%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\documentstyle[12pt]{article}
\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%
\hbadness=10000
\hbadness=10000
\begin{titlepage}
\nopagebreak
\def\thefootnote{\fnsymbol{footnote}}
%%%%%%%%%%%%%%%%%%%%%%%%%% Preprint No. %%%%%%%%%%%
\begin{flushright}
        {\normalsize
DPSU-98-8\\
November, 1998   }\\
\end{flushright}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\vspace{1cm}
\begin{center}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
{\Large Model-independent analysis of soft masses\\
in heterotic string models \\
with anomalous $U(1)$ symmetry}

\vspace{1cm}

{\bf Yoshiharu Kawamura
\footnote[1]{e-mail: ykawamu@gipac.shinshu-u.ac.jp}}

%(and $\cdots$)

\vspace{1cm}
Department of Physics, Shinshu University \\

   Matsumoto, 390-0802 Japan \\

\end{center}
\vspace{1cm}

\nopagebreak
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{abstract}
We study the magnitudes of soft masses
in heterotic string models with anomalous $U(1)$ symmetry 
model-independently.
In most cases, $D$-term contribution to soft scalar masses 
is expected to be comparable to or dominant over other contributions
provided that supersymmetry breaking
is mediated by the gravitational interaction and/or an anomalous
$U(1)$ symmetry and the magnitude of vacuum energy is not more than
of order $m_{3/2}^2 M^2$.
\end{abstract}

\end{titlepage}

\newpage

\section{Introduction}

Superstring theories are powerful candidates for the unification
theory of all forces including gravity.
The supergravity theory (SUGRA) is effectively constructed from 
4-dimensional
(4D) string model using several methods \cite{ST-SG,OrbSG,OrbSG2}. 
The structure of SUGRA is constrained
by gauge symmetries including an anomalous $U(1)$ symmetry
($U(1)_A$) \cite{ST-FI} and stringy symmetries 
such as duality \cite{duality}.

4D string models have several open questions and
two of them are pointed out here.
The first one is what the origin of supersymmetry (SUSY) breaking is.
Although intersting scenarios 
such as SUSY breaking mechanism due to gaugino condensation \cite{gaugino}
and Scherk-Schwarz mechanism \cite{SS}
have been proposed, realistic one has not been identified yet.
The second one is how the vacuum expectation value (VEV) of dilaton
field $S$ is stabilized.
It is difficult to realize the stabilization with a realistic VEV of $S$
using a K\"ahler potential at the tree level alone
without any conspiracy among several terms which 
appear in the superpotential \cite{S-Stab}.
A K\"ahler potential generally receives radiative
corrections as well as non-perturbative ones.
Such corrections may be sizable for the part related to
$S$ \cite{corr1,corr2}.
It is important to solve these enigmas
in order not only to understand the structure of more fundamental theory
at a high energy scale but also to know the complete SUSY particle spectrum
at the weak scale, but it is not an easy task
because of ignorance of the explicit forms of 
fully corrected total K\"ahler potential.
At present, it would be meaningful to get any information on SUSY
particle spectrum model-independently.\footnote{
The stability of $S$ and soft SUSY breaking parameters are discussed
in the dilaton SUSY breaking scenario in Ref.\cite{Casas}.}


In this paper, we study the magnitudes of soft SUSY breaking parameters
in heterotic string models with $U(1)_A$ and derive model-independent
predictions for them
without specifying SUSY breaking mechanism
and the dilaton VEV fixing mechanism.
The idea is based on that in the work by Ref.\cite{ST-soft}.
The soft SUSY breaking terms have been derived from 
$\lq\lq$standard string model" and analyzed 
under the assumption that SUSY is broken by $F$-term condensations 
of the dilaton field and/or moduli fields $M^i$.
We relax this assumption such that 
SUSY is broken by $F$-term condensation of $S$, $M^i$ and/or 
matter fields with non-vanishing $U(1)_A$ charge 
since the scenario based on $U(1)_A$ as a mediator
of SUSY breaking is also possible \cite{DP}.
In particular, we make a comparison of magnitudes between
$D$-term contribution to scalar masses and $F$-term ones and 
a comparison of magnitudes among scalar masses, gaugino masses 
and $A$-parameters.
The features of our analysis are as follows.
The study is carried out in the framework of SUGRA 
model-independently,\footnote{The model-dependent analyses are carried out
in Ref.\cite{DP,model-dep1,model-dep2}.}
i.e., we do not specify SUSY breaking mechanism, extra matter contents, 
the structure of superpotential and
the form of K\"ahler potential related to $S$.
We treat all fields including $S$ and $M^i$ as dynamical fields. 

The paper is organized as follows. 
In the next section, we explain the general structure of SUGRA briefly
with some basic assumptions of SUSY breaking.
We study the magnitudes of soft SUSY breaking parameters 
in heterotic string models with $U(1)_A$ model-independently in section 3. 
Section 4 is devoted to conclusions and some comments.


\section{General structure of SUGRA}

We begin by reviewing the scalar potential in SUGRA \cite{SUGRA,JKY}.  
It is specified by two functions, the total K\"ahler
potential $G(\phi, \bar \phi)$ and the gauge kinetic function
$f_{\alpha \beta}(\phi)$ with $\alpha$, $\beta$  being indices of 
the adjoint representation of the gauge group.   
The former is a sum of the K\"ahler potential $K(\phi, \bar \phi)$ 
and (the logarithm of) the superpotential $W(\phi)$
\begin{equation}
   G(\phi, \bar \phi)=K(\phi, \bar \phi) +M^{2}\ln |W (\phi) /M^{3}|^2
\label{total-Kahler}
\end{equation}
where $M=M_{Pl}/\sqrt{8\pi}$ with $M_{Pl}$ being the Planck mass, and is
referred to as the gravitational scale.
We have denoted scalar fields in the chiral multiplets by $\phi^I$
and their complex conjugate by $\bar \phi_J$.  
The scalar potential is given by
\begin{eqnarray}
   V &=& M^{2}e^{G/M^{2}} (G_I (G^{-1})^I_J G^J-3M^{2})
 + \frac{1}{2} (Re f^{-1})_{\alpha \beta} \hat D^{\alpha} \hat D^{\beta}
\label{scalar-potential}
\end{eqnarray}
where
\begin{eqnarray}
\hat D^\alpha =  G_I ( T^\alpha \phi)^I = (\bar \phi T^\alpha)_J G^J.
\label{hatD}
\end{eqnarray}
Here $G_I=\partial G/\partial \phi^I$, 
$G^J=\partial G/\partial \bar \phi_J$ etc, and
$T^\alpha$ are gauge transformation generators. 
Also in the above, $(Re f^{-1})_{\alpha \beta}$ and  
$(G^{-1})^I_J$ are the inverse matrices of $Re f_{\alpha \beta}$ and  
$G^I_J$, respectively, and a summation over $\alpha$,... and  $I$,... is
understood.  
The last equality in Eq.(\ref{hatD}) comes from the
gauge invariance of the total K\"ahler potential.
The $F$-auxiliary fields of the chiral multiplets are given by
\begin{equation}
   F^I =Me^{G/2M^{2}} (G^{-1})^I_J G^J.
\label{F}
\end{equation}
The $D$-auxiliary fields of the vector multiplets are given by
\begin{equation}
   D^{\alpha} = (Re f^{-1})_{\alpha\beta} \hat{D}^{\beta}.
\label{D}
\end{equation}
Using $F^I$ and $D^\alpha$, the scalar potential is rewritten down by
\begin{eqnarray}
   V &=& V_F + V_D , \nonumber \\
  V_F &\equiv& F_I K^I_J F^J - 3M^{4} e^{G/M^{2}}  ,
\label{VF}\\
   V_D &\equiv& \frac{1}{2} Re f_{\alpha \beta} D^{\alpha} D^{\beta}.
\label{scalar-potential 2}
\end{eqnarray}


Let us next summarize our assumptions on SUSY breaking.  
The gravitino mass $m_{3/2}$ is given by
\begin{equation}
   m_{3/2}= \langle Me^{G/2M^{2}} \rangle
\label{gravitino}
\end{equation}
where $\langle \cdots \rangle$ denotes the VEV.  
As a phase convention, it is taken to be real.  
We identify the gravitino mass with the weak scale in most cases.
It is assumed that SUSY is spontaneously broken by some $F$-term condensations
($\langle F \rangle \neq 0$) for singlet fields under 
the standard model gauge group
and/or some $D$-term condensations ($\langle D \rangle \neq 0$)
for broken gauge symmetries.
We require that the VEVs of $F^I$ and $D^{\alpha}$ should satisfy
\begin{eqnarray}
&~&   \langle (F_I K^I_J F^J)^{1/2} \rangle \leq O(m_{3/2}M) ,
\label{VEV-F} \\
&~&   \langle D^\alpha \rangle \leq O(m_{3/2}M) 
\label{VEV-D}
\end{eqnarray}
for each pair $(I,J)$ in Eq.(\ref{VEV-F}).
%(The summation over $I$ and $J$ is not made in Eq.(\ref{VEV-F}).)
Note that we allow the non-zero vacuum energy $\langle V \rangle$
of order $m_{3/2}^2 M^2$ at this level, which could be canceled
by quantum corrections.


In order to discuss the magnitudes of several quantities, it is
necessary to see consequences of the stationary condition 
$\langle \partial V /\partial \phi^I \rangle =0$.   From
Eq.(\ref{scalar-potential}),  we find
\begin{eqnarray}
 \partial V /\partial \phi^I &=& G_I ( {V_F \over M^2}
   + M^{2}e^{G/M^{2}} ) + M e^{G/2M^{2}} G_{IJ} F^J \nonumber \\
 &~& - F_{I'} G^{I'}_{J'I} F^{J'} 
     - \frac{1}{2} (Re f_{\alpha \beta}),_I D^\alpha D^\beta
\nonumber \\
 &~& + D^\alpha (\bar \phi T^\alpha )_J G^J_I .
\label{VI}
\end{eqnarray}
Taking its VEV and using the stationary condition, we derive the formula
\begin{eqnarray}
 m_{3/2} \langle G_{IJ} \rangle \langle F^J \rangle &=& 
 - \langle G_I \rangle ( {\langle V_F \rangle \over M^2} + m_{3/2}^2 ) 
 + \langle F_{I'} \rangle \langle G^{I'}_{J'I} \rangle 
   \langle F^{J'} \rangle
\nonumber \\
&~& + \frac{1}{2} \langle (Re f_{\alpha \beta}),_I \rangle 
   \langle D^\alpha \rangle \langle D^\beta \rangle 
    - \langle D^\alpha \rangle \langle (\bar \phi T^\alpha )_J \rangle
   \langle G^J_I \rangle .
\label{<VI>}
\end{eqnarray}
We can estimate the magnitude of SUSY mass parameter 
$\mu_{IJ} \equiv m_{3/2} (\langle G_{IJ} \rangle + \langle G_{I} \rangle
\langle G_{J} \rangle/M^2 - \langle G_{I'} (G^{-1})^{I'}_{J'} 
G^{J'}_{IJ} \rangle)$ using Eq.(\ref{<VI>}).
By multiplying $(T^\alpha \phi)^I$ to Eq.(\ref{VI}), 
a heavy-real direction is projected on.  
Using the identities derived from the gauge
invariance of the total K\"ahler potential
\begin{eqnarray}
   & & G_{IJ}(T^\alpha \phi)^J + G_J (T^\alpha )_I^J
   - K^J_I(\bar \phi T^\alpha)_J = 0,
\\
   & & K_{IJ'}^J (T^\alpha \phi)^{J'} + K^J_{J'} (T^\alpha)^{J'}_I
       -[G^{J'} (\bar \phi T^\alpha )_{J'}]_I^J = 0,
\end{eqnarray}
we obtain
\begin{eqnarray}
  {\partial V \over \partial \phi^I} (T^\alpha \phi)^I &=&
      ( {V_F \over M^2} + 2 M^2 e^{G/M^2} ) \hat D^\alpha
       - F_I F^J (\hat{D}^{\alpha})^I_J 
\nonumber \\
 &~& - \frac{1}{2} (Re f_{\beta \gamma}),_I (T^\alpha \phi)^I
       D^\beta D^\gamma
      + (\bar \phi T^\beta)_J G^J_I (T^\alpha \phi)^I D^\beta .
\label{VI2}
\end{eqnarray}
Taking its VEV and using the stationary condition, 
we derive the formula
\begin{eqnarray}
&~& \{ {(M_V^2)^{\alpha\beta} \over 2g_\alpha g_\beta} 
+ ({\langle V_F \rangle \over M^2} + 2 m_{3/2}^2)
 \langle Re f_{\alpha\beta} \rangle \} \langle D^\beta \rangle
       = \langle F_I \rangle \langle F^J \rangle 
         \langle (\hat{D}^{\alpha})^I_J \rangle \nonumber \\
&~& ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 
+ \frac{1}{2} \langle (Re f_{\beta \gamma}),_I \rangle 
   \langle (T^\alpha \phi)^I \rangle 
   \langle D^\beta \rangle \langle D^\gamma \rangle 
\label{<VI2>}
\end{eqnarray}
where $(M_{V}^{2})^{\alpha \beta}=
2g_\alpha g_\beta \langle (\bar \phi T^\beta)_J K^J_I 
(T^\alpha \phi)^I \rangle$
is the mass matrix of the gauge bosons and
$g_\alpha$ and $g_\beta$ are the gauge coupling constants.  
Using Eq.(\ref{<VI2>}), we can estimate the magnitude of
$D$-term condensations $\langle D^\beta \rangle$.

Using the scalar potential and gauge kinetic terms, 
we can obtain formulae of soft
SUSY breaking scalar masses $(m^2)_I^J$, soft SUSY breaking
gaugino masses $M_{\alpha}$ and $A$-parameters $A_{IJK}$ \cite{KMY2,Kawa1},
\begin{eqnarray}
(m^2)^J_I &=& (m^2_F)^J_I + (m^2_D)^J_I , \\
(m^2_F)^J_I &\equiv& (m_{3/2}^2 + {\langle V_F \rangle \over M^2}) 
\langle K^J_I \rangle \nonumber \\
& & + \langle {F}^{I'} \rangle 
\langle ({K}_{I'I}^{I"} (K^{-1})^{J"}_{I"}
{K}^{JJ'}_{J"} - {K}_{I'I}^{J'J}) \rangle \langle {F}_{J'} \rangle + \cdots
\label{mF} ,\\
(m^2_D)^J_I &\equiv&  \sum_{\hat{\alpha}} q^{\hat{\alpha}}_{I} 
\langle D^{\hat{\alpha}} \rangle \langle K^J_I \rangle ,
\label{mD} \\
M_\alpha &=& \langle F^I \rangle \langle (Re f_\alpha)^{-1} \rangle 
\langle f_\alpha,_I \rangle 
\label{Ma} ,\\
A_{IJK} &=& \langle F^{I'} \rangle (\langle f_{IJK},_{I'} \rangle 
+ {\langle K_{I'} \rangle \over M^2} \langle f_{IJK} \rangle
\nonumber \\
&~& - \langle K_{(II'}^{J'} \rangle \langle (K^{-1})_{J'}^{J"} \rangle 
\langle f_{J"JK)} \rangle )
\label{A}
\end{eqnarray}
where the index $\hat{\alpha}$ runs over broken gauge generators,
$Re f_\alpha \equiv Re f_{\alpha\alpha}$ and $f_{IJK}$'s are Yukawa
couplings some of which are moduli-dependent.
The $(I \cdots JK)$ in Eq.(\ref{A}) stands for a cyclic permutation 
among $I$, $J$ and $K$.
The ellipsis in $(m^2_F)^J_I$ stands for extra $F$-term contributions
and so forth.
The $(m^2_D)^J_I$ is a $D$-term contribution
to scalar masses.


\section{Heterotic string model with anomalous $U(1)$}

Effective SUGRA is derived from 4D string models taking a field
theory limit.
In this section, we study soft SUSY breaking parameters
in SUGRA from heterotic string model with $U(1)_A$.\footnote{
Based on the assumption that SUSY is broken by $F$-components of
$S$ and/or a moduli field, properties of
soft SUSY breaking scalar masses have been studied in Ref.\cite{N,KK}.}
Let us explain our starting point and assumptions first.
The gauge group $G=G_{SM} \times U(1)_A$ originates from the 
breakdown of $E_8 \times E_8'$ gauge group.
Here $G_{SM}$ is a standard model gauge group
$SU(3)_C \times SU(2)_L \times U(1)_Y$ and $U(1)_A$ is an anomalous $U(1)$
symmetry.
The anomaly is canceled by the Green-Schwarz mechanism \cite{GS}.
Chiral multiplets are classified into two categories.
One is a set of $G_{SM}$ singlet fields which the dilaton field
$S$, the moduli fields $M^i$ and 
some of matter fields $\phi^m$ belong to.
The other one is a set of $G_{SM}$ non-singlet 
fields $\phi^k$.
We denote two types of matter multiplet as 
$\phi^\lambda = \{\phi^m, \phi^k\}$ .


The dilaton field $S$ transforms as 
$S \rightarrow S-i\delta_{GS}^{A} M \theta(x)$ under $U(1)_A$ 
with a space-time dependent parameter $\theta(x)$.
Here $\delta_{GS}^{A}$ is so-called Green-Schwarz coefficient
of $U(1)_A$ and is given by 
\begin{eqnarray}
  \delta_{GS}^{A} &=& {1 \over 96\pi^2}Tr Q^A 
        = {1 \over 96\pi^2} \sum_{\lambda} q^A_{\lambda} ,
\label{delta_GS}
\end{eqnarray}
where $Q^A$ is a $U(1)_A$ charge operator,
$q^A_{\lambda}$ is a $U(1)_A$ charge of $\phi^{\lambda}$ 
and the Kac-Moody level of $U(1)_A$ is rescaled as $k_A=1$.
We find $|\delta_{GS}^{A}/q^A_m| = O(10^{-1}) \sim O(10^{-2})$
in explicit models \cite{KN,KKK}.

The requirement of $U(1)_A$ gauge invariance yields the form 
of K\"ahler potential $K$ as,
\begin{eqnarray}
   K &=& K(S + {\bar S} + \delta_{GS}^{A} V_A, M^i, {\bar M}^i,
            {\bar \phi}_\mu e^{q^A_\mu V_A}, \phi^\lambda) 
\label{K-st}
\end{eqnarray}
up to the dependence on $G_{SM}$ vector multiplets.
We assume that derivatives of the K\"ahler potential $K$ with respect to
fields including 
moduli fields or matter fields are at most of order unity
in the units where $M$ is taken to be unity.
However we do not specify the magnitude of 
derivatives of $K$ by $S$ alone.
The VEVs of $S$ and $M^i$ are supposed to be
fixed non-vanishing values by some non-perturbative effects.
It is expected that the stabilization of $S$ is
due to the physics at the gravitational scale $M$ 
or at the lower scale than $M$.
Moreover we assume that the VEV is much bigger than 
the weak scale, i.e., $O(m_{3/2}) \ll \langle K_S \rangle$.
The non-trivial transformation property of $S$
under $U(1)_A$ implies that $U(1)_A$ is broken down at some high energy
scale $M_I$.

Hereafter we consider only the case with overall modulus 
field $T$ for simplicity.
It is straightforward to apply our method to more complicated 
situations with multi-moduli fields.
The K\"ahler potential is, in general, written by
\begin{eqnarray}
   K &=& K^{(S)}(S + {\bar S} + \delta_{GS}^{A} V_A) +
K^{(T)}(T + {\bar T}) + K^{(S, T)}
\nonumber\\ 
&~& + \sum_{\lambda, \mu} ( s_{\lambda}^{\mu}(S + {\bar S} 
+ \delta_{GS}^{A} V_A)
+ t_{\lambda}^{\mu}(T + {\bar T}) + u_{\lambda}^{{\mu}(S,T)} ) 
\phi^\lambda {\bar \phi}_\mu + \cdots 
\label{K-st2}
\end{eqnarray}
where $K^{(S, T)}$ and $u_{\lambda}^{{\mu}(S,T)}$ are mixing terms between
$S$ and $T$.
The magnitudes of $\langle K^{(S, T)} \rangle$, 
$\langle s_{\lambda}^{\mu} \rangle$ and 
$\langle u_{\lambda}^{{\mu}(S,T)} \rangle$
are assumed to be $O(\epsilon_1 M^2)$, $O(\epsilon_2)$ and $(\epsilon_3)$
where $\epsilon_n$'s ($n=1,2,3$) are model-dependent parameters 
whose orders are expected not to be more than one.\footnote{
The existence of $s_{\lambda}^{\mu} \phi^\lambda {\bar \phi}_\mu$
term in $K$
and its contribution to soft scalar masses are discussed in 
4D effective theory derived through the standard embedding 
from heterotic M-theory \cite{het/M}.}
We estimate the VEV of derivatives of $K$ 
in the form including $\epsilon_n$.
For example, $\langle K^\mu_{\lambda S} \rangle \leq O(\epsilon_p/M)$
($p=2,3$).
Our consideration is applicable to models in which some of $\phi^\lambda$
are composite fields made of original matter multiplets in string
models if the K\"ahler potential meets the above requirements.
Using the K\"ahler potential (\ref{K-st2}), $\hat{D}^A$ is given by
\begin{eqnarray}
  \hat{D}^A &=& -K_S \delta_{GS}^{A} M 
      + \sum_{\lambda, \mu} K_\lambda^\mu {\bar \phi}_\mu (q^A \phi)^\lambda
      + \cdots .
\label{D-st}
\end{eqnarray}
The breaking scale of $U(1)_A$ defined by 
$M_I \equiv |\langle \phi^m \rangle|$ is estimated as
$M_I = O((\langle K_S \rangle \delta^A_{GS} M/q^A_m)^{1/2})$
from the requirement $\langle D^A \rangle \leq O(m_{3/2} M)$.
We require that $M_I$ should be equal to or be less than $M$,
and then we find that the VEV of $K_S$ has an upper bound such as
$\langle K_S \rangle \leq O(q^A_m M/\delta^A_{GS})$.


The $U(1)_A$ gauge boson mass squared $(M_{V}^{2})^A$ is given by
\begin{eqnarray}
 (M_{V}^{2})^A = 2 g_A^2 \{ \langle K^S_S \rangle (\delta_{GS}^A M)^2
            + \sum_{m, n} q^A_m q^A_n 
          \langle K^{n}_{m} \rangle 
          \langle \phi^m \rangle \langle {\bar \phi}_n \rangle \}
\label{MV2A}
\end{eqnarray}
where $g_A$ is a $U(1)_A$ gauge coupling constant.
The magnitude of $(M_{V}^{2})^A/g_A^2$ is estimated as 
$Max(O(\langle K^S_S \rangle (\delta^A_{GS} M)^2), O(q^{A2}_m M_I^2))$.
We assume that the magnitude of $(M_{V}^{2})^A/g_A^2$ is 
$O(q^{A2}_m M_I^2)$.
It leads to the inequality 
$\langle K^S_S \rangle \leq O((q^{A}_m M_I/\delta^{A}_{GS} M)^2)$.

The formula of soft SUSY breaking scalar masses on $G_{SM}$ non-singlet
fields is given by \cite{KK}
\begin{eqnarray}
(m^2)^k_l &=& (m_{3/2}^2 + {\langle V_F \rangle \over M^2}) 
\langle K^k_l \rangle 
%\nonumber \\
+ \langle {F}^{I} \rangle \langle {F}_{J} \rangle
(\langle R_{Il}^{Jk} \rangle + \langle X_{Il}^{Jk} \rangle) , \\
\langle R_{Il}^{Jk} \rangle &\equiv& 
\langle ({K}_{Il}^{I'} (K^{-1})^{J'}_{I'}
{K}^{kJ}_{J'} - {K}_{Il}^{Jk}) \rangle , \\
\langle X_{Il}^{Jk} \rangle &\equiv& 
 q^A_k ((M^2_V)^A)^{-1} \langle ({\hat D}^A)_I^J \rangle
\langle K^k_l \rangle.
\end{eqnarray}
Here we neglect extra $F$-term contributions and so forth
since they are model-dependent.
The neglect of extra $F$-term contributions is justified 
if Yukawa couplings between heavy and light fields
are small enough and the $R$-parity violation is also tiny enough.
We have used Eq.(\ref{<VI2>}) to derive the part related to 
$D$-term contribution. 
Note that the last term in r.h.s. of Eq.(\ref{<VI2>}) is negligible
when $(M_{V}^{2})^A/g_A^2$ is much bigger than $m_{3/2}^2$.
Using the above mass formula, the magnitudes of 
$\langle R_{Il}^{Jk} \rangle$ and 
$\langle X_{Il}^{Jk} \rangle$ are estimated and given in Table 1.
Here we assume $q^A_k/q^A_m = O(1)$.

\begin{table}
\caption{The magnitudes of $\langle R_{Il}^{Jk} \rangle$ and 
$\langle X_{Il}^{Jk} \rangle$}
\begin{center}
\begin{tabular}{|c|l|l|}
\hline
$(I,J)$ & $\langle R_{Il}^{Jk} \rangle$ & $\langle X_{Il}^{Jk} \rangle$ \\
\hline\hline
$(S,S)$ & $O(\epsilon_p/M^2)$ & 
$Max(O(\langle K^S_{SS} \rangle/\langle K_{S} \rangle), 
O(\epsilon_p/M^2))$ \\ \hline
$(T,T)$ & $O(1/M^2)$ 
& $Max(O(\epsilon_1/(\langle K_{S} \rangle M)), 
O(1/M^2))$ \\ \hline
$(m,m)$ & $O(1/M^2)$ & $O(1/M_I^2)$ \\ \hline
$(S,T)$ & $O(\epsilon_p/M^2)$ 
& $Max(O(\epsilon_1/(\langle K_{S} \rangle M)), 
O(\epsilon_3/M^2))$ \\ \hline
$(S,m)$ & $O(\epsilon_p M_I/M^3)$
& $Max(O(\epsilon_p/(\langle K_{S} \rangle M)), 
O(\epsilon_p/(M M_I))$ \\ \hline
$(T,m)$ & $O(M_I/M^3)$
& $Max(O(\epsilon_3/(\langle K_{S} \rangle M)), 
O(1/(M M_I))$ \\ \hline
\end{tabular}
\end{center}
\end{table}

Now we obtain the following generic features on $(m^2)^k_l$.\\
(1) The order of magnitude of $\langle X_{Il}^{Jk} \rangle$ is
equal to or bigger than that of $\langle R_{Il}^{Jk} \rangle$
except for an off-diagonal part $(I,J)=(S,T)$.
Hence {\it the magnitude of $D$-term contribution is comparable to
or bigger than that of $F$-term contribution except for the universal part}
$(m_{3/2}^2 + \langle V_F \rangle/M^2)\langle K_l^k \rangle$.\\
(2) In case where the magnitude of $\langle F_m \rangle$ is bigger than
$O(m_{3/2} M_I)$ and $M > M_I$, 
we get the inequality $(m^2_D)_k > O(m_{3/2}^2)$ since the magnitude of 
$\langle \hat{D}^A \rangle$ is bigger than $O(m_{3/2}^2)$.\\
(3) In order to get the inequality $O((m^2_F)_k) > O((m^2_D)_k)$, the
following conditions must be satisfied simultaneously,
\begin{eqnarray}
&~& \langle F_T \rangle, \langle F_m \rangle \ll O(m_{3/2} M) ,~~
\langle F_S \rangle = O({m_{3/2} M \over \langle K^S_S \rangle^{1/2}}) 
\nonumber \\
&~& {M^2 \langle K^S_{SS} \rangle \over \langle K_S \rangle
\langle K^S_S \rangle} < O(1) ,~~
{\epsilon_p \over \langle K^S_S \rangle} < O(1) ,~~(p=2,3)
\label{conditions}
\end{eqnarray}
unless an accidental cancellation among
terms in $\langle \hat{D}^A \rangle$ happens.
To fulfill the condition $\langle F_{T,m} \rangle \ll O(m_{3/2} M)$,
a cancellation among various terms including $\langle K_{I} \rangle$ and 
$\langle M^2 W_{I} / W \rangle$ is required. 
Note that the magnitudes of 
$\langle K_{T} \rangle$ and $\langle K_{m} \rangle$
are estimated as $O(M)$ and $O(M_I)$, respectively.


The gauge kinetic function is given by
\begin{eqnarray}
  f_{\alpha\beta} = k_\alpha {S \over M}\delta_{\alpha\beta}
  + \epsilon_\alpha  {T \over M}\delta_{\alpha\beta} 
+ f_{\alpha\beta}^{(m)}(\phi^\lambda) 
\label{f}
\end{eqnarray}
where $k_{\alpha}$'s are Kac-Moody levels
and $\epsilon_\alpha$ is a model-dependent parameter \cite{epsilon}.
The gauge coupling constants $g_\alpha$'s are related to the 
real part of gauge kinetic functions such that 
$g_\alpha^{-2} = \langle Re f_{\alpha\alpha} \rangle$.
The magnitudes of gaugino masses and $A$-parameters
in MSSM particles are estimated using the formulae
\begin{eqnarray}
M_a &=& \langle F^I \rangle \langle h_{aI} \rangle 
\label{Ma2} ,\\
\langle h_{aI} \rangle &\equiv& \langle Re f_a \rangle^{-1} 
\langle f_a,_I \rangle \\
A_{kll'} &=& \langle F^I \rangle \langle a_{kll'I} \rangle 
\label{A2}, \\
\langle a_{kll'I} \rangle &\equiv& \langle f_{kll'},_I \rangle 
+ {\langle K_I \rangle \over M^2} \langle f_{kll'} \rangle
- \langle K_{(kI}^{I'} \rangle \langle (K^{-1})_{I'}^J \rangle 
\langle f_{Jll')} \rangle .
\end{eqnarray}
%where $(kll')$ stands for a cyclic permutation.
The result is given in Table 2.
Here we assume that $g_\alpha^{-2} = O(1)$.

\begin{table}
\caption{The magnitudes of $\langle h_{aI} \rangle$ 
and $\langle a_{kll'I} \rangle$}
\begin{center}
\begin{tabular}{|c|l|l|}
\hline
$I$ & $\langle h_{aI} \rangle$ & $\langle a_{kll'I} \rangle$
\\ \hline\hline
$S$ & $O(1/M)$ & 
$Max(O(\langle K_{S} \rangle/M^2), O(\epsilon_p/M))$ \\ \hline
$T$ & $O(\epsilon_\alpha/M)$ & $O(1/M)$ \\ \hline
$m$ & $O(M_I/M^2)$ & $O(M_I/M^2)$ \\ \hline
\end{tabular}
\end{center}
\end{table}

In case that SUSY is broken by the mixture of $S$, $T$ and matter 
$F$-components such that
%\begin{eqnarray}
$\langle (K^S_S)^{1/2} F_S \rangle$, $\langle F_T \rangle$, 
$\langle F_m \rangle= O(m_{3/2} M)$ ,
%\label{Mix}
%\end{eqnarray}
we get the following relations among soft SUSY breaking parameters
\begin{eqnarray}
(m^2)_k &\geq& (m^2_D)_k = O(m_{3/2}^2 {M^2 \over M_I^2}) \geq
(A_{kll'})^2 = O(m_{3/2}^2) ,
\label{Mix-rel1}\\
(M_a)^2 &=& O(m_{3/2}^2) \cdot Max(O({\langle K^S_S \rangle}^{-1}),
O(\epsilon_\alpha^2), O({M_I^2 \over M^2})) .
\label{Mix-rel2}
\end{eqnarray}


Finally we discuss the three special cases of SUSY breaking scenario.
%i.e., dilaton dominant SUSY breaking scenario,
%moduli dominant SUSY breaking scenario 
%and matter dominant SUSY breaking scenario.

\begin{enumerate}
\item In the dilaton dominant SUSY breaking scenario
\begin{eqnarray}
\langle (K^S_S)^{1/2} F_S \rangle = O(m_{3/2} M) \gg 
\langle F_T \rangle, \langle F_m \rangle ,
\label{S-dom}
\end{eqnarray}
the magnitudes of soft SUSY breaking parameters are estimated as 
\begin{eqnarray}
(m^2)_k &=& O(m_{3/2}^2) \cdot Max(O(1), O({M^2 \langle K^S_{SS} \rangle
\over \langle K^S_S \rangle \langle K_S \rangle}), 
O({\epsilon_p \over \langle K^S_S \rangle})) , \nonumber \\
M_a &=& O({m_{3/2} \over \langle K^S_S \rangle^{1/2}}) , ~~~
A_{kll'} = O(m_{3/2}) \cdot Max(O({\langle K_{S} \rangle \over M}), 
O(\epsilon_p)) . \nonumber
\end{eqnarray}
Hence we have a relation such that $O((m^2)_k)  \geq O((A_{kll'})^2)$.

As discussed in Ref.\cite{model-dep1}, gauginos can be heavier than
scalar fields if $\langle K^S_S \rangle$ is small enough and
$O(M^2 \langle K^S_{SS} \rangle) < O(\langle K_S \rangle)$.
In this case, dangerous flavor changing neutral current 
(FCNC) effects from squark mass non-degeneracy are avoided 
because the radiative correction
due to gauginos dominates in scalar masses at the weak scale.
On the other hand, in Ref.\cite{model-dep2}, 
it is shown that gauginos are much lighter than
scalar fields from the requirement of 
the condition of vanishing vacuum energy 
in the SUGRA version of model proposed in Ref.\cite{BD}.
In appendix, we discuss the relations among the magnitudes of
$\langle K_S \rangle$, $\langle K^S_S \rangle$ and
$\langle K^S_{SS} \rangle$ under some assumptions.

\item In the moduli dominant SUSY breaking scenario
\begin{eqnarray}
\langle F_T \rangle = O(m_{3/2} M) \gg 
\langle (K^S_S)^{1/2} F_S \rangle, \langle F_m \rangle ,
\label{T-dom}
\end{eqnarray}
the magnitudes of soft SUSY breaking parameters are estimated as 
\begin{eqnarray}
(m^2)_k &=& O(m_{3/2}^2) \cdot Max(O(1), O({\epsilon_1 M \over
\langle K_S \rangle})) ,\nonumber \\
M_a &=& O(\epsilon_\alpha m_{3/2}) , ~~~
A_{klm} = O(m_{3/2}) . \nonumber
\end{eqnarray}
Hence we have a relation such that
$O((m^2)_k) \geq O((A_{kll'})^2) \geq O((M_a)^2)$.
The magnitude of $\mu_{TT}$ is estimated as $\mu_{TT} = O(m_{3/2})$.

\item In the matter dominant SUSY breaking scenario
\begin{eqnarray}
\langle F_m \rangle = O(m_{3/2} M) \gg 
\langle (K^S_S)^{1/2} F_S \rangle, \langle F_T \rangle ,
\label{matter-dom}
\end{eqnarray}
the magnitudes of soft SUSY breaking parameters are estimated as 
\begin{eqnarray}
(m^2)_k &=& O(m_{3/2}^2 {M^2 \over M_I^2}) , ~~~
M_a, A_{kll'} = O(m_{3/2} {M_I \over M}) . \nonumber 
\end{eqnarray}
The relation $(m^2)_k \gg O((M_a)^2) = O((A_{kll'})^2)$
is derived when $M \gg M_I$.
The magnitude of $\mu_{mn}$ is estimated as 
$\mu_{mn} = O(m_{3/2} M/M_I)$.
This value is consistent with that in Ref.\cite{DP}.
\end{enumerate}


\section{Conclusions}

We have studied the magnitudes of soft SUSY breaking parameters
in heterotic string models with $G_{SM} \times U(1)_A$,
which originates from the breakdown of $E_8 \times E'_8$, and derive 
model-independent predictions for them
without specifying SUSY breaking mechanism
and the dilaton VEV fixing mechanism.
In particular, we have made a comparison of magnitudes between
$D$-term contribution to scalar masses and $F$-term ones and 
a comparison of magnitudes among scalar masses, gaugino masses 
and $A$-parameters
under the condition that $O(m_{3/2}) \ll \langle K_S \rangle \leq
O(q^A_m M/\delta^A_{GS})$, $(M_V^2)^A/g_A^2 = O(q^{A2}_m M_I^2)$
and $\langle V \rangle \leq O(m_{3/2}^2 M^2)$.
%assumption that the dilaton field is stabilized due to the physics
%at the Planck scale $M_{Pl}$ or at the lower scale than $M_{Pl}$
%but much bigger than $m_{3/2}$.
The order of magnitude of $D$-term contribution of $U(1)_A$
to scalar masses is comparable to
or bigger than that of $F$-term contribution $\langle F^I \rangle
\langle F_J \rangle \langle R_{Il}^{Jk} \rangle$
except for the universal part 
$(m_{3/2}^2 + \langle V_F \rangle/M^2)\langle K_l^k \rangle$.
If the magnitude of $F$-term condensation of matter fields 
$\langle F_m \rangle$ is bigger than
$O(m_{3/2} M_I)$, the magnitude of $D$-term contribution $(m_D^2)_k$ 
is bigger than $O(m_{3/2}^2)$. 
In general, it is difficult to realize the inequality 
$O((m^2_D)_k) < O((m^2_F)_k)$ unless conditions such as 
Eq.(\ref{conditions}) are fulfilled.
We have also discussed relations among soft SUSY breaking
parameters in three special scenarios on SUSY breaking,
i.e., dilaton dominant SUSY breaking scenario,
moduli dominant SUSY breaking scenario 
and matter dominant SUSY breaking scenario.

The $D$-term contribution to scalar masses with different broken
charges as well as the $F$-term contribution from the difference among
modular weights can destroy universality among scalar masses.
The non-degeneracy among squark masses of first and second families
endangers the discussion of the suppression
of FCNC process.
On the other hand, the difference among broken charges
is crucial for the scenario of fermion mass hierarchy 
generation \cite{texture}.
It seems to be difficult to make two discussions compatible.
There are several way outs.
The first one is to construct a model that the fermion mass hierarchy is 
generated due to non-anomalous $U(1)$ symmetries.
In the model, $D$-term contributions of non-anomalous $U(1)$ symmetries
vanish in the dilaton dominant SUSY breaking case
and it is supposed that anomalies from contributions of
the MSSM matter fields are canceled out 
by an addition of extra matter fields.
The second one is to use ``stringy'' symmetries 
for fermion mass generation in the situation with degenerate 
soft scalar masses \cite{texture2}. 
The third one is to use a parameter region that
the radiative correction due to gauginos, which is flavor independent,
dominates in scalar masses 
at the weak scale.
It can be realized when $\langle K^S_S \rangle$ is small enough and
$O(M^2 \langle K^S_{SS} \rangle) < O(\langle K_S \rangle)$.

Finally we give a comment on moduli problem \cite{cosmo}.
If the masses of dilaton or moduli fields are of order of the weak scale,
the standard nucleosynthesis should be modified
because of a huge amount of entropy production.
The dilaton field does not cause dangerous contributions
in the case with $\langle (K^S_S)^{1/2} F_S \rangle = O(m_{3/2} M)$
if the magnitude of $\langle K_S^S \rangle$
is small enough.\footnote{
This possibility has been pointed out
in the last reference in \cite{corr2}.}
%and there exists a mixing among $S$ and $T$ in K\"ahler potential.
Because the magnitudes of $(m_F^2)_S$ is
%, $(m_F^2)_T$ and $(m_F^2)_m$ are 
given by $O(m_{3/2}^2/\langle K_S^S \rangle^2)$.
% up to some factors including $\epsilon_n$, $\langle K_{SS}^S \rangle$, 
%$\langle K_{SS}^{SS} \rangle$ and so forth.


\section*{Acknowledgements}
The author is grateful to T.~Kobayashi, H.~Nakano, H.P.~Nilles and 
M.~Yamaguchi for useful discussions.

\appendix

\section{On derivatives of K\"ahler potential related to dilaton}
\label{app:A}

We discuss the relations among $\langle K_S \rangle$, 
$\langle K^S_S \rangle$ and $\langle K^S_{SS} \rangle$
using SUSY breaking conditions and the stationary conditions
of scalar potential.
We list our assumptions first.
\begin{enumerate}
\item SUSY is broken by the mixture of $S$, $T$ and matter 
$F$-components such that
%\begin{eqnarray}
$\langle (K^S_S)^{1/2} F_S \rangle$, $\langle F_T \rangle$, 
$\langle F_m \rangle= O(m_{3/2} M)$.
%\label{Mix2}
%\end{eqnarray}

\item The magnitude of $\langle K_S \rangle$ is much bigger than 
$O(m_{3/2})$ and it is comparable to  or smaller than 
$O(q^A_m M / \delta^A_{GS})$.
The latter is equivalent to the condition that
the magnitude of $M_I \equiv |\langle \phi^m \rangle|$ is at most $O(M)$.

\item The magnitude of $\langle K^S_S \rangle^{1/2}$ is much bigger than 
those of $\langle K^T_S \rangle$ and $\langle K^m_S \rangle$,
and it is comparable to or smaller than $O(q^A_m M_I/\delta^A_{GS} M)$.
The latter is equivalent to the condition that
the magnitude of $(M_{V}^{2})^A/g_A^2$ is $O(q^{A2}_m M_I^2)$.

\item The magnitude of $\delta^A_{GS} / q^A_m$ is $O(1/10) \sim O(1/100)$.

\item No cancellation happens among terms in 
$\langle K_S \rangle$ and $M^2 \langle W_S \rangle / \langle W \rangle$.
On a later discussion, we relax this assumption.
\end{enumerate}
Under these assumptions, the following relation is derived
\begin{eqnarray}
\langle K^S_S \rangle^{1/2} = O({\langle G_S \rangle \over M})
= O({\langle K_S \rangle \over M} 
    + M {\langle W_S \rangle \over \langle W \rangle})
\label{KSS-rel}
\end{eqnarray}
by the use of the definition (\ref{total-Kahler}).
If $\langle K_S \rangle$ is bigger than 
$M^2 \langle W_S \rangle / \langle W \rangle$, we find that
$\langle K^S_S \rangle^{1/2} 
= O(\langle K_S \rangle / M) \leq O(q^A_m / \delta^{A}_{GS})$.

Further we can get the following relation among $\langle K_S \rangle$, 
$\langle K^S_S \rangle$ and $\langle K^S_{SS} \rangle$
from the stationary conditions (\ref{<VI>}) and (\ref{<VI2>}),
\begin{eqnarray}
{\langle K^S_{SS} \rangle \over \langle K^S_S \rangle} 
    = Max(O({\langle K^S_{S} \rangle \over \langle K_S \rangle}),
          O({\langle K^S_{S} \rangle^{1/2} \over M})).
\label{KSSS-rel}
\end{eqnarray}

Let us consider a typical case with non-perturbative 
superpotential derived from SUSY breaking scenario by gaugino
condensations. 
The non-perturbative superpotential $W_{non}$ is, generally, given by
\begin{eqnarray}
W_{non} = \sum_i a_{i}(\phi^\lambda, T)
 exp({-b_{i} S \over \delta^A_{GS} M})
\label{W_{gaugino}}
\end{eqnarray}
where $a_{i}$'s are some functions of $\phi^\lambda$ and $T$, 
and $b_{i}$'s are model-dependent parameters of $O(q^A_m)$.
Using the second assumption, Eqs.(\ref{KSS-rel}) and (\ref{W_{gaugino}}), 
we get the relation
$\langle K^S_S \rangle^{1/2} = M \langle W_S \rangle / \langle W \rangle
= O(q^A_m / \delta^{A}_{GS})$ 
if $O(\langle W_{non} \rangle) = O(\langle W \rangle)$.
This relation means $M_I = M$ from the third assumption,
and it leads to the relation such that 
$M \langle K^S_S \rangle / \langle K_S \rangle 
= O(q^A_m / \delta^{A}_{GS})$.
%\begin{eqnarray}
%{\langle K^S_S \rangle \over \langle K_S \rangle} 
%    = O({q^A_m  \over \delta^{A}_{GS} M}) .
%\label{KSS-rel3}
%\end{eqnarray}
We obtain the relation $M \langle K^S_{SS} \rangle / \langle K^S_S \rangle
= O(q^A_m  / \delta^{A}_{GS})$ using Eq.(\ref{KSSS-rel}).
Finally we discuss the case where the cancellation happens 
among terms in 
$\langle K_S \rangle$ and $M^2 \langle W_S \rangle / \langle W \rangle$.
Then the magnitude of $\langle K^S_S \rangle^{1/2}$ and
$M \langle K^S_S \rangle / \langle K_S \rangle$ can be smaller than
$O(q^A_m / \delta^{A}_{GS})$.
Hence the magnitude of gaugino masses can be bigger than 
those of scalar masses in case where $\langle K^S_S \rangle$
is small enough 
and $M^2 \langle K^S_{SS} \rangle / \langle K^S_S \rangle < O(1)$.


\begin{thebibliography}{99}

\bibitem{ST-SG}
E.~Witten, Phys. Lett. {\bf B155} (1985) 151;\\
J.P.~Derendinger, L.E.~Ib\'a\~nez and H.P.~Nilles,
Nucl.~Phys. {\bf B267} (1986) 365;\\
L.E.~Ib\'a\~nez and H.P.~Nilles, Phys.~Lett. {\bf B169} (1986) 354;\\
S.~Ferrara, C.~Kounnas and M.~Porrati, Phys.~Lett. {\bf B181} (1986) 263.

\bibitem{OrbSG}
M.~Cveti\u{c}, J.~Louis and B.~Ovrut, 
Phys.~Lett. {\bf B206} (1988) 227;\\
S.~Ferrara, D.~L\"ust and S.~Theisen, Phys.~Lett. {\bf B233} (1989) 147.

\bibitem{OrbSG2}
L.J.~Dixon, V.S.~Kaplunovsky and J.~Louis, 
Nucl.~Phys. {\bf B329} (1990) 27.

\bibitem{ST-FI} 
M.~Dine, N.~Seiberg and E.~Witten, Nucl. Phys. {\bf B289} (1987) 589;\\
J.~Atick, L.~Dixon and A.~Sen, Nucl.~Phys. {\bf B292} (1987) 109;\\
M.~Dine, I.~Ichinose and N.~Seiberg, Nucl.~Phys. {\bf B293} (1987) 253.

\bibitem{duality}
S.~Ferrara, D.~L\"ust, A.~Shapere and S.~Theisen, 
Phys. Lett. {\bf B225} (1989) 363.

\bibitem{gaugino}
J.P.~Derendinger, L.E.~Ib\'a\~nez and H.P.~Nilles, 
Phys. Lett. {\bf B155} (1985) 65;\\
M.~Dine, R.~Rohm, N.~Seiberg and E.~Witten, Phys. Lett. {\bf B156} 
(1985) 55.

\bibitem{SS}
J.~Scherk and J.H.~Schwarz, Nucl. Phys. {\bf B153} (1979) 61, Phys. Lett.
{\bf 82B} (1979) 60.

\bibitem{S-Stab}
N.V.~Krasnikov, Phys. Lett. {\bf B193} (1987) 37;\\
L.~Dixon, in the proceedings of the A.S.P. meetimg, Houston (1990);\\
J.A.~Casas, Z.~Lalak, C.~Mu\~noz and G.G.~Ross, 
Nucl. Phys. {\bf B347} (1990) 243;\\
T.~Taylor, Phys. Lett. {\bf B252} (1990) 59;\\
B.~de Carlos, J.A.~Casas and C.~Mu\~noz, Nucl. Phys. {\bf B399}
(1993) 623.

\bibitem{corr1}
S.H.~Shenker, Proceedings of the Cargese Workshop on Random Surfaces,
Quantum Gravity and Strings, Cargese (France) 1990;\\
T.~Banks and M.~Dine, Phys. Rev. {\bf D50} (1994) 7454.

\bibitem{corr2}
P.~Bin\'etruy, M.K.~Gaillard and Y.Y.~Wu, Nucl. Phys. {\bf B493}
(1997) 27; Phys. Lett. {\bf B412} (1997) 288;\\
T.~Barreiro, B.~de Carlos and E.J.~Copeland,
.

\bibitem{Casas}
J.A.~Casas, Phys. Lett. {\bf B384} (1996) 103.

\bibitem{ST-soft}
A.~Brignole, L.E.~Ib\'a\~nez and C.~Mu\~noz,
Nucl. Phys. {\bf B422} (1994) 125;\\
T.~Kobayashi, D.~Suematsu, K.~Yamada and Y.~Yamagishi, 
Phys. Lett. {\bf B348} (1995) 402;\\
A.~Brignole, L.E.~Ib\'a\~nez, C.~Mu\~noz and C.~Scheich, 
Z. Phys. {\bf C74} (1997) 157.

\bibitem{DP}
G.~Dvali and A.~Pomarol,
Phys. Rev. Lett. {\bf 77} (1996) 3728; Nucl. Phys. {\bf B522} (1998) 3;\\
R.N.~Mohapatra and A.~Riotto, Phys. Rev. {\bf D55} (1997) 1138; 
{\it ibid} 4262.

\bibitem{model-dep1}
N.~Arkani-Hamed, M.~Dine and S.P.~Martin,
Phys. Lett. {\bf B431} (1998) 329.

\bibitem{model-dep2}
T.~Barreiro, B.de~Carlos, J.A.~Casas and J.M.~Moreno,
Preprint SUSX-TH-98-019, IEM-FT-179/98, .

\bibitem{SUGRA}
H.P.~Nilles, Phys. Rep. {\bf C110} (1994) 1.

\bibitem{JKY}
I.~Joichi, Y.~Kawamura and M.~Yamaguchi, 
 Touhoku-Preprint, TU-459, DPSU-9403 (unpublished).

\bibitem{KMY2} 
Y.~Kawamura, H.~Murayama and M.~Yamaguchi, 
Phys. Rev. {\bf D51} (1995) 1337.

\bibitem{Kawa1} 
Y.~Kawamura, Phys.~Rev.~{\bf D53} (1996) 3779.

\bibitem{N} 
H.~Nakano, Report No. KUNS 1257, HE(TH)94/05,  1994
(unpublished);
Prog. Theor. Phys. Supplement {\bf 123} (1996) 387.

\bibitem{KK}
Y.~Kawamura and T.~Kobayashi, 
Phys. Lett. {\bf B375} (1996) 141 [Erratum: {\bf B388} (1996) 867];
Phys. Rev. {\bf D56} (1997) 3844.

\bibitem{GS} 
M.B.~Green and J.H.~Schwarz, Phys. Lett. {\bf B149} (1984) 117;\\
L.E.~Ib\'a\~nez, Phys. Lett. {\bf B303} (1993) 55.

\bibitem{KN} 
T.~Kobayashi and H.~Nakano, Nucl. Phys. {\bf B496} (1997) 103.

\bibitem{KKK}
Y.~Kawamura, T.~Kobayashi and T.~Komatsu,
Phys. Lett. {\bf B400} (1997) 284.

\bibitem{het/M}
A.~Lukas, B.A.~Ovrut and D.~Waldram, ;\\
H.P.~Nilles, M.~Olechowski and M.~Yamaguchi, .

\bibitem{epsilon} 
H.P.~Nilles, Phys. Lett. {\bf 180B} (1986) 240.

\bibitem{BD}
P.~Bin\'etruy and E.~Dudas, Phys. Lett. {\bf B389} (1996) 503.

\bibitem{texture} 
M.~Leurer, Y.~Nir and N.~Seiberg, Nucl.~Phys. {\bf B398} (1993) 319; 
Nucl.~Phys. {\bf B420} (1994) 468;\\
Y.~Nir and N.~Seiberg, Phys.~Lett. {\bf B309} (1993) 337;\\
L.E.~Ib\'a\~nez and G.G.~Ross, Phys. Lett. {\bf B332} (1994) 100;\\
P.~Bin\'etruy and P.~Ramond, Phys. Lett. {\bf B350} (1995) 49;\\
E.~Dudas, S.~Pokorski and C.A.~Savoy, Phys. Lett. {\bf B356} (1995) 45.

\bibitem{texture2}
T.~Kobayashi, Phys. Lett. {\bf B354} (1995) 264; 
Phys. Lett. {\bf B358} (1995) 253.

\bibitem{cosmo}
G.D.~Coughlan, W.~Fischler, E.W.~Kolb, S.~Raby and G.G.~Ross, 
Phys. Lett. {\bf B131}, 59 (1983);\\
B.~de~Carlos, J.A.~Casas, F.~Quvedo and E.~Roulet, 
Phys. Lett. {\bf B318}, 447 (1993);\\
T.~Banks, D.~Kaplan and A.~Nelson, 
Phys.Rev. {\bf D49}, 779 (1994).

\end{thebibliography}


\end{document}



