\documentstyle[11pt,epsf,fleqn]{article}
\setlength{\topmargin}{0mm}
\setlength{\headheight}{0mm}
\setlength{\headsep}{0mm}
\setlength{\textheight}{225mm}
\setlength{\textwidth}{160mm}
\setlength{\oddsidemargin}{0mm}
\setlength{\evensidemargin}{0mm}
%\setlength{\parskip}{2.0ex plus0.5ex minus0.5ex}
\setcounter{page}{1}
\def\refname{Reference}
\renewcommand{\thesection}{\Roman{section}.}
\renewcommand{\thesection}{\arabic{section}.}
\renewcommand{\thesubsection}{\Alph{subsection}.}
\begin{document}
\title{The one-loop correction to the neutral Higgs boson of the minimal  
supersymmetric standard model in explicit CP violation scenario}
\author{S.W. Ham$^{(1)}$, S.K. Oh$^{(1,2)}$, E.J. Yoo$^{(2)}$, C.M. Kim$^{(2)}$, D. Son$^{(1)}$ 
\\
\\
{\it $^{\rm (1)}$ Center for High Energy Physics, Kyungpook National University}\\
{\it Taegu 702-701, Korea} \\
{\it $^{\rm (2)}$ Department of Physics, Konkuk University, Seoul 143-701, Korea}
\\
\\
}
\date{}
\maketitle
%\thispagestyle{empty}
\begin{abstract}
We investigate the neutral Higgs sector of the minimal supersymmetric standard model in explicit CP violation scenario.
Within the context of the effective potential formalism, the masses of the neutral Higgs bosons at the one-loop level are calculated by taking into account the contributions of the following loops of ordinary particles and superpartners: top quark, the scalar top quarks, bottom quark, the scalar bottom quarks, tau lepton, the scalar tau leptons, $W$ boson, the charged Higgs boson, the charginos, $Z$ boson, the scalar and pseudoscalar Higgs bosons, and the neutralinos.
We improve the one-loop radiative corrections to the neutral Higgs sector by explicitly including the contributions of the pseudoscalar Higgs boson loop to the masses of the neutral Higgs bosons at the one-loop level with explicit CP violation for a typical set of parameter values of the MSSM.
\end{abstract}
\vfil
\eject
%***********************************************************************
\section{Introduction}
%***********************************************************************

For new physics beyond the standard model (SM) [1], the enlargement of the Higgs sector is considered as one of indispensable ingredients, especially for supersymmetric models. 
Among various supersymmetric extensions of the SM, the minimal supersymmetric standard model (MSSM) is the simplest one in the sense that it has just two Higgs doublets in its Higgs sector [2].
In principle, in the models with multiple Higgs doublets, the violation of CP symmetry may be accomplished either spontaneously or explicitly by the mixing between the scalar and pseudoscalar Higgs parts [3]. 
In practice, however, it is known that at the tree level the Higgs potential of the MSSM conserves CP symmetry, because the complex phase can always be eliminated by rotating the Higgs fields. Thus, neither explicit nor spontaneous CP violation can happen in the MSSM at the tree level [4].
Even at the one-loop level the scenario of spontaneous CP violation is excluded, because the radiatively corrected Higgs potentential of the MSSM leads to a very light Higgs boson which is unacceptable by the CERN $e^+ e^-$ LEP2 data [4,5].

The remaining possibility at the one-loop level for the MSSM is then explicit CP violation scenario. 
A number of investigations have been devoted to examine explicit CP violation in the MSSM [6]. 
In those investigations it is well observed that the tree-level situation is considerably modified when radiative contributions are included.
In particular, the MSSM Higgs potential with explicit CP violation is found to allow the lightest neutral Higgs boson to possess a mass above the expemental lower bound from the LEP2 data [6]. 
Therefore, the general consensus is that the MSSM at the one-loop level can accommodate explicit CP violation.

Quite recently, Ibrahim and Nath [7] have investigated explicit CP violation scenario, paying their attention to the phenomenological implications of the non-trivial CP phase on the chargino sector. 
They also have computed the scalar-pseudoscalar mixings arising from the neutralino sector, and compared with those from the chargino sector [8]. 
Their calculations are quite exhaustive since almost all the relevant loops are included: top quark, the scalar top quark, bottom quark, the scalar bottom quark, tau lepton, the scalar tau lepton, $W$ boson, the charged Higgs boson, the charginos, $Z$ boson, the neutral Higgs bosons, and the neutralinos. 
They have reported that in the MSSM with explicit CP violation the neutralino exchange corrections to the mixings of the CP-even sector and the CP-odd sector are comparable to the chargino exchange corrections [8]. 

We are interested in the same field, namely, the MSSM at the one-loop level with explicit CP violation. 
Our point of interest is, however, a little different. 
We are mainly concerned in the masses of the neutral Higgs bosons at the presence of explicit CP violation in the MSSM at the one-loop level.  
In order to evaluate the one-loop contributions to the neutral Higgs boson masses, we include all the loops of the relevant particles and superpartners. 
In particular, the contribution of the pseudoscalar Higgs loop is explicitly taken into account.
 
The inclusion of the pseudoscalar Higgs contribution is not only required but also necessary, because there is a non-trivial correlation between the ordinary particles and the corresponding superparticles in the one-loop effective potential.
The running scale in the one-loop effective potential might be determined by irregular values for the calculations of the Higgs boson masses and their mixing elements if the number of ordinary particle loops be different from that of the corresponding superpartner loops.

In the neutral sector of the MSSM, there are $Z$ boson and three neutral Higgs bosons as ordinary particles, and  four neutralinos as superpartners. Thus, the radiative contribution of the pseudoscalar Higgs loops should be included, whenever the loops of $Z$ boson, two scalar Higgs bosons are taken into account together with the loops of four neutralinos. 
The mass matrix for the neutral Higgs bosons at the one-loop level without the pseudoscalar Higgs contribution might lead to the different masses and the different mixing angles among the neutral Higgs bosons.
These differences would also lead to different results for the Higgs productions and their decays.

In this paper, we evaluate the masses of the neutral Higgs bosons in the MSSM with explicit CP violation at the one-loop level.
We take into account the loops of the pseudoscalar Higgs boson as well as all the loops that have been considered in the investigations of Ref. [8], namely, the loops of top quark, the scalar top quarks, bottom quark, the scalar bottom quarks, tau lepton, the scalar tau leptons, $W$ boson, the charged Higgs boson, the charginos, $Z$ boson, the scalar Higgs bosons, and the neutralinos. 
Our investigation is focused on the contributions of the pseudoscalar Higgs loops.
The masses of the neutral Higgs bosons in the MSSM at the one-loop level is 
evidently affected by the inclusion of the pseudoscalar Higgs contribution. 

%***********************************************************************
\section{The MSSM Higgs sector}
%***********************************************************************

A starting point to evaluate the neutral Higgs boson masses in the MSSM with explicit CP violation may be the tree-level Higgs potential, which is given in terms of two Higgs doublets $H_1^T = (H_1^0, H_1^-)$ and $H_2^T = (H_2^+, H_2^0)$ by
\begin{eqnarray}
  V^0 
  &=& {g_2^2\over 8} (H_2^{\dag} \vec\sigma H_2+ H_1^{\dag} \vec\sigma H_1)^2 
     +{g_1^2\over 8}(|H_2|^2 - |H_1|^2)^2       \cr
  & &\mbox{} + m_1^2 |H_1|^2 + m_2^2 |H_2|^2 
     - m_3^2 (H_1^T \epsilon H_2 + {\rm H.c.}) \ ,
\end{eqnarray}
where $g_1$ and $g_2$ are the U(1) and SU(2) gauge coupling constants, respectively, and $m_i^2$ $(i = 1, 2, 3)$ are the soft SUSY breaking masses.
The three Pauli matrices are denoted by $\vec\sigma$, and $\epsilon$ is an antisymmetric $2 \times 2$  matrix with $\epsilon_{12} = 1$.

It is well observed that there can be no CP phase in the above tree-level Higgs potential after rearranging the Higgs fields. 
Thus, the three soft SUSY breaking masses $m_i^2$ $(i = 1, 2, 3)$ are real.
While $m_i^2$ $(i = 1, 2)$ can be either positive or negative, we choose $m_3^2$ to be positive.
On the other hand, in order to accommodate the explicit CP violation scenario, we assume that the coefficients in the soft SUSY breaking terms are in general complex. 
To be specific, the following quantities are assumed to be complex in general: 
The Higgs mixing parameters with mass dimension $\mu$, the U(1) gaugino mass $M_1$, the SU(2) gaugino mass $M_2$, and the trilinear soft SUSY breaking masses $A_t$, $A_b$, and $A_{\tau}$.
The complex phases in these quantities are responsible for the CP violation.
We take the soft SUSY breaking masses $m_Q$, $m_T$, $m_B$, $m_L$, and $m_E$ be real. 

Now, as the electroweak symmetry is broken spontaneously, the Higgs fields develop non-trivial VEVs.
Since we do not consider spontaneous CP violation, only the neutral components of $H_1$ and $H_2$ develop the VEVs $v_1$ and $v_2$, respectively, which are assumed to be real and positive. The ratio of the two VEVs are defined as $\tan \beta = v_2/v_1$.
In terms of $v_1$ and $v_2$, the quark masses are given as $m_t^2 = (h_t v_2)^2$, $m_b^2  = (h_b v_1)^2$, and the gauge boson masses as $m_W^2 = (g_2 v)^2/2$ and $m_Z^2 = (g^2_1 +g^2_2)v^2/2$ with $v^2 = v_1^2 + v_2^2$.

At the same time, the tree-level masses of other particles and superpartners can be calculated.
The scalar top quarks, the scalar bottom quarks, the scalar tau leptons, and the charginos obtain their tree-level masses as follows:
\begin{eqnarray}
        m_{{\tilde t}_{1, 2}}^2 
        &=& m_t^2 + {1 \over 2}(m_Q^2 + m_T^2) 
\mp \left [ {1 \over 4} (m_Q^2 - m_T^2)^2 \right.  \cr
& &\mbox{}\left. + m_t^2 (A_t^2  + \mu^2 \cot^2 \beta 
+ 2 A_t \mu  \cot \beta \cos \phi_t) 
\rule{0mm}{5.5mm} \right ]^{1 \over 2}    \ , \cr
        m_{{\tilde b}_{1, 2}}^2 
        &=& m_b^2 + {1 \over 2} (m_Q^2 + m_B^2) 
\mp \left [ {1 \over 4} (m_Q^2 - m_B^2)^2 \right. \cr
& &\mbox{}\left. + m_b^2 (A_b^2 + \mu^2 \tan^2 \beta  
+ 2 A_b \mu \tan \beta \cos \phi_b) 
\rule{0mm}{5.5mm} \right ]^{1 \over 2}              \ , \cr
        m_{{\tilde \tau}_{1, 2}}^2
        &=& m_t^2 + {1 \over 2}(m_L^2 + m_E^2) 
\mp \left [ {1 \over 4} (m_L^2 - m_E^2)^2 \right.  \cr
& &\mbox{}\left. + m_{\tau}^2 (A_{\tau}^2 + \mu^2 \cot^2 \beta  
+ 2 A_{\tau} \mu \cot \beta 
\cos \phi_{\tau}) \rule{0mm}{5.5mm} \right ]^{1 \over 2}    \ , \cr
        m_{{\tilde \chi}_{1, 2}}^2 
        &=& {1 \over 2} \left \{M_2^2 + \mu^2 
+ 2 m_W^2 \right \} \mp \left [ {1 \over 4} 
\{M_2^2 - \mu^2 \}^2 - m_W^2 \{M_2^2 - \mu^2\} \cos 2\beta \right. \cr
& &\mbox{} \left. + 2 m_W^2 \{(M_2 \sin \beta)^2   
+ (\mu \cos \beta)^2  + M_2 \mu \sin  2 \beta \cos \phi_c\} 
        \rule{0mm}{5.5mm} \right ]^{1\over 2}      \ ,
\end{eqnarray}
where terms of $O(g^4_2)$ are neglected.
Notice that there appear four non-trivial CP phases in the above expressions, which are defined as follows: $\phi_t$ is the relative phase $A_t$ and $\mu$, 
$\phi_b$ is that of $A_b$  and $\mu$, $\phi_{\tau}$ is that of $A_{\tau}$ and $\mu$, and $\phi_c$ is that of $M_2$ and $\mu$.

The tree-level neutralino mass matrix is given as
\begin{equation}
	{\cal M}_{\tilde \chi} = \left ( 
\displaystyle{
\begin{array}{cccc}
M_1 e^{i\phi_1} & 0 & -{g_1\over \sqrt2} H^0_1 & {g_1 \over \sqrt2} H^0_2 \cr
 & & & \cr
0 & M_2 & {g_2 \over \sqrt2} H^0_1 & - {g_2 \over \sqrt2} H^0_2   \cr  
 & & & \cr
- {g_1 \over \sqrt2} H^0_1 & {g_2\over \sqrt2} H^0_1 & 0 & -\mu e^{i\phi_2}  \cr  
 & & & \cr 
{g_1 \over \sqrt2} H^0_2 & - {g_2\over \sqrt2} H^0_2 & -\mu e^{i\phi_2} & 0 
\end{array}  } 
\right )  \ , 
\end{equation}
where $\phi_1$ is the relative phase between $M_1$ and $M_2$ and $\phi_2$ is the relative phase between $M_2$ and $\mu$.
However, $\phi_2$ is identical to $\phi_c$. Thus, the neutralino mass matrix introduces only one more phase.
Consequently, we have in general five non-trivial CP phases: $\phi_t$, $\phi_b$, $\phi_{\tau}$, $\phi_c = \phi_2$ and $\phi_1$.
The above neutralino mass matrix is complex and symmetric, but not Hermitian.
In order to calculate the  neutralino masses, the Hermitian matrix $M_{{\tilde 
\chi}^0}^{\dagger} M_{{\tilde \chi}^0}$ is diagonalized through a similarity transformation.
The tree-level neutralino masses are denoted as $m_{{\tilde \chi}_i^0}^2$ $(i = 1,2,3,4)$, and they are sorted such that $m^2_{{\tilde \chi}_i^0}<m^2_{{\tilde \chi}_j^0}$ for $i<j$.

For the Higgs sector, the tree-level mass of the charged Higgs boson is obtained as 
\[
        m_{C^+}^2 
        = m_1^2 \sin^2 \beta + m_2^2 \cos^2 \beta 
        + m_3^2 \sin 2 \beta + m_W^2 - {m_Z^2 \over 2} \cos^2 2 \beta 
\] 
and the tree-level masses of the three neutral Higgs boson are obtained as
\begin{eqnarray}
        m_{h, H}^2 
        & = & {1 \over 2} (m_1^2 + m_2^2 + m_Z^2) 
        \mp {1\over2} \left [(m_1^2 - m_2^2)^2 +  4 m_3^4 
        + {m_Z^4 \over2} (3 \cos 4\beta + 5)  \right. \cr
        & &\mbox{} \left. +4 m_Z^2 \{(m_1^2 - m_2^2) \cos 2\beta +m_3^2 \sin 2\beta \} 
        \rule{0mm}{4.5mm} \right ]^{1 \over 2}     \ , \cr
        m_A^2 
        &=& m_1^2 \sin^2 \beta + m_2^2 \cos^2 \beta 
        + m_3^2  \sin 2 \beta -{m_Z^2 \over 2} \cos^2 2 \beta   \ ,
\nonumber
\end{eqnarray}
from the tree-level Higgs potential of the MSSM.
Note that there is no CP phase in the expressions for the tree-level Higgs boson masses because the tree-level Higgs potential does not have any CP phase.

Now, let us turn to the one-loop level. 
We employ the effective potential method [9] to estimate radiative corrections to the tree-level Higgs sector.
The Higgs potential at the one-loop level is 
\[
        V = V^0 + V^1 \ ,
\]
where $V^1$ represents the radiative corrections due to the loops, and is calculated by the effective potential method. 
It may conveniently be decomposed as  
\begin{equation}
        V^1 = V^t +V^b +V^{\tau} +V^{\tilde \chi} +V^h +V^{{\tilde \chi}^0} \ ,
\end{equation}
where 
\[
     V^t = {3 \over 32\pi^2} \left \{ \sum_{i = 1}^2 {\cal M}_{{\tilde t}_i}^4
     \left (\log {{\cal M}_{{\tilde t}_i}^2 \over \Lambda^2} - {3\over 2} \right )
     - 2 {\cal M}_t^4 \left (\log {{\cal M}_t^2 \over \Lambda^2}
     - {3\over 2} \right ) \right \}
\]
for the one-loop contributions of top quark and the scalar top quarks, 
\[
     V^b = {3 \over 32\pi^2} \left \{ \sum_{i = 1}^2 {\cal M}_{{\tilde b}_i}^4
        \left (\log {{\cal M}_{{\tilde b}_i}^2 \over \Lambda^2} - {3\over 2} \right )
     - 2 {\cal M}_b^4 \left (\log {{\cal M}_b^2 \over \Lambda^2}
     - {3\over 2} \right ) \right \} 
\]
for those of bottom quark and the scalar bottom quarks, 
\[
     V^{\tau} = {1 \over 32\pi^2} \left \{ \sum_{i = 1}^2 {\cal M}_{{\tilde \tau}_i}^4
        \left (\log {{\cal M}_{{\tilde \tau}_i}^2 \over \Lambda^2} - {3\over 2} \right )
     - 2 {\cal M}_{\tau}^4 \left (\log {{\cal M}_{\tau}^2 \over \Lambda^2}
     - {3\over 2} \right ) \right \}
\]
for those of tau lepton and the scalar tau leptons,
\begin{eqnarray}
        V^{{\tilde \chi}} 
        &=&  {1 \over 32\pi^2} \left  \{ 3 {\cal M}_W^4  \left (\log {{\cal M}_W^2 
        \over \Lambda^2} - {3\over 2} \right ) 
        + {\cal M}_{C^+}^4 \left ( \log  {{\cal M}_{C^+}^2 \over \Lambda^2}
        - {3\over 2} \right ) \right \}  \cr
        & & \mbox{} 
        - {1 \over 16\pi^2}\left \{ \sum_{i = 1}^2 {\cal M}_{{\tilde \chi}_i}^4
        \left (\log {{\cal M}_{{\tilde \chi}_i}^2 \over \Lambda^2} - {3\over 2} \right )   
        \right \} 
\nonumber
\end{eqnarray}
for those of $W$ boson, the charged Higgs boson and the charginos,
\[
        V^h = {3 {\cal  M}_Z^4 \over  64 \pi^2}  
        \left (\log  {{\cal M}_Z^2 \over  \Lambda^2} - {3 \over 2}\right ) 
        + {1 \over 64  \pi^2} \left \{ \sum_{i  = 1}^3 {\cal  M}_{h_i}^4 
        \left( \log {{\cal M}_{h_i}^2 \over \Lambda^2} - {3 \over 2} \right )\right \}        
\]
for those of $Z$ boson and both scalar and pseudoscalar neutral Higgs bosons,
and finally, 
\[
        V^{{\tilde \chi}^0} = -{1 \over 32 \pi^2}\left \{
        \sum_{i = 1}^4 {\cal M}_{{\tilde \chi}^0_i}^4
        \left (\log {{\cal M}_{{\tilde \chi}^0_i}^2 \over \Lambda^2} - {3\over 2} 
        \right )     \right \}  
\]
for those of the neutralinos.

In the above expressions, $\Lambda$ is the renormalization scale in the modified minimal  
subtraction (${\overline {\rm MS}}$) scheme and ${\cal M}$ stands for the 
field-dependent mass matrix of the relevant ordinary particles and their superpartners.
Note that the coefficient of $V^{\tau}$ differs from those of  $V^t$ and $V^b$ 
as the color factor is absent in it. 
We may group the above components into two parts: 
The charged part of $V^1$ consists of $V^t$, $V^b$, $V^{\tau}$, and $V^{\tilde \chi}$, and the neutral part of $V^1$ consists of $V^h$ and $V^{{\tilde \chi}^0}$.
Note that the loops of all the relevant particles and superpartners are included: top quark, bottom quark, tau lepton, $W$ boson, the charged Higgs boson, $Z$ boson, the neutral scalar and pseudoscalar Higgs bosons, and then the scalar top quarks, the scalar bottom quarks, the scalar tau leptons, the charginos, and the neutralinos.

Among others, the top quark and scalar top quark contributions are known to be 
most dominant to radiative corrections to the tree-level Higgs sector for most of the MSSM parameter space.
The contributions of the bottom and scalar bottom quark are not negligible for 
large $\tan \beta$.
It is also known that in the MSSM Higgs sector with explicit CP violation, the tree-level Higgs couplings to fermions and to gauge bosons are significantly modified by the radiative corrections due to the contributions of $W$ boson, the charged Higgs boson, and the charginos [7]. 

From the one-loop effective potential, we obtain two minimum equations 
for $v_1$ and $v_2$. They are: 
%*************************************************************
\begin{eqnarray}
        0 & = & 
m_1^2 - m_3^2 \tan \beta + {m_Z^2 \over 2} \cos 2 \beta 
- {3 m_t^2  \mu K_{{\tilde t}_1}  \over 8 \pi^2  v^2 \sin  2 \beta} f(m_{{\tilde  t}_1}^2, \ 
m_{{\tilde t}_2}^2)   \cr
& &\mbox{} + {3 m_b^2 \over 16 \pi^2 v^2 \cos^2 \beta}
\left \{  m_{{\tilde b}_1}^2  \left (\log  {m_{{\tilde b}_1}^2  \over  \Lambda^2} -  1 \right)  +  
m_{{\tilde b}_2}^2 \left (\log {m_{{\tilde b}_2}^2 \over \Lambda^2} - 1 \right) \right \} \cr
& &\mbox{} - {3 m_b^2 A_b K_{{\tilde b}_2} \over 16 \pi^2 v^2 \cos^2 \beta} f(m_{{\tilde 
b}_1}^2, \ m_{{\tilde b}_2}^2) 
- {3 m_b^4 \{\log(m_b^2 / \Lambda^2) -1\} \over 8 \pi^2 v^2 \cos^2 \beta} 
- {m_{\tau}^2   \mu K_{{\tilde  \tau}_1} \over   8 \pi^2  v^2  \sin 2   \beta} f(m_{{\tilde 
\tau}_1}^2, \ m_{{\tilde \tau}_2}^2)   \cr
& &\mbox{} - {m_W^2 \over 8 \pi^2  v^2} \left \{ m_{{\tilde \chi}_1}^2 \left (\log  {m_{{\tilde 
\chi}_1}^2 \over   \Lambda^2} -  1 \right)   +  m_{{\tilde  \chi}_2}^2 \left   (\log {m_{{\tilde 
\chi}_2}^2 \over \Lambda^2} - 1 \right) \right \} \cr
& &\mbox{}  + {m_W^2  K_{{\tilde \chi}_1}\over  8 \pi^2  v^2} f(m_{{\tilde  \chi}_1}^2, \ 
m_{{\tilde \chi}_2}^2) + {3 m_W^4 \over 16 \pi^2 v^2} \left ( \log {m_W^2 \over \Lambda^2} - 
1 \right) \cr
& &\mbox{} + {m_{C^+}^2 \over  32 \pi^2 v^2} \left  (\log {m_{C^+}^2 \over \Lambda^2}  - 1 
\right ) (2 m_W^2 - m_Z^2 \cos 2 \beta) \cr
& &\mbox{} + {m_Z^2 \over 32 \pi^2 v^2} 
\left \{ m_h^2 \left (\log {m_h^2 \over \Lambda^2} - 1 \right ) 
+ m_H^2 \left (\log {m_H^2 \over \Lambda^2} - 1 \right ) \right \} 
- {K_{h_1} \over 32 \pi^2} \{f(m_h^2, \ m_H^2) - 1 \} \cr
& &\mbox{} - {m_A^2  m_Z^2 \over 32  \pi^2 v^2} \left(\log  {m_A^2 \over \Lambda^2}  - 1 
\right) \cos 2 \beta + {3 m_Z^4 \over 32 \pi^2 v^2}
\left (\log{m_Z^2 \over \Lambda^2} - 1 \right ) \cr
& &\mbox{} + \sum^4_{i = 1} {m_{{\tilde \chi}^0_i}^2  \over 32 \pi^2 v \cos \beta}  \left (\log 
{m_{{\tilde \chi}^0_i}^2 \over \Lambda^2} - 1 \right )
{A_1  m_{{\tilde \chi}^0_i}^6  + B_1 m_{{\tilde \chi}^0_i}^4 + C_1 m_{{\tilde \chi}^0_i}^2 
+ D_1 \over \prod\limits_{j \not= i} (m_{{\tilde \chi}^0_i}^2 - m_{{\tilde \chi}^0_j}^2)}     
\nonumber
\end{eqnarray}
%************************************************
for $v_1$, and 
%************************************************
\begin{eqnarray}
        0 & = & 
m_2^2 - m_3^2 \cot \beta - {m_Z^2 \over 2} \cos 2 \beta 
 - {3 m_t^2 A_t  K_{{\tilde t}_2} \over  16 \pi^2 v^2 \sin^2  \beta} f(m_{{\tilde t}_1}^2,  \ 
m_{{\tilde t}_2}^2) \cr
& &\mbox{} + {3 m_t^2 \over 16 \pi^2 v^2 \sin^2 \beta}
\left \{  m_{{\tilde t}_1}^2  \left  (\log {m_{{\tilde  t}_1}^2 \over   \Lambda^2} - 1  \right)  +  
m_{{\tilde t}_2}^2 \left (\log {m_{{\tilde t}_2}^2 \over \Lambda^2} - 1 \right) \right \} \cr
& &\mbox{} - {3 m_t^4 \{\log (m_t^2 / \Lambda^2)  - 1 \} \over 8 \pi^2 v^2 \sin^2  \beta} - 
{3 m_b^2 \mu K_{{\tilde b}_1}  \over 8  \pi^2 v^2  \sin 2  \beta} f(m_{{\tilde b}_1}^2,  \ 
m_{{\tilde b}_2}^2) 
- {m_{\tau}^2 A_{\tau} K_{{\tilde \tau}_2} \over  16 \pi^2  v^2 \sin^2 \beta}  f(m_{{\tilde 
\tau}_1}^2, \ m_{{\tilde \tau}_2}^2) \cr
& &\mbox{} + {m_{\tau}^2 \over 16 \pi^2 v^2 \sin^2 \beta}
\left \{ m_{{\tilde \tau}_1}^2 \left (\log  {m_{{\tilde \tau}_1}^2 \over \Lambda^2} - 1  \right) +  
m_{{\tilde \tau}_2}^2 \left (\log {m_{{\tilde  \tau}_2}^2 \over \Lambda^2} -  1 \right) \right \} 
\cr
& &\mbox{} - {m_{\tau}^4 \{\log  (m_{\tau}^2 / \Lambda^2) -  1 \} \over 8  \pi^2 v^2 \sin^2 
\beta}
- {m_W^2 \over  8 \pi^2 v^2}  \left \{ m_{{\tilde  \chi}_1}^2 \left  (\log {m_{{\tilde \chi}_1}^2 
\over \Lambda^2} - 1 \right)  +  m_{{\tilde \chi}_2}^2 \left  (\log {m_{{\tilde \chi}_2}^2 \over 
\Lambda^2} - 1 \right) \right \} \cr
& &\mbox{} +  {m_W^2 K_{{\tilde \chi}_2}  \over 8  \pi^2 v^2} f(m_{{\tilde  \chi}_1}^2, \ 
m_{{\tilde \chi}}^2)+ {3 m_W^4 \over 16 \pi^2 v^2} \left ( \log {m_W^2 \over  \Lambda^2} - 1 
\right) \cr
& &\mbox{} + {m_{C^+}^2 \over  32 \pi^2 v^2} \left  (\log {m_{C^+}^2 \over \Lambda^2}  - 1 
\right ) (2 m_W^2 + m_Z^2 \cos 2 \beta) \cr
& &\mbox{} + {m_Z^2 \over 32 \pi^2 v^2} 
\left \{ m_h^2 \left (\log {m_h^2 \over \Lambda^2} - 1 \right ) 
+ m_H^2 \left (\log {m_H^2 \over \Lambda^2} - 1 \right ) \right \} 
- {K_{h_2} \over 32 \pi^2} \{f(m_h^2, \ m_H^2) - 1 \} \cr
& &\mbox{} - {m_A^2  m_Z^2 \over 32  \pi^2 v^2} \left(\log  {m_A^2 \over \Lambda^2}  - 1 
\right) \cos 2 \beta + {3 m_Z^4 \over 32 \pi^2 v^2}
\left (\log{m_Z^2 \over \Lambda^2} - 1 \right )  \cr
& &\mbox{} + \sum^4_{i = 1} {m_{{\tilde \chi}^0_i}^2  \over 32 \pi^2 v  \sin \beta} \left (\log 
{m_{{\tilde \chi}^0_i}^2 \over \Lambda^2} - 1 \right )
{A_2 m_{{\tilde \chi}^0_i}^6  +B_2 m_{{\tilde \chi}^0_i}^4 +C_2 m_{{\tilde \chi}^0_i}^2 
+ D_2 \over \prod\limits_{j \not= i} (m_{{\tilde \chi}^0_i}^2 - m_{{\tilde \chi}^0_j}^2)} 
\nonumber
\end{eqnarray}
%********************************************************************
for $v_2$, where the expressions for $A_i$, $B_i$, $C_i$, and $D_i$ $(i = 1, 2)$ are given in Appendix A and the various short-hand notations are defined as
\begin{eqnarray}
K_{{\tilde t}_1} & = & A_t \cos \phi_t + \mu \cot \beta \ , \cr
K_{{\tilde t}_2} & = & A_t + \mu \cot \beta \cos \phi_t \ , \cr
K_{{\tilde b}_1} & = & A_b \cos \phi_b + \mu \tan \beta  \ , \cr
K_{{\tilde b}_2} & = & A_b + \mu \tan \beta \cos \phi_b \ , \cr
K_{{\tilde \tau}_1} & = & A_{\tau} \cos \phi_{\tau} + \mu \cot \beta \ , \cr
K_{{\tilde \tau}_2} & = & A_{\tau} + \mu \cot \beta \cos \phi_{\tau} \ , \cr
K_{{\tilde \chi}_1} & = & M_2^2 +\mu^2 + 2 M_2 \mu \tan \beta \cos \phi_c \ , \cr
K_{{\tilde \chi}_2} & = & M_2^2 +\mu^2 + 2 M_2 \mu \cot \beta \cos \phi_c \ ,  \cr
K_{h_1} & = & {4 m_Z^2 \over v^2} (2 m_Z^2 \cos 2 \beta + m_1^2 - m_2^2)
+ {4 m_Z^2 \over v^2} (m_Z^2 \sin^2 \beta - m_3^2 \tan \beta) \ , \cr
K_{h_2} & = & \mbox{} - {4 m_Z^2 \over v^2} (2 m_Z^2 \cos 2 \beta + m_1^2 - m_2^2) + {4 m_Z^2 \over v^2} (m_Z^2 \cos^2 \beta - m_3^2 \cot \beta) \ , 
\nonumber
\end{eqnarray}
and
\[
        f(m_1^2, \  m_2^2)  
        = {1  \over  (m_2^2-m_1^2)} \left  \{   m_1^2 \log  {m_1^2 \over 
        \Lambda^2} -m_2^2 \log {m_2^2 \over \Lambda^2} \right \} + 1 \ .
\]

Note that these two minimum equations are not linear with respect to $m_1^2$ nor to $m_2^2$, because the tree-level Higgs boson masses in the logarithmic functions contain them. 
Thus, it is impossible to solve the minimum equations to obtain $m_1^2$ or $m_2^2$ analytically. They will be solved numerically in the next section, to reduce the number of free parameters.

Now, let us redefine the two Higgs doublets as
\begin{eqnarray}
\begin{array}{ccc}
        H_1 & = & \left ( \begin{array}{c}
          v_1 +h_1 + i \sin \beta h_3   \cr
          \sin \beta C^{+ *}
  \end{array} \right )  \ ,  \cr
        H_2 & = & \left ( \begin{array}{c}
          \cos \beta C^+           \cr
          v_2 +h_2 + i \cos \beta h_3
  \end{array} \right ) \ ,  
\end{array}
\end{eqnarray}
where $h_1$, $h_2$, $h_3$ are neutral Higgs fields and $C^+$ is the charged Higgs field.
At the presence of the explicit CP violation, $h_1$, $h_2$, $h_3$ are mixed states of the CP parity. 
By differentiating the Higgs potential at the one-loop level with respect to the neutral Higgs fields, a $3 \times 3$ symmetric mass matrix for them is obtained in the $(h_1, h_2, h_3)$-basis. 
The mass matrix at the one-loop level for the neutral Higgs fields may be decomposed as
\[
        M_{ij} = M^0_{ij} +M^1_{ij}
\]
where $M^0_{ij}$ is obtained from $V^0$, and $M^1_{ij}$ from $V^1$. 
In particular, $M^1_{ij}$ may further be decomposed as 
\[
        M^1_{ij} = M^t_{ij} + M^b_{ij} +M^{\tau}_{ij} 
        +M_{ij}^{\tilde \chi} +M_{ij}^h +M_{ij}^{{\tilde \chi}^0}  
\]
where, as the superscripts suggest, $M^t_{ij}$ represents the radiative contributions from $V^t$. Likewise, $M^b_{ij}$, $M^{\tau}_{ij}$, $M^{\tilde \chi}_{ij}$, $M^h_{ij}$, and $M^{{\tilde \chi}^0}_{ij}$ represent respectively the radiative contributions from $V^b$, $V^{\tau}$, $V^{\tilde \chi}$, $V^h$, and $V^{{\tilde \chi}^0}$. 

The elements of the symmetric tree-level mass matrix $M^0$ in the $(h_1, h_2, h_3)$-basis are easily calculated as
%*********************  tree level *************************************
\begin{eqnarray}
        M_{11}^0 &=& m_3^2 \tan \beta + m_Z^2 \cos^2 \beta  \ ,  \cr
        M_{22}^0 &=& m_3^2 \cot \beta + m_Z^2 \sin^2 \beta  \ ,  \cr
        M_{33}^0 &=& {2 m^2_3 \over \sin2\beta}   \ , \cr
        M_{12}^0 &=&\mbox{} -m_3^2 -m_Z^2 \cos \beta \sin \beta \ ,  \cr
        M_{13}^0 &=& 0 \ , \cr
        M_{23}^0 &=& 0 \ , 
\end{eqnarray}
where $M_{13}^0 = M_{23}^0 = 0$ indicates that there is no mixing between the scalar and the pseudoscalar components at the tree level, since no CP phase is present in the tree-level Higgs potential $V^0$.
We remark that $m_3^2$ in the above expressions is actually a function of both $m_1^2$ and $m_2^2$ through the minimum equations. 
Thus, although $m_1^2$ and $m^2_2$ are absent in $M_{ij}^0$, the tree-level masses should be calculated numerically.

For $M^1_{ij}$, we first calculate $M_{ij}^h$ that comes from $V^h$. 
The result is given as follows:
%*********************  Higgs Self  ******************************
\begin{eqnarray}
        M_{11}^h 
& = & {v^2 \cos^2 \beta K_{h_1}^2 \over 16 \pi^2}
{g(m_h^2, \ m_H^2) \over (m_H^2 - m_h^2)^2}  
- {m_Z^2 \over 8 \pi^2 v^2} (m_3^2 \tan \beta + 4 m_Z^2 \cos^2 \beta) f(m_h^2, \ m_H^2) \cr
& &\mbox{}   + {m_Z^4   \cos^2 \beta   \over 16   \pi^2 v^2}  \log  \left ({m_h^2  m_H^2  \over 
\Lambda^4} \right )  
+ {m_Z^2 \cos^2  \beta K_{h_1} \over  8 \pi^2} {\log  (m_H^2 / m_h^2)  \over (m_H^2  - 
m_h^2)} \cr
& &\mbox{} + {m_Z^4 \cos^2 \beta \cos^2 2 \beta \over 16 \pi^2 v^2} 
\log \left ({m_A^2 \over \Lambda^2} \right ) 
+ {m_Z^4 \cos^2 \beta \over 8 \pi^2 v^2} \log \left ({m_Z^6 \over \Lambda^6} \right ) \ , \cr
 & & \cr
        M_{22}^h 
& = & {v^2 \sin^2 \beta K_{h_2}^2 \over 16 \pi^2}
{g(m_h^2, \ m_H^2) \over (m_H^2 - m_h^2)^2}  
- {m_Z^2 \over 8 \pi^2 v^2} (m_3^2 \cot \beta + 4 m_Z^2 \sin^2 \beta) f(m_h^2, \ m_H^2) \cr
& &\mbox{} + {m_Z^4 \sin^2 \beta \over 16 \pi^2 v^2} \log \left ({m_h^2 m_H^2 \over \Lambda^4} \right )  
+ {m_Z^2 \sin^2  \beta K_{h_2} \over  8 \pi^2}  {\log (m_H^2 /  m_h^2) \over (m_H^2  - m_h^2)} \cr
& &\mbox{} + {m_Z^4 \sin^2 \beta \cos^2 2 \beta \over 16 \pi^2 v^2} 
\log \left ({m_A^2 \over \Lambda^2} \right ) 
+ {m_Z^4 \sin^2 \beta \over 8 \pi^2 v^2} \log \left ({m_Z^6 \over \Lambda^6} \right ) \ , \cr
 & & \cr
        M_{33}^h 
& = &{m_Z^2 \over 8 \pi^2 v^2} 
\left \{ {(m_Z^2 - 2 m_3^2 / \sin 2 \beta) \over (m_H^2 - m_h^2)} - {1 \over 2} \right \}
\left \{m_h^2 \left (\log {m_h^2 \over \Lambda^2} - 1 \right) + m_H^2 \left (\log {m_H^2 \over  \Lambda^2} - 1 \right )  \right \} \cr
& &\mbox{} +  {m_Z^2 \cos 2  \beta \over 8  \pi^2 v^2} (2  m_Z^2 \cos 2  \beta + m_1^2  - 
m_2^2) f (m_h^2, \ m_H^2) \cr
& &\mbox{} + {3 m_Z^2 \cos^2 2 \beta \over 32 \pi^2 v^2}
m_A^2 \left \{ \log \left ({m_A^2 \over \Lambda^2} \right ) - 1 \right \} \ , \cr
 & & \cr
        M_{12}^h 
& = & {v^2 \sin 2 \beta K_{h_1} K_{h_2} \over 32 \pi^2} {g(m_h^2, \ 
m_H^2) \over (m_H^2 - m_h^2)^2}  
+ {m_Z^2 \over 4 \pi^2 v^2} (m_3^2 + m_Z^2 \sin 2 \beta) f(m_h^2, \ m_H^2) \cr
& &\mbox{} +{m_Z^4 \sin 2\beta \over 16 \pi^2 v^2} \log \left ({m_h^2 m_H^2  \over \Lambda^4} \right )  
+ {m_Z^4 \over 4 \pi^2  v^2} {(m_Z^2 \sin 2 \beta -2 m_3^2) \over (m_H^2  - m_h^2)} \log  \left ({m_H^2 \over m_h^2} \right ) \cr
& &\mbox{} - {m_Z^4  \sin 2 \beta  \cos^2 2 \beta  \over 16 \pi^2 v^2}  \log \left ({m_A^2 \over \Lambda^2} \right ) 
+ {m_Z^4 \sin 2 \beta \over 16 \pi^2 v^2} \log \left ({m_Z^6 \over \Lambda^6} \right ) \ ,  \cr
& & \cr
        M_{13}^h & = & 0 \ , \cr
& & \cr
        M_{23}^h & = & 0 \ ,
\nonumber
\end{eqnarray}
where 
\[
        g(m_1^2,m_2^2) 
        = {m_1^2 + m_2^2 \over m_1^2 - m_2^2} \log {m_2^2 \over m_1^2} + 2 \ .
\]
In these expressions, $m_h^2$, and $m_H^2$, and $m_A^2$ are the tree-level masses of the scalar and the pseudoscalar Higgs bosons previously obtained, and they are nonlinear functions of $m_1^2$ and $m_2^2$.
Here, too, we have $M_{13}^h = M_{23}^h = 0$ because there is no CP phase in $V^h$.
In this sense, the explicit CP violation scenario in the Higgs sector of the MSSM may be regarded as the radiative CP violation, which appears only in radiative corrections due to superpartners, via the CP phases in soft SUSY breaking terms.

Next, we calculate $M_{ij}^{{\tilde \chi}^0}$, the contributions of radiative corrections due to the four neutralinos. We obtain 
\begin{equation}
        M_{ij}^{{\tilde \chi}^0}  
= -\sum_{k=1}^4
{m_{{\tilde \chi}^0_k}^2 \over 32 \pi^2} 
\left (\log {m_{{\tilde \chi}^0_k}^2 \over \Lambda^2} - 1 \right ) 
{\partial^2 m_{{\tilde \chi}^0_k}^2 \over \partial h_i \partial h_j}
- \sum_{k=1}^4 {1\over 16  \pi^2} \log{m_{{\tilde \chi}^0_k}^2 \over \Lambda^2} \left 
({\partial m_{{\tilde \chi}^0_k}^2 \over \partial h_i}  \right)  \left( {\partial  m_{{\tilde 
\chi}^0_k}^2 \over \partial h_j} \right )  \ , 
\end{equation}
where the excplicit expression for the first-order derivatives 
$\partial m_{{\tilde \chi}^0_k}^2 / \partial h_i$ $(i = 1,2,3)$ is given by
\[
        {\partial m_{{\tilde \chi}^0_k}^2 \over \partial h_i} 
        = -{\displaystyle 
        A_i m_{{\tilde \chi}^0_k}^6 +B_i m_{{\tilde \chi}^0_k}^4 
        +C_i m_{{\tilde \chi}^0_k}^2 +D_i 
        \over 
        \prod\limits^4_{a \not= k} (m_{{\tilde \chi}^0_k}^2 - m_{{\tilde \chi}^0_a}^2)}
  \\ ,
\]
and the explicit expression for the second-order derivatives $\partial^2 m_{{\tilde \chi}^0_k}^2 /\partial h_i \partial h_j$ is given by
\begin{eqnarray}
        {\partial^2 m_{{\tilde \chi}^0_k}^2 \over \partial h_i \partial h_j}  
        &=&\mbox{} -{A_{ij}m_{{\tilde \chi}^0_k}^6 +B_{ij} m_{{\tilde \chi}^0_k}^4 
        +C_{ij} m_{{\tilde \chi}^0_k}^2 +D_{ij} 
        \over \prod\limits^4_{a \not= k} 
        (m_{{\tilde \chi}^0_k}^2 - m_{{\tilde \chi}^0_a}^2)} \cr
& &\mbox{} 
        + \sum_{a\not=k}^4
{1 \over (m_{{\tilde \chi}^0_k}^2 - m_{{\tilde \chi}^0_a}^2)} 
        \left ({\partial m_{{\tilde \chi}^0_k}^2 \over \partial h_i} 
        {\partial m_{{\tilde \chi}^0_a}^2 \over \partial h_j} 
        +{\partial m_{{\tilde \chi}^0_a}^2 \over \partial h_i} 
        {\partial m_{{\tilde \chi}^0_k}^2 \over \partial h_j} \right ) \ .
\end{eqnarray}
Here, the formulae for $A_i$, $B_i$, $C_i$, and $D_i$ $(i = 1,2)$ are the same as those in the minimum equations, and the formulae for $A_3$, $B_3$, $C_3$, and $D_3$ are given as
\begin{eqnarray}
        A_3 &=& 0   \ , \cr
        B_3 &=&\mbox{} - {4\mu \over v} \left[  M_2 m_W^2 \sin \phi_2 
        -M_1 (m_W^2 - m_Z^2) \sin (\phi_1 + \phi_2) \right] \ , \cr
        C_3 &=&\mbox{} {4\mu\over v}
        \left[ M_2 m_W^2 (M_1^2 +\mu^2) \sin\phi_2 
        +M_1 (m_W^2 -m_Z^2) (M_2^2 +\mu^2) \sin (\phi_1 +\phi_2) \right]  \ , \cr
        D_3 &=&\mbox{} - {4\mu^3 \over v} \left[ M_1^2 M_2 m_W^2 \sin \phi_2  
        -M_1 M_2^2 (m_W^2 -m_Z^2) \sin (\phi_1 +\phi_2) \right] \ .
\nonumber
\end{eqnarray}
The expressions for $A_{ij}$, $B_{ij}$, $C_{ij}$, and $D_{ij}$, $(i,j=1,2,3)$ are given in Appendix B. 
Actually, $A_{ij}$ = 0 for all $i,j$ because four elements of the neutralino mass matrix are zero.
Lastly, the remaining components of the one-loop mass matrices, namely, $M_{ij}^t$, $M_{ij}^b$, $M_{ij}^{\tau}$ and $M_{ij}^{\tilde \chi}$ are also calculated, and the results are given in Appendices C, D, E, and F, respectively. 

In the scenario of explicit CP violation, both $M_{13}$ and $M_{23}$ are responsible for the scalar-pseudoscalar mixing and eventually account for the CP violation.
In particular, $M_{i3}^t$ $(i =1, 2)$ are proportional to $\sin \phi_t$, $M_{i3}^b$ $(i= 1,2)$ to $\sin \phi_b$, $M_{i3}^{\tau}$ $(i=1,2)$ to $\sin\phi_{\tau}$, $M_{i3}^{\tilde \chi}$ $(i = 1, 2)$ to $\sin \phi_c$.
Meanwhile, $M_{i3}^h$ $(i =1,2)$ are zero, as noticed above.
$M_{i3}^{{\tilde \chi}^0}$ $(i=1,2)$ have complicated expressions, but in the mass matrix, one can see that every term is proportional to $\sin (\phi_1 +\phi_2)$ or $\sin \phi_c$ since $B_3$, $C_3$, $D_3$, and $B_{i3}$, $C_{i3}$, $D_{i3}$ $(i = 1,2)$ are proportional to the phases.

%***********************************************************************
\section{Numerical analysis}
%***********************************************************************

Now, we are in a position to investigate the effect of CP phases in the 
one-loop corrected MSSM Higgs sector. 
We first investigate the difference between the case with CP phases and 
the one without them.

%***********************************************************************
\subsection{CP conserving case}
%***********************************************************************
If all of the five CP phases vanish in the Higgs sector at the one-loop level, CP symmetry would be conserved.
In this case, the scalar-pseudoscalar mixing would disappear because both $M_{13}$ and $M_{23}$ of the mass matrix of the neutral Higgs bosons become zero. 
In other words, the mass matrix would become block diagonalized if the CP phases be all absent. 

We first separate the imaginary part from the neutral Higgs sector to obtain the mass of the pseudoscalar Higgs boson at the one-loop level without CP phases.
The result can be written as
\[
        m^2_A = (m^2_A)^0 +(m^2_A)^1
\]
where $(m^2_A)^0$ is the square of the tree-level mass of the pseudoscalar Higgs boson and $(m^2_A)^1$ is the radiative correction to it .
The square of the tree-level mass is given as
\[
         (m^2_A)^0 = {2 m_3^2 \over \sin 2 \beta}
\]
and the radiative corrections is further decomposed as
\[
        (m^2_A)^1 
        = (m^2_A)^t +(m^2_A)^b +(m^2_A)^{\tau} +(m^2_A)^{\tilde \chi}   
        +(m^2_A)^h +(m^2_A)^{{\tilde \chi}^0} \ ,
\]
where each term is obtained essentially from $V^t$, $V^b$, $V^{\tau}$, $V^{\tilde \chi}$, $V^h$, and $V^{{\tilde \chi}^0}$, respectively.

Explicitly, they are expressed as follows:
\begin{eqnarray}
        (m^2_A)^t &=& {3 m_t^2 \mu A_t \over 16 \pi^2  v^2 \sin^3 \beta \cos 
\beta} f(m_{{\tilde t}_1}^2, \ m_{{\tilde t}_2}^2) \ , \cr
        (m^2_A)^b &=& {3 m_b^2 \mu A_b \over 16  \pi^2 v^2 \cos^3 \beta \sin 
\beta} f(m_{{\tilde b}_1}^2, \ m_{{\tilde b}_2}^2)  \ , \cr
        (m^2_A)^{\tau} &=& {3 m_{\tau}^2 \mu A_{\tau} \over 16 \pi^2 v^2 \sin^3 
\beta \cos \beta} f(m_{{\tilde \tau}_1}^2, \ m_{{\tilde \tau}_2}^2) \ , \cr
        (m^2_A)^{\tilde \chi} &=&  {m_W^2 M_2  \mu \over \pi^2  v^2 \sin  2 \beta} 
f(m_{{\tilde \chi}_1}^2, \ m_{{\tilde \chi}}^2) - {m_W^2 m_{C^+}^2 \over  16 \pi^2 v^2} \left \{ 
\log \left ({m_{C^+}^2 \over \Lambda^2}\right ) - 1 \right \} \ , \cr
        (m^2_A)^h &=& {m_Z^2 \over 8 \pi^2 v^2} 
\left \{{m_Z^2 - 2 m_3^2 /  \sin 2 \beta \over m_H^2  - m_h^2} - {1 \over 2}  \right \} \left 
\{m_h^2 \left (\log{m_h^2  \over \Lambda^2} -  1 \right  ) + m_H^2  \left (\log{m_H^2  \over 
\Lambda^2} - 1 \right ) \right \} \cr
& &\mbox{} - {m_Z^2 \cos 2 \beta \over 8 \pi^2 v^2}
\{2 m_Z^2 \cos 2 \beta + m_1^2 - m_2^2 \}  f(m_h^2, \ m_H^2) \cr
& &\mbox{} + {3  m_Z^2 \cos^2 2 \beta  \over 32 \pi^2 v^2}  m_A^2 \left(\log {m_A^2 \over 
\Lambda^2} - 1 \right)  \ ,  \cr
        (m^2_A)^{{\tilde \chi}^0} &=&\mbox{} - \sum^4_{i = 1} {\mu m_{{\tilde \chi}^0_i}^2  \over 
16 \pi^2 v^2 \sin 2 \beta} \left (\log {m_{{\tilde \chi}^0_i}^2 \over \Lambda^2} - 1 \right )
{{\displaystyle {B m_{{\tilde \chi}^0_i}^4 +  C m_{{\tilde \chi}^0_i}^2 + D  \over \prod\limits_{j 
\not= i} (m_{{\tilde \chi}^0_i}^2 - m_{{\tilde \chi}^0_j}^2)}}} \ , 
\nonumber
\end{eqnarray}
where
\begin{eqnarray}
        B & = & M_2 m_W^2 + M_1 (m_Z^2 - m_W^2) \ , \cr
        C & = & M_2 m_W^2 (M_1^2 + \mu^2) 
        + M_1 (m_W^2 - m_Z^2) (M_2^2 + \mu^2) \ , \cr
        D & = & \mu^2 M_1 M_2 \{M_1 m_W^2 + M_2 (m_Z^2 - m_W^2) \} \ .
\nonumber
\end{eqnarray}

Next, we turn to the real part of the neutral Higgs sector to obtain the masses of the two scalar Higgs bosons at the one-loop level without CP phases.
The squared masses of the two scalar Higgs bosons are given by the eigenvalues
of a $2\times2$ mass matrix, and analytically calculated as
\begin{equation}
        m_{h,H}^2 = {1\over 2}\left[ M_{11} +M_{22} 
        \mp \sqrt{ (M_{11} -M_{22})^2 +4M_{12}M_{21} } \right]  \ ,
\end{equation}
where $M_{ij}$ $(i,j = 1,2)$ are the elements of the mass matrix at the one-loop level without CP phases. Thus, they consist of all the loop contributions, namely, $M^t_{ij}$, $M^b_{ij}$, $M^{\tau}_{ij}$, $M_{ij}^{\tilde \chi}$, $M_{ij}^h$, and $M_{ij}^{{\tilde \chi}^0}$, as well as the tree-level values $M^0_{ij}$.
We would not write down the full expressions for $m_{h,H}^2$, since
they are complicated functions of many parameters coming from the soft SUSY 
breaking in the MSSM. 
Moreover, to obtain numercal values for the neutral Higgs boson masses, the minimum equations should also be solved numerically.

Now, for numerical calculations, we need concrete numbers. 
We set $m_t = 175.0$ GeV, $m_b =4.0$ GeV, $m_{\tau} = 1.7$ GeV, $m_W = 80.4$ GeV, and $m_Z = 91.1$ GeV. 
Then, the remaining relevant free parameters are $\Lambda$, $m_1$, $m_2$, $m_3$, $\tan \beta$, $\mu$, $m_Q$, $m_L$, $m_T$, $m_B$, $m_E$, $A_t$, $A_b$, $A_{\tau}$, $M_2$, and $M_1$.
For simplicity, we set $m_Q = m_L$ = 500 GeV, $m_T = m_B = m_E$= 800 GeV, $A_t = A_b = A_{\tau}$= 300 GeV, and $M_2 = M_1$= 200 GeV.
Further, we set $m_3$ = 200 GeV and $\mu$ = 400 GeV.
We are then left with $\Lambda$, $m_1$, $m_2$, and $\tan\beta$. 

Among $\Lambda$, $m_1$, $m_2$, and $\tan \beta$, two of them may be eliminated by using the minimum equations. 
Usually, $m_1$ and $m_2$ may be chosen. However, as aforementioned, the minimum equations are not linear in $m_1$ and $m_2$. 
On the contrary, one can observe that both of the two minimum equations are linear in the running scale $\Lambda$. Hence, we choose one minimum equation, the one for $v_1$, 
and solve it to obtain analytically the running scale $\Lambda$. 
The other minimum equation, the one for $v_2$, is then solved numerically
with respect to $m_1$ by using the bisection method. 
Finally, the remaining parameters, $m_2$ and $\tan \beta$, are fixed at certain values.  

In this way, we calculate the radiatively corrected Higgs boson masses at the one-loop level without CP phases. 
Tables 1 and 2 summarize the numerical results at the one-loop level without CP phases.
The entries of Tables are the elements of the mass matrix of the neutral
Higgs bosons in the $(S_1, S_2, A)$-basis, where all the contributions are fully listed. The unit is (GeV)$^2$. 
Note that there are no columns for the $M_{13}$ and $M_{23}$ because the scalar-pseudoscalar mixing is absent and the mass matrix is
block diagonalized since the CP phases are zero.  
The number in the last row in each column is the sum of the numbers in the 
preceding rows in the same column. 
The last column lists the contributions to $M_{33}$, which are solely responsible for the pseudoscalar Higgs boson, whereas the first three columns account for the $2\times2$ submatrix for the scalar Higgs boson masses.  

In Table 1, we set $m_2$ = 150 GeV and $\tan \beta = 5$, for the range of $|m_1| < 1000$ GeV. We obtain $\Lambda$ =  799.7 GeV and $m_1$ = 456.0 GeV as the solutions for the minimum equations.
From the numbers in Table 1, the mass of the pseudoscalar Higgs boson is given as
\[
        m_A = 471.0 ~~{\rm (GeV)}
\]
and the masses of the two scalar Higgs bosons are calculated as
\begin{eqnarray}
        m_h & = & 121.9 ~~{\rm (GeV)}   \ ,  \cr
        m_H & = & 466.6 ~~{\rm (GeV)}  \nonumber
\end{eqnarray}
at the one-loop level without CP phase. 

We set $m_2$ = 150 GeV and $\tan \beta = 20$ in Table 2.
Here, in Table 2, we obtain $m_1$ = 917.4 GeV and $\Lambda$ = 911.3 GeV from 
the minimum equations.
For this choice of parameter values, the mass of the pseudoscalar Higgs boson is obtained as
\[
        m_A = 935.9 ~~{\rm (GeV)}
\]
and the masses of the two scalar Higgs bosons are calculated as
\begin{eqnarray}
        m_h & = & 120.2 ~~{\rm (GeV)}   \ , \cr
        m_H & = & 925.8 ~~{\rm (GeV)}   \nonumber
\end{eqnarray}
at the one-loop level without CP phase.

We have some comments on the radiative corrections due to the loops of $Z$ boson, the neutral Higgs bosons, and the neutralinos.
One can observe in both Table 1 and Table 2 that the sum of $M_{12}^h$ and $M_{12}^{{\tilde \chi}^0}$ are comparatively large. This implies that the contributions of $Z$ boson, the neutral Higgs bosons (both scalar and pseudoscalar), and neutralinos play important role in the radiative corrections to the mixing between the scalar Higg bosons at the one-loop level.

On the contrary, one can find in Table 1 that, for the mass of the pseudoscalar Higgs boson, the contributions of $Z$ boson, the neutral Higgs boson and neutralino loops are smaller than those of the charged Higgs boson and charginos or those of top quark and scalar top quarks but larger than those of bottom quark and the scalar bottom quarks or those of tau lepton and the scalar tau leptons.
In other words, for the values of parameters as set in Table 1, we have 
$(m^2_A)^{\tilde \chi} > (m^2_A)^t > ((m^2_A)^h + (m^2_A)^{{\tilde \chi}^0}) > (m^2_A)^b > (m^2_A)^{\tau}$.   
The situation is slightly different in Table 2. The contribution of the neutralino sector is smaller than any other ones except for that of the scalar tau sector.

%***********************************************************************
\subsection{CP violating case}
%***********************************************************************

Now, let us consider the CP violating case. If there exist some CP phases in the Higgs potential at the one-loop level, they would produce the scalar-pseudoscalar mixing through non-zero $M_{13}$ and $M_{23}$ and thus generate explicit CP violation in the neutral Higgs sector.
In the MSSM Higgs sector, we show that there are in general five CP phases. 
They are contained in the masses of the scalar top quarks, the scalar bottom quarks, the scalar tau leptons, the charginos, and the neutralinos.

As an illustration, we set $\phi_t = \phi_b = \phi_{\tau} = \phi_c (\phi_2) = \phi_1 = \pi/4$ to see the effects of these CP phases in the mass matrix of the neutral Higgs boson in the $(h_1, h_2, h_3)$-basis. 
We repeat the calculations done for Tables 1 and 2, and show the results in Tables 3 and 4.
Tables 3 and 4 summarize the numerical results at the one-loop level with five CP phases.
The entries of Tables are the elements of the mass matrix of the neutral Higgs bosons in the $(h_1, h_2, h_3)$-basis, where all the contributions are fully listed. The unit is (GeV)$^2$. 
Note that, as the CP phases induce the CP violation through the scalar-pseudoscalar mixing, the columns for the $M_{13}$ and $M_{23}$ are also present in Tables 3 and 4.
The number in the last row in each column is the sum of the numbers in the preceding rows in the same column. 
The values of relevant parameters in Table 3 are the same as those in Table 1, except for $\Lambda$ and $m_1$. The numbers of these two parameters are given as the solutions for the minimum equations. 
Since the minimum equations now contain the CP phases, they would be obtained 
differently from the case of Table 1 where CP phases are absent. 

For Table 3, we obtain $\Lambda$ = 800.1 GeV and $m_1$ = 454.6 GeV as the solutions for the minimum equations. 
The differences in these numbers between the cases of Table 1 and Table 3 are evident though small they might be.
For this choice of parameter values in Table 3, the masses of the three neutral Higgs bosons are calculated as 
\begin{eqnarray}
        && m_{h_1} = 122.2 ~~{\rm (GeV)}   \ , \cr
        && m_{h_2} = 462.6 ~~{\rm (GeV)}   \ , \cr
        && m_{h_3} = 466.0 ~~{\rm (GeV)}   \ ,
\nonumber  
\end{eqnarray}
at the one-loop level with CP phases ($\phi_t = \phi_b = \phi_{\tau} = \phi_c (\phi_2) = \phi_1 = \pi/4$).

Similarly, the values of relevant parameters in Table 4 are the same as those in Table 2, except for $\Lambda$ and $m_1$. For Table 4,  we obtain $\Lambda$ =  912.6 GeV and $m_1$ = 911.9 GeV as the solutions for the minimum equations.
For this choice of parameter values in Table 4, the masses of the three neutral Higgs bosons are calculated as 
\begin{eqnarray}
        && m_{h_1} = 119.0 ~~{\rm (GeV)}   \ , \cr
        && m_{h_2} = 918.3 ~~{\rm (GeV)}   \ , \cr
        && m_{h_3} = 922.4 ~~{\rm (GeV)}   \ ,
\nonumber  
\end{eqnarray}
at the one-loop level with CP phases ($\phi_t = \phi_b = \phi_{\tau} = \phi_c (\phi_2) = \phi_1 = \pi/4$).

In Tables 3 and 4, one can easily notice that, among the contributions to $M_{13}$ and $M_{23}$, the tree-level ones and the Higgs ones vanish:
$M^0_{13} = M^0_{23} =0$ and $M^h_{13} = M^h_{23} =0$. 
This reflects the fact that there is no CP phase in the tree-level Higgs sector of the MSSM.
All others are not zero, accounting for the scalar-pseudoscalar mixing, because the CP phases are present there.  
Actually, one can trace the dependence of the various contributions to $M_{13}$ and $M_{23}$ on the CP phases. 
If $\phi_t = 0$, one would have $M_{13}^t = M_{23}^t = 0$. 
Further, one would have $M_{13}^b = M_{23}^b = 0$ for $\phi_b = 0$.
Likewise, one would have $M_{13}^{\tau} = M_{23}^{\tau} = 0$ for $\phi_{\tau} = 0$, $M_{13}^{\tilde \chi} = M_{23}^{\tilde \chi} = 0$ for $\phi_c (\phi_2) = 0$, and $M_{13}^{{\tilde \chi}^0} = M_{23}^{{\tilde \chi}^0} = 0$ for $\phi_2 = \phi_1 = 0$.
Among the non-zero contributions to $M_{13}$ and $M_{23}$, we find that $M_{23}^{{\tilde \chi}^0}$ of Table 3 and $M_{i3}^{{\tilde \chi}^0}$ $(i =1, 2)$ of Table 4 contribute most dominantly to the scalar-pseudoscalar mixing. Thus, Tables 4 indicate that the radiative corrections due to the neutralino loops contribute most significantly to the CP mixing effects.

In Tables 3 and 4, the contributions of the pseudoscalar Higgs boson loops deserve some detailed discussions.  
The contributions of the pseudoscalar Higgs boson loops are included in all of $M^h_{11}$, $M^h_{22}$, $M^h_{33}$, and $M^h_{12}$. Of course, the contributions of the loops of the scalar Higgs bosons are also included in them.
Without the contributions of the pseudoscalar Higgs boson loops, the values of $M^h_{11}$, $M^h_{22}$, $M^h_{33}$, and $M^h_{12}$ would be changed. 
In particular, if the pseudoscalar Higgs boson is not taken into account, $M^h_{33}$ would be definitely zero. In our calculations, $M^h_{33}$ is not zero. 
However, as $M^h_{33}$ is not negligible and is required to diagonalize the mass matrix of the neutral Higgs bosons at the one-loop level, the contributions of the pseudoscalar Higgs boson loops should be included when one considers the masses of the neutral Higgs bosons, as well as the mixing among them, at the one-loop level.
The same is true of $(m^2_A)^h$ in Tables 1 and 2. Without the contribution of the pseudoscalar Higgs boson loops, the value of $(m^2_A)^h$ would be zero, and consequently $m_A$ at the one-loop level would be changed, even if the change 
would not be significant. 

The contributions of the neutralino loops depend crucially on the CP phase $\phi_1$.
In other words, the CP phase $\phi_1$ occurs only in the expressions for the neutralino contributions. 
Now, in order to examine in more detail the dependence of the contributions of the neutralino loops on $\phi_1$, we plot $M^t_{13}$, $M^{\tilde \chi}_{13}$, $M^{{\tilde \chi}^0}_{13}$, and $M_{13}$ as functions of $\phi_1$ in Fig. 1(a), where the values of other parameters are the same as Table 3.
Likewise, in Fig. 1(b) we plot $M^t_{23}$, $M^{\tilde \chi}_{23}$, $M^{{\tilde \chi}^0}_{23}$, and $M_{23}$ as functions of $\phi_1$, where the values of other parameters are the same as Fig. 1(a).
We do not plot $M_{13}^{\tau}$ and $M_{13}^b$ in Fig. 1(a), nor $M_{23}^{\tau}$ and $M_{23}^b$ in Fig. 1(b), since the tau lepton and scalar tau lepton contributions as well as the bottom quark and scalar bottom quark ones to $M_{13}$ or $M_{23}$ are found to be small for the whole range of $\phi_1$.

We repeat calculations for the parameter values of Table 4 in Fig. 2(a) to plot $M^t_{13}$, $M^{\tilde \chi}_{13}$, $M^{{\tilde \chi}^0}_{13}$, and $M_{13}$ as functions of $\phi_1$.
In Fig. 2(b), we plot $M^t_{23}$, $M^{\tilde \chi}_{23}$, $M^{{\tilde \chi}^0}_{23}$, and $M_{23}$ as functions of $\phi_1$ for the same parameter values as Fig. 2(a).
Here, too, we do not plot $M_{13}^{\tau}$ and $M_{13}^b$ in Fig. 2(a), nor $M_{23}^{\tau}$ and $M_{23}^b$ in Fig. 2(b).

Note that in those four figures $M_{13}^t$ and $M_{23}^t$ as well as $M_{13}^{\tilde \chi}$ and $M_{23}^{\tilde \chi}$ are very stable against $\phi_1$. 
Therefore, the dependence on $\phi_1$ of $M_{13}$ and $M_{23}$ respectively are actually the dependence of the contributions of the neutralino loops, $M^{{\tilde \chi}^0}_{13}$ and $M^{{\tilde \chi}^0}_{23}$, on $\phi_1$, up to constants. 
The dependence on $\phi_1$ of $M^{{\tilde \chi}^0}_{13}$ in Figs. 1(a) and 2(a), as well as that of $M^{{\tilde \chi}^0}_{23}$ in Fig. 2(b) is quite strong, whereas that of $M^{{\tilde \chi}^0}_{23}$ in Fig. 1(b) is rather mild.

For $M_{13}$, the contributions of the neutralino loops are either smaller or larger than other contributions for some values of $\phi_1$. 
For the case of Fig. 1(a) the sizes of those radiative contributions may be sorted as $|M_{13}^{{\tilde \chi}^0}| > M_{13}^t > M_{13}^{\tilde \chi}$ for $0 < \phi_1 < 0.22 \pi$, $0.32 \pi < \phi_1 < 0.52 \pi$, and  $0.74 \pi < \phi_1 < \pi$.
For $M_{23}$, both Figs. 1(b) and 2(b) show that the contributions of the neutralino loops are larger than other contributions for the whole range of 
$0 \le \phi_1 \le \pi$. 
The contributions of the loops of top quark and the scalar top quarks come next, and the contributions of the loops of $W$ boson, the charged Higgs boson, and charginos are almost null for the full range of $\phi_1$.
Therefore, one may note that the contributions of the neutralino loops to $M_{23}$ mixing for the range of $0 \le \phi_1 \le \pi$ are larger than the contributions of other loops.

%***********************************************************************
\section{Conclusions}
%***********************************************************************

In the MSSM, the tree-level neutral Higgs sector may be divided into the scalar part and pseudoscalar part, and there is no mixing between them. 
Any phase that can cause the scalar-pseudoscalar mixing, hence the CP violation, can be absorbed away at the tree level.
At the one-loop level, where explicit CP violation is viable by introducing several CP phases in the effective Higgs potential, the scalar-pseudoscalar mixing inevitably occurs.
The scalar-pseudoscalar mixing is manifested by the non-vanishing $M_{13}$ and
$M_{23}$ matrix elements of the neutral Higgs bosons.
Evidently, these off-diagonal elements affect the masses of the neutral Higgs bosons when the mass matrix is diagonalized.

We have investigated the masses of the neutral Higgs bosons in the MSSM at the 
one-loop level with explicit CP violation. 
In explicit CP violation scenario, five non-trivial CP phases are introduced, from the soft SUSY breaking terms of the MSSM Lagrangian, in the masses of the scalar top quarks, the scalar bottom quarks, the scalar tau leptons, the charginos, and the neutralinos.
These phases penetrate into the mass matrix of the neutral Higgs bosons.
We have considered all the contributions of relevant loops: 
The loops of the pseudoscalar Higgs boson as well as all the loops of top quark, the scalar top quarks, bottom quark, the scalar bottom quarks, tau lepton, the scalar tau leptons, $W$ boson, the charged Higgs boson, the charginos, $Z$ boson, the scalar Higgs bosons, and the neutralinos. 

In the MSSM with explicit CP violation, the mass matrix elements that are responsible for the scalar-pseudoscalar mixing receive large contributions from the four neutralinos for some values of the relevant parameters.
Also, the mass matrix element that becomes the squared mass of the pseudoscalar Higgs boson in the limit of CP conservation can be significantly changed by the contribution of the pseudoscalar loop.

%*****************************************************************
\vskip 0.3 in

\noindent
{\large {\bf Acknowledgments}}
\vskip 0.2 in
\noindent
This work was supported by Korea Research Foundation Grant (2001-050-D00005).

\vskip 0.2 in

\vfil\eject
%**************************************************************
\noindent
{\large {\bf Appendix A}}
\vskip 0.2 in
\noindent
The coefficients that appear in the first derivatives of the neutralino masses with respect to the neutral Higgs fields in the minimum equations are given as follows:
\begin{eqnarray}
        A_1 &=&{4\cos\beta\over v} m_Z^2  \ , \cr
        B_1 &=&\mbox{} - 
        {4\cos\beta\over v} \left[ m_Z^2 (m_Z^2 + \mu^2) +M_1^2 m_W^2 
        -M_2^2 (m_W^2 - m_Z^2) \right] \cr
        & &\mbox{} +{4\sin\beta\over v} \left[ M_2 m_W^2 \mu \cos\phi_2 
        +M_1 \mu (m_W^2 -m_Z^2) \cos (\phi_1 + \phi_2) \right] \ , \cr
        C_1 &=& {4\cos\beta\over v}
        \left[ 2m_Z^4 \mu^2 \sin^2 \beta 
         +M_1^2 m_W^2 (m_W^2 + \mu^2) \right]  \cr
        & &\mbox{} +{4\cos\beta\over 4} (m_W^2  - m_Z^2)
        \left[M_2^2 (m_W^2 - m_Z^2  - \mu^2) 
        -2M_1 M_2 m_W^2 \cos\phi_1 \right]  \cr
        & &\mbox{} -{4\sin\beta\over v} \mu
        \left[ M_2 m_W^2 (M_1^2 + \mu^2) \cos\phi_2
        -M_1 (m_W^2 -m_Z^2) (M_2^2 +\mu^2)\cos (\phi_1 +\phi_2) \right] \ , \cr
        D_1 &=& \mbox{}- {4\sin\beta\over v} \mu^2 \sin2\beta 
        \left[ M_1^2 m_W^4 +M_2^2 (m_W^2 -m_Z^2)^2  
        -2M_1 M_2 m_W^2 (m_W^2 -m_Z^2) \cos\phi_1 \right]  \cr
        & &\mbox{} +{4\sin\beta\over v} \mu^3
        \left[ M_1^2 M_2 m_W^2 \cos\phi_2  
        -M_1 M_2^2 (m_W^2 -m_Z^2) \cos(\phi_1 +\phi_2) \right]  \ ,
\nonumber
\end{eqnarray}
and
\begin{eqnarray}
        A_2 &=& A_1 (\cos\beta \leftrightarrow \sin\beta)  \ , \cr
        B_2 &=& B_1 (\cos\beta \leftrightarrow \sin\beta)  \ , \cr
        C_2 &=& C_1 (\cos\beta \leftrightarrow \sin\beta)  \ , \cr
        D_2 &=& D_1 (\cos\beta \leftrightarrow \sin\beta)  \ .
\nonumber
\end{eqnarray}

\vskip 0.2 in
\noindent
{\large {\bf Appendix B}}
\vskip 0.2 in
\noindent
The coefficients that appear in the second derivatives of the neutralino masses with respect to the neutral Higgs fields in the radiatively corrected mass matrix for the neutral Higgs bosons are given as follows:
\begin{eqnarray}
A_{11} & = & 0  \ , \cr
B_{11} & = &\mbox{} - {8 m_Z^4 \cos^2 \beta \over v^2} 
- {4 M_2 m_W^2 \mu \tan \beta \cos \phi_2 \over v^2} 
+ {4 M_1 \mu (m_W^2 - m_Z^2) \tan \beta \cos(\phi_1 + \phi_2) \over v^2}  \ , \cr
C_{11}  & =  &{8  M_1^2 m_W^4  \cos^2 \beta \over  v^2} 
+  {8 M_2^2 (m_W^2 - m_Z^2)^2 \cos^2 \beta \over v^2} \cr
& &\mbox{} - {16 M_1 M_2 m_W^2 (m_W^2 - m_Z^2) \cos^2 \beta \cos \phi_1 \over v^2}  
+ {4 M_2 m_W^2 \mu (M_1^2 + \mu^2) \tan \beta \cos \phi_2 \over v^2} \cr
& &\mbox{} - {4  M_1 \mu (m_W^2  - m_Z^2) (M_2^2 +  \mu^2) \tan \beta  \cos (\phi_1 + \phi_2)\over v^2}  \ , \cr
D_{11} & = & - {4 M_1^2 M_2  m_W^2 \mu^3 \tan \beta \cos \phi_2 \over v^2} +  {4 
M_1 M_2^2 \mu^3 (m_W^2 - m_Z^2) \tan \beta \cos (\phi_1 + \phi_2) \over v^2}  \  , 
\nonumber
\end{eqnarray}
and
\begin{eqnarray}
A_{22} &=& A_{11} (\cos\beta \leftrightarrow \sin\beta) = 0  \ , \cr
B_{22} &=& B_{11} (\cos\beta \leftrightarrow \sin\beta)  \ , \cr
C_{22} &=& C_{11} (\cos\beta \leftrightarrow \sin\beta)  \ , \cr
D_{22} &=& D_{11} (\cos\beta \leftrightarrow \sin\beta)  \ , 
\nonumber
\end{eqnarray}
and
\begin{eqnarray}
A_{33} & = & 0 \ , \cr
B_{33}  & = &\mbox{}- {8 M_2 m_W^2 \mu \cos \phi_2 \over v^2 \sin 2 \beta} 
+ {8 M_1 \mu (m_W^2 - m_Z^2) \cos (\phi_1+\phi_2) \over v^2 \sin 2 \beta}  \ , \cr
C_{33}  & = &{8 M_2 m_W^2 \mu (M_1^2 + \mu^2) \cos \phi_2
\over v^2 \sin 2 \beta} - {8 M_1 \mu (m_W^2 - m_Z^2) (M_2^2 + \mu^2) 
\cos (\phi_1 + \phi_2) \over v^2 \sin 2 \beta}  \ , \cr 
D_{33} & = &\mbox{}- {8 M_1^2 M_2 m_W^2 \mu^3 \cos \phi_2 \over v^2 \sin 2 \beta} + {8 M_1 M_2^2 \mu^3 (m_W^2 - m_Z^2) \cos (\phi_1 + \phi_2) \over v^2 \sin 2 \beta}  \ , 
\nonumber
\end{eqnarray}
and
\begin{eqnarray}
A_{12} & = & 0  \ , \cr
B_{12} & = & \mbox{} - {4 m_Z^4 \sin 2 \beta \over v^2} 
+ {4 M_2 m_W^2 \mu \cos \phi_2 \over v^2} 
- {4 M_1 \mu (m_W^2 - m_Z^2) \cos (\phi_1 + \phi_2) \over v^2} \ , \cr
C_{12}  & = & {8 m_Z^4 \mu^2 \sin 2 \beta \over v^2} + {4 M_1^2 m_W^4 
\sin 2 \beta \over v^2} + {4 M_2^2 (m_W^2 - m_Z^2)^2 \sin 2 \beta \over v^2}  \cr
& &\mbox{} - {8 M_1 M_2 m_W^2 (m_W^2  - m_Z^2) \sin 2 \beta \cos \phi_1 \over  v^2} - 
{4 M_2 m_W^2 \mu (M_1^2 + \mu^2) \cos \phi_2 \over v^2}  \cr
& &\mbox{} + {4 M_1 \mu (m_W^2 - m_Z^2)  (M_2^2 + \mu^2) \cos (\phi_1 + \phi_2) \over 
v^2} \ , \cr
D_{12} & = &\mbox{} - {8 M_1^2 m_W^4 \mu^2 \sin 2  \beta \over v^2} - {8 M_2^2  \mu^2 (m_W^2 - m_Z^2)^2 \sin 2 \beta \over v^2} \cr 
& &\mbox{} + {16 M_1 M_2 m_W^2 \mu^2 (m_W^2 -  m_Z^2) \sin 2 \beta \cos \phi_1 \over 
v^2} + {4 M_1^2 M_2 m_W^2 \mu^3 \cos \phi_2 \over v^2} \cr
& &\mbox{} - {4 M_1 M_2^2 \mu^3 (m_W^2 - m_Z^2) \cos (\phi_1 + \phi_2) \over v^2}   \ , 
\nonumber
\end{eqnarray}
and
\begin{eqnarray}
A_{13} & = & 0  \ , \cr
B_{13}  & = &\mbox{}- {4 M_2 m_W^2 \mu \cos \beta \sin \phi_2 \over v^2} 
+ {4 M_1 \mu (m_W^2 - m_Z^2) \cos \beta \sin (\phi_1 + \phi_2) \over v^2} \ , \cr
C_{13}  & = & {4 M_2 m_W^2 \mu (M_1^2 + \mu^2) \cos \beta  \sin \phi_2 \over 
v^2} \cr
& &\mbox{} - {4  M_1 \mu (m_W^2  - m_Z^2) (M_2^2  + \mu^2) \cos  \beta \sin (\phi_1 + 
\phi_2) \over v^2} \ , \cr
D_{13}  & = & - {4 M_1^2  M_2 m_W^2 \mu^3 \cos \beta \sin \phi_2 \over  v^2} 
+ {4 M_1 M_2^2 \mu^3 (m_W^2 - m_Z^2) \cos \beta \sin (\phi_1 + \phi_2) \over v^2} \ , 
\nonumber
\end{eqnarray}
and
\begin{eqnarray}
A_{23} &=& A_{13} (\cos\beta \leftrightarrow \sin\beta) = 0 \ , \cr
B_{23} &=& B_{13} (\cos\beta \leftrightarrow \sin\beta) \ , \cr
C_{23} &=& C_{13} (\cos\beta \leftrightarrow \sin\beta) \ , \cr
D_{23} &=& D_{13} (\cos\beta \leftrightarrow \sin\beta) \ .
\nonumber
\end{eqnarray}

%********************** top-stop ****************************************
\vskip 0.2 in
\noindent
{\large {\bf Appendix C}}
\vskip 0.2 in
\noindent
The elements for the mass matrix of the neutral Higgs bosons due to the radiative contributions of the top quark and scalar top quarks are   
\begin{eqnarray}
M_{11}^t & = & {3 m_t^4 \mu^2 K_{{\tilde t}_1}^2
\over 8 \pi^2 v^2 \sin^2 \beta}
{g(m_{{\tilde t}_1}^2, \ m_{{\tilde t}_2}^2) \over
(m_{{\tilde t}_2}^2 - m_{{\tilde t}_1}^2)^2}
+ {3 m_t^2 \mu A_t \cos \phi_t \over 8 \pi^2 v^2 \sin 2 \beta} 
f(m_{{\tilde t}_1}^2, \ m_{{\tilde t}_2}^2)  \ , \cr
& & \cr
M_{22}^t & = & {3 m_t^4 A_t^2 \Delta_{{\tilde t}_2}^2
\over 8 \pi^2 v^2 \sin^2 \beta}
{g(m_{{\tilde t}_1}^2, \ m_{{\tilde t}_2}^2) \over
(m_{{\tilde t}_2}^2 - m_{{\tilde t}_1}^2)^2}
+ {3 m_t^2 \mu A_t \cot \beta \cos \phi_t \over 16 \pi^2 v^2 \sin^2 \beta} f(m_{{\tilde t}_1}^2, 
\ m_{{\tilde t}_2}^2) \cr  
& &\mbox{}+ {3 m_t^4 \over 8 \pi^2 v^2 \sin^2 \beta} 
\log \left ({m_{{\tilde t}_1}^2  m_{{\tilde t}_2}^2 \over m_t^4} \right ) 
 + {3 m_t^4 A_t K_{{\tilde t}_2} \over 4 \pi^2 v^2 \sin^2 \beta}
{\displaystyle \log (m_{{\tilde t}_2}^2 / m_{{\tilde t}_1}^2)
 \over (m_{{\tilde t}_2}^2 - m_{{\tilde t}_1}^2)} \ , \cr
& & \cr
M_{33}^t & = & {3 m_t^4 \mu^2 A_t^2 \sin^2 \phi_t
\over 8 \pi^2 v^2 \sin^4 \beta}
{g(m_{{\tilde t}_1}^2, \ m_{{\tilde t}_2}^2) \over (m_{{\tilde t}_2}^2
- m_{{\tilde t}_1}^2 )^2}
+ {3 m_t^2 \mu A_t \cos \phi_t \over 16 \pi^2 v^2 \sin^3 \beta \cos \beta}
f(m_{{\tilde t}_1}^2, \ m_{{\tilde t}_2}^2)   \ , \cr
& & \cr
M_{12}^t & = & {3 m_t^4 A_t \mu \over 8 \pi^2 v^2 \sin^2 \beta}
{K_{{\tilde t}_1} K_{{\tilde t}_2}  
\over (m_{{\tilde t}_2}^2 - m_{{\tilde t}_1}^2)^2}
{g(m_{{\tilde t}_1}^2, \ m_{{\tilde t}_2}^2)}    
- {3 m_t^2 \mu A_t \cos \phi_t \over 16 \pi^2 v^2 \sin^2 \beta}
f(m_{{\tilde t}_1}^2, \ m_{{\tilde t}_2}^2)  \cr
& &\mbox{} + {3 m_t^4 \mu K_{{\tilde t}_1} \over 8 \pi^2 v^2 \sin^2 \beta}
{\log (m_{{\tilde t}_2}^2 / m_{{\tilde t}_1}^2)
 \over (m_{{\tilde t}_2}^2 - m_{{\tilde t}_1}^2)} \ , \cr
& & \cr
M_{13}^t & = & \mbox{} - {3 m_t^4 \mu^2 A_t K_{{\tilde t}_1}  \sin \phi_t 
\over 8 \pi^2 v^2 \sin^3 \beta}
{g(m_{{\tilde t}_1}^2, \ m_{{\tilde t}_2}^2) \over
(m_{{\tilde t}_2}^2 - m_{{\tilde t}_1}^2)^2 }  
+ {3 m_t^2 \mu A_t \sin \phi_t \over 16 \pi^2 v^2 \sin \beta \tan \beta}
f(m_{{\tilde t}_1}^2, \ m_{{\tilde t}_2}^2)   \ , \cr
& & \cr
M_{23}^t & = & \mbox{} - {3 m_t^4 \mu A_t^2 K_{{\tilde t}_2}  \sin \phi_t 
\over 8 \pi^2  v^2 \sin^3  \beta} {g(m_{{\tilde t}_1}^2,  \ m_{{\tilde  t}_2}^2) \over (m_{{\tilde 
t}_2}^2 - m_{{\tilde t}_1}^2)^2 }  
+ {3 m_t^2 \mu A_t \sin \phi_t \over 16 \pi^2 v^2 \sin \beta} f(m_{{\tilde t}_1}^2, \ m_{{\tilde 
t}_2}^2) \cr
& &\mbox{} - {3 m_t^4 \mu A_t \sin \phi_t \over 8 \pi^2 v^2 \sin^3 \beta} 
{\log (m_{{\tilde t}_2}^2 /  m_{{\tilde t}_1}^2) \over 
(m_{{\tilde t}_2}^2 -m_{{\tilde t}_1}^2)}  \ .
\nonumber
\end{eqnarray}

%***************** bottom-sbottom *******************************************
\vskip 0.2 in
\noindent
{\large {\bf Appendix D}}
\vskip 0.2 in
\noindent
The elements for the mass matrix of the neutral Higgs bosons due to the radiative 
contributions of the bottom quark and scalar bottom quarks are 
\begin{eqnarray}
M_{11}^b & = & {3 m_b^4 A_b^2 K_{{\tilde b}_2}^2
\over 8 \pi^2 v^2 \cos^2 \beta}
{g(m_{{\tilde b}_1}^2, \ m_{{\tilde b}_2}^2) \over
(m_{{\tilde b}_2}^2 - m_{{\tilde b}_1}^2)^2}
+ {3 m_b^2  \mu A_b  \tan \beta \cos  \phi_b \over  16 \pi^2  v^2 \cos^2 \beta}  f(m_{{\tilde 
b}_1}^2, \ m_{{\tilde b}_2}^2)  \cr
& &\mbox{} + {3 m_b^4 \over 8 \pi^2 v^2 \cos^2 \beta} 
\log \left ({m_{{\tilde b}_1}^2  m_{{\tilde b}_2}^2 \over m_b^4} \right )
+ {3 m_b^4 A_b K_{{\tilde b}_2} \over 4 \pi^2 v^2 \cos^2 \beta}
{\log (m_{{\tilde b}_2}^2 / m_{{\tilde b}_1}^2)
 \over (m_{{\tilde b}_2}^2 - m_{{\tilde b}_1}^2)}  \ , \cr
& & \cr
M_{22}^b & = & {3 m_b^4  \mu^2 K_{{\tilde b}_1}^2
\over 8 \pi^2 v^2 \cos^2 \beta}
{g(m_{{\tilde b}_1}^2, \ m_{{\tilde b}_2}^2) \over
(m_{{\tilde b}_2}^2 - m_{{\tilde b}_1}^2)^2}
+ {3 m_b^2 \mu A_b \cos \phi_b \over 8 \pi^2 v^2 \sin 2 \beta}
f(m_{{\tilde b}_1}^2, \ m_{{\tilde b}_2}^2) \ , \cr
& & \cr 
M_{33}^b & = & {3 m_b^4 \mu^2 A_b^2 \sin^2 \phi_b
\over 8 \pi^2 v^2 \cos^4 \beta}
{g(m_{{\tilde b}_1}^2, \ m_{{\tilde b}_2}^2) \over (m_{{\tilde b}_2}^2
- m_{{\tilde b}_1}^2 )^2}
+ {3 m_b^2 \mu A_b \cos \phi_b \over 16 \pi^2 v^2 \cos^3 \beta \sin \beta}
f(m_{{\tilde b}_1}^2, \ m_{{\tilde b}_2}^2)  \ , \cr
& & \cr
M_{12}^b & =  & {3 m_b^4 \mu  A_b K_{{\tilde b}_1} K_{{\tilde  b}_2} 
\over 8 \pi^2 v^2 \cos^2 \beta}
{g(m_{{\tilde b}_1}^2, \ m_{{\tilde b}_2}^2)
\over (m_{{\tilde b}_2}^2 - m_{{\tilde b}_1}^2)^2}  
- {3 m_b^2 \mu A_b \cos \phi_b \over 16 \pi^2 v^2 \cos^2 \beta}
f(m_{{\tilde b}_1}^2, \ m_{{\tilde b}_2}^2)  \cr
& &\mbox{} + {3 m_b^4 \mu K_{{\tilde b}_1} \over 8 \pi^2 v^2 \cos^2 \beta}
{\displaystyle \log (m_{{\tilde b}_2}^2 / m_{{\tilde b}_1}^2)
 \over (m_{{\tilde b}_2}^2 - m_{{\tilde b}_1}^2)}  \ , \cr
& & \cr
M_{13}^b & =  & \mbox{} -  {3 m_b^4  \mu A_b^2 K_{{\tilde  b}_2} \sin 
\phi_b \over 8 \pi^2 v^2 \cos^3 \beta}
{g(m_{{\tilde b}_1}^2, \ m_{{\tilde b}_2}^2) \over (m_{{\tilde b}_2}^2 - m_{{\tilde b}_1}^2)^2 }  
+ {3 m_b^2 \mu A_b \sin \phi_b \over 16 \pi^2 v^2 \cos \beta}
f(m_{{\tilde b}_1}^2, \ m_{{\tilde b}_2}^2) \cr
& &\mbox{} - {3 m_b^4 \mu A_b \sin \phi_b \over 8 \pi^2 v^2 \cos^3 \beta} 
{\log (m_{{\tilde b}_2}^2 / m_{{\tilde b}_1}^2)  \over (m_{{\tilde b}_2}^2 - m_{{\tilde b}_1}^2)}  
 \ , \cr
M_{23}^b & = &\mbox{} - {3 m_b^4 \mu^2 A_b K_{{\tilde b}_1} \sin \phi_b 
\over 8 \pi^2 v^2  \cos^3 \beta} {g(m_{{\tilde  b}_1}^2, \ m_{{\tilde  b}_2}^2) \over (m_{{\tilde 
b}_2}^2 - m_{{\tilde b}_1}^2)^2 }  
+ {3 m_b^2 \mu A_b \tan \beta \sin \phi_b \over 16 \pi^2 v^2 \cos \beta}
f(m_{{\tilde b}_1}^2, \ m_{{\tilde b}_2}^2)  \ .
\nonumber
\end{eqnarray}

%********************** tau-stau ****************************************
\vskip 0.2 in
\noindent
{\large {\bf Appendix E}}
\vskip 0.2 in
\noindent
The elements for the mass matrix of the neutral Higgs bosons due to the radiative contributions of the tau lepton and scalar tau leptons are 
\begin{eqnarray}
M_{11}^{\tau} & = & {m_{\tau}^4 \mu^2 K_{{\tilde \tau}_1}^2
\over 8 \pi^2 v^2 \sin^2 \beta}
{g(m_{{\tilde \tau}_1}^2, \ m_{{\tilde \tau}_2}^2) \over
(m_{{\tilde \tau}_2}^2 - m_{{\tilde \tau}_1}^2)^2}
+ {m_{\tau}^2 \mu A_{\tau} \cos \phi_{\tau} \over 8 \pi^2 v^2 \sin 2 \beta} 
f(m_{{\tilde \tau}_1}^2, \ m_{{\tilde \tau}_2}^2)  \ , \cr
& & \cr
M_{22}^{\tau} & = & {m_{\tau}^4 A_t^2 K_{{\tilde \tau}_2}^2
\over 8 \pi^2 v^2 \sin^2 \beta}
{g(m_{{\tilde \tau}_1}^2, \ m_{{\tilde \tau}_2}^2) \over
(m_{{\tilde \tau}_2}^2 - m_{{\tilde \tau}_1}^2)^2}
+ {m_{\tau}^2  \mu A_{\tau}  \cot \beta   \cos \phi_{\tau} \over  16 \pi^2   v^2 \sin^2 \beta} 
f(m_{{\tilde \tau}_1}^2, \ m_{{\tilde \tau}_2}^2) \cr  
& &\mbox{}+ {m_{\tau}^4 \over 8 \pi^2 v^2 \sin^2 \beta} 
\log \left ({m_{{\tilde \tau}_1}^2  m_{{\tilde \tau}_2}^2 \over m_{\tau}^4} \right ) 
 + {m_{\tau}^4 A_{\tau} K_{{\tilde \tau}_2} \over 4 \pi^2 v^2 \sin^2 \beta}
{\displaystyle \log (m_{{\tilde \tau}_2}^2 / m_{{\tilde \tau}_1}^2)
 \over (m_{{\tilde \tau}_2}^2 - m_{{\tilde \tau}_1}^2)}  \ , \cr
& & \cr
M_{33}^{\tau} & = & {m_{\tau}^4 \mu^2 A_{\tau}^2 \sin^2 \phi_{\tau}
\over 8 \pi^2 v^2 \sin^4 \beta}
{g(m_{{\tilde \tau}_1}^2, \ m_{{\tilde \tau}_2}^2) \over (m_{{\tilde \tau}_2}^2
- m_{{\tilde \tau}_1}^2 )^2}
+ {m_{\tau}^2 \mu A_{\tau} \cos \phi_{\tau} \over 16 \pi^2 v^2 \sin^3 \beta \cos \beta}
f(m_{{\tilde \tau}_1}^2, \ m_{{\tilde \tau}_2}^2)  \ , \cr
& & \cr
M_{12}^{\tau} & = & {m_{\tau}^4 A_{\tau} \mu \over 8 \pi^2 v^2 \sin^2 \beta}
{K_{{\tilde \tau}_1} K_{{\tilde \tau}_2}  
\over (m_{{\tilde \tau}_2}^2 - m_{{\tilde \tau}_1}^2)^2}
{g(m_{{\tilde \tau}_1}^2, \ m_{{\tilde \tau}_2}^2)}    
- {m_{\tau}^2 \mu A_{\tau} \cos \phi_{\tau} \over 16 \pi^2 v^2 \sin^2 \beta}
f(m_{{\tilde \tau}_1}^2, \ m_{{\tilde \tau}_2}^2)  \cr
& &\mbox{} + {m_{\tau}^4 \mu K_{{\tilde \tau}_1} \over 8 \pi^2 v^2 \sin^2 \beta}
{\log (m_{{\tilde \tau}_2}^2 / m_{{\tilde \tau}_1}^2)
 \over (m_{{\tilde \tau}_2}^2 - m_{{\tilde \tau}_1}^2)}  \ , \cr
& & \cr
M_{13}^{\tau} &=& \mbox{} -{m_{\tau}^4 \mu^2  A_{\tau} K_{{\tilde 
\tau}_1} \sin \phi_{\tau} \over 8 \pi^2 v^2 \sin^3 \beta}
{g(m_{{\tilde \tau}_1}^2, \ m_{{\tilde \tau}_2}^2) \over
(m_{{\tilde \tau}_2}^2 - m_{{\tilde \tau}_1}^2)^2 }  
+ {m_{\tau}^2 \mu A_{\tau} \sin \phi_{\tau} \over 16 \pi^2 v^2 \sin \beta \tan \beta}
f(m_{{\tilde \tau}_1}^2, \ m_{{\tilde \tau}_2}^2)   \ , \cr
& & \cr
M_{23}^{\tau} &   = & \mbox{}  -  {m_{\tau}^4 \mu  A_{\tau}^2  K_{{\tilde 
\tau}_2} \sin \phi_{\tau} \over 8  \pi^2 v^2 \sin^3 \beta}  {g(m_{{\tilde \tau}_1}^2, \ m_{{\tilde 
\tau}_2}^2) \over (m_{{\tilde \tau}_2}^2 - m_{{\tilde \tau}_1}^2)^2 }  
+ {m_{\tau}^2  \mu  A_{\tau} \sin   \phi_{\tau} \over 16   \pi^2 v^2  \sin \beta}  f(m_{{\tilde 
\tau}_1}^2, \ m_{{\tilde\tau}_2}^2) \cr
& &\mbox{} - {m_{\tau}^4 \mu A_{\tau} \sin \phi_{\tau} \over 8 \pi^2 v^2 \sin^3 \beta} 
{\log (m_{{\tilde \tau}_2}^2  / m_{{\tilde \tau}_1}^2)  \over (m_{{\tilde \tau}_2}^2  - m_{{\tilde 
\tau}_1}^2)}  \ . 
\nonumber
\end{eqnarray}

%*********************  W, C, X1, X2  ******************************
\vskip 0.2 in
\noindent
{\large {\bf Appendix F}}
\vskip 0.2 in
\noindent
The elements   for the  mass  matrix  of the   neutral Higgs  bosons  due  to  the 
radiative 
contributions of the $W$ boson, charged Higgs boson, and charginos are 
\begin{eqnarray}
M_{11}^{\tilde \chi} & = & \mbox{} - {m_W^4 \cos^2 \beta \over 4 \pi^2 v^2}
{K _{{\tilde \chi}_1}^2 g(m_{{\tilde \chi}_1}^2, \ m_{{\tilde \chi}_2}^2)
\over (m_{{\tilde \chi}_2}^2 - m_{{\tilde \chi}_1}^2)^2}  
+ {m_W^2 M_2 \mu \cos \phi_c \over 4 \pi^2 v^2 \cot \beta}  
f(m_{{\tilde \chi}_1}^2, \ m_{{\tilde \chi}_2}^2) \cr
& &\mbox{} - {m_W^4 \cos^2 \beta \over 2 \pi^2 v^2} 
{K_{{\tilde \chi}_1} \log ({m_{{\tilde  \chi}_2}^2 / m_{{\tilde  \chi}_1}^2}) \over (m_{{\tilde 
\chi}_2}^2 - m_{{\tilde \chi}_1}^2)} 
+ {m_W^4 \cos^2 \beta \over 8 \pi^2 v^2} \log \left ({m_W^6 m_{C^+}^2 \over 
m_{{\tilde \chi}_1}^4 m_{{\tilde \chi}_2}^4} \right ) \cr
& &\mbox{} - {m_W^2 \sin^2 \beta \over 16 \pi^2 v^2} 
\left \{m_{C^+}^2 \log \left ({m_{C^+}^2  \over \Lambda^2} \right ) -  m_{C^+}^2 \right \} \ , 
\cr
 & & \cr
M_{22}^{\tilde \chi} & = & \mbox{} - {m_W^4 \sin^2 \beta \over 4 \pi^2 v^2}
{K_{{\tilde \chi}_2}^2 g(m_{{\tilde \chi}_1}^2, \ m_{{\tilde \chi}_2}^2)
\over (m_{{\tilde \chi}_2}^2 - m_{{\tilde \chi}_1}^2)^2}  
+ {m_W^2 M_2 \mu \cos \phi_c \over 4 \pi^2 v^2 \tan \beta}  
f(m_{{\tilde \chi}_1}^2, \ m_{{\tilde \chi}_2}^2) \cr
& &\mbox{} - {m_W^4 \sin^2 \beta \over 2 \pi^2 v^2} 
{\Delta_{{\tilde \chi}_2} \log ({m_{{\tilde  \chi}_2}^2 / m_{{\tilde  \chi}_1}^2}) \over (m_{{\tilde 
\chi}_2}^2 - m_{{\tilde \chi}_1}^2)} 
+ {m_W^4 \sin^2 \beta \over 8 \pi^2 v^2} \log \left ({m_W^6 m_{C^+}^2 \over 
m_{{\tilde \chi}_1}^4 m_{{\tilde \chi}_2}^4} \right ) \cr
& &\mbox{} - {m_W^2 \cos^2 \beta \over 16 \pi^2 v^2} 
\left \{m_{C^+}^2 \log \left ({m_{C^+}^2 \over \Lambda^2} \right  ) - m_{C^+}^2 \right \}  \ , 
\cr
 & & \cr
M_{33}^{\tilde \chi} & = & \mbox{} - {m_W^4 \over \pi^2 v^2}
{M_2^2 \mu^2 \sin^2 \phi_c \over (m_{{\tilde \chi}_2}^2 - m_{{\tilde \chi}_1}^2)^2} g(m_{{\tilde 
\chi}_1}^2, \ m_{{\tilde \chi}_2}^2)
+ {m_W^2 M_2 \mu \cos \phi_c \over \pi^2 v^2 \sin 2 \beta}
f(m_{{\tilde \chi}_1}^2, \ m_{{\tilde \chi}_2}^2)     \cr
& &\mbox{} - {m_W^2 \over 16 \pi^2 v^2} 
\left \{m_{C^+}^2 \log \left ({m_{C^+}^2 \over \Lambda^2} \right  ) - m_{C^+}^2 \right \}  \ , 
\cr
 & & \cr
M_{12}^{\tilde \chi} & = &\mbox{} - {m_W^4 \over 8 \pi^2 v^2}
{\sin 2 \beta K_{{\tilde \chi}_1} K_{{\tilde \chi}_2}
\over (m_{{\tilde \chi}_2}^2 - m_{{\tilde \chi}_1}^2)^2}
g(m_{{\tilde \chi}_1}^2, \ m_{{\tilde \chi}_2}^2)
+ {m_W^2 M_2 \mu \cos \phi_c \over 4 \pi^2 v^2} 
f(m_{{\tilde \chi}_1}^2, \ m_{{\tilde \chi}_2}^2) \cr
& &\mbox{} - {m_W^4 \over 8 \pi^2 v^2}
{\sin 2 \beta (K_{{\tilde \chi}_1} +K_{{\tilde \chi}_2}) 
\over (m_{{\tilde \chi}_2}^2 - m_{{\tilde \chi}_1}^2)}
\log \left ({m_{{\tilde \chi}_2}^2 \over m_{{\tilde \chi}_1}^2} \right) 
+ {m_W^4 \sin 2 \beta \over 16 \pi^2 v^2}
\log \left ( {m_W^6 m_{C^+}^2 \over m_{{\tilde \chi}_1}^4 m_{{\tilde \chi}_2}^4} \right ) \cr
& &\mbox{} - {m_W^2 \sin 2 \beta \over 32 \pi^2 v^2}
\left \{m_{C^+}^2 \log \left ({m_{C^+}^2 \over \Lambda^2} \right ) - m_{C^+}^2 \right \}     \ 
,  \cr
 & & \cr
M_{13}^{\tilde \chi} & = &\mbox{} - {m_W^4 \over 2 \pi^2 v^2} 
{M_2 \mu   \cos \beta  K_{{\tilde  \chi}_1}  \sin \phi_c   \over (m_{{\tilde  \chi}_2}^2  - 
m_{{\tilde \chi}_1}^2)^2} g(m_{{\tilde \chi}_1}^2, \ m_{{\tilde \chi}_2}^2) 
+ {m_W^2  M_2 \mu  \sin \phi_c  \over 4  \pi^2 v^2  \sec \beta}   f(m_{{\tilde \chi}_1}^2,  \ 
m_{{\tilde \chi}_2}^2) \cr
& &\mbox{}  -  {m_W^4 M_2  \mu  \cos \beta  \sin  \phi_c \over  2  \pi^2 v^2   
(m_{{\tilde 
\chi}_2}^2  -  m_{{\tilde  \chi}_1}^2)}  \log  \left  (  {m_{{\tilde  \chi}_2}^2  \over  m_{{\tilde 
\chi}_1}^2} \right)
 \ , \cr
 & & \cr
M_{23}^{\tilde \chi} & = &\mbox{} - {m_W^4 M_2 \mu \over 2 \pi^2 v^2}
{\sin \beta   K_{{\tilde \chi}_2}   \sin \phi_c  \over  (m_{{\tilde  \chi}_2}^2  -  m_{{\tilde 
\chi}_1}^2)^2} g(m_{{\tilde \chi}_1}^2, \ m_{{\tilde \chi}_2}^2)  
+ {m_W^2  M_2 \mu  \sin  \phi_c \over  4 \pi^2  v^2  \csc \beta}  f(m_{{\tilde \chi}_1}^2,   
\ 
m_{{\tilde \chi}_2}^2) \cr 
& &\mbox{} - {m_W^4 M_2 \mu \over  2 \pi^2 v^2} {\sin \beta \sin  \phi_c \over (m_{{\tilde 
\chi}_2}^2  -  m_{{\tilde  \chi}_1}^2)}  \log  \left  (  {m_{{\tilde  \chi}_2}^2  \over  m_{{\tilde 
\chi}_1}^2} \right) \ .
\nonumber
\end{eqnarray}

\vfil\eject
%******************************************************************
\begin{thebibliography}{99}
%******************************************************************
\bibitem{1} S. Glashow, Nucl. Phys. {\bf 22}, 579 (1961);
    S. Weinberg, Phys. Rev. Lett. {\bf 19}, 1264 (1967);
    A. Salam, in {\it Proc. 8th Nobel Symposium, 367}, edited by N.
    Svartholm (Almquist and Wiksell, Stockholm, 1968).
\bibitem{2} H.P. Nilles, Phys. Rep. {\bf 110}, 1 (1984);
    J.F. Gunion and H.E. Haber, Nucl. Phys. B {\bf 272}, 1 (1986);
    J.F. Gunion, H.E. Haber, G.L. Kane, and S. Dawson,
    {\it The Higgs Hunters' Guide} (Addison-Wesley Pub. Co., Redwood
    City, CA, USA, 1990).
\bibitem{3} S. Weinberg, Phys. Rev. Lett. {\bf 37}, 657 (1976); 
    M. Masip and A. Rasin, Phys. Rev. D {\bf 52}, 3768 (1995); 
    M. Masip and A. Rasin, Nucl. Phys. B {\bf 460}, 449 (1996);
    M. Masip and A. Rasin, Phys. Rev. D {\bf 58}, 035007 (1998).
\bibitem{4} A. Pomarol, Phys. Lett. B {\bf 287}, 331 (1992);
  N. Maekawa, Phys. Lett. B {\bf 282}, 387 (1992);
  N. Haba, Phys. Lett. B {\bf 398}, 305 (1997);
  O. Lebedev, Eur. Phys. J. C {\bf 4}, 363 (1998).
\bibitem{5} A. Pilaftsis, Phys. Lett. B {\bf 435}, 88 (1998);
    A. Pilaftsis, Phys. Rev. D {\bf 58}, 096010 (1998).
    D.A. Demir, Phys. Rev. D {\bf 60}, 055006 (1999).
\bibitem{6} M. Carena, J. Ellis, A. Pilaftsis, and C.E.M. Wagner,  Nucl. Phys. B {\bf 586}, 92 (2000);
    S.Y. Choi, M. Drees, and J.S. Lee, Phys. Lett. B {481}, 57 (2000);
    G.L. Kane and L.T. Wang, Phys. Lett. B {\bf 488}, 383 (2000).
        M. Carena, J. Ellis, A. Pilaftsis, and C.E.M. Wagner,
    Phys. Lett. B {\bf 495}, 155 (2000);
    S.W. Ham, S.K. Oh, E.J. Yoo, and H.K. Lee, J. Phys. G {\bf 27}, 1 (2001);
    A.G. Akeroyd and A. Arhrib, Phys. Rev. D {\bf 64} 095018 (2001);
    S.Y. Choi, K. Hagiwara, and J.S. Lee, Phys. Rev. D {\bf 64}, 032004 (2001);
Phys. Lett. B {\bf 529}, 212 (2002).
\bibitem{7} T. Ibrahim and P. Nath, Phys. Rev. D {63}, 035009 (2001).
\bibitem{8} T. Ibrahim and P. Nath, .
\bibitem{9} S. Coleman and E. Weinberg, Phys. Rev. D {\bf 7}, 1888 (1973).
%******************************************************************
\end{thebibliography}
%******************************************************************
\vfil\eject

%**************************************************************
{\bf Figure Captions}
\vskip 0.3 in
\noindent
Fig. 1(a) : The plot of the $(1,3)$-element of the mass matrix of the neutral Higgs bosons at the one-loop level, $M_{13}$ (solid curve), as a function of $\phi_1$ for  $\phi_t = \phi_b = \phi_{\tau} = \phi_c (\phi_2) = \pi/4$, $m_2$ = 150 GeV and $m_3$ = 200 GeV, $\mu$ = 400 GeV, $m_Q$ = 500 GeV, $m_T$ = 800 GeV, $A_t$ = 300 GeV, and $M_2$ = 200 GeV.
We set $\tan \beta$ = 5. 
These values for the parameters are the same as Table 3, except that $\phi_1$ is taken as a variable. 
We also plot $M_{13}^{{\tilde \chi}^0}$ (dashed curve), $M_{13}^t$ (dot-dashed
curve), and $M_{13}^{\tilde \chi}$ (dotted curve). 

\vskip 0.2 in
\noindent
Fig. 1(b) : The same as Fig. 1(a), except that $M_{23}$ (solid curve), $M_{23}^{{\tilde \chi}^0}$ (dashed curve), $M_{23}^t$ (dot-dashed curve), and $M_{23}^{\tilde \chi}$ (dotted curve) are plotted.

\vskip 0.2 in
\noindent
Fig. 2(a) : The plot of the $(1,3)$-element of the mass matrix of the neutral Higgs bosons at the one-loop level, $M_{13}$ (solid curve), as a function of $\phi_1$, for the parameter values of Table 4: 
$\phi_t = \phi_b = \phi_{\tau} = \phi_c (\phi_2) = \pi/4$;  
 $m_2$ = 150 GeV, $m_3$ = 200 GeV, $\mu$ = 400 GeV, $m_Q$ = 500 GeV, $m_T$ = 800 GeV, $A_t$ = 300 GeV, and $M_2$ = 200 GeV; $\tan \beta$ = 20; 
and $\phi_1$ is taken as a variable.
$M_{13}^{{\tilde \chi}^0}$ (dashed curve), $M_{13}^t$ (dot-dashed curve), and $M_{13}^{\tilde \chi}$ (dotted curve) are also plotted.

\vskip 0.2 in
\noindent
Fig. 2(b) : The same as Fig. 2(a), except that $M_{23}$ (solid curve), $M_{23}^{{\tilde \chi}^0}$ (dashed curve), $M_{23}^t$ (dot-dashed curve), and $M_{23}^{\tilde \chi}$ (dotted curve) are plotted.

\vfil\eject

%************************* TABLE 1, 2 ************************************
%{\bf Table Caption}
%\vskip 0.3 in
%\noindent
\begin{table}[ht]
\caption{The elements of the symmetric mass matrix of the neutral Higgs bosons in the $(S_1, S_2, A)$-basis, at the one-loop level without CP violation.
The unit is (GeV)$^2$.
The values of the relevant parameters are $m_2$ = 150 GeV, $m_3$ = 200 GeV, $\mu$ = 400 GeV, $m_Q$ = 500 GeV, $m_T$ = 800 GeV, $A_t$ = 300 GeV, and $M_2$ = 200 GeV. 
We set $\tan \beta$ = 5. 
The minimum equations yield $\Lambda$ = 799.7 GeV and $m_1$ = 456.0 GeV.
The number in the first row in each column is the tree-level value.
The number in the last row in each column is the sum of all numbers in the preceding rows, representing the value at the one-loop level.
The numbers in each column in between the two rows represent the various loop contributions as decomposed in the one-loop Higgs potential.
The last columnn lists the $M_{33}$ elements, which account solely for the pseudoscalar Higgs boson, whereas the first three columns account for the $2\times2$ mass matrix of the scalar Higgs bosons.} 
\begin{center}
\begin{tabular}{c|c|c|c||c|c} 
\hline 
\hline
$(i, j)$         & (1, 1)      & (2, 2) & (1, 2) & & (3, 3) \\ 
\hline
\hline
$M_{ij}^0$  &   200318.5  & 15962.5  &  $- 41592.5$ & $(m^2_A)^0$ &  208000.0  \\
\hline
$M_{ij}^t$  &  3881.8 &   7008.1 &  $- 384.4$ &   $(m^2_A)^t$ & 4061.3  \\
\hline
$M_{ij}^b$ &  0.2 & 2.5 &  $- 12.3$ &  $(m^2_A)^b$ & 63.9 \\
\hline
$M_{ij}^{\tau}$ & 0.5 & 0.02 & $- 0.1$ &  $(m^2_A)^{\tau}$ & 0.5 \\
\hline
$M_{ij}^{\tilde  \chi}$ &   4502.1 &  $- 61.4$   & 893.0  & $(m^2_A)^{\tilde \chi}$ & 8780.8  
\\
\hline
$M_{ij}^h$ & $- 1268.2$ & $- 1035.7$ & 393.5 &  $(m^2_A)^h$ & $- 1775.5$  \\
\hline
$M_{ij}^{{\tilde \chi}^0}$ &   $ 2819.7$ & 464.0  & 2495.3 & $(m^2_A)^{{\tilde \chi}^0}$  & $ 2670.9$ \\
\hline
$M_{ij}$ & 210254.7& 22340.0 &  $- 38207.5$ & $m_A^2$ & 221801.9  
 
\\
\hline
\hline
\end{tabular}
\end{center}
\caption{The same as Table 1, except for $\tan \beta$ = 20; 
$\Lambda$ = 911.3 GeV and $m_1$ = 917.4 GeV.}
\begin{center}
\begin{tabular}{c|c|c|c||c|c} 
\hline 
\hline
$(i, j)$         & (1, 1)      & (2, 2) & (1, 2)   & &  (3, 3) \\ 
\hline
\hline
$M_{ij}^0$ & 800020.6 & 10260.3 &  $- 40413.0$ &   $(m^2_A)^0$ & 802000.0 \\
\hline
$M_{ij}^t$ & 26929.2  & 6667.9 & $-1023.8$ &  $(m^2_A)^t$ & 27012.4 \\
\hline
$M_{ij}^b$ & 4.1 & 14.4 & $- 314.7$ & $(m^2_A)^b$ & 6310.3 \\
\hline
$M_{ij}^{\tau}$  &  3.1 &   0.008 &  $-  0.2$  &  $(m^2_A)^{\tau}$ & 3.1  \\
\hline
$M_{ij}^{\tilde \chi}$  &  18987.8  & $-  169.8$ & 962.0  & $(m^2_A)^{\tilde \chi}$ & 36948.8 \\
\hline
$M_{ij}^h$ & $- 2349.5$ & $- 1243.9$ &  197.7 &  $(m^2_A)^h$ & $- 8063.5$  \\
\hline
$M_{ij}^{{\tilde \chi}^0}$  & $ 11938.5$  & 458.0 & 4481.2  & $(m^2_A)^{{\tilde \chi}^0}$ & $11689.9$ \\
\hline
$M_{ij}$ &  855533.9 &  15986.9 &  $- 36110.7$    &  $m_A^2$ & 875901.0 \\
\hline
\hline
\end{tabular}
\end{center}
\end{table}
\vfil\eject

%******************  TABLE 3, 4   *******************************
\begin{table}[ht]
\caption{The elements of the symmetric mass matrix of the neutral Higgs bosons in the $(S_1, S_2, A)$-basis, at the one-loop level with CP violation for $\phi_t = \phi_b = \phi_{\tau} = \phi_c (\phi_2) = \phi_1 = \pi/4$.
The unit is (GeV)$^2$.
The values of the relevant parameters are $m_2$ = 150 GeV, $m_3$ = 200 GeV, $\mu$ = 400 GeV, $m_Q$ = 500 GeV, $m_T$ = 800 GeV, $A_t$ = 300 GeV, and $M_2$ = 200 GeV.
We set $\tan \beta$ = 5. 
The minimum equations yield $\Lambda$ = 800.1 GeV and $m_1$ = 454.6 GeV.  
The number in the first row in each column is the tree-level value.
The number in the last row in each column is the sum of all numbers in the preceding rows, representing the value at the one-loop level.
The numbers in each column in between the two rows represent the various loop contributions as decomposed in the one-loop Higgs potential.}
\begin{center}
\begin{tabular}{c|c|c|c|c|c|c} 
\hline 
\hline
$(i, j)$  & (1, 1) & (2, 2) & (3, 3) & (1, 2) & (1, 3) & (2, 3) \\ 
\hline
\hline
$M_{ij}^0$ &  200318.5 & 15962.5  & 208000.0 & $- 41592.5$  &  0.0 &  0.0  \\
\hline
$M_{ij}^t$ &  2768.3 & 7069.2 & 2885.8 & $- 250.4$ & 119.3 & 319.0 \\
\hline
$M_{ij}^b$ & 43.7 & 1.7 & 45.3 & $- 8.7$ &  1.7 & 8.5  \\
\hline
$M_{ij}^{\tau}$ &  0.1 & 0.005 & 0.1 & $- 0.02$ & 0.004 & 0.02 \\
\hline
$M_{ij}^{\tilde \chi}$ & 3343.3 & $- 103.7$ & 6387.3 & 654.3 & 101.9 & 84.3 \\
\hline
$M_{ij}^h$ & $- 1272.2$ & $- 1035.5$ & $- 1765.1$ & 394.7 & 0.0 & 0.0  \\
\hline
$M_{ij}^{{\tilde \chi}^0}$ & $ 1535.8$ & 391.7 & $ 1553.9$ &  3347.4 & $- 44.5$ & 2465.0  \\
\hline
$M_{ij}$ & 206737.5&  22285.9 &  217107.3 & $- 37455.3$ & 178.5 & 2876.9 \\
\hline
\hline
\end{tabular}
\end{center}
\caption{The same as Table 3, except for $\tan \beta$ = 20; 
$\Lambda$ = 912.6 GeV and $m_1$ = 911.9 GeV.}
\begin{center}
\begin{tabular}{c|c|c|c|c|c|c} 
\hline 
\hline
$(i, j)$  & (1, 1) & (2, 2) & (3, 3) & (1, 2) & (1, 3) & (2, 3) \\ 
\hline
\hline
$M_{ij}^0$ & 800020.6 & 10260.3 &  802000.0 & $-40413.0$ &  0.0 &  0.0  \\
\hline
$M_{ij}^t$ & 19193.3 & 6696.0 & 19242.7 &  -724.9 & 55.6 & $ 743.7$ \\
\hline
$M_{ij}^b$ & 4472.7 & 9.9  & 4480.3 &   $-223.2$ & 11.5 & 223.9  \\
\hline
$M_{ij}^{\tau}$ &  0.7 & 0.002 & 0.7 & $-0.04$ & 0.002 & 0.04 \\
\hline
$M_{ij}^{\tilde \chi}$ & 13760.6 & $-182.0$ & 26486.4 & 694.8 & 30.3 & $ 5.4$  \\
\hline
$M_{ij}^h$ & $- 2387.1$ & $- 1243.8$ & $- 7991.7$ & 201.2 & 0.0 &  0.0  \\
\hline
$M_{ij}^{{\tilde \chi}^0}$ &  $ 6426.5$ & 421.5 &   $ 6470.7$ & 2125.7 &  700.0 & 2950.4 \\
\hline
$M_{ij}$ & 841487.3 & 15962.9 & 850689.1 & $- 38339.5 $ &  797.5 & 3923.4 \\
\hline
\hline
\end{tabular}
\end{center}
\end{table}
\vfil\eject
%***********************************************************************
%Figure
\setcounter{figure}{0}
\def\figurename{}{}%
% (FIG 1a)
\renewcommand\thefigure{Fig. 1(a)}
\begin{figure}[t]
\epsfxsize=13cm
\hspace*{2.cm}
\epsffile{fig1a.eps}
\caption[plot]{The plot of the $(1,3)$-element of the mass matrix of the neutral Higgs bosons at the one-loop level, $M_{13}$ (solid curve), as a function of $\phi_1$, for the parameter values of Table 3: 
$\phi_t = \phi_b = \phi_{\tau} = \phi_c (\phi_2) = \pi/4$;  
$m_2$ = 150 GeV, $m_3$ = 200 GeV, $\mu$ = 400 GeV, $m_Q$ = 500 GeV, $m_T$ = 800 GeV, $A_t$ = 300 GeV, and $M_2$ = 200 GeV; 
$\tan \beta$ = 5; 
$\Lambda$ = 878.6 GeV and $m_1$ = 449.6 GeV;
and $\phi_1$ is taken as a variable.
$M_{13}^{{\tilde \chi}^0}$ (dashed curve), $M_{13}^t$ (dot-dashed curve), and $M_{13}^{\tilde \chi}$ (dotted curve) are also plotted.}
\end{figure}

% (FIG 1b)
\renewcommand\thefigure{Fig. 1(b)}
\begin{figure}[t]
\epsfxsize=13cm
\hspace*{2.cm}
\epsffile{fig1b.eps}
\caption[plot]{The same as Fig. 1(a), except that $M_{23}$ (solid curve), $M_{23}^{{\tilde \chi}^0}$ (dashed curve), $M_{23}^t$ (dot-dashed curve), and $M_{23}^{\tilde \chi}$ (dotted curve) are plotted.}
\end{figure}

% (FIG 2a)
\renewcommand\thefigure{Fig. 2(a)}
\begin{figure}[t]
\epsfxsize=13cm
\hspace*{2.cm}
\epsffile{fig2a.eps}
\caption[plot]{The plot of the $(1,3)$-element of the mass matrix of the neutral Higgs bosons at the one-loop level, $M_{13}$ (solid curve), as a function of $\phi_1$, for the parameter values of Table 4: 
$\phi_t = \phi_b = \phi_{\tau} = \phi_c (\phi_2) = \pi/4$;  
$m_2$ = 150 GeV, $m_3$ = 200 GeV, $\mu$ = 400 GeV, $m_Q$ = 500 GeV, $m_T$ = 800 GeV, $A_t$ = 300 GeV, and $M_2$ = 200 GeV; 
$\tan \beta$ = 20; 
$\Lambda$ = 705.0 GeV and $m_1$ = 891.3 GeV;
and $\phi_1$ is taken as a variable.
$M_{13}^{{\tilde \chi}^0}$ (dashed curve), $M_{13}^t$ (dot-dashed curve), and $M_{13}^{\tilde \chi}$ (dotted curve) are also plotted.}
\end{figure}

% (FIG 2b)
\renewcommand\thefigure{Fig. 2(b)}
\begin{figure}[t]
\epsfxsize=13cm
\hspace*{2.cm}
\epsffile{fig2b.eps}
\caption[plot]{The same as Fig. 2(a), except that $M_{23}$ (solid curve), $M_{23}^{{\tilde \chi}^0}$ (dashed curve), $M_{23}^t$ (dot-dashed curve), and $M_{23}^{\tilde \chi}$ (dotted curve) are plotted.}
\end{figure}

%***********************************************************************
\end{document}
%***********************************************************************

