin{2\beta}}\right)\right.\right.\\
\null\hspace{5cm}\left.\left.+2(\lambda_3+\lambda_4)\left(M_{EW}^2-\lambda_P^.s^2+2\sin^2\beta\frac{(A_{ud}+\lambda_P^Ms)s}{\sin{2\beta}}\right)\right]\ln\left<\frac{m_{h}^2m_{H}^2}{\Lambda^{4}}\right>\right.\\
\null\hspace{2cm}\left.+\left[A_{ud}^2-2\lambda_P^.s(A_S+2\kappa^2s)\right]\ln\left<\frac{m_{h_S^0}^2m_{a_S^0}^2}{\Lambda^4}\right>+\ldots\right\}\\
\left(m_{H_{d}}^2\right)^{H^{0}}\simeq\frac{1}{64\pi^2}\left\{\left[A_{ud}^2+4\lambda_.\left(M_{EW}^2+\lambda_P^.s^2+2\sin^2\beta\frac{(A_{ud}+\lambda_P^Ms)s}{\sin{2\beta}}\right)\right.\right.\\
\null\hspace{5cm}\left.\left.+2(\lambda_3+\lambda_4)\left(M_{EW}^2-\lambda_P^.s^2+2\cos^2\beta\frac{(A_{ud}+\lambda_P^Ms)s}{\sin{2\beta}}\right)\right]\ln\left<\frac{m_{h}^2m_{H}^2}{\Lambda^{4}}\right>\right.\\
\null\hspace{2cm}\left.+\left[A_{ud}^2-2\lambda_P^.s(A_S+2\kappa^2s)\right]\ln\left<\frac{m_{h_S^0}^2m_{a_S^0}^2}{\Lambda^4}\right>+\ldots\right\}\\
\left(A_{ud}\right)^{H^{0}}\simeq\frac{1}{32\pi^2}\left\{(\lambda_3+\lambda_4+\lambda_P^.)A_{ud}\ln\left<\frac{m_{h}^2m_{H}^2}{\Lambda^{4}}\right>+(\lambda_P^.A_{ud}+2\lambda_P^MA_S)\ln\left<\frac{m_{h_S^0}^2m_{a_S^0}^2}{\Lambda^4}\right>+\ldots\right\}\\
(\lambda_P^u)^{H^{0}}\simeq\frac{1}{32\pi^2}\left\{\left(\lambda_P^.(2\lambda_.\lambda_3+\lambda_4+\lambda_P^.)+\lambda_P^{M\,2}\right)\ln\left<\frac{m_{h}^2m_{H}^2}{\Lambda^{4}}\right>+2\left(\lambda_P^.(4\kappa^2+\lambda_P^.)+2\lambda_P^{M\,2}\right)\ln\left<\frac{m_{h_S^0}^2m_{a_S^0}^2}{\Lambda^4}\right>+\ldots\right\}\\
(\lambda_P^d)^{H^{0}}\simeq\frac{1}{32\pi^2}\left\{\left(\lambda_P^.(2\lambda_.\lambda_3+\lambda_4+\lambda_P^.)+\lambda_P^{M\,2}\right)\ln\left<\frac{m_{h}^2m_{H}^2}{\Lambda^{4}}\right>+2\left(\lambda_P^.(4\kappa^2+\lambda_P^.)+2\lambda_P^{M\,2}\right)\ln\left<\frac{m_{h_S^0}^2m_{a_S^0}^2}{\Lambda^4}\right>+\ldots\right\}\\
(\lambda_P^M)^{H^{0}}\simeq\frac{1}{32\pi^2}\left\{\lambda_P^M(\lambda_3+\lambda_4+2\lambda_P^.)\ln\left<\frac{m_{h}^2m_{H}^2}{\Lambda^{4}}\right>+2\lambda_P^{M}(\kappa^2+\lambda_P^.)\ln\left<\frac{m_{h_S^0}^2m_{a_S^0}^2}{\Lambda^4}\right>+\ldots\right\}\\
 \end{cases}
\end{equation}}\noi
The logarithms involving ratios of Higgs masses (symbolized by \ldots) are too complicated to write down explicitely. They also appear 
within contributions to the $\mathbb{Z}_3$-violating parameters. We could check however that such contributions to 
$\mathbb{Z}_3$-violating parameters vanished in relevant limits ($m_{h_S^0,a_S^0}^2\ll m_{H^0}^2\ \bigoplus\ \sin{2\beta}\to0$,
$m_{H^0}^2\ll m_{h_S^0,a_S^0}^2$, $m_{a_S^0}^2\to0$).

\noi We skip the computation of corrections to the quartic doublet parameters, here as well as for the neutralino 
contributions: such a task, although straightforward (perturbative calculation of the eigenvalues of a matrix up 
to the fourth order) promises to be technically tedious.

{\bf Neutral-Higgs contributions - nMSSM:}\newline
We can draw some conclusions from the study in the NMSSM case. For simplicity, we will confine here to contributions to pure 
singlet parameters, so that we need only consider ${\cal M}_{H^0}^2(S,H_{u,d}^0=0)$. Obviously, doublet and singlet sectors 
will decouple again, and the doublet sector will generate the same corrections to the $\l_i$'s as the charged Higgs sector. As
for the singlet sector, it is particularly simple in the nMSSM since no dependence in $S$ appears. We thus obtain the 
leading-logarithms to the pure-singlet coefficients:
{\small\begin{equation}\begin{cases}
\left(\lambda_T\right)^{H^{0}}\simeq-\frac{A_{ud}m_{12}^2}{32\pi^2}\left\{\ln\left<\frac{m_{h}^2m_{H}^2}{\Lambda^{4}}\right>-\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\right\}\\
\left(m_{S}^2\right)^{H^{0}}\simeq\frac{1}{32\pi^2}\left\{\left[A_{ud}^2+2\lambda_P^.\left(M_{EW}^2-\lambda_P^.s^2+\frac{A_{ud}s-m_{12}^2}{\sin{2\beta}}\right)\right]\ln\left<\frac{m_{h}^2m_{H}^2}{\Lambda^{4}}\right>+\frac{2\lambda_P^.(A_{ud}s-m_{12}^2)}{\sin{2\beta}}\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\right\}\\
\left(\mu_S^2\right)^{H^{0}}\simeq\frac{A_{ud}^2m_{12}^2\sin^2{2\beta}}{64\pi^2\left(A_{ud}s-m_{12}^2\right)}\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\\
\left(A_{S}\right)^{H^{0}}\simeq-\frac{3\lambda_P^.A^3_{ud}s^2\sin^3{2\beta}}{64\pi^2(A_{ud}s-m_{12}^2)^2}\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\\
\left(\tilde{A}_S\right)^{H^{0}}\simeq-\frac{3\lambda_P^.A_{ud}m_{12}^2\sin{2\beta}}{32\pi^2\left(A_{ud}s-m_{12}^2\right)}\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\\
(\kappa^2)^{H^{0}}\simeq\frac{1}{32\pi^2}\left\{\lambda_P^{.\,2}\ln\left<\frac{m_{h}^2m_{H}^2}{\Lambda^{4}}\right>-\frac{\lambda_P^.A_{ud}^2\sin{2\beta}}{A_{ud}s-m_{12}^2}\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\right\}\\
\left(\kappa^2_S\right)^{H^{0}}\simeq-\frac{A_{ud}^3\lambda_P^.s\sin^5{2\beta}}{64\pi^2(A_{ud}s-m_{12}^2)^2}\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>\\
\left(\tilde{\kappa}^2_S\right)^{H^{0}}\simeq\frac{\lambda_P^.A_{ud}^2\sin^3{2\beta}}{32\pi^2(A_{ud}s-m_{12}^2)^2}(3A_{ud}s-m_{12}^2)\ln\left<\frac{m_{h}^2}{m_{H}^2}\right>
\end{cases}\end{equation}}

{\bf Summary of the analysis:}\newline
Among the potentially large logarithms, we may distinguish among those of the form $\ln\frac{m^2}{\Lambda^2}$, which compare
a given sector to the scale $\Lambda$, typically chosen as the mass of the third-generation squarks, or to another sector, and
those sensitive to hierarchies within a sector $\ln\frac{m_i^2}{m_j^2}$.

In the case of the NMSSM, the logarithms $\ln\frac{m^2}{\Lambda^2}$ obviously appear only in the corrections to the 
$\mathbb{Z}_3$-conserving parameters. Moreover, when logarithms of the type $\ln\frac{m_i^2}{m_j^2}$ appear within 
$\mathbb{Z}_3$-violating parameters, they tend to be balanced by prefactors vanishing in the hierarchical limit 
(typically $\frac{m_i^2m_j^2}{(m_i^2-m_j^2)}$), so that they cannot be regarded as an enhancement factor (contrarily
to when they appear in $\mathbb{Z}_3$-conserving parameters, where the prefactor does not necessarily vanish in this 
limit). One can thus conclude that leading-logarithms preserve the $\mathbb{Z}_3$-induced structure of the potential.

For the nMSSM, the $\mathbb{Z}_3$-symmetry is actually still present at tree-level in all the sectors of the spectrum,
with the exception of the Higgs sector, where it is explicitly violated. Consequently, large logarithms still favour 
the $\mathbb{Z}_3$-conserving terms (even though they are not all present at the classical level), while 
$\mathbb{Z}_3$-violating effects perdure in the Higgs sector. In that case, large logarithms seem likely to destroy 
the classical structure.
