%Subject: so10.tex
%\documentstyle[preprint,aps]{revtex}      % Galley
%\documentstyle[prl,aps,epsf]{revtex}      % Galley
%\documentstyle[aps,prl,epsfig]{revtex}      % Galley
%\documentstyle[aps,prl,twocolumn,tighten,psfig]{revtex}  %PRL
%\documentstyle[twocolumn,prl,aps,epsfig]{revtex}  % PRL
%\documentclass{elsart}                      % PLB
\documentclass[11pt]{article}
%\input{epsf}
\textheight=25 true cm
\textwidth=16 true cm
\oddsidemargin=-0.25cm
\topmargin=-0.5in
\usepackage{psfig}

%\textheight=23 true cm
%\textwidth = 16 true cm
%\oddsidemargin=-0.25cm
%\topmargin=-0.5in
\def\baselinestretch{1.68}
\thispagestyle{empty}
\flushbottom
%\voffset = 15mm
\voffset = 0mm

% Reference Macros:  Enter parameters in order Vol, Page, Year
\def\issue(#1,#2,#3){{\bf #1}, #2 (#3)} % AIP format
%\def\issue(#1,#2,#3){#1 (#3) #2} % PLB format

\def\opcit(#1){ {\em op. cit.}, #1}

\def\APP(#1,#2,#3){Acta Phys.\ Polon.\ \issue(#1,#2,#3)}
\def\ARNPS(#1,#2,#3){Ann.\ Rev.\ Nucl.\ Part.\ Sci.\ \issue(#1,#2,#3)}
\def\CPC(#1,#2,#3){Comp.\ Phys.\ Comm.\ \issue(#1,#2,#3)}
\def\CIP(#1,#2,#3){Comput.\ Phys.\ \issue(#1,#2,#3)}
\def\EPJC(#1,#2,#3){Eur.\ Phys.\ J.\ C\ \issue(#1,#2,#3)}
\def\EPJD(#1,#2,#3){Eur.\ Phys.\ J. Direct\ C\ \issue(#1,#2,#3)}
\def\IEEETNS(#1,#2,#3){IEEE Trans.\ Nucl.\ Sci.\ \issue(#1,#2,#3)}
\def\IJMP(#1,#2,#3){Int.\ J.\ Mod.\ Phys. \issue(#1,#2,#3)}
\def\MPL(#1,#2,#3){Mod.\ Phys.\ Lett.\ \issue(#1,#2,#3)}
\def\NP(#1,#2,#3){Nucl.\ Phys.\ \issue(#1,#2,#3)}
\def\NIM(#1,#2,#3){Nucl.\ Instrum.\ Meth.\ \issue(#1,#2,#3)}
\def\PL(#1,#2,#3){Phys.\ Lett.\ \issue(#1,#2,#3)}
\def\PRD(#1,#2,#3){Phys.\ Rev.\ D \issue(#1,#2,#3)}
\def\PRL(#1,#2,#3){Phys.\ Rev.\ Lett.\ \issue(#1,#2,#3)}
\def\SJNP(#1,#2,#3){Sov.\ J. Nucl.\ Phys.\ \issue(#1,#2,#3)}
\def\ZPC(#1,#2,#3){Zeit.\ Phys.\ C \issue(#1,#2,#3)}

%S. Amato \etal, \NIM(A324,535,1992);\hfil\break

%Typesaving Macros -----------------------------------------------------

\def\be {\begin{equation}}
\def\ee {\end{equation}}
\def\bea {\begin{eqnarray}}
\def\eea {\end{eqnarray}}
\def\n {\nonumber}
\def\bc {\begin{center}}
\def\ec {\end{center}}

\def \mf{{m_{1}}^{2}}
\def \ms{{m_{2}}^{2}}
\def \m3{\left |{m_{3}}\right |}
\def \mths{\left | m_{3}^{2}\right |}
\def \HF{\left |{H_{1}}\right |}
\def \H2{\left |{H_{2}}\right |}
\def \gpsq{{g'}^{2}}
\def \g2sq{{g_{2}}^{2}}
\def \phialpha{{\phi}_{\alpha}}
\def \sumalpha{\sum_{\alpha}}
\def \galpha{{g_{\alpha}}}
\def \MHU{m_{H_d}^2}
\def \MHD{m_{H_u}^2}
\def \MHF{m_{1/2}}
\def \MSX{m_{16}}
\def \MTN{m_{10}}
\def \SU{SU(2)\times U(1)}
\def \MGL{m_{\tilde g}}
\def \MSQ{m_{\tilde q}}
\def \MSL{m_{\tilde l}}
\def \MCH{m_{\tilde \chi^{\pm}}}
\def\lapp{\mathrel{\rlap{\raise.5ex\hbox{$<$}}
                    {\lower.5ex\hbox{$\sim$}}}}
\def\gapp{\mathrel{\rlap{\raise.5ex\hbox{$>$}}
                    {\lower.5ex\hbox{$\sim$}}}}

%------------------------------------------------------------------------
\setcounter{page}{0}

\begin{document}
%\renewcommand{\thefootnote}{\fnsymbol{footnote}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\title{
%\vspace*{0.10truein}
%\begin{flushright}
%\small 
%\end{flushright}
%\vspace*{0.25truein}
Effects of SO(10) D-Term on Yukawa Unification and Unstable Minima of the Supersymmetric Scalar Potential}
\author{ {\large\sl Amitava Datta}
\thanks{Electronic address: adatta@juphys.ernet.in}
\thanks{On leave of absence from Jadavpur University.}\\
Department of Physics, Visva-Bharati, Santiniketan - 731 235, India
\and
{\large\sl Abhijit Samanta}
\thanks{Electronic address: abhijit@juphys.ernet.in}\\
Department of Physics, Jadavpur University, Kolkata - 700 032, India
}
\date{\today}

\maketitle

\begin{abstract}
We study the effects of SO(10) D-terms  on the allowed parameter space 
( APS ) in models with  $t - b - \tau$ and $b - \tau$ Yukawa unifiction.
The former is allowed only for moderate values of the D-term,  
if very precise ( $\le 5\%$ ) unification is required.
Next we constrain the  parameter space by looking for different 
dangerous directions  where the scalar 
potential may be unbounded from below ( UFB1 and UFB3 ) constraints.  
The common trilinear 
coupling $A_0$ plays a significant role in constraing the APS.
For very  precise  $t - b - \tau$ Yukawa unification,
$-\MSX \lapp A_0 \lapp \MSX$ can be probed at the LHC. 
Moreover, an interesting mass hierarchy with very heavy sfermions 
but light gauginos, which is strongly disfavoured in
models without D-terms, becomes fairly common in the presence 
of the D-terms. 
The APS exhibits interesting characteristics if the common sfermion soft
mass $\MSX$ is not the same with the soft mass $\MTN$ for Higgs sector.
In $b - \tau$ unification models with D-terms, the APS allowed 
by Yukawa unification
and REWSB increases and UFB1 constraint become weaker. 
However for $A_0 \lapp 0$, a stronger UFB3 condition still puts, 
for a given $\MSX$, a stringent 
upper bound on the common gaugino mass  and a  lower
bound on the  common soft breaking scalar mass for 
for a given $\MHF$.
\end{abstract}

PACS no: 12.60.Jv, 14.80.Ly, 14.80.Cp
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\pagestyle{plain}   \pagenumbering{arabic}
\setcounter{footnote}{0}
\renewcommand{\thefootnote}{\arabic{footnote}}

%\narrowtext
%\twocolumn

\section{Introduction}

It is quite possible that the Standard Model (SM), is not the ultimate theory of
nature, as is hinted by a number of theoretical shortcomings. One of the
most popular choices for physics beyond SM is supersymmetry (SUSY)
\cite{susy-review}. However, SUSY introduces a plethora of new parameters,
and it is important to constrain them in as many theoretical
ways as possible, in addition to direct searches at colliders. 

One of the
most useful ways to constrain the allowed parameter space (APS) of SUSY
models is to consider the dangerous directions of the scalar potential
where the potential  may be unbounded from below (UFB) or develops
a charge and/or color breaking (CCB) minima \cite{oldufb}. 
Different directions are chosen by
giving vacuum expectation value (VEV) to one or more  
coloured and / or charged scalar fields, while
keeping the VEVs of the other scalars to zero.


In a very interesting  paper which revived interest in
UFB and CCB constraints, Casas {\em et al} \cite{casas} investigated
the effects of such constraints on SUSY models. Though their
formulae  are fairly model-independent, they have carried out
the numerical analysis  for
 moderate values  of $\tan\beta$ (the ratio of the VEVs  of the two Higgs
fields) only, when one can ignore the
effects of b and $\tau$ Yukawa couplings in the relevant
renormalization group equations (RGE's).
Further they  used the standard minimal supergravity
(MSUGRA) assumption of universal soft scalar mass $m_0$ and universal
gaugino mass $m_{1/2}$ at the GUT scale $M_G$, referred to hereafter
as the `conventional scenario', to determine the sparticle
spectrum. Their main result was that within the framework of MSUGRA, a
certain
UFB constraint known as UFB3 with VEV given in the direction of the
slepton field  puts the tightest bound on the SUSY parameter
space that they considered (see eq. (93) of \cite{casas} and the discussions
that follows).

In a recent paper \cite{paper1}, we have extended and complemented the
work of \cite{casas} by looking at the allowed parameter space (APS)
subject to such `potential constraints'
for large values of $\tan\beta$, motivated by possible partial $b$-$\tau$ 
\cite{b-tau,partial} and full
$t$-$b$-$\tau$ Yukawa unification \cite{sotenbr}. Such unifications are
natural consequences of an underlying Grand Unified Theory 
( GUT ), {\em e.g.}, one in which the GUT
group SO(10) breaks directly into the SM gauge group SU(3) $\times$ SU(2)
$\times$ U(1). With a minimal Higgs field content (one {\bf 10}-plet),
containing both the Higgs doublets required to give masses to u and d
type quarks, 
all three Yukawa couplings related to the third generation fermions 
must unify at the GUT scale; if one assumes
more than one {\bf 10}-plet, at least the bottom and the tau Yukawa
couplings must unify.

In addition we require radiative electroweak symmetry breaking 
( REWSB ) \cite{rewsb}. We further require that the lightest neutralino
( $\tilde \chi^0$ ) be the lightest supersymmetric particle ( LSP ). 
All the above
constraints will be used to obtain the allowed parameter space ( APS )
although the last two may not be stated explicitly everywhere.

The salient points of \cite{paper1} are summarised below:
We found that  a) for the common trilinear coupling $A_0 \lapp 0$, 
the requirement of partial $b-\tau$ unification  
in conjunction with the stability condition on the potential yields highly
restrictive sparticle spectra with upper, and in many cases, lower bounds,
stronger than the available experimental lower bounds, on the soft SUSY
breaking common scalar mass and the common gaugino mass ( $\MHF$ ). b) Over a
significant region of the parameter space, the model becomes even more 
restrictive if the common sfermion soft mass ( $\MSX$ ) is different 
from the soft mass for the Higgs sector ( $\MTN$ ).  
c) We also found that the bulk of this restricted 
parameter space can be probed at the LHC.  d) In models 
with full $t-b-\tau$ Yukawa unification with moderate accuracy ( $\sim 10\%$ ), $A_0 \leq 0$ is ruled out from 
potential constraints. For  $A_0 \geq 0$ a reasonably large APS is
obtained although sparticles with masses close to the  currently available
lower bounds are forbidden. 


These UFB and CCB constraints depend crucially on the particle spectra
at the properly chosen scale where the true minimum and the dangerous 
minimum can be reliably evaluated from the tree level potential 
( $V_{tree}$ ) \cite{gamberini,casas}. 
Such spectra, in turn, depend on the boundary conditions at
the GUT scale. In our earlier paper \cite{paper1}, we  assumed 
a common soft breaking ( SB ) mass for all sfermions ( $\MSX$ ) at $M_G$, 
and  a common
mass parameter for both the Higgs fields ($ \MTN$ ). we then 
studied the constraints for 
two sets of boundary conditions: i)
 the SUGRA motivated universal scenario ( $\MSX = \MTN$ ),
and ii) a  nonuniversal
scenario ($\MSX \neq \MTN$ ). 
The second model is motivated by the fact that a common scalar mass at the
Planck scale, generated by the SUGRA mechanism,  may lead to nonuniversal
scalar masses at the GUT scale due
to different running of $m_{10}$ and $m_{16}$, as they belong to
different GUT multiplets \cite{running}.

It is welknown that
the breaking of  SO(10)
to the  lower rank SM group introduces nonzero D-terms at the GUT scale
\cite{d-term}.
These SO(10) breaking D-terms alter the sparticle spectra at the GUT
scale . 
In this paper we focus our attention on  the impact of such 
D-terms on the APS restricted by  Yukawa
unification and  the stability of the potential in both universal
and nonuniversal scenarios. In recent times the phenomenology of
the D-terms have attained wide attention\cite{pheno-d,tata}.

D-terms acquire particular importance in the context of $t$-$b$-$\tau$
unification, as has already been noted in the literature \cite{tata}.           
The new finding of  this paper, which will be elaborated below, is
that while moderate values of the D-term indeed lead to better
unification, larger  magnitudes of this term may worsen it.
We also find that certain generic mass patterns which are very uncommon 
in models without the D-terms become the favoured ones in the presence
of D-terms.

Throughout the paper we ignore the possibility that nonrenormalizable
effective operators may stabilise the potential \cite{nonrenorm}. The
dangerous minima that we encounter in our analysis typically occur at
scales $\lapp 10^8$ GeV where the effects induced by the 
nonrenormalizable operators, which in principle can be significant
in the vicinity of the GUT scale,  are not likely to be very serious.

It has  been pointed out in the literature
that the standard vacuum, though
metastable, may have a lifetime longer than the age of the universe
\cite{claudson}, while the true vacuum is indeed charge and colour
breaking.
If this  be the case, the theory seems to be
acceptable in spite of the existence of the unacceptable UFB minima that
we have analysed. However, the life-time calculation, which is relatively
straightforward for a single scalar field, is much more uncertain in
theories where the potential is a function of many scalar fields. Thus
it is difficult to judge the reliability of these calculations. Moreover,
the constraints obtained by us does not loose their significance even
if the false vacuum idea  happens to be the correct theory. If these
constraints are violated by the expeimental data then that would
automatically lead to the startling conclusion that  we are living on
a false vacuum  and charge and colour symmetry may eventually breakdown.

In section 2 we shall mainly review the results of  \cite{paper1}
and add a few new observations. In section 3 we shall qualitatively
discuss the impact of the D-terms on the APS. The numerical constraints
in the $\MSX - \MHF$ plane will be presented in section 4. Our conclusions
will be summarised in the last section.                          

\section{Constraints obtained without the D-terms}

In this section we summarise the results of  \cite{paper1}. 
Our main motivation is to orient the discussion in such a way that
the circumstances under which the  effects of  the 
D-terms could be most interesting,  may be anticipated. However, we  also 
 take this opportunity to add a few  points
which were not included in  our earlier work. 

The methodology that we follow is the same as that of \cite{paper1}
( see section III ). We start with the Yukawa couplings at the 
weak scale and evolve them to the GUT scale $M_G$ and take the minimum 
value of tan$\beta$ which is sufficient for Yukawa unification to
a certain accuracy. We then check  the REWSB condition, compute 
the  physical sparticle spectrum and test their consistency with the
currently available experimental bounds. Next we impose the UFB conditions
for testing the stability of the potential. Our approach is different 
from the ones which compute the APS for a fixed tan$\beta$. 
If, for example, we fix tan$\beta \sim$ 49, which is quite often
sufficient for $t-b-\tau$ Yukawa unification, we would have obtained
much smaller APS due to the conflict with REWSB condition. On the other hand
by adjusting the value of tan$\beta$ at the minimum  value required 
by full unification we obtained a conservative APS.

 
Now some comments on the accuracy of the Yukawa unification are  in
order.
In analyses based on  currently available techniques, such unification may
indeed appear to be only approximate due to various uncertainties 
( see, e.g. ref \cite{paper1} ).
In order to accommodate such uncertainties, one relaxes the
Yukawa unification condition by a finite amount (5\%, 10\% or 20\%).  It
is interesing to note that quite often the UFB constraints rule out
subtantial parts of the extended APS which open up due to relaxation of
the accuracy of Yukawa unification\cite{paper1}.
The review of this section will mostly concentrate on an unfication 
of modest accuracy ( $\sim 10\%$ )following reference \cite{paper1}.

It is welknown that $\mu < $ 0 , where $\mu$ is the Higgsino mass 
parameter, is not suitable even for $b - \tau$  Yukawa  unification
\cite{tata} since that requires too high a value of tan$\beta$ which
makes the Yukawa couplings diverge during the running from the weak scale 
to the GUT scale.

For  $\mu > $ 0 one still needs  sufficiently large values of tan  
$\beta$ to boost the bottom and the $\tau$ Yukawa couplings at
the weak scale required for unification at $M_G$. Such values are, in general,
 disfavoured by the REWSB condition. The essential reason is that
in this case  $m_{H_{d}}^2$ is driven by large b and $\tau$ Yukawa
couplings to  values lower  
than that of $m_{H_{u}}^2$ and $\mu^2$ becomes negative. For $b - \tau$
Yukawa unification the required value of tan $\beta$ 
, though relatively high,  is not big enough to be in
conflict with the REWSB condition. For still higher values of tan
$\beta$ required by very precise $t - b -\tau$ unification
unification and REWSB can not be reconciled. On the other hand 
if only approximate unification is required, then relatively low
values of tan $\beta$ serve the purpose and a reasonablly large  APS
emerges.

Now we focus on the basic reasons which leads to upper and lower 
bounds on the APS if partial and full unification is required.
For relatively lower values of tan$\beta$ required by partial
unification, REWSB can be accommodated in the phenomenologically 
interesing small $\MSX-\MHF$ region of the parameter space
(viz. for $\MSX,\MHF\sim$ 200 GeV, the required value is 
tan$\beta\sim 30,$ and
for $\MSX,\MHF\sim$ 800 GeV, tan$\beta\sim$ 41 ). Smaller values of the
soft breaking parameters are disfavoured for the  given tan$\beta$. 
The  trend remains unaltered 
irrespective of the choice of the other parameters. 
Since tan$\beta \sim$ 30 can accomodate partial unification 
we find no lower bound  on $\MSX$, $\MHF$ much stronger than the current 
experimental
bounds. For higher values of $\MSX$, however, nontrivial and stronger
lower bounds on $\MHF$ emerges. These bounds due to Yukawa unification
and REWSB are relatively weak for large
negative values of $A_0$ and becomes stronger
as this parameter is algebraically increased ( see figures 6 - 10 of
\cite{paper1} ). On the other hand the UFB constraints are very potent
for large negative values of $A_0$. In this sense the two sets of
constraints are complementary.

Since  $Y_t$ is relatively slowly varying  with respect to
tan$\beta$ compared to
$Y_{\tau}$ and $Y_b$,  for very accurate ( 5 \% ) $t-b-\tau$ unification
we  need higher value of
tan$\beta\sim 48 - 50$ . In this case
the  low $\MSX - \MHF$ region is excluded by REWSB condition leading
to lower bounds much stronger than the experimental ones and the APS
becomes rather narrow.
For example,  
$\MSX=600, \MHF=1000$ for  tan $\beta$ =49.5 gives an idea of lowest
allowed masses in this case.     

If we relax the accuracy of full
unification lower  values  of tan$\beta$ serves the purpose.
Consequently the theoretical lower bounds on sparticle masses also become
 weaker. Thus  full unification with modest accuracy (say 10\%)
tends to yield APS similar to that of $b-\tau$ unification with 
the only difference that tan$\beta$  still happens  to be smoewhat 
larger than that for partial unification, resulting in smaller 
number of allowed  points in the phenomenologically
interesting low $\MSX-\MHF$ region ( compare figures 8 and 11 of 
\cite{paper1} ). For an accuracy $\sim$ 20\% the APS 
is almost the same as $b-\tau$ case. 

For a given $\MSX$, full unification also leads to nontrivial
theoretical lower bounds on $\MHF$  which gets stronger
for larger $\MSX$, as $A_0$ is algebraically increased. This
observation is more or less unaltered even if the constraint of 
universal scalar mass is relaxed ( see figures 11 -12 of 
\cite{paper1} ). This behaviour also
leads to an APS where for a given $\MHF$, there is an upper bound
on $\MSX$.

%In summary, without the D-terms  both partial  unification and full
%unification do not allow a 
%phenomenologically interesting sparticle 
%spectrum, where all sfermions could be much heavier than the gauginos
%including the gluinos.  The only exception
%being a specific nonuniversal scenario in the context of $b -\tau$
%unification (fig 9 of \cite{paper1}). this interesting prediction
%can be tested at the LHC. In contrast
%we shall see that this scenario is one
%of the preferred mass patterns in the presence of D - term.   

For a given $\MSX$ there is always an upper bound on $\MHF$ coming for the
requirement of the neutralino LSP. However, the UFB bounds often impose
stronger upper bounds on $\MHF$. Such bounds  always gets stronger for
large negative values of $A_0$ for reasons associated with the 
running of the Higgs
mass parameters ( see figures 1 - 4 of \cite{paper1} and the diccussions
on
them ). This trend is seen  for both $b -\tau$ and $t- b- \tau$ 
Yukawa unification. Nonunivarsality of the scalar masses
often makes these  upper bounds 
 more sringent. For examle, with $\MTN < \MSX$ the  scenario with
heavy sfermions but lighter gauginos is the only viable scenario   
in the presence of UFB constraints ( see figure 9 of \cite{paper1}).
The UFB bounds become weaker as $A_0$ is algebraically increased.
Clearly the UFB  constraints hyperactive for large negative $A_0$
are complementary to the ones arising
from Yukawa unification and REWSB which are more potent for 
 larger values of $A_0$.    



%For these higher $m_D$ tan$\beta$ has its
%upper limit $\sim 51$. Increase of tan$\beta$ makes failure of REWSB for
%lower part of $\MSX-\MHF$ plane; and violation of neutralino LSP for the
%upper part.
           
\subsection{$t$-$b$-{\Large$\tau$} Unification}   
After the introductory qualitaive remarks, we now present some
quantitative details which will be helpful in appreciating the differences
between the APSs with and without the D-terms. In order to comment on the 
accessibility of the APS at LHC energies we need some benchmark values.
We present,  as  rough guide lines,  the results from \cite{tatadrees}
( see Figure 1d of \cite{tatadrees} )    :
the  region bounded by $\MHF <$ 0.9 TeV and $\MSX <$ 1.5 
TeV can be probed at the LHC . Although for the large
$\MSX$ region the reach in $\MHF$ is somewhat smaller. With progess in
R \& D, however, an expansion of the testable region  is not quite
unlikely. In particular in the large tan$\beta$ scenario some 
very special signals involving third generation sfermions might 
provide useful handles for extending the LHC
reach.


Requiring $t$-$b$-{\Large$\tau$}  
Yukawa unification even with a modest accuracy ($\approx$
10\%) it was found in  \cite{paper1} in the universal  model that  $A_0
\lapp 0$ is ruled out by the UFB1 constraint. Invoking nonuniversality
of the scalar masses does not change this strong conclusion. The running
of $\MHU$ and $\MHD$ for such values of $A_0$ explains this constraint
( see figure 1 of \cite{paper1} and the discussions after eqn 8 ).


However, since this
constraint may be bypassed by  requiring a false vacuum with a large
life time as discussed in the introduction \cite{claudson}, we describe 
briefly the APS due to Yukawa unification alone. 

For large negative value of $A_0$ (e.g., $A_0=-2\MSX$), the unification 
allowed APS is a narrow
band of large $\MSX$ and $\MHF$ values, with $\MSX \gapp 800$ and $\MHF
\gapp 1000$GeV, which are  hardly  interesting even in the context of
SUSY searches at the LHC.


However, for $-\MSX \lapp A_0 \lapp \MSX$, ( see fig. 11  
of \cite{paper1} ), a realtively large APS with smaller
$\MSX$ and $\MHF$, more  interesing for LHC physics, are found (
typically $\MSX \gapp 400$ GeV and $\MHF \gapp 500$ GeV). The UFB1
constraints mainly excludes points with high $\MSX$ and $\MHF$ which
are not very interesting phenomenologically. Nonuniversality with 
$\MTN > \MSX$ leaves APS  qualitatively the same so far as the
phenomenologically interesting low $\MSX$ and $\MHF$ region is concerned.  
(see fig. 12  of \cite{paper1} ). For $\MTN < \MSX$, the APS is similar
except that at a few points with still lower values of $\MHF$  are
allowed. A narrow slice of the APS is testable at LHC.

For larger positive values of  $A_0$, (e.g., $A_0=2\MSX$) 
unification allowed APS decreases drastically. 
A very small region, having a shape  qualitatively similar to that
for $A_0=-2\MSX$, with $\MSX \gapp 500$ and $\MHF \gapp 900$GeV was
allowed. 
                
Scanning over the entire parameter space we have not found any  region
with very heavy sfermions, practically beyond the reach of LHC, along with 
light gaugunos with $\MHF$ close to the  current experimental lower bound.

If we require more accurate  unification ( e.g., 5\%), which occurs
at a relatively higher value of tan$\beta ( \sim 49 )$ that tends to
violate both the REWSB condition and the nutralino LSP condition, the APS
reduces drastically. No point is obtained for large negative $A_0$ (e.g.,
$A_0=-2\MSX$). For $-\MSX \lapp A_0 \lapp \MSX$, a very small region with
$\MSX(\MHF)\gapp$ 500(1000)GeV is obtained. If we increase $A_0$ to large
positive values (e.g., $A_0=2\MSX$)  very few points with
$\MSX(\MHF)\gapp$ 1000(1300)GeV are obtained. Thus unification close to the
ideal situation without D-terms seems to be beyond the perview of LHC searches.

Thus we 
conclude that if we require very precise Yukawa unification without 
D-terms, the APS is practically negligible. The squark and gluino are
expected to
be rather heavy ( $\MSQ, \MGL \gapp$ 2 TeV ), while $\MSL \sim$ 500 GeV
and $\MCH \sim$ 1000 GeV can not be ruled out with absolute certainty
, although the corresponding APS is rather tiny. One of the main results
of
this paper is to show that precise  $t-b -\tau$ Yukawa 
unification and relatively light
 sparticles with masses close to the current experimental lower bounds
are compatible in the presence of D-terms of moderate accuracy.
    
\subsection{$b$-$\tau$ Yukawa unification} 

We first consider  large negative values of
$A_0$ ( $A_0  \lapp -2\MSX$, say ).  In the universal scenario unification
alone
permits a rather large APS including points with  $\MHF$ and $\MSX$ having
values close to the current experimental lower bounds ( fig 6 of 
\cite{paper1} ). The strong
constraints from UFB1 and UFB3 conditions, however, 
lead to a much restricted  APS bounded
from both above and below. In this limited region   $\MSX >$ 400 GeV 
irrespective of $\MHF$. The lower bound on $\MHF$, however, essentially
comes from experiment. The upper bounds on the other hand is so restricted
that almost the entire theoretically
allowed parameter space can be scanned at the LHC. 

For nonuniversal boundary conditions and  $A_0 = -2\MSX$,
an intresting 
sparticle spectrum emerges, which is rather uncommon if full 
unification is required. Only with 
$\MTN < \MSX$, this testable mass hierarchy with
the squarks and sleptons much heavier than the gauginos, is obtained.
The minimum  of $\MHF$ is close to the experimental lower bound 
and the upperbound is $\MHF < $ 1 TeV . It should be emphasised
that this stringent upper bound arises  due to 
strong UFB3 constraints ( fig 9 of \cite{paper1} ).
Thus copious gaugino production at LHC accompanied by unobservable or
much suppressed sfermion production, is a hall mark of this scenario.
For  $\MTN > \MSX$ the requirement of unification restricts the APS so
severely that inspite of the weakening of the  UFB conditions a highly
restricted APS with its bulk accessible to LHC searches is obtained
( fig 10 ).

The APS permitted by unification alone is severely squeezed  for
algebraically larger values
of $A_0$ although the UFB bounds get weaker. For moderate negative values
of $A_0$ (say, $A_0 =$ -$\MSX$ ) or $A_0$ = 0, substantial
fractions of the APSs  can be probed  at the LHC as is indicated by our
bench mark values.  

\section{Impact of D-terms on Yukawa unification and stability of the Potential} 
As has already been mentioned in the introduction, D-terms have 
non-trivial impact on Yukawa unification. The reasons have already
been discussed in the literature \cite{tata}. We repeat them very briefly
for the sake of completeness.

 We shall now focus our attention on an SO(10) SUSY GUT \cite{sotenbr}
 containing all the quarks and leptons of a given generation in a 16
 dimensional multiplet which includes the heavy right handed neutrino.
 In this model the first two
 generations of squarks and sleptons with small Yukawa couplings
 are expected to be highly degenerate at the GUT scale, even
 if the  running of the soft breaking masses between
 $M_P$ and $M_G$\cite{running}  is taken into account. In
 principle nonuniversal masses for the third generation sfermions with
 a larger Yukawa coupling is  possible due to the above
 running. However, we shall assume this intergeneration
 nonuniversility to be small compared to the nonuniversality due to 
 D-terms, which is the main subject of our attention described below. 
Thus the 16 plets belonging to all the three generations are assumed to 
have a common soft breaking mass $\MSX$ at $M_G$.
 
The soft breaking mass of the light higgs bosons at $M_G$ 
may be significantly different from that of the sfermion sector
due to the running between $M_P$ and $M_G$. The light higgs doublets
reside in a 10 plet of SO(10) and hence are
renormalised differently. Moreover they have to couple to other super
heavy GUT fields in order to implement the mass-split between the coloured
higgs bosons and the colour neutral ones responsible for $SU(2)\times
U(1)$ breaking. Unfortunately the magnitude of the resulting
nonuniversility is not calculable without specifying all the couplings of
the higgs bosons, which are not known presently. We parametrize this
uncertainty by taking the Higgs soft breaking mass  at  the GUT scale 
to be $\MTN$ which is different from  $\MSX$.

 Non-universality of the sfermion masses at may still arise at $M_G$ due to
D-term contributions, which appear when SO(10) breaks into a
group of smaller rank\cite{d-term}. In general such contributions could be
different for different members of the 16-plet . However, these
non-universal terms are generation independent, so that no
additional problem due to flavour changing neutral currents arise.   

As a specific example we shall consider the breaking of SO(10) directly to
the SM gauge group \cite{d-term}. The group $SO(10)$ contains $SU(5)\times
U(1)$ as a subgroup.  It is further assumed that the D-terms are linked to
the breaking of this $U(1)$ only.  The squark- slepton masses in this case
are
\bc 
$m_{\tilde Q}^2 = m_{\tilde E}^2 = m_{\tilde U}^2 = m_{16}^2 +
m_D^2$\\ $m_{\tilde D}^2 =m_{\tilde L}^2 =m_{16}^2 -3 m_D^2$\\
$m_{H_{d,u}}^2 =m_{10}^2 \pm 2 m_D^2$\\
\ec    
where the unknown parameter $m_D^2$( the D-term )  can be of either
sign. The mass differences
arise because of the differences in the $U(1)$ quantum numbers of the
sparticles concerned.  As can be readily seen from the above formula for
$m_D >$ 0, the left handed sleptons  and right handed down type
squarks ( belonging to the $\bar 5$ representation of $SU(5)$ ),  
are relatively light compared to the members of the $10$ plet of $SU(5)$.  
The phenomenology of these
D-terms  have been studied by several authors \cite{pheno-d,tata}.

From the above spectrum it is quite  clear that the 
conflict between Yukawa unification and REWSB is softened in the 
presence of the  D-terms. Since  $\MHD$ is somewhat
smaller  than  $\MHU$ at $M_G$ for positive values of the 
D-term,  $\mu^2$ does not become negative
at the weak scale even for tan $\beta$ large enough to produce
$ t - b - \tau $ unification.           

As discussed in  \cite{paper1},  the 
variation of $\MHU$ and $\MHD$ with respect to the common trilinear
coupling $A_0$ is of crucial importance in understanding the UFB
constraints. Here we extend the 
discussion for non-zero values of the D-term, $m_D =$ $\MSX/5$ and
$\MSX/3$, which is  illustrated by fig 1. 
We find that larger $m_D$ drives $\MHD$ to more negative values, while
$\MHU$ is driven to positve values. In addition, it follows
 from REWSB condition  that as
the difference $\MHU - \MHD$ increases, the higgsino mass parameter $\mu$
increases. As a result  the UFB1 constraint becomes weaker 
for large $m_D$ values (see eq. 18 of \cite{casas} ).
Moreover at the GUT scale $m_{L_i}$ becomes smaller for larger $m_D$. From
eq. (32) of \cite{casas} it can be argued that the parameter space where
$\MHD + m_{L_i}^2$ is negative increases, and the model is likely 
to be more succeptible to  the UFB3 codition. 


We next note that though moderate values of $m_D$ leads to better Yukawa
unification, somewhat larger values of this parameter spoils it. Although
the D-terms do not affect the evolution of the Yukawa couplings directly
through the renormalization group (RG) equations, they change the initial
conditions through SUSY radiative corrections to $m_b(m_Z)$ \cite{pierce}.
This is illustrated in figures 2-4, where the unification is shown for
three different values of $m_D$. The choice of
other SUSY paremeters for these figures are as follows: 
\bc
$m_{10}=m_{16}$ = 1500GeV, \ $m_{1/2}$ = 500GeV, \ tan$\beta$=48.5, \
$A_0$ = 0.
\ec 
From figure 2 ($m_D=0$), we see that the accuracy of unification is rather
modest ($\sim 15\%$). As $m_D$ is further increased to $m_{16}/5$ (fig.\
3), the $\tilde b\tilde g$ loop corrections (see eq.\ (8) of
\cite{pierce}) to $m_b(m_Z)$ increases and leads to a better unification.
However, if we increase $m_D$ further to $m_{16}/3$, the accuracy of unification
deteriorates ( fig.\ 4) since $m_b(m_Z)$ suffers a correction which is
too
large. We have checked that this feature holds for a wide choice of SUSY
parameters. We thus conclude that with the inclusion of nonzero D-terms
of moderate magnitude,
$t$-$b$-$\tau$ Yukawa unification within 5\% is possible, while for
much larger values the unification worsens.
 
 Throughout this paper we mainly restrict ourselves to this 5\%
accuracy, which is achievable only in the presence of the D-terms .  For
 unification of smaller accuracy, the APS  is
approximately the same as for the partial $b$-$\tau$ Yukawa unification ,
which  has been studied independently.

\section{Results}    
In this section, we discuss quantitatively the impact of the 
D-terms on the parameter space allowed by $b - \tau$ and $t - b - \tau$
Yukawa unification and  UFB constraints,
in the context of the universal and nonuniversal models.
%(i) SO(10) SUSY GUT with a universal scalar mass $m_{16} = m_{10}$;
%(ii) SO(10) SUSY GUT with nonuniversal scalar masses: $m_{16} \not = m_{10}$. 
% Both these models are 
%discussed in the context of $t$-$b$-$\tau$ and partial  $b$-$\tau$ 
%Yukawa unification. Through out the paper we have also required radiative
%electroweak symmetry breaking and demanded that the lightest neutralino be
%the LSP. Although often these conditions will not be mentioned explicitly. 


\subsection{$t$-$b$-{\Large$\tau$} Unification}
We start our discussion for moderate values of the D - term (e.g., 
$m_D = \MSX/5$) and the universal scenerio. For large negative values 
of $A_0$ (e.g., $A_0=-2\MSX$),
which disfavours REWSB at this large tan$\beta$ ( $\sim$ 49 ), only 
a narrow band of $\MHF$ is allowed by Yukawa unification. However,
these points correspond to rather heavy sparticles (e.g., $\MSX(\MHF)
\gapp$ 1100(1300)GeV ) which are of little interest even for SUSY 
searches at the LHC. Moreover, the tiny APS allowed by the 
unification criterion is ruled out by the UFB conditions.

Non-universality  affects the APS marginally.
If we take $\MTN > \MSX$ (e.g., $\MTN = 1.2\MSX$), $\MHD$ and
$\MHU$ are less negative for smaller $\MSX$ and $\MHF$, 
which helps REWSB. As a result  the APS slightly expands  in the low $\MSX
- \MHF$ region and
the lower limit on $\MHF$ decreases by 200 GeV.
On the contrary, if we decrease $\MTN$ (e.g., $\MTN = 0.8\MSX$), the APS 
reduces drasically and very few points are allowed. Thus no squarks - gluino 
signal is expected at LHC for $A_0 \lapp -2\MSX$ irrespective of the
boundary conditions ( 
universal or non-universal) on  the scalar masses. 
Over a small region 
of the APS somewhat lighter  sleptons ( $\MSL \sim$ 1000GeV ) are
permitted.
%in two bands of $\MHF$, one with 
%$\MHF\sim$ 200GeV with $\MSX\gapp$ 2400 GeV and anther with 
%$\MHF\sim$ 1700GeV with $\MSX\gapp$ 1100 GeV are obtained.

 Relatively large APSs with phenomenologically interesting  values
of $\MHF$ open up
for $-\MSX \lapp A_0 \lapp \MSX$, which is favourable for both 
Yukawa unification and REWSB. This is an interesting 
region accessible to experiments. It may be recalled that without
the D-terms negative $A_0$ was was ruled out even if unification
at the  level of 10\% accuracy was required.

  
We begin with the universal model and moderate $m_D$. 
In fig. 5 we show the $\MSX - \MHF$
plane for $A_0 = -\MSX$. We see, a large APS is allowed by the unification
criterion.  For $\MSX \gapp 1200$GeV, low values of $\MHF$ are quite common.

Scanning over the APS we find that the  lowest gluino mass is
$\MGL$ = 564 GeV, the minimum sfermion masses are $\MSQ \approx$ 1200 GeV,
$\MSL \approx$ 1200 GeV. It should be stressed that this mass pattern with
all sfermions much heavier than the gauginos including the gluino, cannot
be accommodated without the D-terms. In the presence of D-term 
this hierarchy is a distinct possibility for $-\MSX \lapp A_0 \lapp \MSX$.

Requiring Yukawa unification alone,
$\MSX$ can be as low as 700 GeV. However, this requires 
$\MHF \geq$ 1100 GeV, corresponding to: $\MGL \geq$ 2422 GeV, 
$\MSQ \approx$ 2200 GeV, $\MSL \approx$ 829 GeV.

As the potential  constraints are switched on  an interesting upper bound
on $\MHF$ is imposed by the UFB1 constraint ( figure 5) . As a result
practically   over the entire APS, 
the gauginos are required to be significantly lighter than the sfermions. 
Moreover the allowed gaugino masses are accessible to searches at the LHC. Points
corresponding to the low $\MSX$ region are also disallowed, leading
to $\MSX >$ 1.2 TeV.

As $A_0$ is further increased algebraically  the APS
 decreases due to  unification and REWSB constraints. This trend is
similar
to the one seen without the D - Terms. For $A_0 = 0$, we obtain
 an upper limit  $\MSX \sim$ 2400 GeV . However, the lower limits 
on $\MSX$ and
$\MHF$ are relaxed by $\sim$ 200 GeV in comparison to the $A_0 = -\MSX$ case. 
As we further increase the value
of $A_0$ to $A_0 = \MSX$, the APS is almost same with $A_0 = -\MSX$,
although the UFB constraints are much weaker. However, the light gaugino
heavy sfermion spectrum prevails in all cases.
We may conclude that as the absolute value of $A_0$ increases, Yukawa
unification is less restricted while REWSB is somewhat disfavoured.                
When both act in combination, we get a relatively large
APS for $\left |A_0\right | = \MSX$ and a somewhat smaller one
for $A_0 = 0$.

We next focous on the impact of nonuniversality on the 
 negative $A_0$ scenario. The shape of the APS is affected
appreciably ( see figure 6).
As $\MTN$ decreases, $Y_b$ get larger SUSY threshold correctons
than $Y_{\tau}$ and $Y_t$; this disfavours  Yukawa unification.
On the other
hand $\MHU$ and $\MHD$ becomes more negative for even smaller values of
$\MSX$ and $\MHF$, which disfavors REWSB. The overall APS is smaller
compared to the  universal case ( compare figure 5 and 6 ). A large
segment, which could be phenomenologically  interesting according to our
bench mark values, are ruled out.


For  $\MTN < \MSX$, The UFB1 constraint still imposes an upper bound
on the gaugino mass and a stringent lower bound on 
 $\MSX$ ; as in the universal scenario, only  sleptons and
squarks  beyond the reach of LHC are allowed, while the gauginos
are within the striking range of LHC practically over the entire APS.  
As an example, for $\MTN = .8\MSX$, we see from fig. 6,  
$\MSX \gapp 1600$ GeV .                                       
No major change is noted in the APS for $A_0 = 0$ and $A_0 = \MSX$
apart from the fact that the UFB constraints get weaker.

For a different pattern of nonuniversality ( $\MTN > \MSX$ ), 
Yukawa unification alone  narrows down  the  APS considerably. However,
it is seen that regions with simultaneously low values of $\MSX$ and $\MHF$ 
are permitted  in this specific nonuniversal scenario  only. On the otherhand
the heavy sfermion,  light gaugino mass hierarchy,  preferred 
by the other scenarios, is disfavoured.  For $\MTN = 1.2\MSX$ (  fig. 7 )
it is found that $\MSX \gapp$ 800 GeV for $\MHF\gapp$ 300 GeV. 

The unification allowed parameter space, however, is very sensitive to the
UFB conditions which practically rules out the entire APS for negative $A_0$.

We now discuss the impact of larger  D-terms on the parameter space.
For example, with  $m_D = \MSX/3$,  the APS reduces
drastically in the universal as well as in the 
nonuniversal scenario with $\MTN < \MSX$, irrespective of $A_0$. 
This is illustrated in figure 8. and  is in complete agreement with our
qualitative discussion.

Only in the special nonuniversal scenario $\MTN > \MSX$ the APS tends to increase
with increasing $m_D$. However, $m_D$ cannot be increased arbitrarily.
For  $\MTN=1.2\MSX$, the APS  begins to shrink again 
for  $m_D \gapp \MSX/3$ and  we find no point for $m_D = \MSX/2$.
 
As $\MTN$ is increased  further, Yukawa unification  occurs in a 
 narrower APS. This, nevertheless,  is a phenomenologically interesting region
 where  lower  $\MSX-\MHF$ values can be accommodated.
For example, at the point  $\MSX(\MHF) =$400(300),
 $\MTN=1.5\MSX, m_D=\MSX/3$, $A_0$=0 and tan$\beta \sim$ 51, we obtain $\MGL$ = 742
GeV, $\MSQ \approx$ 700 GeV, $\MSL \approx$ 400 GeV and $m_{\tilde \tau_1}$ = 274
GeV. However, we cannot increase $\MTN$ arbitrarily either, the APS reduces
drasically for $\MTN \gapp 1.5 \MSX$.                                 

\subsection{$b$-$\tau$ Yukawa unification}
In our previous work\cite{paper1} without D-terms, we had shown
that the APS is strongly restricted due to Yukawa unification
and UFB constraints. If D-terms are included, 
Yukawa unification and REWSB occur over a larger region of the
parameter space.
This is primarily due to the fact that while Yukawa unification can 
now be accomodated for a relatively low tan$\beta$, REWSB may occur
for tan$\beta$ somewhat higher than that permitted by the $m_D = 0$ case.
This reduces the conflict between unification and REWSB.
As a result $\MHF$ almost as low as that allowed by the LEP bound on the 
chargino mass is permitted over a wide range of $\MSX$. In some cases 
the upper bound on $\MSX$ for a given $\MHF$ is also relaxed.
Similarly  for a fixed $\MSX$, 
the upper bound on $\MHF$ is sometimes relaxed by few hundred GeVs.
Through out this paper we require this  unification to an 
accuracy of $<$ 5\%.

For  large negative values  of $A_0$ ( $A_0 = -2\MSX$ )and $m_D = \MSX/5$ in the
universal scenario, the unification allowed
APS expands significantly compared to the $m_D$ = 0 scenario. Moreover,
the  phenomenologically intriguing scenario with light gauginos  but
sleptons and squrks beyond the reach of LHC ( fig.  9),
which was rather uncommon without the D-terms (see section 2 ),  is now viable.

Without the D-term, the APS was  severely restricted by the UFB conditions
( see figure 6 of \cite{paper1} ) for  large negative values of $A_0$. 
As discussed earlier,  inclusion  of D-term increases the value 
of $\mu$. As a rseult  UFB1 looses its constraining power; lower values of
$\MHF$ are allowed for large $\MSX$ by UFB1. On the other hand 
 as the value of D-term increases, UFB3 becomes slightly more restrictive and the
upper bounds on $\MHF$ for relatively low values of $\MSX$ are now stronger  ( e.g.
for $\MSX$ = 600, 1000  GeV, $\MHF <$ 300, 600 ).



Now, for algebraically larger  $A_0$, the Yukawa unification allowed APS 
generally get more restricted for $m_D = 0$. The same  trend is  observed  
for $A_0 \le 0$ in the presence of the D-terms. For $A_0 > 0$ the APS again
expands. However the UFB 
constraints are  found to be progressively weaker as $A_0$ is
 increased from $A_0 = -2\MSX$. No new features in the APS is noticed.

We next consider the non-unversal scenerio $\MTN \ne \MSX$. 
If we take $\MTN < \MSX$ and  $m_D = \MSX/5 $, 
the unification allowed parameter space 
for $A_0 = -2\MSX$ is more or less the same as in the universal scenario
( see figure 10 ). The entire APS  is, however, ruled out due to very powerful 
constraints  obtained from the UFB3 condition. This conclusion
holds for even larger values of $m_D$ while, as expected, the UFB 
constraints loose their potency for algebraically larger values of $A_0$. 


For $\MTN > \MSX$ and large negative $A_0$ ( $A_0 = - 2\MSX$ )
the unification allowed APS reduces drastically compared to the universal case in
the  $m_D = 0$ scenario ( fig. 10 in \cite{paper1} ). The same trend 
is observed in the moderate $m_D$ scenario. 
Although the APS is significantly  larger than that for $m_D \ne 0$
(  fig 11 ). For a given $\MHF$ ( $\MSX$ ) the upperbound on $\MSX$
($\MHF$ ) gets weaker. Relatively light gluinos consistent with current
bounds are allowed over a larger region of the parameter space.
The UFB constraints restrict the APS further and put rather strong bounds
on $\MHF$ and $\MSX$. A large fraction of this restricted APS
is accessible to tests at LHC energies. 
This interesting spectrum is not preferred by the nonuniversal scenarios.
( fig 10 \& 11 )

The usual reduction of the APS due to unification constraints as $A_0$ is incereased
from $A_0 = - \MSX$ also holds in this nonuniversal scenario.

If we increase $m_D$ further the APS due to Yukawa 
unification reduces for reasons diccussed earlier. 
The UFB1 constraint also gets weaker. On the other hand 
the UFB3 constraints  become rather potent.
For example, i) with $A_0 = - 2\MSX$ and $m_D \gapp \MSX/3$ the entire APS
for $\MTN = \MSX$ or $\MTN < \MSX$ is ruled out. ii) $A_0 = -\MSX$ 
and $m_D \gapp \MSX/2$ the entire APS corresponding to $\MTN = \MSX$ 
or $\MTN < \MSX$ is ruled out. On the other hand for $\MTN > \MSX$ 
the APS is sizable.

\section{Conclusions}

As has been noted in the literature \cite{tata} very precise 
(t - b -$\tau$ ) Yukawa unification ( accuracy $\approx$ 5\% ) 
is possible within the framework of an SO (10) SUSY GUT
only in the presence of SO (10) D-terms, which arise quite naturally.
In this paper we have noted that while this observation is true for modeate values
of the D-term ( $m_D \approx \MSX/5$ say), the APS shrinks for larger $m_D$ (
$\gapp \MSX/3$ ). We have also studied the impact of the UFB constraints
\cite{casas} on the unification allowed parameter space. Our analysis is carried
out within the general framework where all scalars may or may not 
be degenerate at the GUT scale. Our conclusions are summarised as follows.
                                   
i) It was noted in \cite{paper1} that even if full unification with 
modest accuracy
(  $\approx$ 10 \% ) is required, negative values of the trilinear softbreaking
parameter $A_0$ 
are forbidden by the UFB  constraints with out the D-term. 

In this paper we have shown that large negative values of $A_0$ 
continues to be  strongly disfavoured even in the presence of D-terms. 
However, even with very precise unification ( $\approx 5\%$ ), moderate  
negative values of $A_0$ are consistent with  the stability of the 
potential when D-terms are introduced ( see figures 5 - 8 ).
More importantly  this scenario strongly favours relatively heavy sfermions 
with much lighter gauginos( including the gluino) well within the 
striking range of the LHC. The universal as well as the 
nonuniversal scenario with $\MTN < \MSX $ 
prefer this hierarchy. On the other hand the scenario with  $\MTN > \MSX$, 
which disfavours this hierarchy, is practically ruled out by UFB constraints.
We  stress that this  hierarchy is stronly disfavoured without the D-terms
 even if  $t - b - \tau$ Yukawa unification with modest accuracy is required. 

For algebraically larger values of $A_0$ the heavy sfermion, light gaugino
scenerio may be accomodated although the UFB constraints are now weaker.

After a detailed study,
we have found that over a significant region of the APS  with moderate values 
of $m_D$ ( $\approx \MSX/5$ ) gauginos as light as the current experimental 
lower bound ( $\MHF \approx$ 100 GeV ) are allowed. In such cases the 
sfermions are sometimes within the reach of the LHC. In an extreme case
only the gauginos are observable with masses close to the current lower
bounds, but sfermions are beyond the kinematical limit of LHC. 
Without D-terms this is possible only in the specific non-universal 
scenerio ( $ \MTN < \MSX$ ) with $b - \tau$ unification.

As in the $m_D = 0$  case, the UFB1 condition strongly disfavours 
large negative values of $A_0$ ( $A_0 \lapp -2\MSX$ ) and sets experimentally 
testable upper limits on $\MHF$ over the entire APS  (fig. 5 - 8 ). 
For example some of the electroweak gauginos in addition to the LSP are
neccessarily lighter than 1 TeV. The non-universal scenerio with 
$\MTN > \MSX$ is strongly disfavoured by UFB1 condition.

The partial $b - \tau$ unification which takes place at relatively low tan
$\beta$ has usually a larger APS even without the D-terms. 
The APS further increases after inclusion of D-terms. Apart for 
a very special nonuniversal scenario $\MTN < \MSX$, the  mass pattern 
with large $\MSX$ and small $\MHF$ is not permitted in the absence of the
D-terms. In the presence of the D-terms this mass pattern can be well accommodated
in the universal scenario for large negative values of $A_0$, although the
nonuniversal scenarios still disfavour it.

This mass hierarchy is seen for moderately negative values of $A_0$ but are
disfavoured by $A_0 \gapp 0$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


{\em \bf Acknowledgements}:

The work of AD was supported by DST, India (Project No.\ SP/S2/k01/97)
and BRNS, India (Project No.\ 37/4/97 - R \& D II/474). 
AS acknowledges CSIR, India, for his research fellowship.

\begin{thebibliography}{99} \bibitem{susy-review} For reviews see, {\em
e.g.},
  H.P. Nilles, Phys. Rep.  {\bf 110}, 1 (1984);
  P. Nath, R. Arnowitt and A. Chamseddine, Applied N=1 Supergravity, ICTP
    series in Theoretical Physics, vol. I, World Scientific (1984);
  H. Haber and G.L. Kane, Phys. Rep. {\bf 117}, 75 (1985).
\bibitem{oldufb}
  L. Alvarez-Gaum\'e, J. Polchinski and M. Wise, Nucl. Phys. B{\bf 221},
495
   (1983);
  J.M. Frere, D.R.T. Jones and S. Raby, Nucl. Phys. B{\bf 222}, 11 (1983;
  M. Claudson, L.J. Hall and I. Hinchliffe, Nucl. Phys. B{\bf 228}, 501
(1983);
  C. Kounnas {\em et al}, Nucl. Phys. B{\bf 236}, 438 (1984);
  J.P. Derendinger and C.A. Savoy, Nucl. Phys. B{\bf 237}, 307 (1984);
  M. Drees, M. Gl\"uck and K. Grassie, Phys. Lett. B{\bf 157}, 164 (1985);
  J.F. Gunion, H.E. Haber and M. Sher, Nucl. Phys. B{\bf 306}, 1 (1988);
  H. Komatsu, Phys. Lett. B{\bf 215}, 323 (1988). 

\bibitem{casas}
  J.A. Casas, A. Lleyda and C. Mun\~oz, Nucl. Phys. B{\bf 471}, 3 (1996).
                                                                                
\bibitem{paper1}
  A. Datta, A. Kundu and A. Samanta, \PRD(63,015008,2001) 
 \bibitem{b-tau}
  M.S. Chanowitz, J. Ellis and M.K. Gaillard, Nucl. Phys. B{\bf 128}, 506
   (1977);
  A.J. Buras {\em et al}, Nucl. Phys. B{\bf 135}, 66 (1978);
  D.V. Nanopoulos and D.A. Ross, Nucl. Phys. B{\bf 157}, 273 (1979), Phys.
   Lett. B{\bf 108}, 351 (1982);
  J. Ellis, D.V. Nanopoulos and S. Rudaz, Nucl. Phys. B{\bf 202}, 43 (1982).
                                                                                
\bibitem{partial}
  G.F. Giudice and G. Ridolfi, Z. Phys. C{\bf 41}, 447 (1988);
  M. Olechowski and S. Pokorski, Phys. Lett. B{\bf 214}, 393 (1988);
  P.H. Chankowski, Phys. Rev. D{\bf 41}, 2877 (1990);
  M. Drees and M.M. Nojiri, Nucl. Phys. B{\bf 369}, 54 (1992);
  B. Ananthanarayan, G. Lazarides and Q. Shafi, Phys. Lett. B{\bf 300}, 245
    (1993);
  H. Arason {\em et al}, Phys. Rev. Lett. {\bf 67}, 2933 (1991);
  V. Barger, M.S. Berger and P. Ohmann, Phys. Rev. D{\bf 47}, 1093 (1993);
  M. Carena, S. Pokorski and C.E.M. Wagner, Nucl. Phys. B{\bf 406}, 59 (1993);
  M. Carena {\em et al}, Nucl. Phys. B{\bf 426}, 269 (1994);
  R. Rattazzi and U. Sarid in {\protect\cite{sotenbr}}.                        

\bibitem{sotenbr}
  B. Ananthanarayan, G. Lazarides and Q. Shafi, Phys. Rev. D{\bf 44}, 1613
    (1991);
  L.J. Hall, R. Rattazzi and U. Sarid, Phys. Rev. D{\bf 50}, 7048 (1994);
  R. Rattazzi and U. Sarid, Phys. Rev. D{\bf 53}, 1553 (1996).  

\bibitem{rewsb}
  L. E. Ibanez and G. G. Ross, Phys. Lett. B{\bf 110}, 215 (1982);
  J. Ellis, D. V. Nanopoulos and K. Tamvaskis, 
  Phys. Lett. B{\bf 121}, 123 (1983);
  L. Alvarez - Gaume, J. Polchinski and M. Wise, 
  Nucl. Phys. B{\bf 221}, 495 (1983).

\bibitem{gamberini}
  G. Gamberini, G. Ridolfi and F. Zwirner, Nucl. Phys. B{\bf 331}, 331 (1990).  

\bibitem{running}
  P. Moxhay and K. Yamamoto, Nucl. Phys. B{\bf 256}, 130 (1985);
  B. Gato, Nucl. Phys. B{\bf 278}, 189 (1986);
  N. Polonsky and A. Pomarol, Phys. Rev. D{\bf 51}, 6532 (1995).         

\bibitem{d-term}
  M. Drees, Phys. Lett. B{\bf 181}, 279 (1986);
  Y. Kawamura, H. Murayama and M. Yamaguchi, Phys. Rev. D{\bf 51}, 1337 (1995). 
\bibitem{pheno-d}
  H. Murayama {\em et al}, Phys. Lett. B{\bf 371}, 57 (1996);
  R. Rattazzi and U. Sarid in {\protect\cite{sotenbr}};
  A. Datta, A. Datta and M.K. Parida, Phys. Lett. B{\bf 431}, 347 (1998);
  A. Datta {\em et al}, Phys. Rev. D{\bf 61}, 055003 (2000);
  H. Baer {\em et al}, Phys. Rev. D{\bf 63}, 015007 (2001).       
\bibitem{tata}  
See  R. Rattazzi and U. Sarid in {\protect\cite{sotenbr}};
H. Murayama, M. Olechowski and S. Pokorski,
 Phys. Lett. B{\bf 371}, 57 (1996); H. Baer {\it et al} Phys. Rev.  D{\bf 61}, 111701
(2000).  
\bibitem{nonrenorm}
  G. Costa {\em et al}, Nucl. Phys. B{\bf 286}, 325 (1987).    
\bibitem{claudson}
 M. Claudson, L.J. Hall and I. Hinchliffe, Nucl. Phys. B{\bf 228}, 501
(1983);  A. Kusenko and P. Langacker, Phys. Lett. B{\bf 391}, 29 (1997).
\bibitem{tatadrees} 
H. Baer {\em et al}, Phys. Rev.  D{\bf 59}, 055014 (1999).
\bibitem{pierce}           
  D. Pierce {\em et al}, Nucl. Phys. B{\bf 491}, 3 (1997).

\end{thebibliography}
%%%%%%%%%%%%%%%%%%%%%%%%INCLUDES PSFIGURES%%%%%%%%%%%%%%%%%%%%%%%%%
%\twocolumn
\begin{figure}[htb]
%\vspace*{1cm}
\centerline{
\psfig{file=mh2md.ps,width=12cm,angle=270}
}
\caption{\sl{The variation of the Higgs mass parameters $\MHD$ and
            $\MHU$, evaluted at the scale $M_S = \sqrt{m_{t_L} m_{t_R}}$, 
            with the trilinear coupling $A_0$. 
            The solid ( dotted ) lines are for $m_D=\MSX/5 ( \MSX/3 )$.
            The top two lines are for $\MHD$ while the lower pair 
            is for$\MHU$. We have used $m_{16}=m_{10}=m_{1/2}=1$ TeV,
            tan$\beta$ = 45. 
            }}
    \label{fig:figure1}
\end{figure}
%-----------Yukawa unification variation yith m_D----------------------
\begin{figure}[htb]
%\vspace*{1cm}
\centerline{
\psfig{file=yu0.ps,width=12cm,angle=270}
}
\caption{\sl{The variation of Yukawa couplings  
             with renormalization scale Q (GeV).
            From above the lines are for top, bottom and $\tau$ Yukawa
            couplings respectively.
            We have used $m_{16}=m_{10}$=1.5 TeV, $m_{1/2}=0.5$ TeV.
            $A_0=0$ and $m_D$ =0.
            }}
    \label{fig:figure2}
\vspace*{5mm}
\centerline{
\psfig{file=yu5.ps,width=12cm,angle=270}
}
\caption{\sl{The same as Fig. 2, with $m_D=\MSX/5$.
           }}
     \label{fig:figure3}
\end{figure}
%\vspace*{5mm}
\begin{figure}[htb]
\centerline{
\psfig{file=yu3.ps,width=12cm,angle=270}
}
\caption{\sl{The same as Fig. 2, with $m_D=\MSX/3$.
           }}
     \label{fig:figure4}
\end{figure}
%-------$t-b-\tau$ Yukawa Unification-------------------------
\begin{figure}[htb]
%\vspace*{1cm}
\centerline{
\psfig{file=m0p1md5a0_1.ps,width=12cm,angle=270}
}
\caption{\sl{The allowed parameter space in the universal scenario with
             $t-b-\tau$ unification $\le$ 5\%. All the points are 
             allowed by the Yukawa unification criterion; the asterisks 
             are ruled out by UFB1.
             We set $m_D = \MSX/5$ and $A_0=-m_{16}$.
            }}
    \label{fig:figure5}
\vspace*{5mm}
\centerline{
\psfig{file=m0p.8md5a0_1.ps,width=12cm,angle=270}
}
\caption{\sl{The same as Fig. 5, with $\MTN = .8\MSX$.
           }}
     \label{fig:figure6}
\end{figure}
%\vspace*{5mm}
\begin{figure}[htb]
\centerline{
\psfig{file=m0p1.2md5a0_1.ps,width=12cm,angle=270}
}
\caption{\sl{The same as Fig. 5, with $\MTN = 1.2\MSX$.
           }}
     \label{fig:figure7}
\vspace*{5mm}
\centerline{
\psfig{file=m0p1md3a0_1.ps,width=12cm,angle=270}
}
\caption{\sl{The same as Fig. 5, with $m_D = \MSX/3$.
           }}
     \label{fig:figure8}

\end{figure}
%--------------------b -\tau------------------------------------------
\begin{figure}[htb]
%\vspace*{1cm}
\centerline{
\psfig{file=m0p1md5a0_2.ps,width=12cm,angle=270}
}
\caption{\sl{The allowed parameter space in the universal scenario with
             $b-\tau$ unification $\le$ 5\%. All the points are allowed 
             by the Yukawa unification criterion; the asterisks are 
             ruled out by UFB1 and the boxes are ruled out by UFB3.
             We set $m_D = \MSX/5$ and $A_0=-2m_{16}$.
            }}
    \label{fig:figure9}
\vspace*{5mm}
\centerline{
\psfig{file=m0p.6md5a0_2.ps,width=12cm,angle=270}
}
\caption{\sl{The same as Fig. 9, with $\MTN = .6\MSX$.
           }}
     \label{fig:figure10}
\end{figure}
%\vspace*{5mm}
\begin{figure}[htb]
\centerline{
\psfig{file=m0p1.2md5a0_2.ps,width=12cm,angle=270}
}
\caption{\sl{The same as Fig. 9, with $\MTN = 1.2\MSX$.
           }}
     \label{fig:figure11}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}


